SPESPE 18799

Type-Curve Analysis Using the Pressure Integral Methodby T.A. Blasirqame, J.L. Johnston, and W.J. Lee, Texas A&M U.

SPE Members

Copyright1989, society of Petroleum Engineers, Inc.

This paper wee prepared for presentationat the SPE California Regional Meeting held in Hakerafieidi California, APril *7, 198g.

This paper was aalscted for preaentafionby an SPE ProgramCommitteefollowingraview of informationcontained in an abstract submittedrrythe author(e).Gantentaof the papas presented, have not been reviewed by the Sodefy of PetroleumEnginaeraand are subjacfto correctionby the author(a).The material, aa presented doaa not necessarilyrefkany poaltlonof the Societyof PetroleumEnginwrs, Ita offlcera,or membere. Papara presentedat SPE meetingsare aubjactto publicationreview by EditorialCommitteesof the SocIofPetroleumEngineers.ParmieelontocopyIsrasfriofadtoen abatractofnotmorethanS00worde,Illuafrationamaynotbe Cupisd.The abstractshouldcontainconspicuousacknow[adgmof where and by whom the paper ia presented. Write PublicationsManager, SPE, P.O. Sox SS38S6, Richardson,7X 750S?-SS36.Telex, 730989 SPEDAL.

OF ~

This paper presents new type curves and typecurve plotting fr.rnctionswhich use integrationrather than differentiation of well test data. The new type curves require a number ofThese functions provide more unique type curve dimenaionleas variables, some of ~hich are not inmatches for noisy dst% because the integral of common use. To facilitate our later discussion ofthese data yields a much amor)therfunction. The the curves, we define these dimensionless variabletype curves presented in this work were generated in this section. Their origin and their utilityfrom analytical solutions and are applied in discussed in later sections.exactly the same manner as in conventional typecurve analysi.a. The dimensionless wellbore pressure, PD, is

defined as

khdp

The purpose of this paper is to introduce new P~ = ......,..,,.....,...,........(1

type curves and type curve plotting functions that 141.2 qB~allow uriiqueanalysis of noi.aywell test data.These new type curves are based on the integral of The dimensionless time, tD, based on the wellborethe pressure drop function. The motivation forthese new plotting functions and type curve radius is defined assolution? was our discouraging experiences withpoor (non-unique) type curve matches of noisy data, 0’.0002637 kt

‘D =..............s............(2

The solutions plotted on conventional pressure $Kctrw2drop type curves and data analys:a6techniques have

been reported by several authors - for homogeneous and the dimensionless time, tLfD, based on fractuand vertically-fractured reservoirs. Thecorresponding pressure derivatives and matching half-length, Lf, is defined as

techniques have been :i:;ussed extensively in the

petroleum literature. - 0.0002637 kt

‘LfD =.........................(3

This paper discusses development of the new $FctLf2type curve plotting functions and the correspondinganalytical solutions used to generate type curvesfor homogeneous and vertically-fractored The dimensionless pressu”re derivative

reservoirs. We demonstrate the advantages of these function, pDdr irrdefined in two forms, which are

new type curves with examples of noisy test data. identical mathematically. The first form iSWe conclude with a suggested procedure and exampleapplication of these new type curves to field data, dpD

‘Dd= — ...............#.,............(4

d(ln tD)References and illustrations at end of paper.

525

2 Type Curve Analysis Using the Pressure Integral Method SPE 18799

—

Alternative y,

L-2. This section su~marizes analytical expressions1.

pDd “ ‘D (5)used to calculate the dimensionless variables

— .....,e.0.,....,......,,........ defined in the prbvious section. ThesedtD expressions, which are derived in Appendix A, model

radial~ bilinesr and linear, and wellbore storage

We prefer the form in Eq. 5 because it allows us todominated (unit slope line) flow regimes.

obtain pDd readily from general analytical

s,lutions which include wellbore storage, bilinearo. linear flow regimes for vertically-fractured 19rL ?~voirs, and radial flow for homogeneous

The solution which models radial flow is

reservoirs.PD - a [ln(tD) + b] + S ...,.,...........(lo)

A new dimensionless pressure integralfunction, pDi, later is defined as where tLfD is substituted for tD for the fractured

well cases.1 tD

The coefficients in Eq. 10 are definedas follows

PJ)i= - JPD(T) dT .........,...........,,..(6)

tD os = 0.5 for all cases

~~ IFThe dimensionless pressure integral differencefUnCtiOn, pDid, is defined as

Homogeneous19 0.80907 25

pDid = pD - PDi (7) Infinite Conductivity 20 2.20000 10..............,.........,....Uniform Flux20 2.S0907 10

We demonstra~? in Appendix A that Eq. 7 can also beexpressed as The plotting functions (dimensionless

variables) for radial flow are summarized below.

‘pDi

pDid = ‘D — ,,................,..........(7a)

dtD ‘Di 0.5 [ln(tD) +b - 11 + S ....(11)

‘Did 0.5 ..............(12)i.e., the right-hand-sidek of Eqns. 7 and 7a areidentical mathematically. ‘Dd

0.5 ..............(13)

‘Dirl ‘Di..............(14)

We also use two other plotting functions which‘Dir2

1.0 ..............(15)are dimensionless ratios. These functions giveunique resolution to flow regimes such as wellborestorage domination (unit slope line), bilinear and

These functions identify unique character-

linear flow for fractured reservoirs and radialistics of the radial flow regime. For instance,

flow for homogeneous reservoirs. The firstEqns. 10 and 11 indicate that pD and ppi behave

dimensionless ratio, pDirl, i.ssimilar to that very similarly. This me,ansthat pDi could be used

defined by Onur and Reynolds16 and isas a semilog plotting fuixkion; this could be auseful alternative for very noisy pressure data.

PDi Eqns. 12 and 13 predict that a horizontal line

pDirl . — ..............................(8) with a plotting function value of 0.5 will be seen

2 ‘Didon graphs of both poid and pDd vs. tD during radial

flow . This behavior is well known for the

This function is useful in analyzing cases with dimensionless pressure derivative, pDd.

radial flow because, as tD -> W, pDirl approaches

pDi .The remarkable fact is that both pDd and pDid

have exactly the same horizontal (constant Value)

The other dimensionless ratio combines the behavior during radial flow. This suggests that a

dimensionless pressure integral difference type curve plot of pDid may be similar in shape tofunction, pDid, with the conventional dimensionless a pDd type curve for other flow regimes. This

pressure derivative function, pDd. This function

ispossibility will be addressed specifically later inthis paper when other phenomena such as wellborestorage and the early time effects of fractured

‘Didwells (bilinear and linear flow) are added to these

‘Dir2 = —solutions.

...............,.,..............(9)

‘DdEq. 14 is’our equivalent of the ratio function

defined by Onpr and Reynolds 16 for theEq. 9 requires accurate and noise free data to dimensionless pressure derivative. Note that’thisyield the correct derivatives and subsequent ratio function is equivale~t to PDi for radialresolution to particular flow regimes. Thisfunction can be used to identify uniquely the

flow . This suggests that if the early data are

existence of a particular flow regime.affected by wellbore storage or fractured welleffects, the later behavior of the pDirl ratio

SPE

.,

18799 T.A. Blasingame, J.L. Johnston and W.J. Lee

function during radial flow should be the same asP~~ ●

Eq. 15 stetes that the second ratio function,

‘Dir2’ has a value of 1.0 for radial flow. The

practical application of Eq. 15 is that pDir2 will

lie on a horizontal line with a numerical value ofunity during radial flow cm a type curve.

