RAmey Agarwal, CD

7/25/2019 RAmey Agarwal, CD

1/10

3s3f

Annulus Unloading Rates as Influenced

Storage and Skin Effect

HENRYJ. RAM EY, JR.

RAMG. AGARWAL

MEMBERS A ME

ABSTRACT

The m od ern t rend in w ell test ing (bu ildup or

d ra w dow n ) ha s been tow a rd acqu is it ion

a nd

ana lys i s

of shor t-t im e da ta . Pressu re d a ta ear ly in a test a re

usua lly d is tor ted by severa l ~actors tha t m ask the

con ven t ion al s tra igh t line. S om e o/ tbe fa ctors a re

w ellbore storage and var ious sk in effects such as

those due to per fora t ions, pa r t ia l penetra t ion ,

non -Darcy f low , or w el l st imu la t ion ef fects.

Recen t ly , Agarw al et a l. 1 p resen ted a f undamenta l

s tudy of the im por tance of w el lbore storage w ith a

sk in e fect to short -t im e transien t f low . Th is paper

/ u r ther extend s the concept of ana ly zing short -t im e

w el l tes t d a ta to includ e solu t ions of cer ta in

d r i l ls tem test prob lems and of cases w herein the

storage constan t, CD, undergoes an abrupt change

f rom

on e

constan t va lue to another . An exam ple of

the la t ter case is change in storage ty pe f rom

compression

to liqu id

level va r ia t ions w hen

tubinghead pressure d rops to a tm ospher ic du r ing

prod uct ion . The purpose of the presen t paper is to:

(1 ) p resen t tabu la r and graph ica l resu lts for the

sand face f low ra te, q~f

and the annu lus un load ing

ra te, qa , as a fract ion of the constan t su r face ra te,

q, and (2 ) i l lu st ra te severa l pract ica l w ell tes t

si tua t ions tha t requ ire

such a solu t ion . Resu lts

includ e a range of va lues of tbe storage constan t ,

CD ,

and the sk in ef fect , s , u sefu l for w ell test

problems.

Annu lus un load ing or s torage has been show n to

be an im portan t phy sica l ef fect tha t of ten con trols

ea rly w ei t es t beha vior .

i + ;- Q*,)r l .,

As a rest i t

O\

Lb .- =.u -J .

i t appears tha t in terpreta t ions of short -t im e w el l

test da ta can be m ad e w ith a grea ter rel iabil i ty ,

and solu t ions to other storage-d om ina ted problem s

can be obta ined easily . Techn iques presen ted in

th is paper shou ld enab le the users to ana ly ze

certa in shor t-t ime w el l test d a ta tha t cou ld otherw ise

be regard ed as useless.

Origin a l m anu scr ip t r ec e ived in Soc ie t y of Pe t ro le um Engin eer s

office J uly 23 , 197 1 , Revised m an u scr ip t r ece ived Ap ril 1 7 , 1 972 .

Paper (SPE 3 53 S) was presen t ed a t t h e SPE 46 t h Ann ua l Fall

Mee t in g, h e ld in New Or lea n s , Oc t . 3 -6 , 1 9 71 . @ Copyr igh t 19 72

Am er ican In s t it u t e of Min in g,

Met allu rgic a l, and Pe t ro le um

En gin eer s , k .

p re fe ren ces given a t end of PaPer .

Th is paper will be p rin t ed in Tra n sac t ion s volum e 253 , wh ich

will c ove r 19 72 .

I

STANFORD U.

STANFORD, CALI F.

I

AMOCO PRODUCTION CO.

TULSA, OKLA.

INTRODUCTION

In a

recent

pa per, Aga rw a l

et a l. 1

presen t ed a

st udy of t he im port a nce of w ellbore st ora ge w it h a

skin effect t o short -t ime t ra nsient flow . They a lso

present ed a n a na lyt ica l expression for t h e fra ct ion

of t h e const a nt surfa ce ra t e,

q,

produced from t h e

annulus

(

dpWD

dpWD *

C D ~

= E r

D

)

Alt hough t he r igorous soi~ t ion (hm wrsior i ht tegra i)

a nd long- a nd sh ort -t ime a pproxima t e forms w ere

discussed, neit her t a bula r nor gra phica l result s of

PZLJD

t he a nnulus unloa ding ra t e, C D

d tD

w e re g iv en .

I t n ow a ppea rs t ha t such solut ion s- a re useful in

cert ain drillst em t est problem s a nd in ca ses w herein

t he st ora ge const a nt , CD , cha nges dur ing a w ell

t est . An exa mple is cha nge in st ora ge t ype from

compression t o liq uid level cha nge w hen t ubinghea d

pressur e d rops t o a t mosph eric durin g pr oduct ion .

