Advances in PracticalWell-Test AnalysisH.J. Ramey Jr., WE, Stanford U

hrtroductlon

The1980’sproduced important advances inweff-test analysis, The large number of newand sometimes competitive methods, how-ever, also pr.duced confusion. The min ob-jective of this paper is to consider the currentstate ofpractkd wef3-testanalysis methcds.

Often, new studies produce conclusionsthat time proves incomplete or partly untrue.The storage log-log type curve was initiallypresented as a method to anafyze short-timedata. This was later found to be impossible.But the diagnostic value of log-log curveswas far more important Smn shon-tkmeanal-ysis. Later, the derivative was added to thetype curve, leading to the conclusions thatHomer curves were no kmgerneces$my andthat shom-dme analysis was now possible.Neither conclusion is entirdy correct, yetthe diagnostic value of the derivativeremains.

Another major development was com-puter-aided interpretation. The computewas necesswy to dtierentiate data andhelped to prepare the large number of graphsrequired for modem interpretation. An im-portant breakthrough resulted with the de-velopment of nonlinear regression forspecific models and the ability to considerrate variation. Results of an interpretationconld be used to simulate test data, and the”“simulatedand field,data cou2dbe compared.The regression coefficient or confidenceliit provided a quantitative measure of theagreement betwem field data and the modelchosen. Results were also used to determinewhere a correct straight liie would be cma Homer buifdup graph. This procedureproved that fmdmg a Horner straight-lineslope with precision was diilcult. Wide-Wread use of electronic pressiue gauges andcomputer data acquisition created new prob-lems for a well-test analyst. A new problemrevealed by type-curve analysis is present.ed here.

Background

Well tests were originally pm-formed to de-termine the quality of a welf or to permitestimation of producing rates at differentproducing pressures. Well-test interpretation

CoPy@ht1992Societyd PetroleumEnQlneers

650

evolved into a sophkticated field of findingwhether poor producing quality was a resultof weUdtunage, poor formation permeabfi-ity, or 10wformation pressure—thus, a lowdriving force. Tests were also performed totind static formation pressuie for material-babmce studies of gas or od in place.

fn the 1950’s, Homerl and MiUer et al 2started the first em of modem well-test anal-ysis: straight-line methods. Semifog graphsof shut-in pressure vs. the Iogarithm of shut-in time, or the ratio of prcducing time plusshut-in time, to shut-in time generated astraight line with a reciprocal slope that wasrelated to eff~tive permeability. Thestraight lime could be extrapolated to longshut-in times to estimate static pressure.

Moore et al 3 and van Everdingen andHurst+ had pr.sviously presented the ccm-cept of wellbore sqmage and had taught use-ful analytical methods to solve well-testproblems. fn ,1953; Hurst5 and van Ever-dingen6 presbnted the concept of theweUbore-sHn effect (factor). Ftiy, Maf-thews et al. 7 presented an analytical metl-.od to correct extrapolated static pressmesfor a well in various drainage boundaries.

By 1967, pressure-buildup methcds wereso imps’tant that the SPE publisbed the fustmonograph on weU-test analysis by Mat-thews and Russell. 8 This important bcok isstiUone of the most popular volumes in theSPE Monograph Series, with 31,542 copiessold between 1967 and 1991. The Matthewsand RusseU monograph, however, signaledthe end of the wraight-~ne era of weU-teS$.mdysis. AUconstant-rate well-test soluticmsin a slab reservoti bad a semilog straight I.tieof the same S1OP6,and it was not possibleto identify the proper straight line when twoor more straight segnients appeared in a sil-gle weU test.

Surprisingly, the 1949 van Everdingenand Hurst4 paper contained a log-log typecurve for the weUbore-storage problem thatstarted the second era of weU-test analysis.Theis9 described log-log graphic methods,as welf as the Homer semilog pressme-buifdop graph. Theis’ work was consideredhistorical and was reprinted by the SPE. 10

Like works on straight-line methods, log-log graphing approaches were proposed bytwo separate research groups 11.12at about

.lune 1992. JPT

sy~D@inguikhed

the same time. lle log-log procedure af-lowed comparison of field data and an ana-lytical nwdel throughout the entire timedomain and could be used to fmd the starttime of a semiIog straight line. New log-logtype CUIV.Swere produced rapidly for awide variety of flow mcdels (hydraulic frac-tures, dud-pxosity systems, etc.). The sec-ond SPE well-test monograph byEarlougher13 reviews these developmentsthoroughly.

A 1976 studytd reviewed hnportant fin-dingsof the previous decade and stressedpractical applications of new methods thatwere not previously described. At that time,three log-log versions of the storage and skinproblem had appeared, and several typecuwes for fractured wells had been present-ed Many weU-test analysts threw up theirhands at the prdifeiation of type cuv.s. The1976 study concluded that a major advan-tage of the log-log type curve w that it wasusually possible to identify the flow modeland to fmd the martof a wnifog straight linefor the appropriate model. It was recom-mended that a Homer straight line shoufdstitl be used as the fti basis of analysiswhere possible,

The 1976 study pointed out problems withfracture type cumes (short apparent fracturelengths for large jobs) and other existingworries on selection of an appropriate typecurve, One major ad~antage of the under-standing that the new methods brought wasthat it was possible to correct bad test dataand, in many cases, to fdJ in missing data.

Problems identifr?d in the 1976 smdy havesince been solved and hue breakthroughspresented, The purpose of this study is topresent usetil, practical meth0d3 for welf-test analysis and design, Important new in-formation includes selection of an indu8~-standard storage type CUIVC,development ofderivative methods, solution of the t3nite-fmcture<onductivity problem, and development of computer-aided interpretation anddesign.