In this section we will consider a generalform of a power function to model the bilinear andlinear flow regimes in a fractured well, The powerfunction model is also used to develop relationsfor the wellbore etorage dominated flow regime(unit SIOPe line). This model is

PD = at bD ,..............................., (16)

‘here ‘LfD is substituted for tD for the fractured

well cases. The coefficients in l%q.16 are definedas follows:

Bilinear F10W5 2.45083/~ 1/4

Linear Flow20 G 1/2

Wellbore Storage21 I/cD 1

Functions to evaluate these dimensionless variablesare derived in Appendix A. The results forbilinear, linear, and wellbore storage dominatedflow regimes are given in the tables below.

Bilinear E’lowRegime:

P~ 2.45083/~ tLfD1’4 ,..(17)

‘Di 0.80 pD .............(18)

‘Did 0.20 pD ...!.........(19)

‘Dd 0.25 pn .............(20)

‘Dirl 2,0 ...............(21)

%Lr2 0.8 ...............(22)

These functions uniquely identify the bilinearflow regime which OCCUSN at very early times in thetlow of wells with finite or infinite conductivityvettical fractures.

Linear Flow.Regihle:

P~ ~)tLfD1/2

..........(23)

%i 0.6667 pD ...........(24)

‘Did 0.3333 pD ...........(25)

‘Dd 0.50 pD .............(26)

‘Dirl 1.0 ..............(27)

‘L)ir2 0.6667 .............(28)

These functions uniquely identify the linearflow rOgiMS which occurs relatively early infractured wells with high fracture conductivity.

Wellbore Storage Dominated Flow Regime:

Pfj tD/cD ..............(29)

‘Di0.50 pD .............(30)

‘Did0.50 pD .............(31)

‘Dd pD ..........,....(32)

‘Dirl0.5 ...............(33)

‘Dir20,5 ...............(34)

These functions”uniquely identify the wellborestorage dominated flow regime which occurs at veryearly times.

NOte that pDi, pDid and pDd are fractional

multiples of the pD function durin”gthese flow

regimes. This means that we would expect thesefunctions to be parallel on type curves.

Note also that PDir. and PDir2 are constant

during these flow regimes. This results in ahorizontal line on the type curve for each of thesefunctions. These horizontal lines can be used asreference lines in much the same way that theconventional pressure derivative function. pDd# in

cuzrently used to evaluate radial flow data’.

The ratios PDirl and PDir2 can be verY

powerful analysis tools because of their uniqueresolution of a particular flow regime, but thesefunctions require teat data that is acctmate andfree of noise.

In this section we will develop type curvesusing the new plotting functions for homogeneousand vertically-fractured well systems. Thehomogeneous reservoir and uniform flux and infiniteconductivity fractured well caaes will include thee:fects of wellbore storage. The finiteconductivity vertical fracture cases will notinclude the effects of wellbore storage in thiswork.

~:

The type curve solutions for this case areshown in Fig. 1. This type curve was generatedfrom the “continuous line source- solution (Eq. B-6). Note the similarity between the pD and pDi

solutions. These solutions are slightly non-

parallel at early times (tD<102), but later these

functions bkcome parallel, as predicted by Eqns. 10and 11. Note also that the pDd and pDid solutions

become essentially equivalent for tD>102, as

predicted by Eqns. 11 and 12.

Finally, we note the behavior of the pDirl and

pDir2 solutions in l?ig.1. The PDir. solution

starts out slightly above the pDi SOlutiOnp but for

tD>102 these solutions become equivalent, as

predicted by Eq. 14. The PDir2 solution starts

out slightly below it long time value of 1.0; but

reachea this value for tD>102. This long time

-behavior of pDir2=l.0 is predicted by Eq. 15.

4 Type Curve Analysis Using the pressure Integral Method SPE 1879

The major conclusion that is reached from Fig.1 is that the analytical solutions for radial flow,given in Eqns. 10-15, are verified. This meansthat the solutions we propose for the type curveanalysis of radial flow data are valid. We willlook at the radial flow behavior of the infiniteconductivity and uniform flux vertical fracturecases later in this section.

~:

The type curve solutions for tk,i~case arepresented in Figs. 2,3 and 4. These type.curveswere generated from the ‘cylindrical source-

2s>3 curves and fromsolution, Eq. B-9, for the CDe

the infinite conductivity vertically-fractured well

solution, Eq. B-II, for the CDe2sC3 curves. This.

approach was suggested by Gringarten, et als forthe construction of the PD type curve. One

alternative for the curves where .5<CDe2s<3 would

be to use the van Everdingen and Meyer23

solution,Eq. B-10, for near wellbore stimulation. Eq. B-5is the Laplace space relation that was used to addthe effezts of skin and tiellborestorage to thesolutions stated above.

The PDi and pDid solutions are shown in Fig.9Z.. This type curve is analogous to the pD and pDd

..solutions presented by Bourdet, et al.lu Thiscurve is used to analyze in exactly the same manneras the Bourdet, et al solutions. The mostinteresting feature of this figure is the~dispersed” behavior of the pDid Solution. This

behavior doea not allow us to observe a distinctend of wellbore storage distortion or, morepractically, the start of the semilog straightline. However, we have observed a correlation ofthe start of the semilog straight line with’themaximum value of the pDid SOIUtiOn. This

correlation predicts that the start of the semilogstraight line occurs about 1 log cycle in timeafter the maximum value of the pDid Solution is

observed. We have placed a line Indicating thestart of the semilog straight line (radial flow) onFig. 2. The analyst can use this line to betterinterpret data on the semilog plot.

Fig. 3 shows the behavior of the,pDi and PDir.

solutions. This type curve is analogous to the one

presented by Onur and Reynolds16 for the pD

solution and the pD/(2pDd) ratio solution. The

pDirl solution was shown earlier to be equivalent

to the pDi solut$on during radial flow (whicJh

occurs after wellbore storage distortion effects -end) . This behavior is predicted by Eq. 14+and isevident in Fig. 3.

The pDir- solution also has been shown to

exhibit a unique behavior during the wellborestorage dominated flow regime. This is the flowregime that occurs when the pD solution exhibits a

‘unit slope linen behavior. The pLirl solution is

single valued at 0“.5-duringthis flow regimeaccordingto Eq. 33. This means that duringwellbore storage dominated flow the PDirl solution

will appear as a horizontal line a value of 0.5 onthe type curve. This behavior is evident on Fig. 3

for the type curves where CDe2s>10 , This behavior

also exists for the type curves where CDe2s<10, but

occurs somewhat earlier than tD/CD=O.l, which is

the minimum x-axis value.

Fig. 4 shows the behavior of the pDi and pDir2

solutions. Recall that the pDir2 solution

incorporates both the pDid and pDd solutions. The

pDir2 solution was shown earlier to be single-

vdued at 1.0 during radial flow (which occursafter wellbore storage distortion effects end)according to Eq. 15. The PDir2 solution exhibits a

horizontal line on the type curve at a value of1.0. This behavior is evident on Fig. 4 for the

type curves where CDe102s<10 . This behavior also

exists for the type curves where CDe2s>1010, but

this effect occurs later than tD/CD=104, which is

the maximum x-axis value.

The PDir2 solution also has been shown to

exhibit exactly the same behavior as the PDirl

solution during the wellbore storage dominated flowregime according to Eqns. 33 and 34. This meansthat during wellbore storage dominated flow thepDir2 solution will also appear as a horizontal

line on the type curve with a value of 0.5. Thisbehavior is evident on Fig. 4 for the type curves

where CDe2s>10. This behavior also exists for the

2s<10, but occurs somewhattype curves where CDe

earlier than tD/Cp=O.lt which is the minimum x-axis

value.

:

The type curve solutions for this case arepzesented in Figs. 5,6 and 7. These type curveswere generated,using the infinite conductivityvertically-fractured well solution, Eq. B-n, given

by Gringarten, Ramey and Raghavan. 20 Eqns. B-5 andB-14 to B-17 were used to add wellbore storageeffects to the consta:m rate solution, Eq. B-n.The solutions obtained using this method comparedwell with those obtained”by Alagoa, Bourdet and

Ayoub12 using numerical simulation.