Th e purpose of t his st udy is t o (1) presen t t a bula r

a nd gra phica l result s for t he sa ndfa ce flow ra te a nd

t he a nnulus unloa ding ra te a nd (2) illust ra te severa l

pra ct ica l w ell t est sit ua t ions t ha t req uire t h e

solutions.

TH E CLAS SI C WE LLB ORE

S TORAG E P ROB LE M 2

The problem t o be considered IS on e of fiow of a

slight ly compressible fluid in a n idea l ra dia l flow

system.

The diffusivit y eq ua t ion for fluid flow in

t erms of dimensionless va ria bles is

The init ia l a nd out er bounda ry cond it ions a re

PD(

D >

0)=0(2)

*CD = C (dim en s ion less s t o rage c on s t an t u sed in Ref. 1 ).

OCTOBER, 1972 J F


2/10

lim

{

t)

D(r~Y ~

}

=0 . . . . .

(3)

r + cn

D

w hile t he inner bounda ry condit ion is

a nd

E q. 4 st a t es t ha t t he dimensionless w ellbore

unload ing ra te,

%. %

Plus

t h e d im en sion les s

sandface rate,

q~, / q,

must equa l unit y. From E q. 4:

apD

()

dpwD

= - =1-CD7

(6)

q

brn m=,

dtn

u

u.

where q~~ is

t he sa ndfa ce flow ra t e a nd q is t he

const a nt surfa ce flow ra t e. E q . 6 ca n be rew rit t en

in rerms of a nnulus unloa ding ra te, qa, as

~a

dpwD

.

Cn

(7)

E q . 5 int roduces a st ea dy-st a t e skin effect a nd,

t hus, a pressure

drop a t t he sa ndfa ce t ha t is

proport iona l t o t he sa ndfa ce flow ra te. The pressure

w it hin t he w ellbore is represent ed by p WD , w hile

pressure on t he forma t ion side of t he skin effect is

repr esent ed by

pD .

The w ellbore unloa ding

or

storage

const a nt , C

9

is t ha t defined by va n

E verdingen a nd H urst . Tha t is,

c

. . . . . . . . .

(8)

CD =

2@hc r2

tw

C represent s t he volume of w ellbore fluid unloa ded

or st ored, cubic cent imet ers per a tmosphere. S tora ge

ma y be by vir t ue of eit her compressibilit y or a

ch an gin g liq uid level.

Finally,

pD, fD

a nd t D a re defined in t he usua l

4 5 4

PD(rD>tD)=

r

D=~

21-t kh (pi - pr t)

.

w

. . . . . . . . . .

kt

D=

vtr:

. . .

(9)

. . .

(lo)

. . .

(11)

B ot h cylindrica l a nd line source solut ions of t he

dimensionless flow ing pressure, P WD , in t he form

of rea l inversion

int egra ls, w er e repor t ed by

Aga rwa I

et

a l, 1 They pr esent ed a n a na lysis for t he

pressure, P WD , in

t he w e ll bor e,

giving bot h t abula r

a nd gra phica l result s.

The ma in effect of t he

=,,=1 lh n re c? n rn gY is ~Q C~USC ~h c SU~Q UCe f~t ~ t o

------- .. -----

cha nge a s t he a nnulus unloa ds t o supply a const a nt

surfa ce ra t e, g. They a lso present ed solut ions for

dPwD

t he a nnulus unloa ding ra t e, C D

d tn

in t heir E qs.

26t hrough 31repr oduced here a s E &. 12t hrough 17.

PwD .~j? ~e-u2tD du)/

CD dt

D

o

({

[uCDJO()

- (1 - CDSU2)J l(u)] 2

+ [UCDYO(U)-

(I - CDSU2)Y1(U)12})

. . . . . . . . . . . . . . . . . .

(12)

E q. 12 ha s been obt a ined by convert ing J a eger s

qI i=~riczI SQ2TCC

.qo l u t i on 3 t o fluid flow nomen-

-.. .

cla ture. J a eRer a lso provided short - a nd long-t ime

approximate

forms.

S + o:

dpwD

=l -

CD dt

D

F& short t ime a nd skin ;ffect ,

tD

@2

+

CDs

15G CDS2

+ 0(ty)

. .

. . (13)

F or short t ime a nd a skin effect , s = O:

dpwD

[

D

t

=l-~+Q(~-

CD dt

3

D

CD n CD

CD

2

+ o(t:/2). . .(14)

For long t ime:

2

dpwD CD

+ CD

_=

CD dt

~(% - 1)

cD(1b:2cD)2;Oge;:;. ,] + o~O~D)

D

D

. . . . . . . . . . . . . .

