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WATERFLOODING

UNDER

FRACTURING CONDITIONS

O Q

O O

E.J.L. Koning



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v

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WATERFLOODING UNDER FRACTURING CONDITIONS

PROEFSCHRI FT

t er ver kr i j gi ng van de gr aad van doct or i n de t echni sche wet enschappen aan

de Techni sche Uni ver si t ei t Del f t , op gezag van de r ector magni f i cus,

Pro f .

Dr s. P. A. Schenck, i n het openbaar t e ver dedi gen voor een comm ss i e,

aangewezen door het Col l ege van Dekanen, op di nsdag 27 sept ember 1988 t e

10. 00 uur

door

Er i c J an Leonar dus Koni ng

Doct or andus i n de W skunde en Nat uur wet enschappen

geboren t e Haar l em



- I I -

Di t pr oef schr i f t i s goedgekeur d door de pr omot or pr of . i r . H. J . de Haan



- I l l -



- V -

CONTENTS Page

CHAPTER ONE

I NTRODUCTI ON 1

1. 1. Wat er f l oodi ng 2

1. 2. Wat er f l oodi ng under f r act ur i ng condi t i ons 2

1. 2. 1. Scope and def i ni t i on 2

1. 2. 2. Obj ect i ves of t hesi s 3

1. 2. 3. Pr evi ous wor k 3

1. 2. 4. New el ement s i n t hesi s 5

1. 3. Or gani sat i on 6

Ref erences 7

CHAPTER TWO

THE PORO- AND THERMO- ELASTI C STRESS FI ELD AROUND A WELLBORE 9



- VI -

2. 8. Ot her appl i cat i ons 73

2. 9. Concl usi ons 73

Li st of symbol s 75

Ref er ences 77

Appendi x 2- A - Basi c equat i ons 78

Appendi x 2- B - The par t i cul ar st r ess sol ut i on i n cyl i ndr i cal coor di nat es 81

Appendi x 2- C - The compl et e st r ess sol ut i on i n cyl i ndr i cal coor di nat es 84

Appendi x 2- D - Asympt ot i c expansi ons of t he st r ess sol ut i on 89

Appendi x 2- E - Si mpl i f i ed sol ut i on met hod 93

Appendi x 2- F - Numer i cal met hod t o eval uat e Ao — 97

CTT

Appendi x 2- G - Sol ut i on f or t he pr essur e di st r i but i on 101

CHAPTER THREE



- VI I -

3. 4. 3 Two f i el d cases 133

3. 5 Fract ur e pr opagat i on i n a pat t er n f l ood.

Ef f ect on sweep ef f i ci ency 139

3. 5. 1 Zer o voi dage i n t he absence of r eser voi r st r ess changes 139

3. 5. 2 Zer o voi dage wi t h r eservoi r st r ess changes 145

3. 5. 3 Gener al f l oodi ng condi t i ons and t he use of

a r eser voi r si mul at or 147

3. 7 Concl usi ons 148

Li st of symbol s 150

Ref er ences 152

Appendi x 3-A Cal cul at i on of por o- el ast i c st r esses i n el l i pt i cal

coor di nat es 154

Appendi x 3- B A numer i cal met hod f or cal cul at i ng por o-

and t her moel ast i c st r ess changes 163

Appendi x 3- C Cal cul at i on of t her moel ast i c st r esses

and of t he axes of t he el l i pt i cal f l ui d f r ont s 168



- VI I I -

Appendi x 4- A Sol ut i on f or di mensi onl ess pr essur e f unct i on i n Lapl ace

space 198

Appendi x 4- B Rel at i onshi p bet ween f r actur e cl osure const ant and

f r act ur e l engt h 202

CHAPTER FIVE

A PRACTI CAL APPROACH TO WATERFLOODI NG UNDER FRACTURI NG CONDI TI ONS 207

Summar y 208

5. 1 I nt r oduct i on 209

5. 2 An exampl e 209

5. 3 Condi t i ons f or a successf ul pr ocess 210

5. 4 Pr el i m nar y i nvest i gat i ons 210

5. 5 Basi c dat a gat her i ng 214

5. 5. 1 Measur ement s of i n- si t u st r ess, f r acture or i ent at i on and



- 1 -

CHAPTER ONE

I NTRODUCTI ON

1. 1. Wat er f l oodi ng

1. 2. Wat er f l oodi ng under f r act ur i ng condi t i ons

1. 2. 1. Scope and def i ni t i on

1. 2. 2. Obj ect i ves of t hes i s

1. 2. 3. Pr evi ous wor k

1. 2. 4. New el ement s i n t hesi s

1. 3. Or gani sat i on



- 2 -

INTRODUCTION

1. 1 WATERFLOODI NG

When an oi l f i el d i s expl oi t ed by s i mpl y pr oduc i ng oi l and gas f r om a

number of we l l s , t he r eser voi r pr essur e , i n many cases , dr ops r at her qui ck l y

and so does t he pr oduct i on r a t e .

T he r e f o r e , wat er i s of t en i nj ec ted i nt o t he r eser voi r t o mai nt a i n the

r es er v oi r pr es s ur e. The i nj ec t i on wel l s ar e l oc at ed at c ar ef ul l y c hos en

poi nt s so that as much oi l as poss i bl e i s di spl aced by t he wat er t o t he

pr oduct i on we l l s be f or e wat er s t ar t s t o br eak t hr ough i n t he pr oducer s . The

pr oc es s of r ec over i ng oi l by wat er - i nj ec t i on i s c al l ed wat er f l oodi ng. T he

degr ee t o whi ch t he wat er i s capabl e of sweepi ng the o i l t o t he pr oducer s

wi t hout bypas s i ng i t , i s cal l ed t h e s weep ef f i c i enc y.

Even i n t he opt i mal case not a l l t he o i l can be pr oduced by

wat er f l ood i ng. A cer t ai n amount of oi l a l ways r emai ns t r apped i ns i de t he

por es of t he r ock by capi l l ar y f o rces . T hi s so- cal l ed r es i dual oi l may be as



- 3 -

1. 2. 2 Obj ect i ves of t hesi s

Al t hough an i ncrease i n i nj ect i vi t y i s f avour abl e, excessi ve l at er al

gr owt h of t he f r act ur e may adver sel y af f ect t he sweep ef f i ci ency of wat er

i nj ect i on and r esul t i n pr emat ur e wat er br eakt hr ough. Mor eover, i f t he

f r act ur e i s ver t i cal , excessi ve ver t i cal gr owt h may est abl i sh communi cat i on

wi t h ot her r eser voi r s r esul t i ng i n l oss of i nj ect i on wat er or even wor se,

l oss of oi l .

I t i s t he pur pose of t hi s t hes i s t o:

a) i nvest i gat e t he mechani sm of f r acture i ni t i at i on and f r act ur e pr opagat i on

under t he i nf l uence of cont i nued wat er i nj ect i on,

b) eval uat e t he ef f ect of f r actur e gr owt h on sweep ef f i ci ency,

c) i mpr ove t he met hods f or det er m ni ng f r act ur e di mensi ons,

d) gi ve r ul es f or desi gni ng t he pr ocess of wat er f l oodi ng under f r act ur i ng

condi t i ons i n such a way t hat sweep ef f i ci ency i s uni mpai r ed and ver t i cal

f ract ure growt h i s l i m t ed.

1. 2. 3 Pr evi ous wor k



- 4 -

I n the ear l y l i t er at ur e on wat er f l oodi ng, st udi es appear ed on t he

ef f ect of f r act ur es wi t h a f i xed l engt h on sweep ef f i ci ency usi ng physi cal

.

2

'

3

model exper i ment s

4

A maj or st ep f or war d was t he const r uct i on by Hagoor t et al . of a

numer i cal model t hat coul d si mul at e t he gr owt h of a ver t i cal f r act ur e of

const ant hei ght i n a si mpl e, ver t i cal l y homogeneous r eser voi r . They st udi ed

f r act ur e pr opagat i on as a f unct i on of r eser voi r and i nj ect i on/ pr oduct i on

condi t i ons . One of t he i mpor t ant concl usi ons of t hi s st udy was t hat t he

l eak- of f f r om t he f r actur e i nt o t he reser voi r shoul d essent i al l y be model l ed

as t wo- di mensi onal i n t he pl ane of t he reser voi r . Ther ef or e pr evi ousl y

devel oped anal yt i cal model s wi t h a one- di mensi onal descr i pt i on of l eak- of f

ar e gener al l y i nadequat e f or model l i ng wat er f l ood- i nduced f r act ur es.

Later ,

Hagoor t pr esent ed a br oad and i nnovat i ve st udy on t he subj ect i n

hi s t hesi s "Wat er f l ood- i nduced Hydr aul i c Fract ur i ng" . Apar t f rom t he

numer i cal s i mul at i on model , anal yt i cal cal cul at i ons of sweep ef f i c i ency f or

a 5-spot cont ai ni ng a f r act ur ed i nj ector wi t h a f i xed f r act ur e l engt h wer e

pr esent ed. The cal cul at i ons wer e al so ext ended t o st r at i f i ed r eser voi r s. The



- 5 -

1. 2. 4 New el ement s i n t hesi s

The t hesi s pr esent ed her e t akes t he wor k of Hagoor t and Per ki ns &

Gonzal ez as a s t ar t i ng poi nt . The mai n new el ement s ar e :

- A compl et e anal y t i cal desc r i pt i on of t he s t r es s f i el d sur r oundi ng an

unf r act ur ed we l l bor e. The theor y of t hr ee- di mens i ona l por o - and t her mo-

el as t i c i t y i s used t o cal cul at e t he ef f ec t of pr es sur e and t emper at ur e

changes on r eser voi r rock s t r ess .

The cal cul at i on of t he s t r es s f i el d i s used t o eval uat e t he pr essu re at

whi ch f r ac t ur es can be i ni t i at ed.

- An ana l y t i ca l model of f r act ur e pr opagat i on wi t h a compl et e t wo-

di mens i onal des c r i pt i on of f l ui d l eak- of f i nt o t he r es er v oi r .

- A f ocus i ng on t he mode of pr opagat i on f or whi ch the f l ui d f l ow ar ound t he

f r ac t ur e i s pseudo - r adi al . Thi s means t hat t he pr essur e t r ans i ent s a re

t r a vel l i ng r adi al l y i nt o t he r e ser v oi r .

I t i s shown t hat t hi s mode of f r act ur e pr opagat i on does not i nf l uence t he

sweep ef f i c i ency.



- 6 -

1. 3 ORGANI SATI ON

Thi s t hesi s i s wr i t t en i n f i ve sel f - cont ai ned chapt er s t hat can be r ead

i ndependent l y of t he ot her s.

Chapt er 2 deal s wi t h t he cal cul at i on of t he st r ess f i el d ar ound an

unf r act ur ed wel l bor e. The numer i cal met hod pr esent ed her e was publ i shed

ear l i er i n Ref . 7. The anal yt i cal met hods wi l l be publ i shed i n Ref . 8.

Chapt er 3 descr i bes t he model l i ng of f r act ur e pr opagat i on and i t s

ef f ect on sweep ef f i ci ency. Most of t hi s chapt er was publ i shed i n Ref . 9.

Chapt er 4 descr i bes t he met hod of det er m ni ng f r act ur e l engt h f r om a

f al l - of f t est . Thi s chapt er was publ i shed i n Ref . 10. Recent l y, an ext ens i on

of t he met hod t oget her wi t h a f i el d appl i cat i on was publ i shed i n Ref . 11.

Fi nal l y, Chapt er 5 concl udes t hi s t hesi s wi t h pr act i cal desi gn rul es

f or f i el d i mpl ement at i on.



- 7 -

1. Howar d, G. G. & Fast , C. R. , Hydr aul i c Fractur i ng.

SPE Monograph Vol ume 2, 1970.

2.

Cr awf or d, P. B. & Col l i ns , R. E. , Est i mat ed ef f ect of ver t i cal f r actur es

on secondary r ecover y.

Trans.

AI ME

(1954),

201, p. 192.

3. Dyes, A. B. , Kemp, C. E. & Caudl e, B. H. , Ef f ect of f r act ur e on sweep- out

pat t er n.

Trans.

AI ME

(1958),

213, p. 245.

4. Hagoor t , J . , Weat her i l l , B. D. & Set t ar i , A. , Model l i ng the pr opagat i on

of wat er f l ood- i nduced f r act ur es.

SPEJ ( Aug.

1980) ,

p. 293.

5. Hagoor t , J . , Wat er f l ood- i nduced hydr aul i c f r actur i ng.

PhD. Thesi s , Del f t Techni cal Uni ver s i t y, 1981.

6. Per ki ns, T. K. & Gonzal ez, J . A. , The ef f ect of t her moel ast i c st r esses on

i nj ect i on wel l f ract ur i ng.



- 9 -

CHAPTER TWO

THE PORO- AND THERMO- ELASTI C STRESS FI ELD AROUND A WELLBORE

Summar y

2. 1.

I nt r oduct i on

2. 2. Sol ut i on f or t he st r ess f i el d

2. 2. 1. Assumpt i ons

2. 2. 2.

Not at i on and basi c equat i ons

2. 2. 3. Sol ut i on wi t h Goodi er ' s di spl acement pot ent i al

2. 2. 4. The par t i cul ar sol ut i on i n cyl i ndr i cal coor di nat es

2. 2. 5. Compl et e f or mul at i on of t he pr obl em

2. 2. 6. Met hod of sol ut i on

2. 2. 7. Asympt ot i c expansi on of t he st r ess sol ut i on

2. 2. 8.

Si mpl i f i ed sol ut i on met hod

2. 2. 9. Numer i cal met hod t o eval uat e Ao —

0T



- 10 -

SUMMARY

Changes i n r eser voi r pr essur e or t emper at ur e can change t he st r ess

f i el d i n t he reser voi r r ock. Such changes i n t he st r ess f i el d i nf l uence t he

condi t i ons under whi ch f r act ur i ng of t he r eser voi r rock can occur . They al so

i nf l uence t he geomet r y and di r ect i on of pr opagat i on of i nduced f r act ur es.

Thi s chapt er pr esent s new anal yt i cal and numer i cal met hods f or

cal cul at i ng t he st r ess f i el d ar ound a s i ngl e ver t i cal wel l i n an i nf i ni t e

r eser voi r . Por o- and ther moel ast i c var i at i ons i n the st r ess f i el d resul t i ng

f r om axi symmet r i c changes i n reser voi r pr essur e and t emper at ur e ar e

i ncor por at ed. The met hods ar e gener al i n t he sense t hat no use i s made of

t he assumpt i on of pl ane st r ai n. The f or mul ae pr esent ed al l ow appl i cat i on of

ar bi t r ar y axi symmet r i c pr essur e and t emper at ur e f i el ds.

Si mpl e anal yt i cal f or mul ae f or t her moel ast i c st r ess var i at i ons ar e

pr esent ed f or t emper at ur e di st r i but i ons wi t h a step pr of i l e. Si mpl e

anal yt i cal f or mul ae for por o- el ast i c st r ess var i at i ons ar e pr esent ed for

quasi ,

st eady- st at e pr essur e pr of i l es i nc l udi ng di scont i nui t i es i n f l ui d



- 11 -

THE PORO- AND THERMO- ELASTI C STRESS FI ELD AROUND A WELLBORE

2. 1 I NTRODUCTI ON

I n t hi s chapt er an anal ysi s i s made of t he f act or s t hat i nf l uence t he

str ess f i el d i n t he reser voi r rock surr oundi ng a wel l bor e. I n pr i nc i pl e,

knowl edge of t hi s st r ess f i el d al l ows one t o det er m ne t he downhol e f l ui d

i nj ect i on pr essur e at whi ch t ensi l e f ai l ur e or f r actur i ng of t he rock

occur s .

Thi s so- cal l ed f r acture i ni t i at i on pr essur e i s an i mpor t ant par amet er

i n sel ect i ng a sui t abl e downhol e pr essur e dur i ng i nj ect i on. The i nj ect i on

pr essur e shoul d be l ower i f f r acturi ng i s t o be pr event ed, f or i nst ance t o

avoi d communi cat i on wi t h ot her r eser voi r zones. Or , i t shoul d be hi gher i f

f r actur i ng i s des i r ed, f or i nstance t o i ncr ease the i nj ect i on capaci t y of

t he wel l .

Si nce t he concept of str ess i s gener al l y consi der ed t o be a di f f i cul t



- 12 -

The r el at i on gi vi ng t he st r ess r esul t i ng f r om an appl i ed def or mat i on

or t he def or mat i on resul t i ng f r om an appl i ed st r ess i s cal l ed t he

st r ess/ st r ai n re l at i on. I n t hi s work we cons i der onl y r el at i ons of a l i near

f or m and t her ef or e our anal ysi s f al l s wi t hi n the f r amewor k of t he t heor y of

l i near el as t i ci t y.

When f l ui d i s i nj ected f r om a wel l bor e i nt o a r eser voi r a cer t ai n

pr essur e i s r equi r ed t o squeeze the f l ui d i nt o t he por es of t he reser voi r

r ock. Thi s f l ui d pr essur e i nsi de the wel l bor e exer t s a r adi al l y out war d

f or ce (or equi val ent l y a radi al st r ess or r adi al l oad) on t he sur r oundi ng

r eser voi r r ock. For si mpl i ci t y, we consi der her e an open hol e wi t hout a

casi ng. As a r esul t of t he radi al l oad t he r ock i s def or med ( expandi ng

r adi al l y out ward) and st r esses are gener at ed i n the r ock.

Nor mal l y t he rock i s al r eady i n a st at e of st r ess bef or e t he wel l i s

dr i l l ed. These so- cal l ed t ectoni c st r esses r esul t f r om t he wei ght of the

over bur den and f r om t ect oni c movement of t he ear t h' s cr ust . The r adi al

expansi on of t he bor ehol e now r esul t s i n a decr ease i n the hor i zont al st r ess

t hat i s t angent i al t o the bor ehol e ci r cum er ence. I f t he f l ui d pr essur e i s



- 13 -

shoul d be smal l wi t h r espect t o the i n- si t u t ectoni c st r esses so that non

l i near ef f ect s on mat er i al const ant s such as Young' s modul us and Poi sson' s

r at i o may be negl ected i f t he l at t er ar e taken at t hei r i n- s i t u val ues.

The l i near i t y of t he t heor y enabl es us t o cal cul at e t he st r esses

i nduced by por o- el ast i c ef f ect s, t her moel ast i c ef f ect s and wel l bor e l oadi ng

separat el y. The combi ned ef f ect i s t hen obt ai ned by si mpl y addi ng t he

r espect i ve st r ess component s.

I n t he past por o- el ast i c ef f ect s have been i ncl uded i n t he cal cul at i on

2-4

of t he st r ess f i el d ar ound a wel l bor e . However , t hese wor ks rel i ed on t he

assumpt i on t hat t he i nduced por o- el ast i c def or mat i ons occur onl y i n t he

hor i zont al pl ane of t he reser voi r . Thi s so- cal l ed assumpt i on of pl ane st r ai n

al l ows t he ver t i cal var i at i on of t he st r esses t o be negl ected and r esul t s i n

a cons i der abl e s i mpl i f i cat i on of t he di f f er ent i al equat i ons i nvol ved.

Al t hough i n most cases t he assumpt i on of pl ane st r ai n i s accept abl e

when cal cul at i ng t he st r esses i nduced by t he l oadi ng of t he wel l bor e

( Sect i on 2. 2. 7 of t hi s

work),

t hi s assumpt i on i s gener al l y not val i d when

poro- or t her moel ast i c st r esses ar e consi der ed.



- 14 -

I n Sect i ons 2. 4 and 2. 5 t hese expr essi ons ar e used i n t he gener al

f or mul ae of Sect i on 2. 2 t o der i ve cl osed f or m anal yt i cal expr essi ons f or t he

poro-

and t her moel ast i c change i n t angent i al st r ess. I n Sect i on 2. 6 an

anal ys i s of por o- and t her moel ast i c ef f ects on f ract ure i ni t i at i on pr essur e

i s pr esent ed. Bot h open and cased hol e ar e consi der ed. I n Sect i on 2. 7 t he

change i n t angent i al s t r ess i s cal cul at ed f or t wo r eal i s t i c f i el d cases. I n

Sect i on 2. 8 ot her appl i cat i ons of t he sol ut i on for t he st r ess f i el d ar e

di scussed. Fi nal l y, i n Sect i on 2. 9 t he concl usi ons are pr esent ed.

2. 2 SOLUTI ON FOR THE STRESS FI ELD

2. 2. 1 Assumpt i ons

I n a reser voi r of axi al symmet r y t he f r actur e i ni t i at i on pr essur e for

ver t i cal f r actures depends on the tangent i al st r ess f i el d of t he r eser voi r

r ock. The f ol l owi ng assumpt i ons have been made i n cal cul at i ng t hi s st r ess

f i e l d .



- 15 -



- 16 -

Aa. . = change i n t ot al st r ess t ensor wi t h respect t o the st r ess st at e at

Ap = AT = 0.

3

Ao = t r ace of st r ess t ensor = Z A a.

1- 1

X1

c. . = st r ai n t ensor

5. . = Kr onecker del t a ( 1 i f i =j , 0 i f i / j )

i ]

J J

E = Young' s modul us of bul k r eser voi r r ock

v = Poi sson' s r at i o of bul k r eser voi r r ock

a = l i near t her mal expansi on coef f i ci ent

a = l i near por o- el ast i c expansi on coef f i c i ent

The l i near por o- el ast i c expansi on coef f i c i ent i s def i ned as:

%

=

3

( c

b - v

=

i r

( 1

- ^

} (2

-

2 )

wher e c, and c ar e t he compr essi bi l i t i es of t he bul k r eser voi r r ock and t he



-

17 -

oAT =•a Ap + a AT (2. 3)

p

r

T

so that Eq. (2. 1) becomes:

e. .

= è [ (1+u) Aa. . - uAoS. . ] - aAÏ 6. . ( 2. 4)

13 E 13 13 13

The si gn convent i on i n Eq. (2. 1) and (2. 4) takes compressi ve st resses and

strai ns as posi t i ve.

Eq. (2. 4) can be i nverted to express the stresses i n terms of the

str ai ns.

I f the strai ns are then expressed i n terms of a di spl acement vector

and the stresses are substi tuted i nto the equat i ons of equi l i br i um three

di f f erent i al equat i ons i n the three components of the di spl acement vector

resul t .

For gi ven Ap and AT and for gi ven boundary condi t i ons f or the

stresses t he sol ut i on f or the stress f i el d i s determned. The devel opment of

the di f f erent i al equat i ons i s gi ven i n Appendi x 2-A.

2. 2. 3 Sol ut i on w th Goodi er ' s di spl acement potent i al



- 18 -

and t he f ol l owi ng gener i c not at i on has been i nt r oduced:

AAT = A Ap + A AT ( 2. 9)

p T

A and A ar e cal l ed the por o- and t her moel ast i c const ant s, r espect i vel y.

They ar e def i ned as:

Ef t

1 2

c

A = T-

2

= T

2

- ^ (1

a

) (2 . 10 )

p 1-u l-v c '

b

and

E a

T

A

m = ( 2. 11)

T 1-ü

2. 2. 4 The par t i cul ar sol ut i on i n cyl i ndr i cal coor di nat es

I n t he r est of t hi s chapt er i t i s assumed t hat t he pr essur e and



- 19 -

Two l i m t i ng cases can now be consi der ed. I n the f i r st case we assume

t hat t he ver t i c al var i at i on of AT i ns i de t he r es er voi r z one i s s mal l .

I f f ur t her mor e t he penet r at i on dept h of t he pr essur e and t emper at ure

var i at i ons ar e smal l wi t h r espec t t o t he r eser voi r hei ght , t he onl y

di sp l acement wi l l be i n t he hor i zont al pl ane and:

w = - 3 0 = 0 ( 2. 16)

z

Then ( 2. 13) can be r eadi l y i nt egr at ed, gi v i ng

r

u = - 9

<t>

= T ^ a - ƒ dr ' r '

A T ( r '

, z)

( 2. 17)r l - i ) r

r

w

and fo r t he s t resses f rom

( 2 . 1 5 ) :

A

r

A a = + d r r AT r z



- 20 -

Ac - = Aa. - = AAT( r , z)

rT 0T '

( 2. 20)

Aa - = Aa - = 0 .

zT rzT

We cal l ( 2. 19) the condi t i on of ver t i cal s t r ai n.

Fr om t he above i t i s cl ear t hat dur i ng f l ui d i nj ect i on i nt o a

r eser voi r t he condi t i ons f or por o- and t her moel ast i c st r esses can change

f r om pl ane s t r ai n to ver t i cal s t r ai n. I n Sect i ons 2. 4 and 2. 5 of thi s

chapt er t hi s wi l l be demonst r at ed wi t h expl i ci t exampl es.

I n Appendi x 2- B t he f ul l ( r , z) dependence of t he par t i cul ar sol ut i on

i s obt ai ned f r om ( 2. 13) and ( 2. 15) wi t h the hel p of modi f i ed Bessel f uct i ons

and Four i er i nt egr al t r ansf or ms.

The par t i cul ar sol ut i on does not gener al l y sat i sf y al l pr escri bed

boundar y condi t i ons on t he st r esses. By maki ng use of t he l i near i t y of t he

t heor y, t hi s can be achi eved by addi ng an appr opr i at e sol ut i on of t he

homogeneous equat i ons of el ast i ci t y t o t he par t i cul ar sol ut i on.



- 21 -

IMPERMEABLE

PERMEABLE

~ r i

r 1



- 22 -

Ao = Ap

r w

Aa =0

r z

Aa = 0

r

Aa =0

r z

} « - v | .| < f

( 2. 21)

} r .

V

|.|»f

Si nce we ar e consi der i ng an i nf i ni t e el ast i c medi um we have t he addi t i onal

boundary condi t i ons :

1i m Ao . . = 0

( 2. 22)

r - » ^

lint Aa. . = 0

I z l - »

1J



- 23 -

Ao = Ap

Ao° - O

r z

( 2. 25)

Ao° = O

r

Ao° = Or z

} « - « „ . | . | » f

l i m A o .

.

= l i m Ao. . = 0

Pr obl em ( 2. 25) i s r el at i vel y wel l known and was f i rst sol ved by Tr ant er .

7

Hi s r esul t was extended by Kehl e t o account f or t he shear st r esses exer t ed

on the f or mat i on by t he f r i ct i onal f or ce of t he packer s. Kehl e f ound t hat

f or r eal i s t i c cases t hi s ef f ect can be negl ected.

The sol ut i on to pr obl em ( 2. 24) has, t o our knowl edge, not been

publ i shed i n the l i t er at ur e. We have det er m ned Ao. - by decomposi ng i t i nt o

t he part i cul ar sol ut i on and a sol ut i on to t he homogeneous equat i ons of



- 24 -

TABLE 2- 1 - DECOMPOSI TI ON OF STRESS TENSOR

Aa.

.

= Aa. . - + Aa. .

l j

I J T

i ]

Aa. . = Aa. . - + Aa. . -

l j T 1 } T l j T

; Aa.

.

sat i sf i es compl et e wel l bor e boundary

condi t i ons, Eq. ( 2. 21)

Aa. - sat i sf i es " t r act

I J T

boundar y condi t i ons, Eq. ( 2. 24)

; Aa. - sat i s f i es " t ract i on- f ree" wel l bor e

I J T

Component

Di f f erent i al equat i ons ( d. e. )

Gener at i ng pot ent i al

Aa°.

I D

o

Aa

homogeneous

d. e . of

el as t i c i t y L ove pot ent i al

homoge neo us

d. e. of

el as t i c i t y L ove pot ent i al



- 25 -

Bessel f unct i ons are made. I n t he cal cul at i on of t he i nt egr al s i t i s assumed

t hat t he axi symmet r i c f unct i on AT i s const ant over t he reser voi r hei ght and

zer o out s i de of the reservoi r , i . e.

AT = AT( r ) . H( | - | z | ) ( 2. 27)

wher e h i s t he r eser voi r hei ght and H t he st ep f unct i on def i ned as:

H( a- x) = 1 x < a

( 2. 28)

= 0 x > a

I f t he Bessel f unct i ons are expanded i n power s of r/ h or r ' / h ( whi chever i s

t he smal l est ) t he par t i cul ar sol ut i on f or t he t angent i al st r ess becomes t o

l owest or der :

A( J

0T

=

" 2 2 '

dr

'

r

' AT( r ' ) . N( z , j , r ) + AAT( r ) . H( | - | z| )



- 26 -

I f i n t he sol ut i on for Ao„- t he Bessel f unct i ons cont ai ni ng r and r

ÖT * w

ar e expanded, t he r esul t becomes t o l owest or der i n r / h and r / h:

A

° * T

=

^

A S

HT <

2

'

32

>

wher e

AS

HT = 4 '

d r

'

r

' ^(

r

' ) t

2

'

+

, 3/ 2

+

,

2' , 2, 3/ 2* <

2

'

33

>

r ( z + r ' ) ( z + r ' )

w + —

As f or ( 2. 29) no gener al cr i t er i a can be gi ven f or t he accur acy of

( 2. 32) .

Fur t her di scussi on appear s i n Sect i ons 2. 4 and 2. 5.

I f Tr ant er ' s sol ut i on f or

La

a

i s expanded t o l owest or der i n r / h and

a w

t he t er m wi t h t he modi f i ed Bessel f uncti on K ( kr ) ( see Appendi x 2- D) i s

negl ect ed, we f i nd:

Ap r 2



- 27 -

d

U

+

U

-

M( z, - .

r ) =

~ jji

+ — ( 2. 38)

[ u /

+ 1] ' [ u_ +

1 ]

J / 2

I n F i g. 2. 3 a pl ot of

Aa.

