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Wellbore Effects in Injection Well TestingRobert C. Earlougher, Jr., SPE-AIME, Marathon Oil Co.K. M. Kersch, SPE-AIME, Marathon Oil Co.H. J. Ramey, Jr ., SPE-AIME, Stanford U.

IntroductionInjectivity or pressure falloff tests in injection wellsare commonly used to investigate formation properties.. Ideally, such tests provide information aboutformation permeability, skin factor, distance to fluidbanks, and distance to boundaries. 1 ,2 Wellbore storage3- 9 and its effects on transient testing have beendescribed in the literature. The storage of fluids inthe wellbore due to compression or changing liquiqlevel causes transient test:> to act differently at shorttimes than they would in the absence of wellbore storage effects; long-time behavior is essentially unaffected. Wellbore storage is a general term encompassing the more specific terms, "afterflow" (pressurebuildup) and "unloading" (pressure falloff and drawdown); we use only the general term in this paper.The specific term commonly applied to injectivity testsis wellbore storage.We show here that under certain circumstanceswellbore storage effects, in particular changing wellbore storage, can make test interpretation for formation characteristics practically impossible. We includefieid data illustrating this problem, provide an explanation of the behavior of these data, make suggestions for injection well testing and test analysis, andillustrate the analysis technique. The problem of wellbore storage effects in injection well testing is muchtoo broad and complex to be treated exhaustively inthis paper, but we feel that the material presented isdetailed enough to identify, illustrate, and at leastpartially solve the problem.

We first encountered and finally recognized thisprQblem as a result of a series of injectivity and falloff tests on several wells. The purpose of the testingwas to locate and estimate the distance to fluid banks.Fig. 1 shows data from one ofthese falloff tests, froma 1,OOO-ft-deep water injection well. Pressure dataare from a permanently installed surface-recordingdown-hole gauge. Fig. 1 has several bends that mightbe interpreted as banks, boundaries, interferencefrom adjacent injection or product ion wells, commingled zones, etc. I t is not difficult to pick four, five,or even six different slopes from this particular test.As a result of the multiple slope changes, there weregreat differences of opinion about how to interpretthese data; test interpretation was never satisfactoryto all involved.Fig. 2 shows injectivity test data for the same well.This figure also has several pends that might be interpreted as banks, boundaries, interference, etc. (Thestraight line in Fig. 2 'is used later in an examplecalculation.) Detailed analysis of these two figuresshows that they do not have the same sequence ofslopes and that the slopes do not change at the sametime. There are one or two places where the slopesdo appear to be the same, but these slopes do notoccur at equivalent times. Furthermore, we expectthe general appearance of the injectivity and the falloff tests to be the same1 ;clearly, the shapes of thesetwo curves are quite different. The falloff curve generally has a small slope and is fairly flat in the period

A case history is used to illustrate that wellbore storage can act differently in injectionwell falloff and injectivity tests. This can cause test curves to have shapes characteristicof mobility banks and thus render some falloff tests useless. Injectivity tests, however,may be interpretable.

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:LDP1hr.= 760 ps im = /0 5 psi / CYCLE iSTllRT OF CORRECTc9 SMILOG STRAIGHTo LINE

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, ,, O O ~ O , O : : : - , - - - - - - - - ! O ! - ; - . I - - - - - ~ I - - - - - - ; '1 .0FLOW TIME, t , HR.Fig. 2-lnjectivity test for the well of Fig. 1.

o2001-- pws =/9 4 ps i

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I ,1 0 0 0 ~ , , - - - - - - 1 I - - - - - - - - - , l 1 0 : - - - - - - - - - : : ! 1 0 0SHUT-IN TIME, ilt, HR.Fig. I-Pressure falloff test in water inject ionwell in the I llinois basin.