(15)

SOCIETY OF PET ROLELIM ES GISEERS J O URXAL


3/10

As show n in t he ea rlier pa per,l E q . 15 ma y a lso

be w r it t en in t erms of t he P D (t D ) funct ion for tD

> 100:

dpwD

-CD(1 -

2Q(PDOD) + s)

CD;=

D

2t;

C(t -CD+S)

+DD

()

logetD

+0

2t;

~3 16)

D

C ompa r ison w it h J a eger s funct ions in t erms of

fluid flow nom encla ture, indica tes t ha t

dpwD

CD=

Ns>&5tD) . . . . . .

. (17)

JL

U L

D

u

t ha t pr ovid ed

E q . 12 in

t he form of a cylindrica l

source

solut ion. The line source solut ion for

dP wD

ma y a lso be obt a ined by different ia t ing

D d tD

t he line source

sOIUt iOtI

for pWD , w it h KSp Ct tO ~D.

m

PwD . ~

J(

2

fl

~ ~-U

tD

D

)

J o(U)

du

o

/(

[1 -

-i-

UpCDs+ g u2cDYo(u)f

[

)

~ U2CDJO(U)]2. . . . . .(18)

S oiut ion of E q . - C --

.~ ;- -{{-P + h.. b~e~

lL lU1 Zi?io

a.% r -,,=./. . ..-

pKSente~

b y C ooper

et

a l.4 a nd by va n P oollen a nd

Weber .5 B ot h references t a bula t e a funct ion H / H .

vs ~ for five va lues of a . Th e cor r es pon d en ce

bet w een symbolism of Refs. 4 a nd 5 a nd t h is st udy

is show n in Ta ble 1.

dPwD

Ta bles 2 t o 5 p resent 1- C D

d tD

or q= f/ q

a s

a funct ion of t D for a ra nge of va lues of C D a nd s

useful for w ell t est problems. This in forma tion w a s

obt a ined by numerica lly eva lua ting t he inversion

in teg ra l for t he cy lin dr ica l source s olu t ion p re se nt e d

TABLE lCORRESPONDENCE BETWEEN SYMBOLISM USED

BY OTHER INVESTIGATORS4.5

Cooper et 0/.4

van Pool Ion and Weber5

This Study

(~2/r~l S

[2 1 2 ) A-L

, / ( @

a

.

\ r ~ r~PJ Y C~

kh t

.

/i r:

P

tD

FD

~/ a = $

kt

r~

(@t :

H/H.

H/HO

CD S&a

df D

Note that symbals used in Ref. 1 are common to ground water

hydrology.

OCTOBEB, 1972

[

1

Pw D

TABLE 2

l-CD-

VS tD FOR S= 0, CYLINDRICAL

SOURCE WELL

1 10

1 0 2

103 lo~

m5

I

[

TABLE 3 1.

1

wD

CD %J KLt WtLL

VS fD FOR s=+lO, CYLINDRICAL

----- ...-. ,

w; 103 m ,o~

4,1 Q, QC57*C 0,000V7* O.oocme 0.000010 0.000001 0.000000

n.a 0.019237 0.901v.I ..eo Qt*. 0.00.0,9 0.000002 0...00..

c= o.rme?:e 0.0.., >s a.ooc47+ c.oc.... e c.00.w03 0.000 . . .

. . 1.0 0.094$1, 0.0 C**07 0.000S4S 0.0000.5 0.00000+ 0.00000,

a .Q c.tb. mt .. Qsa*z2 c.. c,.se o.cootah o.oo.onv ..0.0 ..2

B,9

9. X 1>.. O.o*. **a 0.00.s2, 0.000.53 0.000.45 0.0.000s

ls. s

_o, se,8,, ,.,,.706 O.,,, *23 o.m.act, 0.30 .0.. ..000 ,0.

s9.0

0,1,90s. 0.,:9 ..3 0.0,7,., ..40072. 0..00,7, 0.0000,7

. 31..

QJ97A9 +:. -V.13.W77 Q.o.o*~a o.oo.t~. 0.000.1. 0.0000.2

. Lfi O.4 .Gsswee .9. ?S1V12 0.07$153 O,ooe lob 0.000.15 0.0000.1

.. ZnR.o

.S.99T 1.1 V.?wwo

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. . . . JSW9. O

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A19w.w-..u%wu ..-*. *.* ..* 0...0?87 O.*. r*tl 0.061?s3

.._ .auno.n 0. C 9*9ST Q,wv+vs a.w*7.6 0.*9695* 0.70.29, 0., 1713s

XCCSQ,9 W59+9? blS9W0 0. S MV9 ..994927 0.43,092 2.2.06.6

_ti.J u.& a.a,sss *.5SW*, .O. ***9$0 ..$,,.82 ,....7,. . . . . . . . .

