SPESociety of PetroIeurn Engineers

SPE 19817

Use and Misuse of the SuperpositionTime Function in Well Test AnalysisH. Cinco-Ley and F. Samaniego V., Pemex/UNAMSPE Members

Copyright 1989, Society of Petroleum Engineers, Inc.

This paper was prepared for presentation at the 64th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in San Antonio, TX, October 8-11, 1989.

This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper,as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflectany position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Societyof Petroleum Engineers. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgmentof where and by whom the paper is presented. Write Publications Manager, SPE, P.O. Box 833836, Richardson, TX 75083-3836. Telex. 730989 SPEDAL.

ABSTRACT

The superposition time function has beenused as a tool to analyze transient pressure data measured under the influence ofa variable flow rate. This function isusually defined assuming that radial flowequations are valid, however, in practicethere are cases that exhibit several flowregimes (i.e. fractured wells, partiallypenetrating wells, etc.). The present workexamines the limitations of the superposition time concept as applied to builduptests. It appears that, regardless of theflow regimes exhibited by the well reser-voir system the derivative function withrespect to the radial flow superpositiontime for a buildup test follows, at earlytime, the drawdown curve for the pressurefirst derivative function t*dp/dt, then,after a transition period, it follows thedrawdown curve for the pressure secondderivative function t 2 * abs(d2 p/dt 2 ).INTRODUCTION

Well testing has proved to be one of themost reliable tools to evaluate flowcharacteristics of a well-reservoir flowsystem1 . 2 A large number of publicationson this subject was presented in the lastfour decades.The original theory for pressure transient test analysis in the petroleum industry was developed for constantwell flow rate conditions3 . 4 Later severalauthors 5 - 12 presented methods to take intoaccount the rate variations in well test in-terpretation. More sofisticated techniquesof interpretation were developed recently13-16 to take advantage of advances in thetechnology to measure flow rate and pressuresimultaneously with good resolution.

477

The application of the pressure derivative17function t*dp/dt for type curve matching1S-20 and flow regime identification20- 22 hasbecome a standard for well test interpreta-tion in the last few years. Several pressurederivative type curves are now available;most of them were developed for drawdowntests and are applied to the analysis ofpressure buildup tests through the use ofthe superposition time concept. It hasbeen suggested that this concept can takeinto account the variation of the flowrate before shut-in; however, experiencehas shown that this technique producesdistortions in the calculation of thederivative function when the pressure dataare under the influence of a flow regimeother than radial.

The objective of the present work is toexamine the advantages and limitations ofthe application of the superposition timeconcept on the interpretation of pressurebuild-up tests through the use of specificgraphs of analysis (pws vs f(q,t)) and typecurve analysis of the derivative function.

ANALYSIS OF VARIABLE RATE PRESSURE DRAWDOWNDATA.

Let us consider a pressure drawdown testunder variable flow rate conditions (Fig.l)where the flowing bottomhole pressure is afunction of both flow rate and time. As men-tioned before, the original theory for interpretation assumes constant flow rate condi-tions; hence it is necessary to take intoconsideration the variation of the flow rate.

Generally speaking, the methods of interpretat ion for a test with variable rate involvea correction of pressure (Fig.2) or/and a

2USE AND MISUSE OF THE SUPERPOSITION TIME FUNCTIONIN WELL TEST ANALYSIS SPE 19817

correction of the time scale (Fig.3). BO~htypes of corrections are based on the pr~nciple of superposition and ca~ be referredas deconvolution and convolut~on, res~ectively. The former represents a techn~quethat does not assume the flow model ,and thelater is a method based on a predef~ned re-servoir model.

In cases where there is a skin effect a co-rrection is necessary in both pressure andtime. For this case a graph of (p~-P~~)/qNvs L q~*f(t-t~) is used as shown in Fig. 4and Table 1; here the reservoir parametersand well condition (damage) are determined.The function f depends on the reservoir flowmodel, that is, it is represented by log(t),t 1 / 4 , t 1/ 2 , 1/t1/ 2 , for radial, bilinear,linear and spherical flows, respectivelyor more general the function f can be represented by a Po-to relationship correspondingto a given reservoir system.