Storage Type Cuwe

The tmditiond e“gineefing dilemma is thatthe wc of two methods lmsolve one problemyields differmt answers. Well-test analysisis replete. with this probIem. Papers offen

JPT ● June 1992

Author --

say somethiog like c‘the data were analyzedby tie Smith method yielding 20 md, theJones metbcd yielding 2 darcies, and tie vonSchultz type curve yielding 0.15 md. Theaverage is. .“ Remarkably, the threedifferent methods are usuaffy differentgraphs of the same solution.

The log-log type curves described for thetirst time in tie J3arlougher13 monographwere considered controversial by the SPEBoard of Drectors. An early SPE bead de-cision was not to pubfish fuff-sc.afelog-logtype curves because this would indicate SPEapproval of log-log type curves or a partic-UIZ type curve. Fortunately, this decisionwas reversed.

Gringarten et al. 15ended the controversyover tie best form of the welfbore-storageand skin-effect type curve h 1978. Theirtype cum. of the log of dmensiotdess pra-sure vs. the log of tD/CD, with CDea asa parameter, became the industry sta.ndasd.Their original type curve combined radialflow snd fracture flow results and indicatedthe effect of pmdncing time On buildups.The type curve was a remarkable improve-ment that was accepted immediately. So oneproblem identified in Ref. 14 was solved.

A monnmentalfy important aspect of theGringarten et al 15 shtdy is that it camefrom the service side of the petroleum h-dustry. Traditionally, the service industrymarketed hardware and was not strong ininnovation in pressure-mnsient analysis.The Gringarten et al. study signaled aremarkable change. Part of the change wascomected with bardwace. The quamz~stalpressure gauge offered the possibility ofperhaps eight-digit pressure measurements.incredible results were.possible. One coulddetect the moon passing overhead twice aday with a pressure gauge 10,000 ft deepin a shut-in well. The market for sensitivepressure measurements exploded and so didimprovements in other types of pressuremeasurements in welIs.

Two other factors were also changingwell-test analysis. Rapid developments incomputer chips made small, large-storagecomputers possible and perhaps fueled de-velopments in applied mathematics and com-puter science. A statisticzd economist inGermany offered a three-page paper on in-

verting Laplace transforms, 16The Stehfestalgorithm has had a major impact on petr-oleum engineering welf-test analysis andgroundwater-hydrolysis pump-test analysis.

Sometimes, it is easier to evaluate a sO-Iution in Laplace space than in red spaceor time. Some current computer programsstore models in Laplace form and even doregression in Laplace space.

Jn summary, Gringarten et af. 15present-ed a convenient type curve for model andsemifog straight-line identification, but amajor step remained-combining pressuresnd pressur~derivative type curves.

Derivative Methods

Bourdet et al. 17.18presented a major devel-opment in the mid-1980’s. They advocatedsuperimposing the log-time de.-ivstiveof thestorage and skin solution on the acceptedform 15of the storage type curve. ‘f@ Ob-served that the combination of the two typecurves could lead to a unique match withfield data that eliminated the need for aHomer buildup graph. They also obsemdthat &ta tien before a’semilog straight finecould often be interpreted. As is often thecaw, initi8Jclaims for this new methcd werenot entirely correct, and some major advan-tages of a derivative graph were not yetevident.

My initial impression of the derivativeprwedure was that high-precision data wasrequired to make such a procedure feasible.Actually, the key was development of goodnumerical procedures to differentiate fielddata. Bourdon-type data tslen in the 1960’s,as ,welf as high-precision liquid-level dataand gas-purged capillary tube data hsve beensuccessfdly differentiated.

Practical aspects of derivative use are dis-cussed later. One final important step madesince 1976 concerns computer-aided in-terpretation. ‘fbc varie~ of type curves andsem.iloggraphs involved in welf-test analysisand dtiferentiation of field data were per-fectly suited to a PC with appropriatesoftware.

ComputewAided Interpretation

Probably every research effort involved inwell-test analysis began some work oncomputer-aided intapretation in the 1960’s.

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Interpretations ranged from computer plot-ting of buildup graphs to interactive numer-ical simulators capable of consideringcertain formation heterogmeities, The fxstmodem effon to come to my attention wasinitiated by A.C. Gringarten in 1978. Heproduced a sophisticated program that usedlog-log ty~ curves for model identification,progressed to Homer analysis, and eventu-ally verkied the interpretation by comparingfield data with a simulation that used the in-terpreted parameters. .

The Gringarten program was developedfor field use with a portable computer forweflsite test analysis. This approach starteda new oilfield service industry. The nextmajor development was application of non-line% regression in Laplace space by Rosaand Home. 19Their approach avoided hid-and-error matching of field data with amcdel simulation and provided a quantitativemeasure of the msults<onfldence limits.

A major benetit of computer analysis isthat it allows us to correct, ffiter, and selecta manageable data set tiom thousands ofdata points recorded today and to performmany tedious operations, such as differen-tiation after numerous time adjustments. Thecomputer frees the anafyst to think and to

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my various possible interpretative mcdels toseek the best possible interpretation.

The importance of computer-aided in-trepremtive methods cannot be overstated.The point is not just that the computer saveslabor, but that it permits a quantum jumpin the accuracy of the interpretation. 1wouldnot interpret a well test today without com-puter aid simply because I wish to do thebest job possible. This cannot be accom-plished by findiog a sendlog straight line ona Homer graph.