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maoatio L. Q O.Q09Q

9. S9?%.9. _Q,,?,,*O 0. V*V8,V 0.*** 921 0.92 S*68

lcluOnO.a 1.*09Q9 G

G,SS?92%

.0,,.,.,,, ,..$$ 9,0 0.s99 ., o.9@0 .06_

,>

>n9400ao.n 1.000000 I.aooeoo 0. S 99*97 0.99.,7s 0.*..7.5 o.9sbc.2a

_,& .hO&QCfQ - , ,Ic9000 0., $ $$, ,.,,O ,90 C. OW.9, 0..9.915. ,

-Q4 SlwS___ 9,.vYM.w&_ 3, V?*O. *.*****.

4 s5


4/10

by E q . 12. * Ta bles 2 t hrough 4 represent result s

forposit ive or zer o skin effect w hile Ta ble 5 show s

result s for nega t ive skin effect , -5. I t should be

point ed out t ha t result s for nega t ive skin effect

w ere found using E q . 12, s = O a nd t ; a nd C ;

va lues t ha t a re e2s t imes t he t D a nd C D t a bula t ed.

As a n exa mple for s = -5; t ;= e

-lot D a nd C~ =

e

-lOCDO

P ort ions of t he result s a re sh ow n on Figs. 1

t hrough 4. Fig. 1 presen t s a sem ilog gra ph of g~ j/q

a s a funct ion of dimensionless t ime,

t D , for C D

= 1,000 a nd skin effect , s, a s a pa ra m et er . Fig. 2

presen t s a simila r gra ph but for C D = 10,000. S uch

gra phs for ot h er va lues of C D ca n be prepa red

ut ilizing da t a presen t ed in Ta bles 2 t hrough 5.

S emilog gra ph s (Figs. I a nd 2) or log-log gra phs

(not sh ow n h ere) ca n be used for t y pe curve

ma t ching purposes. Fig. 3 presen t s

q~j / q

a s a

funct ion of t D for va rious C D a nd s va lues.

In spect ion of Fig. 3 revea ls t ha t considera t ion of

t he skin effect is ext rem ely im por t a nt . On e t hing

t his figure show s is t ha t skin effect does n ot a ffect

t h e genera l s-sha ped na ture of t h e curves.

Fina lly , F ig.

4

present s a not h er semilog gra ph

where

q~j/ q is

plot t ed a s a

funct ion of fD /C ~ for

s = O. S uch gra phs do t end t o br ing curves t oget her .

This form is simila r t o t ha t of Refs. 4 a nd 5.

TH E S TORAG E CONS TANT AS

FU NC TI ON OF P RE SS URE

The w ellbore storage problem a s or igina lly

visua lized by va n E verdingen a nd H urst 2 considered

a const a nt volume of fluid st ored in t he a nnulus per

unit of pressure cha n ge in t he sa ndfa ce pressure,

pw D .

Ra mey6 poin t ed out

t h a t t his effect ca n

result

either

from compression or liquid level

cha nge. B ot h effect s ca n t a ke pla ce in a single

wellbore.

Consider

a shut -in oil w ell. The pressure pi is

grea t enough t ha t ca singhea d pressure, pcs > pa t m

(a tmosph er ic pressure). I f t he w ell is produced a t a

const a nt surfa ce ra te q, fluid w ill unloa d from t he

a nnulus by expa n sion of liq uid. This ca n be

cha ra ct er ized by a st ora ge effect , C D ~ . B ut w h en

k

VS tD FOR s=+20, CYLINDRICAL

;OURCE WELL

10 ,

t. t

0.2

o.&

,.0

4..?

,.0

0.9

an ,9

. -s.0.

199. Q

-----

.,Qa

. .

av. . . ,s

?Oeo..

. ..-. W, I

Ccoo.

acoo-o. *

:?.0.0,.

JSQ

*OCOOO.O

..AU 1

0..0.0.,

0.00004

0.000>,7

.00.,

0 .0 00 . .7

.0 0? 0 6.

..00. Q?.

0.00.0,,

..0,.s.,

0. OM , ,.

e.orl. r

0 . , 71 .,7

0.110?9

0.,,.,,,

0..7V...

0 .? lw

0.*961

0.W7

0, 993*

0.996233

0.99KJ .2

0.590%?

0.?5929

O.m

0.%YW6

o.9 V?C.9

0,939952

0.5%976

o.m

0.599935

o. 9%sa

0.93 9?3

o . a>w

O.lam

0.8511%

0.%W2

0,93155>

O.+$iw

0.975679

0.385393

o,99z011

O.9@16

0.WF34O

o.99wlb

0.%9522

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0.9%901

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0.15W61

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[email protected]

0.59w6

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0.w9e7

o. 99?493

t h e ca singhea d pressut e pc~ = P a t m, t h e liquid

level w ill st a rt t o drop. This per iod

ca n

be

ch a ra ct er ized by st ora ge effect ,

recogn ize a new st ora ge problem.

c~~.