-wa:::> 600InInwa:Q. soOW...Jo'!'4001

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0.8 to 2 hours. In contrast, the injectivity curve isextremely steep from about 0.15 to about 1.3 hours.The major problem in analyzing these tests, then,lies in resolving the obvious differences. Our firstreaction was to discount the injectivity test becauseof possible rate variations, casing leaks, interferencefrom other wells, and other well known demons. Weknew, however, that the rate was controlled within afew percent during the injectivity test. Casing leaksand interference should affect both the falloff test aridthe injectivity test in the same way, so it is illogicalto discount the injectivity test and not the falloff testfor these reasons. Furthermore, subsequent testing inthis well ~ in other wells indicated that the falloffand injectivity tests were almost always different,regardless of formation characteristics and the natureof the injected fluids. Thus, we concluded that thedifference in these curves must be due to a real phys-ical phenomenon and should be explainable.Identification of the ProblemAfter studying many tests, we realized that the flatportion of the falloff curve always begins at aboutthe time the wellhead pressure becomes atmosphericor goes on vacuum. Fig. 3 shows an extreme exampleof this behavior. This falloff test is for a water injection well completed in a 1,500-ft-deep sand. Notethat the pressure data are essentially constant at about670 psi from 25 minutes to 92 minutes. The hydrostatic head due to a 1,500-ft column of water is about670 psi. The falloff curve plateau in Fig. 3 is muchflatter than that in Fig. 1. We shall give the reasonfor this later.The significance of the plateau in the falloff curveis that the liquid level should begin to fall in the tubingat this point - about 20 to 40 minutes after shut-infor Figs. 1 and 3.This observation indicates that .wellbore storagechanges from fluid compression to falling liquid levelSduring the falloff test. But for an injectivity test, thepressure starts at a low level and increases, so well,,:bore storage changes from rising liquid level to fluidcompressions during the test. (Agarwal and Ramey4explain the theore tica l behavior for these twosituations.)Thus, it is clear that the changing type of wellborestorage is different for the falloff and injectivity tests.

For falloff, first there is expansion of fluid in thewellbore, then falling liquid level. For injection, thissequence is reversed. This results in transient responsecurves of different shapes for the two tests. Examination of the pertinent differential equation shows that,for a constant wellbore storage, falloff and injectivitytests (or for that matter, buildup and drawdown tests)on the same well must look alike and must reveal thesame information.1 ,4 Consideration of changing wellbore storage type indicates that this is not necessarilyso. The theoretical discussion in the following sectionverifies that the behavior in Figs. 1, 2, and 3 is indeeddue to the sequence of changes of wellbore storage.Changing WellboreStorage-Theoretical DiscussionSimple, constant wellbore storage effects were rec-NOVEMBER, 1973 1245

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ognized by vanEverdingen and Hurst" in 1949. Littlewas done about wellbore storage in the petroleumliterature until the mid-1960's. Agarwal et al6 established that a log-log plot (type curve) of IIp vs Ilt hasa unit slope in the wellbore-storage-clominated period.Ramey9 has explained the importance and use of thetype-curve approach in test analysis.Fortunately, many pressure transient tests experience a relatively constant wellbore storage. An example of such a falloff test, taken from Ref. 7, isshown in Figs. 4 and 5. Fig. 4 is a semilog plot ofpressure falloff data for a water injection well in a"dump flood." (By dump flood, we mean that waterwas injected at atmospheric or below atmosphericwellhead pressure.) As a result, this falloff test immediately experienced falling-liquid-Ievel wellborestorage. The type curve that matches the falloff datain Fig. 5 is from Fig. 1 in the Agarwal et al. paper. 6We used Ramey's technique9 to obtain this match.The match demonstrates that this falloff test acts witha single, simple wellbore storage coefficient. We indicate where the semilog straight line should start inFig. 5. Wellbore storage dominates the falloff behavior of this well for about 8 hours. Nevertheless,this test can be analyzed by normal, semilog methods;the straight line is shown in Fig. 4. This part iculartest has been analyzed by these methods and by aregression technique that included the effects of wellbore storage. (See Ref. 7 for a detailed analysis.)The long period of falling-liquid-Ievel wellborestorage indicated in Figs. 4 and 5 is easy to see, understand, and account for in test analysis. The problemsof changing wellbore storage are somewhat different.In the falloff tests of Figs. 1 and 3, we must con-