/ Ap ver sus 2z/ d i s pr esent ed f or var i ous

val ues of 2r / d. Poi sson' s r at i o was t aken as 0. 2. The pl ot shows

Ao

a

up to

w a

f i r st or der ( Eq. ( 2. 34) at r = r ) and up t o second or der ( Eq.

(2 . 3 7 ) ) .

I t

i s shown t hat f or 2r / d < 0. 1 suf f i ci ent accur acy i s mai nt ai ned i f t he

second t er m i s negl ect ed. Fur t hermore, f or 2r / d < 0. 01 t he f unct i on N/ 2

w

behaves as a st ep f unct i on so t hat ( 2. 37) becomes:

i o

«

( t

„ '

- -

*\, W

•=

f

|. |»f

2r

}

— r<

0. 01 ( 2. 39)

o 8

The r esul t

AC T

(r ) = - Ap i s the same as t he wel l - known resul t obt ai ned

0

w w

under pl ane st r ai n condi t i ons. I n the l at t er case i t i s assumed t hat t he

l oaded i nt erval d i s i nf i ni tel y l ong.



-

28 -

o.o

- 0 2

-0.4

-0.6

_ - 5 U = L = «

2r

w

/h 0.01

2 r

w

/h = 0.1

2 r

w

/h = 1.0

UP TO FIRST ORDER

UP TO

SECOND

ORDER



- 29 -

We now observe t hat f or | z| < h/ 2 Eq. ( 2. 40) consi st s of t he wel l - known

pl ane st r ai n expr essi on ( 2. 18) pl us t he term AS - .

Taki ng, f or conveni ence, t he di st ance bet ween t he packers as bei ng

equal t o the r eser voi r hei ght ( h=d) , we have f or t he compl et e t angent i al

st r ess f i el d near t he wel l bor e:

Aa

n

= Aa

n

- + Ao„- + La

a

9 6T

0T

6

r 2 r

= (

7

i ) {AS

HT "

A p

w

}

" ~2

S

d r , r , A T

(

r

' ) ( 2. 41)

r r

w

+ AAT( r ) + AS

HT

| z | < |

r 2 "

= ( — AS - + AS - | z | > r

r HT HT ' ' 2

fo r {

2r / h < 0. 01

r - r

w



- 30 -

wher e

AS

v

- ( z ) = - 2AS

HÏ

( z ) ( 2 . 4 4 )

Fr om compar i son wi t h ( 2. 18) AS - can be i nt er pr et ed as an apparent change i n

t he ver t i cal f ar - f i el d st r ess . When pl ane st r ai n sol ut i ons ar e used t hi s

change must be i ncorporated as t he boundar y condi t i on:

l i m

r-*oo

A(7

z

= A S

t

-

VT

- co < z < + » ( 2. 45)

I n Chapt er 3 t he above i nt er pr et at i on i s used i n cal cul at i ng t he por o-

el ast i c st r esses ar ound a ver t i cal f r actur e. The por o- el ast i c st r esses at

t he f r actur e wal l ar e cal cul at ed i n pl ane st r ai n. Eq. ( 2. 42) wi t h a s l i ght l y

adapt ed ver si on of ( 2. 33) i s t hen used t o account f or devi at i ons f r om pl ane

st r ai n.

2. 2. 9 Numer i cal met hod t o eval uat e Aa z



- 31 -

H

2

T

D

= er f c Wr

D

} R < 1, | z | < |

2 ( l - R

D

)

„ 2

A

D

T

D

+

D

= er f c { ~ — R < 1, | z | > f ( 2. 46)

2 / ( r

D

( l - R

D

2

, )

= 0 R > 1, -o» < z < »

wher e

T

D

T

D

R

=

=

=

T - T

r e s

T . . - T

m j r e s

4a t M

s s

2 2

h M

r

r

( 2. 47)



- 32 -

heat absor bed by i nj ect i on f l ui d = heat gi ven of f by t he r eser voi r

or

2

qt M AT = it R h M AT ( 2. 48)

w c r '

At R t he t emper atur e di f f er ence tends t o zer o ( Eq. (2. 46) ) .

The assumpt i on of one di mensi onal ver t i cal heat conduct i on i s j ust i f i ed i f :

M R

Pe =

Ï TTT

>>

x

( 2

-

4 9

>

s s

wher e Pe i s t he di mensi onl ess Pecl et number . Condi t i on ( 2. 49) means t hat t he

r adi al vel oci t y of t he t emper at ur e f r ont i s much gr eat er t han t he ver t i cal

vel oci t y of t he t emper at ur e t r ansi ent s i n cap and base r ock. Ther ef or e, wi t h

i sot r opi c t her mal conduct i vi t i es and appr oxi mat el y equal t her mal

di f f usi vi t i es i n t he r eser voi r and cap and base r ock, r adi al t emper at ur e

t r ansi ent s may be negl ect ed. Usi ng t he expr essi on f or R i n ( 2. 47) , ( 2. 49)

c



- 33 -



- 34 -

where

Ap = p ( r , t ) - p

k.

X

h

q

n

= f l ui d mobi l i t y (X = —)

M

= r eser voi r hei ght

= i nj ec t i on r at e

= hydr aul i c di f f us i vi t y (r? =

k

0MC

f c

t = in je ct io n time

00 - g

Q

Ei

=

exponent i al i nt egr al

(-

Ei ( - x)

= ƒ ds)

x

2

For r /47jt < 0.02 (2 .51) can be approximated by:

2 * ^ - - l n f - (2.52)

q R

e

with

R = 1.5 /(Tj t) (2 .53)

e



7.0

- 35 -

6.0 -

UNE-SOURCE

QUASI STEADY-STATE



- 36 -

t hat i ncompr ess i bl e f l ui d di spl aces s l i ght l y compr essi bl e oi l i n a pi st on

l i ke manner ( Fi g. 2. 6) . The f l ooded zone consi st s of a col d r egi on wi t h

const ant f l ui d mobi l i t y and a war m r egi on wi t h constant f l ui d mobi l i t y. The

f ol l owi ng exact sol ut i on has been obt ai ned:

2 2

27Th . l

n

c 1 _ F 1 1

, F

V

„ . ,

F

x

Ap, = r — I n — + r — I n — - ~ r — exp ( T~7) . E i ( - 7 - )

q * ! X r X R 2 X ^ 4rj t'

v

4 r j t '

2

2

R R R

T

4 p

2

=

x ;

l n

r - 2 ^

e x p

w

) > E l (

" ^

J (2

*

56)

R

2

2

2 ? r h

. 1 1

.

_ F

. r „

— Ap

3

= - - T- exp ( T ) - E i ( - T )

wi t h Ap. , X. , i = , 2, 3 the pr essur e di f f er ences and mobi l i t i es i n t he col d

f l ooded zone, t he war m f l ooded zone and t he oi l zone, r espect i vel y, T i s t he

hydr aul i c di f f us i vi t y i n t he oi l zone. R and R„ ar e t he r adi i of t he

J J



- 37 -

INJECTION FLUID INCOMPRESSIBLE

OIL SLIGHTLY COMPRESSIBLE



- 38 -

2. 4 ANALYTI CAL SOLUTI ON FOR THERMO- ELASTI C STRESS VARI ATI ONS

Cal cul at i on of st r esses f or t emper at ur e f i el d wi t h st ep pr of i l e

I t was shown i n Sect i on 2. 3. 1 t hat a st ep pr of i l e i s a good

appr oxi mat i on t o the t emper at ur e di st r i but i on i nsi de t he reser voi r pr ovi ded

t hat

2

4a t M

r

D

= —| 1- < 0. 05 ( 2. 59)

h M

r

Fur t her mor e, i f we negl ect t he t emper at ur e change i n cap and base r ock due

t o conduct i on, we si mpl y have:

A T ( r , z ) =

AT. H( ^

- | z| ) . H( R - r ) ( 2. 60)

wi t h H t he st ep f unct i on def i ned i n ( 2. 28) .



- 3 9 -

0TD A„AT

T

r

D

=

t

c

2

D

=

R

c

h

D ~ 2R

c

( 2. 63)

z

Af t er i nt egr at i on there r esul t s :

r

„

2

,

r

~

2

— wD 1 wD 1 i i

A a

* T D "

(

- >

A S

H T D

+

I < > ^D ' W " Ï

N

< W

1 ) +

V D *

r

< 1

( 2. 64a)

2



- 40 -

A f ew wel l bor e r adi i out i nt o t he reser voi r Aa

f l r n

„ = (r / r „) AS„„„

0TD wD' D HTD

becomes ver y smal l and Ao.

m

„ becomes i dent i cal to t he par t i cul ar sol ut i on

0TD

Range of val i di t y of asympt ot i c expr essi ons

To i nvest i gat e t he r ange of val i di t y of ( 2. 64) we have f i r st compar ed

t he anal yt i cal sol ut i on f or Ao. (Eq. ( 2. 64) mnus Ao. ) wi t h a numer i cal

eval uat i on of Ao. . The l at t er i s obt ai ned usi ng the met hod descr i bed i n

Appendi x 2- F.

The r esul t s ar e shown i n Fi gs . 2. 7a- 2. 7f . Her e Ao

r t m

„ was made

3

0TD

di mensi onl ess wi t h r espect t o | A T | r at her t han AT and t her ef or e, si nce t he

di mensi onl ess change i n st r ess i s negat i ve f or r < 1, a cool ed r eser voi r i s

r epr esent ed. For compar i son, t he anal yt i cal pl ane st r ai n sol ut i on i s al so

shown ( see Eq. ( 2. 73) bel ow) .

From t he f i gur es t her e i s excel l ent agr eement bet ween t he anal yt i cal

and numer i cal sol ut i on wi t hi n t he cool ed r egi on. For smal l er di mensi onl ess



- 41 -

1.0

- 0 . 6

-0.8

-1.0

0.8

0.6

0.4

0.2

0.0

0.2

0.4

-

-

-

_

h

0

=10 .

z

D

=0 .0

NUMERICAL SOLUTION

PLANE STRAIN SOLUTION (ANALYTICAL)

ASYMPTOTIC SOLUTION (ANALY TICAL)

0. 0 0.2 0.4 0 .6 0.8 1.0 1.2 1.4 1.6

DIMENSIONLESS RADIAL DISTANCE FROM THE WELLBORE

1.8

2.0

FIG.

2.7a CHANGE IN TANGENTIAL STRESS INDUCED BY A STEP PROFILE



-

42

1.0

0.8

0.6

0.4

0.2

0.0

- 0 . 2

- 0 . 4

-0.6

0.8

T

t . 0

0 . 0

- 1 . 0 "-

NUMERICAL SOLUTION

PLANE STRAIN SOLUTION (ANAL YTICAL)

ASYMPTOTIC SOLUTION (ANA LYTICA L)

_ 1 _ _1_

0 . 0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6


1.8 2.0

F I G .

2.7C

CHANGE

IN

TANGENTIAL STRESS INDUCED

BY A

STEP PROFILE



- 43 -

NUMERICAL SOLUTION

PLANE STRAIN SOLUTION (A NAL YTICAL )

ASYMPTOTIC SOLUTION (ANALYTICAL)

J _

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6


1.8

2.0

F IG. 2.7e CHANGE IN TANGENTIAL STRESS INDUCED BY A STEP PROFILE



44 -

<

Z

LU

O

20.0

16.0 -

12.0 -

OC

O

cc

o:

Ld 8.0

<

4.0

0.0

h„ =1.0

r

w0

= 0.006

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

2.0

FIG.

2.7 g RELATIVE ERROR OF ASYMPTOTIC SOLUTION



- 45 -

wi t h AS f r om ( 2. 65a) . The par t i cul ar sol ut i on can be r egar ded as t he

super posi t i on of t he sol ut i on f or a cyl i ndr i cal i ncl usi on wi t h r adi us R and

t emper at ur e AT and t he sol ut i on f or an i ncl usi on wi t h r adi us r and

w

t emper at ur e - AT. Si nce the anal yt i cal sol ut i on f or Ao

a

__ i s accur at e at

r > R f or h > 10 or 2R / h < 0. 1, we can si m l ar l y concl ude t hat ( 2. 68) i s

accur at e at l east f or 2r / h < 0. 1.

w

For Aff „ we have at t he wel l bor e:

0TD

S T D

(r

wD> "

A£

W

r

wD>

+

A f f

Ö T D

( r

w D

)

=

2 A S

H T D

+

« ' V ^D' <

2

'

69

>

Fr om t he si m l ar i t y bet ween ( 2. 69) and ( 2. 68)

Aa

f l

i s al so accur at e at

l east f or 2 r / h < 0. 1. Thi s i s conf i r med by t he f act t hat t he f unct i onal

w

r el at i onshi p bet ween r and h i n ( 2. 69) i s t he same as t hat bet ween r and d

w w

i n the f i r s t t erm of Ao . ( Eq. ( 2. 37) ) . A conser vat i ve cri t er i on f or t he

o

accur acy of t he anal yt i cal sol ut i on f or

Aa

f l

i s 2r / d < 0. 1.

a w

We concl ude t hat when t he use of t emper at ur e di st r i but i on ( 2. 60) i s



- 46 -

h » 1 ( pl ane st r ai n)

AS

HTD

=

°

2

D

=

°

( 3 )

^HTD

=

° l

Z

D

l '

h

D

( b )

} h

Q

> 100

AS

HTD

=

°

'

2

D I

*

h

D

( C ) ( 2

'

701

A

S„

mT

, = T | z j t h„ ( d)

HTD 4 ' D' D '

} 10 < h < 100

A

S„„ „ = " 7 | z j

4

h

n

(e )

HTD 4 ' D' D

0TD

=

2

A" - -

0TD " 4

Z

D = °

I

2

D I

f h

D

( a )

r 2

}(-T

2

) « 1 ( b) ( 2. 71)

D



- 47 -

h « 1 (ver t i cal st rai n)

AS.

HTD 2

AS =0

HTD

Ao

0TD

Aa

0TD

= 1

= 0

Aa

r t m

( r „) = 2

ÖTD wD

*WW

=

°

< h.

> h.

< h.

> h.

< h.

> h.

( 2. 74)

r

„

2

wD

( — ) « 1

D

( 2. 75)

( 2. 76)

We see f r om t hese l i m t s t hat i nsi de the reser voi r t he st r ess changes



- 48 -



- 49 -

R = 1. 5 / (r j t )

e

When ( 2. 77) i s subst i t ut ed i nt o t he por o- el ast i c count er par t of ( 2. 61) t he

i nt egr al s can be easi l y eval uat ed.

We i nt r oduce t he di mensi onl ess var i abl es:

z , h

Z

D

=

R '

f t

D

=

2Re e

. - 2ffhX

A

- „ „„

A

% > D = a T ^

A

°ö p <

2

-

7 9

>

HpD qA Hp

Af t er i nt egr at i on t here r esul t s :

r 2



- 50 -

/ 2

wi t h asi nh x = l n( x + / ( x +

1 ) ) ,

N def i ned i n ( 2. 65b) and H t he step

f unct i on i n

( 2 . 2 8 ) .

At t he wel l bor e ( 2. 80) becomes:

S

P

D

( r

wD> "

A

eD

( r

wD>-

H(

V K I '

+ 2 A S

H

P

D "

<

2

'

8 2

>

As f or t he t her moel ast i c case, a f ew wel l bor e r adi i out i nt o t he

oi r Aa

a

_ = (r _ / r _) As„ _ become

0pD wD D HpD

i dent i cal to t he par t i cul ar sol ut i on Ao,

r eser voi r

Aa

a

„ = (r

n

/r

n

)

As„ „ becomes ver y smal l and

Aa

n

_ becomes

0pD wD D HpD

J

0pD

0pD*

Range of val i di t y of asympt ot i c expr essi ons

As i n the pr evi ous sect i on, we have f i r st compar ed t he anal yt i cal

sol ut i on f or Aa

f l

wi t h the numer i cal one usi ng t he met hod gi ven i n Appendi x

2- F. The r esul t s are shown i n

F i gs .

2. 10a- 2. 10f . The f i gur es show t hat t her e

i s excel l ent agr eement bet ween t he anal yt i cal and numer i cal sol ut i ons bot h

wi t hi n and out si de t he r eser voi r . I n f act , t he cur ves f or t he numer i cal and

anal yt i cal sol ut i on coi nci de compl et el y i n Fi gs . 2. 10a and 2. 10b and al most



5 1 -

r

wD

» 0 . 0 0 3

h

D

« 1 . 0

z „ = 0 . 0

\

V .

_i_

PRESSURE

STRESS (NUMERICAL)

STRESS (ANALYTICAL)

STRESS (PLANE STRAIN)

_ l _

_ 1 _

0.0 0 0.0 5 0.10 0.15

0.20

0.25 0.30 0.35 0.40 0.45 0.50

F I G .

2. 10 a DIMENSIONLESS PORO -ELA STIC STRESS CHANGES



5 2

-

b

<

10.0

9.0

8.0

7.0

6.0

a

a.

* 5.0

4.0

3.0

2.0

1.0

0.0

0.0003

w D

h„ =0.1

z

D

=0.0

V

PRESSURE

STRESS (NUMERICAL)

STRESS (ANALYTICAL)


0 . 0 0 0.05

0.10

0.15

0.20 0.25

0 .30

0.35 0.40 0.45

F I G . 2.10c DIMENSIONLESS PORO-ELASTIC STRESS CHANGES



- 53 -

10.0

9.0

8.0 It

7.0

6.0

5.0

4.0

3.0

2.0

1.0

0.0

r

w 0

= 0 . 0 0 0 0 3

h

D

■ 0 . 0 1

z

D

= 0 . 0

1 ^

PRESSURE

STRESS (NUMERICAL)

STRESS (ANALYTICAL)


i _ _ 1 _

_ 1 _ _ l _

_ 1 _

0.0 0 0.05 0.10 0.15 0.20 0 .25 0.30 0.3 5 0.4 0

0.45 0.50

F IG.

2. 10 e DIMENSIONLESS PORO -ELA STIC STRESS CHANGES



- 54 -

i n

0 8

OS

0.4

1 i r i

\

*

\

\ l 1

■ -

A \

^ ~ "* ** - ^ \ Ai

^**^^ \ . Ml

\

— I —

I I I

N/2

■

h

0

-l O

• h

0

* 1.0

K^- h

0

=

01

m ^

M i " "N .

1"& t ^ ■* „

i V ' **-*



- 55 -

f unct i on i s t he t her moel ast i c count er par t of t he f unct i on — U as can be

seen by compari ng ( 2. 84) wi t h ( 2. 69) and

( 2. 65a) .

From t he f i gur e i t can be

deduced t hat t he behavi our of t he t wo f unct i ons i s ver y si m l ar except t hat

t he f unct i on ( 2. 85) has a weaker dependence on di mensi onl ess r eser voi r

hei ght and r esembl es t her ef or e mor e cl osel y a st ep f unct i on. Fr om t he

excel l ent agr eement wi t h the numer i cal cal cul at i ons and f r om t he si m l ar i t y

wi t h t he t her moel ast i c case, we concl ude t hat t he anal yt i cal expr essi ons

f or Ao„ _ and Aa

a

_ ar e accur at e for al l h_ wi t h the r est r i c t i on t hat 2r / h

0pD 0pD D w

< 0. 1. Agai n, t hi s r est r i ct i on may be somewhat conservat i ve.

The pl ane st r ai n and ver t i cal s t rai n l i m t s

As i n t he t her moel ast i c case t wo l i m t i ng si t uat i ons can be

consi der ed.

h » 1 ( pl ane st r ai n)



- 56 -

HTD

HpD

A p

D

( r

w D >

= 0 . 4 5

= 0 . 1 2 5

h

for h

D

=

ir

c

h

fo r h„ = ——

D 2R

e

A

VD

( r

wD

)

, , .

h

Ap„(r

n

)

= 2

'

Z

D I

< h

D

D wD

( 2. 90)

A p „ ( r

n

)

=

°

I

2

D I

>

D

*l> wD

These l i m t i ng val ues ar e at t ai ned at much smal l er di mensi onl ess r eser voi r

hei ght s t han i n t he cor r espondi ng t her moel ast i c case ( Eq. ( 2. 74) and

Eq. (2. 76) ) . For compar i son:

= 0. 1 ( a)

( 2. 91)

= 0. 1 ( b)

wher e ( 2. 91) was t aken at z = 0.

Pi g.

2. 12 shows AÖ. „( r l / Ap fr

\

= Aa, . ( r ) / (A Ap( r ) ) f or



- 57 -

_ ^

ho = 10.

h

D

= 0.1

h

0

= 0 .001



- 58 -

A a

r 2

w

0p2D

" T '

4S

HpD

+

A

t

>

2 D

(

" -

H (

2 " 1

Z

D

2 2

- 1 ) }

Ï ^ Q ( ^ r ) J ( ^ - ^ ) Q (« .f ,

V

1

X

l h

Sp3D

=

£ >

AS

HpD

+

A

P 3 D

( r )

'

H (

I -

M >

R < r < R

c - - F

( 2 . 9 2 )

\

-<■ '!"»

<- ï r * <^

2

-' l ^

1D

<V * i» ï ' K ' r

1 )

!

R

p

2 X

!

X

l , l

X

l h

%

v



- 59 -

wher e

2ffhX

AS = AS

HpD qA Hp

4

Ap

i D

( r

w

J N ( Z

' I ' V ~ Ï

Q( Z

' t '

r

w

5 +

4

( 1

" x j>

Q

<

Z

' I ' V

+

Ï

(

x j " xj > <" « ' f ' V

+

4 xj

Q

<

2

' I ' V

( 2. 93)

h

Z

+

Z

-

Q( z , - , r ) = as i nh — + as i nh —

2i rhX

bo. -r. ~ —» ba

Q

. i = 1, 2, 3, 4.

0pi D qA 0pi

and the ot her f unct i ons def i ned i n ( 2. 58) , ( 2. 31) and ( 2 .30 ) .

Agai n, at t he wel l bor e we f i nd:



- 60 -

Gener al expr essi on f or f r actur e i ni t i at i on pr essur e

To eval uat e ( 2. 95) we wr i t e:

p( r ) = Ap( r ) + p.

w w l

W *

A

W

+

°8i

( 2

'

96)

A

v v

=

V

( r

w >

+ A

v

r

w

>

+

S T

(

V

wi t h p. t he i ni t i al r eser voi r pr essur e and

a„.

the i ni t i al t angent i al s t r ess

ar ound t he wel l bor e at pr essur e p. and i ni t i al r eser voi r t emper at ur e T

" i

r

res

Fr om Sect i ons 2. 2, 2. 4 and 2. 5 we have:

AffflmU fZ) = A

m

AT(r ) + 2AS„

m

( z ) (a )

0T w T w HT



- 61 -

I . Sudden change t o new i nj ect i on r at e or t emper at ur e

i nj ect i on t ook pl ace at a const ant r at e q and a const ant t emper at ur e AT f or

a cer t ai n t i me t . As a consequence, t her e i s an appar ent change i n f ar f i el d

st r esses by an amount AS + AS . I f t he i nj ect i on r at e i s now i ncr eased,

at what pr essure wi l l f r act ur i ng occur ? To answer t hi s quest i on we have t o

t ake a cl oser l ook at

( 2. 97) .

Eq. ( 2. 97b) consi st s of a rapi dl y var yi ng par t

A Ap( r ) and a sl owl y var yi ng par t 2AS . That i s, i f Ap i s changed i nt o Ap'

wi t hout t he pr essur e pr of i l e bet ween r and R bei ng changed consi der abl y,

AS„ wi l l r emai n appr oxi mat el y const ant but A Ap wi l l become A Ap' . For

Hp p p

pr act i cal pur poses t her ef or e ( 2. 98) and ( 2. 99) can be used as a f r act ur i ng

cr i t er i on when changi ng t o a new r at e wi t h AS hel d const ant and eval uat ed

at t he t i me j ust bef ore t hi s change i s made. AS„

m

and AS„ can be cal cul at ed

HT Hp

f r om t he expr essi ons i n Sect i ons 2. 4 and 2. 5. The same r easoni ng appl i es

when a change i s made t o a new i nj ect i on t emperat ur e so t hat AT ■+ AT' . Eq.

( 2. 98) can be used wi t h t he t er m A„AT repl aced by A„AT* but wi t h AS

T T HT

eval uat ed j ust bef or e t he change and at t he ol d i nj ect i on t emperat ur e. I n



- 62 -

i n t he wel l at t he same pr essur e and t emper at ur e as t he r eser voi r , we have,

usi ng t he sol ut i on ( 2- E- 7) f r om Appendi x 2- E wi t h C = S . and

1

2 Hi

C

l

= r

w

( P

r

S

Hi

) :

°e i

= 2S

Hi "

p

i

( 2

'

1 0 0 )

Subst i t ut i on i nt o ( 2. 98) gi ves :

2( S

Hi

+

AS

Hp

+

AS

HT

)

- V i * V

T

+ °T

p

f i ■

E

—

j

<

2

-

l 0 l

>

p

I f t he penet r at i on dept hs of bot h t he pr essur e and t emper at ur e

var i at i ons ar e smal l wi t h r espect t o t he reser voi r hei ght , we have

AS„ =

AS„

m

= 0 and

Hp HT

2S

Hi - V i * V

T

* °T

p

f i

=

V ^

( 2

-

1 0 2 )



- 63 -

For hor i zont al t ect oni c st r esses of unequal magni t ude a„. i s gi ven

by :

°0i -

S

hi

+ S

Hi

+ 2 ( S

hi "

S

Hi >

C O S 2 9

" ^

( 2

-

1 05

>

wi t h S

L

. t he l east compr essi ve t ectoni c s t r ess, S„. t he i nt er medi at e

hi Hi

t ect oni c st r ess and 9 measur ed f r om t he di r ect i on of S„. .

Hi

The f r acture i ni t i at i on pr essur e i s obt ai ned by subst i t ut i ng t he

m ni mum of ( 2. 105) i nt o ( 2. 98) whi ch r esul t s i n:

3S. . - S„. + 2AS„ + 2AS

t i m

- A p. + A AT + o

m

_ h l Hi H

P

HT £ i T T

( 2 <1 0 6 )

P

The ef f ect of a casi ng

I n t he above anal ysi s t he pr esence of a casi ng i s not consi der ed. I f a

casi ng i s pr esent and i t i s assumed t o be r i gi d so t hat no l oad can be

t r ansf er r ed f r om t he wel l bor e t o t he f or mat i on, t he cont r i but i on of ( 2. 97c)



- 64 -

Est i mat e of hor i zont al t ectoni c str ess

Most of t he t i me no i nf or mat i on on t he hor i zont al and ver t i cal

t ect oni c st r esses i s avai l abl e. I f , however , t he reser voi r i s known to be

si t uat ed i n a t ect oni cal l y r el axed ar ea, t he ver t i cal st r ess shoul d

appr oxi mat el y equal t he pr essur e of t he over bur den. Fur t her mor e, t he

hor i zont al st r esses shoul d be equal i n magni t ude. Assum ng no l at er al

st r ai ns we have f rom Eq. ( 2- A- l ) wi t h e =

e

= 0:

E

CS

Hi "

°

( S

Hi

+ S

Vi

) ]

"

a A ï =

°

( 2

-

1 0 8 )

wher e S„ . i s the i ni t i al ver t i cal tectoni c s t r ess .

Vi

Sol vi ng f or S„ . gi ves :

Hi

s„.

= r

2

- s„. +

A AT

HI

1- Ü

Vi

( 2. 109)

= 7 - S„. + A p. + A

m

( T - T, )

1-y Vi p i T res dep



- 65 -

TABLE 2- 1I - I NPUT DATA

Sandst one L i mest one

r eser voi r r eser voi r

I nj ect i on r at e, m / d

Ti me of i nj ect i on, days

Reser voi r hei ght , m

2 -15

Ef f ect i ve per meabi l i t y t o wat er , m *10

" " t o oi l ,

Col d wat er vi scosi t y, mPa. s

War m " " ,

Oi l vi scosi t y , "

Connat e wat er sat ur at i on

Res i dual oi l sat ur at i on

Por os i t y

- 1 - 4

8000

730

120

250

1000

1. 0

0. 3

0. 3

0. 12

0. 25

0. 24

60

730

50

1. 0

4. 0

0. 7

0. 4

2. 6

0. 30

0. 25

0. 24



- 66 -

Hi gh- per meabi l i t y sandst one r eser voi r

The f i r st case i s a hi gh- per meabi l i t y sandst one r eser voi r i nt o whi ch

col d wat er i s i nj ect ed at a t emper at ur e 70 C bel ow t he r eser voi r

t emper at ur e. The change i n t angent i al st r ess was f i r st cal cul at ed

numer i cal l y usi ng t he met hod gi ven i n Appendi x 2- F. The t emperat ur e

di st r i but i on i s cal cul at ed f rom Lauwer i er ' s so l ut i on and t he pr essur e f r om

expr essi on

( 2. 56) .

I t shoul d be bor ne i n m nd t hat t he numer i cal met hod cal cul at es t he

par t i cul ar sol ut i on to t he tangent i al st r ess and t her ef or e t he boundar y

condi t i ons at t he wel l bor e ar e not i ncor por at ed i n t hi s met hod. The

numer i cal r esul t s ar e t her ef or e appl i cabl e onl y t o t he r egi on wher e

r

w 2

( 7V « 1.