The PDi and pDid solutions are shown in Fig.

5. This type curve is analogous to the pD and pDd

solutions presented by Alagoa, Bourdet and Ayoub.12

The pDid solution appears to approach its radial

flow form of 0.5, as given by Eq. 12, ‘or ‘LfD>3”This agrees with previous observations of t“hepD

and pDd Solutions where pseudoradial flow is

estimated to start at about tLfD=3. This type

curve can be used to match test data in exactly thesame way as conventional pressure drop andderivative methods are used in ref. 12.

Fig. 6 shows the behavior of the pDi and pDirl

solutions. This type curve is%;nalogous to the one

presented by Onur and Reynolds for fracturedwells. The PDirl solution was shown earlier to be

equivalent to the PDi solution by Eq. 14 fOr radial

528

I SPE

g—

18799 T.A. Blasingame, J.L. Johnston and W.J. Lee 5

flow (which occurs after wellbore storagedistortion effects end). This behavior is notevident in Fig. 6 because the type curve scale is

‘runcated at ‘LfD-lO” ‘owever’ at ‘DLf-10 all ‘fthe curves are converging to the PDi solution and

Eq. 14 sheuld become valid shortly thereafter.

The PDirl solution during the wellbore storage

dominated flow regime is single-valued at 0.5according to Eq. 33. This means that a horizontalline at 0.5 will be evident on all of the type

Curves ‘here cDf>0, during this flow regime. This

behavior is evident on Fig. 6 for the type curves

‘here cDf>O”l” This behavior also exists for the

type curves where CDf<0.1, but occurs earlier than

-4‘LfD=10

, which is the minimum x-axis value.

The pDirl solution also has been shown to

exhibit behavior unique to fractured wells. Thisbehavior occurs for the formation linear flowregime which occurs in wells with high fractureconductivities. The pDir- solution is single-

valued at 1,0 during this flow regime according toEq. 27. This causes a horizontal line with a valueof 1.0 to appear on the type curve. This behavioris evident on Fig. 6 only for the CDf=O type curve

because this is the only case where the linear flowregime is not distorted by wellbore storage. Thisbehavior ends at approximately tLfD=O.O1, which

agrees well with the value of t~fD=0.016 given by

ref. 20.

Fig . 7 shows the behavi~r of the pDi and pDir2

solutions. The PDir2 solution was shown earlier to

exhibit a horizontal line on the type curve at avalue of 1.0 by Eq. 15. This behavior is not quiteevident on Fig. 7, although all of the type curvesare converging to 1.0 at tLfD=lO, which is the

maximum x-axis value,

The PDir2 solution also has been shown to

exhibit exactly the same behavior as the pDirl

solution during the wellbore storage dominated flowregime according to Eqns. 33 and 34. This resultpredicts a horizontal line at 0.5 for the type

curves ‘here cDf>0 during this flow regime. This

behavior i% evident on Fig. 7 for the type curves

‘here cDf>0”03” This behavior also exists for the

type curves where CDfcO.03, but occurs earlier than

-4‘LfD=10

, which is the minimum x-axis value.

The pDir2 solution alsc has been shown to

exhibit behavior unique to the formation linearflow regime that occurs in fractured Yells. ThepDir2 solution is single-valued at 0.67 during

this flow regime according to Eq. 28. This causesa horizontal line with a value of 0.67 on the typecurve. This behavior is evident on Fig. 7 only forthe CDf-O type curve because this is the only caae

where the linear flow regime is not distorted bywellbore storage. This behavior also ends atapproximately tLfD=O.Olf which agrees well with the

value ‘f ‘LfD-0..016given by ref. 20.

The type curve solutions for this case arepresented in Figs. 8,9 and 10. These type curvesolutions were generated in exactly the same manneraa those for the infinite conductivity vertically-fractured well caae. In fact; there are onlysubtle differences.in the infinite conducti~?ity anduniform flux solutions. The choice of which ofthese models to use in analysia is generally basedon experience in a particular region, rather thanzigorous theoretical considerations.

There are no major differences in the behaviorof the pDi and pDid solutions for the infinite

conductivity and uniform flux cases. The infiniteconductivity solution. is shown in Fig. 5 and theuniform flux solution is shown in Fig. 8.

There is a subtle difference in the pDirl

solutions for the infinite conductivity and uniformflux cases shown in ‘Lgs. 6 and 9, respectively.In general, the pDirl solutions for the uniform

flux case are less than the correspmding solutionsfor the infinite conductivity case, for the samedimensionless wellbore storage coefficient, CDf.

Also, for the two solutions with no wellborestorage, the uniform flux solution exhibits a muchlonger formation linear flow period. This isexpected because the formation linear flow solutiongiven by Eq. 12 is the early time solution for theuniform flux case and is only an approximation for

the infinite conductivity case.20 The end of

formation linear flow from Fig. 9 occurs atapproximately t~fD=O.l for the uniform flux case.

The end of linear flow from ref. 20 is estimated to

‘cCur at ‘LfD-0”16’

The only significant difference in the PDir2

solutions for the infinite conductivity end uniformflux cases, shown in Figs. 7 and 10t reSPSCtiVSJYe

concerns the duration of linear flow. Again, theuniform flux solution exhibits a Wuch lo:lgerformation linear flow period. The end of fOrmStiOnlinear flow from Fig. 10 is approximately at

‘LfD=0.1 for the uniform flux case, which again

compares we”llthe estimate of tLfD=0.16 fro; ref.

20.

The type curve solutions for this case arepresented in Figs. 11, 12 and 13. These typeCurves were generated using the SolutiOn of

Rodriguez, Home and Cinco-Ley6 for,a fullypenetrating fracture of finite conductivity.

The PDi and pDid solutions for this case are

shown in Fig. 11. The solutions on this type curveexperience several flow regimes over time. Thefirst flow regime encountered is that of fracture-formation bilinear flow. Eqns. 18 and 19 predictthat PDi and pDid are fractional multiples of PD

during bilinear flow. This means that pD, pDi and

PDid are parallel during the bilinear flow regime.

The end of bilinear flow is indicated on thiS typecurve.

The next flow regime encountered is that offormation linear flow. This flow regti only

occurs ‘or cfD>lOO” Eqns. 24 ●nd 2S prediot tbet

.Da

6 Type Curve Analysis Using the Pressure Integral MethodS!?E 18799

PDi and pDLd are alao fractional multiples of pn l~QF~during linear flow. This tceansthat pD, pDi and

pDid are also parallel during the linear flOWregime. From Fig. 11, the end of linear flowappears to occur at about t~fD-o.015. This agrees

20with previous observations which predict the endof linear flow at tLfD=0.016 for an infinite

conductivity vertical fracture.

Fig. 12 shows the behavior of the pDLrl

solution. This type curve is analogous to the one

presented by Onur and Reynolds 16 for fracturedwalls . This curve is plotted on this scale becausethe PDirl solution only ranges from 1 to 10.

During bilinear flow the pDirl solution is single-

valued at 2.0 according to Eq. 21. This behavioris exhibited in Fig. 12 by all but the very lowestand highest CfD cases. The very low CfD case

exhibits bilinear flow only up to tLfD-5x10-5.

This means that we should not expect to see pDirl

exhibit a single value in the tLfD -10-5 range,

which is the mhimum tLfD value on the type curve.

The very high conductivity cases {CfD>lOO) only

exhibit bilinear flow for tLfD<10

‘5, which b off-

scale on the type curve.

During linear flow the pDirl solution is

single-valued at l.O.according to Eq. 27. Thisbehavior is exhibited in Fig. 12 only by theLnfinite conductivity case. .Ref. 20 ahowed thatthe linear flow solution is a valid approximationfor t,heinfinite conductivity case at early times.The pDirl solution also exhibits a unique behavior

during the radial flow regime. In this case, thePDi and PDirl solutions are equal accordingto Eq.14. Though t:.+pDi solution is not sho~ on Fig.

