on the determination of the pressure-dependent characteristics of a reservoir through transient...

7/25/2019 On the Determination of the Pressure-Dependent Characteristics of a Reservoir Through Transient Pressure Testing

1/12

.

SPE

SPE 19774

On the Determination of the Pressure-Dependent Characteristics

of a Reservoir Through Transient Pressure Testing

F. Samani ego and H, Ci nco-Ley,Pemex/ UNAM

SPE Members

z

CopyrightlSSe, society of PetroleumEngineers, Inc.

Thispefwr was preparedfor presentationatthe 64th AnnualTechnicaiConferenceand Exhibitionof the Societyof PetroleumEngineersheld in San Antonio,TX, OctoberS-1 1, 1SSS,

This paper was eelacwd forpresentationby an SPE Program Committee followingreview of infomrefii containedin an abstractaubmitfedby Ihe author(a).Contentsof the paper,

as presented, have notMen reviewed by the SocleIy of PetroleumEngineersand we aubjeot10correctionby the author(s).The material, aa presented, does nothecaeeerilyretfect

anypoail anofthe societyof PetroleumEngineara,itaoftkara, IX members. Pepere presentedat SPE meafingare sub@c to publicdrrmreviewby EdlforialCommltteeeof the Society

ofPetrc4aumEngineers,P missionfocapyis reetrictedtoanabafrectot nc4moretherr300worda. Itfustrationemsynof becopiad.The abarmcfehouidcontainconepkuouagck~

of where and by whom the paper la presented, Write Publkafiona Manager, SPE,P.O. Sox S3SSS6,Rlchardeorr,TX 750SS-3SSS.Tetax, 7S09S9 SPEDAL.

ABSTRACT

not al ways ri gi dand nondeformabl e.1 2 3

This

probl ~

i s ussual l yhandl edby means of properl ychosen

A practi cal procedurei s presentedto determ nethe

averageHpropert i es.Thi s methodonl y reducesthe

pressuredependentcharacteri sti csof a reservoi rf rom

errors i nvol vedand generall y does not total 1y el l m

transi entpressureanal si s. Thesecharacteri sti csare

i natetheseerrors.

xpressedi n terms of k p)fi l - ~p)}or i f porosit yi s

consi deredconstant i n terms of pert neabii ty. Ex-

A revi ewof the 1i teraturei ndi catesthat most of

pressi onsare deri vedf or f l owi n stress sensi ti ve

theef fort towardthe sol uti onof thi s pressurede-

formati onsof pressuredependent l i qui df l owand of

pendent f l owprobl emhas been focusedon the di rect

real gas f l ow that al l owthrai h the anal ys~sof

probl em(i . e. ,

predi ct i ngthe pressurebehavi orof .t he

drawdownor bui l dup tests, for i qui df l owand the reservoi rf romthe know edgeof perti nentr eservoi r

anal ysi sof theset ests and mul ti pl e- ratetests for

parameters).Cl assi cal treatmentssol vet he di f f usi v~

gas wel l s, the determnati onof the stresssensi ti ve

equati oo,whi chassumes that thedi f fusi vi tyi s a cons

characteri sti csof the produci ngformati on,The best

tant i ndependentof pressure.Whenboth pressureand

way to obtai nt he stresssensi t i vechardcteri sti caof

propertychangesare smal l , the constantproperty

the reservoi r i s to carry out a drawdowntest at a

assumpti oni s j usti f i ed.

If,

i nstead, rock and f l ui d

hi gh rate, that resul ts i n an i mport antpressure

propertychangesare i mportantover the pressurerang

decrement,al l cw ngtheesti mati onof the pressure

of i nterest, thesechangescannot be negl ectedand a

dependent f ormati onparameter i n a w de rangeof

vari abl epropertysoluti onshoul dbe obtai ned.