Thus w e

CD

()c~l~

D

D

[(

c

t)][

wnl yc~l) D

1

D2 - CD1 ~

L

.,=

L Cm J

JJc-

[

c

~ (S,CD2, ~

1

; PWD >

PWD1

2 dtD

. . . . . . . . . . . . . .

(21)

PwD

Thus t h e a nnulus unloa ding ra t e, C D , is a

d tD

com pon en t of t he solut ion of t he problem of cha nging

s tor a ge con st a nt .

Fig. 5 show s ~ w D vs t D for a n exa m ple

case

w here C D ~

= 1,000 up t o t D = 500 a nd P WD 1 =

0.4585; a nd C D ~ = 10,000 t h erea ft er . S kin effect is

a ssumed t o be negligible. Not e t ha t t here is show n

1.0

0.8

0.6

0.4

0.2

0.0

1

6

OIMENSIONIISSTIME, to

FI G . 2

C OMP UTE D [1 C D (dfJ wD /dt D )] VS fD FOR

VARI OU S S K I N E FFE C TS , C D = 10,000,

1. 0

0.8

(55

0.4

0.2

0.0

FIG. 3

C OMP U TE D [1 -

C D (d~ wD /dt D )] VS t D FOR

VARI OU S S TORAG E AND S K I N E FFE CTS .

OCTOBER, 1 97 2

a discont inuit y in slope a t t D = 500 due t o a n

instantaneous

ch a nge in

t h e va lue of st ora ge

const a nt from C D 1 t o .CD 2. Rea l beha vior w ould be

~ r , eo t~ , e~ .

~ ~ e ~ ur~ [jon

Of t he st ora ge effect is

cont rolled by t he second st ora ge const a nt , C D 2,

a nd t he beh a vior is a s if t h e st ora ge const a nt w er e

C D * t h rou gh ou t .

Th e dura tion of st ora ge effect , for

zero sk in ,

ca n be com put ed by t he follow in g

eq ua t ion . 15

t _>&)c . . . . . . . . . . . . .(23

J)- JJ

Now consider t he opposit e ca se w h ere C D 1 =

10,000 t o t D = 500 a nd P WD 1 = 0.04956 a nd C D 2 =

1,000 t herea ft er ; s = O. Fig. 6 presen t s t his ca se.

Not e t ha t t h er e is sh ow n

a

discont inuit y in slope a t

t D = 500, a nd dura tion of st ora ge is a ga in con t rolled

by C D 2 Inspect ion of F ig. 6 a lso revea ls t ha t t h e

va lue of st ora ge con st an t could ch a ng e fr om 10,000

t o 1,000 a t pw D 2 = 4.6773 a s show n. In t his even t ,

PwD would jum P ra pidly t o h e ra dia l lOw line CD

O. This could expla in t he peculia r beh a vior of t h e

un dersa t ura t ed oil buildup da t a of exa mple in

Appendix B , pa ge 134 of t h e

Pressure

Buildup

Monogra ph ,7 point ed out by Ra mey.8 F or such a

case

p W D2

should correspon d t o t he bubble-point

pressure for t h e oil in t h e a nnulus.

I t sh ould be a ppa rent t h a t norma l P VT beha vior

\

-__.

y

m

.m

K mm

~

i. 0

iO 10 2

O.ok _

.01

0.1

FIG. 4 C OMP U TE D (1 -

C D(d Pw D /d tD )] Vs tD /CD

FOR VARI OU S S TORAG E E FFE CTS , S = O.

10

1.0

0.1

,01

SKIN-O

CD

o

ID .500

I

1

102

103

104 1?

106

n AmlclOF ll KC TIMC 1..

Imt.

man

9.u.

aa r , ,ss.,

FI G . 5

COMP UTE D fJ w D VS t D FOR I NC RE AS I NG

S TORAG E E FFE C T, s = O.

4s 7


6/10

w ill ca use t he st ora ge const a nt , C D , t o be a

fun ct ion of pressure. S uch solut ions C D = /(pw D),

w ill n ot be discussed fur t h er in t his pa per.

AN I MP OR TANT D ST P ROB LE M

C ~ n~ ider t he D ST problem as follow s. F or ma t ion

poL

is shut in a t a pressure pi.

B U C

Pi


7/10

CD =

d .

F=

b=

.

H .