Well-test analysis is often referred to, asthe inverse problem. The answer is “four,”what is the question. Questions might bewhat is tvq plus two or what does a goIferyell when he hits a ball into a group? Theream many possl%leanswers. Gringarten de-scribed a logical approach to tbe inverseproblem for weU-test analysis. We se$k thesimplest model that expkins the well-testdata. Computer-aided interpretation helpswith the nommiqueness problem. Many P*sible models can be tested and statistical

June 1992- JPT

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measures used to select the best fit of themodels tested.

Perhaps the best way to illustratelhe newapproach to welf-test analysis is thro@ anexample.

Simulated Buildup

The purpose of this example is to show thatHomer buildup graphs often yield poor es-timates of permeability and skin. Thus, anexact simulation was prepared with knowncorred formation parameters. Table 1presents input formation and fluid parame-ters and simulated pressure-buildup &ta.

Ffg. 1 is a log-log type curve showing aunit-slope storage liiq a semilog straightline should occur at a shut-in time of about10 hours. Fig. 2 is a Homer buldup graphwhere the semilog straight line staris at ashut-in tirneof 10 hours. ThestmigMlinewas selected by magnifying the straight line

JPT o June 1992

portion. The last nine buildup points appearto tit a straight line exactly.

The results of the Homer analysis areapermeability of37.6 md, a skin eEectof6.2,and a P* of 2,827 psi. These results do notmatch theentered values of48md, skin of10, and inbiaI pressure of 2,g10 psi. How-ever, the results appear reasonable and rep-resent results of a “field” data analysiswhere correct values are not known. Type-curve analysis indicated that the wellborestorage and skin modeI was appropriate.

The resulting parameters can also be usedwith the stortwe and skin model to simulatethe test, and he result compared with thedata ori@rally interpreted. Fig. 3 shows thatsuch a. comparison is not good. Next, theconfidence limits of the original interpre@-tion are computed. Table 2 gives the result.Tbe permeabfity from the Homer interpre-tation has an error of +32%. Actually, errorfor the storage model will always be nega-

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tive. Bothpemneability andskin are eithercorrect or underestimated.

Finally, tbe buildup data in Table 1 wereinterpreted witbnonlinear regression19 andthe wellbore storage and skin model. Theregression produced the input parameters(permeability, skin, and initial pressure)exactly.

The results of the preceding exercise zetypical of my experience with field data ir-&pretation~ FoE cases that appetm to havealmost ideal log-log storage @curves andHomer straight lines, a test of the pamme-ters by simulating the field data oftenproduces pm WTeement.Con6dence limitsof the Homer parameters are ‘often in the50% to 150% error range with skin-effecterrors larger than permeability estimates.Subsequent nonlinear regression producesnew parameters with confidence limits re-duced 10-fold. Agreement between fielddata and model-simulated results is often ex-ceUent.

The lame confidence fim.its for Horner

653

analysis w-ma surprise. I previously thoughtthat accuracy of a permw.bilby estimate wasdetermined by the accuracy of graphdifferentiation or perhaps +5%. The prob-lem is more complex. The apparent Homerstraight line is rarely ttie correct one.

Another informative exercise is to plot thecorrect straight line on the Homer graph

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after regression. I’ve tried this many timesin the forlorn hope that 1 could learn howto pick the correct Homer straight line. Myonly conclusion is that I do not know howto plot a Homer straight line. It is not wiseto depend on stmigh-line hand methcdswhen regression with a proper model is pos-sible. AIIotherconclusion is that the potentialerror in p.mmabiii~ and skin tlom a Homeranalysis is often much luger than 1 previ-ously thought, and both parameters &e al-ways underestimated.

There are practical wcasions when aHomer graph mux be used. Perhaps a prop-er model is not obvious, or computer soft-ware is not available, but beware thepotential lack of accuracy.

Another observation is that regressionoften changes the skin effect more than thepermeability. Sometimes, an apparent nega-tive skin value from a Homer graph wilf befound to be positive during regression. FI-mlly, limited experience with interpretationof successive tests on the samewell indicamsthat regression reduces wwiation in apparentpermeability from nm to run.

TM confidence limits from regressionrepresent how field data and an analyticalmodel match and how sensitive regressionis to a particular parameter. ConfidenceIiiits provide a quantitative measure of the

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match between field data and a model andcan be used to rank annual welkest interpre-

tations for a given well. Poor confidencelimits have indicated the existence of newflow models not obvious from straight-lineor log-log methods.

An example of a field test that suggesteda new probIem is presented next.

Field Example

This example concerns a low-rate pumpingwell producing fmm about 13,000 ft. Apressure-buildup test was run by shutdng inthe pump and measuring the liquid level inthe anmdus withs hefium-purged capillarytube and precision surface pressure meas-urement. Tfds is not a typical example ofa field test, but it illustrates the power ofmodem analytical methods.

Table 3 presents formation and fluid pa-rameters and buildup data. The purpose ofthe test was to determine why the oif ratefrom this well (36 STWD) was so muchlower than that of offse~,wells. F%. 4 is alog-log type curve that n unlike any otherf%e seen. However, it appears that a semi-Iog straight line probably will occur afterabout 3,000 hours.

Fig. 5 is a Homer graph that appmemlyhas no straight line. The early shape of thebuildup is odd and is reminiscent of a hump

ing buldup caused by commingled zones,This was a reasomble interpretation. Thethickness of 165 tl is large, and several por-tions of the interval were perforated.

Returning to the log-log type curve (Fig.4), the shape is simiktr to that of a duaf-porosity or communicating layered system.Tbe unit slope storage line, however, isshort and breaks far too abmptfy for ally-thinz exce!i increasing wellbore storage.Fu&xmo~c, wellbore ~torage is usually he$rSt idcntifmble phenomenon at shut.in, sowhat caused the rapid pressure rise?