pressure. I n an al ternatemanner, the stress sensi ti ve

characteri sti cscan be esti matedthroughthe anal ysi s

Raghavanetal , , deri veda f l owequati onconsi deri

ofa bui l duptest precededby a producti onperi odat that rockand fl ui dproperti esvary w th pressure. Thi

a hi gh rate. Further i mprovementmay be attai nedwhen

equati onwhen expressedas a funct i onof a pseudopres

a combi nati onof both tests, a drawdownand a bu l dup

sure mp), resembl esthe di f fusi vi tyequati on, They

test i s used. I t i s al so demonstratedthat to ci rcum

studi edpressuredrawdowntests i n radi al bounded

vent the probl emof esti mati onof the pressurede- reservoi rsproducedat a constantmass rate. Subseque

pendent parameterat earl y ti mes duri ngthe test, due studi es5 6furt her i nvesti gatedtheappl i cati onof the

to the i nf l uenceof wel l borestorageand wel l bore

mp) methodto drawdown,bui l dup, i nj ecti onand fal l -

damage, the convol uti onmethodsof anal i ssthat

1

of f testi ng, Garg has al so studi edthi s probl em

make use of both the sandfacefl owrate an the bott om

Ostensenhas al so presentedsol uti onsfor the pressu

-hol epressurecan be used. The proposedmethod i s dependentf l owprobl em ngas reservoi rs.Pedrosaghas

testedthroughthe anal ysi sof simul atedtests, w th

presentedan anal yti cal sol uti onfor the pressure

errors i n esti matedresul ts of k(p) f l l - $(p)}or forma-

dependentf l owpropert yprobl em

ti onpermeabi l i tyof l ess than 5 percent. Al soa di s-

cussi onof an extensi onof theAgarwal s pseudo- ti me

Methods for sol vi ngthe i nverseprobl em(I . e. ,

concept for pressuredependent. i qui df l owi r a stress

i denti f yi nga pressuredependentreservoi rf romtest

sensi ti veformati oni s presented,

data, and eval uati ngreservo~rparameters)are much

l essnumerousthansol uti onsto the di rect probl em

I NTRODUCTI ON

In

order to properl ypredi ct the reservoi rbehavi or

i t i s i mport antt o i dent i f ythe pressured~pendent

I t has been l cmgrecogni zedthat porousmedi a are

characteri sti csfthe reservoi rearl y i n i ts l i fe.

Thi s pressuredependent i deti t i f i cati onas al so been

Referencesand i l l ust; ati ont end of paper

foundof i nteresti n theanal ysl s of wal l tests

9


2/12

2 OF A RESERVOI RTHROUGHTRA

.

ONTHE DETERMNATIONOF THE PRESSUREDEPENDENTCHARACTERISTI CS

1

[ENTPRESSURETESTI NG

SPE 1. .

i t has been foundconveni ent* to expressEq. 1 i f

termsof a normal i zedDseudorwessurem(D) defi nedbv~

fol l ow nga f racturesti mul tati oni o.The ef fect of the

pressuredependentcharacter i st i cson the behavi or

~f as reservoi rshas been di scussedby Vai rogset

~1y , and Vai rogsand Rhoades. 2These studi es

concl udedthat the pressuredependenteff ects become

more i mportantas the formati onpermeabi l i tyde-

creases.

In other words, permeabi l i tyreducti oni s

proport i onatel ygreater i n l ow- permeabi l i tyhan i n

hi gh- permeabi l i tyormati ons.

Recentl y,methodshave beepproposedfor the esti ma-

ti onof rel ati vepermeabi l i tyat i n- si tucondi ti ons

191s. ~ssenti al l ythe same

usi ngwel l test anal ysi s13 1

proceduremay be fol l owedto esti matet he pressure

dependentcharacteri st i csofa reservoi r through

transi entpressureanal ysi s.

The purposeof thi s paper i s to present a methodto

determ nethe pressuredependentcharacteri st i csof a

reservoi rthroughtr ansi entpressureanal ysi s, i n terr

of k(p) / (1- $(p) )or i f porosi tyi s consi deredconstant

i n terms of permeabi l i ty.Expressi onsare deri vedf or

f l owi n stresssensi ti veformati onsof pressurede-

pendent l i qui dfl owand of real gas fl ow that al l ow

throughtheanal ysi sof drawdownor bui l duptests, or

nul ti pl e- ratetests for gas wel l s, the determnati on

of the stress sensi ti vecharacteri sti csof thepro-

duci ngf orm i on.

BASICMETHOD

PressureDependentLiw d Fl ow

The basi c case for drawdowntesti ngi s that of con-

stant sand- f acemass f l owrate i n a radi al system I t

has been demonstratedthat the transi entwel l behavi or

for f l owi n a pressuredependent systemcan be expres

ed for al l %practi calpurposesby5*:

.