12, this behavior does occur.

Fig. 13 shows the behavior of the,pDir2

6olutSon. This curve was plotted on a semilogscale because the pDir2 solution only varies

between 0.67 and 1.0. The PDir2 solution was shown

earlier to exhibit a horizontal line on the typecurve at a value of 1.0 by Eq. 15. This beha~ioris evident on Fig. 13, but only for very large

t~mes (tLfD>103). This radial flow behavior

~stek,lishesthe upper limit of the pDir2 Solution.

The ~Dir2 solution aLso exhibits unique

behavior for the linear flow regime. During linearE1OW the pD”ir2 solution is single-valued at 0.67

according to Eq. 28. This behavior is exhibited inFig. 13 only by the infinite conductivity case andestablishes the lower limit of the pDir2 solution.

During bilinear flow the pDir2 solution is single-

valued ●t 0.8 according to Eq. 22. This behavioris exhibited in Fig. 13 by most of the curves withthe exceptions again being the very lowest andhighest fracture conductivity cases. Thejustification for these exceptions was givenearlier for pDirl during bilinear flow and this

reasoning still holds for pDir2.

In this section we will analyze the behaviorof a simulated well test using the conventionalpressure derivative solution, pDd, and the new

pressure integral difference solution, pnid. The

purpose of this section is to verify the use of thepressure integral solution for type curve analysis.We intend to achieve this verification by applyingthe new methods of pressuze integral analysis to acase that can not be analyzed using conventionalpressure derivative analyais.

In order to illustrate the advantagea of thepDid solution, we have added 5% random error to the

logarithm of the pD solution for a homogeneous

reservoir with CDe2s=103. We will use both Eqns. 7

and 7a to compute the pDid solution so that the

most practical equation for application to fielddata can be determined. All of the derivativesrequired in this section were computed using themethod proposed in ref. 13.

Fig. 14 shows the behavior of the pDd solution

for 5% error in the logarithm of PD. Note the

extremely erratic behavior of the calculated pDd

solution compared with the analytical pDd solution.

It h conceivable that several matches of thecalculated solution could be obtained and even ifwe consider an average range of scatter, we stillaee one-half of a log cycle variation on the y-axis. Analysis of this data using the pDd solution

could easily yield as much as 100% error in the y-axis match.

The pDid solution computed using Eq. 7 is

shown in Fig. 15. Eq. 7 calculates pDid as a

difference of pD and Pn{d. This means that if PD

is noisy (as it is in t~is example) and pDi is

smooth (from integration it will be at leastsmoother than pD), then the resultin9 pDid sohwion

will still reflect the noise in p-. This is quite

obvious in Fig. 15. However, the-pDid solution

computed using Eq. 7 and shown in Fig. 15 is muchsmoother than the PDd solution in Fi9. 14. Also,

the error in the computed pDid solution tends to

oscillate gently about the analytical solution forpDid* whfch was not true for the pDd solution.

The pDid solution computed using Eq. 7a is “

shown in Fig. 16. Eq. 7a calculates pDid using the

derivative of the integral solution pDi. We note

that the comparison of pDid computed using Eq. 7a

agrees very well with the analytical pDid SOlutiOn.

Fig. 16 not only juitifies the computation of thepDid solution using Eq. 7a, but the excellent

agreement between the calculated and analyticalpDid solutions forms the basis for performing

integral type curve analysia. Based on w~at isobserved in Fig. 16,”we recommend that pDid be

computed using Eq. 7a.

I---

SPE 18799 T.A. Blasingame, J.L. Johnston and W.J. Lee

The most important conclusion in this sectionand perhaps in this paper is that accurate analysisof noisy well rest data is possible using integraltype curve analysis. This means that data gatheredfrom relatively low precision devices such asbourdon tube pressure bombs and acoustic wellsounders may be analyzed with the accuracy andresolution only thought possible for high precision?.eviceasuch as quartz crystal gauges.

In this section we will describe how to usethe new type curve solutions. The objective ofthis section is to outline a procedure to applythese new type curves, including recommendations onhow to obtain the pressure drop derivative andintegral functions.

The following general approach is proposed toa?ply the field well test data to the new typecurves:

1. Calculate the pressure drop function, dpw,

and the time function, t:

a. drawdowni. dpw = pi - pwf

ii,t=tb. buildup

i. dpw = pws - pwf

ii. t = dte = dt/(1 + dt/tp)

2. Calculate the pressure drop derivativefunction, dpw,d, using numerical

techniques such as the Clark and Van

Golf-Racht13

method, finite differenceapproximations or least squares piecewisepolynomials. The formula for dpw,d is

ddpw,d = t —(dpw)

dt

3. Calculate the pressure drop integralfunction, dpw,i, using numerical

integration techniques such as thetrapezoidal rule or the new logarithmicintegration technique developed inAppendix A. ~he formula for dpw,i is

dpw, i = ~ ~tdpwdt

to

4. Calculate the pressure drop integraldifference function, dpw,id, using the

derivative techniques outlined in step 2.

The formula for-dpw,id is

d

‘PW, id = t -(alpW,i)

dt

5. Calculate the pressure integral ratiofunctions, PDirl and PDLr2, as follows:

‘%, ia. PDi=l = —

2dpW,id

WI

‘PW, id

b. ‘Dir2.—

‘pw,d

6. Make the following 3-inch log cycle plotsusing the time function, t, as prescribedin step 1:

a. dpw and dpw,d vs. t.

b. dpwti and dpw,id VS. t.

c. dpw,i and pDirl vs. t.

d. dpw,i andpf)~r2 vs. t.

7. Obtain the pressure and time match pointsfrom the type curves and interpret asfollows:

a. Pressure match point

qB~ PDM

k = 141.2 — (— ‘M.P.

h dpwM

M is dpw orwhere pDM is pD or pDi and dpw

dpwi .

b. Time match point

i. Wellbore storage and skin case

k tCD = 0.0002637 — (— ‘M.P..

@Lctrw’ tD/CD

ii. Fractured well case

k t

‘f2 = 0.0002637 — (— ‘M.P.

$Pct ‘DLf

8. Obtain the type2~rve family parameter

values (eg. CDe‘ cDf and cfD) from the

data match. In particular, “theskin factor, ~, for the homogeneousreservoir case is calculated from the

CDe2s parameter as follows:

‘1

s-- ~n(c e2s,c ~D D

2

In this section we will apply the new type“curves to a field example. This example is for apressure buildup test performed on a well In aninfinite-acting homogeneous reservoir.

The data for this example are from Bourdet, et

do where the data were analyzed using pre.asuredrop and pressure drop derivative analysis. Thedata are distorted by wellbore storage, but alsoexhibit a semilog straight line (radial flow) afte~about 4 hours. The reservoir data and results fronprevious analyses are given in Table 1.

14

8

—

Type Curve Analysis Using the Pressure Integral Method-...

The pD ~tyle type curve graph fOr this case is

shown in Fig. 17. The type curves on this graphare for an infinite-acting homogeneous reservoirwith wellbore storage and skin effects included.Both the conventional type curve plottingfunctions, pD and pDd, are shown as well as the new

plotting functions, pDi and pDid. The type curve

matching results obtained uSin9 the PDi and pDid

vs. tD/CD type curves are:

CDe2s = 5X109 (by interpolation)

tD/cD 5 10.Q..

t = 0.70 hrPDi = 1.0

dpw,i = 57 psi

The pressure match point is solved forpermeabil~ty. This calculation is

k= 141.2

= 141.2

- 10.68

WW PDi

— (—)Mp

h dpw “ “

(174)(1.06)(2.5) 1.0(—)

(107) 57

md.