Fig. 6 is a magnitied Cartesian graph ofshut-in press~s. Within the first &y, thepressure @quid level ti the anmdus) roserapidly about 200 psi, then rose linearly intime for 300 hours! Thus, there appears tobe two wellbore-storage periods in Fig. 6.This strange behavior led to the folIowinganalysis,

Fig. 6 indicates that the liquid level in themmulus rose rapidly on the &t day of shut-in. Then there was a period of cwnstan-ratewellbore storage that lasted more than 10days. What happened? The purpose of thetest was to fmd the cause of a low pumpingrate. pump et%ciencybad not been checked,and perhaps, was not considered. How couldthe pressure rise so rapidly at shut-i”?Perhaps the pump valves were in bad con-

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654 June 1992. JPT

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dition and ~eaking. This could dump the tub- t.h p.. –pwfi where Pwf + is the W=Jreing liquid into the annulus. How much of after the sudden rise caused by dumping tub-a pressure change woufd this came? ing liquid. Fig. 7 is a log-log type curve

A useful rule of thumb is that the pipe ID where the flowing pressure at 3,912 hoursin in. z =bbl fluid/1,000 ft. Hence, 13,000 is changed to 1,952 psi (the pw)+), and theft of 2-in. -ID tubing would contain 52 bbl fmt three points after shut-in are deleted.of liquid. The annular space between 8-in. Compared with Fig. 4, Fig. 7 looks likeand 2.5-ire pipes would contain about 53 a classic wellbore-storage and skin-effectbbl/1,000 ft. Thus, all the tubing liquid type curve for a homogeneous system. Fg.could cause a liquid rise in the annulus of 8, a Homer graph for the adjusted data, ap-about 1,000 ft or a pressure rise of about pears to be a classic storage/skin buildup for300 psi. This rough calculation indicates that a homogeneous system. Neither a com-the rapid pressure rise was caused by the mingled nor a dual-porosity system is in-tubing liquid draining into the ammlus, dicated.which suggess that both the standing and Beause the interpretation seemed reason-traveling valves of the pump.were bad. ~S able, confidence hits were sought for theis a pressure-buildup analysis where a major Homer analysis, Table 4 gives the result.conclusion is that the pump was bad. in Table 4, the large confidence intervals for

We have a hypothesis, What do we do the Homer analysis are remarkable.about it? If tie pump had not been bad, liq- Fig. 9 presents field data vs. a sinxdationuid would have risen in the anmdus during with the parameters developed from thebuildup and all analytical methods would Homer analysis. The simulation does nothave worked. A leaky pump means that all match the field data. Thus, a nonfineartubing fluid was dropped into the well at regression19 on the adjusted field data wasshut-in. This is a slug-test injection at shut- made and a good match was found. Tablein. Known information on slug tests20 im 5 presents the results, and Fig. 10 shows thedicates that tbe slug pressure rise on injec- field data vs. a simulation with tie regres-sion should die rapidly. sion parameters. A comparison of Tables 4

Perhaps pressure rise should be referenced and 5 shows a reduction in contldence linksto the pressure after dumping the tubhg of more than a factor of 10. A comparisonfluid. That is, type curves should be the of Figs. 9 and 10 shows an important im-pressure difference of pw. –pwf+, rather provement in tie agreement of field and

‘iImportant newinformation includesselection of an industry-standard storage typecurve; development ofderhrative methods,solution of the finite-fracture-conducthrityproblem, anddevelopment ofcomputer-aidedinterpretation anddesign.”

simulated results. Fig. 11 is a pressure-derivative graph for the same data. It looksreasonable, except that there is a fairly rapiddrop to the 0.5 derivative value. More wiUbe said about derivative type cume$ later.

This field example is more complex thandiscussed here. An analytical justificationfor the pressure adjustment is presented intie Appendix. But, a conclusion sindlar tothat for the previously discussed simulatedexample remaios: it is mandatory to use in-teI’pretedresults to simulate test data and tocompare the results of the simulation withthe field data to be able to place credencein a Homer analysis.

Regression and simulation are necessawto perform a proper modem analysis, butti.~ssion with determinm”on of confidencelimits is of major iinporta.nce. When I be-gan to train field engineers in well-test anal-ysis 35 years ago, I stressed that the findnecessary step was to comment on the qual-ity of the test. Was the rate constant, wasthe shut-in long enough, etc.? This qualita-tive comment provides a basis for estimatingcomparative merit when reviewing resultsof sequential tests on the same well. Thecotildence limits provide a quantitativemeasure of quality. To do a regression ana-ysis and not record the confidence limits isa terrible omission. ‘fhk contidencdinhreview led me to the shocking conclusion

Fig. 9—Comparison of adjusted field data with results of a Fig. 10—Adjusted field data vs. simulation with regressionsimulation made from parameters from a Horner analysis. parameters.