D(19tD) = *( l ntD+ 0. 80907)

(1)

where t and m are the def i ni ti onsfor the di mensi on-

RQesspsedopres ure, gi venby Eqs. 2 and 3:

Bk(pi ) t

D=

$(Pi )H(Pi)ct (Pi ) r;

(2)

I

. ,

h(pi )(l+i pi l

{mpi )- mp, r)}

D(rD?tD)=

(3)

a~qi~Pi )

and r and mp) are the di mensi onl essdi stanceand

8seudpressuredef i nedby Eqs. 4 and 5:

(4)

I

p

k(p)i(d

m p =

dp

Pm{1-$(P)}B(P)

(5)

In

Eqs. 2 and 3 a. and B are uni t conversi onconstant

def i nedi n Tabl e 1.

I q. 6:

{i - $(Pi )} P(Pi )

nl (p)=

mp)

(6)

k(pi )p(pi )

I hen,

)- 1. 1513U i u(pi )

l (pwf )=m(pl

0 k(pi )h(pt)

k(pi )t

l og(

)+l og(~)+0,86859s} (7)

4(Pi )H(Pi )ct(Pi ) rf i

Y

Aerey i s the exponenti al of the Eul er constant,

~qual to 1. 781.

The sl opemof the sem l ogstrai ght l i ne i s def i ~ed

as:

am pwf

qj~(pi)

n..

-l.1513ao

8

slog t

k pi h pi

Thi s sl opecanbe wri tt enas:

am

1

wwf

qi P(Pi )

*

apwf

, _= -1. 1513ao~(pi )h

alogt

9

Deri vi ngthe def i ni ti onofm(p) gi venby Eq. 6 we

obtai n:

-{l-

pi l ~ pi , k pwf p pwfm pwf

10

~Pwf

k(pi )p(pi )

{ l - 4(Pwf )}P(Pwf )

; .

Substi tuti ngEq. 10 i nto Eq. 9 we obtai n:

~, ,

k pwf

=-1.1513a

qi F(Pi )

Ll (Pwf )

**

:

i-4 pwf }

h pi [l-+ pi l P pwf

1

J wf )

(

11

alOg t

Eq, 11 i s the expressi onthat al l owsan, esti matl on

he pressuredependentparameterk(p ) f i l - $(p. - )}at

any f l ow ngti me, I t i s assumedi n tf l f sequati ~

the thi cknessh(p ) and porosi t $(P ) may beestl i m

f

d f romother sou~ces, l i kewel l og~i ng. I t has b~

demonstratedthat currentl yavai l abl etechni quespro

10


3/12

-.


4/12

ON THE DETERMNATIONOF THE PRESSUREDEpENDENTCHARACTERI STI CS

- .

4

OF A RESERVOI RTHROUGHTRANSI ENTPRESSURETESTI NG

SPE :

mere t and m are the defi ni ti onsfor the di mensi on-

l essti f l eanddi mensi onl esspseudopressuregi venby

; qs. 22 and 23:

Bk pi)t

D=

$P(Pi )ct (Pi ) r~

(22)

Tsch{mpi )- mr , t) }

nD(rDStD)=

(23)

agpscTqsc

Substi tuti ngthe di mensi onl essgroupsgi venby Eqs.

?2and 23 i ntoEq. 21, we get:

p~cT qsc

k pi t

n Pwf =m Pi -1.1513a

{ 109

g hT5c

OI J (Pi )ct(Pi )r~

~l og($)+0.86859(s+Dqsc)}

(24)

Thi s drawdownequati oni s sim l ar to the pressure

dependentl i qui df l owequati onal readydi scussed(Eq. 7)

Fol l ow ngthe same procedureprevi ousl yout l i nedf or

the l i qui dcase, the expressi onthat al l ows an esti -

mati onof the pressuredependent permeabi l i tyat any

t i mecan be wri ttenas:

Tq

Sc

J (Pwf )z(Pwf )

k pwf =-0.5?25ag _

25

Sc

apwf

alogt

The eval uati oni n thi s equati onof groupU(P )z(p )

nd of the sl ope ap / al ogt can be made fol l ow~fgth f

amemethoddi scuss~&for the pressuredependent l i qui d

l owprobl em

Fol l ow ngthe same procedureal readyoutl i nedf or the

rawdowncase, and the di scussi onregardi ngpressure

ui l duptest for the pressuredependent l i qui df l ow

robl em the expressi onthat al l owsan esti mati onof

he

pressuredependentpermeabi l i tyk(pws) at any ti me

an be wri tt en:

pscTqsc ,

(pws)=

0 5725a

P(PW5)Z(PW5)

g hTsc

(26)

apws

( ta +Ata)

a109(-&)

where t

i s the produci ngti me for real gas fl ow

(or f or pressuredependent 1i ui d f l owas di scuss-

d i n l ast secti on), expresse i n terms of pseudo-

t i me, as def i nedby Agarwall :

t

=

(27)

ap Bat

. Fi nal l y,At i s the shuti ~nti meexpressedas a

funct i ono~pseudo- t i me.

Of tengas wel l s produceunder vari abl erate con-

di t i ons. Consi deri ngDarcy f l ow Eq. 24 may be wrl

for thi s case as:

mPi ) -mPwf )

Q : (@% )l o9(t- tJ -1

qN

=1. 1513aghTscj =l qN

+1.1513a

g : C+S

Sc

(28)

where

k(pi )

46

C=l og + l og(

4P ( Pi )ct(Pi )r~ Y

In general , the responseof a speci f i crate step

N, ofa transi entw ti pl e rate test i s a strai ght

l i neon a graphof

~=1

qj-qj-~)109(t-tj-~)

pwf )versus:

Once agai n, fol l ow ngthe sameprocedureprevi ou

di scussed, the expressi onthat al l ows an esti mati o

of the pressuredependentpermeabi l i tyk(pwf ) at a

ti mecan be wri tt en:

pscT ,

l J (Pwf )z(Pwf )

k(pwf )=-0.5725a

g hTsc apwf

(

~)

a[~_l qj-qj-l lOg t tj ~ l

(29)

EXAMPLEOF APPLI CATI ON

Several si mul atedcases were run to test the val

i ty of the proposedmethod. The pressuredependent

data used i n themodel was that of Raghavanet al ,

becauseof several data sets6, i t shows thebi gges

pressuredependencyof the formati onpropert i esw

respect to pressure. Fi gs, 1 presents the f l ui d

densi tyand vi scosi tyas a functi onof pressure. F

2and 3 showthe vari ati onof permeabi l i tyand por

i ty w th respect to pressure, respecti vel y.

Tabl e 2showsdata of i nterestr egardi ngthe fou

di f f erentruns used to test the method, Eventough

vari ati onof porosi tyw th respect to pressurei s

very i mport ant i t was consi deredi n the simul ati on

thesecases.

Fi g. 4 shows a seml ogari thmcgraphof the pres

drawdowndata, Throughconventi onalanal ysi stech.

12

. .


5/12

*ci77A . ean,, *F*A ,, ... - ,,

f-n.-.. -. .,-- . . . .

ques,24 2s

i t canbe demonstratedthat the approx-

ate start of the seml ogstrai ght l i ne i s at about

hours. Fi g. 5 shows an enl argementof the pressure

ta af ter 10 hours whi ch i s neededto bett er def i ne

i nstantaneoussl opeap al ogt. Tabl e3 shows the

l cul at i onsthat are i nvo~vedin the esti mati onof

e pressuredependentparameterusi ngthe data of

n 3. The cont i nuouscurveof Fi g. 6 shows the i nput

ta to themodel ofk(p) / {1- $(p)}, and i ndi vi dual

bol srepresent the esti mati onof the pressure

nsi t i veparameter throughthe use of Eqs. 11 and 14

pressuredrawdownand pressurebui l duptests,

pecti vel y(Tabl e2). I t can be seen f romresul ts

thi s f i gure that themaxi mumerror i n the esti mati on

the pressuredependentparameter i s l ess than 5

rcent, I t i s expectedthat the accuracyof the esti -

es of the pressuredependentparametershown i n

bl e3 w l l i ncreaseas the conti nui tyof the data

dal so i ncreases.