The time match point is solved for thedimensionless wellbore storage coefficient,

This calculation is

k tCD = 0.0002637 — (— ‘M.P.

4ylctEw2 tnlcn

CD “

(10.68) 0.70- 0.0002637 (—)

(0.25)(2.5)(4.2x10-6)(0.29)2 10.0

- 893

The family curve for this case, cDe2s, can be

solved for the skin factor, S, using the estimateof the dimensionless wellbore storage coefficientthat we just obtained. This calculation is

1 ,.,.

~ = - ln(CDe2s/CD)

2

1

= - ln(5x10g/893)2

= 7.77

The results of this analysis cornp~~ very well with

those obtained by Bourdet, et al.

The type curve graph for the PDirl and PDir2

plotting functions is shown i.nFig. 18. The typecurves on this graph are for the same systemdescribed earlier. Note the excellent agreementbetween the pDirl data and type curves. The

probable reason that the early pDirl data does not

SPE 18”;99‘1

lie exactly on 0,5, as prescribed by Eq. 33, islikely due to the numerical differentiation of thePDi data using the method in ref. 13. Perhaps more

robust differentiation methods are necessary, butwe believe that the excellent match of the laterpDirl data verifies the utility of this plotting

function.

Note also that the PDir2 data behaves somewhat

erratically, as we would expect from observing thepDd data on Fig. 170 However, we still obtain a

good match of the PDir2 data and type curve. We

observe that pDir2 is reasonably close to 0.5, as

predicted.by Eq. 34, during wellbore storagedominated flow and we aee the that pDir2 bends over

and starts into radial flow at about tD/CD-75.

This example at least verifies the use of pD1r2 as

a plotting function for field data. At most, itprovides us with a high resolution plottingZunction to identify characteristics uniaue to thebehavior of a parti~ular flow regime. -

The pDir. and PDir2 functions can also be used

quantitatively to estimate the properties of an x-axis match (eg. tD/CD) and the type curve family

-2s-parameter (eg. Cne ) just like conventional type

curve analysis. ‘Since pDirl and pDir2 are

dimensionless ratios that m the y-axis, there isno quantitative analysis of the y-axis. In termsof the current example, the exact same matchingparameters as the pD and Pn{ analysis were obtained

and the analysis of-these ~&ameters would beredundant.

A practical alternative type curve couldincorporate pDir. and pDir2 separately with pDi.

Then pDir. and PDir2 could be used to refine the

pDi data match in the asme way that pDd is

currently used to refine a pD data match. This iS

our justification for the format of the type curvespresented in ~he ‘Development of the New TypeCurves* section.

In this paper we have introduced new plottingfunctions for the type curve analyais of transientwell test data in homogeneous and vertically-fractured wells. The plotting functions weredeveloped to smooth the test data by integration.These new type curves can be applied to data frompressure buildup tests if the equivalent time

introduced by Agarwa18 is used.

The.results in the paper indicate that the newintegral plotting functions could be applied in anautomatic history matching scheme for noisy testdata. We also believe that the integral functionscould be applied successfully to the analysis ofpost-transient well test data.

The results of this study give the followingconclusions.

1. The new type curve integral differenceplotting function gave a much smoother datacurve and more unique type curve match thanthe conventional pressure derivative datacurve for an example set of noisy teut data.

SPE 18799 ‘T.A. Blasingame, J.L. Johnston and W.J. Lee 9

2. Two new dimensionless ratio plottingfunctions were introduced and verified toexist on type curve solutions for thebilinear, linear, radial and wellborestorage dominate< flow regimes.

CD -

cDf =

cfD -P~ =

pDd “

P~i =

pDid -

P~irl=

‘D~r2=

PDO =

‘D =

‘LfD =

Dimensionless wellbore storage

coefficientDimensionless wellbore storage

coefficient based on f’racturehalf-lengthDimensionless fracture conductivity

Diifiensionlesspressure defined by Eq. 1

Dimensionless pressure derivative

function defined by Eqns. 4 and 5Dimensionless pressure integral function

defined by Eq. 6Dimensionless pressure integral

difference function defined by Eqns. 1 andlbFirst dimensionless pressure integral

ratio defined by Eq. 8Second dimensionless pressur% integral

ratio defined by Eq. 9Dimensionless phase redistribution

pressureDimensionless time defined by Eq. 2

Dimensionless time based on fracture

half-length as defined by Eq. 3

*1~ VariaMss

a ,=

b=

Ct =f(x) =

h=k=

‘f =Pi =

Pwf -

P-

:W -rw =

:-

‘P -dt =

dte -

x=

-+ =

P-

Arbitrary constant used in variousequationsArbitrary constant used in variousequations

Total compressibility, psia-1

Arbitrary function used in the derivationof the logarithmic integral formula,Eq. C-2Reservoir net pay thickness, ftReservoir permeability, mdFracture half-length, ft

Initial reservoir pressure, psia

Flowing bottomhole pressure, psia

Shut-In bottomhole pre8sure, psia

Wellbore pressure drop, psi

Uellbore radius, ft

Laplace transform parameterTime, hrTOtal production time, hr

Shut-In time, hr

Agarwa18 effective time function, hr

Arbitrary independent axis variable usedto derive the logarithmic integrationformula

Reservoir porosity, fraction

Fluid ViSCOSity, Cp

d = Derivative functionD = Dimensionlessi = Integral functionid = Integral difference functionWf = Flowing yell conditionswa = Shut-in well conditions

1.

2.

3.

4.

5.

6.

7.

8.

9..-

10.

11.

12.

1I1

van Everdingen, A.F. and Hurst, W.: ‘iTheapplication of the Laplace Transformation toFlow Problems in Reservoirs,n Trans., AIMS(1949) 186, 305-324.

Agarwal, R.G., A1-Hussainy, R., and Ramey, H.J.Jr.: ‘An Investigation of Wellbore Storage andSkin Effects in Unsteady Liquid Flow: I.Analytical Treatment,” SPEJ (Sept. 1970) 278-290.

Gringarten, A.C. et al: “A Comparison BetweenDifferent Skin and Wellbore Storage Type Curvesfor Early-Time Transient Analysis,” paper SPE8205 presented at the 1979 SPE AnnualTechnical Conference and Exhibition, Las Vegas,Sept. 23-26.

Cinco-Ley, H., Samaniego-V., F. and Dominguez-A., N. : ‘Transient Pressure Behavior for a WellWith A Finite-Conductivity Vertical Fracturet”SPEJ (Aug. 1978) 253-264.

Cinco-Ley, H. and Samani.ego-V., F.: “TransientPressure Analysis for Fractured Wells,” JPT(Sept. 1981) 1749-66.

Rodriguezt F., Home, R.N. and Cinco-Ley, H.:“Partially Penetrating Vertical Fractures:Pressure Transient Behavior of a Finite-;onductivity Fracture, ‘1paper SPE 13057presented at the 1984 SPE Annual TechnicalConference and Exhibition, Houston, Sept. 16-19, 1984.

Bourdet, D. and Gringarten, A.C.: “Determin-ation of Fissure Volume and Block Size in (Fractured Reservoirs by Type-Curve Analysis,”paper SPE 9293 presented at the 1980 SPE AnnualI’ethnicalConference and Exhibition, Dallas,Sept. 21-24.

kgarwal, R.G.: “A New Method to Account forProducing Time Effects When Drawdown Type:urves Are Used to Analyze Pressure Buildup and>ther Test Data,” paper SPE 9289 presented atthe 1980 SPE Annual Technical Conference andExhibition, Dallae, Sept. 21-24.

Fair, W.B.: ‘Pressure Buildup Amalysis WithWellbore Phase Redistribution,m SPEJ (April1981) 259-270.

Bourdet, D. et al: “A New Set of Type CurvesSimplifies Well Test Analysis,” World 011(my 1983) 95-106.

