SPESPE 17547

Application of Equivalent Drawdown Time in Well Testingby S.C, Swift, Swift Engineering Co./OGCl

SPE Member

Copyright198S Society of Petroleum Engineers

Th18naper was prepared for presentationat the SPE Rocky MountainRegional Meeting, held in Casper, WY, May 11-13, 19S8.

Th: oar waa selected for preaematlonby an SPE Program Committeefollcwingreview of informationcontainedIn an abstractsubmittedby theauth Contentsof the paper, aa presented, have not been reviewed by the Society of Petrolaum Engineersand are subjectto correctionby theauthor,., The material, aa praemted, does notneceeaarllyreflectany posllionof the Societyof PetroleumEngineers,itaofficara,or members. Paperepresentedat SPE meetingsare tiubjectto publicationreview by EditorialCommifteeaof the SOcletyof Petroleum Englneere. Permlaslonto copy iarestrictedto an abetractof not more than 300 worda. IIluatratlonemay not be wpled. The abstract shouldcontainConaplcuouaacknowledgmentofwhere and by whomthe paper lapresented,Write PubllcationaManager, SPE, P.O. sax 833S3(+,Rlchardeon,TX 75083-3S36. Telex, 730989 SPEDAL.

ABSTRACT The most important tool used for step o,,e is type curveanalysis. The tool best fitted for step two depends on step one.

The concept of equivalent drawdown time was described by Obviously type curves (especially type curve analysis) play aRam Agarwal in 1980. In reality it has been In common use In critical role in drawdown analysis. Examples of tools used forwell test anaiysis since at Ieasf 1950. However, in 1988, step two include semilog plots for radial flow behavior, squareapplication of the concept Is still not widely understood. The root plots for iinear flow behavior, cartesian plots forpurpose of the present paper Is to encourage its application by pseudusteady flow behavior, and type curve matching for moreclarifying its role in well test analysis. complex behaviors such as storage/linear transitions, finite

conductivity fractures and dual flow systems.INTRODUCTION

Equivalent drawdown time was Iiteraily invented to allowDrawdown test analysis is easy. Analysis of tests involving proper utilization of type curves on tests which involve more

more than one rate change is difficult. Equivalent drawdown than one rate change. Although type curves fostered thetime makes the difficult into the easy. creation of equivalent drawdown time, the possibility of much

wider application was immediately recognized. When properly

If that sounds too good to be true, maybe it is, but it’s almost applied, the use of equivalent drawdown time reduces the

truel In the real world it comes out more like this. Drawdown anaiysis of any complex flow to anaiysis of a drawdown test.

test analysis is difficult. Analysis of tests involving more than Therefore any tool used for drawdown analysis, applies

one rate change is so complicated that in makes drawdown test directly to complex data which has been transformed to

analysls look easy by comparison. Equivalent drawdown time equivalent drawdown data.makes the analysis of more complex tests no more difficultthan drawdown test analysis, In the process it gives a simpler In short, Ateq can be used for any kind of analysis of

perspective of well test anaiysis than is possible without it, complex flow data, not just for type cuwe applications. Forexarripie, it could be used to make a radial flow analysis of a

By reducing it to two simple steps, equivalent drawdown Horner rate history or a iinear flow anaiysis of a generaltime make the analysis of compiex flow tests easier to multiple rate history.understand. The two steps are:

An infinite number of different possible behavioral1. Convert the complex data to drawdown data. mechanisms exist. Likewise, an infinite number of different2. Analyze the converted data like a drawdown test, rate histories are possible. The number of different

combinations of behaviors and rate histories is bewildering. IfObviously, a thorough understanding of drawdown behavior equivalent drawdown time is used, we need only understand

is critical to make this plan of attack work. Drawdown anaiysis each mechanism as it pertains to drawdown analysis. Theitself invoives two steps: effects of muitiple rate changes are then factored out of the

process, removing unnecessary clutter.1. Determine the mechanism(s) that is controlling the

behavior of the well. A NEW PERSPECTIVE2. Quantify the parameters reiative tG ihat controlling

rv”.hanism. When any step two technique is applied to a complex flow testbv first transforming it to equivalent drawdown data, thisapplication might at f;rst appear’ to be a new method. However,

References and illustrations at end of paper,

595

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APPUCA’TION OF EQUIVALENT Dit is usually simply a new perspective on an oldertechnique. For example when Agarwai’s equivalent drawdownlima is used for semlioo radial flow analvsls It may have the. ...—-——————.——appearance of a new ‘method, But it Is best thought of as astandard Horner plot viewed from a new perspective.

Indeed, It will be demonstrated that this most reveredInstitution of well test anaiysis, the Horner plot is, and aiwayshas been, an application of equivalent drawdown time. Mostpeople accept the fact that there are several different ways toconstruct a Horner piot, but regardless of whether you useHorner Ratio (extrapolate to the right) or Horner Time(extrapolate to the ieff) you are still making a Horner plot.

in the future, strictly for convenience, you may choose tO

make a semibg plot using Agarwal’s equivalent drawdown time.This is not a sacrilege- It is, like Horner Ratio and HornerTime, just another way to make a Horner plot. I will leave It toyour own conscience as to whether you choose to use such asemilog plot. I wili even tolerate a difference of opinfon as towhether doing so constitutes a new method. But, as for me, andmy house, a Horner plot using Agarwal’s equivalent drawdowntime is just a simpier perspective on the same old Horner plot.

Likewise, when equivalent drawdown time iS applied to anarbitrary rate history by using superposition, it really isjust a new perspective on multirate anaiysis. Even if thearbitrary rate history reflects the sandface rate caused bywellbore storage, it Is still only a new perspective on oldermethods.

There are few problems in weil test anaiysis that have neverbeen previously addressed, Equivalent drawdown time mayoccasionally provide a basis for improving existing methods,as was the case for type curve analysis. Even so, the mostfundamental value of equivalent drawdown time is that itprovides a simpler, more easily understood perspective ofexisting methods of analysis.

PURPOSE

The purpose of this paper is to encourage more widespreadunderstanding of the perspective offered by equivalentdrawdown tim~. it will attempt to accomplish this objective bypresenting a generalized derivation of equivalent drawdowntime and examining various ways which the concept can beapplied, and the considerations which must be made in so doing.

So what’s new? Probably nothingl Superposition stilirequires addition, radiai flow analysis still requires a semiiogpiot, and the sun stiii rises in the east. Even the credit for the“new perspective belongs to Agarwai, not to the presentauthor. There is a remote chance that the perspective offeredin this paper is siightiy more generai than the one taken byAgarwai (hindsight is 20/201) However, my own hope is thatthis paper might promote a wider understanding, and anassociated wider application, of the concept of equivalentdrawdown time within the industry. If this comes to pass, the, Ii wiil be satisfied that my effort was not in vain,

THEORY

The concept of equivalent drawdown time was introduced tothe weii testing literature in 1980 by Ram Agarwai. Itaddressed a very specific probiem. Buiidup type curves basedoniy on shutin time (At) can be misleading, They often exhibitbehavior that has the appearance of radial flow, even whenIittie or no radial fiow effects are present. Agarwai solved thisproblem eiegantiy by demonstrating that the actual buiiduptime (At) shouid be repiaced with an equivalent drawdowntime (Ateq).

kWDOWN TIME IN WELL TESTING SPE 1754For cases where there are two ecfual but OPPosite rate

changes (Horner rate history) and when both trarisients are inradiai flow this equivalent time is the product of the producingtime (tp) and the buildup time (At) divided by the sum of thesame two quantities:

AteqA = tp “At/ (tp+At) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...0. (1)

Equivalent drawdown time should be cieariy distinguishedfrom two other time related concepts of weii testin9, (1) reaigas pseudotime and (2) effective flow time.

Real gas pseudotime corrects time for noticeable variationsin properties of reai gases which are caused by large pressurechanges. This concept was also introduced by Agarwal, and isoften confused with equivalent drawdown time. Both oonceptsare occasionally referred to as Agarwai time. The two oonoeptsare, in fact, totaiiy different. They can be applied separately,or in combination, The present paper wiii totally neglect anycorrections for real gas pseudotime in order to focus on theproperties of equivalent drawdown time. Effective fiow time isa parameter which is used, quite successfully, to model a moreccmplex fiow history as a less oomplex Horner flow history.

GENERAL FORM

Equivalent drawdown time can be defined in generai as:

Ateq . tu “ ~l{X(i=l,N):(Ri*[~((t-ti)/tu )-f((tR -ti)/tU )])} . . . .. . . . . .. . . .. . . . .. ..( 2)

Where:

~(:v~sm arbitrary function of x and ~ 1(y) is its

{X.=1 ,N): (xi)} is the summation of XI over i from 1 to,

RI is the vaiue of the i-th rate change divided by areference rate, qR,

t is time measured from any arbitrary point in anyarbitrary units,

ti is the time of the i-th rate change,

tu is the unit of measurement for dimensionless time,end

tR is a reference time from which Ateq is to be measured.

Note that it is assumed that the potentiai, PR, at the referencetime, tR, is known. Aithough it is not required for equivalenttime, a compiete transformation from (AMP) to (Ateq tAp)requires that the change in potential be known. This change inpotentiai is measured from PR. See Appendix A for thederivation and a detailed explanation of terms.

KEY POiNTS

Application of Equation 2 requires:

1.

2.

A known rate history (number of rate changes, N;magnitude of the rate changes, Aqi; and the time of therate changes, ti).

A known function for the weii behavior and its inverse.

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SPE 17547 SAMUEL C. SWlfT

3. Three additional parameters:

A. Reference Time, tR,B. Reference Rate, qR, andC Characteristic Time, tlr,

Both the reference time, tR, and the reference rate, qR, arecompletely arbitrary. Both also fix associated values ofpotential. The reference level of potential, PR, Is the potentialwhich existed at the time tR. The unit of measurement fordimensionless potential, Pu, is fixed by qR.

Note that the parameters tR and qR tTIUSt alWaYS ~ specified.Whether or not the value of tu must be known explicitly,depends on the nature of the function, t, and the type ofequivalent drawdown time desired; regular or dimensionless.If tu is required, then the solution becomes trail and error interms of !U. It is generally not required to know Pu, a priori.

APPLICATION

The above discussion defines everything necessary tocalculate equivalent drawdown time. However, meaningfulapplication requires an appreciation for the role each itemplays.

