spe 021513 (mccray) decl curve an for var pressure drop var flowrate

8/11/2019 SPE 021513 (MCcray) Decl Curve an for Var Pressure Drop Var Flowrate

1/17

cop~hhl t 99\, S&et y of Pe rdeum Engi neers, Inc.

Th is f mp er WA 8 pr ep et i Ps vp ms en lat im a? d m SPE Qat Ted mb gy Sy mp os iu m h el d i n H ws to n, Tex as , J an uar y 23-24,1691.

TM p ap rx w as $el em ed f or p res en lal ic m b y m SPE Pr og ram Com mi tr to e f ol lo wi ng m ti ew o f i nl or met km c on lei ned In an Mr nc 4 w bm ir ted b y t he au th or (s ), C%n mn ,s o f Um p ap er , ac

p mw nt ed , h am n ot b ean r ev kw ed

gmesoc*vol Pl rk e o urn I%qi neem t nd are sub ject 10 C&mot ion by t ie eul hor(t ). The mat eri el ,

: p resen ted , does not necosaan ly r el lec r any

p os lr im _I of rh o S od al y o f Pet deu m n gl neem , 11so lf io er c, m m em ber s, Pap er a p res en ted at SPE m eet in gs ar e w bj ac t t o p ub li do n r av bw b y Ed it or ki l c om mi tt ees o f t he So dar y o f

Perml wm &@neers, Perml mbn t o copy i s res tdcl ed 10 a n ab$t ract o not m ore t hnn 303 words . Il [ut l ari ons may no be copi ed. The abs tract shou ld c onl ai n .%ns pl cw ous

~~t~ *m @ @ f im me paper i s Pmmmt ad. wri te puMi c.at as hkwer, SPE, P.O Sex S3S9SS. Ri durd$on, TX T5C8S-21S6, Tel ex, 730369 SPEOAL

The motivationfortheworkdescribedin this paperarosefrom

formed intoan cquivslcmconstantrateca.wfor bothgas and liquid

a need to analyze production dcclinc data where the flowing

flow data. Camacho9indcpndcntly vcnficd that this equivalent

bottomhole pressure varies significantly, The varirtncc of the

constant rate formulation is exact for the constant pressure

bottomhole [email protected] theexponential

J

ecline m c1for conventional dcclinc curve analysis (scmilog

production of a sli htly comprtssiblc liquid during boundary

ominri:ti flowcon mons,

plots arid type curves). Using pressure nonmalizcd flow rate

rather than flow rate usuallydoes not remedy this problem. The

McCrayto sought to develop a method to transformvariablc-

rncthmi wc present uscs a rigorous superposition function to

rate/vanablc pressure drop data into an equivalent constant

account for the varianceof rstc and pressure during production.

pressure

case. In doing this, McC raydcvclopcd a recursion

This furwdon is the constant rate analog for vanablc.rate flow

formula to compute an cquivdcnt time for constant wellbore

during post-tmnsicnt conditions and can bc used to develop a

pressure production, tcp,that could bc used with pressure drop

normalizedflowrate to performdcclincCUTVCnalysisusing type

WC b

61%J1U

M@wtJ@xan EnLmms

Decline Curve Analysis for Variable Pressu, J Drop/Variable Flowrate Systems

by T.A. Blaslngame, T,L. McCray, and W,J. Lee, Texas A&M U.

This is a preprint -- subject to correction.


2/17

constant pressure analog for the dcclinc curve analysis of field

2

DeclineCurveAnalysisfor VariablePressureDrop/VariableFlowrateSystcn?s

SpE 21513

Thecomputationalformulaearc givenin AppcndiccsBandC

and wc will verify each. Thcsc include the fcilowing rccuraion

formulae; the integralmethodproposedby McC ray10nd the 2-

and 3-pointbrickwarddtifcmncomethodsdcvclopcdin thiswork

The recursion relations for this part of the verification arc

dcvclopcdinAppendixB andsummarizedinAppendixC,

We willalsouse theboundarydominatedflowrelationswhich

result from

7

tsating the constant rate and constant pressure

anal tics SOIUor..,This dcvclopmcntand thepertinentrelations

ior t is psxtof theverificationamgiveninAppendixB.