Bourdet, D., Ay~ab, J.A. and Pirard, Y.M.:*Use of Pressure Derivative in Well Teathterpretation ,n paper SPE 127,77piesentedat the 1984 California Regional MeetingtLong Beach, April 11-13.

Alagoa, A:, Bourdet, D. and Ayoub, J.A.: “Howto Simplify The Analysis of Fractured WellTests,” World Oil (Oct. 1985).

w

13. Clark, D.G. and Van Golf-Racht, T,D.:‘Pressure-Derivative Approach to Transient TeatAnalyais: A High-Permeability North SeaReservoir Example,” JPT (Nov. 1985) 2023-2039.

14. Puthigai, S.K, and Tiab, D.: !rApplicationofP’D Function to Vertically Fractured Wells-Field Cases,” paper SPE 11028 presented at the1982 SPE Annual Technical Conference andExhibition, New Orleana, Sept. 26-29.

15. Tiab, S. and Puthigai, S.K.: “Pressure-Derivative Type Curves for Vertically FracturedWells,” SPEFE (March 1988) 156-158.

In this section we will”prove that theintegral difference dimensionless variable, pDid#

given by Eqns. 7 and 7a are identicalmathematically. This proof requires that wecombine the pDi and pDid definitions given by EqnS.

6 and ?a and solve for the derivative term in Eq.7a. This gives -

16. Onur, M. and Reynolds, A.C.: “A New Approach d dlfor Constructing Derivative Type Curves for

‘D

Well Test Analysis,” SFEFE (March 1988) 197-— (PDi) - ‘[— ~ PD(T) dTl

206. dtD‘tD ‘D “0

17. Onur, M,t Yeht N-S. and Reynolds, A.C.: ‘tNew pD 1Derivative Type Curves for Well Test Analysis,”paper SPE 1681O presented at the 1987 SPE

.—-— ~t\D(T) dT

Annual Technical Conference and Exhibition, 20Dallas, 27-30. ‘D ‘D

18. Duong, A.N.: 11ANew Set of Type Curves for 1Well Test Interpretation Using the Pressure = — [pD ‘Ljt\D(T) dT]Derivative Ratio,” paper SPE 16812 presented atthe 1987 SPE Annual Technical Conference and ‘D ‘D 0Exhibition, Dallas, Sept. 27-30.

119. Earlougher, R.C. Jr.: Advances in Well Test = — [PD - PDil ..................(A-l)

Analysis, Henry L. Doherty Series, SP!?,Richardson, TX (1977) 5. ‘D

20. Gringarten, A.C., Remey, H.J. Jr. and If we multiply Eq. A-1 by tD we obtainRaghavan, R: ‘Unsteady-State PressureDistributions Created by a.Well With a Single dInfinite-Conductivity Vertical Fracture,” SPEJ(Aug. 1974) 347-360. ‘D ‘(pDi) = [pD - PD~] ..................(A-2)

dt.21. Lee, W.J.: Well Testing, Textbook Series, SPE,

u

Richardson, TX .(1982) 1. which proves that Eqns. 7 and 7a are exactly

22. Stehfest, H: ‘Numerical Inversion of Laplaceequivalent mathematically.

Transforms,” Communications of the ACM (Jan.1970) 13, No. 1, Algorithm 368.:

23. van Everdingen, A.!?.and Meyer, L.J.:“Analyeis of Buildup Curves-Obtained After WellTreatment,t~ JPT (April 1971) 513-524; Trans.?AIWE, 2S1.

24. Guillot, A,Y. and Home, R.N.: “UsingSimultaneous Downhole Flow-Rate and PressureMeasurements To Improve Analysis of WellTests,” SPEFE (June 1986) 217-226.

25. Roumboutsos, A. and Stewart, G.: ‘A DirectDeconvolution or Convolution Algorithm for WellTeat Analysis,” paper SPE 18157 presented atthe 19138SPE Annual Technical Conference andExhibition, Houston, Oct. 2-5.

26. Hornbeck, R.W.: Numerical Methods, Prent+ceHall/Quantum Pub., Englewood Cliffs, NJ (1975).

The solution which models radial flow is19

pD=a [ln(tD) +bl +S ...................(A-3)

‘here ‘LfD is substituted for tD for the fractured

well cases. The coefficients in Eq. 10 are definedas follows:

a = 0.5 for all caaes

Homogeneous19 0.80907 25

Infinite Conductivity 20 2.20000 10

Uniform Flux20 2.80907 10

The dimensionless pressure integral funCtiOn, pDi,

for this case ie derived by combining Eqns. 6 and-A-3 . This function is

I Pf)i=a [ln(tD) +b-1] +S ..............(A-4)

The dimensionless pressure integral differencefunction, pDid, is obtained by subtracting Eq. 4-4

I

63”

.-

SPE 18799 T.A. Blasingame, J.L. Johnston and w..J. Lee

from Eq. A-3, as prescribed by Eq. 7. This resultis - b Poi

b

pDid = a = 0.5 (A-5) “—P~ . . . . . , . , . . . . . . . . . . . . . . . . . . . (A-n).............,......,....,..

b+l

The dimensionless pressure derivativefunction, pDd, is derived by combining Eqns. 5 and

where pD iS defined by Eq. A-9 and pDi is defined

A-3 . This result isby Eq. A-10.

The dimensionless pressure derivative, pDdt is‘Dd - a = 0.5 ...................,......,..(A-6)

derived by combining Eqns. 5 and A-9. This result

The dimensionless ratios defined in Eqns. 8is

and 9 are derived now for radial flow. The first bratiov PDirl$ is obtained by combining Eqns. 8, A-4 ‘Dd

=abtD

and A-5. This result is =bp D ........,..............,......(A-12)

‘Dirl = pDi ......,.,....,.................(A-7) where pD is defined by Eq. A-9.

and the second ratio~ PDir2# is obtained by The dimensionless ratios defined by Eqns. 8

combining Eqns. 9, A-5 and A-6. This result ia and 9 are derived below. The first ratio function,pDirl, is obtained by combining Eqns. 8, A-10 and

‘Dir2 - 1. . . . . . . . . , . . . . . . . . . . . . . . . . . . . , . . . (A-8) A-II . This result is

P~~rl = l/2b ............................(A-13).

The second ratio, PDir2, is obtainedby combiningIn this section we will consider the general

form of a power function to model the bilinear andEqns. 9, A-n and A-12. This reeult is

linear flow regimes in a fractured well. The powerfunction model is alao used to develop relations 1

for the wellbore storage dominated flow regime ‘Dir2- — ...........,.................(A-14)

(unit slope line.). This model is b+l

pD - at b (A-9) Note that PDi, PDid and PDd are simply fractionalD

. . . . . . . . . , , . . . . . . . . . . . . . . . . . . . .multiples of pD during these flow regimes. This

‘here ‘LfDis substituted for tD for the fractured means that we would expect all these functions to

well cases. The coefficients in Eq. A-9 are be parallel to one another when plotted on a type

defined as follows: curve.

Q&’ a b Note also that the pDirl and poirz functions

are constant during these flow regimes. This

Bilinear F10W5 2.45083/~ 1/4 results in a horizontal line en, thetype curve for

Linear Flow20’ G.

each of these functions, the values of which depend

112 ‘ solely on the b coefficient tabulated above. These

Wellbore Storage21 horizontal lines can be used as reference lines in

I/cD 1 much the same way that the conventional pressurederivative function, pDd, is currently used to

The following derivations of the plotting functions identify the radial flow regime.using Eq. A-9 will use the a and b coefficientssymbolically.