PARAMETERS

The characteristic time, !U, is simply a unit of timemeasurement that is especially appropriate to the testsituation under consideration- just as the size of the “cup” issignificant when one requests a ha!f of a cup of coffee, Unlikethe tangible “cup”, tu is difficult to visualize. It can be thoughtof as a “time constant”, in hours, determined by theparameters of the problem. Its exact mathematical definitionIS in Appandix A. It Is also the time unit that is normally usedfor dimensionless time. In fact, if Eq 2 is divided by tu oneobtains the dimensionless equivalent drawdown time, AtDeq.Conventional Horner Time is a dimensionless equivalent time.

The characteristic potential, Pu, is simply a unit of potentialmeasurement that is especially appropriate to the testsituation under consideration (similar to !U for time),Characteristic potential does not appear explicitly in theequation for A!eq, but it is linked by definition to the referencerate, qR (see Appendix A). Reference rate is required In theequivalent time equation in that it is the denominator of each ofthe Ri coefficients. It is totally arbitrary but is often the valueof the rate change which occurred at tR (Note, however, that itis not required that tR be selected at a point in time where arate change occurred), Reference rate can be selected as aconveniently round number, e.g. 100 BOPD, 1 MMCFD, etc. Ifunity is used then Ateq is tied to the unit response function.

The reference time, tR, is totally arbitrary. It would mostoften be chosen as the time of the most recent rate change,since the effect of that rate change dominates the potentialbehavior for some period of time thereafter. The value of thepotential at tR Is the reference potential, PR, If the change inpotential (PR- Pw or AP) is required (e.g. for type curves)then PR must be known. For Agarwal equivalent the, AteqA,

tR is selected at the time of shutin, For conventional HornerTime, AtDeqH, tR is selected at the beginning of flow. HornerTime is the same as Horner Ratio except that for Horner Ratiothe reference rate is the production rate (+q), Horner Timeuses the change in rate (-q) at shutin as the reference rate.

RATE HISTORY

The number of rate changes, N, must be known. If N Is orwthen Ateq reduces to test time, AlthoWh this Is a trivial case ofAteq, it does cover five common aIMly!iCd procedures;drawdown, negative drawdown, buildup following extendedproduction, falloff following extended injection, anddesuperposition. For desuperposition the change In potential Isthe projected change in potential rather than the observedchange In potential. If N is two and the rate changes are equalbut opposite (Aql =-Aq2) then It is called a Horner ratehistory, For multirate analysis N Is two or more and themagnitude of the rate changes are arbitrary.

The time of each rate change, ti, must be known as well as themagnitude of each rate change, Aq, The coefficients, Ri, areobtained by dividing (normalizing) each rate change by thereference rate.

THE FUNCTION

Probably the most difficult to understand aspect of Atsq is

the role of the function, ~. However, the difficulty is notsomething unique to the concept of equivalent drawdown time.Determining (or at least assuming) what function describesthe behavior of a well is the first step in sny type of analys!s.

In general the required function is the characteristicdrawdown behavior of the system, so that if an idea Idrawdown test were run at the reference rate , qR, the changein potential (AP) as a function of drawdown time (At) wouldbe prec!sely expressed as:

AP = f(At) . .. . . .. . . .. . . .. . . .. . . . .. . .. . . .. .. .. .. . .. . . . . .. . . .. . . .. .. . . .. . . . .. . . . .. . .(3)

Perhaps you have noticed that our discussion has becomesomewhat circular. We must know the characteristic responsefunction of the system in order to correct our actualobservations, in order to faithfully reflect the characteristicresponse of the system- that’s life! The characteristicresponse function is only revealed directly, when it Is possibleto run an Ideai drawdown test. For any more complex test,including real drawdowns, the observed data is distorted bysuperposition of multiple “images” of the characteristicresponse function. Equivalent drawdown time can remove thisdistortion, but only if we are skilful enough to select theproper function.

Therefore, in a general sense, the selection of j is alwaystrial and error. We must choose a function, make thecorrection (calculate equivalent drawdown time) and check tosee if the corrected function (transformed data) is the same asthe assumed function. That’s the bad news. The good news isthat we know something about what functions might beappropriate. In fact, petroleum engineers have been pickingsuch functions for many years with at least some success.

With any luck at all, ~ should be proportional to somedimensionless solution to the diffusivity equation, so that Eq 3becomes:

Ap=f(At)=pu*pD(tD)=pu*pD(At/tlJ),..........................( 4 )

Where PD(tD) iS a dimensionless SOIUtiOrr.tO the diffusivityequation.

There are an infinite number of solutions to the diffusivityequation. The following list, while only a finite sampling of aninfinite number of possibilities, represent the ones that aremost useful in practice.

597

1. Steady flow solution, PDs2. Pseudosteady flow solution, PDP3. Linear flow solution, PDLI

(Infinite reservoir)4. Exponential integrai solution , PDEI

(infinite reservoir)5. Cyllndricai weiibore soiution, PDci

(Infinite reservoir)6. Fractured welibore solution, PDFi

(infinite reservoir)7. Pseudosteady wellbore flow soiution, PDPCi

(Cylindrical wellbore)(infinite reservoir)

Aithough the above list Is iimited, it is stiii too iong to coverin a single paper. Therefore, lets cut it down some. The firsttwo are very simpie solutions, so we wili assume that theireffects are well known (a dangerous assumption). PDPCidiffers from PDCi oniy in that it has a compiex (self-inferred) flow history. It therefore can, if desired, be reducedto PDCi by applying equivalent drawdown time. The remainingfour are all reiated in some way to two simple functions:

PDLi=(2/rr)*d(tD) ................................. ...,.,,,,,.,,,.,,, ......(5)

PDLN=(l/2)*in(4 *tDN) ..... .............................. .... ........( 6 )

Where n is a transcendental constant approximated as 3.1416and d(x) represents the square root of x. Aiso in indicates thenaturai logarithm and Y is a transcendental constantapproximated as 1.7811. PDLi is the exact solution for iinearfiow. PDLN is the equation on which semiiog anaiysis is based.Since common usage equates radial flow with “semliogstraight”, Eq 6 constitutes the definition of radiai fiow. itshould be noted that PDLN is not a solution to the diffusivityequation, but an approximation to PDEi, PDCI, PDFi, etc.which oniy applies for iarge dimensionless times.

These two behaviors, linear fiow and radiai fiow, arefundamental. If ii were possible to ?onduct an ideai drawdowntest (instantaneous change in sand face rate) on a reai weii, ailsuch tests wouid exhibit iinear flow initialiy. Also, many realwelis exhibit a period of radiai flow behavior which foilowsweiibore (inner boundary) effects and precedes outerboundary effects, These two simpie functions are consideredextensively in Appendix B.

The uitimate advantage of the concept of equivalent drawdowntime is that the basic procedure is the same regardless of thecomplexity of the function ~. Whether using a simpie functioniike PDLI, a siightiy more complicated one iike PDCi or a verycomplex one invoiving inner boundary effects, dual fiowsystems and outer boundary effects, the concept and basicprocedures are unchanged, They do, however, become morecomplicated to perform.

Any function that successfully transforms complex fiow datato equivalent drawdown data must accurately describe thebehavior of the system under consideration during an ideaidrawdown, it must do so over the reai time intervai beingconsidered- test time or time of projected performance.Unfortunately this single reai time intervai has associatedwith it, N different dimensl~vrless time intervals- one for eachindividual rate change. These intervals are of equai iength butspan distinctly different vaiues of dimensionless time.

One fundamental fact is ciear. Unless the function selectedprovides at ieast a reasonable approximation to theideai drawdown behavior over ali of these N dimensionlessIntervals, it will not succeed.

For this reason’, it is erroneous to assume that equivalenttime based on the assumption of any particular behavior can beused to interpret a different kind of behavior. For example, ifAtsqA is used and the transformed data had a peculiarity thatappeared to be duai flow system behavior, it would bepremature to conciude that the test data is from a dual flowsystem. This couid oniy be substantiated if a function wasseiected which faithfully portrays dual system behavior andthe resuiting transformation confirmed that the peculiarity isa property of the fundamental drawdown behavior and not Justa distortion caused by muitlple rate changes.

A final property of ali functions which are appropriate toequivalent drawdown transformations is that they are zero foraii negative or zero arguments. if a rate change has not yetoccurred it has no effect on potential. This saems trlviai, but itmust be accounted for in the equations. Mathematlcaiiy, Eq 5and Eq 6 are incomplete. They must explicitly be forced to zerofor negative and for zero arguments. That is the reason for thepound sign (#) in some later equations.

EQLfiVALENT TiMES ALREADY iN COMMON USE

Although Equation 2 is somewhat abstract, careful study ofthe factors in its derivation immediately give rise to someimportant applications. One such application is to demonstratethat several equivalent times in common use are reaiiy speciaicases of ti Ie generai equation.

The most obvious is Agarwal’s equivalent time (AteqA) Muchiess weii imown is Tandem linear equivalent time. (AteqT),which differs from Agarwai’s equivalent time oniy in thatiinear flow behavior is assumed,

Very weii known, but not often thought of as equivalent timesare Horner Ratio ([tp +At]/At) and Horner Time(At/[tp+At]), Both of these familiar quantities are speciaicases of general equivalent time, Eq 2. Both aredimensionless equivalent drawdown times which take theirreference time, tR, as the beginning of the flow period andassume radial fiow behavior. Horner Time takes -q as thereference rate where Horner Ratio takes +q, Since HornerTime, Horner Ratio and Agarwai’s equivalent time share acommon rate history and a common assumed functionalbehavior, and differ oniy in the arbitrary parameters, it isobvious that identical results shouid be expected from radiaiflow (semiiog) anaiysls using any one of the three parameters.

Corresponding to Horner Ratio (but assuming linear flowbehavior is present rather than radial) is tandem roottime([d(tp +At)-d(At)]’2). The square root ofthis, parameter is routineiy used for iinear fiow (square root)anaiysis of two equal and opposite rate changes, Tabie 1summarizes the defining properties cf these equivalent times:

Agarwal’s AteqA is a Speciai case for: (1) two equai andopposite rate changes, (2) both transients in radial flow, and(3) using the time of shutin as the reference time and thechange in rate at shu!in as the reference rate. This derivationis in the appendix and the resuit, as expected, is Equation 1.

Agarwai’s equivalent time is properly used oniy whenappiied to two equai and opposite rate changes and then only ifboth transients are in radiai flow. it is weii known that theassumption of two equai and opposite rate changes is anadequate modei for many rate histories that are in fact morecomplex, and that if the rate changes are iarge (especially ifthey are recent; that is to say, near the end of the flow period)then probiems can arise. Much iess weil known are theprobiems which arise from a faise assumption that thetransients are in radiai flow.

—

SE 17547 SAMUEL C. SWIFI’ I.