sider what happens when the fluid level starts to fallin the middle of the test, after a period of compressivewellbore storage. This case is specifically excluded byHazebroek et al.8 in their treatment of falloff in waterinjectionwells.Fig. 6 shows schematic log-log and semilog graphsof pressure falloff behavior when the wellbore storagecoefficient increases stepwise from Cl to C2 at timeIlt l . This corresponds to wellbore storage changingfrom fluid compression to falling liquid level, a common example of increasing wellbore storage. Thelight solid curves in Fig. 6 show pressure falloff forconstant wellbore storage coefficients. The left curveis for a low wellbore storage coefficient, Cl, such asfluid compression storage; the right curve is for a highwellbore storage coefficient, C2 , as might be causedby a falling fluid level. The heavy solid curve, goingfrom Cl to C2 , is the falloff curve that occurs whenwellbore storage changes at time Ilt l Note the distinct flattening in both the log-log and semilog graphs.I f the test is run long enough, data fall on the semilogstraight line after time Ilt2 The dashed curve showsthe location of the correct semilog straight line. InFig. 6, where the storage change occurs early, thesemilog straight line is not reached until very latein the test.Fig. 7 also shows an increasing wellbore storagesituation. In this case, the semilog straight line isnearly reached while wellbore storage is still due tofluid compression. Then, the wellbore storage coefficient increases to falling liquid level. The falloff curve(the heavy solid line) nearly reaches the semilogstraight line (the dashed curve), then flattens anddeparts from this line as the wellbore storage in-

5 0 0 1 - - - ~ - '- ----' --'

100.0.0"------:o..-----c:!'ooSHUT'IN TIME,At,HR.

Fig. 4-Semilog plot of a pressure falloff test for adump flood; falling-fluid-level wellborestorage only. (After Ref. 7.)

SEMILOG

r--CORRECTSTRAIGHTLINE

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pw,=428psi;;..../400

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SHUT-IN TIME, At , HR.Fig. 5-Log-log plot of the fal lof f test inFig. 4. (After Ref. 7.)

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creases at time .6.t1 After long enough shut-in time,the semilog straight line is reached at time .6.t2 Certain features in Figs. 6 and 7 are importantand can help us recognize increasing wellbore storage.The most obvious feature is the flattening of the falloff curve. This is clear in both the semilog and loglog graphs in Fig. 6. However, in Fig. 7, flatteningis easier to see in the semilog graph than in the loglog graph. When dealing with real data, the flatteningis frequently impossible to see on a log-log graphunder these circumstances. This is because of noise inthe data and because of the insensitivity of the log-logscale in the vicinity of the semilog straight line (thedashed curve).The beginning of the correct semilog straight linecan sometimes be identified on a semilog graph. Figs.6 and 7 both show that the falloff curve flattens asstorage begins to change, then steepens, and thenbends to a smaller slope. This slope represents thecorrect semilog straight line unless boundary or interference effects, or both, have become important bythis time and are causing the pressure to approachfinal static pressure. Unfortunately, we know of nocertain diagnostic method to differentiate betweenthese situations.Decreasing wellbore storage behaves quite differently from increasing wellbore storage. Fig. 8 showsthe theoretical response to an injectivity test in whichwellbore storage decreases from a rising liquid level,C2 , to fluid compression, Cl ' As in the previousfigures, the heavy solid curve indicates the observedpressure response, the dashed line is the semilogstraight line, and the light, solid lines are for the two

constant wellbore storage situations. Storage changesfrom rising liquid level to fluid compression at timet1 ; the semilog straight line is reached at time t2 The contrast between Fig. 8 and Figs. 6 and 7 isapparent. The interval from t1 (when the storagechanges) to t2 (when the semilog straight line isreached) is relatively short. The injectivity curve issteep during this time. This response is apparent inFig.2.

It is clear from Figs. 6, 7, and 8 that the semilogstraight line should be reached much earlier whenwellbore storage decreases than when it increases.Thus, the injectivity test should be superior to thefalloff test when wellbore storage can change duringthe test.Fig. 9, a log-log plot of the data of Figs. 1 and 2,demonstrates that the material of Figs. 6, 7, and 8does apply to actual testing situations. The solid linesin this figure are the theoretical curves for the twoconstant wellbore storage cases: fluid compression(left) and changing liquid level (right). The injectivitytest shows a rapid pressure increase from the risingliquid-level storage curve to the semilog straight line.The falloff test shows a long period of changing storage from the compression storage curve to the fallingfluid-level storage curve. Only the last few falloffdata points approach the semilog straight line; significantly, these points reach the semilog straight lineat a much later time than do the injectivity test points.Fig. 9 demonstrates that the tests of Figs. 1 and 2follow the theoretical behavior predicted in Figs. 6,7, and 8. Our experience with other tests indicatesthat this usually happens under these circumstances.