A general approch for the analysis of draw-down tests is presented in Table 2 and in-cludes, first, the estimation of the unitflow rate pressure response (influence func-tion) through the deconvolution process(Table 3) followed by a diagnostic of flowregime based on the pressure derivativefunctions (Figs. 5 and 6 and Table 4) andfinally the application of the specificgraph of analysis, (pi-pwf)/qn versusLqi*f(t-ti). The last step is well docu-

mented in references 5, 6, 9, 10, and 13-16.

ANALYSIS OF PRESSURE BUILDUP DATA.

The pressure buildup is the most frequentlyused test because the bottomhole pressure ismeasured under constant flow rate (q=O)conditions (Fig. 7), theoretically. It can beshown17that, for a buildup test, the earlyshutin time pressure data are dominated bythe last flow rate, the middle time datadepend on both flow rate variation and pro-ducing time and the long time data depend ex-clusively on the cummulative production du-ring the flowing period (Fig. 8). Hence theneed to know the flow rate history beforeshutin for a proper analysis.

Conventional Techniques of Analysis.Conventional methods of interpretation (Hor-ner and M-D-H) assume that the flow rate be-fore shutin is constant and the flow regimeexhibited by the reservoir system is radial.For an infinite acting reservoir the M-D-Hmethod produces a straight line in a graph ofpws versus Log t at the beginning of thetest, then the data deviate because thistechnique does not take into account theeffect of producing time (Fig. 9). ,The Hornermethod considers the effect of t p ~n such away that a graph of P~s versus Log (~t/tp+~t)produces a straight line that goes throughall of data free of wellbore storage effects.In other words the Horner time includes a"correction" for the producing time effect.other types of graph also have been ~sed toconsider flow regimes other than rad1al, suchas P~s vs (tp+~t)1/2 - (~t)1/2 , P~s vs (tp +

~t)1/4 _(~t)1/4, P~s vs (~t) 1/2_(tp+~t) 1/2for linear, bilinear and spherical flow, res-pectively. However, a flow diagnostic.pro~essmust be carried out for a proper app11cat10nof any of these types of graph.

The Superposition Time Graph.For the case of variable flow rate beforeshut in the buildup pressure can be expressedas:

qh) f' (tp+~t-T) dT( 1 )

where f'is the time derivative of the unitflow rate pressure response of the well-reservoir system. If the flow rate history is dis-cretised Eq.1 becomes:

Np~s(~t) = p~ - L q~ (f(tp+~t-t~-d -

i=l

the summation is called "superposition time"t sup and depends on the flow regime that dom-inates the pressure behavior of the system.Sometimes the summmation of the superpositiontime includes the flow rate ratio q~/qN andthe simplified form of the function f (SeeTable 5), in such a way that Eq.2 is given by:

N q~p_s(~t) = p~ - m(qN) L (g(tp+~t-t~_1)-

i=l qNg(tp+~t-td )

(3 )

This equation shows that a graph of pws ver-sus the summation yields a straight line ofslope -m and intercept p~ (Fig.10). The slopeis a function of the last flow rate qN andand depends on the reservoir paramete~s. TheHorner method is a special case of th~s graph,that is the superposition time reduces to theHorner time group when the flow is radial andthe flow rate before shutin is constant.

It was mentioned that the determination ofthe nature of the function g (i.e. log(t),t 1 / 2 t 1 / 4 t- 1 / 2 ) requires a flow diagnosisproc~ss th~Ough the first or second deriva-tive functions. The beginning and the end ofthe proper straight line can be found asshown in Fig. 11. Let us assume that a flowregime j detected, begins at time tb~ and endsat time te~. The start of the straight lineportion in the superposition time graph occursat t sup correspontig to ~t=tb~ and t~eoretically ends at t sup for tp+~t=te3' Th1S lastpoint will depend on both the flow rate his-tory and the flow model exhibited by the re-servoir.

The superposition time can also be defined byusing a Po-to reservoir mode1 13 , thus:

478

SPE 19817H~eeR CINCO-LEY AND FERNANDO SAMANIEGO-V.