The r esul t s af t er t wo year s of i nj ect i on ar e shown i n Fi gs . 2. 13. a-

2. 13c. As t he r eservoi r has good per meabi l i t y t he change i n pr essur e and t he

cor r espondi ng por o- el ast i c st r ess i ncrease ar e smal l . The t her moel ast i c

st r ess r educt i on i nduced by t he l ar ge degr ee of cool i ng i s cl ear l y

dom nat i ng. Because of conduct i on, cool i ng and a cor r espondi ng l ower i ng of

6 7 -



80 . 0

60.0 -

40 . 0 -

20.0

-60.0

- 8 0 . 0

- 2 0 . 0 -

- 4 0 . 0

20 . 0

- 15.0

- 10.0

- 5.0

0.0

- 5 . 0

- 10 . 0

- -15.0

O

JQ

- 2 0 . 0

0.0 20.0 40.0

60.0 80 .0 100.0 120.0 140.0 160.0 180.0 20 0.0 22 0.0 240.0

RADIAL DISTANCE FROM THE WELLBORE ( m )

F IG.

2. 13 a SANDSTONE RESERVOIR



68 -

4 0 . 0

2 0 . 0 -

- i r

T

r

T r

AT

40.0

20 .0

0.0

0.0

-2 0 . 0 -

- - 2 0 . 0

O

O

-4 0 . 0 -

- 4 0 . 0

- 6 0 . 0 -

- 6 0 . 0

- 8 0 . 0

_L_

_1_

- 8 0 . 0

0.0 2 0.0 40.0 60.0 80 .0 100.0 120.0 140.0 160.0 180.0

VERTICAL DISTANCE FROM THE CENTRE

OF THE RESERVOIR (m )

FIG.

2.13c SANDSTONE RESERVOIR, 20 m FROM THE WELL

2 0 0 . 0 2 2 0 . 0 2 4 0 . 0

6 9



80.0

60.0 -

40.0 -

20.0

0.0

- 2 0 . 0

-

- 4 0 . 0

-

- 6 0 . 0

-

- 8 0 . 0

1

-

-

1

T 1 — 1 I"

I

1 1 I

1 -T— — r —' - r -

i f

11

11

/

1

/

1

/

1

/

1

* 1

*

1

1

-. . i I 1

-1 1

ANALYTICAL

NUMERICAL

.)

..1

-

-

0 .0

20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0

200.0 2 20 .0 240 .0

RADIAL DISTANCE FT?OM THE WELLBORE (m)

FIG. 2.14b SANDSTONE RESERVOIR.COMPA RISON OF ANALYTICAL AND NUMERICAL RESULTS.



- 70 -

r educt i on. The r eason i s t hat af t er t wo year s of i nj ect i on t he radi us of t he

col d f r ont ( 175 m) i s qui t e l ar ge wi t h respect t o hal f t he reser voi r hei ght

( 60

m) ,

gi vi ng a r at her smal l di mensi onl ess r eser voi r hei ght of

h/ 2R = 0. 34.

c

I n

F i gs .

2. 14a- 2. 14c t he numer i cal cal cul at i ons ar e compar ed w t h t he

anal yt i cal cal cul at i ons of

Aa.

and

Ao

f l

. The agr eement i s per f ect f or t he

change i n por o- el ast i c st r ess. Ther e i s a di f f er ence f or t he t her moel ast i c

st r ess because t he anal yt i cal sol ut i on uses a temper at ur e st ep pr of i l e

i nst ead of Lauwer i er ' s sol ut i on. Ther ef or e, t he zone of st r ess decrease i n

cap and base r ock, r esul t i ng f r om conduct i on, i s not pr oper l y r epr esent ed.

Over al l t he agr eement i s ver y sat i sf act or y.

Low- per meabi l i t y l i mest one r eser voi r

The second case i s a l ow- per meabi l i t y l i mest one r eser voi r i nt o whi ch

col d wat er i s i nj ect ed at a t emper at ur e 30 C bel ow t he r eser voi r

t emper at ur e. The par t i cul ar sol ut i on for t he change i n t angent i al st r ess was

cal cul at ed numer i cal l y. The resul t s af t er t wo year s of i nj ect i on ar e shown



- 71 -

120.0

100.0 \-

80 . 0 -

60.0 -

40.0 -

20.0 -

0.0

-20.0

- 4 0 . 0

120.0

100.0

80 . 0

- 60.0

- 40 .0

- 20.0

0.0

- - 2 0 . 0

-40 .0

0.0 4.0 8.0 12.0 16.0 20 .0

24 .0

28.0 32.0

RADIAL DISTANCE FROM THE WELLBORE

( m )

36.0 40.0

FIG. 2.15a LIMESTONE RESERVOIR

- 72 -



40.0

O

X I

V)

(/)

ui

cc

\—

to

z

u i

o

z

<

o

z

<

30.0

20.0 -

10.0

0.0

-10 .0



- 73 -

2. 8 OTHER APPLI CATI ONS

Al t hough i n t hi s chapt er t he sol ut i on f or t he str ess f i el d i s appl i ed

t o t he st r ess st at e ar ound a wat er i nj ect i on

wel l ,

ot her appl i cat i ons ar e

poss i bl e. For i nst ance, i t i s al so poss i bl e t o use t he f or mul ae t o cal cul at e

t he st r ess f i el d ar ound a steam i nj ect i on

wel l .

I n t he l at t er case t he use

of t he Mar x- Langenhei m t emper at ur e di st r i but i on i s mor e appr opr i at e t han

t hat of Lauwer i er . The st r ess f i el d ar ound a pr oducti on wel l can al so be

cal cul at ed wi t h t hese new f or mul ae. Fi nal l y, t he der i ved Goodi er and Love

pot ent i al f unct i ons can be used t o cal cul at e t he di spl acement s r at her t han

t he st r esses .

2. 9 CONCLUSI ONS

1. A si mpl e anal yti cal expr essi on has been der i ved f or t he t angent i al st r ess

r esul t i ng f r om r adi al l oadi ng of t he wel l bor e over an i nt er val d. The



- 74 -

4. If the lateral pressure and temperature penetration depths are large with

respect to the reservoir height the poro- and thermo-elastic stress

changes may become twice as large as predicted by the plane strain

solutions.

5. Simple analytical expressions have been derived for the calculation of

the fracture initiation pressure for a vertical fracture. The effects of

poro- and thermo-elastic changes in rock stress have been incorporated.

6. Cooling of the reservoir following the injection of cold water may induce

thermal fracturing. If the radius of the cold front is still small with

respect to the reservoir height fracturing occurs first near the top and

bottom of the reservoir.

7.

If the thermo-elastic decrease in reservoir rock stress is dominating

with respect to the poro-elastic increase, steep stress gradients may be

created at the vertical boundaries of the reser voir. Thermal f ractures



- 75 -

LI ST OF SYMBOLS

A gener i c el as t i c cons t ant

A por o- el as t i c cons t ant

A t her mo- e l as t i c const ant

c compr ess i bi l i t y of bul k r eser voi r r ock

c c ompr es s i bi l i t y of r oc k gr ai ns

d di st ance bet ween packer s

e t r ac e of s t r ai n t ens or

E Young' s modul us

h r eser voi r hei ght

M heat c apac i t y of f l ui d- f i l l ed r es er voi r r o ck

M heat capaci t y of cap and base r ock

M heat capac i t y of i nj ect i on wat er

R r adi us of col d f r ont

c

R„ r adi us of f l ood f r ont

F

R ef f ec t i v e ext er i or r adi us



- 76 -

Aa t r ace of st r ess t ensor

a

t ensi l e st r engt h of r eser voi r rock

Subscri pt s

T gener i c f or combi ned por o- and t her moel ast i c

T t her moel ast i c

p por o- el ast i c

D di mensi onl ess

Super scri pt s

o sol ut i on t o homogeneous equat i ons of el ast i ci t y

ps pl ane st r ai n



- 77 -

REFERENCES

1. Ti moshenko, S. P. & Goodi er , J . N. , Theor y of El ast i c i t y.

McGr aw Hi l l Book Company I nc. , New Yor k, 1951.

2.

Geer t sma, J . , Pr obl ems of r ock mechani cs i n pet r ol eum pr oduct i on

engi neer i ng.

Proc. Fi r st Congr . of t he I nt l . Soc. of Rock Mech. , Li sbon 1966,

Vol . I , p. 585.

3. Medl i n, W L. & Masse, L . , Labor at or y i nvest i gat i on of f r actur e i ni t i at i on

pr essur e and or i ent at i on.

SPE- 6087, 1976.

4. Ri ce, J . R. & Cl ear y, M. P. , Some basi c st r ess di f f us i on pr obl ems f or

f l ui d- sat ur at ed el ast i c por ous medi a wi t h compr essi bl e const i t uent s.

Revi ews of Geophysi cs and Space Physi cs , 1_4, No. 2 ( May

1976) ,

p.

227.

5. Per ki ns, T. K. & Gonzal ez, J . A. , Changes i n ear t h st r esses around a

wel l bor e caused by r adi al l y s ymmet r i cal pr essur e and t emper at ur e

gr adi ent s .



- 78 -

APPENDI X

2- A

BASI C EQUATI ONS

The l i near s t r es s / s t r ai n r el at i ons

f or

combi ned por o-

and

t her mo-

el as t i c def or mat i ons

ar e :

e. . = è [ ( l +u) Ao. . - uAof i . . ] -

aAT 6. .

( 2- A- l )

i ]

E

v

l ] i ] i ]

wher e Ao.

. i s

t he change

i n

t ot al st r ess wi t h r espect

t o

t he i ni t i al s t r es s

s ta te

at AT =

0. The t r ace

Ao

i s gi ven by:

Ao

= £ Aa..

i - 1

"

The combi ned pr ess ur e and t emper at ure ef f ect has been gener i cal l y denot ed

a s :



- 79 -

gi ves :

Aa. .

= - T77I "T £

3

-

u

-

+ 3

-

u

-

+

T^T"

3

^

u

^

6

-

■

+

T T" AT 8.

.

(2-A-6 )

13 2( l +o) 1 ] 3 1 l - 2o k k 13 l - 2u 13

wher e summat i on over r epeat ed i ndi ces i s underst ood.

Subst i t ut i on i nt o the equi l i br i um condi t i ons f or t he s t r esses:

3 .A a . . = 0 (2 -A -7 )

1 ID

r e s u l t s i n :

- ( l - 2 u ) 3 , 3 , u . - 3 . 3 , u , + 2 ( 1 + 0 ) a 3 .A T = 0 ( 2 - A - 8 )

k k i i k k i

v

Thi s i s a set of t hr ee di f f er ent i al equat i ons i n t he three component s of t he

di spl acement vect or. A par t i cul ar sol ut i on t o ( 2- A- 8) can be f ound by

12

i nt r oduci ng Goodi er ' s di spl acement pot ent i al :



- 80 -

The vol ume di l at at i on gener at ed by t hi s pot ent i al i s f rom ( 2- A- l l )

e = 3. 3. 0 = - 7 ^ aAT ( 2- A- 13)

k k 1- v

v

'

Subst i t ut i on i nt o ( 2- A- 4) gi ves as a par t i cul ar sol ut i on for t he st r esses:

Ao. - = 77- e. . + AAT 6. . ( 2- A- 14)

l j T 1+u l ] l ]

wher e t he f ol l owi ng gener i c not at i on has been i nt r oduced:

AAT = A Ap + A

m

AT ( 2- A- 15)

p

r

T '

Ea Eap T

wi t h A = - —

c

, A„ = - — . A and A

m

ar e cal l ed t he por o- and t her moel ast i c

p 1- u T

1-v

p T

r

constants,

r espect i vel y.

Expr essi ng ( 2- A- 14) i n t er ms of t he di spl acement pot ent i al gi ves:

Ao. - = - 7 - 3. 3. 0 + AAT 6. . ( 2- A- 16)



- 81 -

APPENDI X 2- B

THE PARTI CULAR STRESS SOLUTI ON I N CYLI NDRI CAL COORDI NATES

When AT has axi al symmet r y, i . e.

AT = AT( r , z ) ( 2- B- l )

t he di spl acement vect or has onl y a radi al and a ver t i cal component :

U

l

= U

'

U

2

=

°'

U

3

= W

(2-B -2 )

whi ch can be expr essed i n ter ms of t he di spl acement pot ent i al as :

u = - 3 0 ; w = - 3 0 ( 2- B- 3)

r z

Usi ng t he r el at i onshi p bet ween di spl acement component s and st r ai ns i n

12

cyl i ndr i cal coor di nat es , we f i nd:



- 82 -

A sol ut i on t o ( 2- B- 6) can be obt ai ned wi t h modi f i ed Bessel f unct i ons i n t he

f o r m

j

+00 OS

0 ( r , z ) = - ƒ d r ' ƒ d z* ƒ dk A T ( r ' , z ' ) r ' c o s k ( z - z ' ) I ( k r ' ) K ( k r )

ir

o o

r -o» o

w

( 2 - B - 7 )

C D

+0 0 00

+ - ƒ d r ' ƒ d z ' ƒ dk A T ( r ' , z ' ) r ' c o s k ( z - z ' ) K ( k r ' ) I ( k r )

n

o o

r -e» o

w i t h m = - —

a.

l - o

Usi ng t he f ol l owi ng pr oper t i es of t he modi f i ed Bessel f unct i ons:

K

Q

( X ) = - K^x)

K (x )

K, ( x) = - K (x )

1 O X



- 83 -

r

+• » K (k r )

A a „ - = ƒ d r ' ƒ dz ' ƒ dk A T ( r , z ) r k c o s k ( z - z ) I ( k r ' ) - ^-j

& 1 !

ir

o kr

r

-o» o

w

• +» » I , ( k r )

+

- ƒ d r ' ƒ dz ' ƒ dk

A T ( r , z ) r k c o s k ( z - z )

K ( k r ' ) -

i

—;

ff

o

kr

r

-o» o

+ AAT(r ,z ) (2 -B -10 )

A

° r z T

=

f '

d r

'

*

d z

' '

d k

A T ( r , z ) r k

2

s i n k ( z - z )

I ( k r ' ) K (k r)

r

-o» o

w

( 2 - B - l l )

O D +00 00

A

— 2

ƒ

d r ' ƒ dz ' ƒ dk

A T ( r , z ) r k s i n k ( z - z )

K ( k r ' ) I (kr)

r

-oo o

From ( 2- B- 5) and (2- B- 6) we have:

Aa = Ao - + Ao

f l

- + Aa - = 2AAT ( 2-B- 12)

rT 0T zT * '



- 84 -

APPENDI X 2-C

THE COMPLETE STRESS SOLUTI ON I N CYLI NDRI CAL COORDI NATES

For gi ven AT t he st r esses ar e det er m ned by t he boundar y condi t i ons at

t he wel l bor e and at i nf i ni t y. The wel l bor e- model and t he cor r espondi ng

boundar y condi t i ons ar e di scussed i n Sect i on 2. 2. 5. The pr obl em i s sol ved by

decomposi ng t he st r ess t ensor as f ol l ows:

Ao. . = Aa. - + Ao? . ( 2- C- l )

13 13T i ]

wher e Ao. . - i s t he sol ut i on t o the " t r act i on- f r ee" wel l bor e pr obl em and Ao. .

13T

r

13

i s t he sol ut i on to t he homogeneous equat i ons of el ast i ci t y sati sf yi ng

boundar y condi t i ons of uni f or m r adi al wel l bor e l oadi ng over a l i m t ed

i nt er val .

The boundar y condi t i ons f or t he "t r act i on- f r ee" wel l bor e pr obl em ar e

gi ven by:



- 85 -

a

a

= 3 ( uV

2

- - 3 ) * ( 2- C- 5)

B

z r r

v

a = 3 ( ( l - u) V

2

- 3

2

) $

r z r z

wi t h V

2

= 3

2

+ - 3 + 3

2

(2 -C -6 )

r r r z

v

Si nce ( 2- C- 2) must be sat i sf i ed f or ar bi t r ar y axi symmet r i c f unct i ons

) ,

i t i s cl ear that Ao. - al so <

l j T

dependence i t i s conveni ent t o wr i t e:

A T ( r , z ) ,

i t i s c l ear t hat Ao. - al so depends on

A T ( r , z ) .

To i sol ate t hi s

(2 -C-7 )

00 +00

Ac - = ƒ d r ' ƒ d z ' A T ( r ' , z ' ) AS - ( r , r ' , z , z ' ) ( a)

-* J —00 J

W

go +00

A o ° - = ƒ d r ' ƒ d z ' A T ( r ' , z ' ) Ao"° - ( r , r \ z , z ' ) ( b )

J j - _ 0 0 -"

W

The pr obl em i s sol ved i f Aa. - can be obt ai ned f r om a Love f unct i on such

I J T

- 86 -



00

$ ( r , r ' , z , z ' ) = ƒ dk[ B( r ' , k) K ( kr ) + C( r ' , k) k r K

n

( k r ) ] - r

L

( 2 - C - 9 )

o

l

,

o

Us i ng t he pr oper t i es of modi f i ed Bessel f unc t i ons :

( 3

2

+ - 3 ) K ( kr ) = k

2

K (kr)

r r r ' o o

K*(x) = - K, ( x) ( 2- C- 10)

o 1

K (x)

K

l (

x) = - K

o

( x) -

-

l r

where t he pr i me means di f f erent i at i on w th respect to x, Eq. ( 2- C- 9) can be

ver i f i ed by di rect substi tut i on i nto

( 2- C- 4) .

From (2- C- 5) we obt ai n:

Ao°- = - ƒ dk[B( r' , k) F (kr) + C( r ' , k) P ( kr) ] cosk( z- z' )

o

( 2- C- l l )



-

87

A

2 V V

B F

n

( k r ) + C F „ ( k r ) = - r ' k K ( k r ' H l (kr ) , ]

l

v

w' 2 w

ir

o o w kr

w

( 2 - C - 1 4 )

A 2

I

l

( k r

w

)

B F

3

( k r

w

)

+

CF

4

(kr

w

, = - r'k K^kr') — —

Solving for B and C gives:

A 2 1

r

V V

I

l

{kr

w

)

B = - r ' k K (kr ' ) „,,

t

{[ I (kr )

-.

—] F „ ( k r )

-,

—F „ ( k r )}

ir o D(kr ) o w kr 4 w' kr 2 w'

w w w

A 2 1 , ^ ^ ' w '

I

l

( k r

w

)

C = - - r ' k K ( k r * ) — r — - {[I (kr ) ] F. (kr ) F

n

( k r )}

ir o D(kr ) o w kr 3 w kr 1 w

w w w

( 2 - C - 1 5 )

w i t h

D ( k r ) = F

n

( k r ) F. (kr ) - F . ( k r ) F, (kr )

- 88 -



ST

=

Ao

ei

+

A a

ö ï ( 2 - C - 1 9 )

The r emai nder of t he pr obl em consi st s of sol vi ng t he homogeneous

equat i ons of el ast i ci t y f or a wel l bor e subj ected to a r adi al l oad Ap over

an i nt er val d.

The boundar y condi t i ons ar e:

Aa = Ap , Aa =0 r = r , I z I < r

r w r z w ' ' - 2

Aa° = 0 , Ao°

z

=0 r - r

w

, | z | > f ( 2- C- 20)

l i m Aa. . = l i m Aa.

.

= 0

r-»»

x

3 |

z

| - x»

1 D

Pr obl em ( 2- C- 20) was sol ved by Tr ant er i n a manner anal ogous t o t hat

used above. Usi ng our not at i on and t aki ng compr essi ve st r esses as posi t i ve,

hi s r esul t f or t he t angent i al st r ess becomes:

- 89 -



APPENDI X 2- D

ASYMPTOTI C EXPANSI ONS OF THE STRESS SOLUTI ON

Asympt ot i c expansi on f or t he par t i cul ar sol ut i on

We consi der t he case t hat AT i s const ant over t he r eser voi r hei ght and

zero out s i de of t he r eser voi r :

AT( r , z ) = AT( r ) . H( | - | z | ) ( 2- D- l )

wi t h H t he st ep f unct i on as i n ( 2. 28) .

The i nt egr al over z' i n t he expr essi on ( 2- B- 10) f or Ac, - can t hen be

car r i ed out , resul t i ng i n:

A

r

- 2

K

l

( k r )

Ao^-

= - J ƒ dr ' ƒ dk r ' AT( r ' ) F( k, z, h) k I

0

( kr ' )

k f

r o

w

• °° I ( kr )

+ - ƒ dr ' ƒ dk r ' AT( r ' ) F( k, z, h) k

2

K (kr*)

— ,

( 2- D- 2)

it

o kr

- 90 -



r °»

A( 7

0T

=

" f f

X d r

'

S

d k

7"

A

^

r

' >

F

(

k

'

z

'

h

) ^( k r )

r o

w

o as

+ \ - ƒ dr ' ƒ dk r ' AT( r ' ) F( k, z, h) k

2

K ( kr ' ) + AAT( r , z )

2 f f O

r o

The i nt egr al s over k can now be ef f ect ed, r esul t i ng i n:

Aa

6Ï

=

" 2 2

5

d r

'

r

'

A

T( r ' ) . N( z , | , r )

r r

w

(2-D-6)

(2-D-7)

, ^

+ A A Ï

r z

A

+ r ' ) ' z + r ' )

oo

Z 2

+

f / dr ' r ' AT( r ' ) {_ , ^

/ 2 +

^

2 +

' }

+

AAT( r , z )

wi t h

z z

N(z,

^ , r )

' ' . 2

A

2, 1/ 2 . 2

A

2. 1/ 2

- 91 -



Asympt ot i c expansi on f or Ag. -

Usi ng ( 2- D- l ) t he i nt egr al over z ' i n ( 2- C- 17) can be car r i ed out ,

resul t i ng i n:

<■

"> k K (kr ' )

Ao°-

= - J ƒ dr ' ƒ dk AT( r ' ) r ' F( k, z, h)

9

- —

r o ( kr ) D(k r )

w w w

K (k r )

(

k r

W( kr

w

) - ( l -2i >) K

Q

( k r ) }

( 2- D- l l )

wi t h W def i ned i n ( 2- C- 18) and F i n

( 2- D- 3) .

Apar t f r om ( 2- D- 5) we need t he addi t i onal asympt ot i c expansi ons:

K

o

( x ) * - i n - - 7

K. ( x) -» -

1 x

} X « 1

(2-D-12)

wher e 7 = 0. 5772 i s Eul er ' s cons tant .

- 92 -



Eval uat i on of t he k- i nt egr al r esul t s i n

r 2

Af f

0T

= (

7

i )

A S

HT

( Z )

( 2- D- 15)

wher e

AS

HÏ

( z ) = f ƒ dr ' AT( r ' ) r ' {

2

'

+

+ 2

' ~

2 3 / 2

> ( 2- D- 16,

r ( z

L

+ r ' ) ( z + r ' )

w + -

Asympt ot i c expansi on f or Ao.

I f i n ( 2- C- 21) we expand t he Bessel f unct i ons cont ai ni ng r t o l owest

or der i n r / h and i f f ur t her mor e t he t er m wi t h K ( kr ) i s negl ect ed we have,

w o

usi ng ( 2- D- 13. a) :

A

2

Ap r »

Acr° = - — -

j!L

- ƒ dk F( k, z ,d) k K, ( kr )

a it r 1

o

- 93 -



APPENDI X 2- E

SI MPL I FI ED SOLUTI ON METHOD

Accor di ng t o Sect i on 2. 2. 7 i f 2r / h < 0. 01 t he f unct i on N/ 2 becomes a

st ep f unct i on and t he f unct i on M becomes negl i gi b l y smal l . N and M ar e

def i ned i n ( 2- D- 8) and ( 2- D- 10) r espec t i vel y . Under t hi s condi t i on and c l ose

t o t he wel l bor e ( r - r ) we have f r o m ( 2- D- 7) and ( 2- D- 9) f or t he par t i c ul ar

s ol ut i on:

Ao

m = T / dr ' r ' AT( r ' ) + AS - ( z

r T 2 HT

r r

w

HT

V

i

i h

| z| < 2

1*1 >2

( 2 - E - l )

Aa„ z = - r ƒ dr ' r ' AT( r ' ) + AAT( r ) + AS - ( z ) | z | < £

0T

2

r r

HT'

w

- 94 -



2r / h < 0. 01, wher e f or si mpl i ci t y t he l oaded i nt er val i s t aken as bei ng

equal t o the r eser voi r hei ght

( h=d) .

Thi s suggest s t hat we can sol ve t he

compl et e st r ess pr obl em usi ng pl ane st r ai n sol ut i ons onl y and ( 2- E- 3) as a

boundar y condi t i on at i nf i ni t y. I n the f ol l owi ng we demonst r at e t hat t hi s i s

i ndeed t he case.

The compl et e sol ut i on f or t he str ess f i el d consi st s of t he sum of t he

par t i cul ar sol ut i on and a sol ut i on t o the homogeneous equat i ons of

el ast i ci t y.

We now wr i t e t hi s as:

Ac.

.

= Aa. . - + Ao? . ( 2- E- 4)

i] 13T ij

I f we sol ve t he pr obl em i n pl ane st r ai n and i ncor por at e ( 2- E- 3) we have t he

f ol l owi ng boundar y condi t i ons:

( a )

o = Ap

r w

I

I h

r

= V H

K

2

- 95 -



r

(2 -E -7b )

where t he const ant s C and C- are det er m ned by t he boundar y condi t i ons.

Boundar y condi t i on ( 2- E- 5c) gi ves

C„ = A S -

2 HT

and appl i cat i on of boundar y condi t i on ( 2- E- 5a) t o ( 2- E- 6a) and ( 2- E- 7a)

gi ves :

: = r (Ap - AS - )

1 w *w HT

i

i h

(2 -E -8 )

wher eas

(2 -E -5b ) ,

( 2- E- 6c) and ( 2- E- 7a) gi ve:

C_ = - r AS -

1 w HT

1* 1

> 2

(2 -E -9 )

Addi ng t he sol ut i ons t oget her we have f i nal l y:

- 96 -



Aa

z

-

= AAT( r ) + AS

y

- ( z ) | z | < |

= ^

v ?

( z ) | z| >§

(2-E-13)

wher e

AS

v

- ( z ) = - 2AS

HÏ

( z ) (2 -E -14)

Her e, As - can be i nt er pr et ed as an appar ent change i n t he ver t i cal f ar -

f i el d st r ess whi ch has al so t o be i ncor por at ed as a boundar y condi t i on at

i n f i n i t y ( r

-*

«) when sol vi ng i n pl ane st r ai n.

- 97 -



APPENDI X 2- F

A NUMERI CAL METHOD TO EVALUATE

Ao.-

0

T

I n Appendi x 2- A i t was shown t hat t he par t i cul ar sol ut i on Aa

a

- coul d

0T

be obt ai ned f r om t he di spl acement pot ent i al :

0( x, y, z) = ƒ dx' dy' dz'

QT

(

X

' ' Y ' '

z

' ? ( 2- F- l )

wi t h

2 2 2 1/ 2

R = f ( x- x' ) + ( y- y' ) + ( z - z ' ) ] ' ( 2- F- 2)

I f aT has axi al symmet r y ( 2- F- l ) can be t r ansf or med i nt o cyl i ndr i cal

coor di nat es and t he i nt egr al over t he azi mut h can be eval uat ed l eadi ng t o:

OO +00

0( r , z) = f - ƒ ƒ dz ' AT( r ' , z ' ) {^~)

1/2

X K( X) ( 2- F- 3)

Z7T r

r -<*>

- 98 -



2 2

wher e X' = 1- X and E t he compl et e el l i pt i c i nt egr al of t he second ki nd,

( 2- F- 5) becomes:

OP +OP

A

— r ' 1/ 2 —

Af f

0T

=

"

Ti

!

d r

' '

d z

'

A T

<

r

' '

z

' )

( ~£)

MK ( X )

- u

E ( X ) }

+

AAT( r , z )

r

-o» r

w

( 2 - F - 7 )

wi t h

2 2 2

r ' - r + ( z- z ' )

« = 2

S

'

2

( 2 - F - 8 )

( r - r ' ) + ( z- z ' )

The i nt egr and i n ( 2- F- 7) has a s i ngul ar i t y i n r =r ' , z=z' . I n order t o

eval uat e ( 2- F- 7) numer i cal l y t hi s s i ngul ar i t y has t o be i sol at ed and

eval uat ed anal yt i cal l y. For t hi s pur pose, we wr i t e the sol ut i on to Poi sson' s

equat i on ( 2. 13) i n t er ms of or di nar y Bessel f unct i ons:

OB 09 00 •

# ( r , z )

= - ƒ

dz '

ƒ

dr '

ƒ dk

AT( r ' , z ' ) r '

e ' ' J

( k r )

J

( k r ' ) ( 2- F- 9)

'

i _

o o

-o» r o

- 99 -



Fr om ( 2- F- 12) t he cont r i but i on of t he si ngul ar i t y t o the i nt egr al s i n

( 2- F- 7) can now be wr i t t en:

r +e

z+ e

- _ - kl z- z ' l

J

l

( k r )

6( «. , «_ ) = - f ƒ dr ' ƒ

£

dz ' ƒ dk AT( r •, z' ) r ' k e

K | Z z

' -^r J ( kr ' )

r _ 6

l

Z

'

6

2

(2 -F -13)

I f the r egi on of i nt egrat i on i s suf f i c i ent smal l so that AT i s essent i al l y

const ant i n thi s r egi on, t he i nt egr al over z' can be eval uat ed.