H O =

Jo =

1 . =

J1

k =

KO =

K1 =

f i w D

PD =

P a t m =

P

Cs =

pi =

p . =

q=

qa =

9s/ =

D =

r=

r c =

rd p =

rs =

r

w .

s=

s

t

T.

tD =

u=

Y. =

Y1 =

a=

B:

y .

p=

T=

dimension less w ellbore st ora ge. const a nt

(see E q . 8)

I ? u sed in Ref. 1

d iff er en t ia l op er a t or

a fun ct ion defined by J a eger3

forma tion t hickness, cm

corresponds to hea t t ra nsfer coefficient a s

used by J a egers (see E q . 23)

fluid hea d in w ell a t t ime

t (see

Ta ble 1)

ir t it ia i fiuid hea d in w eii (see Ta bie 1)

B essel funct ion of first kind, order zero

B essel funct ion of first kind, order on e

f or m a t ion p er m ea b il it y

modified B essel funct ion of second kind,

or der z er o

modified B essel funct ion of second kind,

or der on e

dim ensionless pressure drop w it hin t he

wellbore

dim en sion less pr essu re d rop on t he form at ion

.= ;J -$ QL; n eg{eet

=Iue . =-... -.. -s.

a t m os ph er ic pr es su re

ca sin g pr es su re

in it ia l for ma t ion pr es su re

pressure a bove t he D ST va lve

sur fa ce flow ra te

a nnulus un loa din g ra te

sa ndfa ce ra ce

d im en sion less r a diu s,

r / r w

r ad ia l d is ta n ce

ra dius of ca sing, cm (see Ta ble I)

ra dius of drill pipe, cm (see Ta ble 1)

ra dius of sa nd screen or open hole, cm (see

Ta ble 1)

w ellbore ra dius, cm

skin fa ct or , dimensionless (in t he ma in t ext )

va r ia ble of La pla ce

t ra nsform (in t h e

Appendices)

skin fa ct or , dimensionless (in t h e Appen-

dices)

corresponds t o C C th (in Ta ble 1)

t im e, sec

cor res pon ds t o t ra n sm issibilit y,

k b/ p

(in

~ a bie ~ )

dimension less t ime (see E q . 1)

variable

of

integration

B essel funct ion of second kin d, order zero

B essel funct ion of secon d kind, order on e

corresponds t o l/C D a s used by J aeger

3

(see E q. 23)

corresponds t o 1/2C D (in Ta ble 1)

cor respon ds t o tD/2cf)in Ta ble 1)

E uler s con st a nt , 0.5772

viscosit y, cp

dimensionless t ime (see E q . 23)

d =

pa r t ia l op er a t or

@ = porosit y , fra ct ion of bulk volume

SUBSCRIPTS

a=

a t m =

Cs =

D .

i=

r=

Sf

=

t=

w =

Wf .

refers

to

annulus

refers t o a t m os ph er ic con dit ion

r efer s t o ca sin gh ea d con dit ion

d im en sion less q ua n tit y

r efer s t o in it ia l r es er voir con dit ion

refers t o a ra dia l loca tion

refers t o condit ion s a t sa ndfa ce

refers t o t ime

refers t o con dit ions a t w ellbore ra dius

refer s t o fiow ing con dit ion a t w eiibore

1.

2.

3.

4.

5.

6.

7.

8.

radius

R E F E R E N C E S

Aga rw a l, R. G ., A1-H ussa iny, R. a n d Ra mey, H . J ., J r.:

An In vest iga tion of Wellbore S t ora ge a nd S kin E ffect

in U nst ea dy Liquid Flow : I . Ana ly t ica l Trea t ment , )$

Sot. Pet . En g. J . (S ep t. , 19 70) 2 79-290.

va n E ver din gen , A. F . a nd H ur st , .%: { Tfn e Applica tion

of t he La pla ce Tra nsforma tion t o Flow P roblems,~

T r a n s . AI ME (1949) Vol. 186, 305-324.

J a eger , J . C.:

C onduct ion of H ea t in a n I nfin it e

Region B oun ded I nt erna lly by C ircula r C ylinder of a

P e rf ect C on du ct or ,

$ Au sf. j, P jys.

(1956)VO1.~o.

2,167.

C ooper , H . H ., J r ., B redeh oeft , J . D . a nd P apa dopulos,

I . S .:

Response of Finit e - Dia met er Well t o a n

frrst a nt a n eous C ha rge of Wa t er , Wa t er Resources

Resea rch (1967) Vol. 3, No. 1, 263-269.

va n P oollen, H . K . a n d Webet , J . D .: D a t a Ana lysis

for H igh I nflux Wells, pa per SP E 3017 presen t ed a t

45t h An nua l Fa ll Meet ing, H oust on, Oct . 4-7, 1970.

Ra mey, H . J ., J r .:

,tNon-D arcY Flow a nd Wellbore

S tor age E ffect s in P r essur e B uildu p a nd D ra w dow n of

G as Wells, J .

Pet . T ech . (F eb ., 1965) 223-233.