3

Hence the derivative of pressure with respectto t aup can be expressed as:

dp_s

NP_.(~t) = Pi - mpo E qi (Po(tpo+~to-toi_1)

i=1

dP""a d~ t(4) =

Here mpo comes from the definition of Po dt aup(see Table 5). The application of the super-position time graph requires a trial and errorprocedure to find the relationship between toand t that produces a straight line.

N qi.E ----

i=1 qN

1 1{-------- - ----------}tp+~t-ti. t",-+:6 t-t:i._1

(6 )

At early shutin times this equation becomes:'

At large values of shutin time Eq.6 reduces to

where Q is the cummulative production duringthe flow period. According to the instantaneou!source theory17 the time derivative of the presure buildup at long times is: .

(8 )

( 7 )

dP_a

d~t

24 Q

dP~a

d~ t

24 Q

~t ------

=

=

dtaup

d p_a

Thus, as mentioned by Bourdet et al. 23 , thederivative of the shutin pressure with respectto the superposition time approaches the firstderivative function t plfor pressure drawdowncorresponding to the last flow rate.

Drawdown Type Curve Matching.The application of the type curve analysistechnique as a diagnostic process allows the analyst to determine the start of the semilog dp~sstraight line and the detection of reservoirheterogeneities1S, 19. Usually, drawdown typecurves (pressure drop and time derivative of~ressure) are used to analyze pressure buildupdata because of its simplicity as compared toa buidup type curve that involves the producin~time as an additional parameter match. The ap-plication of drawdown type curves is valid un-der certain condition, that is, the producingtime must be large as compared to the shut intime20 (tp >10 ~t)i if this limitation is notsatisfied then data should be co-rrected. Tomatch the drawdown type curves a correctionon the time scale can be made by using the"effective time" teff defined by Agarwal 21based on the radial flow equations.This correc- dtsuption is similar to the one involved in theHorner graph and yields excellent results ifthe drawdown data before shutin are free ofwellbore storage and the flow exhibited by thereservoir is radial. It should be mentionedthat the effective time method can not be usedfor the pressure derivative analysis to correctthe time scale. d p_a

( 9 )

here P(qN) is the pressure drawdown correspoding to rate qN. A combination of Eqs. 8 and 9gives:

The first and second derivative functions fordifferent flow regimes in terms of real varia-bles are as follows 17 :

(10)[~t]2 ----------=d P_s

Equations 7 and 10 are valid for any flow re-gime, thus, it can be stated that the super-position time pressure derivative of buildupdata behaves, at early time, as the drawdownfirst derivative function and at large shutintimes follows the drawdown second derivativefunction.

Therefore, it appears that the superpositiontime derivative of the pressure buildup atlarge values of time approaches the drawdownsecond derivative function defined in refer-ence 17.

I dtsup

tp+~t-ti_1Ln {-------------}

tp+~t-ti

N qit sup = E

i=1 qN

The definition of the superposition time, assuggested by Bourdet et al., is based on theradial flow equations and is given by:

A proper application of some of the methodsalready discussed requires a diagnosis of theflow regimes exhibited by the reservoir duringthe test. The process becomes complex if theflow rate changed during the producing period.There are two techniques that allows the flowregime identification under these conditions:a) the superposition time pressure derivative23 and b) the instantaneous source method17 .Although the application of these techniquesis well documented, there are some aspectsrelated to the first method that deserve tobe analyzed.

Another method used in the analysis of pressure d ~ tbuildup data to match the drawdown pressuredrop type curves involves the desuperpositionof the drawdown effects as suggested by Ragha-van 22 . This technique assumes constant flowrate during the producing period and requiresthe inital pressure and the bottomhole flowingpressure before shutin.

( 5 )

479

4USE AND MISUSE OF THE SUPERPOSITION TIME FUNCTIONIN WELL TEST ANALYSIS SPE 19817

Wellbore Storage and Pseudo-steady State Flow

CONCLUSIONS

From the material discussed in this work thefollowing conclusions can be presented:

1. The interpretation of pressure buildup datacan be carried out through the use of bothspecific graph of analysis and type curvematching.