6

The dimensionless pressure integral , pDi, is

derived by combining Eqns. 6 and A-9. Theresulting function is The development of the new type curves

requires the analytical solution for each reservoir

a system of interest. The most straightforwardapproach is to use the Laplace transform of the

‘Di- — tDb –- solution and an inve~~ion algoti.thmsuch as the one

s b+l proposed by Stehfest This is straightforward1 because the differenti~tion and integration of the

“—P~ .............................(A-IO) dimensionless pressure function, pD, can beb+l performed as multiplication and diviaiont

respectively, of pD in Lsplace space, ,PD(S1, by thewhere pD is defined by Eq. A-9. The dimensionless Laplace parameter, s.pressure integral difference, pDid, is obtained by

subtracting Eq. A-10 from Eq. A-9 *S in Eq. ‘!or Specifically, these relations are

differentiating Eq. A-10 as in Eq. 7a. This resultis d(pD)

ab — - L-l(s PD(S)} .....,................(B-1)

b dtD‘Did - — ‘D

b+~

---

12 Type Curve Analysis Using the Pressure Integral Method SPE 18799

and

‘D1

j pDdtD= L-l{ ‘pD(S)) .................(B-2)

o s

Also, the’effects of wellbore storage andphase redistribution can be added directly to theLaplace transform of the solution. This isdemonstrated in the section below.

Fairg derived a general relationship inLaplace space to add the effects of wellbore8torage, skin and phase redistribution to theconstant rate dimensionless pressure solution, p9D.

This relation is.—

[spgD(s) + s] [1 + CDS2POD(S)I

pD(s) = ....(B-3)

S[l + CDS(SPSD(S) + s)]

where pD(s) is the Laplace transform of the

dimensionless pressure distorted by wellborestorage, skin andlor phase redistribution. In Eq.B-3, PSD(S) is the Laplace transfozmof the

constant rate solution with no distorting effects.When none of these effects are present Eq. (B-3)reduces to the trivial form

pD (s) = PSD(S) ...................,........(B-4)

In this work the effect of phasereciistribution is not considered (POD(S)-O). Eq.

B-3 then reduces to the form

[spSD(s) + s]

PD(S) = . . . . . . . . . ..(B-5)

S[l + CDS(SPSD(S) + s)]

Eq. B-5 was used in this work to generate the typecurves which have wellbore storage and skineffects.

The solution for an infinite-acting

homogeneous reservoir iS1g

.‘Dz

psD =-Ei(-—)+S .....................(B-6)

4tD

Eq. (B-6) is known as the “continuous line source”

solution and is valid for tD/rD2 > 25, The so

called “log approximation to Eq. (B-6) is19

1

Psn = - [ln(tD/r2D) + 0.80907] + S ........(B-7)

2

Eq. B-7 is valid for”tD/r2D > 100 or with less than

1* ● rror for tD/r2D > 10.

The Laplace space solutions for a well in aninfinite homogeneous reservcir are given below.

,Line Source Solution:

This is the Laplace transform of Eq. B-6 and~al,2

1PsD(s) = - ko(~) .......................(B-8)

s

Cylindrical Source Solution:

This is the Laplace space solution for a well

with a finite radius wellbore. This solution isl

ps.(s) = - - . . . . . . . . . . . . . . . . . ..(B-9)

s3/2kl(~)

Eq. B-9 degenerates into Eq. B-8 for large times(ie, as s->0). Eqns. B-5 and B-9 were used togenerate type curves (Figs. 2-4) for thehomogeneous reservoir case

Matrix-Acid Solution:

This solution was proposed by van Everdingen*.

and Meyer&a to represent a well that had beenstimulated with acid in the near weUbore region.The solution is applicable to wells with negativeskin factors in the O to -3 range. The Laplacespace solution is

PSD(S) - ..(B-1O)

S3’2 kl(~)) + S2 ko(fi)/2

Eq. B-10 also degenerates into Eq. B-8 for largetimes.

Gringarten, Ramey and Raghavan20

developed asolution for the dimensionless pressure, Pn, at the

wellbore of a vertically fractured well in-aninfinite homogeneous reservoir. The fracture isconsidered to have either infinite conductivity (nopressure drop along the fracture from the fracturetip to the wellbore) or uniform flux across thefracture face along the entire length of thefracture, The uniform flux case has a variablepressure drop along the fracture antiappears tobehave as a variab$e:condpctivity verticalfracture.The real space solution is

1 l-XD

P~~ =-- [erf(_ )

2 2-

1+xD

+ erf( )1

2-

..-

SPE 18799 T.A. Blasingame, J.L. Johnston and W.J. Lee 13

I-xD -(l-xD)z

- (----) Ei(---------)

44tLfD

l+XD -(1+XD)2

- (----) Ei(---------) ............(B-11)

44tLfD

where XD = O for the uniform flux case and XD =

0.732 for the infinite conductivity case. We havenot obtained the analytical Laplace transform ofEq. B-n and have no evidence that it exists inclosed form.

Since Eq. B-n was derived for an infinite,homogeneous reservoir, the solution does eventuallyexhibit a radial-like or ‘pseudoradial” flowregime. For pseudoradial flow, Eq. B-n reduces tothe equation of a semilog straight-line, analogousto the infinite-acting homogeneous reservoir case.The form of the equations are the same for theinfinite conductivity and uniform flux verticalfracture cases, but the coefficients of theseequations are different. The pseudoradial flow

20relation for the infinite conductivity case is

1PsD ‘,- [ln(tLfD) + 2.200001 .............(B-12)

2

The pseudoradial flow relation for the uniform flux

case is20

1psD = - [ln(t~fD) + 2.80907] .............(B-13)

2

Rodriguez, Home and Cinco-Ley6 developed asolution for a partially penetrating, finiteconductivity vertical fracture in an infinitehomogeneous reservoir. This solution can be usedto predict the behavior of a fully penetrating .finite conductivity vertical fracture withoutwellbore storage effects. The solution is verytedious and is not repeated here. Figs . 11-13 aretype curves based on this solution.

The problem addresaed in this section is toinclude wellbore storage in the solutions for theinfinite conductivity and uniform flux verticalfracture cases. However, since closed form Laplacetransforms of these solutions are not available, wemust somehow bring the constant rate solutions intoLaplace apace. This can be accomplished bytabulating the constant rate solution and bringingthese tabulated data into Laplace space.

Gulliot and Horne24 25and Roumboutsos and Stewartproposed methods to take the Laplace transform oftabulated data. Although these methods aresomewhat similar in both design and implementation,the Roumboutsos and Stewart method appears to bemore robust.

Roumboutsos and Stewart developed a formula totake the Laplace transform of tabulated data thatassumes that the data are piecewise linear. TheRoumboutsos and Stewart formula is

1 -St1

n -Sti-l -Sti

PSD(S) = - [ml (l-e ) + Zmi(e -e )

92i= 2

-Stn- ~

+mne 1 . . . . . . . . . . . . . . . . . . . . (B-14)

where

‘sD(tD,i) - psD(tD,i-l)

‘i =.............(B-15)

‘D,i - ‘D,i-l

to~o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-16)

and

tn ...................................(B-17)~oa

Eq. B-17 is used only to derive Eq. B-14 and not tocalculate the coefficient in Eq. B-15. Also, Eq.

B-14 requires the use of a numerical inversion

algorithm such as the one proposed by Stehfest.22

This method appears to have two nume~icalinstabilities. First, the accuracy of the resultsdepends somewhat on how finely the data table isdivided. We found that 30 to 40 points per logcycle of data gave results which compare well withthe analytical solution for the test cage dihcussedbelow. Second, regardless of the data gridrefinement, the method oscillates at large times.However, the severity of the oscillations doesappear to depend on the grid refinement.