ACTI CALASPECTS

We have seen i n the previ ousexampl eof appl i cat i on

at probl emsare present for theappl i cati onof the

oposedmethodat short ti mes due to i nf l uenceof

ects suchas wel l borestorageandwel l boredamage.

thi s respect i t i s of i mportanceto keep i nmnd,

poi ntedout by Serr aet al . l s, that drawdownand

i l dupresul tsare complementary.The drawdownanal y-

s yi el ds goodesti matesof the pressuredependent

rameter( k($)~l - $(p)}or k(p)) at l owval ues of

reand the bui l dupanal ysi syi el ds goodesti mate

the parameterat hi gh val ues of pressure. Conse-

entl y, by combi ni ngdrawdownand bui l duptest resul t

can obtai na gooddef i ni ti onof the pressure

pendentparameter. The best way to obtai nthe stress

nsi ti vecharacteri sti csof the reservoi r i s to carry

t a drawdowntest at a hi gh rate, that resul ts i n

i mportantpressuredecrement,al l ow ngthe esti ma-

onof the pressuredependent formati onparameter i n

w de rangeof pressure. Gi ?cethi s test i s concl uded,

i s recommendedto carry outs bui l duptest to

pl ementthe drawdownresul ts.

Anotherway to ci rcumventthi s probl emof esti mati on

the pressuredependent parameterat earl y t i mes

ri ngthe test, due to the i nfl uenceof wel l bore

orageand wel l boredamage, i s to appl y themethods

convol ut i onanal ysi sthat make use of the sand-

ratemeasurements26 27 280

Thesemethods requi rethat both the sandfacef l ow

rate q(t) and the downho~epressurep ( t) be meas-

red. The mai n advantagei s that the e~ti mati onof

he pressuredependentparametercanbe carr i edout

n a w der pressurerange, that i ncl udesthe hi gh-

ressuresthat prevai l sat short ti meaf ter the wel l

s openedto producti on,and that are under the

nf l uenceof wel l borestorageef fects, I t seems that

he sol uti onpresentedby Stewart , Wttmannand

euni er26, f urtherdi scussedby Strel tsova2gcoul d

e used for pressuredrawdowntest anal ysi s. The rate

(t) i s approxi matedby a seri es of l i near segments

nnecti ngthe nodes of the i nterval sas i l l ustrated

n Fi g. 7, Basedon the f i ndi ngsof theseauthors and

n the resul tsof Samani egoand Ci nco~Ow th regard

thevari abl erate pressuredependent l i qui df l ow

robl emthe fol l ow ngexpressi oncan be wri tten:

Am(Pwf )

B(Pi )

=

1.1513a

q t

0 k pi h pi fed t

2. 246f3k(pi )

+1Og(

)+0, 86859s}

30

$(Pi )V(Pi )ct(Pi )r ~

where

fcd(t )=l og(t - tn- l )-

J - ( l - : %

2.3026

N-1 qi (t)

t- ti ~

+z

l og ( - )

31

i=l q(t)

t - t i

q(t)=qi-l+~t-i - ~) i-l


6/12

ONTHE DETERMNATIONOF THE PRESSUREDEPENDENTCHARACTERI STI CS

5

OF A RESERVOI RTHROUGHTRANSI ENTPRESSURETESTI NG

SPE 19774

van Everdi ngenand Hurst ski n factor.

or two cases. The f i rst consi dersa short drawdown

test w th whol e sandfacef l owhi story pri or to cessati ol

of sandfacefl ow Thi s my be the case for a test

speci al l ydesi gnedto i nvesti gatethe pressuredependen-

cyof a produci ngf6rmati onor f or a dri l l stemtest.

Stewartet al . 2s al so consi der the caseof a wel l that

l asbeenproduci ngfor consi derabl eti meand compl ete

i nformati onon the f l owhi storypri or to shut- i ni s not

~vai l abl e.Theyassume that the wel l has been produci ng

steadi l ybefore i t i s shut- i nat the surf acefor a

w i i l dupest. I n thi s test the producti onl oggi ngdevi c[

i s set i n posi ti onduri ngt he constant rate peri odand

measurementsstart w th the steadyrate f l owand the

f l ow ngbottomhol epressure.

.