Ma t t hew a , C . S . a nd Russeii, I l. G .: P ressure B uildup

and

Fl ow Test s i n Wel l s Mon ogr aph S er ies, S ociet y

of P et roleum E ngineers, D alla s (1967) Vol. 1, 134.

Ra mev, H . i., J r .: -.

1.1 *, ---- n --- T-+ a ..

>no~ -l~ e B Vell .C3SL L/aLiY . ..=.

pret a~ ion in - t h; P resen ce of S kin E ffect a nd Wellbore

S t ora ge, J , Pet . T ech . (J a n . , 1970) 97-104.

AP PE ND IX A

S OLU TI ON FOR C ONS TANT RATE WITH

S TORAG E AND S K I N E FF E C T U S ING AN

I BI TE G RATE D MATE RI AL B AL ANC E

ON WE LL B OR E

C AS E 1

C D is const a nt .

a2pD

-+ ~ ~ = ~ . . .. ..

PD(rD,) =O. . . . .

. . . . .

(A-1)

(A-2)

OCTOBER, 19 72

4 5 9


8/10

1 im

r+~

pD(rD, tD) =

o . . . . . . . (A-3)

D

+

D

pD

1( )

CDPWD-

dtD = tD . .

(A-4)

Z)~ D rD = l

o

Not e t ha t E q . A-4 is t he t ime int egra l of E q . 4 in

t he ma in t ext .

The La pla ce t ra nsfor m of E q. A-1 yields

;D

+ L%= .:

br~ rD &D D

- pD(rD,O). (A-6)

S ubst it ut ing E q. A-2 in E q. A-6, w e get

a2;D

+ ~ ~ -s~ = O. .. (A-7)

&~ r D &D D

The solut ion of E q . A-7 is:

FD

= AI O

D@

-I-

KO( rDfi). .

(A-8)

To sa tisfy E q. A-3, t he a rbit ra ry const a nt A must

be set eq ua l t o zero;

FD

=BKO(rD@) . . . . . . . . . (A-9)

(

1 . . (A-13)

B = + -

)(

)

D S P WD -

Th e La pla ce t ra nsform of E q. A-5 pr ovides

~wD=[=Dq] ---(A-14)

rD=l

S ubst it ut ing E qs. A-9 a nd A-10 in E q. A-14, w e

ge t

FWD=

[

KO(fi)i-S ~ Kl(fi)]oA-15)

(

1

5wD= -

S

S olving for j7w D, w e obt ain

Fw D = ~

s

. . . . . . . . . . . . . . .

(A-17)

w here K O a nd K I a re t h e modified B essel funct ions

of t he second kind, of zero a nd unit orders. The

rea l inversion int egra l solut ion t o E q . A-17 is:

.

dpD

w

-BfiK1(Dfi) . . . . . . .

(A-1O)

PWD = ~

1[

2

.

&D

1/

U tD du

(1-e )

YTo

I f w e a pply t h e L a pla ce t ra nsform t o E q . A-4, w e

ge t

u

~D

1

CD

S;wD

-

=. . . .

(A-12)

&D

s

r =1

D

To eva lua te t h e a rbit ra ry const a nt , w e subst it ut e

E q . A-10 in E q . A-12 a nd obt a in

*Note t h a t in t h e Appen dic es , S = sk in fa c t or an d s = va r ia b le

of Laplac e t r a n s fo rm .

(u3 [uC-J(u) - (1-CDSU2)J1(U)12

(--(~ -1)-~., .

J

[

+ UCDYO(u) -

(1-cDsu2)Yl(q2~) >

. . . . . . . . . . . . . . .

(A-18)

w h ere lo(u) a nd J I (u) a re t he B essel funct ions of

t h e first kind, of zer o a nd unit or der s, a nd y . (u)

a nd YI (u) a r e t he B essel funct ions of t he secon d

kind of t he r espect ive order s.

CAS E 2

C D is a va r ia ble. C D is C D 1 fr om ~ wD j t o P WD I,

t hen is C D 2. Not e t ha t only E q . A-4 of C a se 1 w ill

change.

4 6 0

SOCIETY OF PETROLEUM EXGIXEERS J O URSAL


9/10

. . . . . . . .

(A-26)

t,

f

dtD -- (A-19)

o

$Pw,

j~>~D =

,D ,,ii,,

o

D D-

where

o

wD1

CD

wD

For PWD > PWDI,

wD1

:w.

fD~dpwD + jcD2dpwD

wD1

Jjlpwl)l + CDPWD -

wD1

..

wD1 , 1

cl)) + w,

D

ap

-J( )

~ ~ rD= D-

. . .

(A-22)

,. --T.