.IPT. June 1992 655

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Fig. ii -DerlvatWe type curve for adjusted field example data.

that 1did not know whereto draw a semilogstraight line on a Homer graph to determinethe Fest interpretation. - “

Adjust[ng DataFig. 11 is a type curve for the acljmted FieldExample data that can be used to male im-portant observations about the adjusbnent ofbuildup data, The pressure (upper) curvead derivative (lower) curve both sw afonga unit-slope line. This is the wellbore-storagetime period when constant ffow rate causesa linear reb.don between pressure and time,The pressure curve is a pressure-differencegraph-static preszure less the flowing prez.sure just befo~ the well was shut in. Theflowing pressure is the weak link. 14 Ref.14 discusses reasons why the flowing pres-sure may be measured correctly, but not bethe proper flowing pressure for the typecurve. If the flowing pressure is too low,the pressure difference wiUbe @a large andthe early field M will approach aunit-slopeline from above. If the flowing pressure istoo high, the pressure difference will be toolow and early data points wiU approach aunit-slope line from below. Neither eventwiU affect the pressure derivative. Thus, apowerful feature of the derivative type curveis that it aids diagnosis of a flowing-pressureerror. Tbe correct flowing pressure can befound by a Cartesian graph of static pres-sure (Ref. 14).

Another problem is that the effective shut-in time maybe later than the recorded shut-in time, Most master valves cannot be closedinstantaneously. I have observed gas I@shut-ins that Tequired 15 @nutes to close the

656

master valve, yet I could stifl hear noisecaused by fluid movement imi& the pipe.Sometimes, this problem is obvious. Datafrom the service company log flowing pres.sure for some time, and then log the shut-intime. The shnt-in pressures continue at theflowing level for some minuta before a riseisappment. As before, wellbore storage maybe used to correct the shut-in the by a !inearexmapolation to tie flowing pressure. A de-rivative graph, such as Fig. 11, however,can be of great use to delmmine whether thisprobIem exists, even if a flowing-pressurehistory is not available.

If the shut-in time is recorded before theeffective shut-in, the apparent shut-in pres-sures do not chamge for some time. ‘Tiepressure derivative will be zero. Then, thederivative wilf increase rapidIy towmd thecorrect values on the unit-slope line. Resultia late rapid rise in the early derivative anda sharp maximum usually indicate that theshut-in time reqoires adjustment. This isanother step where computer aid is neces-sw. ~~g~g the shut-in time requiresmodification of the producing time, re, ad-dition of the correction to all shut-in tmms,and redifferentiation. This is far too muchwork to be done by hand, The tidal workto produce Fig. 11 to see the problem is toomuch work to be done by hand. Computeraid is mandato~.

Bourdet et al. 17originally indicated thatproper differentiation was necessary, butthey dtd not describe. the method. Obviousdifferentiation metlmds such as spfine fittingand numerical differentiation dld not workweU. Simple differencing could producenoise in the differential curie. Finally, ade-

quate methods to produce the differentialcurve were develcqxd, but computer aid wasrequired. Both dhTemntiation and adjust-ment of data absolutely require computeraid. Computer software is like the slide ruleof the 1940’s. A god anatyst uses it tomanipulate data-and not as a black box forinterpretation,

Combmtions of improper flowing pres-sure and shut-in the exist. Some derivativetype cumes show the pressure-differencecurve approaching a unit slope from aboveand the derivative type curve approaching aunit slope tiom below. A major feature ofthe Bourdet et al. 17 derivative type curveis that both the pressurexfifference andpressure-derivative curves start with a unitslope, which they must do for the weUbore-storage and skfn-flow model.

Ref. 14 idicates that simiIar cmm.ctionsmay be made for fracture flow mcdels. Stat-ic pressure should be a linear fumdon of thesquare root of time. This is correct for thesimple linear flow fracture models.

Fractured Wefls

The 1976study 14 reported that tests onhydratically fractured wells ofien matchedthe early tiacture type curves21 but yieldedapparent fracture lengths of 10 ft when de-sign lengths were more than 1,000 ft. Aniarly study by Holditch and Morse22 at-tributed this to high-velocity flow in a fmc.~. But wOrkby key et al M indicatedthat finite-fracture permeability-width wasa more Siely cause of the problem. Theyused a finite-element solution. Pratsz4 prO-vided the key to this problem in a classic

June 1992. JPT.—. . . . ..-.

I

‘1

1‘2

(a) tp (b) tP+

Fig. A-l-Leaky pump problem.

study on steady-state flow. In Aug. 1978,Cinco-f.ey et al. X presented a troly clazsicsemianalytic solution of this problem thathas become a standard for verification oftinite-difference solutions. Another impor-tant study was offered the next year byAgzrwal etaL~ The C1nco-LeyetaL25study solved the fracture well-test problcmcited in Ref. 14. Itisamagrdficent study.

Conclusions

A number of important findings since 1976have brought well-tesf analysis closer to ful-ti13ingthe expectations of analysts since the1950’s. Type curves, including presmre-derivative and Enite<onductivity fracturetype curves, and mmputer-aided interpr-tation have given modem analysts the toolstodoremarkable btterpretative work. Ar-tificial intelligence programs for modelselection arealready rumdng.27 Regressionwith confidence limits on model parametersand final simulation of a test provide proofthat an interpretation was reasonable. As aresult, the following conclusion appearjustified.

1. It is often inmossible to find a correctand observable Homer straight line that pro-vides parameters that will generate the fielddata teawnably when simulated. Regressionwith dldata.a ndapmpermodel i$neces-sq, and confidence limits shmdd bsrecorded as a measure of the qmlity of theanalysis.

2. Pressure-builduIJ data freauentlv re-quire adju.mnent of tb~flowing p;emu;, theshut-in time, or both.

JPT . June 1992

3. Pressure-derivative type mmves aresensitive throughout the time domain andpermit identitlcation of events not evidenton either log-log or semilog premue gmpim.