Fi . I

shows the log-log

behaviorof the

qo

functionsvcraus

t

Dan tcp~function for tic cssc of a wellccntcrcd in abounded

cimulaxreservoir(r~=NP), Duc to thenumberof mcthcdsbeing

cansidcm wewill discussthe transientand boundarydominated

flow behavior separately, First wc note that, during early times

(transient flow), all of the tcpp methods yield a good

approximation to tic

q~(t~)

sohmon, cxccpt at very early times

(t@20). This

impliesthat allof thesemethodsyielda reasonable

approximation to the analytical solution during transient flow,

Obviously,theanalyticalsolutionfor tmundarydominatedflow is

not valid during transient flow as shown by the deviation of this

solutionand the transientflow sohstion.

Now if wc consider the late time (boundarydominated flow)

portion of Fi&.1 (rD>3x105),wc find that virtuallyall of the

to D

mcthds agrc.cwry closdy

with the

qD(lD)

Solution.Athough

&ls

scale

prccludcsveryCIOSCnspection,it dots appearthat t D2 for

thedcnvativcmcth(xl1dots showsigni lcantdeviation,

f%iswill

bc invcstigsttcdmore closely when these dara arc rcplottcd on a

scmilog

q~

gmphin Fig. 2.

Fig. 2 is

a

rcplot of Fig. 1 using a scmilogscale for the

qD

fLSnctiOnSnd a cancsian scsdcfor the tDfunctions. Wc nOtChat

the

scsulta

fordcnvativc method1 do beginto diverge from those

r f the Othermcthtsds,which cIwly ovcr]ay thecorrect solution,

Fig, 2 sug~sta that dtivativc rrscthod1 should not be used in

L

racticc,but that theintegraltncthw dcnvativcmcthcx 2, andthe

undarydominatd flowmethodshouldgiveaccuratewsults.

Of these, the boundarydominatti flowmethod is the easiest

to apply since it dots not rcqtlirc recursion calculations, but

functions. Wc will demonstrate the ap~licationof the boundary

dominatedflowmcthcxion a simulatedhquidproductioIIaqucncc

anda fic dcaacfor a gasWCIIhathasbeenanalyzedpreviouslyin

the literature.ZM

In this sectionwcwill applythe tcP-tcrransformdescribediri

theprevioussectionto a simulatedproductionsqucrtcc in anOil

WCI1.Themscrvoirdataand flowhistoryarcgivenin Table 1.

TABLE 1

WellimdReservdr Parameters

(Well Centered in alloundedCircular Resemoir)

B,

RB/STB

1,00

cl,psia-l

15.0X 1o-I5

h,

ft

4

0.::

\i, Cp

rW,

ft

02:

re, ft

745 (40

acre)

k,

md

1,0

s

pi, psia 48~

CA

31.62

Usingthe WC] ndrcscswoirparametersgivenaboveandEqs.A-3

toA-5,wccomputethem

and b

paramctwsandwcobtain

m=

2,3&/09x10-z@S~/D/D

b

=

32.8948

pSi&~@

Flom History

flowscqucncc

;

r, days q,STBfD

180

50

180

const~t

p~

p~,

psia

constantrate

2000


3/17

SPE21513

T.A.B usingamc,T.L. McCrayandW,J,La

3

I

correctconstant pressure solution. It can be seen in Fig. 4 that

prcs: xc drop normalization does not yickl a constant nrcssure

I analogsolution,

Clearly, wc must usc other techniques which are more

rigorousthan psessumdrop normalizationfor field applica{{,onsf

decline curve analysis,

The methodof choice will be the onc

proposed by Blasmgamc and Lec7 whit: converts variablc-

ratchriablc rcssure tip data to the equiwdcm constant rare

case. From t is anal sis wc will obtain them andb parameters

quired by Eq. B-1 or transformationto an equivalent constant

pressure system,

Fig, 5 showsthe cartesianplot of alp/q vs. tcr(=Q/q) rquircd

to determine them and b parameters. m is the slope of this plot

and

b

is the intcrcc t. Although thcte is somedata scatter, it is

1kar that them an

b

psramctcrado rcprmcnt a best fit trend of

the dam. Therefore, the step of determining the m and b

parametersis illustrtttcdas a simple and straightforwardprocess.