There are In faot many special cases of Equation 2 which am and for linear flow are:in oommon use, although they often are not thought of ajequivalent time. obviously, drawdown analysis (radial, linear, AteqL={~(i=f,N):

or even pseudosteady reservoir limit behavior) are all special Ri*[d#(t-ti)-d# (tR-ti)]}h2oases where N is one. As Agarwai pointed out, the concept of

.. . . . . . . . . . . . . . . . . . . . . . . . . . . (11)

aquivaient time is not limited to type curve analysis. Once the wheretransformation to equivalent drawdown time is made, the databecomes, in every way, drawdown data so that semi-log o Xso ....................................,.,,.. (12)pbts, square root plots and cartesian plots may be wiled if 4#(x) = {appropriate. Appropriate in this case means both that the weii d(x) X>ois exhibiting a particular behavior and that the transformation

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

was made using that same behavior, Hopefully, at this point, the abstract function j(x) hasassumed some more concrete meaning. it has in fact been

Classical Horner Tim% At/(tp+At) , is a dimensionless assumed to be equai to common dimensionless potentialequivalent drawdown time, exactly the same as Agarwai’s functions, The function for radial flow is PDLN and theexcept that the reference time is at the beginning of the function for ikrear flow is PDLI.drawdown, the reference rate 1sthe finai rate change, -q , andthe dimensionless form is used. if +q is used as the reference The real power of Equation 2 is that ~(x) is not restrictedrate, then the famiiiar form, (tp+At)/At,is obtained. oniy to simple soiutlons to the diffusivity equation, By

inserting more com Iicated solutions (models) one can

Note that both of these forms are oommonly used for semilog fconsider any case wh h may be deemed to be appropriate, e.g.anaiysls. They generally can not be used for type curve the finite cylindrical wellbore (PDCI), infinite conductivityanalysis, because the (Initial) potentiai at the reference time fracture (PDFi), finite capacity fracture with biiinear flow,Is not known. Indeed, the whole key to Agarwal’s disoovery of dual porosity reservoirs, etc.AteqA was the giving Of fuil mathematical resPect to theoommonly used reference potentiai - the (known) potentiai in Although it is beyond the soope of this paper to mnsider anythe weil at shutin. of these cases in detaii, one thing should be obvious. Extension

of a special case equivalent drawdown time to a case where itLinear fiow anaiysis is more important than commoniy does not apply is at best risky, and in many cases totally

recognized. This iack of recognition is at least partially due to wrong. For example A!eqA assumes radial flow for alimisusing pressure versus square root of time plots for transients and a Horner rate history. if it is appiied to aanalysis, when the more compiioated tandem root time plots Horner rate history, but iinear fiow behavior, it works fairiywere required. Much has been said and written about how the well, but not exactiy, If the flow is linear but tha rate historyHorner rate history must be considered in radial fiow analysis compiex it can be totaiiy misleading.of buiidup data. The simpler single rate change (Miiler, Dyesand Hutchinson history) is not adequate In many cases. This is Agarwai’s equivalent time can be used generally, as a basiseven more true in linear f!ow than in radial flow. for examining unknown data. However, if the data is seen to be

in anything other than radial flow, then Equation 2This discussion couid obviously be extendad to include many should be used to determine the proper equivalent time for the

other possible combinations of dimensionless and dimensional suspacted behavior. The appearance of non-radiai flowequivalent times, different fiow behaviors, and different behavior subjected to a radiai fiow transformation can be veryreference times. Some of the resuits are interesting, and may misleading For exampie, a real resewoir with dual porosityeven be in use by some experts in weii testing. However, the behavior will probabiy not iook like known dual porosityabove discussion covers the most oommoniy seen special cases soiutions if AteqA is used. Furthermore, if storage is presentand further discussion is beyond the scope of this paper. aiso, the radiai flow transformation wiil distort the data in a

different manner. The bottcm iine is common sense- don’tMULTiRATE ANALYSiS FROM THE NEW PERSPECTIVE expect radial flow transformations to work on non-radial flow

data. Check it out, it might work adequately in some cases, butThe most obvious application of Equation Z beyond those It certainiy wiil not in others.

cases already discussed is to consider more complicated ratehistories. For exampie, when attempting to analyze an extended FLOW HiSTORY PARTiALLY SELF-iNFERREDdrawdown test wifh major disruptions included in the fiowperiod, it is obvious that the Horner rate history does not One soiution to the diffusivity equation is encountered‘pply. For these cases it is convenient to write equivalent sufficiently often as to deserve speciai consideration, the sodrawdown time for an arbitrary fiow history, but stiil tailed storage and skin solution (PDPCI) This solution wasassuming that ali transients have a common functional introduced by Agarwai and Ramey in 1970 and almostbehavior. simultaneously by McKinley. After considerable debate, it was

These cases for radial flow and iinear fiow are derived in thegeneraiiy agreed that in most cases skin couid be adequatelyaccounted for by using the effective weiibore radius, rwa, or

appendix. The i’esuits for radiai fiow are: rw*eA(-s). This debate took several years and resulted in thepresentation of the same basic soiution on numerous log-log

AtsqR= {tU*W4}*{rI(i=l, N): formats (e.g. l-Agarwal&Ramey; 2-McKinley; 3-(t-ti) ‘Ri#/(tR -ti)ARi# } .....................................( 7 ) Eariougher; 4- Gringarten). All of these formats were

essentiai evolutionary steps. Unfortunately many petroieumwhere: engineers view these different type curves as fundamentally

different methods rather than just different ways ofII(i=l ,N):(xi)=xl ●X2”X3*...*XN ... . . .. . . .. .. . . . . .. . . .. . . . . . .. . . . . ..( 8 ) presenting the same information. Each of these presentations (

as weil as others) have particular advantages and(Y*tu/4)’Ri for x s O...............................(9) disadvantages and may be the most useful for a given test

xARi# = { anaiysis.xARi forx>O ,..,,.,,!..,,..,.. ........... (lo)

599

APPUCATION OF EQUIVALENT DRAWDOWN TfME IN WELL TESTING SPE 175. ..

This function, PDPCI, can be used in equation 2, however the SUMMARY OF APPLICATIONSmathematics, including the parameters, become somewhatcumbersome. A more satisfying approach is to back out the Defining the rate history and choosing reference quantities Isstorage effects by treating the afterflow as finite rate changes straightforward. Choosing the best function for calculatingat the sand face. When the weilbcre storage behavior is simple, equivalent drawdown time is the major consideration.it Is easy to make a welibcre modei which calculates theunknown sandface rate from known surface rates. For Because of the special emphasis placed on radiai fiowexampie, if constant weilbcre storage Is assumed the sandface behavior, AteqR, and its speclai Horner rate form, AtsqAt willrates are easiiy catcuiated. When these rate changes are taken be the most common forms used. They wiil probably be theInto amount the transformed potential data reduces to the choice for “first looks” at test data of unknown character. Ifweiibcre behavior without storage, in this case PDCi. This is type curve analysis then confirms the presence of radial flowtrue however, only to the extent that the underlying reservoir behavior, quantification can proceed in the conventionalbehavior conforms to PDCi. Fortunately, it works in many manner, using Horner plots or radial superposition plots forcases. Even a fractured weli (PDFi) conforms closeiy to PDCi.The only significant deviation Is in the transition from PDLI to

radial flow analysis. Because of its Convenience, AteqR wiii

PDLN; and even there the error is less than 10%.gradually become the parameter of choice for radial fiowanalysis.

This process Is illustrated in the Waterton #1 example. It If the first look reveals nomadial flow, the type curvecan obviously be extended to more complicated cases(continuously changing storage or discretely changing storage

should be redone using AteqL. if linear fiow is confirmed

for instance) by simpie modifications of the procedure ofanalysis can proceed using Tandem root time or ikrear

calculating sandface rate from surface rate,superposition piots. Here again, because it is more convenient,AteqL will gradually become the paramqter of choice for linearflew anaiysis.

It is interesting to note that Akq R wcrks very PcorlY forthis application. Because PDLN behaves peculiarly (iiteraliyunreail) for short drawdcwn times, AteqL Is a much better

if the flow behavicr is more complex than simpie ilnear or

choice when dealing with afterflow corrections. In fact theradial fiow, every effort shcuid be made to find the drawdcwnfunction which properly transforms the data. The complex

auihor believes that AteqL is a preferred tool for “first lock” function of choice is PDCi, and it wiii give a satisfactoryusage, but most authorities would regard that as a personal“iinear flow bias” (to which i proudly confess!). So i will stop

performance in most cases. However, more complicatedfunctions which accurately refiect inner boundary effects,

short of recommending it for general use, However, it doesn’thurt to check it just in case. It might just be that the weird

compiex reservoir effects and outer boundary effects wiil

behavior seen on a given test is a distortion caused by muitipleoccasionally be necessary.

imaging of PDLI rather than some super sophisticatedfundamental behavior. In fact, if this effect as well as minor

EXAMPLES:

wellbore perturbations caused by phase redistribution andcrossfiow are faithfully eliminated, those seven PD functions

Two greatly different exampies are considered. Both are

previously considered are adequate In the vast majority ofgenerated data (not real weiis), however the points made are

cases- I did not say all Ibased on personal experience of the author in dealing with realwell tests.

BEHAViOR SELF-INFERRED The first example, Waterton #1 could be (almost) correctly

The applications discussed above involve a known functionalevaluated uskrg only a data type curve based on Agarwal’s

behavior. However Equation 2 is not iimited to any particularequivalent time and a semiiog plot based on the standard

function. This gives rise to a particularity important possibleHorner parameter. Indeed some would point out that use of oniy

application where the observed behavior itself might be usedthe Horner analysis arrives at the correct interpretation.

to deduce the actual functional nature of the drawdownEven so, in the opinion of the authcr, use of the general form of

behavior of a real well: totaily independent of any assumedequivalent drawdown time for both type curve analysis and

(guessed) functional behavior. Indeed this is possible in someradial fiow (semilog) analysis provides a more Iogicaliy

cases, as is illustrated in the Sombrero #1 exampie.consistent approach and a higher degree of confidence in theresuits.