Us i ng:

CO

ƒ dk

J , (k r )

J ( kr ' ) = - r ' < r

1 o r

o

1

2r '

r ' = r ( 2- F- 14)

=0 r ' > r

- 100 -



Usi ng

l i m {K(X)

- £ I n

- i £- }

= 0

(2-F-18)

x

2

+i

x

"

x

we have:

l i m

I

= ; - - ( ;

I n 16 - x l nx}

(2-F-19)

-*n 2 IT 2

e

2°

wth

x = /

2

2

(2-F-20)

•(«2 + « )

F i n a l l y

we

have

t o

l owest order

i n e e :

- 101 -



APPENDI X 2- G

SOLUTI ON FOR THE PRESSURE DI STRI BUTI ON

We assume that i n an i nf i ni t e reser voi r s l i ght l y compr essi bl e oi l i s

di spl aced pi st on- l i ke by i ncompr essi bl e i nj ecti on f l ui d. The f l ooded zone

consi st s of t wo r egi ons of di f f er ent constant mobi l i t y. I n the col d r egi on

t he mobi l i t y i s det er m ned by the vi scosi t y of i nj ect i on f l ui d at t he

i nj ect i on t emper at ur e and i n t he war m r egi on by t he vi scosi t y of i nj ect i on

f l ui d at t he or i gi nal reservoi r t emper at ur e ( Fi g. 4) . Thi s l eads to the

f ol l owi ng set of di f f erent i al equat i ons ,

- 3 ( r 3 ) Ap, = 0 r < r < R

r r r l w — - c

- 3 ( r 3 ) Ap„ = 0 R < r < R„ ( 2- G- l )

r r r *2 c - - P

- 3 ( r3 ) Ap, = - 3 Ap. R < r < »

r r r

r

3 17 t 3 F - -

- 102 -



r espect i vel y:

M

w t

1 / 2

R

c

= C

M~ hj r

]

r

( 2- G- 3]

R =

r 1 at , l / 2

F 0( 1- S - S ) hf f

J

or wc

wher e S i s t he r esi dual oi l sat ur at i on i n t he f l ooded zone, S i s t he

or wc

connat e wat er sat ur at i on and 0 i s t he porosi t y. The ot her symbol s are

def i ned i n Sect i on 2. 3. 1.

Sol ut i ons t o ( 2- G- l ) are sought i n t he f or m

( 2- G- 4)

Ap = A l nr + B; Ap, = C l nr + D

2

A p

3

=

'

Ei

(

t ^ >

+ G

wi t h Ei t he exponent i al i nt egr al :

-

103

-



CHAPTER THREE

ANALYTI CAL MODELLI NG OF FRACTURE PROPAGATI ON

CONTENTS

Summar y

3. 1 I nt r oduct i on

3. 2 Fr act ur e pr opagat i on i n an i nf i ni t e r eser voi r i n t he absence

of r eser voi r st r ess changes


3. 2. 2 One- di mensi onal l eak- of f

3. 2. 3 Two- di mensi onal l eak- of f - Pseudo- r adi al sol ut i on

3. 2. 4 Two- di mens i onal l eak- of f - El l i pt i cal sol ut i on

3. 3 The ef f ect of poro- and t her moel ast i c st r ess changes on

f r act ur e pr opagat i on pr essur e

3. 3. 1 Def i ni t i on of f r actur e pr opagat i on pr essur e

3. 3. 2 Anal yt i cal cal cul at i on of por o- el ast i c s t r ess

- 104 -



Appendi x 3- A Cal cul at i on of por o- el ast i c st r esses i n el l i pt i cal coor di nat es

Appendi x 3- B A numer i cal met hod f or cal cul at i ng por o-

and t her moel ast i c st r ess changes

Appendi x 3- C Cal cul at i on of t her moel ast i c st r esses

and of the axes of the el l i pt i cal f l ui d f ront s

- 105 -



SUMMARY

Anal yt i cal model l i ng of wat er f l ood- i nduced f r actur e pr opagat i on i s

di scussed. A model i s pr esent ed wi t h a compl et e t wo- di mensi onal descr i pt i on

of f l ui d l eak- of f i nto the r eservoi r . A di mens i onl ess i nj ect i on rat e i s

def i ned and i t i s shown t hat f or val ues of t hi s number smal l er t han 0. 61 t he

f r act ur e pr opagat es wi t h pseudo- r adi al l eak- of f . Thi s means t hat t he

pr essur e t r ans i ent s are t ravel l i ng radi al l y i n t he pl ane of t he r eser voi r .

An appr oxi mat e t hr ee- di mensi onal cal cul at i on of por o- el ast i c st r ess changes

at t he f r act ur e f ace i s per f or med anal yt i cal l y f or a f r act ur e sur r ounded by

a pseudo- r adi al pr essure pro f i l e i ncl udi ng el l i pt i cal di scont i nui t i es i n

f l ui d mobi l i t y. The r esul t s are compar ed wi t h numer i cal cal cul at i ons and ar e

shown t o be cor r ect f or r at i os of f r actur e hal f - l engt h t o r eser voi r hei ght

smal l er t han 10. The numer i cal met hod can be easi l y i ncor por at ed i nt o

numer i cal f r act ur e/ r eser voi r si mul at or s such as devel oped i n t he past . I f a

t hermal si mul at or i s used, t he same met hod can be empl oyed t o cal cul at e

t her moel ast i c st r ess changes. An anal yt i cal model f or wat er f l ood- i nduced

f r act ur e gr owt h wi t h pseudo- r adi al l eak- of f under t he i nf l uence of por o- and

- 106 -



ANALYTI CAL MODELLI NG OF FRACTURE PROPAGATI ON

3. 1 I NTRODUCTI ON

I n t he pr evi ous chapt er a st udy was made of t he condi t i ons under whi ch

f r act ur es ar e creat ed at an i nj ect i on

wel l .

Thi s chapt er i nvest i gat es t he

f act or s t hat i nf l uence t he gr owt h of such an i nduced f r act ur e. Knowl edge of

t hese f act or s may hel p us t o deci de whet her f r acturi ng of i nj ect i on wel l s i s

benef i ci al or det r i ment al t o t he per f or mance of t he di spl acement pr ocess i n

a gi ven r eser voi r .

Unl i m t ed f r actur e gr owt h can have a number of undesi r abl e

consequences . F i r s t , i f a f racture grows rapi dl y i nt o the reser voi r i t wi l l

di st or t t he geomet r y of t he di spl acement f r ont s. Dependi ng on t he posi t i on

of t he pr oduct i on wel l s t hi s may r esul t i n pr emat ur e br eakt hr ough of

i nj ect i on f l ui d and consequent l y i n a poor sweeep ef f i ci ency of t he pr ocess.

Second, i f a f r actur e pr opagat es ver t i cal l y i nto anot her r eser voi r , l oss of

i nj ect i on f l ui d may occur . Thi r d, the successful appl i cat i on of a t er t i ar y

- 107 -



t he f ormat i on dur i ng pumpi ng i s al ways model l ed as one- di mensi onal

per pendi cul ar t o t he f r act ur e sur f ace.

Fract ur es extendi ng f r om an i nj ect i on wel l ar e nor mal l y not cr eat ed

wi t h speci al l y desi gned f l ui ds. They ar e usual l y i nduced wi t h t he same f l ui d

t hat i s i nj ect ed f or t he pur pose of cont r ol l i ng t he di spl acement pr ocess i n

t he r eser voi r ; most of the t i me thi s i s j ust wat er . Qui t e of t en, f r actur i ng

occur s uni nt ent i onal l y when t he wel l i s br ought t o i t s r equi r ed i nj ect i on

pot ent i al .

Si nce her e the i nj ect i on rate of t he f l ui d and i t s vi scosi t y ar e

usual l y much l ower t han i n a t ypi cal st i mul at i on j ob, and si nce no addi t i ves

ar e pr esent t o m ni m se t he l eak- of f , an i nj ect i on wel l f r actur e gr ows mor e

sl owl y t han a f r actur e i n a st i mul at i on j ob. As a r esul t , t he l eak- of f has

an essent i al l y t wo- di mensi onal f l ow pat t er n i n t he pl ane of t he r eser voi r

r at her t han one- di mensi onal . Fur t her mor e no speci al f l ui d r heol ogy needs t o

be consi der ed and t he vi scous pressur e dr op al ong t he f r act ur e i s

negl i gi bl e.

I n cont r ast to st i mul at i on j ob f r act ur es, whi ch ret ai n a f i xed l engt h

af t er t he j ob has been compl et ed, i nj ect i on wel l f r act ur es keep pr opagat i ng

- 108 -



el ement ,

t he f r act ur e becom ng st at i onar y f or bal anced i nj ect i on and

pr oduct i on.

Hagoor t et al . di d not cons i der the ef f ect of f l ui d mobi l i t y r at i os

di f f er ent f r om uni t y on f r actur e pr opagat i on and sweep ef f i c i ency.

Fur t her mor e, t he ef f ect of t her moel ast i c st r esses on f r act ur e pr opagat i on

wer e not i ncl uded, whi l e the ef f ect of por o- el ast i c st r esses wer e i ncl uded

onl y i n a ver y s i mpl i s t i c way.

3

Recent l y, Per ki ns and Gonzal ez pr esent ed a sem - anal yt i cal model of a

f r ac tur e ext endi ng f rom a si ngl e i nj ect i on wel l i nto an i nf i ni t e r eser voi r .

Leak- of f was model l ed i n t wo di mensi ons. I n addi t i on, t her moel ast i c changes

i n r eser voi r r ock str ess and thei r ef f ect on f r act ur e pr opagat i on pr essur e

wer e i ncor por at ed. I t was shown that cool i ng of t he r eser voi r r ock f ol l owi ng

i nj ect i on of col d wat er may cause f r act ur es t o become ver y l ong. I n t hei r

work Per ki ns and Gonzal ez di d not pr ovi de a sat i sf act ory met hod of

cal cul at i ng t he ef f ect of por o- el ast i c st r ess changes on f r acture

pr opagat i on, nor di d t hey consi der t he ef f ect of near by pr oducer s on

f r actur e gr owt h.

4

- 109 -



on sweep ef f i ci ency i n a pat t er n f l ood i s consi der ed. The use of a numer i cal

r eser voi r si mul at or and t he i ncor por at i on of t he numer i cal met hod of Sect i on

3. 3 f or cal cul at i ng changes i n rock st r ess i s di scussed. F i nal l y, i n Sect i on

3. 6 t he concl usi ons ar e pr esent ed.

3. 2 Fr act ur e pr opagat i on i n an i nf i ni t e r eser voi r i n the absence of

r eser voi r st r ess changes


I n t he const r uct i on of our si mpl e anal yt i cal f r act ur e pr opagat i on model

we use t he f ol l owi ng assumpt i ons:

1. A ver t i cal f r actur e wi t h a r ectangul ar sur f ace ar ea ext ends l at er al l y

f rom a s i ngl e wel l i n an i nf i ni t e reser voi r . The f r actur e hei ght i s

const ant and i s t he same as t he r eser voi r hei ght . The f r act ur e shape i s

el l i pt i cal i n hor i zont al cros s- sect i ons ( F i g. 3. 1) .

2.

The f r actur e has i nf i ni t e conduct i vi t y, i . e. t he f l ui d pr essur e dr op

al ong t he f r act ur e can be negl ect ed.

- 110 -



- I l l -



wher e

p. = i ni t i al r e ser voi r pr e ss ur e

l

Ap

=

P

f

- p. wi t h

p

f l ui d pr essur e i ns i de f r ac tu re

k

V

0MC

f c

k

=

per meabi l i t y

u -

vi scos i t y

c = t ot al c ompr es s i bi l i t y

= por os i t y

t

=

t i me s i nce start

of

i nj ect i on

2

2

—s

e r f c

=

compl ement ar y er r or f unct i on ( er f c( x)

=

-j -

ƒ e ds )

x

Appl yi ng Dar cy

' s l aw t o

( 3. 1) gi ves ,

f or t he

one- di mensi onal l eak- of f

per

f r ac tu re a rea :

k 3 E

=

k _ Ap_

1_

- 112 -



2

x

wi t h

a

an, as yet , unknown const ant . Thi s gi ves r ( x) =

—

. Subs t i t ut i ng t hi s

a

i nt o ( 3. 4) and i nt egr at i ng gi ves

a

2j r khA

P

( 3

'

6 )

conf i r m ng t hat a i s i ndeed const ant and that ( 3. 5) i s cor r ect .

Def i ni ng t he di mensi onl ess quant i t i es:

L

D

=

T i ^ y

a n d q

D

= i ^ z ;

(3

-

7 )

( 3. 5) and ( 3. 6) gi ve, f or t he f r actur e pr opagat i on:

L

D

=

H

q

D

(3 .8 )

We shal l cal l ( 3. 8) t he Car t er sol ut i on si nce the above anal ys i s i s s i m l ar

t o t hat gi ven by Car t er i n Ref . 5. The di f f er ence i s t hat Car t er consi der ed

t he compl et e vol ume bal ance:

- 113 -



r eser voi r r ock st r ess ar e negl ect ed, t hi s pr essur e i s const ant and equal s

t he i n- si t u hor i zont al r eser voi r r ock- st r ess S . Rock mechani cal changes i n

pr opagat i on pr essur e wi l l be di scussed i n Sect i on 3. 3.

The assumpt i on of one- di mensi onal l eak- of f per pendi cul ar t o t he

f r a c t u r e i s j u s t i f i e d i f

3 E

/ IE

3x

/

3y

« 1 ( 3. 12)

wher e p = p( x, y) i s t he por e pr essur e di st r i but i on ar ound t he f r act ur e.

By usi ng ( 3. 1) and account i ng f or t he t i me si nce l eak- of f began, t he

sur r oundi ng pr essure pr of i l e becomes:

p ( x , y , t ) - p

i

= Ap er f c ( - )

2/ ( r ? ( t - ) )

a

T

L y

= Ap er f c ( 2—) ( 3. 13)

- 114 -



I n F i g. 3. 2 t he pr essur e penet r at i on f ront ( 3. 15) i s pl ot t ed f or

var i ous val ues of L . Condi t i on ( 3. 17) means t hat t he vel oci t y of f r act ur e

pr opagat i on i s much gr eat er t han t he vel oci t y wi t h whi ch t he pr essur e

di st ur bance t r avel s i nt o the r eser voi r .

3. 2. 3 Two- di mensi onal l eak- of f - _Pseudo- r adi al sol ut i on

I f t he f r act ur e pr opagat es much mor e sl ow y t han t he pr essur e

di st ur bance, we have

L

D

« 1 ( 3. 18)

and Car t er ' s model i s cer t ai nl y not val i d.

When condi t i on ( 3. 18) i s sat i sf i ed, i t may be assumed t hat t he f r act ur e

behaves quasi - st at i cal l y. That i s, t he pr essur e pr of i l e may be det er m ned by

t r eat i ng L as a const ant and maki ng t he subst i t ut i on L ■ L( t ) af t er t he

pr essur e sol ut i on has been f ound. I n addi t i on, t he pr essur e penet r at i on

f r ont may be t aken t o move r adi al l y out war d i nt o t he r eser voi r wi t h r espect

- 115 -



1.0

0.8

L

D

=L/SQRT(77t)

L

D

=5.

L

0

=10.

L

D

=50.

L

o

=10O.

0.6

0.4

- 116 -



Agai n L behaves accor di ng t o Eq. ( 3 . 5 ) , but now wi t h a much smal l er a. We

shal l cal l ( 3. 21) t he 2- D pseudo- r adi al sol ut i on. Fr om ( 3. 10) and

( 3 . 2 0 ) ,

t he di mensi onl ess r at e of change of f r act ur e vol ume becomes:

A V

f D

=

f * f < ? >

eXP(

-^

5 (3

*

22)

3. 2. 4 Two- di mensi onal l eak of f - El l i gt £cal _sol ut i on

_ _ _

Hagoor t et a l . obser ved t hat , i n an i nf i ni t e reser voi r , L i s al ways

pr opor t i onal t o t he squar e r oot of t i me, even i n t he r egi on i nt er medi at e

4

bet ween ( 3. 17) and ( 3. 18) . I n Ref . 2 and al so i n Hagoor t ' s t hesi s , a gr aph

i s pr esent ed of L vs. q . Thi s gr aph was cal cul at ed numer i cal l y wi t h a

numer i cal r eser voi r si mul at or coupl ed t o an anal yt i cal f r act ur e model . I n

t he f ol l owi ng we gi ve an anal yt i cal sol ut i on of L as a f unct i on of q t hat

al so hol ds i n the i nt er medi at e regi on.

7

Gr i ngar t en et al . gave t he f ol l owi ng anal yt i cal expr essi on f or t he

const ant r at e sol ut i on of t he pr essur e i n a st at i onar y i nf i ni t e conduct i vi t y

- 117 -



Usi ng t he pr oper t i es of t he er r or f uncti on and t he exponent i al i nt egr al :

er f ( x) -» 1

} x » 1 ( 3. 27)

Ei ( - x) -» 0

er f ( x) - ^ x

Ei ( - x) -* - Jn(yx)

} x « 1

wher e 7 = 0. 5772, i t can be shown f r om ( 3. 26) t hat Car t er ' s sol ut i on ( 3. 8)

i s r et r i eved f or L » 1 and t hat t he 2- D pseudo- r adi al sol ut i on ( 3. 21) i s

r et r i eved f or L « 1.

I n Fi g. 3. 3, Eq. ( 3. 26) , whi ch we cal l the 2- D el l i pt i cal sol ut i on, i s

shown t oget her wi t h the Car t er and t he 2- D pseudo- r adi al sol ut i on. I n Fi g.

3. 4 t he 2- D el l i pt i cal sol ut i on i s compar ed wi t h t he gr aph i n Hagoor t ' s

t hesi s on p. 132. The agr eement i s ever ywher e wi t hi n 10%

7

Accor di ng t o Gr i ngar t en et al . , the l i near f l ow per i od i s val i d f or

- L18 -

T 1 1 I I I I I 1 1—I M I N I 1 1 1 l /l I I |J



L

D

=L/SQRT(77t )

q

D

= q / x / 2 7 r k h A p

- 119 -



10

Iff r

_

_

,.

-

-

-

_

-

-

-

#

#

•

1 1 — r -

*

.*

*

/

/ /

■ i — r T T r r

#

S f

•• /

//

//

//

1 1 i y y i i i

S '

s '

s /

sf '

• • • #

/ '

9

,

-

-

-

- 120 -



Fi g. 3. 3 cl ear l y demonst r at es t he i mpor t ance of t he way l eak- of f i s

model l ed t o t he pr edi ct i on of f r act ur e l engt h. For q < 0. 61 t he cor r ect

l eak- of f model i s 2- D pseudo- r adi al . I f , i nst ead, t he 1- D Car t er model i s

used, a f r act ur e l engt h i s pr edi ct ed t hat i s t oo l ar ge by 86% or mor e. The

di f f er ence becomes r api dl y l ar ger as q decr eases.

3. 3 The ef f ect of por o- and t her moel ast i c st r ess changes on f r act ur e

pr opagat i on pr essur e

3. 3. 1 Def i ni t i on of t he f r act ur e gr ogagat i on gr essur e

I n t he f ol l owi ng we anal yse t he f act or s t hat i nf l uence the f r act ur e

pr opagat i on pr essur e. Agai n we consi der t he wedge- shaped f r act ur e of

Fi g. 3. 1.

I f t he f l ui d pr essur e i n t he f r actur e and t he hor i zont al st r ess i n t he

r eser voi r rock are uni f or m over t he f r actur e sur f ace, t he pr opagat i on

pr essur e i s gi ven by :

- 121 -



P

f

= S

H

( 3. 33)

I f was shown i n Chapt er 2 that changes i n r eser voi r t emper at ur e or

pr essur e af f ect t he st at e of st r ess of t he reser voi r r ock. When the

r eser voi r i s cool ed, t he rock t ends t o cont r act and a t her moel ast i c

decr ease of S_ r esul t s. Si m l ar l y, heat i ng causes S t o i ncr ease. When t he

H H

r eser voi r pr essur e r i ses t he rock mat r i x t ends t o expand, r esul t i ng i n a

por o- el ast i c i ncrease i n S .

H

One of t he i mpor t ant r esul t s of Chapt er 2 i s t hat t he var i at i on i n

hor i zont al st r ess f or a gi ven wel l bore pr essure or t emper at ur e depends on

t he r at i o of t he r eser voi r hei ght and t he penet r at i on dept h of t he pr essur e

or t emper at ur e f r ont . The st at e of def or mat i on or st r ai n of t he r eser voi r

rock r anges f rom str ai n onl y i n a hor i zont al pl ane ( pl ane st r ai n) f or smal l

penet r at i on dept hs t o s t r ai n onl y i n a ver t i cal di r ect i on ( vert i cal s t r ai n)

f or l ar ge penet r at i on dept hs. For a cer t ai n wel l bor e pr essur e or t emper at ur e

t he change i n hor i zont al st r ess under ver t i cal st r ai n condi t i ons may be

l ar ger t han t he change i n st r ess under pl ane st r ai n condi t i ons by as much as

- 122 -



assumed t hat an el l i pt i cal ar ea i n whi ch the change i n pr essur e i s uni f or m

sur r ounds t he f r act ur e. Whi l e a t emper at ur e pr of i l e wi t h t he shape of a st ep

f unct i on i n a l at er al di r ect i on i s qui t e real i s t i c under cer t ai n condi t i ons ,

such a pr of i l e i s hi ghl y unr eal i s t i c f or a pr essur e f i el d cr eat ed by f l ui d

f l ow. I n the f ol l owi ng we cal cul at e t he por o- el ast i c st r ess changes at t he

f r act ur e f ace usi ng a mor e r eal i st i c appr oach.

*

Empl oyi ng t he el l i pt i cal coor di nat e system ( Fi g. 3. 5) :

x = L cosh |

COSJ J

( 3. 35)

y = L si nh £ si nrj

t he f ol l owi ng expr essi on f or t he st eady- st at e pr essur e pr of i l e sur r oundi ng

g

an i nf i ni t e conduct i vi t y f r act ur e was der i ved by Muskat ,

Ap (

* > * i S h

l n

<L coshg +

I

si nh i >

( 3

'

36

>

- 123 -



- 124 -



The corr espondi ng poro- el ast i c st r ess changes a t t h e f r ac tu re f ace are

der i ved i n Appendi x 3- A. When t he ass umpt i on of pl ane s t r ai n i s made, the

di mensi onl ess st r ess changes become:

A a

( P

^

= J Ap r o) - 7

( 3. 39a)

ypD 2 *D 4

Ao

( p

= J Ap f o) + 7 ( 3. 39b)

xpD

2 * D ' 4 '

wher e

Aa » , .

A xp 2)rkh

Ao „ = .

xpD

A qu

( 3. 40)

wi t h

an

anal ogous def i ni t i on

f or t he

y- component

and

Ap„( o) =

P (

° * 2j rkh ( 3. 41)

r

D qu

The super scr i pt ps denot es pl ane st r ai n. A i s t he por o- el ast i c const ant as

def i ned

i n t he

same appendi x.

- 125 -



Q( u) = 2 asi nh j - ( 3. 44)

wi t h h t he r eser voi r hei ght . The super scr i pt ( axi al ) denot es axi symmet r i e.

Equat i on ( 3. 43) gi ves t he appar ent change i n f ar - f i el d st r ess at t he

ver t i cal cent re of the r eservoi r .

The di mensi onl ess st eady- st at e pr essur e dr op bet ween t wo concent r i c

ci r cl es wi t h radi i r and r i s gi ven by:

Ap £

a x i a l )

= l n ^ ( 3. 45)

The di mensi onl ess st eady- st at e pr essur e dr op bet ween t wo conf ocal el l i pses

wi t h maj or axes a , a and m nor axes b , b can be obt ai ned f r om ( 3. 45) by

r epl aci ng r and r wi t h an "aver age r adi us"

a. + b.

r . - - ^ — - i = 1, 2 ( 3. 46)

so t hat we have

- 126 -



AS

HpD

=

2

A P

D

( 0 ) +

4

Q (

V ï

Q(

f >

( 3

'

4 9 )

wi t h

2R

Ap

D

( o) = I n - ^ ( 3. 50)

As i s di scussed i n Appendi x 3- A t he addi t i onal boundar y condi t i on at

i nf i ni t y i s sat i s f i ed i f AS„ i s s i mpl y added to ( 3. 39) . The modi f i ed

HpD

por o- el ast i c st r ess changes at t he f r act ur e f ace t her ef or e become:

Aa

y

P

D

=

2

A P

D

( 0 )

- Ï

+ A S

H

P

D

( 3

'

5 1 a )

Ao „ =

T;

Ap„( o) + 7 + AS„ „ ( 3. 51b)

xpD 2 * D

V

4 HpD

w i t h AS

IT

„ f r o m ( 3 . 4 9 )

HpD

2

- 1 2 7 -



27rh . , . . , 1 . ,

d

F *

b

F . __ 1 , , 3 . 0 i / ( T ? t ) , , .

v

—

A p

2

a)

- r

m e

L cosh i + L s i n h |

)

+

r

in

(

y

+

^ >

(b)

2 j F r

2jrh . .

,

v

1 . 3. 0 • ( nt ) ,

%

— A p

3

( { ) = - l n (

L cosh£

J ^

s

J

n h £

) (c)

wi t h Ap. , X. , i = , 2, 3 t he pr essur e change and f l ui d mobi l i t y i n t he col d

f l ooded, war m f l ooded and oi l zone r espect i vel y, T i s t he hydr aul i c

di f f us i vi t y i n t he oi l zone, a „ and b _ ar e t he maj or and m nor axes of

c ,F c ,F

J

t he el l i pt i cal col d and f l ood f r ont , respect i vel y.

I t i s shown i n Appendi x 3- A t hat t he cor r espondi ng st r ess changes i n

pl ane st r ai n become:

Ao

<P

!

}

-

}

A

P i n

( o) - RypD 2 *1D

A o

( P

^ =

\

Ap f o) + R

xpD 2 ' I D

( a )

( 3. 54)

( b )

- 128 -



(a xia l) , , • i, i

AS = i

Ap

( a x i a l )

( r ) - ^ Q( r )

HpD 2

y

l D

V

w

1

4

u v

w'

( 3. 57)

1

X

l 1

\ \ 1 \

+ 4 ( i - T

2

) Q( »

c

> ♦ 4 ( ^ " xj ) Q( R

P

) ♦ 4 ^ Q( R

e

)

wi t h R def i ned i n ( 3. 37) , Q i n ( 3. 44) and R , R t he r adi us of t he col d and

f l ood f r ont r espect i vel y. Fol l owi ng ( 3. 46) we make t he subst i t ut i ons:

a

„ +

b

«

R . - >

C

^

F

,

Cf F

( 3. 58)

c, F 2 '

A p

Dl

X i a l , ( r

w

)

"

A p

Dl

( 0 ) f r

°

m

( 3

'

5 3 a )

The modi f i ed por o- el ast i c st r ess changes at t he f r act ur e f ace become:

- 129 -



3. 3. 3 Numer i cal £al cul at i on_of _gor o- el ast i c st r ess_changes_at t he f r act ur e

wal l

We have devel oped a ver y si mpl e numer i cal met hod f or ver i f yi ng t he

anal yt i cal expr essi ons f or t he por o- el ast i c str ess changes at t he f r act ur e

wal l .

The met hod consi st s of di vi di ng t he reser voi r i nt o Car t esi an gr i d

bl ocks wi t h a const ant pr essur e wi t hi n each bl ock. Anal yt i cal expr essi ons

can be der i ved f or t he str esses exer t ed by an i ndi vi dual bl ock. The st r esses

at t he f r act ur e f ace ar e cal cul at ed by summ ng t he cont r i but i ons of al l

bl ocks. The met hod i s out l i ned i n Appendi x 3- B. I n Tabl e 3- 1 t he r esul t s of

numer i cal cal cul at i ons ar e pr esent ed t oget her wi t h t hose of anal yti cal

cal cul at i ons f r om expr essi ons ( 3. 39) and ( 3. 51) . The r ange of di mensi onl ess

r eser voi r hei ght s consi der ed i s 0. 01 £ h S 1. 0 wi t h h = h/ 2R . For l ar ger

val ues of h t he pl ane st r ai n expr essi ons ar e suf f i c i ent l y accur at e.

I t i s shown that t he anal yt i cal r esul t s f or Ao „ ar e wi t hi n 1% and f or

ypD

Ao ^ wi t hi n 5% of t he numer i cal r esul t s f or L/ h £ 1. 0. For L/ h > 1. 0 t he

xpD

anal yti cal r esul t s become l ess accur at e, t he di f f er ence wi t h t he numer i cal

r esul t s bei ng 10% f or Ao _ and 7% f or Ao

n

when L/ h = 10 and I r = 0. 01.

xpD ypD D

-

130 -



TABLE

3- 1 -

CALCULATION

OF

PORO -ELASTIC STR ESSES

AT

FRACTURE WALL

COMPARISON

OF

NUMERICAL

AND

ANALYTICAL RESULS

N um . A n a l . A n a l .

Num.