Fig. 6 showsthe log-logplot of Ap/q vetmssrc,that could be used

for typecurve matchingon constantrate typecurves. Fig. 6 also

shows that the concept of using Aplq and fcr appears to also be

validfor transientflow, giventhe agreementbetweenthe constant

rate and constant pressure base case (pW~3000 psia) during

transientflow (rcrc50days).

The nextstep is tousc them and b parameters in Eq. B-1 to

convert from tcr (constant rate analog time) to

tcp

(constant

pressure analog time), This is also a simpleand straightforward

procedure. Once tc is computed,a log-log plot of q/@ vs. Gp is

iade. Fig.7 is suc a plot andwcimmcxiiatclynote that all cases

overlay the same trend during both transient and boundary

dominatedflow. obviously, the analyticalsolution for boundary

dominatedflow(exponentialdecline)will not agreewith transient

flowsolution.

Fig, 7 represents the endpoint of our effort to determine art

quivalent anstant pressuretransformationfor variablc-rate/vari-

~blcpressuredrop flowdata, Wc aresatisfiedthatthis is a logical

and consistentprocedurethat shouldyield accurate rcsuhs when

applied to field data, The verification of this method is that all

cases overlay the base case @~-30fXt ]sia), where ~IAPand t

were used as the plotting functions for I$Cbase case. At this

TABLE 2

Well~Reservoir par~ters

(Assumed Geometry: Well Centered in

a Bounded Circular Reservoir)

B, R13/MSCF

0.70942

et, psia-l

1.870

X 104

h, ft

0.;:

Cp

0,02167

Fw,ft

0,354

-5,30 *

~, md

0,0786S *

p.~ psia 710

pi, psia

4175

C*

31,62

G, Bscf ( ef,2)

3.360

G, Bscf (rcf,3)

3.035

*Averageof valuesobtainedfromref. 2 and3.

Fromthe resultsof ref. 8 wchave

ma =

2,O5536X1O-3s@lSCFP/D

b. =

1.3094psi/MscF/D

G= 2.6281 Bscf

Fig. 8, which is a log-log plot of

@a/q

versus tc~,a,is taken

directly fromrcf, 8 and shownhere for complctcmess.Wc htwc

includedthe computedresponseduting boundarydominatedflow

as prescribedby Eq. A-1, T hc A pa /4 and tcr,a variables arc the

pseudotimc and pseudopressureas defined and computed in ref.

8, This nomenclature may seem awkward, but defining these

variablesin thismannerallowsus to usc liquidflowquations for

analysis, Therefore, anyequationswcpreaemarevalid for either

liquid or gas flow as long as the correct time and pressure

variablesarc used.

Wc note that the boundarydominated flow sohstiondoss not

model the transient flowbehaviorof the data in Fig. 8. This is

cxpectcdandwc onlystatethisobservationforcompletcncss,Wc


4/17

4

DcclincCur-wAnalysisfor VariablePressureDrop/VaririblcFlowrateSystetm

SPE21513

And finally,for gasweil test analysisEq. B-1hxomes

(5),

We haveusedEq. 5 toeomjwtcthe tc~ functionsusedin Fig.

f

. Note that Fig,9 is a log-log lotot

q AP~

versus tcp,~and that

h

he the boundary dominated ow sOIWiOn(Computccfq14Pa

function) agrees very wcli with the data during boundary

dominated flow but not during transient flow. This is expected

andwe shouldnot be conccrncdaboutthisdifference.

OncewehavecreatedFig, 9 using the

q/Apa

versus rCPadata,

wc will want omate} this data upon the Fetkovichl typecurve,

Fig. 10reprc%ms this typecurvematch. Note that thedata agree

with the t

r

curve during the transition from transient to

boundary ominated flow and throu hou; boundary dominated

h

low , The sc am it y

of data fortc ~


5/17

SPE21513

T,A. Blnsirsgime,T,L. McCrayandW.J.Lee

5

The fourth method developed was a rigorous identity which

equates the kxtndary dominated solutions for constant rate and

constant pressure production. The resulting two-parameter

relation @q. B-1) may bc used for dimensionless solutions or

field data applications. When the m and b parameters are

determined using the methods developed in ref, 7, data scatter

shouldhaveIittlceffecton theVSIUCSf theparametersbecausea

best fit trend is established. These characteristics make the

boundarydominated flow method the most usefulproduct of this

work.