. (A-L>)

D

D (*-24)

The La pla ce t ra nsform of E q. A-24 pr ovides

; .l(,l -

C D ) + ~ w D cD

&;D\

s

\< JrD=~ .. ~ ~ . A-25)

S ubst it ut ing E q. A-10 in E q . A-26, w e get

1

- DsfiwD pwl(CDl - CD)(A.27)

~=s

@Kl(@

S ubst it ut ing t he va lue of

B

in E q . A-15, w e get

[

1

FWD

1

- @FwD- PWDJ

Dl -

CD2)

s

[Ko @) + SFKJq

(A

2t3)

L tiKl(~) J -

F act or in g ~ WD a nd r ea rr an gin g,

.

1

PWn=

..-

S

wD1 ( CD 1 -

D)

. . . . . . . . . . . . . . .

(A-29)

E q. A-29 ca n be in ver t ed dir ect ly since bot h

pa rt s a re know n.

P WD ( S >D ~ C D ) = pw D ( cD 2Yt D ) +

[

1

wDl(ycD@,~

. . . . . . . . . . . . . . .

(A-30)

AP P E ND I X B

D RI L LS TE M TE ST ANALYS I S WH ERE

P ROD U C E D FLU I D D OE S NOT FI LL

DRILLSTRING

C onsider t ha t t he pi the tool is opened and

formation

OCTOBER, 1 9 7 2

4 6 1


10/10

fluid ent er s t h e drillst r ing. As fluid ent ers , t he

sa ndfa ce pr essur e,

pw j, increases d u e , t o he

hydra ulic hea d of pr oduced fluld. At som e t ime, t h e

hyriraidic

L--J .:11 .N...h ... h A .A p:~ ~ ~ ~ c~ ~fi

ncaa

WUL iappuau..

, yl, a..-

w ill cea se. J a eger3 ha s con sidered a n a na logous

h ea t con du ct ion pr oblem .

a2pD

apD aPD

+1

. . . . . . .

.(B-l)

~ rD arD atD

P D (r D , o)= o . . .. 2)-2)

lim

r ~mPD(rD, tD) =

.(B-3)

D

dpwD bpD

or

D=

. .

.(B-4)

dtD &

DD=l

wD= [PD-S : ]rD=, ~~B-

D

pwD(0)= - (B-6)

The La pla ce t ra nsform of E q. B -1 pr ovides

a2; D

+1

a; D .

s~

pD( rD , O) . (B -7)

&-~ r D &D D

S ubst it ut ing E q. B -2 a nd solving, w e get

E q . B -3 req uires t ha t t he a rbit ra ry const a nt A must

be z er o.

;D

= BKO(rD@ . . . . . . . . . .(B-9)

d;D

=

-B@Kl(rD@) . . . . . .(B-1o)

drD

The La pla ce t ra nsform of E q. B -4 yields

p5D)

- CDPWD(~) .

(R-II)

CDSPWD

[ a

~.

rD=l

D

S ubst it ut e E q . B -10 a nd E q . B -6 t o obt a in

CD~5~D

- 1] = -B~~K1(@ -(B-12)

-f r s; q

B=

D~ . w~

. .

(B-13)

@Kl(@

The La pla ce t ra nsfor m of E q . B -5 is

WD = [,D s(~ )l_l - (B -14)

D D -

S ubst it ut ing E q s. B -9 a nd B -10 int o E q.

;WD=

B [Ko(ia +

s@qF l

S ubst it ut e E q.

B-13

t o ob ta in

-CD

[S;m

- q

~wD .

[KO(@)

fiK@)

+

s@l(@] . . . .

S olving for ~ w ~ :

~ w D =

C D [K o( @ +

S{SK1(

w]

@l @ +Cf [KO +S@l @]

(B-17)

. . . . . . . . . . . . . . . .

The a nsw er to t h is problem is given by t he

in ver sion i nt eg ra l,

B -14,

. .

(B-15)

. .

(B -16)

/u (.UC ~ (U) \-. /.. \l

-2

(

[L

- ( MD~u2)d I(

U J

Do

\

+ [UCDYO(U)-

)

(1-CDSU2)Y1(U)12}

. . . . . . . . . . . . .

B-18

where p. is t he pr essur e a t a da t um inside t he D S T

a t t ime zer o,

Pi

- Pwf

dpwD

=CD ......

(B-19)

P i - o

dtD

S ee E q . 26 of Ref. 1. This solut ion ma y be obt a ined

fr om J a eger s fun ct ion

3

F(b, a ,d

or

F(S,

I /C D, t D).

The dimensionless pressure used in t he a bove

deriva tion is different fr om t he usua l P D. Tha t is,

Pi

- r,t

D=pi-pO -

(B -20)

+ + +

SOCIETY OF PETROLEUM EXCINEERS J OURNAL

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