LOG-LOG

Fig. 7-Log-log and semilog theoretical falloff curvesfo r a step increase in wellbore storage. Storagechanges just before the semilog straigrt. line is reached.

LOG-LOG

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method given by Ramey.3We calculate the dimensionless wellbore storagecoefficient6:

5=+1

CHANGING LlDUIO LEVELSTORAGE

COMPRESSIVESTORAGE

CORRECT STRAIGHT LINEON SEMILOG GRAPH

';;CoCo...

w 100 < ~ ~ OJ: cOUw 0a::::>UlUl A FALLOFF,!:i.p=PJoIS- P",fC. 0 INJEcT/vlrr , t::.p=p",rPws

5.6146 CCD = --=--=------:-271"hepC tTw2(5.6146)(0.01)

CD = 271"(16)(0.22)(7 X 10-6)(0.29)2 = 4,310.On Fig. 9, it is indicated that the falloff data startwith CD = 320. The dimensionless storage coefficient

changes from 320 to 4,310 during the test. At thestart of the falloff, the top 116 ft of the tubing contained gas at a pressure of 457 psi. The combinedeffect of trapped gas and liquid-level change causedthe value of CD to be 320. Liquid compression alonein the tubing would result in CD = 5.2.In regard to the falloff test, a straight line can beidentified near the end of the test on Fig. 1. However, it is very short, and to measure the slope withaccuracy would be difficult.In this case, one normally might not expect storage to be a large factor in the test. The well had only1,000 ft of 2-in. tubing with a bottom-hole packer!But a falling liquid level caused wellbore storage todominate this test.Finally, because of the shortage of analyzable dataduring the falloff, we note that no mobility bank isevident in this test.

1 0 ~ O , O : ; ; - ' - . . L - - - - - - - - f ~ - - - - - - - L . - - - - - - - J , O .TESTING TIME, HR,

Fig. 9-Log-log plot of falloff and injectivitydata of Figs. 1 and 2.

Testing and Analysis MethodWe consider both constant and changing wellborestorage situations. Transient tests with constant wellbore storage are much easier to recognize and interpret than tests with changing wellbore storage.Constant Wellbore StorageWhen only fluid-compression wellbore storage occursin a falloff or injectivity test, wellbore storage haslittle effect on test results and analysis. This commonly happens in injection projects with reservoirpressure high enough to support a fluid column tothe surface at all times. This situation, essentially theideal one normally treated in transient testing theory,is discussed adequately in the literature.I, 8When liquid level starts to fall immediately on

qw = 100 STB/DBw = 1.0 RB/STB

Example CalculationThe falloff and injection tests shown in Figs. 1, 2,and 9 are analyzed in detail to illustrate the analysistechnique. First we inspect the injectivity test on thelog-log type curve (Fig. 9). The early injection dataclearly form a unit slope straight line indicating wellbore storage is important during the first half hourof injection. Fig. 9 shows that the injection data reachthe correct semilog straight line at about 1.5 hours ofinjection. Thus, the start of the correct straight linecan be identified on Fig. 2. We have used this approach to draw the straight line shown in Fig. 2.Fewer data points are plotted on Fig. 9 than on Fig.2 only to aid clarity of presentation.Known:

JLw=lcph = 16 ftep = 0.22Ct = 7 X 10-6psi- ITw = 0.29 ft

depth = 995 ft2-in. EUE tubing with packer set at 979 ft.

A conventional analysis of the semilog straightline from Fig. 2 gives lk= 162.6 ' : : : (1)_ (100)(1)(1) _k - 162.2 (105)(16) - 9.7 md

s = 1.151 [(PIhr-Pws) -log ( k )m epfLCtTw2+ 3.23] . (2)s = 1.151 [ ( 7 6 0 1 ~ 5 1 9 4 )

- log CO.22)(1)(7 710-6)(0.29)2)+ 3.23] = 0.9.