Applications:

Wc recommendusingthe methodsprcscntcdin this work thr

the type curve analysis of variable-rate/variable pressure drop

productiondata, The methcd is relatively simpleand should be

applicable to a wide range of WCI1est problems, including the

analysisof gas welltestdatademonstratedin thiswork.

Conclusions:

1,

2.

3.

The recursion fonnulae discussed in this work should not be

appliedin practicedue to problemsassociatedwith the erratic

natureof fielddata,whichcouldcausepoor results.

The boundarydominatedflowmethodis themethodof ckaicc

to transform the constant rate analog time function ir,to a

constant pressure analog time function, This

mcthid

is

consistent,easy to apply,and shouldgive accuratercsuitsfor

a widerangeof problcmtypes.

The boundary dominated transform method can be used co

model constit wellborepressureproductionbehaviorcxaciy

during boundary dominated flow and should give accurate

resultsduring transientflow.

NO a

Dimensionless Variables

bD

= dimensionlessconstantdefinedby Eq,B-4

CA = dirrmsionless shapefactor

= dimensionlessconstantdefinedbyFA.B-3

Cp =

constantpressureor constantpressureanalog

D=

dirncnsionlcssvariable

mp = matchpointona typecurve

Wc gratefully acknowledge the assistance of Elizatwth

BarbozattndJemniferJohnstonfor their hcIpin thepreparationof

dds manuscript.

1.

2.

3.

4.

5.

6.

7.

Fetkovich, M.J.: Dcclinc Curve Analysis Using Type

Curves,JPT (June 1980) 1065-77,

Fetkovich,M.J., et ULDecline-CurveAnalysisUsingType

Curves.-CaseHistories,SPEFE (Dec. 1987) 637-56.

Frairn, M.L. and Wattcnbmgcr, R,A.: Gas Reservoir

Decline-CusvcAnalysisUsingTypeCurvesWithRealGas

PseudopressureandNormalizedTime,WEFE (&c, 1987)

671-82,

Ehlig-Economides, C.A. and Ramcy,H,J., Jr.: Transient

Rate Dcclinc Analysis for Wells Produced at Constant

prCSSUR,

:PEJ

(Fcbo1981)98-104,

Ehlig-Economides, C.A. and Ramey, H,J,, Jr,: pressure

Buildup for Wells produced at a Constantpressure,PEJ

(FcIJ. 1981)105-114.

Blasingamc, T,A. and Lee, W. J.: Properties of

HomogeneousRcscrvoi.m,NaturallyFracturedReservoirs,

andHydraulicallyFracturedReservoirs fromDeclineCurve

Analysis, paper SPE 15018 presented at the 1986 SPE

PcrrnianBasinOil andGas RecoveryConfcrencc,Midland,

TX, March 13-14, 1986.

INasingamc,T.A. and Lee,W.J.: Variable-RateReservoir

Limits Testing, paper SPE 1.5028presented at the 1986

SW? Permian Basin Oil and Gas Recovery Conference,


6/17

6

DeclineCurveAnalysisfor VariablepressureI)rop/VaricblcF mvratesystems

SPE21513

4(0

=mtcr+b

N)

where

(t) (time in Days)

--1- q(t)dr =

Cr f?(r) *

q(t)

(A-1)

(A-2)

m=5.6l5--@---

@hcfi

(A-3)

()

=70.6 n~

eTCAr~2

(A-4)

and

,

rw = rw e-s

(A-5)

We will also need the general solutionfor a well producing at a

constantpressureduringboundarydominatedflow. Thk solution

is given by Ehlig-Economides and Ramey~,Sand later by

BlasingameandLec.c This so utionis

&)cprow)

(A-6)

Ourobjecciveis to develop a general time function that allowsus

to use Eq. A-6 to model a variable-rate/vanable-pressur~ drop

process,

Thisgeneralrelationis

(A-7)

where tcp is the

time

at which the constant pressure solution is

valid for a general variable-rate/variable-pressuredrop response.

In this sense,ICPs an unknownwhichmustlx dctcrmincri.