The author’s numerical skiils are ver~ primitive, but this The second example, Sombrero #1, Illustrates many of theexample illustrates several things to my satisfaction. shortcomings of attempting to use methods which fail to take

1. There is no free iunch - meaning:into account the complex nature of the rate history andor theeffects of non-radial flow behavior. In addition, it aiso shows

1.a Extrapolation and interpolation techniques arethat non-radial flow behavior can be (in the author’sexperience and cpinion often Is) important in high rate

critical, (international oii) production as weil as tight gas production

1.b The oniy time this method Is guaranteed to work isin North America. In fact, non-radial flew effects seem to obeyMurphy’s law - they tend to occur in the most unexpected

when you don’t need it, namely if you observe theentire possible range of time values.

places ( anywherel ) and at the worst pcssible times (whencash ficw predictions have been based on the assumption of

2. Even using a iimited range of data and very limitedmore graduai radial declines).

numericai skills this technique makes it possible to Icokthough the clutter caused by complex rate histories and see

Understanding of non-radial fiow behaviors cannot predict

the vague outiine of the underlying drawdown behavior.prcduct prices. However, such an understanding can provide a

This view can in some cases be a mere reaiistic viewreasonable basis for anticipating many declines in production

than that obtained by presupposing a certain functionalrate. This knowiedge can be critical in managing the impact of

behavior. (e.g. radlai, iinear, biiinear, dual porosity,these declines, whether they occur in tight gas sands of North

storage, steady state, pseudosteady state, etc.)America or high rate oil production from fractured reservoirsanywhere in the world.

JE 17547 SAMUEL C. WI/WI’—

These two examples illustrate expanded applications that are PU = -0.86857*m ..................,,..,,. .,.,.....!. ....................... (14)possible using the generalized form ot equivalent drawdowntime. Each case first examines the type curve to determine Note that technicality PU is negative since -q was selected aswhat flow mechanism(s) are controlling the well behavior in the reference rate. in order to insure a proper (positive)each time region. After identifying the controlling mechanism,the appropriate analytical tool for that behavior is then

value for permeability, the qfl must be signed in the definitionof Pu. The reference rate for this example is negative (qR=-

examined; semilog plot for radial flow; square root plot forlinear flow.

q). When the reference rate is positive then the slope, m, isnegative so that Eq 14 still applies.

This analysis was done on a personal computer using 2.standard application programs.

Completion efficiency is derived from the point in timewhere the semilog straight line crosses the potential existing

WATERTON #1: PREFRAC TEST ON TIGHT GAS WELLin the well just prior to tR, (AtR), The characteristic time,tu , fOr this teSt is 4*AtR/V Or 2.245 ~AtR.

Table #2 shows the data for Waterton #1. Note that effectsof the variations of gas properties are ignored for simplicity.

tu = 2.2458*AtR .............................................................. ( 15 )

Both pseudopressure and pseudotime may be required in manyreal cases, The use of those concepts would only obscure the

3. if the test is a shutln test, as is this example, the

basic purpose of this example, to illustrate the application ofextrapolated potentiai at infinite shutin time assumlp. radlai

generalized equivalent drawdown time. fiow describes aii transients, P*R, Is often of Interest. It canbe calculated from the equation of the semilog straight line

The obvious approach to this probiem wouid be to use AbqR evaiuated at the limit of AteqR as the shutin time approaches

and correct for each rate change as it occurs at the sand face. infinity .This limit is aiways the value of AteqR evaiuated at tR,

However, as is pointed out irr Appendix B - Special Case for [AteqR(tR)].

Stepwise ShutIn, AteqR does some undesirable mathematicaltricks when each stepwlse change in rate is inciuded. However, AteqR(tR)=(t-tl )hR1’(t.t2)h R2=...*

the desired effect can be obtained by using two transformations (t-tN-l )~RN. I .................. ................................ (16)which do work, AteqLA and A~qR . The first transformation iSAteqL applied to each Stepwh R3t0 change caused by afterflow, For a Horner rate history this is tp; for this example it is 512hence the additional “A” subscript indicating that afterfiow is hours. The equation for the semiiog straight iine is:inciuded. This transformation is defined for aii times but isvaiid only if the behavior is linear flow. Table #3 shows the pw = pR.(l/2)*pU*in( 4*AteqR/tU) .................. . ..........( 1 7)

procedure used to obtain At@A.Therefore the extrapolated potential is:

The second transformation is for radial flow behavior, but isoniy valid for zero afterflow. The approximate type cuwe for P*R = pR-(l/2)*pu *ifl{4*[AteqR(tR) ]/tU} .................( 18)this data can be obtained by using the first transformation todefine the eariy time behavior and the second to define thelatter (radiai flow) part . Both are shown on Figure #1. Since

Bear in mind the sign of qR and therefore PU f“. (his e)(afTIpieis negative,

this is generated data, the actuai type curve for a singie ratechange (AteqX) is shown for reference. ( Too bad we can’t do SOMBRERO #l : EXTENDED EVALUATION OFthat for reai data ). The type curve defined as the “upper” iNTERNATIONAL OiL DISCOVERYenvelope of these approximations clearly represents areasonable facsimile of the underlying resewoir behavior. Tabie 4 shows the derived test data for this example. It was

WATERTON #1generated using an (assumed 1) piecewise linear

: TYPE CURVE ANALYSIS dimensionless potentiai function. S-ch a function is definitelynot a solution to the diffusivity equation, but represents pure

The behavior of the data using AkqR Cieariy shows Storage iinear fiow, followed by pure biiinear fiow, foliowed by abehavior from 0.001 hr to 0.02 hr, foilowed by a storage- second pure iinear fiow period.radial transition extending to 1 hr, foiiowed by radial fiow. Assuch one wouid expect that a semiiog anaiysis would yield vaiid As such it has a basic similarity to a f(acture system ofradiai flow parameters (k and S) when applied to the iatter limited extent, draining a larger reserve c.>ntained in tightportion of the data. Also, type curve matching of normai type rocks where some pressure drop occurs in the fracturecurves or of “derivative group” type curves might be appiiedto the data beyond 0.02 hrs using any of the commoniy

system. It is fundamentally simiiar to the finite capacityfracture soiutions, commonly used in analysls of tight gas

availabie theoretical type cuwes which exhibit storage-radiai production.transitions. However, be warned that the shape of thetransition using AteqA wiii in generai not be the same as the SOMBRERO #1 : TYPE CURVE ANALYSISdrawdown transition since pure radiai fiow does not inciudepseudosteady weiibore flow effectsi Figure 3 shows the Type Curve for this weii. Included are

WATERTON #1 : RADIAL FLOW ANALYSiSseverai standard equivalent times. This figure couid easily beinterpreted as an example of why equivalent drawdown time

Figure two is a semilog piot USin9 AteqR. Quantification ofdoesn’t work- it’s too confusing. However the things that makethe plot complicated are integrai parts of well behavior, and as

radial fiow parameters can proceed in typical fashion. such can be present in most any real test data. A much more

10reasonable attitude toward Fig 3 is that iife is complicated but

The slope (m) of the semiiog straight reveals the if you reaily understand equivalent drawdown time, it can be aquality of the reservoir. The characteristic potential (Pu) for vaiuabie asset in explaining complex flow data.this test (based on qR = -q) is -2*m/in(l O) or -0.87 *m,Conventional reservoir quaiity parameters (transmissibility, The first three parameters plotted ciearly illustrate that

permeability thickness and permeability) are then derived neither At, nor Ateq A, nor Ateq T Mkibiy refiect thefrom the definition of Pu. underlying drawdown behavior. They aii appear to indicated

601

a APPLICATION OF EQUIVALE~ DRAWDOWN TIME IN WELL TESTING SPE 17547. .. _. —... . . . . -.

some radial flow followed by something, maybe the beginnings For a Horner rate history this Is tp; for this example it isof depletion effects. These implications are totally erroneous. 63.83 hours. The equation for the square root straight line is:They fail because none faithfully reflects the actual ratehistory of the weii. The first time parameter, At, assumes no pW=p R-[2*pLJ/d(@/d( tlJ)]*d(AteqL) ,,,..,...,..,,..,.,.,, (2 1 )effect on potential from prior rate changes- that is prior tothe main one under czrnskferation. The other two (AfaqA and Therefore the extrapolated potentiai is:A!eqT) fail primarily because they use the Horner ratehistory, and for this case that is an inappropriate model. A p* L=p R-[2*pu/d(~)/d( tu)]*d(AteqL[tR ]).,,., . . . .. .. ..(22)minor effect is that neither AfeqA nor AteqT is the apprOfXiate

behavior function. Bear in mind the sign of qR, and therefore Pu, for this exampleis negative.

This iast reason is the primary explanation of why AfeqL,and especially AteqR, fail. Aithou9h they eccurateiY refiect the Note that for this exarnpie two linear flow behaviors arerate history, they faii to use the appropriate functional seen, or nearty seen. The most important in terms of iong termbehavior. it is worth notin9 that AfeqL WO~S much better behavior is the one which is being approached in the latterthan AteqR. ( i am biased for linear flow, beoause in mY portion of the data. This wouid represent the behavior of theexperience it works better in the many of the welis that i test- fracture face for a massive fracture or the behavior of theeven high rate oii wells. In fact, one could justifiably argue matrix for a naturaiiy fractured reservoir. if the square rootthat the additional improvement gained by the Seif inferred straight iine is seiected at the end of this data then thefunction, Ataqs is not worth the extra troubie.) characteristic iinear fiow parameter and the ioss in potentiai

relative to this region is obtained. If the eariy line is seiectedHowever, one of the main points of this example is that then these same parameters are obtained for the near weiibore

sometimes the observed behavior, AP versus At can be used to region.obtain a better view of the underlying characteristic behaviorthat can be obtained from any a priori assumed behavior. Sure CONCLUSIONSenough, Ateqs is the best of aii the data sets shown. Except, ofcourse, Ateqx, which is eXact, and would not be available f?r a The foiiowing conclusions are drawn.reai weill

1. Equivalent drawdown time is a powerful tool forEither AteqL or Ateqs give a reasonably good view of the reducing complex fiow histories to equivalent

actual ideai drawdown behavior of this well. drawdown histories.

SOMBRERO #1 : LINEAR FLOW ANALYSiS A In theory, any rate history can be so transformed.

Figure 4 is a square root plot using Ateqs. Quantification Of B. in practice, it is essential that the functional

linear flow parameters can proceed in typicai fashion. nature of the characteristic response function ofthe system be defined. This definition can be from

1. The slope (mL) of the square root straight iine reveals a priori knowiedge, an assumed behavior or from

the iinear flow parameter, Pu/d(tLr). This parameter shows extensive observation.

the combined effect of cross-sectional area to fiow andpermeability. The characteristic linear fiow parameter 2. Equivalent drawdown time depends strongiy on the

[P@d(tu )] fOr this tC%t (based on qR = -q) iS ‘d(7t)*ITIL/2 nature of this functional behavior of the system.

or -0m86623*mL. Conventional linear fiow parameters(cross-sectional area to flow or fracture iength) are then

3. Some correction for the effects of muitipie rate changes

derived from the definition of Plj/d(tu) if the other requiredis essential unless the final rate change is totaliy

parameters are known.dominates the potentiai changes. To be certain that thisis so, the finai rate change must be separated from the

PIJ/4(tu) = -0.88623*mL ................................... ..........( 1 9)next most recent rate change by more than 100 timesthe duration of the test.