'The duration of the transition period betweenthe first derivative behavior and the secondderivative behavior depends, according to Eq.5, on the flow rate history and producing timeThe deviation from the first derivative behav-ior occurs at approximately 6.t= 0.05 t p Herea 5% difference between the curves is consid-ered. The superposition time derivative fOllOWJSthe second derivative curve after 6. t= 2tp Hence, the transition period lasts for abouttwo log cycles.

ype curves involving a combination of the fir tand the second derivative functions can be de-veloped. Figures 12 and 14 show this type ofgraph for a well with wellbore storage and skiteffects and for a well with an infinite conducI~ivity vertical fracture, respectively. SimilaItype curves can be constructed for finite con-ductivity fractures and double porosity reser-voirs.

(11 )

(12)

(14)

(13)

(15)o

t 6. p' = AJ. t 1 / 2

Linear Flow

t 6. p' = A",s t

Radial Flow

Spherical Flow

Bilinear Flow

t 6. p' = Asph t- 1 / 23

t 2 abs{6.p' ') = --- Asph t- 1 / 24

t 2 abs{6.p")

't 6. p' = A r

t 2 abs{6.pll) = Ar

2. The application of the drawdown pressurederivative type curves has limitations in thein the analysis of pressure buildup data atintermediate values of shutin time.

3. It appears that, regardless of the reser-voir model, the superposition time pressurederivative of buildup data follows, at earlytime the behavior of the drawdown pressurederi~ative function and after a transitionperiod follows the behavior of the drawdownsecond derivative function.

According to Eqs. 11 through 15 the first de-rivative is, in general, not equal to the sec-ond derivative function, except for the radialflow case.As a consequence of the above discussion itcan be said that, regardless of the flow modelthe analysis of pressure buildup data can beperformed through the use of type curve match-ing of the superposition time derivative; how-ever two sets of drawdown type curves are re-quired: the first and the second derivativefunction type curves.Figure 12 presents the first and the deriva-tive fun~tion type curves for radial flow un~ 4. The superposition time based on radial flowder the 1nfluence of wellbore storage an~ sk1n' can be applied to analyze data measured afterIt can be seen that they are completly d1ffer- a variable rate flowing period.ent at early time, but both sets of type curve~

approache~ a single line when we~lbor7 st~rage 5. A general approach involving a combinationeffects d1ssapear. If the produc1ng t1me 1~ of sets of first and second derivative func-large, it is expected. that the pressure bU1ld ,tion type curves can be used to analyze an en-up data m~tch the ent1~e dr~wdo~n type cu~ve; Itire buildup test . This approach is ba~ed onhowever, 1f the produc1ng t1me 1S small (1.e. the superposition time derivative of bU1ldupflowing pressure before shutin is sti~l.affec- data.ted by wellbore storage) the superpos1t10ntime derivative follows at ,early time t~e.first6. New type curves for the superposition timederivative type curve and after a trans1t10n pressure derivative are presented.period follows the second derivative curve (seeFig.13 ).

NOMENCLATURE

rateunit flow rate pressure responsetime derivative of the unit flowpressure response.function, Eq. 3slope in a variable rate test.dimensionless pressureinitial pressureflowing bottomhole pressureshut in bottomhole pressure

=

=

=

=

=

9mpDpipwf =pws =

The analysis of pressure buildup tests throughthe aplication of the superposition time deri- fvative can lead to serious errors of interpre- fl'tation, such are the cases where the reservoirexhibits flow regimes other than radial. Forinstance, if the system is dominated by linearflow during the entire test the analyst canerroneously conclude that the system exhibitsdouble porosity behavior because the superpo-sition time derivative shows two parallelstraight lines of half slope.

480

SPE 19817 HEBER CINCO-LEY AND FERNANDO SAMANIEGO-V. 5

REFERENCES

subscripts

b = beginning, bilinearD = dimensionlesse = endf = flowingi = initial, ith period1 = linear,s = shutin:sph = sphericalw = wellbore

1. Matthews, C. S. and Russell, D. G.: "Pressure Buildup and Flow Tests in wells",Monograph Series, Society of PetroleumEngineers,Dallas (1967) 1.

2. Earlougher, R. C., Jr.: "Advances in WellTest Analysis", Monograph Series, Societyof Petroleum Engineers, Dallas (1977) 5.