McCray10proposed [he following rcl.~tionas a defining refwion

fortcp

tp

We

need

to prove theright-h~.,ld-side(RHS) of ~., A-8. This is

done by integratingEq. A-7 so it is the sameformas the RHSof

W,.A-8. This gives

or

CombiningEqns.A-7andA-11gives

(A-11)

(A-12)

Notice that the right-hand-sides of Eqs. A-10 and A-12 arc

identical, This result proves that Eq. A-8 is exact for boundary

dominatedflow,

AppendixB: J% @Jl

rv

130ml~

Theobjectiveof thisswion is todevelopa methodtocomputethe

rcpfunction. A relatively simplerelation is obtainedby quating

Eqs.A-1and A-7 and solvingfortcp.Thisgives

~ Exp(~@), *

or

%/2=$ql

+f pcr )

(B-1)

or in termsof dimcnsimlessvariables

(

mq. .d

c/)D=mD

)

(R-2)

where


7/17

SPE21513

T,A.31asingamc,T,L,McCrayandW,J, lee

7

I

And combiningEqns. B-8,B-9, and B-11, and solving for AtCP,i

Integral hiethod

gives

McCmylo proposed to computethetc functionby approximation

i

[

.~ @(ti) Q(ti.$ 4Q(ti-1)+~

of the integral in Eq. A-8 using t c trapezoidal rule, This

repl %(ti) Ap(ri.2) @(ti-1) AP(ti)

1

(B-13)

essentiallyresults in a recursionformulawherethe tcpfunciion

is

computeda:

Cp= f

At.p,i

Appendix C:

~~

i= 1

The AtCP,iermsare computedusingindividualtrapumidpanelsof

In the calculations, the constant pressure dimensionless rate

*C q(1)

solutionis definedas

function. For an individualtrapezoidwchave

APO)

qcpD = ~

(c-1)

[

Theequivalentdimensionlesstimeis

Ii .% ?L * + ~ti-l)

2 @(ti) &(ti. 1)

1

tcpD = i~l AtcpD,i

(c-2)

also

where theZMCpDjregiven for theintegralmethodas

[

/i - Q(tJ Q(ti.])

.. .

AP(ri) Wi-1)

1

2[%%+1

AtcPD,i~ ~ ~-

for a givenpane .

[

PD,i FD,i-l

1

(c-3)

Combiningand solvingfor AtcP,igives andfol dxivative method1 astD,i pfi,i

AtcPD,i = tD,i -

[

J

@2iL. aLIL

PD,i-1

(c-4)

t~p,i =

~(ti) A~ti-

and forderivativemethod2 as

[

&,. 4U.L

1

[

,- pll,i tD,i-2 . 4tD,i- 1&

AlcpD,I

@(ti) AP(~i.] )

2 PD,i-2 PD,i-1 PE,;

(B-7)

1

(c-5)

Derivative

Metlwds

Othermethods,which are basedon thedcnvativc of Eq.A.7, can

bedevelopedtocompute the tcpfunctionusingEq.B-6.Differen-

tiationof Eq.A-7 withrespectto rCPyields

q(t)

()

- d Q(O

dtcp Ap(i)

I

(B-8)

We can alsou~eihcImund.uydominatedflowmethodto compute

the tcpDfunct]on, In this case, tcp is obtained using E+ R-2,

B-3 and B-4.

[

I

I


8/17

o

0

o

Y

u

11,,,

j

1

... . . . . . . . . .. , , . . . .. ,, ,. .. . . . . . . .

//

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

..,,,..,,. . . . . . . .

o

-

.,,,,.,,.

...,,,,,,

z

v

,,,,

2

o ~ ............

E

.

Lf i t l l

I 1 1

1,,,,,

I

1

1,,1,,

I t

9

,......-.=

--


9/17

m

,,,,,,,,.,,,,,,..,,,,.

.W-)

cd


10/17

o

o

%

a

d

cu~

-

n /

~twl

p>

-i+

c l

.

CO

a~a;

(

.........

.$J ~ ,............... . . .d. . , d, ,

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

Wo

yc&

k

-/

:

/If

%

\

(Q

..-

5?