Note that technicality PU is negative since -q was seiected as 4.the reference rate. In order to insure a proper (positive)

Agarwai’s equivalent time (Horner rate history; radiaifiow assumed):

value for permeability, the qR must be signed in the definition 4.a Applies to a specific rate history and then onlyof Pu. The reference rate for this example is negative (qR=- when ail transients are in radiai fiow,q). When the reference rate is positive then the slope, m, is 4,b Works perfectiy within these restrictionsnegative so that Eq 19 stiil appiies. 4,c Has widespread and important application because

2. Loss in potential between the observation point and theof the importance of radial fiow behavior and ofthe flexibility of the Horner rate history in

region causing the linear fiow is simpiy the difference betweenthe intercept of the iinear fiow straight iine and PR.

satisfactorily approximating other rate histories.

5. Compiex rate histories andlor other fiow behaviors3. if the test is a shutin test, as is this exampie, the demand other forms of equivalent time.

extrapolated potential at infinite shutin time, assuming linearflow describes ali transients, P*L, is often of interest. it can 6. Generalized equivalent drawdown time is easiiybe calculated from the equation of the square root straight iine simplified to speciai cases if the functional form is

evaluated at the iimit of AteqL as the shutin time approaches assumed.infinity,This iimit is aiways the value of AkqL evaluated at tR,

[AteqL(tR)].7’. Under favorable conditions generalized equivalent

drawdown time can be used to obtain functional form if

Ateq L(tFr)=[R I “d(t-tl )*R2*~(!-t2)*... *rate history is known.

FtN.l*~(t-t&J.l )]A2 , .. .. .. . . .. . .. . . . .., .!!.,.,..,.,,,,, . . . .. . .. . (20)

--.

PE 17547 SAMUEL C. SWIFT 9

Cross-sectional area to flow at the well

Ct

f(x)

fl(x)

f-’(Y)

h

i

k

m

ML

N

PD

PDCI

PDEI

PDP

PDLI

PDLN

F’Ds

Pm

Pm

PDUT

Total compressibility of the system

An arbitray function of x

The arbitrary function of x associated with thei-th rate change

The inverse function of t(x) , so that x = f-l (f(x)).

Formation thickness

An index value referring to a particular ratechange. If the maximum value of i is 1 then itmay be thought of as indicating the “initial”condition.

Formation permeability

Slope of semiiog (radial flow) straight iine.

Slope of square root (Linear flow)straight line.

Total number of rate changes in a known ratehistory.

Dimensionless Potentiai change measured inunique units (Pu).

PD for a right circular Cyiinder located in anInfinite resewolr and produced at constantrate.

PD for the Exponential Integral solution.

PD for Pseudosteady flow,

PD for a right circular Cylinder located in aninfinite reservoir and produced at constant ratewith initiai P seudosteady fiow from thewellbore (Storage).

PD for iinear fiow.

The semilog straight iine function, (1/2) (ln(4tD /V))

PD for Steady flow.

Dimensionless Potentlai change measured fromPw@tl , in unique units (Pu). If there is onlyone change in rate then FDU is Identicai to PD.However if move than one rate change Isinvoived, then PDU is referenced to anarbitrary rate qu. Note that this processrequires the introduction of the coefficients, Ri.

Dimensionless Potential change measured fromPR , in unique units (Pu).

Totai sum of the Dimensionless Potentialchanges caused by each of the severai ratechanges, Aqi

f% The value of potential in the well at the

Pw

Pw@tl

PIJ

PIJl

Ri

RI

~

qi

w

m

rw

rw A

t

ti

tD

tDu

!DRi

tDui

tp

LtR

tlJ

tl

x

particular time, tR. The ValUe of tR (andtherefore the value of PR ) is totally arbitrary.

The value of potential in the well at any time, t.

The value of potential in the well at theparticular time, tl. For a drawdown test, this isthe initial (static; quiescent) potential.

The unit of change for dimensionless potential.This notation is used when the referenoe rate isfixed at some arbitrary value.

The unit of change for dimensionless potentialbased on the first rate change, Aql.

Ratio of the I-th change in rate (Aqi) to thereference change in rate (qU).

Ratio of the first change in rate (Aql ) to thereference change In rate ( qu ),

Production rate during drawdown,

Production rate foiiowing the i-th rate change.

The magnitude of the arbitrary referencechange in rate.

Arbitrary reference rate.

Welibore radius for a well having a rightcircular cyiinder for a welibore.

Arealiy equivalent weilbore area, Aw/(2rcrw)

Real time in any unit measured from any fixedpoint of time.

The arbitrary time which at which the i.th ratechange occurs.

Dimensionless Time change measured in uniqueunits (tu).

Dimensionless Time measured from thebeginning of flow (for a single rate change) andin unique units (tit).

Dimensionless Time from the i-th rate changeto the reference time, tR, and measured inunique units (tU).

Dimensionless Time from the i-th rate changeto the current time, t , and measured in uniqueunits (tu),

Producing time for buildup; injection time forfaiioff,

Arbitrary reference time; fixes PR.

Characteristic time; the unit of dimensionlesstime; 0* f.L’Ct’rwAA2fk

Time at which the first rate change occurred

Arbitrary (normaliy independent) variable, e.gy = f(x)

603

10 APPLICATION OF EQUIVALENT DRAWDOWN TIME IN WELL TESTfNG SPE 1754

Y

B

Ap

At

AtDcqH

AtDeqHR

Atsq

AteqA

AteqL

AteqLA

AtsqR

Ateqs

AteqT

AtR

Aqi

Aql

Arbitrary (normally dependant) variable, e.gy = ~(x) but also seen as x = tl( y )

Formation volume factor, reservoir volume persurface volume.

Change in potential; PR-Pw

Test time, simple flow time for a drawdown;shutin time for buildup; etc.

Special dimensionless equivalent time forHorner rate history; radial flow assumed; qR=-q; tR=O; same as Horner time,

Special dimensionless equivalent time forHorner rate history; radial flow aSSUfTIed; qR=q;

tR=O; same as Horner ratio.

General equivalent drawdown time with norestrictions as to rate history or as to the wellbehavior.

Special equivalent drawdown time with twoequal and opposite rate changes and with radialflow assumed for all transients, This is themost widely known form of A~q.

Special equivalent drawdown time with anarbitrary rate history but with linear flowassumed for all transients.

Same as AteqL except afterflow rates arecalculated and rate changes due to afterflow areincluded.

Special equivalent drawdown time with anarbitrary rate history but with radial flowassumed for all transients.

Special equivalent drawdown time with anarbitrary rate history but with the functionalbehavior Sslf defined. While this sounds Idealfor all cases, in reality the entire range of thefunction Is never known from actual data, sothat some extrapolation is always required, Thissometimes gives rise to Intolerable errors, Itthe old story, the more data the better theinterpretation, assuming the data to be reliable.

Special equivalent drawdown time with twoequal and opposite rate changes and with linearflow assumed for all transients (Tandem linearrate changes).

Equivalent time at which the semilog straightline crosses the value of PR.

The magnitude of the change in rate at the i.thrate change

The magnitude of the change in rate at the firstrate change. For a drawdown test, this is qminus zero, or simply, q.

X(i=l ,N):(xi) Summation of xi over i, from 1 to N.

0 Porosity

% Transcendental mathematical constant;approximately 3.1416.

— . . -.

l_l(l=l,N):(xI) Repeated multiplication of XI over 1,from 1 to N.

k Fluid viscosity

Y Transcendental mathematical constant;approximately 1.7811

ACKNOWLEDGEMENTS

This paper would never have been written were it not forthose students who participated in my couree, “PressureTransient Analysls in Tight Rock” held In Calgary December7-11, 1987. Their Interest, enthusiasm and comprehensionwere an inspiration to me, To each of them go my sincerethanks.

REFERENCES

1.

2,

3.

4.

5.

6.

7.

8,

9.

10.

11,

Miller, C, C., Dyes, A,B., and Hutchinson, C,A.,Jr.: “TheEstimation of Permeability and Reservoir Pressure FromBottom Hole Pressure Build-Up Characterlstlcs,”Trarr.,AlME(1950 )189, 91-104

Horner, D.R.:”Pressure Build-Up in Wells,” Proc,, ThirdWorld Pet. Cong., The Hauge. (1951) Sec.11,503-523

Scott, J.O.:”The Effect of Vertical Fractures on TransientPressure Behavior of Wells,” J. Pet. Tech (Decl 963)1365-1369; Trans., AlME,228.

Matthews,C.S, and Russell, D.G,:Pressure Bu//dup and HowTests in We//s,Monograph Series, SPE, Dallas(l 967) 1.

Millheim, Keith K. and Cichowitz, Leo : “Testing andAnalyzing Low-Permeability Fractured Gas Wells,”SPE1 768 J,Pet. Tech. (Febl 968) 193-198;Trarrs,,AlME,243

Agarwal, Ram G., A1-Hussalny, Rafi, and Ramey,H.J.,Jr.:”An Investigation of Wellbore Storage and SkinEffect in Unsteady Liquid Flow: 1. Analytical Treatment:Sot. Pet, Eng. J.(Septl 970) 279-290; Trarrs.,AIME,949

Theory and Practice of Testing of Gas Wells, third edition,Energy Resources Consewation Board, Calgary, Alta.(1975)

Earlougher, R.C, Jr.: Advances in Well Test Analysis,Monograph Series, SPE, Dallas(l 977) 5.

Agarwal, Ram G.:’’’Real Gas Pseudo.Time’ A New Functionfor Pressure Buildup Analysis of MHF Gas Wells,” paper8279 presented at the 1979 SPE Annual TechnicalConference and Exposition, Las Vegas, Sept23-26.

Agarwal, Ram G.:”A New Method to Account for ProducingTime Effects When Drawdown Type Curves are Used toAnalyze Pressure Buildup and Other Test Data,” paper9289 presented at the 1980 SPE Annual TechnicalConference and Exposition, Dallas, Sept21 -24.