3. Horner, D. R.: "Pressure Buildup in Wells"Proc., Third World Petroleum Congress, TheHague (1951) Sec. II, 503-523.

4. Miller, C. C., Dyes, A. B. and Hutchinson,C. A.: "The Estimation of Permeability andReservoir Pressure from Bottom Hole Pres-sure Build-Up Characteristics", Trans.AIME (1950) Vol.189.

5. Russell, D. G.: "Determination of FormationCharacteristics from Two-Rate Flow Tests",J. of Pet. Tech. (Dec. 1963)1347-1355.

6. Odeh, A. S. and Jones, L. G.: "PressureDrawdown Analysis, Variable-Rate Case", J.of Pet. Tech. (Aug.1965)960-964.

7. Gladfelter, R. E., Tracy, G. W. and Wilsey,L. W.: "Selecting Wells Which Will Respondto Production-stimulation Treatment", Drilling and Product. Pract .. API (1955) 117-129

8. Winestock. A. G. and Colpitts, G. P. : "Advances in Estimating Gas Well Deliverability",J. Cnd. Pet. Tech. (July-Sept., 1965)111-119.

9. Odeh, A. S. and Jones, L. G.:"Two-Rate FlowTest, Variable-Rate Case- Application toGas-Lift and Pumping Wells", J. Pet. Tech.(Jan.1974)93-99.

10.Earlougher, R. C., Jr.:"Variable Flow RateReservoir Limit Testing", J. Pet. Tech.(Dec.1972)1423-1429.

11.0deh, A. S. and Selig, F.:"Pressure Buildup Analysis, Variable-Rate Case", J. Pet.Tech. (July 1963) 790-794.

12.Bostic, J.N., Agarwal, R.G. and Carter, R.D.:"Combined Analysis of postfracturingPerformance and Pressure Buildup Data forEvaluating an MHF Gas Well", J. Pet. Tech.(Oct.1980)1711-1719.

13.Fetkovich, M.J. and Vienot, M.E.:"RateNormalization of Buildup Pressure By UsingAfterflow Data", J. Pet. Tech. (Dec.1984)2211-2224.

14.Stewart, G., Wittmann, M.J. and Meunier,D.:"Afterflow Measurement and Deconvolutionin Well Test Analysis", Paper SPE 12174,presented at the 58th Annual Technical Con-ference and Exhibition of SPE of AIME, SanFrancisco, Ca., Oct.5-8, 1983.

15.Meunier, D., Wittmann, M.J. and Stewart, G."Interpretation of Pressure Buildup TestUsing In-Situ Measurement of Afterflow", J.Pet. Tech. (Jan.1985)143-152.

16.Kucuk, F. and Ayestaran, L.:"Analysis ofSimultaneously Measured Pressure and Sand-face Flow Rate in Transient Well Testing",J. Pet. Tech. (Feb.1985)323-334.

17.Cinco-Ley, H. Kuchuk, F., Ayoub, J., Sama-niego-V., F. and Ayestaran, L.: "Analysisof Pressure Tests Through the Use of Ins-tantaneous Source Response Concepts", PaperSPE 15476 presented at the 61st Annual Technical Conference and Exhibition of SPE ofAIME, New Orleans, LA, October 5-8, 1986.

18.Bourdet, D., Whittle, T. M., Douglas, A. A.and Pirard, Y. M.:" A New Set of Type Curve~

Sim~lifies Well Test Analysis", World Oil,Apr11, 1984.

19.Gringarten, A. C.: "Type Curve Analysis:What It Can and Cannot Do", J. Pet. Tech.(Jan. 1987)11-13.

20.Gringarten, A. C., Bourdet, D. P., Landel,P. A. and Kniazeff, V.: "A Comparison Be-tween Different Different Skin & WellboreStorage Type Curves for Early Time TransienAnalysis", Paper SPE 8205, Sept. 1979.

21.Agarwal, R. G.: "A New Method to Account fo]Producing Time Effects When Drawdown TypeCurves Are Used to Analyze Pressure Buildupand Other Test Data", Paper SPE 9289 presen-ted at the 1980 SPE Annual Fall Conferenceand Exhibition, Dallas, TX, spt. 21-24.