A n a l . A n a l .

h

2R

e

0. 01

0. 1

L/ h

0. 01

0.1

1.0

10

0. 01

0. 1

1.0

xpD

6. 82

5. 66

4. 32

2. 28

4. 56

3. 40

2. 06

xpD

6. 82

5. 66

4. 42

2. 51

4. 56

3. 41

2. 16

Ao<P^

xpD

4. 85

3. 70

2. 55

1. 40

3. 70

2. 55

1. 40

ypD

6. 32

5. 16

3. 93

2. 17

4. 06

2. 90

1. 67

ypD

6. 32

5. 16

3. 92

2. 01

4. 06

2. 91

1. 66

Aa

( P

^

ypD

4. 35

3. 20

2. 05

0. 901

3. 20

2. 05

0. 901

Ap

D

( o )

9. 21

6. 91

4. 60

2. 30

6. 91

4. 60

2. 30

- 131 -



The numer i cal met hod used i n Ref . 3 di f f er s f r om t he one we used i n

t hat i t d i vi des t he reservoi r i nt o smal l adj acent cyl i nder s r at her t han i nt o

smal l par al l el epi peds. The anal yt i cal r esul t s f or t he str esses exer t ed by

one cyl i nder i s known and t he cal cul at i on i s per f or med by summ ng over al l

cont r i but i ng cyl i nder s .

Si nce our met hod uses Car t esi an gr i d bl ocks i t has t he advant age

t hat i t can be easi l y i ncor por at ed i nt o numer i cal f r actur e/ r eser voi r

si mul at or s such as devel oped i n Ref . 2 and r ef i ned i n Ref . 10. The

appl i cat i on f or such si mul at or s wi l l be di scussed f ur t her i n t he next

sec t i on.

3. 4 Fr act ur e pr opagat i on i n an i nf i ni t e reser voi r under t he i nf l uence of

r eser voi r st r ess changes

I n t he f ol l owi ng we const r uct an anal yt i cal f r act ur e pr opagat i on

model t hat i ncor por at es t he ef f ect of por o- and t her moel ast i c changes i n

r eser voi r r ock st r ess on the f r acture pr opagat i on pr essur e.

- 132 -



These assumpt i ons enabl e p. i n ( 3. 32) t o be cal cul at ed f r om Ap ( o)

i n ( 3. 53a) . S„ can be cal cul at ed f r om ( 3. 34) wi t h Aa f r om ( 3. 59) and Aa „

' H yp yT

f r om t he expr essi on i n Appendi x 3- C. At ever y t i me t t he maj or and m nor

axes of t he el l i pt i cal col d and f l ood f r ont s can be cal cul at ed f r om a heat

and vol ume bal ance r espect i vel y as was s hown i n Ref . 3 and i s agai n r epeat ed

i n Appendi x 3- C. Subst i t ut i on of p and S i nt o ( 3. 32) t hen l eads t o a non

£ a.

l i near al gebr ai c equat i on f or t he f r acture hal f - l engt h for ever y t i me t .

Thi s equat i on i s sol ved wi t h a si mpl e Newt on i t er at i on pr ocedur e.

Thi s model i s si m l ar t o t he one pr oposed by Per ki ns and Gonzal ez i n

Ref .

3. The mai n di f f er ence i s i n t he cal cul at i on of t he por o- el ast i c

st resses . Per ki ns and Gonzal ez have cal cul at ed Aa f r om t he f or mul ae f or

yp

Aa i n Appendi x 3- C af t er maki ng t he subst i t ut i ons AT

Ap and A -* A .

They do not st at e expl i ci t y how Ap i s def i ned. Thi s pr ocedur e, however ,

i mpl i es t hat t he f r act ur e i s assumed t o be sur r ounded by an el l i pt i cal zone

of uni f or m change i n pr essur e Ap. We have t aken t he por o- el ast i c st r esses

f r om t he anal ys i s i n the pr evi ous sect i on. Her e t he por o- el ast i c st r esses

- 133 -



4a t M

r

D

= " f " - * j < 0. 05

(3. 61,

h M

r

where a i s t he t her mal di f f usi vi t y of t he cap and base r ock and M , M

ar e the vol umet r i c heat capaci t i es of t he f l ui d- f i l l ed r eser voi r r ock and

of t he sur r oundi ng f or mat i on, r espect i vel y.

We def i ne ( 3. 61) al so as a cr i t er i on i n the case of el l i pt i c symmet r y.

3. 4. 3 Two_f i el d_cases

Fracture propagat i on has been eval uat ed f or t wo real i st i c f i el d

cases.

Bot h cases deal wi t h wat er f l oodi ng i n whi ch t he t emper at ur e of t he

i nj ect i on wat er i s l ower t han the or i gi nal r eser voi r . t emper at ur e. The t wo

cases cor r espond t o t he ones consi der ed i n Chapt er 2, sect i on 2. 7. The i nput

dat a ar e gi ven i n Tabl e 3- I I .

Hi cj h- germeabi l Hy sandst one_r eser voi r

- 134 -



TABLE 3- I I - I NPUT DATA

Sandst one

r eser voi r

8000

730

500

450

120

250

1000

1.0

0.3

0.3

0. 12

Li mest one

r eser voi r

60

730

220

160

50

1. 0

4. 0

0. 7

0. 4

2. 6

0. 30

I nj ect i on r at e, m / d

Ti me of i nj ect i on, days

I ni t i al hor i zont al reservoi r s t r ess, bar

I ni t i al reservoi r pr essur e, bar

Reser voi r hei ght , m

2 -15

Ef f ect i ve per meabi l i t y t o wat er , m *10

»

o i l

f

Col d wat er vi scosi t y, mPa. s

War m " " ,

Oi l vi scosi t y , "

Connat e wat er sat ur at i on

- 135 -

500

100



4 9 0 - INITIAL HORIZONTAL STRESS = 500 bar

o

a:

(A

CO

U i

IX.

O L

y

o

I

O

O

m

480

470

460

450

INITIAL PRESSURE = 450 bar

gup

FRAC LENGTH

MAJOR AXIS,COLD FRONT

MINOR AXIS, COLD FRONT

INJECTION RATE = 80 00 m V d

20 40 60

TIME OF INJECTION (days)

8 0

8 0

6 0

40 5

20

100

F IG.

3 .6 FRACTURE GROWTH IN SANDSTONE RESERVOIR

- 136 -



di f f us i vi t y i n the oi l zone gi ves f or L af t er 100 days: L = 0. 004, hence

( 3. 30) i s sat i sf i ed. As an addi t i onal check we can appl y ( 3. 31) . From

Fi g.

3. 6 t he aver age bot t omhol e pr essur e, p. , i s 460 bar , t hus dur i ng

pr opagat i on Ap = p

f

- p. = 10 bar . For a conser vat i ve check, i t may be assumed

t hat t he r eser voi r cont ai ns onl y col d wat er . Thi s gi ves q = 0. 49 and 3. 0

exp( - l / q ) = 0. 39. Assumpt i on 4 i s t her ef or e j ust i f i ed. Assum ng a t ypi cal

f r actur e hal f - wi dt h w of 10 m t hen, f rom ( 3. 22) , t he t i me at whi ch

AV = 0. 01 i s gi ven by t ( AV = 0. 01) i s 3. 4 hour s. Assumpt i on 3 i s t her ef or e

j ust i f i ed. From ( 3. 61) we have that at t = 100

days,

T = 0. 003. The

assumpt i on of a shar p t emper at ur e f r ont i s t her ef or e al so consi st ent .

Low- ger meabi l i t v^ l i mest one_r eser voi r

The second f i el d case t hat we have eval uat ed i s a l ow per meabi l i t y

l i mest one r eser voi r i nt o whi ch col d wat er i s i nj ected at a t emper at ur e 30 C

bel ow t he r eser voi r t emperat ur e. The r esul t s ar e shown i n F i gs . 3. 8 and 3. 9.

As, f or t he sandst one r eser voi r t he pr esence of a f r act ur e wi t h L = 0. 3 m was

-

137

234

INITIAL PRESSURE = 160 bar BHP

20



~ 232

o

X I

. INITIAL HORIZONTAL STRESS

= 220 bar

a:

V)

(/>

U l

ce

a .

O

x

O

t

O

CD

230

228

226

224

FRAC LENGTH

MAJOR AXIS.COLD FRONT

MINOR AXIS.COLD FRONT

INJECTION RATE

= 60 nf/d

^

4 0 60

TIME OF INJECTION (days)

F I G .

3.8

FRACTURE GROWTH

IN

UMESTONE RESERVOIR

80

16

12

8

100

- 138 -



t = 100 days we have f r om ( 3. 61) r = 0. 017. The model i s t her ef or e

consi st ent under t he gi ven condi t i ons.

Moni t or i ng of i nj ect i vi t y_ i ndex

For t he sandst one r eser voi r S„ cont i nuousl y decr eases because of

H

cool i ng. As a r esul t t he i nj ect i on pr essur e r equi r ed t o pr opagat e t he

f r act ur e decr eases f r om 477 bar i ni t i al l y t o 450 bar af t er 100

days.

The

i nj ect i vi t y i ndex ( I I ) , def i ned as q/ Ap wi t h Ap = p

f

- p. , cor r espondi ngl y

3 3

i ncreases f r om 296 m / d. bar t o 1000 m / d. bar , a f actor of 3. 4.

For t he l i mest one r eser voi r t he combi ned ef f ect of an i ncr easi ng S and

a decr easi ng resi st ance t o f r act ur e pr opagat i on ( r i ght - hand si de of ( 3. 32) )

r esul t s i n an al most const ant i nj ect i on pr essur e dur i ng t he 100 days of

pr opagat i on. Consequent l y t he I I i s al most const ant over t hi s per i od wi t h a

sl i ght t endency t o i ncrease.

I f t he i nj ect i on wel l i s unf r actur ed and t he i nj ect i on f l ui d has a

l ower mobi l i t y t han the reser voi r f l ui d t he I I decl i nes cont i nuousl y. Thi s

i s a consequence of more i nj ect i on pr essur e bei ng r equi r ed as t he l ow

- 139 -



3. 5 FRACTURE PROPAGATI ON I N A PATTERN FLOOD. EFFECT ON SWEEP EFFI CI ENCY

3. 5. 1 Zer o voi dage i n t he absence of r eser voi r st r ess changes

Cal cul at i on of stabl e f r act ur e l engt h

I ni t i al l y, t he f ract ure pr opagat es as i n an i nf i ni t e reser voi r . Dur i ng

t hi s st age t he f r act ur e pr opagat i on depends on t he di mensi onl ess i nj ect i on

rate and on t he hydraul i c di f f us i vi t y i n t he r eser voi r ( Sect i on 3. 2) .

Af t er a whi l e, however , t he t r ansi ent s of t he i nj ect i on and pr oduct i on

wel l s wi l l meet and t he f r act ur e gr owt h wi l l be mai nl y det er m ned by t he

voi dage r epl acement i n t he pat t er n. For uni t mobi l i t y r at i o t he f r act ur e

becomes st abl e i f i nj ecti on and pr oduct i on ar e i n bal ance ( zer o voi dage) .

W t h a voi dage r epl acement r at i o gr eat er t han one ( over i nj ect i on) t he

f r act ur e cont i nues t o gr ow wher eas wi t h a voi dage r epl acement r at i o l ess

t han one ( under i nj ect i on) i t wi l l st op gr owi ng and c l ose.

For zer o voi dage and uni t mobi l i t y r at i o, t he i nj ect i on and pr oduct i on

t r ansi ent s meet at a const ant pr essur e boundary wher e t he pr essur e r emai ns

at the i ni t i al reservoi r pr essur e.

- 140 -



( h

/

/

/

\

\

/ \

/

/

X

A

\

-f)

/d

\

\

/

/

/

- 141 -



For a f r actur ed i nj ector we use ( 3. 62) and subst i t ut e L/ 2 f or r :

^ ^

.

ln >08

f

(3<63)

qu L

wher e / A was r epl aced by i t s equi val ent , d, t he di st ance bet ween i nj ect or

and near est pr oducer .

We can now sol ve ( 3. 63) f or t he st abl e f r act ur e l engt h at zer o voi dage

and uni t mobi l i t y rat i o:

J = 1. 08 exp (- ZT) ( 3. 64)

d q

D

wher e q , t he di mensi onl ess i nj ecti on r at e, i s def i ned as i n

( 3 . 7 ) :

<3„ = o

,

u A ( 3. 65)

D 2f f kh Ap

v

wher e Ap = p

f

- p. and p i s t he f r act ur e pr opagat i on pr essur e whi ch f or

l onger f r act ur es, accor di ng t o Sect i on 3. 2, equal s S , t he hor i zont al r ock

- 142 -



pr i or t o t he onset of st eady- st at e, t hi s mode of f r acture pr opagat i on i s not

expect ed t o i nf l uence t he sweep ef f i ci ency.

To pr ove thi s we l ook at t he si t uat i on f or a non- pr opagat i ng f r act ur e

wi t h a const ant l engt h throughout i nj ect i on.

I n Ref . 7 i t i s shown that f or a f r act ur e wi t h const ant l engt h pseudo-

r adi al f l ow i s reached at t i me:

t = 3. 0

L

2

/ TJ

( 3. 68)

wher e t he subscr i pt pr denot es pseudo- r adi al f l ow.

Accor di ng t o the conj ect ur e above t her e i s no ef f ect on sweep-

ef f i c i ency i f pseudo- r adi al f l ow i s r eached bef or e t he onset of st eady- st at e

or i f t < t . Fr om ( 3. 67) and (3. 68) we t hen f i nd:pr ss

L/ d < 0. 21 ( 3. 69)

4

Hagoor t cal cul at ed sweep ef f i ci enci es f or a 5- spot wi t h a f r actur e of

- 143 -



0.8

0.4

3

O

1 4 4 -



l : U

Cl f t

\J.O

o c

U.Ö .

- ° 0 4

U.*T

u

° 0.2

0

1.0

n o

U. o

ex

>-

-

\ M

, \

\ -\

N

A

\ \

\ x

^<^X^

A

« u i /

J-*~*

-,; » , . ~"

m

'

- 145 -



The f r act ur e l engt h at ( war m wat er br eakt hr ough can sti l l be

cal cul at ed f r om ( 3. 64) pr ovi ded t he cor r ect r eser voi r pr essur e and

mobi l i t i es ar e used.

Let us denot e t he f i xed bot t omhol e pr essure i n. t he pr oducer s by p . For

t he pr oducer s t o del i ver war m wat er at r at e q at br eakt hr ough t he reservoi r

pressure has to be (Eq. 3. 62) :

p = p + - 2 - ( f ) I n 0. 54 — ( 3. 70)

^ *w 2irh k' ww r

v

w

wher e q = q f or a 5-spot and q = 1/ 3 q f or a 9-spot wi t h equal r at es f or

P P

cor ner and si de wel l s, q i s t he i nj ecti on r at e. The subscri pt denot es warm

wat er .

I f at wat er br eakt hr ough t he const ant pr essure squar e sur r oundi ng t he

i nj ector i s mai nl y f i l l ed wi t h col d wat er , t he f r actur e l engt h at

br eakt hr ough can be obt ai ned f r om ( 3. 64) pr ovi ded q i s t aken as:

q^ =

9

• (£) ( 3. 71)

- 146 -



wi t h Aa , Aa _ t he por o- and t her moel ast i c changes i n r ock st r ess at t he

yp yT

e

f ract ure f ace.

Aa can be cal cul at ed wi t h t he met hods i n Sect i on 3. 3. 2. I t i s

yp

appr oxi mat el y gi ven by:

Aa = c. A . ( p. - p) ( 3. 74)

yp p f

wher e c has a val ue i n t he r ange 0. 5 < c < 1. 0» dependi ng on t he r at i o of

h/ d wi t h h t he r eser voi r hei ght . I n most cases c wi l l be cl ose t o 0. 5.

Taki ng c = 0. 5, subst i t ut i ng i nt o ( 3. 73) and sol vi ng f or p , we f i nd:

2S + 2Aa - A p

P

f

= —

~ 2^H

*"

( c = 0>5)

( 3. 75)

I f we assume t hat t he col d- wat er f r ont has become mor e or l ess ci r cul ar by

t he t i me war m wat er br eaks t hr ough, Aa can be cal cul at ed f r om Eq. ( 2. 67)

i n Chapt er 2:

- 147 -



Of cour se, t he model shoul d be modi f i ed such t hat t he cal cul at i on of

t he pr essur e r i se bet ween t he steady- st at e val ue f or R and t he f r act ur e

t akes onl y the di scont i nui t i es i n f l ui d mobi l i t y cont ai ned wi t h R i nt o

account .

For exampl e, no pr essure r i se bet ween R and t he f l ood f r ont R

shoul d be cal cul at ed i f R_ > R .

F e

I n a si m l ar way t he model can be used t o det er m ne t he maxi mum

f r actur e pr opagat i on pr essur e t hat wi l l occur i n t he per i od f r om f r act ur e

i ni t i at i on unt i l wat er br eakthr ough. Thi s i s i mpor t ant f or t he eval uat i on of

ver t i cal f r act ur e cont ai nment ( see Chapt er 5, sect i on

5 . 6 . 1 ) .

3. 5. 3 Gener al f l oodi ng condi t i ons and t he use of a reser voi r si mul at or

To anal yse f r act ur e gr owt h and sweep ef f i ci ency f or l ar ger

di mensi onl ess i nj ecti on rates or a voi dage repl acement r at i o di f f er ent f r om

2

one, a numer i cal s i mul at or i s requi r ed. Hagoor t et a l . wer e the f i r st t o

coupl e an anal yt i cal model of a pr opagat i ng f r act ur e t o a numer i cal

r eser voi r si mul at or . The ef f ect of r eser voi r st r ess changes on f r act ur e

pr opagat i on were not t aken i nt o account except f or a ver y much si mpl i f i ed

- 148 -



3. 6 Concl usi ons

1. An anal yt i cal model of f r acture gr owt h i n an i nf i ni t e r eser voi r and i n

t he absence of r eser voi r st r ess changes has been pr esent ed. The onl y r ock

mechani cal par amet er i ncor por at ed i n t hi s model i s t he hor i zont al

r eservoi r s t r ess . I t i s shown that l eak- of f f rom the f r actur e i nt o t he

r eser voi r i s one- di mensi onal per pendi cul ar t o t he f r acture f ace f or

di mensi onl ess i nj ect i on r at es gr eat er t han 4. 5. The l eak- of f i s 2- D

pseudo- r adi al f or di mensi onl ess i nj ect i on rates smal l er t han 0. 61. I n t he

l at t er case a one- di mensi onal descr i pt i on wi l l l ead t o an over est i mat i on

of f r act ur e l engt h by a f act or of t wo or mor e.

2.

The t hr ee- di mensi onal por o- el ast i c st r ess changes at t he f r act ur e f ace

i nduced by a 2- D pseudo- r adi al pr essur e pr of i l e that i ncl udes el l i pt i cal

di scont i nui t i es i n f l ui d mobi l i t y have been cal cul at ed anal yt i cal l y. For

smal l r at i os of r eser voi r hei ght t o pr essur e penet r at i on dept h the 3- D

st r ess changes ar e si gni f i cant l y l ar ger t han t hose cal cul at ed under an

assumpt i on of 2- D pl ane st r ai n.

3. A numer i cal met hod has been pr esent ed f or cal cul at i ng por o- and t her mo-

- 149 -



7.

The numer i cal met hod f or cal cul at i ng por o- and t her moel ast i c st r esses as

pr esent ed her e can be i ncor por at ed i nt o a numer i cal f r act ur e/ r eservoi r

si mul at or . For t he st udy of f r act ur e pr opagat i on under mor e gener al

f l oodi ng condi t i ons such a si mul at or can be a power f ul t o o l .

- 150 -



Li st of symbol s

a maj or axi s of el l i pse

A ar ea of const ant pr essur e squar e

A por o- el ast i c const ant

P

A t her moel ast i c const ant

c compr essi bi l i t y of bul k r ock

c compr essi bi l i t y of r ock gr ai ns

c t ot al por e compr ess i bi l i t y

e t r ace of st r ai n t ensor

e b / a

c c c

e

F V

a

F

er f er r or f unct i on

er f c compl ement ar y er r or f unct i on

E Young' s modul us

Ei exponent al i nt egr al

g.

.

met r i c t ensor


- 151 -



Greek

a

l i near por o- el ast i c expansi on coef f i c i ent

a t her mal di f f usi vi t y of cap and base r ock

a l i near t her mal expansi on coef f i ci ent

5. . Kr onecker del t a

J O

c. . s t r ai n t ensor

I D

7 } hydraul i c di f f us i vi t y

X f l ui d mobi l i t y

u

vi scos i t y

u Poi sson' s rat i o

0 por osi t y

£ el l i pt i cal coordi nat e

a.

.

st r ess t ensor

T di mensi onl ess heat i nj ect i on t i me

- 152 -



REFERENCES

1. Veat ch, R. W J r . , Over vi ew of cur r ent hydr aul i c f r acturi ng desi gn and

t r eat ment t echnol ogy - Par t I .

J PT ( Apr i l 1983) , pp. 677- 687.

2. Hagoor t , J . , Weat her i l l , B. D. & Set t ar i , A. , Model l i ng t he pr opagat i on

of wat er f l ood- i nduced f r act ur es.

SPEJ ( Aug.

1980) ,

pp. 293- 303.

3. Per k i ns, T. K. & Gonzal ez, J . A. , The ef f ect of t her moel ast i c st r esses on

i nj ect i on wel l f ract ur i ng.

SPEJ ( Feb. 1985) , pp. 78- 88.

4.

Hagoort , J . , Wat er f l ood- i nduced hydr aul i c f r acturi ng.

Ph. D. Thesi s , Del f t Techni cal Uni ver s i t y, 1981.

5. Car t er , R. D. , Appendi x t o "Opt i mum f l ui d char acter i st i cs f or f r actur e

ext ensi on" by G. C. Howard and G. R. Fast ,

Dr i l l ,

and Pr od. Pr ac , API

(1957),

p. 267.

6. Kucuk, F. & Br i gham E. W , Trans i ent f l ow i n el l i pt i cal sys tems.

SPEJ ( J une 1981) , pp. 309- 314.

- 153 -



13.

Wei nber g, S. , Gr avi t at i on and cosmol ogy - Pr i nci pl es and appl i cat i ons of

t he gener al theory of r el at i vi t y.

W l ey & sons, New Yor k

(1972),

chapt er I V.

14.

Adl er , R. , Baz i n, M. & Schi f f er , M. , I nt roduct i on t o gener al r el at i vi t y.

McGr aw- Hi l l

(1975),

second edi t i on, p. 149.

15.

Nowacki , W , Ther moel ast i c i t y.

Engl i sh edi t i on, Per gamon Pr ess

(1962),

p. 50.

- 154 -



APPENDI X 3- A

CALCULATI ON OF PORO- ELASTI C STRESSES I N ELLI PTI CAL COORDI NATES

The por o- el ast i c l i near st r ess - s t r ai n rel at i ons i n a gener al

cur vi l i near coor di nat e syst em ar e gi ven by:

E

E<1

Aa.

. = 7T~ («• •

+

7~T~

e g. .) +

T H ?

-

Ap g. . (3-A-l)

i ] 1+u i ] l - 2o ' 13 l -2u ' 13 '

wher e a i s the l i near por o- el ast i c expansi on coef f i c i ent gi ven by:

o = ( — J ( 1 - -

2

) ( 3 - A - 2 )

p E c,

b

and c and c, ar e the compr essi bi l i t i es of t he gr ai ns and of t he bul k mat r i x

- 155 -



Ea

A = —

E

(3 -A -6 )

p 1-ü

The scal ar 0 i s cal l ed Goodi er ' s di spl acement pot ent i al and sat i sf i es:

k kJ

0.

k

= ( 9

*

f /

) .

k

= " m Ap ( 3- A- 7)

wi th m = *- *- a .

(1 -u) p

The covar i ant der i vat i ve

é

. . i s gi ven by

; i ]

2

0 . . = — 7 — : <t> - T. . ~ <t> (3-A-8)

; i ] - i

a

: i ] - k

3x 3x 3x

k

wi t h T. . bei ng t he af f i ne connect i on. The l at t er i s r el at ed t o the met r i c

13

13

t ensor by :

- 156 -



12

and t he el ement s of t he af f i ne connect i on ar e gi ven by :

_m

_ 1

m

mm h _ m

m dx

( 3- A- 12)

, 3h h 3h

_m _ _,m _ 1_ m _n _ _ra m

mn nm h . n mm , 2 , n

i 9x h dxn

wher e m n ar e di f f er ent and r epeat ed i ndi ces ar e not summed. I n a conf ocal

el l i pt i al coordi nat e sys tem ( £ , T J ) t he met r i c t ensor i s gi ven by

L

2

—

(cosh2{

-

COS2TJ)

0

• i

3

"

{

, »

L

( cosh2£ - cos2rj ) ( 3- A- 13)

- 157 -



u

* ^

"i

\-t

2

"2

From

( 3- A- 5) , ( 3- A- 8) ,

( 3- A- 14) and ( 3- A- 15) we have

A. _ _ J _ r i _ 3_é _ k_ s i nh2j 3£ L_ s i n2q Mi .

a A n

Aa

{p~ 1+u

l

h

2 ^2 2

h

4 3* 2

h

4 3r,

J +

V

P

Af l

_ _ I _

f

l _ j f i . L ! si nh2$ 30. _ l £ si nh2r? 30,

OT

"

1 + ü

V

Bv

2 2

h

4 3

*

2

h

4 3

" P

. E , 1 3 f t L s in2 ?? 30 L

2

s i n h 2 { 3 0 , , „

A <

W TT

V

(~2 Jïfr, - T "^4 if " F ~ 9?

(3

~

A

-

16)

- 158 -



The el l i pse wi t h coor di nat e £ separ at es t he i nf i ni t e r eser voi r i nt o t wo

r egi ons.

The i nner r egi on f or ms t he ar ea t hat i s af f ect ed by t he change i n

pr essur e. We have t o def i ne a separ at e pot ent i al i n t he out er r egi on to

ensur e cont i nui t y of t he di spl acement s acr oss £ .

We seek a s ol ut i on i n t he f or m

mr

2

* *i

mT

2

* !

a

" 2

J d É

l ' <3{

2

cosh2$

2

Ap( É

2

) d{

2

- =U - Ap( ncos2r j

o o

-2{

+ K e cos2r j + K cos h2{

COS2TJ

( 3- A- 21)

0 = K e~

2

' cos2r ? + K £ ( 3- A- 22)

By usi ng ( 3- A- 18) i t can be shown by di r ect subst i t ut i on t hat ( 3- A- 21) i s a

sol ut i on of ( 3- A- 19) .

Accor di ng t o the di scussi on l eadi ng t o ( 3. 34) , t he f ol l owi ng boundary

condi t i ons must be i mposed on t he di spl acement s:

- 159 -



r

2

A

r

2

K

3

=

" T i dT

A P

U=0

+

1 ^

A P

h = l

C O s h 2

*e (3-A-25)

e

K

4

= - — / d{ COs h2{ A p U) d*

o

From ( 3- A- 16) and ( 3- A- 17) we have:

A

%P

=

V

P

"

A

°i p ( 3- A- 26)

We obt ai n f r om ( 3- A- 16) and

( 3- A- 21) :

Aa

Ép

(

*

=0)

=

2

V

P ( 0 )

" ï ^

f ï

K

l "

ï f c ? 2

K

2

<

3

"

A

"

27

>

0 £ r> < 2ff

From ( 3. 36) we have:

- 160 -



- 2{ e

2

(1

e S)

ï4"

(3_A

-

31)

e

Put t i ng Aa

( p

^

}

= Aa „( {=0) we f i nal l y have f r om (3-A-26) , ( 3- A- 29) and

xpD rj pD

s

J

( 3- A- 31) :

( 3- A- 32)

Aa

( p s )

=

X

Ap ( o) - ^ (—-)

ypD 2

y

D

v

' 2

v

l +e '

Aa

( p S)

= i Ap ( o) +

l

( - - ) -

xpD 2 * V ' 2

l

l +e

;

c

e

Si nce i n ( 3. 38) i t was assumed that a = b = R ( t ) , wi t h R gi ven i n

e e e e

( 3. 37) ,

we have t hat e = 1 and ( 3. 39a) and ( 3. 39b) r esul t f r om

( 3- A- 32) .

e

As di scussed i n the t ext , we must modi f y ( 3- A- 32) t o account f or

devi at i ons f rom pl ane st r ai n condi t i ons. Fol l owi ng Chapt er 2, sect i on 2. 2. 8,

t hi s i s done by i mposi ng t he addi t i onal boundar y condi t i on:

- 161 -



0

4

= K <

4 )

e"

2£

cos2rj + K^

4)

J

where

Ap = {

Ap O £ { <, l

1 c

A p

2 *c " * * *F (3-A-34)

Ap

3

£

p

< | ^

e

0 f * £

e

£ and £ are the elliptical coordinates of the elliptical temperature and

flood fronts respectively and i=l,2,3 for £ in the cold water, warm water

and oil zone, respectively. The potential functions satisfy:

2 2

3 0. 3 0.