We compared this method to the analyticalsolution for a homogeneous reservoir with wellborestorage. Eq. B-6 was used t.ogenerated thetabulated data for the l?oumboutsos and Stewartformula, Eq. B-14. The analytical solution wasgenerated using Eqns. B-5 and B-8. We compared theRoumboutsos and Stewart formula and the analyticalsolution, analyzed the error, and have presentedthe results in Fig. 19. Note that all of theerrors in the calculation of the dimensionlesspressure, pD, are less than 1%. Similar behavior

is also noted for the dimensionless pressureintegral function, pDid, on this figure. However,

we note that the dimensionless pressure derivativefunction, pDd, behaves very poorly for tD/CD>20.

The poor analytical performance of PDd forced

us to evaluate this function numerically, using theinverted pD solution. The error analyses for this

case are shown in Fig. 20. Also included in Fig.20 is the pDid function calculated usin9 numerical

integration. The error in both the pDd and pDid

functions was reduced significantly by evaluatingthe derivative and integral functions numerically.This leads us to recommend the that derivatives andintegrals be evaluated numerically rather than fromthe Laplace inversion solution when the Roumboutsosand Stewart method is used.

Since the objective of this section was tofind a method to add wellbore storage to theconstant rate solutions for the infiniteconductivity and uniform flux vertically-fracturedwell cases, we believe that the error analysis forthe homogeneous case verifies the use of theRoumboutsos and Stewart method to include wellborestoraae for these cases.

6s7

:

14 Type Curve Analysis Using the Pressure Integral Method SPE 18799

- Specifically, Eqns. B-5, B-II and B-14 to B-17 Combining Eqns, C-3 and C-5 giveswere used to generate the type curves for theinfinite conductivity and uniform flux fracture Xn 1

cases. The type curves for the infinite ~f(x)dx =;conductivity case are shown in Figs. 5-7.

---- (f(xi)xi - f(xi-l)xi-l) (C-6)The type

curves for the uniform flux case are shown in Figs. ‘1 i-l bi+l

8-10. The results obtained for these casescompared favorably with the results Alagoa, Bourdet For a piecewise log-linear functionaland Ayoub 12 generated using numerical simulation. approximation, the bi coefficient is evaluated as

follows

FOR ~ 1. Set up equations for the partition~.~

log f(xi) E bi log Xi + log ai

Hornbe.ck26 gives a simple derivation of the .......(C-7)trapezoidal rule which can be used to integrate 109 f(xi-l) = bi 109 Xi-l + lo9 aiunevenly spaced data. This result is

Xn2. Solve for b.

in 1

jf(x) dx=- Z(xi-xi-l) (f(xi)+f(xi-l)) ..(c-l) lo9(Yi/Yi-1)

‘1 2 i=l bi = ....................(c-8)

lo9(xi/xi-1)where

For the first partition (xl-xO), we can estimate bj

‘0=0 ............. .....................(C-la)using the following strategies

f(xo) = o ............,...,...............(C-lb)a. set bl = b2

We will now develop a log space analog to the b. obtain bl from the first few points using

trapezoidal rule. This relation will integrate the least squares.data assuming a piecewise log linear behavior ofthe data. The logarithmic integration will more Or, we can just calculate the first partial sum,accurately represent data which extends over

11’using the trapezoidal rule rather than the log

several log cycles, as well test data usually does. integration formula, Eq. c-6.

We will start this derivation by consideringthe logarithmic or “pow$r law” function torepresent the piecewise nature of the data. Thisfunction is given as

f(x) = axb .,,........................,.,,.(c-2)

If we consider our data to be piecewise and we Table 1

intend to integrate piece by piece, we note thatReservoirPropertiesfor the Bourdet,et al

10Example

Xn

~f(x) dx= : Ii Reservoirporosity,!$ 0.25 (fraction)Reservoir net pay, h

‘1 i=l107 ft.

Reservoir permeability, k ● 10.78 md.Skin factor, S + 7.7

n Wellbore radius, rw‘i

0.29 ft.

. z J f(x) ...................(C-3)Dimensionless Wellbore Storage

Coefficient ,CD 879i=l ‘i-1 Total Compress iblity, Ct 4.2 x 10-6 psia-l

orOil FVF, B. 1.06 RB/STB

oil viscosity,&o

jx’i(X) dx

2.5 Cp

Ii = .......,. ..................4)-4) Oii production rate, q 174 STB/DTotal production time, t

‘i-1P

15.33 hr

* Permeability value is average of semilog and

:ombining Eqns. C-2 and C-4 gives convent io,ml type curve analysis.+ Skin factor value is from convention’1 type

curve analysis.

Ii = ai ~x~b~x

‘i-1

1=— (f(xi)xi - f(xi-l)xi-l) ..:......(C-s)

bi+l

...

= 18799

II

.—---L--.-l------= “’ 1

ma mrnd fwr-4a01d emmazd SaeIUOT=UaTO

I

&

-i -~ Ts

CUOTWJ.d bulaaol.aembwsa SC.WJOICU.WTO

—-----F-=

SRE 18799

-=

.

.

.

-=

.

Ma

10●

10‘

,.4

i

1/‘Dirl - 0’5

WM - ‘Fig. 9 - NOW Poi and Poirl TYPUcurve solut~on~ ‘or

Iim,cgeneous Reservoirs with a UnMomFlux Wrtical

Fracture ana Nellbore Storage Effects.

‘Ig “ 11- ‘e”% and ppid TW cUWe sol.tion~ ‘oxImm09enemaReservoirs with a Finite ConductivityVerfical Fracture.

“’rrTTm

Fi9. 10 - NCW PDi and PDi=2 TYWJ c.r.e *lvtiOns fOr

HcmwgenrnusReservoirs with a Onifom FIM VerticalFracture and Nellbore storage Effects.

“.~. :. :.s *. a 10” # ~n d w’ ~.

‘LfDFig. 12 - New pDirl Type Curve f-al.tiona for ltomgemaus

Neservoira with a Finite Conductivity Vertical

*

,,

,, ,,

‘“l---=Q93-

030-

Om -

‘s

‘k\~;‘m “-q

on

o.m

0,ssrI

m$-.”.

!

%0.10.51.03.05.Q

;:

lti500

!

“fimice

\

J

.—

‘LfD

Fig. 13 - New pDi=2 TYP@ CUrW sO1ut ions for HomogeneousReservoirs with a Finite Conductivity verticalFracture

3 — Data curve

‘-- TyQeCurve

~z 1:

&

&

8u

~s ,e-l

S*+ , ,,,,, r , , ,,4,,,

la+, , ,,,,,, , , rn’q-

1 la 162 103 1

riq<15 - Typa Cwxe Matchof pDid Calculated Usin.qEq. 7 for

pD Data with 5 percent IWmdofr,Logarithmic Err.az

(Honwgeneovs Reservoir, CDe2s-103).

10

10-2 I1 ,,,, r

10-1,,,,,, r , 8,,,,, I t ,,,,,,

1,

10 Id I❑3

Fig. 14 - Type Curve Match of p f’JZPD Data with 5 PercentRandom Logarithmic Srr% {Homogeneous RemxvOLr,

cDe’5-,03,

la

— Data Curve

‘-- Wpe Curve 1

IIe-z I , , ,,,,,, , , t,,,,,

Id, ,,, ,,,, , ,,, ,,,, ,

1 la d ma‘D/cD

‘ig, 16 - * cur= Mtch of PDid Calculated usin9 Eq. 7e

for PD Data with S percent Random LcqarithMic Error

Uk+uageneous Resez.mxr, C~=ZS-103).

= 18799

0 .. ia

.

cuOTxwn.3 6uwJ01d aanceeid ssetuovuma

“.

‘m.

i“.

lull,,, * 1,,,,, , t 1,,,,, , e * 1

‘m.

m“

J=’..

.

7a.

---

---/‘---

. .

, , ! I“’’i’’’-l’’’’ l’’”.

7 Ym g ?m~mti”:

~‘lam , l“’’I’’’’ 1’’’’ 1’’”1. .: y-7 ‘?m Tm

m~--~.

..&z!