~

Fromthe resul tsof thi s study, i t can be concl uded

that a practi cal procedurehas been presentedto deter-

ni nethe pressuredependentcharacteri sti csof reservoi l

i n terms of k(p)f ll - +(p)}or i f porosi tyi s consi dered

: onstant i n terms of permeabi l i ty.Thi s procedurei s

essenti al l ybasedon themethods recentl yproposedby

11-Khal i f ahet al \ 3andSerr aet al l qf or theesti mati on

j f rel ati vepermeabi l i tyat i n- si tucondi ti onsusi ng

ti el ltest anal ysi s. Expressi onsare deri vedf or f l ow

i n stress sensi ti vef ormati onsof pressuredependent

I i qui dand,of real gas fl ow that al l owthroughthe

anal ysi sof drawdownand bui l duptests for l i qui df l ow

Endtheanal ysi sof these testsand mul ti pl e- ratetests

For gas wel l s, the determnati onof the stress sensi ti w

characteri sti csof the reservoi r.The proposedmethod

i s testedthroughthe anal ysi sofsimql ated tests, w th

wrors i n esti matedresul ts of k(p)f ll - $(p)}l ess than

5 percent. An extensi onof Agarwal s pseudo-ti meconcep

for pressure-dependent i qui df l owi n a stress sensi ti w

formati onhas been presented. I t i s al so demonstr ated

that earl y ti mepressuredependentparameteresti mati on

canbe carri edout throught he convol uti onmethodof

anal ysi s.

ACKNOWLEDGMENTS

.

The authors thankM J . Economes for encouragi ngthel

to conti nuethei r researchai med to f i nda sol uti onto

the esti mati onof the pressuredependentcharacteri sti c

of a reservoi r.

NOMENCLATURE

Ct =

D

=

k

=

mp) =

~D .

P=

Pwf =

P

Ws =

q=

r=

w =

systemtotal compressi bi l i ty

turbul enttermcoef f i ci ent

permeabi l i ty

pseudopressurefor press~redependent l i qui d

fl ow Eq, 5, or real gas potenti al or pseudo-

pressurefor real gas fl ow Eq. 19,

dimensi onl esspseudopressurefor pressure

dependent l i qui df l ow Eq. 3, di mensi onl ess

pseudopressurefor real gas f l ow Eq. 23,

pressure

f l ow ngbottomhol epressure

shut- i nbottcmhol epressure

f l owrate

radi us

wel l bordradi us

s=

t

=

t; (P) =

t

ap =

D =

Ata =

T=

z

=

P. =

$=

0=

ti me

pseudo-ti mefor pressuredependent l i qui d

f l ow Eq. 17, or pseudo- ti mefor real gas

fl ow al so gi venby Eq. , 17.

produci ngti me expressedi n terms of pseudo-

ti me, Eq. 27.

di mensi onl essti me, Eq. 2 for pressure

dependent l i qui df l owand Eq, 22 for real

gas fl ow

shut- i nt i me expressedi n terms of pseudo-

ti me

temperature

real gas devi ati onfactor

vi scosi ty

porosi ty

densi ty

Subscri pts

c.

= convol uti on

d = drawdown

D

= di mensi onl ess

f

= f l ow ng

9

= gas

i

= i ni t i al

j

= i ndex

N

= l ast rate i nterval i n a mul ti pl e- ratefl ow

test

o = l i qui d(oi l )

Sc

= standardcondi t i ons

w

=wel l bore

REFERENCES

1.

2.

3.

4.

5.

6.

7,

Mei . nzer,O. E. : Compressi bi l i tynd El asti ci tyof

Artesi anAqui fers,Econ. Geol . , 23(1928)263- 271

J acob, C. E. : Onthe Fl owof Water i n an El asti c

Artesi anAqui fer, Trans. Amer. Geophys, Uni on

(1940)574-586.

J acob, C. E. : Engi neeri ngydraul i cs, (H. Rouse,

Ed. ), J ohnWl ey and Sons Co. , I nc. , NewYork

(1950)321-386.

Raghavan, R. , Scorer, D. T. , andM l l er, F, G, : An

I nvesti gati onby Numeri calMethodsof the Ef f ect

of PressureDependentRockand Fl ui dProperti es

on Wel l Fl owTests, Sot. Pet. Eng. J . (J une,

1972)267- 276.

SamaniegoV. F. :

An I nvesti gati onof Transi ent

Fl owof Reservoi r Fl ui dsConsi deri ngPressure

DependentRockand Flui dPropert i es,PhD

di ssertati on,StanfordUni versi ty,StanfordCa.

(1974)

Samani egoV. F. , Bri ghai n,WE. , and M l l er, F. G. :

An I nvesti gati onof Transi ent Fl owof Reservoi r

Fl ui dsConsi deri ngPressureDependentRockand

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140-150; Trans. , AI ME,163.