- 163 -



APPENDI X 3- B

A NUMERI CAL METHOD FOR CALCULATI NG PORO- AND THERMO- ELASTI C STRESS CHANGES

I n Car t esi an coor di nat es

(

x

i'

x

2

'

x

^

t n e e <

J

u a t i o n f o r t n e

di spl acement

pot ent i al ( 3- A- 7) becomes

a

2

a

2

a

2

( 1 + 2

+

2

}

* = " " P ( 3- B- l )

9x

l

3x

2

3X

3

whi ch has a sol ut i on

*

=

4Ï '

d x

i

d x

2

d x

3 Ap( x , x' , x ) | ( 3- B- 2)

wher e

2 2 2 1/ 2

R =- [ ( x

1

- x| ) + ( x

2

- x ) +

x

3

- x )

] ' (3 -B -3 )

- 164 -



2

.

girlr K*

r

v

2 +

(

v

k

2

) 2

+

( v

k

3

) 2 ]

1 / 2 }

+

V

P ( X

1'

X

2'

X

3

} 5

i j

( 3

"

B _ 5 )

The i nt egr al s i n ( 3- B- 5) can be shi f t ed t o obt ai n:

3 l

*

2

*

3

a

2

2 2

. ƒ d k

x

ƒ d k

2

ƒ d k

3 i r 5 r

[ x ^ - k ^

+

( x

2

- x - k

2

)

- a

L

- a

2

- a

3

i

D

( x

3

- x

3

- k

3

)

2

] "

1 / 2

} + A A p (

X ; L

, x

2

x

3

) 8 ( 3 - B - 6 )

- 165 -



. V

a

2 V

a

3

A

.

U

2

_ a

2

U

3

+a

3

- at an + at an —

V

a

i

r

- l ,

+

2,

+

3 V

a

i

r

- l ,

+

2 , - 3

,

U

2

+a

2

U

3'

a

3 ,

U

2

+a

2

U

3

+a

3

+ at an at an —

V

a

i

r

- l , - 2

/ +

3 V

a

i ' - 1 , - 2 , - 3

where

- 2 - 2 - 2 1 /2

r

±l ,±2 ,±3 "

[ (

V V

+ {

V

a

2

) +

< V

a

3

} ] ( 3

"

B

"

9 )

I can be obt ai ned f r om I by maki ng t he i nt er changes:

u «-* u1 2

a *-* a

1 2

(3 -B -10)

I f t he pr essur e di st r i but i on Ap i s symmet r i cal i n t he l i ne x =0, t hen

- 166 -

2 1/2



cosh{ + sinhg = v+ (v -1) v > 1

where

[ ( , W ,

2

-

4

x A V

/ 2

+

x

2

2

+Xl

2

+

L

2

1/2

v

= { i

l

\

?

i j

(3-B-12)

2L

In

the

calculation

we

have multiplied

the

edges

a,#a of the

grid

blocks by a constant factor in a direction away from the fr acture, hence

across each grid block there is approximately the same pressure drop.

In Fig. 3-B-l a schematic representation of the gridblock layout is

shown.

The

dots represent

the

gridblock centres

at

which

Ap is

calculated

from

(3.36).

For

illustration purposes

the

edges

of the

gridblocks

in the

figure ar e multiplied by a factor of 1.5 in a direction away from the

fracture.

- 167 -



- 168



APPEND X 3-C

CALCULATI ON OF THERMO- ELASTI C STRESSES AND OF THE AXES OF THE ELLI PTI CAL

FLUI D FRONT

We def i ne the fol l ow ng di mensi onl ess quant i t i es:

xT yT

xTD

=

AJ T

'

yTD " A AT (3- C- l )

T

J

T

where A„ i s t he thermo- el ast i c constant def i ned as:

T

Eo

A

m

= , (3- C- 2)

T 1-U

w th a the l i near thermal expansi on coef f i ci ent . AT i s the temperature

di f f erence between i nj ect i on temperature and reservoi r temperature. For an

- 169 -

2



V. =

ir L

h s i n h £ . c o s h £ . i = c , F ( 3 - C - 5 )

and

M

v

c

■

sr

q t 3

-

c

'

6 )

r

V

F * « I -B ' -S )

q t ( 3

-

C

"

7 )

or wc

S i n c e a . = L co s h £ . an d b . =

L

s i n h £ .

r

i t f o l l o w s t h a t

i i i i

a

i

=

2 " * i * 7?7'

"i ■ I <•*! " 7FT'

i = c , F ( 3 - C - 8 )

w h e r e

2V. 2V. 2 .

X / Z

- 170 -



-

171

-



CHAPTER FOUR

A PRESSURE FALL- OFF TEST FOR DETERM NI NG FRACTURE DI MENSI ONS

Summar y


4. 2 Cal cul at i on of t he pr essur e f al l - of f wi t h a c l os i ng f r actur e


4. 2. 2 I nt egr al equat i on f or t he di mensi onl ess pressur e f unct i on

4. 2. 3 Sol ut i on f or t he di mensi onl ess pressur e f unct i on

4. 3 Anal ys i s of a pr essur e f al l - of f test

4. 3. 1 Four met hods t o det er m ne f r act ur e l engt h

4. 3. 2 Di scussi on

4. 4 Concl usi ons

Li st of symbol s

- 172 -



SUMMARY

Pressure transient theory for a water-injection well in the presence of

a closing f r acture and a discontinuity in fluid mobility is developed in

elliptical coordinates. Solutions are obtained using the Laplace transform

and numerical inversion. From the results, it is concluded that a pressure

fall-off test with a closing fr acture in principle provides f our diff eren t

methods for determin ing the fracture length. The first method is based on

rock mechanical principles only. The second method makes use of formation

linear flow. The third method analyses the transition of the pressure

transients from the inner fluid region to the outer region in conjunction

with a heat or volume balance. The fourth method analyses the skin as seen

during pseudo-radial flow in the outer fluid zone.

- 173 -



A PRESSURE FALL- OFF TEST FOR DETERM NI NG FRACTURE DI MENSI ONS

4. 1 I NTRODUCTI ON

I n t he pr evi ous chapt er t he pr opagat i on of wat er f l ood- i nduced f r act ur es

was st udi ed wi t h r el at i vel y si mpl e model s. Two i mpor t ant concl usi ons f r om

t hi s st udy are a) t he convent i onal Car t er model of one- di mensi onal l eak- of f

per pendi cul ar t o the f r act ur e i s gener al l y i nadequat e, b) changes i n

r eser voi r pr essur e and t emper at ur e can have a si gni f i cant ef f ect on t he

r eser voi r r ock st r ess and t her ef or e on the f r act ur e pr opagat i on pr essur e.

A quant i t at i ve pr edi ct i on of f r act ur e l engt h wi t h t hese model s shoul d

be t r eat ed wi t h some car e. Thi s i s because, on t he one hand, t hey r el y on

many si mpl i f yi ng assumpt i ons and, on t he ot her hand, a great number of i nput

dat a ar e r equi r ed. A met hod of det er m ni ng t he di mensi ons of a gr owi ng

f r act ur e at cer t ai n moment s dur i ng i t s pr opagat i on i s t her ef or e usef ul i n

- 174 -



In Ref . 3 a method was proposed for analysing the dimensions of

minifracs from a recording of the pressure data immediately after pumping

has stopped. Since this method relies on the Carter leak-off model both

during fracture propagation and during fracture closure and since,

f urther more, the fluid leak-off is assumed to be independent of pr ess ur e,

this method is in general not suited to analyse the dimensions of

waterflood-induced fractures.

An important first step towards the analysis of a fall-off test for

4

closing waterf lood-induced fr actures was made by Hagoort in his thesis . By

applying a simple rock mechanical model for the fracture he was able to

relate f ractur e closure to early-time f luid f low. He presented an analytical

expres sion for the theoretical pres sur e r esponse which is valid as long as

formation fluid flow is still linear perpendicular to the fracture and

before the fracture has completely closed.

This chapter presents an extension of Hagoort's model to account for

different fracture geometries, transition from early time linear to late

time pseudo-radial flow, pressure response during and after closure and the

- 175 -



4. 2 CALCULATI ON OF THE PRESSURE FALL- OFF W TH A CLOSI NG FRACTURE


The t r ansi ent pr essur e response i n t he f r act ur ed i nj ect i on wel l i s

cal cul at ed usi ng the f ol l owi ng assumpt i ons.

1. The f r act ur e has i nf i ni t e conduct i vi t y dur i ng pr opagat i on, dur i ng cl osur e

and af t er c l osur e. The assumpt i on of i nf i ni t e conduct i vi t y af t er c l osur e

means t hat a channel of ver y hi gh per meabi l i t y r emai ns where t he f r act ur e

occur r ed. Thi s coul d be caused by erosi on or by a m smat ch of t he

f r actur e f aces . The ef f ect of t hi s condi t i on and devi at i ons f rom i t wi l l

be di scussed i n t he anal ysi s of t he r esul t s .

2.

Dur i ng t he pr essur e decl i ne t he l engt h and hei ght of t he f r act ur e r emai n

const ant . The f r actur e vol ume di m ni shes accor di ng t o:

dV dp

- dT

= C

f ^

( 4

'

X)

- 176 -



l engt h by subst i t ut i ng L = L( t ) , p

f

( t ) = p

f

= const ant and sol vi ng f or L( t )

( L = f ract ur e hal f - l engt h, t i s i nj ect i on

t i me) .

Thi s met hod was shown t o be

cor r ect by compar i son wi t h t he resul t s f r om a numer i cal f r act ur e/ r eser voi r

s i mul at or as pr esent ed i n Ref . 4.

Conver sel y, t he pr essur e di st r i but i on sur r oundi ng a pr opagat i ng

f r act ur e at t i me t can be obt ai ned by eval uat i ng t he const ant r at e/ f i xed

f r actur e l engt h sol ut i on f or t i me t and f ract ure hal f - l engt h L ( t ) .

Suppose t he f r act ur e has a hal f - l engt h L( t

.

)

when t he wel l i s shut - i n

SO

at t i me t . . Gener al i si ng t he above r esul t t o t he case of var yi ng rates

sh

gi ves f or t he pr essur e i n t he f r actur e at t :

fi

pf

(t

sh> ■ ' - f "

k

6 p

f

( t

"

X ) d X

o o

. ( 4. 2)

sh

wher e 5p. = p, - p. and q. i s t he tot al l eak- of f f rom t he f r actur e i nt o t he

f f ï

4

er

r eser voi r . 5p_ i s t he constant r at e sol ut i on f or an i nf i ni t e conduct i vi t y

f r actur e of hal f - l engt h L( t .

)

and f or an i nj ect i on r at e q . Duhamel ' s

* sh

J

o



- 178 -



u

= vi scos i t y

h = r eser voi r hei ght

p . = pr essure i n t he f r act ur e at t he moment of shut - i n ( p

.

i s a const ant )

sh , sh

n, = (- ) , = hydr aul i c di f f us i vi t y i n i nner regi on

1

0MC

1

J

0 = por osi t y

c = t ot al compr essi bi l i t y

cr

We mul t i pl y Eq. ( 4. 7) wi t h - 1 and add

Sp .

( t ) = p . - p. t o bot h

- _ E

Sli S O

1

s i des .

Usi ng Eq. ( 4. 8) and — 5p = - — Ap, we have, af t er t he i nt r oduct i on

of t he di mensi onl ess var i abl es

P

f D

( At

D>

=

Pf D Df sh) ' "

P

f D

( t

D( sh)

+ A

V

+

'

D[1

-

c

f

D

axl

p

f D

(

V

]

i c f r r P f D

(At

D

-

x

D

)dX

D

o D D

( 0

^

A t

D

< A t

D ( c l )

) ( 4

'

1 0 )

- 179 -

4. 2. 1 Sol ut i on f or t he di mensi onl ess pr essur e f unct i on



I f t he const ant r at e sol ut i on i s known, Eqs. ( 4. 12) and ( 4. 13)

det er m ne t he wel l bor e pr essur e r esponse dur i ng and af t er cl osur e,

r espect i vel y. An anal yt i cal sol ut i on for the constant r at e i nf i ni t e

conduct i vi t y f r actur e was pr esent ed by Gr i ngar t en et al . i n Ref . 2. I n Ref .

6 t he same pr obl em was sol ved i n el l i pt i cal coor di nat es i n Lapl ace space

7

usi ng Mat hi eu f unct i ons, Recent l y , t hi s appr oach was ext ended t o i ncl ude

t he pr esence of an el l i pt i al di s cont i nui t y i n f l ui d mobi l i t y. I t i s t hi s

met hod t hat we have adopt ed i n sol vi ng Eqs. ( 4. 12) and

( 4. 13) .

Some det ai l s

ar e gi ven i n Appendi x 4- A. For an el l i pt i cal coor di nat e syst em see

Fi g.

4. 1. Eq. ( 4. 12) i s sol ved usi ng the Lapl ace tr ansf or m The sol ut i on i n

g

Lapl ace space i s i nver t ed numer i cal l y wi t h t he St ehf est al gor i t hm .

Eq. ( 4. 13) i s sol ved by det er m ni ng - r r—p, _ , . .

A

, —r p. _ i n Lapl ace space and

n ' D

i nver t i ng wi t h t he St ehf est al gor i t hm f or 0 < At < At , . . The i nt egr al i n

t he f i r st t er m on t he r i ght - hand s i de of Eq. ( 4. 13) i s t hen i nt egr at ed

numer i cal l y. The second t er m i s agai n cal cul at ed i n Lapl ace space and

- 180 -



- 181 -

10

i



A t

D

FIG. 4.2 TYPE CURVE FOR A PRESSURE F A LL -O F F TEST W ITH A CLOSING FRACTURE

- 182 -



d ( P

D

)

d ( In At

n

)

4n= 0.5

- 183 -

Pseudo- r adi al f l ow i s r eached i nsi de t he i nner zone when t he der i vat i ve



r eaches t he val ue 0. 5. Af t er t r ansi t i on t o the out er zone t he const ant val ue

0. 5

K

i s r eached. F i g. 4. 4 shows a l og- l og pl ot of t he l ogar i t hm c

der i vat i ve of p, „ vs At f or var i ous val ues of { . Thi s pl ot shows t hat i f

r

f D D o

t he f r actur e i s c l ose to the f ront ( smal l £ ) , pseudo- r adi al f l ow i s not

at t ai ned i n t he i nner r egi on.

4. 3 ANALYSI S OF A PRESSURE FALL- OFF TEST

4. 3. 1 Four met hods t o det er m ne f r act ur e l engt h

Based on t he concept ual r esul t s gi ven i n Fi gs . 4. 2-4. 4 the f ol l owi ng

phi l osophy emer ges f or a pr essur e f al l - of f t est . F i r s t , shut t he wel l i n

wi t h a downhol e shut - of f t ool t o avoi d t he f r act ur e cl osur e bei ng masked by

wel l bor e st or age. Si nce ear l y- t i me' dat a ar e i mpor t ant , a pr essur e gauge wi t h

a hi gh sampl i ng r at e shoul d be used t o r ecor d t he pr essur e. Then make a

- 184 -

9

whi ch 2L/ h

f

« 1, the Perki ns-Kern- Nordgren model (PKN) , f or f ractures w th



2L/h » 1 and, el l i psoi dal f ractures for the i ntermedi ate case, h i s the

f racture hei ght at the wel l bore. Fi g. 4.5 shows the CGK and PKN geometr i es,

F i g. 4.6 shows the el l i psoi dal geomet ry.

The f racture l ength can be determned graphi cal l y as f ol l ows. Est i mate

the f racture hei ght h , f orma new di mensi onl ess cl osure constant ,

C

f D . 3,

n

2,

( 4

'

1 6 )

h

f

( l -o )

where v and E are Poi sson' s rat i o and Young' s modul us, respecti vel y. The

superscri pt ( rm denotes rock mechani cal and serves t o di st i ngui sh thi s

def i ni t i on of the di mensi onl ess cl osure constant f rom that i n ( 4 . 9 ) .

dp

f

C, = - q/

-

TT i s determned f romthe l i near pl ot of p vs At . Enter Fi g. 4. 7

t at

and determne 2L/h for a part i cul ar f racture geometry. Thi s t hen gi ves a

val ue for L.

Al ternat i vel y, i f h i s not known, a certai n val ue for the rat i o 2L/ h

f

- 185



X

CGK-MODEL

- 186 -



Z-AXIS

- 187 -



10'

10 r

10"

10

10"

-1 1 r

i — r - n - n —

CGK

PKN

ELLIPSOID

1 I I I ï

/.-

.

i r" i ' i \ i > i i |

- 188 -

From t he per meabi l i t y of t he sem - l og anal ysi s and f r om t he f l ui d and



r el at i ve per meabi l i t y dat a

( MA 0 C

) can be est i mat ed. Then f r om Eq. ( 4. 17)

t he quant i t y 2hL i s obt ai ned. Al t hough Eq. ( 4. 17) i s convent i onal l y der i ved

f or a r ectangul ar f r actur e ar ea, i t can be used f or a f r actur e ar ea of any

geomet r y i f 2hL i s r epl aced by t he cor r espondi ng expr essi on f or t hi s ar ea.

We t her ef or e have

and

h = h ( CGK and PKN) ( 4. 18)

h = | h

f

( el l i ps oi d) ( 4. 19)

I f t he f r actur e cl oses very t i ght l y, f or mat i on l i near f l ow cannot

occur. The di mensi onl ess pr essur e wi l l behave di f f er ent l y i n t hi s case t han

i ndi cat ed i n our t ype- cur ves and t he anal ysi s descr i bed above cannot be

appl i ed.

Met hod 2b

- 189 -

Not e t hat Eq. ( 4. 20) does not consi der t he pr esence of an el l i pt i cal



di scont i nui t y i n f l ui d mobi l i t y. I t can ther ef or e be used onl y when i

l ar ge enough f or l i near f l ow t o occur i n t he i nner r egi on.

Met hod 3. Tr ansi t i on f l ow

The di mensi onl ess der i vat i ve pl ot of Fi g. 4. 4 can be compar ed wi t h a

s i m l ar pl ot of t he f i el d dat a to est i mat e t he pos i t i on of t he mobi l i t y

di scont i nui t y £ . Si nce the onset and l engt h of t he t r ansi t i on zone and i t s

hei ght above or bel ow t he l i ne wi t h const ant val ue 0. 5 K depend on t he

eccent r i c i t y of t he mobi l i t y cont i nui t y, a mat ch wi t h the f i el d dat a i n t hi s

r egi on gi ves a val ue f or £ • I n t he usual way, a mat ch al so pr ovi des a val ue

f or L si nce the cor r espondi ng At and At ar e known.

I f | has been det er m ned, a heat or vol ume bal ance, dependi ng on t he

t ype of mobi l i t y di scont i nui t y, can then be used t o obt ai n an addi t i onal

est i mat e of L. Fol l owi ng Chapt er 3, Appendi x 3- D, L can be obt ai ned f r om

V. . 1/ 2

- 190 -

Met hod 4. Pseudo- r adi al f l ow



I f pseudo- r adi al f l ow i n t he out er f l ui d zone has been r eached, t he

ski n as seen i n the out er zone, can be det er m ned f r om

1. 5

• ( *

At )

A

*

f

( A t

> ■ 5 * r

{ l n

1

+ s }

<

4

-

2 5

>

2 w

wher e X„h i s det er m ned f r om t he sl ope of a Hor ner pl ot . The compl et e

expr essi on f or t he pr essur e dr op i n t he f r actur e i s gi ven by:

n

. a + b . 3. <V( T) , At )

* f

<

A t

>

«

*

< t

ln

T^

+

t

ln

a

+ b

^ <

4

2 6

>

1 2 o o

wher e a , b are t he maj or and m nor

axes,

r espect i vey, of t he el l i pt i cal

di scont i nui t y i n f l ui d mobi l i t y. I t f ol l ows that t he ski n i s rel ated t o

f r actur e hal f - l engt h accor di ng t o:

X_ a + b a + b

_ 2 , o o , o o , . __ ,

- 191 -

wi t h R , t he r adi us of t he mobi l i t y di scont i nui t y g i ven by:



V. 1/ 2

R = (-J) i = c,F (4.31)

o jrh

Of cour se, i f pseudo- r adi al f l ow has al r eady been reached i n t he i nner

f l ui d zone, t hen a Hor ner pl ot al so gi ves X h and t he ski n as seen i n t he

i nner zone can be det er m ned. Thi s ski n i s r el at ed t o f r act ur e l engt h i n t he

usual way:

L = 2r exp( - S) ( 4. 32)

As f or met hod 3, met hod 4 i s appl i cabl e bef or e cl osur e or af t er

cl osur e,

i f a channel of ver y hi gh per meabi l i t y r emai ns where t he f r act ur e

occur r ed.

4. 3. 2 Di scussi on

- 192 -

leak-off at the tip will produce an appreciable pressure gradient only in



the fracture near the tip. Therefore, the pressure at the wellbore does not

differ very much from the average pressure in the fracture and the foregoing

closure analysis is valid. For a similar conclusion/ see Ref. 3.

The choice between substitution of Eq. (4.18) or Eq. (4.19) in methods

2a and 2b applies when the fracture does not extend over the whole reservoir

height. For this reason the early-time type-curve analysis must be

restr icted to the period of linear f low perpendicular to the f racture. If

the fracture covers the whole reservoir height, h is always equal to the

reservoir height. Then the type-curve analysis in methods 2a or 2b may be

extended beyond the linear flow regime into the elliptical flow regime. The

types curves must then be generated with the method given in Appendix 4-A.

The an alysis of methods 3 and 4 relies on a sharp elliptical

temperature or flood front. If the temperature of the injection fluid is

different from that of the reservoir, according to Chapter 3, Section 3.4.2,

a sharp front prevails if:

2

- 193 -

4. 4 CONCLUSI ONS



I n pr i nc i pl e, a pr essur e f al l - of f test pr ovi des f our met hods f or

det er m ni ng f r actur e l engt h.

1. I n t he f i r s t f l ow per i od t he f ract ur e cl oses accor di ng t o i t s

compr essi bi l i t y. The pr essur e var i es l i near l y wi t h t i me. By usi ng rock

mechani cal pr i nci pl es, t he s l ope of t he str ai ght l i ne gi ves a val ue for

t he f r actur e l engt h.

2.

The second f l ow per i od i s det er m ned by ei t her l i near f or mat i on f l ow or a

combi nat i on of f r actur e c l osur e and l i near f or mat i on f l ow. I n t he f i r st

case t he usual squar e- r oot t i me anal ysi s and i n t he second case t ype-

cur ve mat chi ng r esul t i n a val ue f or t he f r act ur e l engt h.

3. The thi r d f l ow per i od i s det er m ned by the tr ansi t i on f r om t he i nner

f l ui d r egi on t o the out er f l ui d r egi on. The l engt h, pr essur e l evel and

onset of t hi s t r ansi t i on f l ow can be used t o obt ai n a t ype- cur ve mat ch

wi t h a l ogar i t hm c der i vat i ve pl ot . The mat ch pr ovi des a val ue bot h f or

the el l i pt i al coor di nat e, £ , of the f l ui d f ront and f or the f ract ure

- 194 -

LI ST OF SYMBOLS



A Four i er coef f i c i ent of Mat hi eu f unct i onso

c t ot al por e compr ess i bi l i t y

Ce_ modi f i ed Mat hi eu f unct i on of t he f i r st ki nd

2n

C, f r act ur e cl osur e const ant

E Young' s modul us

E( k) compl et e el l i pt i c i nt egr al of t he second ki nd

Fek„ modi f i ed Mat hi eu f unct i on of t he t hi r d ki nd

zn

h f or mat i on t hi ckness

k per meabi l i t y

L f r actur e hal f - l engt h

M heat capaci t y of f l ui d- f i l l ed r eser voi r rock

M heat capaci t y of cap and base r ock

M heat capaci t y of i nj ect i on f l ui d

p.

i ni t i al reservoi r pr essur e

p, f l ui d pr essur e i n the f r actur e

- 195 -

Gr eek



a t hermal di f f us i v i t y of cap and base r ock

i j

di f f us i vi t y

K mobi l i t y r at i o

X mobi l i t y of i nner and out er f l ui d r egi on r esp .

1

/

2

X s/ 4

t f v i s cos i t y

v Poi s s on' s r at i o

0 por os i t y

£ el l i pt i al c oor di nat e

S di f f us i vi t y r at i o

Subsc r i pt s

1 i nner r egi on

2 out er r egi on

c t emper at ur e f r ont

D di mensi onl ess

- 196 -

REFERENCES



1. Cl ar k , K. K. , Tr ans i ent pr essur e test i ng of f r actur ed wat er i nj ect i on

wel l s .

J PT, ( J une 1968) , pp. 6.

2. Gr i ngar t en, A. C. , Ramey, H. J . J r . & Raghavan, R. , Unst eady- st at e pressur e

di st r i but i on cr eat ed by a wel l wi t h a s i ngl e i nf i ni t e conduct i vi t y

ver t i cal f r act ur e.

SPEJ ( Aug. 1974) , pp. 347- 360.

3. Nol t e, K. G. , Det er m nat i on of f r actur e par amet er s f r om f r actur i ng

pr essur e decl i ne.

SPE 8341, 1979.

4.

Hagoor t , J . , Wat er f l ood- i nduced hydr aul i c f r actur i ng.

Ph. D. Thesi s , Del f t Techni cal Uni ver s i t y, 1981.

5. Van Ever di ngen, A. F. & Hur st , W , The appl i cat i on of Lapl ace

t r ansf or mat i on t o f l ow pr obl ems i n r eser voi r s.

Tr ansact i ons AI ME ( Dec. 1949) , pp. 305- 324.

197 -

14. Pal mer , I . D. & Car r ol l , H. B. J r . , 3D hydr aul i c f r actur e pr opagat i on i n



t he pr esence of st r ess var i at i ons.

SPE/ DOE 10849, 1982.

- 198 -

APPENDI X 4- A



( 4 - A - l )

SOLUTI ON FOR DI MENSI ONLESS PRESSURE FUNCTI ON I N LAPLACE SPACE

Sol ut i on t o di f f us i vi t y equat i on i n el l i pt i cal coor di nat es

*

Empl oyi ng a conf ocal el l i pt i cal coor di nat e system!

x = L cosh £ cos 17

y = L si nh £ si n r\

t he di mensi onl ess di f f usi vi t y equat i on becomes:

3 2 p

D

3 2 p

D 1 „

9P

D

—

J -

+ —

Y ~ 2

(cosn 2

* ~

c o s 2 r

» )

J £~

( 4 - A - 2 )

3£ 3rj D

- 199 -

Thi s equat i on has t he f ol l owi ng gener al sol ut i on t hat i s per i odi c i n 77



wi t h per i od J T:

P

D

(*,r?) = Z £C

2n

Ce

2n

( £ , - X ) c e

2n

( 7j , - X) +

F

2 n

P e k

2

n

(

* ' ~

X ) C e

2n

( T ?

' ~

X, }

f

4

"

6

)

n=0

wher e X = - , C. and F ar e const ant s t hat are det erm ned by t he boundar y

condi t i ons.

Ce„ , ce_ , Fek. ar e Mat hi eu f unct i ons wi t h the f ol l owi ng

, „ 2n 2n 2ndef i ni t i ons :

ce (7/ , X) = Z A ( X) cos2r rj

*

n

r =0

r

OB

ce_

( T J , - X )

= ( - l )

n

Z

( - l )

r

A

n

(X)

cos2rr?2n _ 2r

r =0

ce (0, X) ce ( f , X) »

T

_

Ce . U, - X) = ( - i ) »-

2

^— | S - 2 _ Z

<- l )

r

A

2n

( X ) .

2 n

[ A

2n

( X ) ]

2

r =0

2 r

- 200 -

2ir

i

ce (»j,-X) dr? = (-1)° 2JT A

n

(X) (4-A-9)



0

Solution to the composite problem in elliptical coordinates

The Laplace transf orm of the constant rate solution p in the presen ce

7

of an elliptical discontinuity in fluid mobility is given by :

PfD = "

{ 1 6 X 2

*

^ ^

A

o

n ( X , [ C

2 n

C e

2 n ^ w ' -

X > + F

2 n

F e k

2 n < * w ' "

X

>

]

^ <

4

"

A

-

1 0

>

n=0

wher e X = s/4 . The functions Ce_ and Fek_ are defined in

(4-A-7).

£ is

2n 2n

v

'

s

w

the coordinate of the inner boundary. The prime denotes differentiation with

respect to £.

The constants C. and P. are obtained from:

2n 2n

m

A

22

A

33 -

A

23

A

32

^ n det(A) -

( 4 A 1 1 }

- 201 -

A

3 2

= KFek'

n

( {

o

, - X )



A

33

=

"

F e k

2n

( £

o ' -

q )

wher e q = XJ and K, $ and £ are gi ven i n Eq.

( 4. 14) .

I n Ref . 7 the

const ant s A t o A ar e mul t i pl i ed by a per i odi c Mat hi eu f unct i on. Si nce

t hese per i odi c f unct i ons drop out i n Eq,

( 4 - A - l l ) .

we have om t t ed t hem

her e.

Sol ut i on wi t h c l os i ng f r actur e

Equat i on ( 4. 12) i s a convol ut i on- t ype i nt egr al equat i on that al so

occur s i n the anal ysi s of or di nar y wel l bor e st or age pr obl ems . Taki ng t

Lapl ace t r ansf or m of Eq. ( 4. 12) and sol vi ng for p . gi ves :

~cr .

v

P

f D

<

s

>

P f D<

S) =

, „ 2- cr ,

|

( 4

"

A

"

1 4 )

1 +C

f D

S P

f D

( S )

- 202 -

APPENDI X 4- B



( 4 - B - l )

RELATI ONSHI P BETWEEN FRACTURE CLOSURE CONSTANT AND FRACTURE LENGTH

Assum ng a uni f or m pr essur e p

f

i nsi de the f r act ur e, we have:

dV

f

dV

f

dp

f

dt ~

=

dp^ ' dt ~

so that f r om Eq. ( 4 . 1 ) ,

dVf

C

f

= — (4-B-2)

For a f r act ur e of t he CGK t ype ( Fi g. 4. 5) t he vol ume i s gi ven by,

V = h it w( o) L ( 4- B- 3)

- 203 -

gi vi ng,



„PKN

( 1- u )

T

.