Swift, Samuel C.: Practlca/ We// Testing, O G C 1,Tulsa (1984)

604

SPE 17547 SAMUEL C. SWIH 11

APPENDIX A: GENERAL CASE DERIVATION

SfMPLE FLOW TEST

For a simple flow test (drawdown test):

f%= i(tD) ....................!.,................,,. ,,, ,,..,,,.., .......... (A-1 )

Where:

PD is the Dimensionless Potential.

tD is the Dlmenslonless Time.

Dimensionless Potential is the change in actual potentiai,measured in characteristic units for the particular testconditions under consideration; or:

pD = (P@tl-Pw)/PLi 1....................................... (A-2)

Where: Pw@tl is a constant equal to the pdential in theweii at t = tl, before the initial change in rate occurred.

Pw is the potentiai in the weli at any time, a variable.

PU 1 is the characteristic potential change for thesep[ ticuiar test conditions; or

pul =Aql*13/(2*n*k* h/~) ................................ (A-3)

WhereAqt is change in volumetric rate,

B is formation volume factor,

k is permeability,

h is thickness, and

~ is viscosity.

Likewise, Dimensionless Time is the change in actuaitime, measured in units which are characteristic of theparilcular test conditions under consideration; or:

tD = (t-tl)/tu......................................................!.(A-4)

Where tl is the time at which the initiai change in rateoccurred - a constant.

t is time measured from an arbitrary reference, avariabie.

tu is the characteristic time change for theseparticular test conditions; or:

tlj = O*p*Ct*rw A/k ................m.........oo.(A......oo..(A-5)

Where a is fractional porosity,

p is Viwrsity,

ct is totai compressibility,

rwA is areaiiy equivalent weiibore radius, and

k is permeability.

This oan be written as:

PDIJ = Ri”fh)u) ,,..,,..,..,,,,.,,.....................................(A-e)

Where

FtI = Aq~lqu ,............................................................ (A-7)

Where the additional subscript “U” is added to indicate thatall changes are measured from the initlai potentiai but thatthese changes are measured in dimensionless units whichare referenced to an arbitrary rate change, qu.

PDu = (f’w@tl-pw)/pu ................................... (A-8)

Where:

pu = qu *B/( 2*z*k*h/p) ................................(A.9)

Use of the subscript “U” on tD u is SOleiy fOrconsistency.

A commoniy seen example of the function , f ,isthe functionfor the semilog straight line which several solutions to thediffusivity equation approach for sufficiently iarge vaiues oftime.

pD = PDLN(tDU)= (1 /2)*in(4*tD U/V) ..........................................(A-l o)

Where, V is Euler’s constant, 1.7811 and theiipp?OXhWitiOfl’ !S vaiid fOr valUeS of tDu greater than 20.

COMPLEX FLOW TEST

For a complex flow test:

PDUT = Z(i=l .N):[Ri*~i(tDUi) ]........... ................... ..(A.l 1)

Where:

RI = Aq~qu .. . . .. . . .. . . .. . . .. .. .. . . .. . . .. . . .. . .. .. ..o. .. .. .. .. .. . .. . .. . (A-1 2)

Aqi = qi.q(i.l) .........................................................(A. ~3)

qi is production rate foiiowing the i-th rate change, and

qu is the arbitrary rate used in Pu.

N is the total number of rate changes

i is an index designating a particular rate change, and

X(i=l ,N): designates a summation from i=l to i=N.

Note that for rate changes within a singie weilbore, tu isnormaliy constant, so that:

tDLJi u (t-ti)/tlj .......................................................(A.l4)

if it is assumed that there is no difference in functional formfor aii rate changes:

fi(x) = f(x) ....................................................!. . . . . . . . . . (A-15)

So that:

pDUT = Z(i=l.N):[Ri'i( tDUi)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(A.le)

Note that for a simple flow test, Aql was (qO) or simply q,the productbn rate. For a comptex test each rate change causesa potential change which Is proportional to the magnitude ofthat particular rate change. Analogy to the simple case ispresewed by using the arbitrary rate, qu, both to normalizethe coefficients of the time functions (Ri) and to fix the valueof PIJ.

EQUIVALENT DRAWDOWN TIME

The concept of equivalent drawdown time, as used in welltesting, involves two separate ideas. First is the classicalmathematical concept of equivalency - attempting to reduce acomplex problem to an equivalent simple one. Second, isrecognizing that the reference level for potential, from whichthe change in potential is measured, is arbitrary.

Consider the second idea fi,-1. In order to measure potential(or anything eise for that matter) it is necessanr to define thestarting point (reference level or value) for the quantity andthe units in which changes from that ievel wiil be measured.For ordinary dimensionless potentiai the reference vaiue isPw@tl (the quiescent condition) and the unit is PU. However,both the reference potential level and the units of itsmeasurement are totaliy and separately, arbitrary. The unitof measurement for potential is fixed by seiecting anarbitrary reference rate. The reference level of potentiaifrom which changes in potential are to be measured is fixed byseiecting a reference time, tR.

in order to emphasize that the reference conditions arearbitrary, some redefinition of the quantities used to measuredimensionless potentiai is required. The unit of measurementwhich has been previously defined is Pu. in the simpie case,Pu, was determined by the rate, q. But in the more generaicase, PU, was determined by the arbitraty rate, qU. This waspossible only because of the introduction of the coefficients,R i, which normalize each actual rate change, Aqi, to thearbitrary rate, qu. Therefore no change is required in PU asiong as it is fully understood that qu, and therefore PU, istotaiiy arbitrary, and that the rmrmaiizing coefficients, Ri,are required in the summation.

The effect of the use of an arbitrary reference level ofpotentiai is more subtle. instead of measuring changes inpotential from the initiai condition they wiil now be measuredfrom the value of potentiai that occurred at the arbitrarytime, tR. This potential is defined as the potential in the weli attR, or, pR.

The finai effect of both the arbitrary reference rate and the(independently) arbitrary reference Ievei of potential ismanifested in the reference dimensioniess potentiai:

PDR= (PR.Pw)/PU ................................. ......................(A.l 7)

PDR= [(Pw@tl-Pw)-(Pw@tl -PR)]/PIJ ........................(A- 18)

PDR= pDUT(tDUi).pDUT(tDRi) .....................................(A.l9)

Where tDIJi is ordinary dimensionless time evaiuated at anytime, t:

tDui=(t.ti)/tU .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . ...(A.2O)

and tDRi iS dimensionless tiITIe evaluated at the referenCe time,tR:

tDRi=(tR-ti)/tlJ...........................o...................................(A.2l)

In general the dimensionless pressure terms in thepreceding equation invoive multiple rate changes, andtherefore are obtained from the standard superpositionequation for the complex case derived eariier:

I pDIJT(tDIJi)=~(i=t,N):Ri*f(tDui)......................,0,.., ,..,. (A-22)

I and,

I pDuT(tDRi)=~(i=i,hJ):Fti*f(k)Ri).................................(A-23)

I Therefore:

I pDR*~(i=l,N):Ri*[~( tDUi)-~(tDRi)] ...... ......... ....... .....(A-24)

If we now return to the first idea invoived in equivalentdrawdown time, we must ask if the compiex case shown abovecan be modeled as an equivalent Simple case by USin9 AtDeq.Obviousiy if this is possible then:

I pDR=~(AtDeq) ................................................................. (A-25)

I So that ecfuivaient drawdown time, if it exists, is the soiutir-m

I to:

I i(AtDeq)=~(i=l, N): Ri*[~(tDUi)-t(tDRi) ] ...................(A-26)

I Proceeding with the soiution,

AtDeq=f-l(z(i=l, fd):

Ri*[f(tDui).f(tDRi) ]} . .. .. .. .. .. .. . . .. . . .. . . .. . . . .. . . .. .. .. ...(A.27)

Where ~-1 is the inverse function off , or x = fll { ~[ x ] }.Or by using the definition of AtDeq:

Ateq = tU*~”l{Z(i=l ,N):Ri*[~((t-ti)/tlJ )-f((t13 -ti)/tlJ )]} . . .. . . . .. . . .. . .. .. . .. ...(A-26)

This is the general form of equivalent drawdown time for anarbitrary rate history and an arbitrary weil behavior,

I APPENDIX B: SPECIAL CASES

I AGARWAL EQUIVALENT TIME

The development done by Agarwai was in fact a speciai case ofAteq for a “Horner” rate history and assuming “radiai” flowbehavior. in terms of the generai Ateq development, theseassumptions are:

IN=2 ........................................................! .00..>.,..!(B-l )

I Aql = +q....................................................................(B.2)

Aq2 = -q .. . .. . .. . .. . .. . .. . .. . . . . .. . .. . . ...!. . . . .. . . . . . .. .. . . . . .. . .. . .. . . . . ..(B-3)

@ = .q .. . .. . .. . . . . .. . .. . . . . .. . .. . . . .. . . .. . .. . . . . .." .. . . ..`". "..." ".. ..""(B"4)

I R1 = +q )/(-q) = -1 ................................(B-5)

I R2 = (-q )/(-q) = +1 ,..........(B.6)................(B-6)

ltR= tz-tl = tp.....................o..........(B.7)

tl = o.......................................................`............(B.8)

tz = tp ....................................................................(B.9)

t-tz = At . . . .. . .. .. . .. . . . .. . . . .. .. . .. . .. . .. . . .. . .. . .. .. .. .. . . ... . . .. .. . .. ..(B.lo)

I

606

SPE17547 SAMUELC.SWIFI’ 13

t-tl = tp+At ............................................................(B.l 1)

of(x)={

Xso ......................4 B.12)

[In(l 0)/2] *[logl O(4*XIV)] x>O..,.....................(B-13)

V = 1.7811 ...........................................................(B-14)

~-l(y) ={ V/4}* {10 A[2”y/ln(10)l} ..............................(B-15)

Note that either of these approximations may do very strangeihii~~s for ti rms greater ihaii O but less than 20 ● tu (20time constants).