4. Modem analytical procedures usecomputer-aided methods for the foUowingsteps (1) to inspect pressure and derivativetype curves and to fiter and adjust data asnmdefl (2) to tind a Homer s’uaightline fornonfine~-regression input parameters; (3)to tind confidence limits and to simulate thetest with the final parameters after regres-sion (4) to compare field data and modelsimulation in Ainensiordess coordinates and(5) if the restdt is not gccd, to select the nextmodel in order of complexity and repeatprevious steps. Artificial intelligence soft-ware can also aid in the selection of propermodels.

Nomenclature

B = FVF, RWSTBcc = system compressibility, psi – 1c = wellbore-storage constant, ft3

CD = dimensionless wellbore-storageComtznt

dc = casing ID, ftd, = tubing OD, fth = formation thickness, ftk = penneabilky, md

L, = Height 1 in Fig. A-1, ftL2 = Height 2 in Fig. A-1, ft~ = sIope of wdogline,

psilcyclep = pressure, psi

p+ = Homer extrapolated pressure,psi

~~?heconfidence limitsprovide a quantitativemeasure of quaIitymTo do a regressionanalysis and not recordthe confidence Hrnits isa terrible omission.”

AP = P~, –PW. psiPD= dimensionless presmre change

defined by Eq. A-7

PDSL = dimensiO~ess press~eflnction defined by Eq. A-8

pi = initial pressure, psi

p,,f = flowing wellbore pressure, psip~f+ = flOwing weUbO~ p~$sure at

shut-in, psi

pm = 3hut-in wellbore pressure, psiq = surface flow rate, STB/D

rw = wellbore radius, fts = skin, dimensionlesst = time, hours

At = shut-in time, hoursrD = dimensionless time

AtD = dimensionless shut-in timetp = producing time before shut-in,

hoursy = viscosity, cp

PO = density, lbm/ft3.$ = porosity, fraction

Acknowledgments

Portions of tiis work were funded by U.S.Department of Energy Grant No. DE-AS07-841D12529. Computer time was pro-vided by Stznford U. Software was providedby Scientific Softwaro-lntercomp fnc. andGarret Computing Systems Inc.

References

1. Homer, D.R.: “Pressure Build-Up inWelJs,,,Proc., Third W.dd Pet. Cong.,EJ.Bill &L), Leiden (1951)If, 503-22.

2. ML1l.r,C.C., Dyes, A.B., and Hutcldnson,C.A. Jr.: “E.sdmationof Perme.biliq andResewoirPressurefromBottom-HoleFTes-

657

sw Build-Up Cbaracteristim,,, Tram.,AIME (1950) 189, 91-104,

3, Moore, T.V., Sdikbuis, R.J., and Hunt,W.: ‘,The Detmnimfion of Pwmeab!Jhyfrom Field Dam,,, API Bulletin 211, API,Ddfas (1933).

4, van Everding.sn, A.F. and Hurst, W.: CLAPplication of the Lapl,ace Transfmmation mFlow Problem in Reservoirs, 3> Trans.,AfME (1949) 1S6, 305-24.

5. Hnrst, W.: C<Establishment of the Skin Ef-fect and Its Impediment to Fluid Flow fntoa Wellbore,,, Pet Enc. (Oct. 1953)23, B-6.

6, mu Everdingen, A.F.: “The Skin Effect andlts Inftuencc m the Productive Capacity ofa WeU,” Trans. , AIME (1953)198, 171-76.

7. Matfhews, C.S., Bmms,F., and H&ebumk,P.: “A Methcd for Determinationof AvcmgePress.= in a Bounded Reservoir,” Tram.,MME (1954) 201, 182-91.

8. Matthews, C,S. and RussefJ,D.G.: Prm$ureBuildup m?dFfow Tars in Welfs, Momg@Series, SPE, Richardson, TX (1967) 1.

9. Theis, C.V.: “The F@lationdip BetweentheLowering of fbe Pie.zcmwfricSurface and tieRate and DuratiQRof Discha~e of a WellUsing Gmund-Wa.ferSbxage,,, Trans., AGU(1935) ff, 519-24.

10. Theis, C.V.: “The R&tionsbip BefweentheLowering of OwFiezcmetric Surface and theRate and Dumfkm of Discharge of a Wellf.hillg Gmtmd-Wa,ti Storage,,, PressureT,am;ent Testing M.thod$, Reprint Series,SPE, Richardson, TX (19S0) 14,27-32.

11. Md(inky. R.M.: “Weftbore TrammissihilhvFromA&t30w-Dominaied pressureBuitdu;Data, ‘SJPT (July 1971) S63-7Z Trans.,AIME, Z51.

12, Agamwd, R.G,, A1.Huwiny, R,, andRamey, H.J, Jr. : “h ?.nvmdg,tim of wett.boreStorageandSkinEffeclinUmteadyLiq-uid F30w:1. .4ndYticalTrearmmr,,3SPEI(Sept. 1970) 279-9~ Tram,, AIME, 249.

13. Earlouxher, R.C. Jr.: .ddvunce$in WellZwAMly&, Monozraph Series, SPE, Richarc-xm. TX (1977) 5.

14. hey, H.J. Jr,: “Pracficaf Use of ModernW.U Test Analysis, ” Pressure TrmL?mTesting A&hmfs, Reprint SW@ SPE,Richardson, TX (1980) 14, 46-67.

15. Gtigtien, A,C, el al., “A Cornptism. B=rween Different Ski” and Wellbore StorageTw Curves for Early-Time Tmnsimt ,4nal-Ysi%”PPI SPE 8205 prese.b2d at the 1979SPE Annual Tecbniczl Conference and Ex-hihiti.m, Las Vegas, Sept. Z-26.