8g

(

,, ,-i

G

1~ u

2

.... ,. , ,, , ,, , , . , .. . . . . . . . . . . . .. , , . ., ,, ,

m

,,.,,

...,,

,.,,,,,,,,,.,,..,...,..,.,,,

r:

Q

,.1

,, ..,,.,,,,..,,, ,,, ,,, ,,, ,,, ,,, .,,,,

%., ~a

CQ

.

w

g :-

Lg

-z T.

3

=$&

~

\; - -

IIUQ

CA

..,.,,,.,,,,., .,.,,,.,.,,..,,,,4,,,,,,,

%

m

.,-

% -

%g

u

IL w

J

d

,,................., .....................

-o

TY

.-

%

.-

-


11/17

m

1

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

, ,, , , ,4

t-a

.

::/

W-io

y~

~

c1.

/

.,

,, .,,. . . . . . . . . . . . .,. ...., . ,,,..

r:

,

LA

0?

8g

m ~

IL w

&

... .......,,,,,.

m

L-9

0

;:

~-%

/

>

........

. ...4.... . ... ,,

A)/

. ..... , ,., ,,, ,, ,,, ,,, ,,, ,,,

(u

.?-

VI

cb@

.

8$

%g

II w

Y

CA

., ,.,,,,,,, ,,,,.,,.,..,,,,,.,,,,,,

1-

t-

/

H :

,.,,

/,:

...,,,,.,,, ,,, ,,, ,, ,,, ,,, .,

\

~-

; g~

I fiq

2

J

C&

.,,,,

. . . . .. . . . . .. . . .. .,, , , ,,.,,

Z

~ ~~

8

m

, -

K.

;i$ g

J

V*

....,,,.,,,,,,,

2,,.,,,,,,

.n.

%

d)

G

c)

r-l

%

-3

E

I-.4

WJ

T1

.-

.ij-

-

&

8

.-


12/17

.

0

~(jo

1000

1500

2000 2500

120

I

3000

1

1

I

I

~ 120

~p=+() ltl psia ~

j)o-.

.. ..-..---.--.-_______...........

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

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

........

...._.-...-.__

~

80-

. - - - .+.._-. _-. . _ ..... ... . . -. -+...................................+............................

~

~

2 40

............. ... . -- _. ______

q=50 S?B/l) ~?.~z~ p~a pWf=15U0 psia ~

@W;] ; (ficw 2) ~ (fl~v~ @ ;

i

\

o

i

500

1

1000

i

1500

2000

2500

3000

L (=Q/q),days

\

F@ITe 5- ~afi~sja~

plot of Ap/q versus ta for the liquid simulation cases.


13/17

i

,,,,,,,,.,..,,,,,,,

.,

,,,,,,,,,,,,

u


14/17

to

--23

%

.,,,,.,,,,,,,,,,,

C2

(J

.

%

c l .

. .,, ,.,,,,,,,,,,,,,.

. .

- r-

-t)

F*

,0--

{

II

,,,.,,.........................,,,.

cd

.

K

/

............ ,,,,,,.,


15/17

$5

.-

-&)

c

10


16/17

sPE 21513

.

10

101

102

103

104

d---

*

1

n

1 1

1

1

q7J01

6

4

1

2

{

--------------------------------s,

----

--- - -*.-.----------------------------------------------------------------------------------------00

-

6

/

\

~ f

4

4

Bmmdaiiy Dominated Flow Solution

@APa=(l/ba) El:p[-(rnJbJtcP,.l

-). \:

or

2

@Pa=[% 1.,,, + baJ-l

............ .................................+........................................ ................................................... ...... ...... ...... ..................

10-]

6

Results of Gas

~aterial J3alance Iterative

4-

solution ( 131asinmrne and Lee8~

ma =2.05536x10-3 Psi/MSCF~~

2-

ba =1.3094 @./lvlSCF/D

2

G=2.6281 Bscf [tW,.= (bJmJ ~[1 + ([email protected]&J I

~o-2

i

1

I1

I J

10-2

10 101 102 103

4

10

t

.P,a,days

Figure 9- Plot of q/Apa versus tCP,aor d~.taof Fetkovich, et aL2

t

~P,afunction computed using IC,, and the equation shown.

.


17/17

102

101

10

10-]

~o-2

r

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spe 021513 (mccray) decl curve an for var pressure drop var flowrate

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