22.Raghavan, R.: "Effect of Producing Time onType Curve Analysis", J. Pet. Tech. (June,1980)1053-1064.

flow rateflow rate during Nth rate periodcummulative productiontimeproducing timeshutin time

q =qN =Q =t =tp =

t =

481

TABLE 1

SPECIFIC GRAPH OFANALYSIS

p Ctl .. QCtl'"FLOW MODEL

~P w US L Q .FCtl~ 1

SLOPE AND INTERCEPTWELL AND RESERUOIR PARAMETERS

TABLE 3

ESTIMATION OF THE UNITFLOW-RATE RESPONSE

PRESSURE AND FLOW RATEDATA

~DECONUOLUTION

IMPULSE

~INFLUENCE FUNCTION

DERIUATIUES

TABLE 5-SLOPE OF THE SUPERPOSITIONTIME GRAPH BASED ON Po -10 MODELS

TABLE 2

GENERAL METHODOLOGYOF ANALYSIS

-ESTIMATION OF UNIT FLOW-RATE RESPONSE.

-DIAGNOSIS OF FLOW REGIMES

-APPLICATION OF SPECIFICGRAPHS OF ANALYSIS.

TABLE 4

FLOW DIAGNOSISINFLUENCE FUNCTION

DERIUATIUES

~

TYPE OF FLOWDURATION

TABLE 6. SI PREFERRED UNITS, CUSTOMARY UNITS, AND UNITCONVERSION CONSTANTS USED IN THESE SYSTEMS.

Model

Linear

Bi linearRadial

Spherical

Parameteror variable 51 Preferred Units Customary Units

MPD

IJm 2 md"LqN BIJL m ft

kbh q m'/D STB/D Dr Mscf/D

IJ Pa.s cpB m3 /m 3 RB/STB

"qN BIJ ~ fraction fractionkh -1 .-1

CI Pa ps,

P KPa psihours hours

"spqN BIJ"o="OL="ob 1842 141.2

kr -. -4W B 3.6x10 2.637x10

482

qq

Logt If6p ~

TIM EFig. l-Varlable flow rate test.

(6.t)carr

TIM EFig. 3-Time correction for variable rate.

UELL80RE STORAGEPSEUOO-STEAOY-STATE

LINEAR

8 ILINEAR

--------- RADIAL~-1"2~ SPHERICALLog t

Fig. 5-Flrst derivative graph of diagnosis.

483

q

1\..

..

Q.'StnIIIa:

N...

[]\D.J

TIM EFig. 2-Pressure correction for variable rate.

't = f (q .. tcarr

TIM EFig. 4-Pressure and time corrections.

LINEAR

8 ILINEAR

--------- RADIAL~-1"2~ SPHERICALLog t

Fig. 6-Second derivative graph of diagnosis.

Horner

q

HTt::~---------------------

1-------7>-b.t

TIM EFig. 7-Pressure buildup for constant rate.

t corr

Fig. 9-A comparison of Horner and MDH graphs.

q

HQ)

6t >2tp

3 P

M EFig. 8-Pressure bUildup for variable rata.

Fig. 10-Superposltlon time graph.

~EGION OF UALIOITV OF. THESUPE~POSIT ON TIME G~APH

--~l!.t

Fig. ll-Beglnnlng and end of straight line.

484

10 '

10 10 :a10 'to/CD

101

CD E2S ~I

'\/1040~1/~1- I\. r\102

I , \

1O~ \/- Vv 10 H7 ~ \ 1\~ fIRST DERIVATIVE IA~ v 106 L.'-. "'ll\. ~ \ \ ,

!~~03 "R I' ,,\ ~\lI.~""

~ f\ 1\riI iiiI~ ~ , ,

I.J!! / iii 1/ SECOND DERIVATIVE1.1 /{Oll :0

/ 10 fII 110 I1i106 1020" 'I

,

-o0..oo.......

en(01octC'J**o

of-'

Fig. 12-Type curves for radial flow with skin and wellbore storage.

~dfCt)

6tFig. 13-Schematic of match of pressure buildup derivative.

485

10

10 t10

.-..-

---~V ~ ....~~~~ ~~/---~

/~

,

-

~ 17,IP ~

""'/ I--~~ ....... I.