Defining A@, subject to the abOVeassumption as AteqA:

AteqA={tlJ}*{~/4}*{ioA[2*y/[fl( lo)]} (B-16)

Y = ~(1-l,2):Ri*[~( tDUi)-~(tDRi)] (B-17)

Y= -f([t-o]/tu)+f([tp-o]/tlJ )+f([t-tPl/tu )-t([tP-tPl/tu) (B-16)

Y = [ln(l O)/2]*{- lloglo(4*[t-o]/ttJ/v)]+ [loglo(4*[tp-o] /tlJ/v)]+ [loglo(4*[t-tp] /tlf/v)]- [0]} .............................................(B.l9)

Clearly:

[2*y/ln(l O)]=[tog10(4*tp*At/( tp+At)/tLt/V)] .,......,............... (B-2o)

Or:

(10 A[2*y/ln(10)])={4*tp*At/(tp+At) /tU/Y } ,..,, .,.,0,..,.... .,.,,,.,,,...!!. (B-21 )

And:

At~qA=(tLJ}*{~/4)*

{4*tp*At/(tp+At)lt@4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(B-22)

So that, finaliy:

AteqA = tp ● At /(tp+At) . . . . . . . . . . . ...(B. . . . . . . . . . . . . . . . . . . . ..(B-23)

EQUIVALENT TIME- RADIAL

If the flow history is arbitrary but “radial” flow behavior isassumed:

X50,..,............(B.......(B-24)f(x) = {0

[ln(l O)/2]*[ioglo(4*x/Y)] x> O .. .. .. . . .. . . .. . . .. . .. . . ..(B-25)

Y= I.7811 ..,,.,...,,,,...,.., ................(B-26)

~-l (y) s {W4}*{10A[2*y/in( lO)l} . . . . . .. . ....!.. .. . . . . .. . . .(B-27)

Note that either of these approximations may do very strangethings for tImes greater than O but less than20 ● tu ( 20 time constants ).

Deflnlng A~q, subject to the above assumption as AbqR:

AteqA ={tu}*{Y/4}*{i oA[2*y/tn(i 0)]} ....................(B.28)

y = X(i=l ,N):Ri*[logl O#([t-ti]ltU)-lOgl O#([tR-ti]/tU)] ... . . .. .. .. .. .. . ..o....(B-29)

X < 0.......................(B-3O)loglo#(x)=l’

[ln(l O)/2]”[loglo(4*x/v)] X>o ,,,,.,, ........31).....(B.31)

v= 1,7811 ................................(B-32)

or:

Y s X(i=l, N):logl O{(t-tij’Ri#/(tR-ti)ARi#} .... .... .................(B-33)

where:

(V ● tu / 4)’Ri for XSO.,.........(B.............(B.34)xARi#={

xfiR\ for x>O...........................(B-35)

Note that both time difference terms in equation 6-31 are keptseparate in order to avoid oonfusion in ihe application of theRi# function. Either, or both, arguments may be less than orequal to zero. So that:

AteqR = {tu*y/4}*{~(i=l, N):(t-ti)ARi#/(tR -ti) AR i# } ..................................................(B.36)

wherel_l(i=l,N):(xi) = XI*;.2*X3*...*XN .. .. . . .. . . .. .. .. .. .. . . .. . . . ...!.. (B-37)

and(Y ● tu / 4)ARi for XSO.,........................(B-34)

xARi#={xARi for x>O,..........................(B-35)

EQUIVALENT’ TIME- LINEAR

If the flow history is arbitrary but “linear” flow behavior isassumed:

X<o .. .. . ..o. . .. .. . .. . . ... . . ... .. .. . ..(l3-38)f(x) ={ 0

2 ●SQRT(x/rr) X <0 .......... .......................(B-39)

f-l(Y) = 7c*(y/2)A2 .,,,,.,.,,..,,., ..............................(B-4o)

ARBITRARY RATE HISTORY

Defining Ateq, subject to the &x3Vt3 aSSUIT@On %5 AteqL:

AteqL = {tlJ}*(n}*{[y/2 ]A2} ....................4B..............4B-41 )

Y = {2/SO RT(fi*tu)} ●{Z(i=l ,N):Ri*[DSQT#(t-ti)-DSQT# (tR-ti)]} .. ....... .................(B-42)

o X<o . . .. .. .. . . .. .. .. .. .. . . .. . . . .. . . ..(B-43)DSQT#(x)=(

sQRT(4”x/7c) X>o ..................................(B.44)

or:

OuI

APPLICATION OF EQUIVALENT DRAWDOWN TIME IN WELL TESTING SPE 17547

AtaqL={z(\=l ,N):RI*[SQRT#(t-ti)-SQRT# (tR-ii)]}A2 ..,...,...,...,., ......(B-45)

where:

o X’so .....................,......0......(B-46)SQRT#(x)={

SQRT(X) X>o.,,,.,..............................(B-47)

HORNER RATE HISTORY

For this special case:

N=2 ..................................................o...............(B.46)

Aql = +q.....................................,,....0..,..,,.., ............(B-49)

Aq2 = .q ..................................................................(B.5O)

%t = .q ..................................................................(B.5l)

R1 = (+q)!(-q) = -1 .......................................(B-52)

R2 = (-q)/( -q) = +1.......................................(B-53)

tR = tz-tl = tp .......................................(B-54)

tl = O..................................................................(B.55)

tz = tp ...............am.,,,... ..,..,, .,,..,,.,,,..,,, ...................(B-56)

t-t2 = At................................................................. ,(B-57)

t-tl = tp+At ..............................................................(B.56)

AteqL = {[-l] *[ SQRT(tp+At)-SQRT( tp)]+[l]”[SQRT(At)]}A2 ....................................(B-59)

Or:

AteqL=(SQRT(tp)+SQRT(At)-SQRT(tP+At))A2................. . ..(B-60)

STEPWISE SHUTIN

An Interesting special case arises when a shutin test occursas a stepwise process. This might be a surface shutin withwellbore storage (constant or changing ; or an intentionallystepwise process). In this Jase an interesting complicationarises due to the fact that the coefficients of the time terms(usually 4/tuPf) do not quite disappear.

For this case the rate history is broken into two parts. It isassumed that there are N total rate changes, but only M ratechanges occur prior to the beginning of the test. Note that therate change marking the beginning of the test falls in thesecond group ( i = M+l ). Also:

tR = t(M+l) ...................................................................(B.6l)

@ =-X(i=l, M):(Aqi)(Negative of rate prior to stwtin) . . . . .. .. .. .. . . .. .. .. . . . .. . . ..(B-62)

So that:

Ateq=tu*~-l {Z(i=l,M):Ri*[ f((t”ti)/tu )-t((tR-ti)/tu)l+X(i=M+l ,N):Ri*[~((t-ti) /tU)]} .... .. .. .. .. .. . .. . .. .. ..(B-63)

The behavior of the parameter, tu , is tied to themathematical properties of the function, ~ , For the PDLIfunction which is weil behaved near t=tR, the parameter, tu ,falls out of the equation. So that the above equation reduces toEq (B-43). Note that splitting the summation has no effect.Unfortunately, the wideiy used and more familiar logarithmicapproximation, PDLN is not so kind.

In the case of the logarithmic approximation, the summationmust be split yet a second time, to see what Is OCCurring.Let tLbe the time of the iast rate change which occurs before thetime, t, that the current measurement measurement ofpotential is made. Then the sum from L+l to N is zero so thatthe summation In Eq(B-64) is truncated at L.

Ateq ~ tU*~-l(X(i=l ,M):Ri●[i((t-ti)/tU )-~((tR-ti)/tU)]+2(i=M+l ,L):Ri*[j((t-ti) /tU)]} .,, .,.,!,....,.,.. ,. .,,,.., (B-64)

When ~ is assumed to be PDLN:

AteqR= {ll(i=l,M): [(t”ti)/(tR”ti) ]ARi}●{~(i=M+l ,L):(t-ti)ARi}●{(tU*Y/4)fi[l -Z(i=M+l ,L) :Ri]} .....(B..................(B-65)

This last term is the probleml For the special case where theexponent, [1 -X(i=M+l ,L):Ri] , Is zero (after the ‘enfY’ ofafterflow) this term setties down and behaves. However priorto that time the value of tu affects tke value of equivalent time.Furthermore this effect changes with time (as L Increases),ieaving no effect only when the true sandface rate is zero.

As if this were not bad enough, when the sandface ratechanges continuously some of the dimensionless times are suchthat the function PDLN may be a poor or intolerableapproximation. PCLN is a vaiid approximation oniy for tD >20, it is poor for W4 < tD e 20 and it becomes physicallyintolerable fOr tD c W4 , giving pressure change contributionwith the opposite sign of the real changes.

Therefore, AteqR, is a poor candidate for analyzin9 ratechanges which are influenced by afterflow.

Author’s note: This incidentally explains why Horner plotson buiidup data in tight weiis (infiuencecf by oniy iinear flowand storage) appear to break over to a fotaily meaninglesssemilog straight iine. This break is not caused by astorage/radial transition but rather by “misiocation” of thestorage affected data. if one contemplates the known behavior ofstorage/linear transitions, no such break should be seen.Therefore such a break is frequently interpreted as finaiiygetting to radial flow. The type curve for these cases is selfjustifying when the speciai case of At~qR , narnelY, A~qAt is

used, since the same errors occur on the iog-iog plot. A veryabrupt transition is seen which implies radiai fiow withsevere damage. if AteqL is used then Only linear fiow is seen.This is a ciassic case of two distinctly different weii behaviorscausing neariy identicai observed data. Either couid be correct,but the physically correct one is normaliy indicated by thecircumstances, not by the data. That is why krmwiedge andexperience wiii always be a critical element of well testinterpretation.

However, AteqR often does become weil behaved again foriong shutins. in these circumstances afterflow approaches zeroand tD ( hopefully but not necessarily ) is much greater than20. if the iimit of Eq B-62 is considered as t approaches 00then t-ti = t-tL ; and the exponent approaches zero, so that:

linlit(t=_)Ateq R=l/i(H(i=l,L): [(tR-ti)lARi) . .. .. . . .. . . .. . . .. . . . ..(B-66)

608

SPE17547 SAMUEL C. SWIFT 15.-

Which fora Horner rate Mstory, L=l, reduces to:

lirnit(t*co)Ate~R = tp ... ,,..,0.,,,.,.,,,, ..,. ,,, ,.,, .,....,,. ,,, ,.,, .,, ,.,.. .,(B-67)

Whlchclearly shows that If the flow time was insufficient tosee radial flow effects, no amount of buildup time will giveradial flow parameters.

APPENDIX C: NUMERICAL METHODS

The procedure used Is primitive numerical analysis (sincethe author lacks extensive expertise in the sublect). It isIncluded hereto clari~the actual procedure used, but also inthe hope that It will shortly be brought somewhere near thestate of the art In application of numerical procedures.

My first attempt Involved the following steps:

1. Assume that the obsewed potential change versus theactual time change was a first approximation to theunknown function. (Note obvious problems are themethods of extrapolation to times before and after theobserved data and to a lesser extent interpolation, I used(in log-log space) linear !merpolaticn and linearextrapolation from the nearest two points.)

2, Calculate equivalent time using known rate history andthe previous approximation to ~(x).

3, Use Ap versus the transformed Ateq as the ungradedapproximation to ~(x).