16. Sfehfest, H,: ,’AJgorithm368 Numerica2Jn-vemicmof Laplace Transforms,,, Commwd-can’onsof the A CM (Jan, 1970) 13, No. 1,47-49.

17. Bourdet, D. et al.: “A New Set of QpeCurves Simplifi.% Well Teti Amlysis; zWorfd Oil (k@ 1983) 95-1o6.

IS, Bourdet, D. .? al: ‘%,termetirw Well Testsin Fractwed Reservoirs,’, Ww”ti Oil (Oct.1983> 77-S7.

19. Rosa; A“.J.and Home, R.N.: “AmonxdedTw-C~e Mtihg in WeU Test AmiysisUsing Laplace Space Determination of Pa-rameter Gradi.mt$,>s paper SPE 12131P,esen~ at tie 1983SPE AnnuaI TechnicalConferenceand Exbibitim, San Francisco,Oct. 5-s,

20. Ramey, H.J. Jr., Agarud, R.G., andMmOn,L: ‘‘Analysisof ‘Slug Test, OFDST J7]owPericx2Dafa,y>J. Cdn. Pet, Td. (Jtiy-Sept,1975) 1-11.

21. Grin@en, A.C,, Ramey, H.J. Jr., andRaghavan,R.: “Applied Press”r. .4mtysisfor Fracmred WeUs,99JPT (J.lY 1975)887-92; Trans., AfME, 259.

22. Holditch,S.A. and Morse. R,A.: “The Ef-fects of Non-Damv Flow & the Behavior ofHydmulicafly Fra&wedWeUs,”JPT (Oct.1976) 1169-78.

23. Ramey, H.J. Jr, et .1.: “pressure TransientTestinE of Hy&auficdly-Fractured Wetk, ,>Proc.,-AmericanNwle& sec. Topiw2Me&ingon Energyand MineralResourceRecov-ery, Colorado School of blinm, Gokfen(1977).

24. Prafs, M.: .’Effect of Verdcaf Fractures mReservoir Behavior—fmompressible FluidCase,” SPEJ (June 1961) 105-lfi Tram.,.klhfE, 222.

25. Cinco-LeY. H.. Samaniezo-V.. F.. andDomingu&-A.,N.: “Tram~entPike E&havior for a W.U Wifha Finhe-CmductivhyVertical Fracture,,, SPEJ (Aug. 197S)253-64.

26. A#trwl, R.G., Carter, R.D. , md Pcd.tock,C.B,: “EvacuationandPerformancePredic-tion of Low-PermeabilityGas wells sdmn.fatedbyMzsive HydradicFcacb.uing,’SJPT(Mach 1979)362-72 Trans. , m, 267.

27. Akin. O.F. and Home. R.N.: “US, ofk-thiciaf intelligencein WetJ-TestAmtysis,,,JPT (iMarch 1990) 342-49,

Appendlx—The Leaky Pump ProbJem

Fig. A-1 is a schematic of Ibe leaky pumpproblem. Fig. A-la shows conditions at theinstant of shutting in the weJl by stoppingthe pwmp at time r~ The tubing is full, thebottomhole flowing pressure is PWP andthere is a liquid level in the mmdw.

Fig. A-lb shows conditions a moment af-ter time tp. Both the standing and travelingvalves m the pump leak, snd oil dmim mpid-Iy from the tubing info the annulus. The liq-uid level in the tubing drops L, feet, andthe Jiquid level in the anmdus rises L2 feet.A volumetric bakmce is

>fL1 =;(d~ -d,?)L2. (A-I)

Let PObe the density of wellbore oil. Thepressure gradient in the liquid is

dp/dL=pe/144 psilft. . . . . . . . (A-2)

The wellbore storage during drawdown isbased on tie anrmlar volume,

;(d&d;)

c1 =(%,/144)

, . . . . . . . ..(A-3)

wbiJe weUbore storage during pressurebui2dup after pump leakage is based on thetotaf anmdus and tabing storage,

()C2= >? /(po/144). .(A-4)

The dimensionless forms of fbese storagecoefficients are

18(d~–d~)CD, = —

p.h.$ctr~(A-5)

Igd:

and CDZ = — . . . . . . . ..(AQp.hdctr;

The sudden dmimge of Jiquid from thetop LI feet of tubing is a slug injection intothe weJJ at the instant of shut-in. Duringdrawdown, producing pressures sre givenby the welJbore-storage and skin solutionwith a storage coefficient of CD1. Becauseof the sudden emptying of the tubing, weJ2-bore storage: changes abmptfy from CD, toCm at shut-m, However, it is nece?suy tosuperpose a slug injection ofithe quantity ofliquid drained from L1 feet of tubing atshut-in. The conventional wellbore-storageand skin dimensionless wellbore preswre is

*4&#’’-p)’PEl(s>crJJrJ) = —

. . . . . . . . . . . . . . ..(A-7)

and the slug dimensionless pressure is

JW2($GA3)=‘Pwf+-p)(Pwf+ -Pwf)

. . . . . . . . . . . . . . ..(A-S)

The shut-in wellbore pre8sure is

141.2qBp(Pi ‘Pws) = ~

XIPD(S+CD1 jtp+At)–PD(S, CD2At)]

- (Pwf+ ‘Pnf)PDSL($ Cm ,AtD) ,

. . . . . . . . . . . . . . ..(A-9)

and tie flowing pres8ure at shut-in is

kh

‘(pi ‘pw/)‘pD(~!CDI ,tp)141.2qB&

. . . . . . . . . . . . . ..(A-IO)

The conventional log-log type curve forbuildup is based on the pressure differeme(Pws–Pwf):

kh

‘(p.. ‘P.f) ‘PD(~,CDI,fP)141.2qBp

‘PD(X CD1 ,tp +M +PD($ CD2 sAf)

kh+— (P~f+ ‘Pwf)PDSL(S,cDZ.AO.141.2qBp

. . . . . . . . . . . . . ..(A-11)

Ordinarily, the first two terms on the right~%: A-II are assumed to be equar becauseshut-m times are resticted to less than 10%of the producing time. 2ftbis were done, theresuJt wotdd sdU be a superposition of tbewellbore-storage and skin problem and antistzrdaneous slug injection, There wmdd beaninstmmmems jump in pressure at shut-iu,

Let us derive lbe pre.smre from the differ-ence between the static pressure and Pwf+,an instant after shut-in.