2

, , „ « , »

C

f

= f f - *-£

*■

L h

f

( 4 - B - 8 )

For an el l i ps oi dal f r act ur e ( Fi g. 4. 6) t he vol ume i s gi ven by :

v

£

- 1 '

i ts " f V

wi t h a and b the maj or and m nor axi s r espect i vel y of t he el l i pt i cal

f r act ur e ar ea. E( k) i s t he compl et e el l i pt i cal i nt egr al of t he second ki nd

and

2 b

2

k = 1 - S j ( 4- B- 10)

a

Fr om Eqs. ( 4- B- 9) and ( 4- B- 10) we have:

- 204 -

S

H

=

S

H1

M ï ï



S = S Iz l > -

H °H 2 ' ' 2

14

t he wi dt h equat i on ( 4- B- 7) becomes ,

w(o)

=

ü = i h

f

.

• Pf - Sni ) "

\

( S ^- S ^t c o s - V j - u I n C

1

^

1

- " >) }]

( 4- B- 13)

( 4- B- 14)

wher e

u =

z~

( 4- B- 15)

h

f

However , f r om ( 4- B- 14) ,

- 205 -

From Eqs. (4-B-5), (4-B-8), (4-B-ll) and (4-B-12) a plot of C** ' vs



2L/h, can be made for the various fr acture geometries (Fig. 4. 7) . This plot

can be used to determine L or h as described in method 1 in the text.

- 207 -

CHAPTER FIVE



A

PRACTI CAL APPROACH TO WATERFLOODI NG UNDER FRACTURI NG CONDI TI ONS

Summar y


5. 2 An exampl e

5. 3 Condi t i ons f or a successf ul pr ocess

5. 4 Pr el i m nar y i nvest i gat i ons

5. 5 Basi c dat a gat her i ng

5. 5. 1 Measur ement s of i n- si t u st r ess, f r actur e or i ent at i on

and el ast i c modul i

5. 5. 2 I nj ect i vi t y tes t and f al l - of f tes t i ng

5. 5. 3 Mat chi ng wi t h pr opagat i on model

5. 6 Det er m nat i on of opt i mal r eser voi r pr essur e, i nj ect i on rate

and wel l pat t er n

- 208 -

SUMMARY



A practical approach to waterf looding under fr acturing con ditions is

presented. It is shown that injecting under fracturing conditions has a large

potential for a reduction in the total number of wells . The process can, in

many cases, be designed in such a way that sweep efficiency is unimpaired and

vertical fracture containment is ensured. Design formulae are presented for

calculating the optimum reservoir pressure, injection rate and well pattern.

The case of a reservoir at hydrostatic pressure with a typical gradient for

the horizontal reservoir rock stress of 0.17 bar/m is investigated in detail

for unit mobility ratio. With the stress in cap and base rock exceeding that

in the r eservoir by 14% or more, the most attractive options ar e a f ractured

7 spot and a fractured

9 spot

with a maximum reduction in the number of wells

of 40% and 3 3%, respectively. For more moderate str ess contr asts the r eservoir

pres sur e has to be lowered to ensure ver tical containment. With the horizontal

rock stress in cap and base rock exceeding that in the reservoir by only 9% a

fr actured 13-spot at a lower reservoir press ure results in a 16% reduction in

- 209 -

A PRACTI CAL APPROACH TO WATBRFLOODI NG UNDER FRACTURI NG CONDI TI ONS



5. 1 I NTRODUCTI ON

I n the pr evi ous chapt er s t he physi cs of f r actur e i ni t i at i on and

pr opagat i on wer e descr i bed i n det ai l . A met hod f or moni t or i ng f r act ur e l engt h

by conduct i ng a f al l - of f t est was di scussed.

Thi s chapt er shows how t he concept s devel oped so f ar can be appl i ed i n

pr ac t i ce.

I t i s out l i ned what t he advant ages may be of wat er f l oodi ng under

f r act ur i ng condi t i ons and how t he pr ocess shoul d be desi gned i n order t o

cont r ol t he f r acture gr owt h.

5. 2 AN EXAMPLE

- 210 -

These problems do not occur with fracturing of the injectors. Injection



into the water layer is generally no problem and injection at the fracture

propagation pressure ensures that the fracture stays open.

Suppose that at the prevailing reservoir pressure the maximum

production rate for a typical producer equals approximately the maximum

injection rate for a typical (non-fractured) injector. Therefore, to avoid

pressure depletion of the reservoir, as many injectors are required as there

are producers. This makes an arrangement of confined five-spots the most

suitable well pattern (Fig. 5 . 1).

If,

by injecting under fracturing conditions, the injection rate can be

increased by, for example, a factor of three, an inverted nine-spot would be

possible (Fig. 5 . 1) . For a certain total field off take only a third of the

injectors would be required compared to the five-spot pattern (for

simplicity's sake we discard deviations from symmetry near the field

boundaries).

- 2 1 1 -



v<p

"VS*

A P

v ,

J/«p V«o

^

f

V

* P X V jp g

*?

4 P

' * p

- 212 -

wher e:



q = i nj ect i on r at e

h = r eser voi r hei ght

p = f r act ur e pr opagat i on pr essur e

*

p.

= pr evai l i ng r eser voi r pr essur e

k/u =

mobi l i t y of i nj ect i on f l ui d

I f q i s l ess t han 0. 61, t he pr essur e t r ans i ent s t r avel r adi al l y i nt o t he

r eser voi r bef or e a s i t uat i on of st eady- state i s reached. Ther ef or e, f r actur e

gr owt h wi l l have no ef f ect on sweep ef f i c i ency, r egar dl ess of f r act ur e

or i ent at i on. I t i s assumed t hat , at t he pr evai l i ng r eservoi r pr essur e p. ,

i nj ect i on and pr oduct i on ar e bal anced so t hat af t er t he onset of st eady-

st at e t he f r act ur e har dl y gr ows any f ur t her .

jnay

The maxi mum i nj ect i on r at e, q , for i nj ect i ng under f r act ur i ng

condi t i ons at r eser voi r pr essur e p. , t hus becomes:

maX

- 213 -

I ni t i al l y, of cour se, the wel l spaci ng s t i l l has t o be det er m ned.



However , t he l ogar i t hm c t er m i n ( 5. 3) has a typi cal val ue of about 7. I n a

l i mest one r eser voi r t he wel l s. ar e most l y st i mul at ed wi t h an aci d t r eat ment

gi vi ng a typi cal ski n of S = - 2. On t he other hand t he wel l may have s ome

mechani cal damage gi vi ng f or i nst ance a ski n of S = +1.

Prom ( 5. 2) , wi t h p = S , and ( 5 . 3 ) , a t ypi cal i mpr ovement i n

£ H

i nj ect i vi t y by i nj ect i ng under f r actur i ng condi t i ons i s found to be:

max

<

s =

-

2

>

3 K

^ziz

K 5

<

s =

1

>

<

5

-

4

>

~max

q

wher e t he l ogar i t hm c t erm was t aken as 7.

We see f r om ( 5. 4) t hat i nj ect i ng under f r actur i ng condi t i ons, as f ar as

uni mpai r ed sweep ef f i ci ency i s concer ned, r esul t s, i n most cases, i n a

consi der abl e i mpr ovement i n i nj ect i vi t y.

W t h regar d t o ver t i cal f r actur e cont ai nment t he si t uat i on i s more

di f f i cul t . I f cap and base rock cons i s t of thi ck shal es , t her e i s a good



- 215 -

W t h the ar r ay soni c l oggi ng t ool t he dynam c el ast i c modul i of t he

r ock can be det er m ned . I deal l y, one shoul d l i ke to use these f or a di r ect



cal cul at i on of t he i n- s i t u st r ess. I t shoul d be not ed however , t hat

gener al l y t hese const ant s ar e di f f er ent f r om t hose measur ed under st at i c

condi t i ons .

The l at t er ar e mor e rel evant f or i nvest i gat i ng t he st r ess st at e

i n a r ock. Qui t e apar t f r om t hi s di f f er ence, i t i s not r ecommended t o

comput e t he i n- si t u str esses usi ng t he i n- si t u val ues of t he el ast i c modul i

si nce t hi s i gnor es t he bur i al hi st or y of t he f or mat i on ( see Chapt er 2,

sec t i on 2. 6) . However , i f i n t he same f i el d t he di f f er ence i n t he dynam c

el ast i c modul i f or t wo adj acent l ayer s i s measur ed as uni f or m acr oss t he

f i el d, t hen, most l i kel y, t he same hol ds f or t he di f f er ence i n s t r ess .

Ther ef or e, soni c l oggi ng can be a power f ul met hod f or i nvest i gat i ng t he

l at er al uni f orm t y of s t ress cont r as ts ac ross t he f i el d.

5. 5. 2 I nj ect i vi t y t est and fal l - of f t es t i ng

I n t he same wel l i n whi ch t he s t r ess measur ement s wer e made an

i nj ect i vi t y t est shoul d be carr i ed out t o det er m ne the char acter i s t i cs of

- 216 -

A day or so bef or e a f al l - of f tes t wi l l be car r i ed out i nj ect i on i s

st opped t o l ower a hi gh- r esol ut i on pr essur e gauge and spi nner i nt o t he hol e.



Af t er i nj ect i on has r esumed a spi nner sur vey i s car r i ed out t o see whet her

t he f l ow di st r i but i on i s di f f er ent or t he same as t he base case i ndi cat i ng

par t i al or compl et e extensi on, r espect i vel y, of t he f r act ur e t owar ds cap and

base r ock.

The pr essure f al l - of f can be anal ysed wi t h t he met hod of Chapt er 4 t o

det er m ne t he di mensi ons of t he i nduced f r acture. I f t he act ual cl osi ng of

t he f r act ur e i s obser ved as a peak on a l ogar i t hm c pr essur e der i vat i ve

pl ot ,

t he hor i zont al r eser voi r r ock st r ess can be det er m ned. A compar i son

wi t h t he i ni t i al s t r ess as det er m ned i n t he m crof r actur e t est t hen

i ndi cat es possi bl e por o- and t her moel ast i c changes.

I f kh has si gni f i cant l y i ncreased compar ed t o the ref er ence f al l - of f

case, t he f r act ur e has most l i kel y pr opagated i nt o anot her per meabl e zone.

I f t he per i od of af t er f l ow f r om t he wel l bor e i nt o t he r eser voi r i s

expect ed t o be l ar ge wi t h r espect t o t he per i od of f r act ur e cl osur e one

shoul d i deal l y use a downhol e shut - of f devi ce t o shut t he wel l i n. As an

al t er nat i ve, however , si mul t aneous pressur e and spi nner dat a t oget her wi t h

- 217 -

b) t he maxi mum i nj ect i on pr essure may be est abl i shed f or whi ch no such



5. 5. 3 Mat chi ng wi t h pr opagat i on model

I f the f r actur e hal f - l engt h obt ai ned f rom t he f al l - of f t est at t he end

of t he i nj ect i on t es t sat i s f i es :

L„ = /,

L

, . T < 0. 58 ( 5. 5)

D / ( Vsh>

wher e n„ i s t he hydr aul i c di f f us i vi t y i n t he out er f l ui d zone and t . t he

2 sn

t ot al i nj ect i on t i me, t hen accor di ng t o Chapt er 3, aect i on 3. 2. 4, f l ow

around t he f r act ur e was pseudo- r adi al . Thi s means t hat no ef f ect on sweep

ef f i c i ency i s expected i f i nj ect i on and pr oduct i on i n the pat t er n ar e

bal anced. I t al so means t hat i t shoul d be possi bl e t o mat ch t he pr opagat i on

pr essur e as a f unct i on of t i me wi t h t he anal yt i cal model of Chapt er 3,

- 218 -

wher e:



S = hor i zont al r eser voi r r ock st r ess at reser voi r pr essur e p

H

La

= por o- el ast i c st r ess change at f r act ur e f ace

yp

Ac „ = t her moel ast i c st r ess change at f r act ur e f ace

yT

K = cr i t i cal s t r ess i nt ens i t y f act or

L = f r acture hal f - l engt h

The por o- el ast i c st r ess change i s gi ven by:

A o

y

P

= c

' V

( p

f "

p) ( 5>7)

wher e A i s t he poro- el ast i c const ant and c i s a number i n t he r ange

0 < c < 1. 0.

The act ual val ue of c depends on the r at i o of t he pr essur e penet r at i on

dept h t o t he r eser voi r hei ght and on t he r at i o of t he pr essur e penet r at i on

dept h t o t he f r act ur e l engt h ( see Chapt er 3, sect i on 3 . 3 . 2 ) .

- 219 -

I f i n t he i n- s i t u st r ess t est t he magni t ude of t he hor i zont al r eser voi r

r ock st r ess was det er m ned t o be S . at t he pr evai l i ng r eser voi r pr essur e



p. ,

we have f or t he r eser voi r rock st r ess at r eser voi r pr essur e p:

S

H "

S

Hi

+ A

p

A P ( 5

'

9

>

wher e Ap = p - p. .

The f r act ur e pr opagat i on pr essur e, p. at r eser voi r pr essur e p. can be

obt ai ned f rom ( 5. 8) by r epl aci ng S„ wi t h S

TI

. and p wi t h p. . We t hus f i nd

H Hi l

f r om ( 5. 8) and ( 5. 9) f or t he di f f er ence i n pr opagat i on pr essur e at r eser voi r

pr essur e p and p. :

( 1- c) A

P f

= p

f

+

T^T^

•

Ap ( 5

-

10)

P

Si nce cA < 1, accor di ng t o ( 5. 10) , a l ower r eser voi r pr essur e r esul t s i n a

l ower f r act ur e pr opagat i on pr essur e.

- 220 -

I n i t i a l l y / of cour se, t he wel l spaci ng i s not known and t her ef or e t he

model shoul d

be run f or an

expect ed r ange

of

wel l spaci ngs.



I t shoul d

be

not ed t hat t he cal cul at ed maxi mum pr opagat i on pr essur e,

p

f

, may

never

be

r eached dur i ng t he act ual i nj ect i vi t y t est ,

f or

i nst ance

because ver t i cal pr opagat i on i nt o anot her r eser voi r occur r ed. On t he ot her

hand p may al so be exceeded i n t he t est because no si t uat i on of bal anced

i nj ect i on

and

pr oduct i on coul d

be

cr eat ed

and

t her ef or e

no

st eady- st at e

coul d occur .

When t he maxi mum propagat i on pr essur e

i s

det er m ned f r om t he model

t he

corr espondi ng val ue f or c can al so be obt ai ned. From ( 5. 10) t he expect ed

- m a x

i. i.1.

maxi mum pr opagat i on pr essur e,

p. , at

r eser voi r pr essur e

p

can then

be

obt ai ned f r om

p

accor di ng

t o:

(1-c)

A Ap

- max

max p

, c .. n

,

P

f

= P

f

+

1-C A

( 5

'

1 1 }

As was di scussed above,

c

t ypi cal l y r anges bet ween

0. 4 and

0. 7.

For c = 0. 5

we have:

- 221 -

The s i gni f i cance of ( 5. 14) or ( 5. 15) i s t hat f or any r eser voi r pr essur e

bel ow p ver t i cal f r act ur e cont ai nment i s guar ant eed. Thi s means, f or



c/b

i nst ance, t hat i f t he i ni t i al st r ess cont r ast bet ween S„. and S i s notH .

l ar ge enough t o cont ai n t he f r act ur e so t hat t her ef or e p, > S , a l ower

r H

r eser voi r pr essur e can be det er m ned f rom ( 5. 14) at whi ch ver t i cal

pr opagat i on i nt o cap and base r ock wi l l not occur . On t he ot her hand, f or

l ar ge i ni t i al st r ess cont r ast s, ( 5. 14) may be used t o det er m ne t he r i se i n

r eser voi r pr essur e t hat i s al l owed bef or e f r act ur i ng i nt o cap and base rock

occur s .

I f t he ver t i cal f r actur e gr owt h was moni t or ed dur i ng the i nj ect i vi t y

t est t hen t he maxi mum i nj ect i on pr essur e may have been est abl i shed f or whi ch

t he f r act ur e pr opagat ed i nt o cap and base rock but not f ar enough to

est abl i sh communi cat i on wi t h ot her r eser voi r s. I n t hat case t hi s pressur ec/b

can be used i n ( 5. 13) rather t han S .

H

5. 6. 2 Det er m nat i on of maxi mum i nj ect i on r at e

- 222 -

cor r espondi ng bot t omhol e pr essur e i n t he pr oduct i on wel l . Fr om t he PI the

maxi mum pr oduct i on r at e q at r eser voi r pr essur e p can be obt ai ned.



Dependi ng on the rat i o of pr oduct i on wel l s t o f r act ur ed i nj ect i on

wel l s ,

R, at r eser voi r pr essur e p, wi t h R gi ven by

_ - max

R =

3

( 5. 17)

t he appr opr i at e wel l pat t er n at r eser voi r pr essur e p, wi t h p < p , can be

det er m ned. Subsequent l y t he r educt i on i n the number of wel l s compar ed t o

i nj ect i on at non- f r actur i ng condi t i ons can be cal cul at ed. F i nal l y, t he

f r act ur ed wel l pat t er n and t he r eser voi r pr essur e bel ow p t hat gi ve t he

maxi mum r educt i on i n t he number of wel l s can be sel ect ed.

A det ai l ed_exampl e

Let us r et ur n t o our exampl e i n Sect i on 5. 2 i n mor e det ai l . I t was

assumed t hat at t he pr evai l i ng r eser voi r pr essur e, p. , t he maxi mum i nj ect i on

- 223 -

Ap = p- p. . We have al so cal cul at ed t he r educt i on i n the number of wel l s t hat

can be obt ai ned by i nj ecti ng under f r acturi ng condi t i ons at r eser voi r



P

pr essur e p compar ed t o t he number of non- f r act ur ed wel l s r equi r ed at

r eser voi r pr essur e p. .

We i nt r oduce the f ol l owi ng def i ni t i ons:

N = t ot al number of non- f r act ur ed wel l s at r eser voi r pressur e p. ,

r equi r ed f or a cer t ai n f i el d of f t ake Q.

N = t ot al number of wel l s , r equi r ed f or a cer t ai n f i el d of f t ake Q, whi l e

i nj ect i ng under f r actur i ng condi t i ons at r eser voi r pr essur e p.

= r at i o of r equi r ed number of pr oducer s at r eser voi r pr essur e p t o t he

r equi r ed number of pr oducer s at r eser voi r pr essur e p. .

F. = r at i o of r equi r ed number of f r act ur ed i nj ectors at r eser voi r pr essur e

p t o t he r equi r ed number of pr oducer s at r eser voi r pr essur e p. .

Ap = di mensi onl ess change i n r eser voi r pr essur e def i ned as:

A

P n

= r ^ = ,

A P

, ( 5. 19)

D

( P f

i n

- P i )

( P i _ P w )

- 224 -

The m ni mum pr opagat i on pressur e p

f

/ ent er s the cal cul at i ons because i t

was used t o det er m ne t he maxi mum al l owabl e i nj ect i on rat e under f r act ur i ng



condi t i ons i n accor dance wi t h Sect i on 5. 6. 2. I n t r ansl at i ng p

f

t o

di f f er ent r eser voi r pr essur es Eq. ( 5. 10) was used i n Sect i on 5. 6. 1 wi t h

c = 0. 5 and A = 0. 5 as t ypi cal val ues.

P

I t shoul d be not ed t hat t he r educt i on i n t he number of wel l s i s

det er m ned by compar i son wi t h t he opt i mal non- f r act ur ed case. I t may not

al ways be possi bl e t o oper at e t he reser voi r at t he cor r espondi ng opt i mal

r eservoi r pr essure i n whi ch case t he r educt i on i n t he number of wel l s as

cal cul at ed i n ( 5. 23) i s t oo pessi m s t i c .

The r at i o of pr oducer s t o f r act ur ed i nj ect or s and t he r educt i on i n t he

number of wel l s as a f unct i on of t he di mensi onl ess change i n r eser voi r

pr essur e have been pl ot t ed i n Pi g. 5. 2 . The dot t ed cur ves r epr esent

Eq. ( 5. 22) and ( 5. 23) . For cer t ai n val ues of Ap t he r at i o R t akes on

non- i nt eger val ues. For i nst ance, at Ap = 0. 1, R equal s 2. 5. Thi s means

t hat t he i nj ect i on rate i nt o a f r actur ed i nj ector bal ances t he pr oduct i on

r at e of 2. 5 pr oducer s. Thi s i nj ect i on r at e i s t he maxi mum as cal cul at ed

- 225 -

DIMENSIONLESS CHANGE IN RESERVOIR PRESSURE

0-4 -0 -2 0 0 0-2 0-4 0-6 0-8



- 226 -



INVERTED 5-SPOT

R = 1

/

INV E RTE D 7 -S P O T

R = 2

/

- 227 -

Fi g. 5. 2 cl ear l y shows t hat wat er f l oodi ng under f r act ur i ng condi t i ons,

as f ar as no i mpai r ment of sweep ef f i ci ency i s concer ned, of f er s an enor mous



scope f or t he r educt i on i n t he number of wel l s. The r educt i on i n t he number

of wel l s f or non- i nt eger R r eaches a maxi mum of 42%at Ap = 0. 45. Thi s

cur ve has a maxi mum si nce accor di ng t o Bq. ( 5. 20) and ( 5. 21) f ewer pr oducer s

and mor e i nj ect or s ar e requi r ed as t he r eser voi r pr essur e i ncr eases and vi ce

ver sa.

*

I f a st r i ct l y regul ar wel l pat t er n i s adher ed t o t he maxi mum r educt i on

i n t he number of wel l s i s 40% Thi s r educt i on can be obt ai ned wi t h a

fractured

5-spot

or a f r actur ed

7-spot

wi t h t he l at t er bei ng oper at ed at a

l ower l evel of t he reser voi r pr essur e.

As di scussed i n Sect i on 5. 6. 1 ver t i cal f r actur e cont ai nment i s onl y

ensur ed f or r eser voi r pr essur es p such t hat p < p . I n t he t er m nol ogy of

Fi g.

5. 2 t hi s means t hat we must def i ne a maxi mum di mensi onl ess change i n

r eser voi r pr essur e Ap . The onl y cases t hat sat i sf y bot h t he condi t i ons of

ver t i cal cont ai nment and uni mpai r ed sweep ef f i ci ency ar e l ocat ed i n t he par t

of t he gr aph f or whi ch Ap < Ap

-

228 -

Si nce we had assumed t hat the I I and PI are equal and that the

pr evai l i ng r eser voi r pr essur e,

p. ,

equal s

the

opt i mum pr essur e

f or

i nj ect i ng



under non- f r act ur i ng condi t i ons

we

have

(see

Appendi x

5-A, Eq.

( 5 - A - 2 ) ) :

m n

,

P<=

+

P.

p

i

=

( 5. 27)

wi t h

p the

m ni mum bot t omhol e pr essur e

i n a

t ypi cal pr oducer .

Assum ng

a

gr adi ent

f or S . of 0. 17

bar/m

and

assum ng t hat

the

wel l s

ar e pr oduced

by

g a s l i f t

so

that

the

bot t omhol e pr essur e

has a

typi cal

gr adi ent

of 0. 016

bar/m

we

f i n d

for p. ,

fr om ( 5. 27)

and

( 5. 25)

a

gr adi ent

of 0. 10 bar /m i . e. the reser voi r pr essur e, p. , i s pract i cal l y hydr os ta t i c .

For

a

t ypi cal r ange

i n

gr adi ent s

f or the

stress

i n cap and

base rock,

we

, .

, -

A

max

f i n d

f or Ap :

S°

/ b

/ dept h

( b a r / m

s

c / b

/ s

H

/

H

Ap

max

- 229 -

maxi mum r educt i on i n the number of wel l s of 40% The most sui t abl e wel l

pat t er n i n t hi s case i s t he i nver t ed 7-spot because i t r equi r es a l ower



r eser voi r pr essur e t han the 5- spot .

We see f r om t he t abl e above t hat Ap i s extr emel y sensi t i ve t o t he

act ual st r ess cont r ast . Thi s emphasi zes t he i mpor t ance of accur at e st r ess

dat a f or t he opt i mal desi gn of wat er f l oodi ng under f r acturi ng condi t i ons.

Comgar i son wi t h_Hagoor t ' s_work

Hagoor t , i n hi s t hesi s , compar ed the conduct i vi t i es f or f r act ur ed and

non- f r actured f i ve- spot s as a f unct i on of L/ d wi t h L t he stabl e f r act ur e

hal f - l engt h dur i ng st eady- st at e and d t he wel l spaci ng. The conduct i vi t y of

a pat t er n i s def i ned as

C =

a

( 5. 28)

p . - p

wi wp

wi t h q t he st eady- st at e i nj ecti on r at e i nt o t he pat t er n and p . - p t he

w wp

- 230 -

conductivity a relatively high reservoir pressure is required which would

result in most cases in a loss of vertical fr acture containment.



Hagoort also considered a doubly fr actured 5-spot in which both the

injector, and producer are fr actured. Of course, for L/d = 0.21 for both

injector and producer a doubly fractured 5-spot is possible at the same

reservoir pressur e as the non-fr actured 5-spot (equal fractured II and PI ).

With the same conditions as considered in Fig. 5.2 only a third of the

injectors and a third of the producers are required as compared to

non -fr acturing, giving a reduction in the total number of wells of 67% .

Therefore, the fact that mainly 9 spots and 7-spots appear as viable options

in Fig. 5.2, strongly reflects the fact that only the injector is fractured.

5.7 FULL-SCALE IMPLEMENTATION

In the previous section it was shown how the reservoir pressure can be

determined for which the reduction in the number of wells is optimal while

- 231 -

c/b

to be constr ained to a maximum S , the horizontal rock stress in cap and

H



base rock. This maximum may be slightly higher if sufficient containment at

this higher pressure was inferred from the injectivity test. Suppose that

initially the reservoir press ure is lowered to a level p with p larger than

p , the reservoir pressure that permits fr acture growth at the fr acture

c/b

propagation pressure S . Since the injection pressur e is constrained to a

n

maximum that is lower than the fracture propagation pressure at reservoir

pressure p, the injectivity will be reduced. This results in a situation of

temporar y underinjection until the reservoir pressure reaches p and the

fr acture can propagate. Therefor e with a constrained injection pr ess ur e, the

reservoir pressure will adjust itself to the level that is required to keep

the fracture open. With the fracture open, the injection rate may exceed the

maximum rate for which sweep eff iciency is unimpaired. This would r esult in

temporary overinjection and therefore in a rise in reservoir pressure which

would eventually reduce the injectivity again. Nevertheless, the fracture

may grow too long in this period and, depending on its healing ability,

could permanently impair sweep efficiency.

- 232 -

If in the casing sequence a casing shoe is set into the cap rock, a

leak-off test may be performed, after drilling out the shoe, to measure the



for mation stren gth. If this is done across the field, additional inf ormation

on lateral consistency of f ormation strength is obtained.

5.8 SPECIAL APPLICATIONS

Line drive

If there is an indication that the direction of minimum in-situ str ess

is the same across the field, this circumstance may be exploited to set up a

line drive pattern. Alternate rows of injectors and producers with the rows

parallel to the fracture orientation could result in a very high sweep

efficiency.

Such an indication of rectilinear horizontal stress trajectories could

come from measurements of microfrac orientation together with, for instance,

a clear trend in the orientation of a fault system.

- 233 -

Poor - qual i t y i nj ect i on wat er

For i nj ect i on wat er of poor qual i t y consi der at i on may be gi ven t o



f r actur i ng t he i nj ector r at her t han t o f i l t er i ng t he wat er t o mai nt ai n

i n j e c t i v i t y .

The cont i nuous pl uggi ng of new ar eas of t he f r act ur e f ace woul d, of

cour se, r esul t i n cont i nui ng f r actur e gr owt h. The ef f ect on sweep ef f i c i ency

can agai n be checked wi t h t he si mul at or of Ref . 12. I t i ncor por at es a model

f or descr i bi ng per meabi l i t y det er i or at i on as a f unct i on of wat er qual i t y and

por e vol umes i nj ect ed.

I t i s of i nt erest t o not e that the pr essur e f al l - of f t echni que of

Chapt er 4, t oget her wi t h deconvol ut i on as descr i bed i n Ref . 6, can be used

t o cal cul at e t he ski n at t he f r actur e f ace. Sever al such cal cul at i ons i n

t i me can t hen be used t o val i dat e t he pl uggi ng model .

5. 9 CONCLUSI ONS

- 234 -

5) The fact that 7-spots and 9 spots are the most attractive options

reflects the fact that only the injectors are fractured. If both



injector s and producer s are f ractured the maximum reduction in the number

of wells is 67% with the fr actures in the producer s of the same length as

the stable f racture length at the in jectors. For a stress contr ast of 14%

or more a doubly fractured 5-spot would be the preferable pattern in this

case.

6) In-situ stress measurements should be performed in the reservoir and cap

and base rock. The stress data are essential for an optimal design of the

process.

7) An injectivity test should be carried out to establish a trend in

fr acture propagation pres sur e. Regular fall-off testing can be used to

determine the change in fracture dimensions with time. A match with the

analytical propagation model results in values for the thermo- and poro-

elastic constants. From the model the maximum fracture propagation

- 235 -

LI ST OF SYMBOLS



A por o- el as t i c cons t ant

P

C conduc t i v i t y

c c oef f i c i ent of pr opor t i onal i t y i n por o- el as t i c s t r es s c hange

d di s t ance f r om i nj ec to r t o near es t pr oducer


I I i nj ect i vi t y i ndex

F r at i o of t he number of pr oducer s at r eser voi r pr essur e p t o t he

number of pr oducers at r eser voi r pr essur e p.