4, Repeat steps 2 and 3 until it converged,

Although this worked after a fashion, it tended to clump thedata about vertical lines. After about 10 iterations the changefrom iteration to iteration increased, and so did the absoluteerror ( for this example I knew the answer ),

My next attempt was to modify step three to:

3. Use Ap versus a weighed average of the old and new valuesof Ateq , Afeq:app= (M*Ateq:old + Ateq:new)l(M+l ) asthe ungraded approximation to ~(x). M equal to 1 and 2were tried.

The results were only slightly better, with clumping stillobvious.

My final attempt was to use step 3, as modified with M = 1(a simple average), but to smooth the new average with a leastsquares fit straight line thru 5 points nearest to Ap. The newapproximation to Ateq was then taken to be the value on thisline at Ap, After 13 iterations (and much soul searching aboutdeadlines and doing less than the best possible job) this lookedgood enough to me,

609

-.——TABLE 1

EQUIVALENT DRAWDOWN itMES IN COMMONUSE

N-m. ... .. .. ... .. .. ... .. . .... ..Symbol . . .. . . . ..~ . . . ... . ... . ...qR tR,,, .. . .. .. R.sun . ...,...

Agarwal(TandemRadial) A&A Rarflal(PDLN) -q tp tp’At/(tp+At)EquivalentTlma

AganvalTandamLinear AteqT Llnaar(PDLl) -q Ip [d(tp)+~(At)-~( Ip+At)]A2Equlvalant Time

CammonTandemUnear AteqT Llnear(PDLl) +q O [i(tp+At)-d(Al)]~2EqulvalantTime

Homer Tlma At~H Radlal(PDLN) -q O At/[lP+At]

Hornar Ratio A*R Radlal(PDLN) +q O [tP+At]/Al

Nota that both of tha commonly used Hornar parameter are actually equivalentdrawdowntlmaa retarancadto the lime that flow was Initlalad.

TABLE 2WELL TEST DATA FOR WA7ERTON #l

PARAM~ERS

0.045 v. 0.671: 0.::43 [$)) ~: 0.00012 ( pall) CM 0.00313 (bbl/pal)0- 0.693 RbbUMCF rw - 0.316 (ft)

RATE HISTORYi tl(hr) Aql ( MCFD )1 0 10002 12 -2503 24 -2504 36 -2505 48 -250

Observed Potentlela

ET(hr) BHP(psla)

0.00 4255.006.00 2302,69 ●*’”’** CALCULATE FROM POTENTIAL BEHAVIOR ‘*’****

12.00 2029.001S.00 2356.0524.00 2314.56 teff n 46+36+24+12 - 120hr30!00 2756,3136.00 2783.2S At(hr) Ap(psl) AteqA(hr) AteqT(hr) AtaqR(hr) AtaqL(hr) AteqLA(hr)42.00 3274.51

46.00000 3338.704a.00020 3339.16

0.000000.000200.000500.001000.002000.005000.010000.020000.050000.100000.200000.500001.000002.00000

0.00 0.00000 0.00000 0.00000 0.00000 0.000000.46 0.00020 0.00020 0.00020 0.00020 0.00000

48.00050 3339.80 1.10 0.00050 0.00050 0.00050 0.000s0 0.0000048.00100 3340.83 2,13 0,00100 0.00100 0.00100 0.00100 0.00002

E 46.00200 3342.81 4.11 0.00200 0.00199 0.00200 0.00201 0.00C06— 48.00500 3348.33 9.63 0.00500 0.00497 0.00500 0.00503 0.00035s

I46.01000 3356.56 17.68 0,01000 0.00991 0.01001 0.01006 0.0012548.02000 3370,93 32.23 0.02000 0.01974 0.02003 0.02023 0.0046346.05000 3404.69 65.99 0.0499S 0.04696 0,05017 0,05092 0.0209746.10000 3444.77 106,07 0.09992 0.09713 0.10069 0.10259 0.0613348.20000 3497,58 158.88 0.19967 0.19192 0.20277 0.20730 0.1705848,50000 3562.76 244.06 0.49793 0.46626 0,51718 0,52833 0.5774449.00000 3654.17 315.47 0.99174 0. S1096 1.06S00 1.07754 1.4117550.00000 3730!07 391.37 1.96721 1.75113 2.26623 2,20453 3.4710253.00000 3S36.66 5.00000 49S.18 4,60000 4.04062 8.55794 5.64S96 11.7226958.00000 3923.62 10.00000 585.12 9.23077 7.37109 15.58745 11.24003 30.3620766.00000 4013.29 20.00000 674.59 17.14266 12.91891 37,78200 21,25617 80.1963196.00000 4116.69 4S.00000 777.99 34,26571 24.21794 105,0000 42.02279 272.0291

SPE610

17543

TABLE 3WA’7ERTON#l OATA@RRECTED FOR AF7ERFLCIW

Note Out:

(I ET (hr]

123456

:910111213141516171819202122

0,000012.000024.000036,000040.000048.000248.000648.001048.002048.005048.010048.020048.050048.100048.2ooo48.500049.000050.000053.000058.000068.000096.0000

qFf -.250 rncfdfU-lE+40: (Th18reduces AleqPDCiAto 4teqLA)Storage prior 10 flmf Shutfn is ignoredStorage aflerflnal shutin Ia constan! al 0.00452 rwflpdAIIer shulin qsf h glvon by((Ap[l+l) .Apl)/(AI[i+l]-All)0.00452.24} . qsurfaceIf thesloragacapacityoftlw wllbore changas,C (0.01M52)canh changedat anyetepLhdlofAteqtAas ETgoesto - is 512hr

13HP(psla) Al (hr) AP (Psi) @ (mCfd)Aql (mcfd) RI AteqLA (hr)

3336,7o3339,163339,603340,633342.81334s.333356.583370,933404.693444.773497.583562.763654.173730.073836.683923.824013.294118.69

0.00000.00020.00050.00100.00200.00500.01000.02000.05000.10000.20000.5000t.000o2.00005.0000

10.000020.000048.0000

0.000.46f.lo2.134.119.63

17.8832.2365.99

106.07158.66244.06315.47391.37496.18585.12674.59777.99

!000,0750,0500.0250.0246.0231 0224,3215.5190.6179,1155,6122.1

67.057.330,815.5

8.23.91.91.00.4

1000,00-250.00-250.00-250.00

.2.01.16.98

.6.72-8.77

-16,92.20.51.23,29.33.68.35.11-29.70-28.5o.15.32

.7.28

.4.37

-4.00001.00001.0000 lU_l E+401.00000.0081 0.0000000.0679 0.0000010.0269 0.0000040.0351 0.0000160.0637 0.0000650,0821 0.0003540.0932 0.00130.1348 0.00460.1405 0.02100.1168 0.06130.1080 0,17080.0613 0.57740.0291 1,41170.0175 3.4710

-1,96 0.0079 11.7229.0.92 0,0037 30,3021-0.57 0.0023 80.1963.0.40 0.0016 272,0291

h= 36o (II)

P= 0,69 (Cp)0. 1.3

I123

ET(hr) i3HP(psia)

1440,00 3839.241440,01 3854,411440.02 32!00.471440.05 3872,141440.10 3077,011440.20 3801,901440.50 3067,971441,00 3692,161442,00 3696.101445,00 3900.151450,00 3909.281464.00 3925,1t1476.00 3934,611408.00 3942.331512,00 3954,781524.00 3959.981536.00 3984.741560.00 3973.191584.00 3960.551008.00 3967.10

TABLE 4SOMBRERO #l ❑UILDUPT EST DATA

PARAMETERS

0. 0.19ct - 0.000125 ( psi.1)

W. 0.316 ( It )

RATE HISTORY

Ii(hr)o

1439.751440

POTENTIAL BEHAVIOR

A!(hr)

0,000.010.020,050.100.200.501.002,005,00

10,0024.oo36.0046.oo72.oo64.oO98.00

120.00144,00t88.00

qRm 25oOO (BPDtR = 1440 (hr)PR - 4255 (pSt )

. . . . . . . . . . . . . . . . .Ip (hr) . 208.2

Aql ( SPD )5000

20000-25000

Number of

13

Ap(psi) AlaqX(hr) AteqA(hr) AteqT(hr) AleqR(hr) AteqL(hr) AleqS(fIr)

0.00 0.0000 0.0000 0.0000 0.0000 0.0000 0.000015.17 0.0095 0,0100 0.0100 0.0097 0.0085 0.000421.23 0.0166 0,0200 0.0199 0,0160 0.0160 0.018232.90 0.0446 0.0500 0.0497 0,0432 0.0343 0.042737,77 0.0690 0.1000 0.0991 0.0784 0.0589 0.075342,86 0.1123 0.1999 0.1974 0.1250 0.09s1 C.119148.73 0.1912 0.4991 0,4917 0.207652.92

0.1705 0.19050.2856 0,9985 0.970s 0.2759 0.2529 0.2817

56.07 0.3544 1.9S62 1.9178 0,3448 0.3708SO.91 0.4664

0.35214,9147 4,7405 0.4374

70.020.6239

0.01400.5640

9.6647 9.0941 0,5119 0.9501 0.9290S5,88 1.8432 22,1550 21,2241 0,s157 1.0806 1.930095.37 2.8037 32,0025 28,1871 0.80S4 2.2174 2.8S44

103.09 3.6283 41.1469 34.7994 0.7079 2.7oOO 3.3S36115,52 5.4932 57,60S0 50.7796 0,7683 3.6019 4.5790120.75 8.0015 85,0425 51.303S 0!7093 4.0156 5.0804125.51 6.4840 72.0125 55.9792 0.S096 4.4121 5.5498133.95 7.3082 84.7232 69.3870 0.8443 5.1614 6.3549141.32 8.2204 98.0222 76.9587 0.8733 5.S024 7.0622147.S6 6,9998 106.1324 03.6953 0.S961 6.5233 7.7616155,66220.42 i:348.52 50492.S8 100

611 SPE 17547

1000

AteqX

100

~

s

10

1.001 .01 .1 1 10 100

Ateq (hr]

Fig.1- Typa curve for WatertorI #1

4400

4000

3300

3600

3400

32U0.1 1 10 100 1000

M-R (l@

Fig. 2- Samilog plot for Vifaterton#1

AteqR

100 ,J

, ●

m

1

.01 .1 1 10 100

4000❑El

1•B~

ElQ

❑

3900 ‘4

p A

#

I P*L:4255

Ateqs* = [0.2*d(1440)+0.8*~(0.25)]”2 =63.63

38000 1 2 3

Fig. 3-Type curve for Sombrero #1

dAte@(~hr)

fig. 4- Square root plot for Sombrero #1