658 June 1992. JPT

kh*(P., –Pwf+)’ —

141.2qB&

x [(l%–Pwj) -(Pwf + –w)].. . . . . . . . . . . . . ..(A-12)

With Eqs. A-8, A-n, and A-12,

kh‘(lhv$ ‘pwf+)=pD(s,cD1 @141.2qBp

‘PD(s, CDI ,1P+~) +PD(s, CD2 M

X[PDSJSZDZ,LW-1]. (.4-13)

The last term initia31yWOUMbe zero andwould approach the constant value.

=(pwf+–p.})=

(Pwf+ –Pwf) _ –(Pwf+ –pwf)—

141.2qBp – 2m ‘

kh 2.303

..,,.., . . . . . . ..(A-14)

where m=162.6~. . . . . . (A-15)

Consider the last term on the right in Eq.A-14. The slug test dimensiotiess pressure,

PDSZ (s, CDZ,ZW, is less than 0.001 fortD/CD 2103 (see Ref. 20). This is approx-imately the dme flat the sendlog mmigbtlinestats for tie we31bore-storageand sti case.Thus,

[PDSL(S3CD,,A0-1]- -1 .(A-16)

and is constantby the time of the start ofthe semi30g straight line. Thus, the skin ef-fect found by use of Pwf + as a new flow-ing pressure shotid be increased by

(Pwf+ –Pwy)Ap, =

2m,... (A. 17)17)

2.303

For large values of CDe2s, say >1010,PDSL – 1 ch~ges from O to – 1 in the three

log cycles before tD/CD = 103. This is the

maximum and dedine periixl of the pressurederivative with resped to log time. Thus, theslug injection shotid affwt the derivativetype cume on the approach to 0.5. The de-rivative should drop rapidly. This effectwould probably not be noticeable becausethe slug-test dnensimdess pressure dropsrapid3y to less than 0.1 for tD/CD between10 and HM.In most cases, the skin ccxmibu-tion caused by use ofpwf+ would be con-stant long before the semilog straight linestats.

Finally, some comments on the effect ofa leaky pump are required. During produ-ction,bad check valves would reduce pumpefflciencv and make the surface oil rate lessthan the&Ip displacement rate. This wotidnot affect anything else. The wellborestorage wotid sd3fbe based on the znnularvolume, and the skin effect would be thecorrect skin effwt. A bad pump wou3dcausea low oiI rate even though the skin effectwas negative.

Although this problem was identified byseeking an explanation for the strznge 10.3-Iog type curve in Fig. 4, it could have beenrecognized by reflection on pumping weUproblems. But in my experience, well-test-amlysis research often requires experimentat ildl scale. Do not reject anomalous data.Seek causes.

Loren Krase and R.G. Agarwa3 have ad-vised me that the derivation presented here isapproximate. I agree. A Discussion that pM-vides the rigorous derivation wilf be present-ed in the new future.

S1 Metric Conversion Factors

bbl X 1.589873 E-01 = m,q x 1.0* E–03 = Pas1? X 3,M8* E-01 = m

md X 9,869233 E–C-4 = IImz

psi x 6,894757 E+cu . !@.psi-l x 1,450377 E-01 = I&a-i

This PWr 1$SPE 20592. DIstlngulshedAuthor Series w-11.1s am wwd, d.wipwe prese.wlons that wmwizethe stale or the an I. an am of technology’ by describingrecent developmem for readers who W. not wckllsts 1.the topics dlmssed. Wm.. by Individuals recog.lzed asexperts I. the ma, !hese articles provideWY rd. rmc.s1. momdefinitive work and present specific de!alls only 10illustrate :he technology. P.rposw. To Worm the g.ner?.lreadmtiP of recent advances 1“ varbu m= d FSkOhU~

mglnearing, A mfibwnd anthology, SPE DMngulsJedAulhorS$riO$ M. 1987-D6c 1932,19 w41atle flOm SPE’SBook Order DePt.

Jm

,.

Henry & Rimie$lJr.. is the:,K61een.and C8dtOit BeaCprtietwtir of.p.~ro,Ieum engln,e.erlnsj,at Stanford u.:a@dan .,internationalconsultant: BefO~,J0irdng:s@3i3t0rd in1966,”IW worked.for Mobil 011“during

1952-63, with an as51gn133ent to theChinese Petroleum Co,qi In Taiwan In1962, and for Texes “A&M‘U. during1963-66. He hoIds BSand PhD degreesin chemical engineering from Purdue U.Ramey sewed on the 1972-75 SPEBoard of Directors and the 1999-91Western Regional Meeting #an@g9,committee. A Distinguished Member”since 1983, Ramey has ,received,the.Lester C. Uren Award (1.973), the John:Franklin Carll Award (1975), a!d !!IFAWthony F. Lucas Gold Medal (f 983). .,

JPT . June 1992 6s.9