F. r at i o of t he number of f r ac t ur ed i nj ec t or s at r eser voi r p ressu r e

l

p to t he number of pr oducers at r eservo i r pr essur e p.

k per meabi l i t y

K c r i t i c al s t r es s i nt ens i t y f act o r

L f r ac t ur e hal f - l engt h

N t ot al number of non- f r act ur ed we l l s at r eservoi r pr essur e p .

N t ot al number of pr oducers at r eser voi r pr essur e p.

- 2 3 6 -

g ma x i mu m p r o d u c t i o n r a t e a t r e s e r v o i r p r e s s u r e p .

q ma x i mu m p r o d u c t i o n r a t e a t r e s e r v o i r p r e s s u r e p



r w e l l b o r e r a d i u sw

R p e n e t r a t i o n d e p t h o f p r e s s u r e t r a n s i e n t s

R r a t i o o f t h e n u mb e r o f p r o d u c e r s t o t h e n u mb e r o f f r a c t u r e d

i n j e c t o r s a t r e s e r v o i r p r e s s u r e p

S

I T

. i n i t i a l h o r i z o n t a l r e s e r v o i r r o c k s t r e s s a t r e s e r v o i r p r e s s u r e

H i

r

p

i

S h o r i z o n t a l r e s e r v o i r r o c k s t r e s s a t r e s e r v o i r p r e s s u r e p

c / b

S h o r i z o n t a l r o c k s t r e s s i n c a p a n d b a s e r o c k

H

G r e e k

X f l u i d mo b i l i t y

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

- 237 -

REFERENCES



1) Br eckel s, I . M & van Eekel en,

H A. M ,

Rel at i onshi p bet ween hor i zont al

st r ess and dept h i n sedi ment ar y basi ns.

J PT ( Sept ember 1982) , p. 2191.

2) Abou- Sayed, A. A. , Br echt er , C. E. & Cl i f t on, R. J . , I n- s i t u st ress

measur ement s by hydr of r act ur i ng: A f r act ur e mechani cs appr oach.

J . Geoph. Res. ( J une 1978) , p. 2851.

3) Daneshy, A. A. , Sl usher , G. L . , Chi shol m P. T. & Magee, D. A. , I n- s i t u

st r ess measur ement s dur i ng dr i l l i ng.

J PT ( August

1986) ,

p. 891.

4) Gr i f f i n, K. W , I nduced f r actur e or i ent at i on det er m nat i on i n t he Kupar uk

r es er v oi r .

SPE

14261,

1985.

5) Schl umberger Br ochur e, M- 090033, Apr i l 1986.

6) Newber r y, B. , Nel son, R. , Cannon, D. & Ahmed, U. , Pr edi ct i on of ver t i cal

hydr aul i c f r act ur e m gr at i on usi ng compr essi onal and shear wave

- 238 -

APPENDI X 5- A



DETERM NATI ON OF THE RATI O OF PRODUCERS TO FRACTURED I NJ ECTORS

AND THE REDUCTI ON I N THE NUMBER OF WELLS

We make t he f ol l owi ng assumpt i ons:

1) We consi der uni t mobi l i t y r at i o and i nj ectors and pr oducer s wi t h

i dent i cal t ypi cal pr oper t i es so t hat :

I I = PI ( 5- A- l )

Let p' denot e t he r eser voi r pr essur e at whi ch t he maxi mum i nj ect i on rat e

i nt o a non- f r act ur ed i nj ector equal s t he maxi mum pr oduct i on r at e f r om a

pr oducer . I t f ol l ows f r om ( 5- A- l ) t hat :

mi n

- 239 -

The maxi mum i nj ect i on r at e i nt o a non- f r act ur ed i nj ect or and t he maxi mum

pr oduct i on r at e ar e gi ven by, r espect i vel y:



~max _ m n , __

, _

. . .

q = ( P

f

~ P

L

) - I I ( 5- A- 4)

q = ( p. - p ) . PI ( 5- A- 5)

p l w

From (( 5- A- l ) t o ( 5- A- 3) we have:

~

ma x

, c > e»

q = q ( 5- A- 6)

Si nce t he opt i mum non- f r act ur ed wel l pat t er n i s a 5-spot we have f or t he

t ot al number of wel l s , N, at r eser voi r pr essur e p.

N = 2N ( 5- A- 7)

P

i n whi ch N i s t he t ot al number of pr oducer s at r eser voi r pr essure

p.

.

- 240 -

—m n mm . __ . _

p.

= p. + 0. 33 Ap ( 5- A- 10)



wi t h Ap = p - p. .

W t h t he above assumpt i ons we can now cal cul at e t he r equi r ed quant i t i es*.

We have f or t he maxi mum pr oduct i on r ate at r eser voi r pr essur e p:

q

p

= (P - P

w

) • PI ( 5- A- l l )

so t hat

q = q + Ap . PI ( 5- A- 12)

P P

wi t h q f r om

( 5 - A - 5 ) .

P

Fr om t he def i ni t i on of F i n Sect i on 5. 6. 3 i n t he t ext and f rom

P

( 5- A- 12) we have:

q,

- 241 -

q

maX

= 0. 61 27ThX ( p™

n

-

p")

(5-A-17)



Fr om ( 5- A- 8) and ( 5- A- 10) we have:

- max _

g

max _

0 >g l 2 f f hX 0 # 6 g A p

(5-A-18)

so t hat f r om ( 5- A- 4) and ( 5 - A - 9 ) :

- max

3

= 3 - 1. 98 Ap„ ( 5- A- 19)

~max D

q

Fr om t he def i ni t i on of F. i n Sect i on 5. 6. 3 we have usi ng ( 5- A- 6) and

( 5- A- 19) :

q j max

F. = —£ - =

a

= (3 - 1. 98 Ap, J ( 5- A- 20)

l - max - max D

q q

The rat i o of pr oducer s t o f r actur ed i nj ectors as a f unct i on of t he

- 243 -

WATERFLOODI NG UNDER FRACTURI NG CONDI TI ONS



SUMMARY

When an oi l f i el d i s expl oi t ed by si mpl y pr oduci ng oi l and gas f r om a

number of wel l s, t he r eser voi r pr essur e, i n many cases, dr ops r at her qui ckl y

and so does t he pr oduct i on r at e. Ther ef or e, wat er i s of t en i nj ected t o

mai nt ai n t he reser voi r pr essur e. The i nj ect i on wel l s ar e l ocat ed i n such a

way t hat as much oi l as possi bl e i s dr i ven t o t he pr oducer s bef or e t hey

exper i ence wat er br eakt hr ough.

Recover i ng oi l by wat er i nj ect i on i s cal l ed wat er f l oodi ng. The

ef f i ci ency wi t h whi ch t he wat er sweeps t he oi l t o the pr oducer s wi t hout

bypassi ng i t i s cal l ed t he sweep ef f i c i ency.

A maj or savi ng of t he expl oi t at i on cost s of an oi l f i el d can be obt ai ned

by a r educt i on i n the number of wel l s. The r equi r ed number of i nj ect or s can

be r educed by i ncreasi ng t hei r i nj ecti on capaci t y. An ef f ect i ve way of doi ng

- 244 -

The obj ect i ves of t hi s t hes i s ar e t o:

a) i nvest i gat e t he mechani sms of f r act ur e i ni t i at i on and pr opagat i on under



t he i nf l uence of cont i nued wat er i nj ect i on,

b) eval uat e t he ef f ect of f r act ur e gr owt h on sweep ef f i ci ency,

c ) i mpr ove t he met hods f or det er m ni ng f r act ur e di mensi ons,

d) gi ve r ul es f or desi gni ng t he pr ocess of wat er f l oodi ng under f r act ur i ng

condi t i ons i n such a way t hat sweep ef f i ci ency i s uni mpai r ed and ver t i cal

f ract ure growt h i s l i m t ed.

Thi s t hesi s consi st s of f i ve sel f - cont ai ned chapt er s. I n Chapt er 1 a

gener al i nt r oduct i on i s gi ven, t oget her wi t h a sur vey of t he wor k t hat was

done i n t he past and of t he new el ement s cont r i but ed by t hi s t hesi s. I n

Chapt er 2 a compl et e, anal yt i cal sol ut i on i s gi ven for t he st r ess f i el d i n

t he reser voi r r ock sur r oundi ng an unf r act ur ed

wel l .

Thr ee- di mensi onal t heory

of por o- and t her moel ast i ci t y has been used t o cal cul at e t he ef f ect of

changes i n pr essur e and t emper at ur e on the rock s t r ess . I t i s shown t hat t he

commonl y used r educt i on of t he st r ess cal cul at i ons t o t wo hor i zont al -

- 245 -

Speci a l a t t ent i on i s f ocused on t he mode of f r act ur e pr opagat i on i n

whi ch t he pr essur e t r ans i ent s t r avel r adi al l y - symmet r i cal l y i n t he pl ane of



t he r ese r voi r . I t i s shown t hat when t he i nj ec t i on ra t e i s sma l l e nough t o

es t abl i s h pr opagat i on wi t h t hi s ps eudo- r adi al f l ow, t he s weep ef f i c i enc y of

wat er i nj ec t i on i s not a f f ec t ed. Thr ee- di mens i onal t heor y of por o - and

t her moel as t i c i t y has been us ed f or an anal yt i c al c al c ul at i on of t he ef f ec t

of pr essur e and t emperat ur e changes on t he st r ess f i el d ar ound a pr opagat i ng

f r ac t ur e wi t h ps eudo- r adi al f l ow. The ef f ec t of a c hange i n t he s t r es s f i el d

on t he f r ac t ur e pr opagat i on vel oc i t y i s i l l us t r at ed wi t h t wo e xampl e s .

F ur t he r , a numer i cal met hod i s gi ven f or t he cal cul at i on of pr e s sur e - and

t emper at ur e- dependent changes i n r eser voi r r ock st r ess that c an eas i l y be

i ncor porat ed i nt o numer i ca l f r ac t ur e gr owt h s i mul at i on model s such as

devel oped i n t he pas t .

Chapt er 4 d i scusses a met hod f or det erm ni ng t he d i mens i ons of an

i nduc ed f r ac t ur e f r om a pr es s ur e f al l - of f t es t . T he t ec hni que c ons i s t s of

measur i ng t he decr eas i ng f l ui d pr essur e downho l e dur i ng an i nt e r r upt i on of

t he i nj ec t i on. F r om t he behavi our of t he f al l i ng pr es s ur e wi t h t i me,

- 246 -

i nj ec t i on bal anc i ng t he t ot al pr oduc t i on. Thi s quant i t y shoul d be l ower

t han t he hor i zont al r ock s t r ess i n cap and base r ock i n or der t o pr event



ver t i c al f r ac t ur e pr opagat i on. I f i t i s t oo hi gh, t he r es er voi r pr es s ur e

must be l owered such t hat t he corr espondi ng decr ease i n t he hor i zont a l

r eser voi r r ock s t r ess r esul t s i n a max i mum f r ac t ur e pr opaga t i on pr essu re

t hat pr ev ent s v er t i c al pr opagat i on.

- Des i gn cal cul at i ons t o es t abl i sh a ) t he max i mum r eser voi r pr essur e at

whi ch ver t i cal cont ai nment i s ensu red, b ) t he max i mum i nj ec t i on ra te at

whi ch sweep ef f i c i ency i s uni mpa i r ed, c ) t he opt i mal wel l pat t e r n and

r eservo i r pr essur e , d ) t he r educt i on i n t he number of we l l s compar ed t o

i nj ec t i on under non- f r ac t ur i ng c ondi t i ons .

I t i s shown i n a t ypi cal exampl e t hat wi t h the hor i z ont al r ock s t r ess

i n cap and base rock exceedi ng t hat i n t he reser voi r by 14% or mor e t he

r educ t i on i n t he number of wel l s by f r ac t ur i ng t he i nj ec t i on wel l s r anges

f r om 33% t o a max i mum of 40%. I f t he pr oduc t i on wel l s ar e a l so f r ac t ur ed

wi t h s t andar d f r ac t ur e s t i mul at i on t echni ques and i f t hese f r ac t ur es have

t he same l engt h as t he s t abi l i sed f r ac t ur e at t he i nj ec t i on we l l t he

-

247

-

WATERSTÜWING BIJ SPLIJTINGSCONPITIES



SAMENVATTI NG

Wor dt een ol i evel d ont gonnen door eenvoudi gweg ol i e en gas t e

pr oducer en ui t een aant al put t en dan zal dat veel al l ei den t ot dal i ng van de

r eser voi r dr uk en, di ent engevol ge, van de pr odukt i esnel hei d. Daar om wordt

vaak wat er gei nj ect eerd om de r eser voi r dr uk zoveel mogel i j k t e handhaven. De

i nj ect i eput t en wor den zodani g gepl aat st dat zoveel mogel i j k ol i e naar de

pr odukt i eput t en wor dt gest uwd voor dat wat er hi er i n door br eekt . Ol i ewi nni ng

door wat er i nj ect i e wor dt wat er st uwi ng genoemd. De f r act i e van het r eser voi r -

vol ume dat ui t ei ndel i j k door het geï nj ect eer de wat er best r eken wor dt , heet

het veegver mogen.

Een van de gr oot st e bespar i ngen op de ont gi nni ngskost en van een

ol i evel d kan ber ei kt wor den met een ver m nder i ng van het aant al put t en. Het

ver ei st e aant al i nj ect i eput t en kan ver m nder d wor den door een ver gr ot i ng van

hun i nj ect i ecapaci t ei t . Een ef f ect i eve mani er om di t t e ber ei ken i s door het

- 248 -

c ) het ver bet er en van de met hoden om de scheur af met i ngen t e bepal en,

d) het geven van een een voor schr i f t om t ot een ont wer p t e komen van



wat er st uwi ng bi j spl i j t i ngscondi t i es voor een gegeven vel d waar i n het

veegver mogen onaanget ast bl i j f t en de ver t i kal e s cheur ui t br ei di ng beper kt

i s .

Di t pr oef schr i f t best aat ui t vi j f op z i chzel f s taande hoof dst ukken. I n

Hoof dst uk 1 wor dt een al gemene i nl ei di ng gegeven samen met een over zi cht van

het werk dat i n het ver l eden i s gedaan en de ni euwe el ement en di e met di t

pr oef schr i f t wor den t oegevoegd. I n Hoof dst uk 2 wor dt een vol l edi ge,

anal yt i sche opl ossi ng gegeven voor het spanni ngsvel d i n het r eser voi r

gest eent e om een ongescheur de put . Dr i e- di mensi onal e por o- en t her mo-

el ast i ci t ei t st heor i e zi j n gebr ui kt om het ef f ect van dr uk- en t emper at uur s

ver ander i ngen op de gest eentespanni ng t e ber ekenen. Aanget oond wor dt dat de

vaak gebr ui kt e r educt i e van de spanni ngsber ekeni ngen t ot het hor i zont al e

vl ak t ot een er nst i ge onder schat t i ng van de spanni ngsver ander i ngen kan

l ei den. Het ber ekende spanni ngsvel d i n het gest eent e wor dt gebr ui kt om de

- 249 -

Aanget oond wordt dat wanneer de i nj ect i esnel hei d l aag genoeg i s om scheur -

ui t br ei di ng met deze zogenaamde pseudo- r adi al e st r om ng te bewer kst el l i gen,

het veegver mogen van

water i nj ect i e

ni et wor dt bei nvl oed.



Dr i e- di mensi onal e por o- en ther moel ast i c i t ei t st heor i e z i j n gebr ui kt

voor een anal yt i sche berekeni ng van het ef f ect van dr uk- en t emper at uur s

ver ander i ngen op het spanni ngsvel d r ond een zi ch ui t br ei dende s cheur met

pseudo- r adi al e st r om ng. Het ef f ect van de met het spanni ngsvel d

ver anderende snel hei d van de scheur ui t br ei di ng wor dt geï l l ust r eerd aan de

hand van t wee voor beel den. Verder wordt een numer i eke met hode behandel d voor

het berekenen van dr uk- en t emperat uur saf hankel i j ke spanni ngsver ander i ngen

di e eenvoudi g i ngebouwd kan worden i n numer i eke s cheurgr oei - si mul at i e-

model l en zoal s di e i n het ver l eden zi j n ont wi kkel d.

Hoof dst uk 4 behandel t een met hode om de af met i ngen van een aangebr acht e

scheur t e bepal en met een zogenaamde dr ukdal i ngst est . De t echni ek best aat

ui t het met en van de dr uk onder i n de put t i j dens een onder br eki ng van de

i nj ect i e.

Aanget oond wor dt dat i n het dr ukver l oop vi er peri oden zi j n t e

onderschei den di e el k onaf hankel i j ke gegevens opl everen over de

scheur gr oot t e. Bovendi en kan de i n- si t u gest eent espanni ng bepaal d wor den

- 250 -

l agen om ver t i kal e scheur ui t br ei di ng t e voor komen. I n het geval van een t e

hoge waar de moet de r eser voi r dr uk ver l aagd wor den zodat de overeenkomst i ge



af name i n de hor i zont al e r eser voi r - gest eent espanni ng een maxi mum

scheur ui t br ei di ngsdr uk geef t di e ver t i kal e ui t br ei di ng ver hi nder t .

- Ont wer pber ekeni ngen voor het vast st el l en van a) maxi mum r eservoi r dr uk

waar bi j ver t i kal e scheur ui t br ei di ng beper kt bl i j f t , b) maxi mum

i nj ect i esnel hei d waar bi j het veegver mogen onaanget ast bl i j f t , c) het

opt i mal e put pat r oon en de over eenkomst i ge r eser voi r dr uk, d) de

ver m nder i ng van het aant al put t en t en opzi cht e van het geval waar i n

gei nj ecteer d wor dt bi j ni et - spl i j t i ngscondi t i es .

I n een kar akt er i st i ek voorbeel d wor dt aanget oond dat wanneer de

hor i zont al e gest eent espanni ng i n de aangr enzende l agen di e i n het

r eser voi r over t r ef t met 14%of meer , de ver m nder i ng i n het aant al put t en

r ei kt van 33% t ot een maxi mum van 40% Wanneer de pr odukt i eput t en ook

gescheur d wor den met st andaar d scheur - st i mul at i e t echni eken en wanneer

deze s cheur en dezel f de af met i ng hebben al s de gest abi l i seer de scheur bi j

de i nj ect i eput dan kan de ver m nder i ng i n het aant al put t en een maxi mum

- 251 -

ACKNOWLEDGEMENTS



Most of t hi s t hesi s i s based on t he wor k I di d dur i ng my l ast t wo year s

at Koni nkl i j ke/ Shel l Expl or at i e en Pr odukt i e Labor at or i um ( KSEPL) .

I compl et ed t hi s t hesi s dur i ng subsequent l eaves whi l e al r eady wor ki ng f or

Pet r ol eum Devel opment Oman ( PDO) .

I shoul d l i ke to t hank t he Management of KSEPL f or per m ssi on t o

publ i sh t hi s wor k and f or pr ovi di ng me wi t h t he oppor t uni t y t o pr oduce t hi s

t hes i s .

I am especi al l y i ndebt ed t o Hel mut Ni ko of KSEPL, wi t h whom I spent

many enj oyabl e and st i mul at i ng hour s di scussi ng t he subj ect . Thr oughout

t hose t wo years Hel mut was an ent husi ast i c cr i t i c and suppor t er . Hel mut has

made several cont r i but i ons t o t hi s work and wr ot e t he comput er progr amme f or

t he numer i cal i nver si on i n Chapt er 4.

I am gr at ef ul t o J oost v. d. Bur gh, J an Geer t sma and Ri k Dr ent h, wi t h

whom I had numer ous i nst r uct i ve di scussi ons.

I am al so i ndebt ed t o my successor at KSEPL, Ben Di kken, who read my

STELLINGEN

behorende bij het proefschrift "Waterflooding under Fracturing

Conditions", E.J.L. Koning, September 1988.

1. De vereenvoudigde, een-dimensionale modellering van vloeistofuitstroming

die algemeen in scheurstimulatie ontwerppr ogramma's wordt toegepast (zie

bijv. Howard h Fast) kan tot een optimistische voorspelling van de

scheurlengte lei den. Het verdient dan ook aanbeveling deze beschr ijving



uit te breiden naar twee dimensies.

Howard, G.G.

i

Fast, C.R., Hydraulic Fracturing,

SPE Monograph, Volume 2, 1970.

2. Sommige oplossingen voor problemen in de poro- en thermo-ela sticiteits -

theorie met gemengde randvoorwaarden (zie bijv. Smith en Hagoort) kunnen

sterk vereenvoudigd worden door een met de Goodier verplaatsings

potentiaal gegenereerde oplossing te superponeren op een in de homogene

elasticiteitstheorie reeds bekende oplossing.

Smith, M.B., Stimulation design for short, precise

hydraulic fractures - MHF, SPE 10313, 1981.

Hagoort, J., Waterflood-induced hydraulic

fracturing, Hfdst. 4, proefschrift, TH Delft, 1981.

3. De twee-dimensionale covariante Poisson vergelijking voor Goodier's

verplaatsingspotentiaal genereert alleen op vlakke twee-dimensionale

ruimtes een particuliere oplossing voor de thermo-elasticiteits-

vergelijkingen. Hetzelfde geldt ook in drie dimensies.

4. De door Nolte ontwikkelde methode om de drukdaling na een mini-

scheurstimulatie te analyseren berust op enkele niet zonder meer

toelaatbare veronderstellingen. Het verdient dan ook aanbeveling het

geldigheidsgebied van deze veel gebruikte methode te bepalen.

Nolte, K.G., Determination of fracture parameters

from fracturing pressure decline, SPE 8341, 1979.

- 2 -

5. De door Ramey en Agarwal gegeven oplossing voor de bodemdruk tijdens

productie met een ver anderende boorgatvuil ing (wellbore stor age) is

onjuist.

Ramey, H.J., Jr & Agarwal, R.G., Annulus unloading

rates as influenced by wellbore storage and skin

effect, SPEJ (Oct. 1972) 453.

6. De bewering van Oake dat gedurende veranderingen in de reservoirdruk de

- 3 -

9. Bij het testen van een geavanceerde oliewinningsmethode op veld-schaal

is succes sterk afhankelijk van nauwe samenwerking tussen de

verschillen de betrokken disciplines, met name tussen die in de resear ch

en die in het veld.

Koning, E.J.L., Mentzer, E., t, Heemskerk, J.,

Evaluation of a pilot polymer flood in the

Marmul field, Oman, SPE 18092, 1988.



totale verticale reservoirgesteentespanning onveranderd blijft is alleen

juist indien de laterale uitgestrektheid van de drukverandering groot is

ten opzichte van de reservoirhoogte.

Dake,

L.P., Fundamentals of Reservoir Engineering,

Elsevier Scient. Publ. Comp., 1976, p. 4.

7. Bij het bepalen van de in-situ gesteentespanning door het maken van

micro-scheuren kan het nodig zijn poro-elastische effecten op de scheur-

sluitingsdruk in rekening te brengen. De in dit proefschrift gegeven

oplossing voor het poro-elastische spanningsveld kan hiervoor worden

gebruikt.

Dit proefschrift, Hoofdstuk 3, p. 127.

8. Deconvolutie van drukdalingsdata voor een zich sluitende scheur of voor

een scheur met boorgat- en/of scheurvulling (wellbore/fr acture s torage)

heeft het voordeel boven ander e methoden dat het expliciet de skin aan

het scheuroppervlak kan opleveren.

Koning, E.J.L.

t

Niko, H., Application of a special

falloff test in a fractured North-Sea injector,

SPE 16392, 1986.

10.

In reservoirs met een matig hoge olieviscositeit kan polymeerstuwing

onder splijtingscondities een grote besparing in het aantal putten

opleveren.

11. Tijdens polymeerstuwing in sterk heterogene reservoirs kan het

stapsgewijs reduceren van de polymeerviscositeit volgens de methode van

Claridge een verslechtering van de olieopbrengst opleveren ten opzichte

van voortdurende injectie met de oorspronkelijke viscositeit. Een zinvol

ontwer p van dit reductieproces is daarom alleen mogelijk indien een goed

model van de reservoirheterogeniteiten beschikbaar is.

Claridge, E.L., Control of viscous finger ing in

enhanced oil recovery processes: effect of

heterogeneities. SPE 7662, 1978.

12 .

Gezien de stormachtige ontwikkeling in de digitalisering van muzikale

infor matie, waarbij voornamelijk gebruik gemaakt wordt van toetsen borden

in combinatie met computertechnieken, is het muziekstudenten aan te

bevelen piano als hoofdvak en informatica als bijvak te kiezen.

13. Het in Oman ingestelde verbod op een huwelijk tussen Omani's en

buitenlanders is in strijd met de geest van de Islam.

14 .

Het toenemende aantal jeep-achtige voertuigen onder stadsbewoners

suggereert een markt voor de terreinfiets.

- 2 -

5. De door Ramey en Agarwal gegeven oplossing voor de bodemdruk tijdens

productie met een veranderende boorgatvulling (wellbore storage) is

onjuist.

Ramey, H.J., Jr 4 Agarwal, R.G., Annulus unloading

rates as influenced by wellbore storage and skin

effect, SPEJ (Oct. 1972) 453.

6. De bewering van Dake dat gedurende veranderingen in de reservoirdruk de

totale verticale reservoirgesteentespanning onveranderd blijft is alleen

- 3 -

9. Bij het testen van een geavanceerde oliewinningsmethode op veld-schaal

is succes sterk afhankelijk van nauwe samenwerking tussen de

verschillende betrokken disciplines, met name tussen die in de research

en die in het veld.

Koning, E.J.L., Mentzer, E., & Heemskerk, J.,

Evaluation of a pilot polymer flood in the

Marmul field, Oman, SPE 18092, 1988.

10 . In reservoirs met een matig hoge olieviscositeit kan polymeerstuwing



juist indien de laterale uitgestrektheid van de drukverandering groot is

ten opzichte van de reservoirhoogte.

Dake, L.P., Fundamentals of Reservoir Engineering,

Elsevier Scient. Publ. Comp., 1978, p. 4.

7. Bij het bepalen van de in-situ gesteentespanning door het maken van

micro-scheuren kan het nodig zijn poro-elastische effecten op de scheur-

sluitingsdr uk in rekening te brengen . De in dit proefschrift gegeven

oplossing voor het poro-elastische spanningsveld kan hiervoor worden

gebruikt.

Dit proefschrift. Hoofdstuk 3, p. 127.

8. Deconvolutie van dr ukdalingsdata voor een zich sluitende scheur of voor

een scheur met boorgat- en/of scheurvulling (wellbore/fracture storage)

heeft het voordeel boven andere methoden dat het expliciet de skin aan

het scheuroppervlak kan opleveren.

Koning, E. J.L. f, Niko, H., Application of a special

falloff test in a fractured North-Sea injector,

SPE 16392, 1986.

onder splijtingscondities een grote besparing in het aantal putten

opleveren.

11. Tijdens polymeerstuwing in sterk heterogene reservoirs kan het

stapsgewijs reduceren van de polyoeerviscositeit volgens de methode van

Claridge een verslechtering van de olieopbrengst opleveren ten opzichte

van voortdurende injectie met de oorspronkelijke viscositeit. Een zinvol

ontwer p van dit reductieproces is daarom alleen mogelijk indien een goed

model van de reservoirheterogeniteiten beschikbaar is.

Claridge, E.L., Control of viscous f ingering in

enhanced oil recovery proces ses: effect of

heterogeneities. SPE

7662,

1978.

12 . Gezien de stormachtige ontwikkeling in de digitalisering van muzikale

informatie, waarbij voornamelijk gebruik gemaakt wordt van toetsenborden

in combinatie met computertechnieken, is het muziekstudenten aan te

bevelen piano als hoofdvak en informatica als bijvak te kiezen.

13 . Het in Oman ingestelde verbod op een huwelijk tussen Omani's en

buitenlanders is in strijd met de geest van de Islam.

14. Het toenemende aantal jeep-achtige voertuigen onder stadsbewoners

suggereert een markt voor de terreinfiets.

STELLINGEN

behorende bij het proefschrift "Waterflooding under Fracturing

Conditions", E.J.L. Koning, September 19B8.

1. De vereenvoudigde, een-dimensionale modellering van vloeistofuitstroming

die algemeen in scheurstimulatie ontwerpprogramma's wordt toegepast (zie

bijv. Howard & Fast) kan tot een optimistische voorspelling van de

scheurlengte leiden. Het verdient dan ook aanbeveling deze beschrijving



uit te breiden naar twee dimensies.

Howard, G.G. & Fast, C.R., Hydraulic Fracturing,

SPE Monograph, Volume 2, 1970.

2. Sommige oplossingen voor problemen in de poro- en thermo-elasticiteits-

theorie met gemengde randvoorwaarden (zie bijv. Smith en Hagoort) kunnen

sterk vereenvoudigd worden door een met de Goodier verplaatsings

potentiaal gegener eerde oplossing te superponeren op een in de homogene

elasticiteitstheorie reeds bekende oplossing.

Smith, H.B., Stimulation design for short, precise

hydraulic fractures - KHF, SPE 10313, 1981.

Hagoort, J., Waterflood-induced hydraulic

fracturing, Hfdst. 4, proefschrift, TH Delft, 1981.

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