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~s; gvoi rs, Adv. Water Resources(March, 1985)

- *


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Permeabi l i tyonGas ProductI onand Wl l Testi ng,

SPE Formati onEvaluati onJ . ( J une, 1986)227- 235.

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Stress-Sensi ti veFormati ons,paper SPE 15115,

presentedat the 56thCal i forni aRegi onalMeeti ng

of SPE, Oakl and, CA. , Apri l 2- 4, 1986.

0. Economes, M J . , personal communi cati on,Dowel l -

Schl umberger,Houston, Tex.(1985)

1. Vairogs,J. ,

HearnTC. L. , Darei ng, D.W, and

Rhoades, V.W:Ef f ectof Rock Stresson Gas

Producti onFromLow-Permeabi l i tyReservoi rs,

J . Pet. Tech(Sept.1971)1161-1167.

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Tests i n Formati onsHavingStress-Sensi t i ve

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Pl aceDetermnati onof Reservoi rRel ati vePermea-

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Test Anal ysi s for Sol ut i on- Gas-Dri veReservoi rs;

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5. Serra, K. V. ,

Peres, A. A.M , and Reynol ds,A. C. :

Wel l Test Anal ysi s for Sol ut i on- Gas-Dri ve

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D. F. , (19BB)

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Real Gas Pseudo-Time-ANewFunct i on

for PressureBui l dupAnal ysi sof M i FGas Wel l s,

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( J anuary1985)143-152.


8/12

TABLE 1.

S1 PREFERREDUNI TS, CUSTOMRY UNI TS, ANDUNI T CONVERSION

CONSTANTSUSED I NTHESESYSTEMS.

Parameter

or vari abl e

S1 PreferredUni ts CustomaryUni ts

k

h

q

P

B

Ct

P

t

; 9

W*

m

i ns/D

Pas

m3/ m

fr acti on

Pa-l

KPa

hours

4.5751X1O 4

3. 6x10- 9

md

f t

STB/ DorMscf / D

Cp

RB/STB

f racti on

psi - l

psi

hours

5. 03332x10q

2.637x10-4

TABLE 2. PARAMETERSOF I NTERESTREGARDI NGTHE SIMULATI ONSOF EXANPLE1

Run

re/rw

Type of

Skin

Uel l bore

Di mensi onl ess

Test

Factor, s

Storage, CD

Rate, q.

1

103

Drawdown

5

10

0.011

2

~~3

Bui l dup

5 0

0.011

3

5X103

Drawdown

10

0 0.047

4

5X103

Drawdown

o

0 0. 100

TABLE3. ESTIMATI ONOF THE PRESSUREPE?ENDENTPARAMETER

k p /{ 1-O p ]

FORPRESSUREDATAOF RUN3.

apwf ,

Pwf

(Pwf )m

hJ J m

(Pw)m

q

(ho~rs) (psi a)

(psi a)

(Cp)

(l b/ ft3)

(psi a/ cycl e)

k(p)

m

(red)

47. 67

1082. 92

1070. 62

/ 1. 50

5.433

1058. 32

50. 267

138.76

97, 2

119.17

1027. 50

1022. 01 5. 402 50. 256

14300

1016. 52

138.69

97. 1

715.02

920.01

911.12

5.360

50. 230

142 . 39

93. 8

953,36

902.22

16

$


9/12

S?E 19774

*

51.00

50.50

500 lax)

154M

2mo

2X

2

0

i=

u

is

L

-9-

v

.

1

160

; 40

120

100 L

80

/

60

500 low 1

p,

pslo

P

PWCJ

/

2000

25

0.290

?

o.zo~

.

2ml

p , psia

.


10/12

s?E S5774

.

200

1Scn

10CM

o

0

on

o

0

0

0

0

0

o

o

0

o

o

Fi g,4-Semllog grsph of preecura dr~wdown data, Run 3 (Tablo 3).

18


11/12

r-

.

x

0

c

z

Pwf

Ei

,

psia

.

Q

o

I o

I

o

4

e

o

t

o


12/12

q(t)

2

z

.

1-

RUN TYf%TEST

/

A 4

DRAWDOWN ~

/

p , Psi o

- - - - ACTUAL FLW RATE

LINEARFLOWRATE APPROXlhlATION

b

qi.]

--------j;--w ~

II

~

/

qf)

I

o 1

t]

tl.1

t[ tN.]

tN

on the determination of the pressure-dependent characteristics of a reservoir through transient...

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