Although it is not essential, we can make someuse of data from the storage-controlled region. Fromany point on the unit slope straight line in Fig. 9 (forexample, 415 psi at 1.0 hour), we can estimate thevolume of water stored per unit bottom-hole pressurechange, the wellbore storage coefficient:

C = (100 STB/D)(1.0 RB/STB)(415 psi/1.0 hr)(24 hr/D)RB= 0 . 0 1 0 -.pSI

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shut-in, as would be the case in a dump flood, wellbore storage effects can last for a significant periodof time. Commonly, storage effects last several hours;but in extreme cases they may last up to several days.In this situation, falloff and injectivity tests will beequally obscured by wellbore storage. Both testsusually reqUire a long time to reach the semilogstraight line. This time may be so long that boundaryeffects may occur, masking the semilog straight lineand rendering the test useless. In any case, there isno particular advantage to running one test or theother, except that falloff tests are usually easier tocontrol than injectivity tests.One possible way to prevent the falling-liquid-levelstorage problem is to inject at such a high rate thata high surface pressure is maintained. Then, compressive liquid wellbore storage occurs, and it maybe possible to interpret an injection or falloff testunder these circumstances. Unfortunately, by increasing the injection pressure in this manner, one runsthe risk of fracturing, or getting into the changingwellbore storage situation.Changing WeUbore StorageI f falling-fluid-level wellbore storage commences atsome time after shut-in for falloff, the situationsshown in Figs. 1, 3, 6, 7, and 9 occur. In our experience, storage can change any time from a few minutesto more than 4 or 5 hours after a falloff test is started.A falloff test experiencing increasing wellbore storagemay not be interpretable. This is because of the unlikely appearance of the semilog straight line duringthe compressive wellbore storage period. In some circumstances the compressive storage lasts long enoughfor the semilog straight line to develop; then the testmay be analyzed. I f the test is run long enough, thesemilog straight line may eventually appear duringthe second storage period. Unfortunately, boundaryeffects may appear at about the time the semilogstraight line starts.

In changing wellbore storage situations, the injectivity test usually has more potential than the fallofftest, because storage decreases in this test. Figs. 2,8, and 9 indicate that the semilog straight line isreached quite rapidly when wellbore storage decreases, so an injectivity test should be analyzable.The injectivity test, however, is sensitive to injectionrate. I f it is not possible to maintain a constant rate,such a test can still be analyzed by using superposition techniques after wellbore storage effects havedied out. In extremely difficult situations, it may beworthwhile to run a two-rate injection test, with theinjection rate either increased or reduced during thetest.1. 2 This procedure may avoid the large, changingliquid-level wellbore storage coefficient.Other Wellbore Storage EffectsAlthough the preceding material deals mainly withinjection well analysis, in our experience wellboreeffects cause much unusual behavior, in both production well testing and injection well testing. Becausethere are many similarities between production andinjection well testing, all material in this paper alsoapplies to buildup (analogous to injectivity) and draw-NOVEMBER, 1973

down (analogous to falloff) tests. The indicated analogies apply only to the sequence of storage changes,not to normal test analysis methods.We have observed many varieties of unusual wellbore behavior:1. Gas may compress in the annulus or tubing.This results in storage continuously decreasing aswellbore pressure increases.2. Gas may expand in the tubing or annulus. This

results in storage continuously increasing as pressuredecreases. The test in Fig. 1 is an example.3. Falling liquid level is accompanied by a constant pressure on the top of the liquid column. Thispressure may be atmospheric (wellhead open to theatmosphere) or vacuum (wellhead closed). In thelatter case, the pressure over the liquid column is thevapor pressure of the liquid in the wellbore at wellbore temperature.4. I f a well is shut in at a header some distancefrom the well, the surface lines can drain into thewell as the fluid level starts to fall. This is actuallywhat occurred at the plateau in the test shown inFig. 3.5. I f injection starts from a header, surface leadlines may have to be filled before or as the wellbore fills.6. Wellbore storage effect may take the form ofphase segregation in the wellbore. Similar effects arecaused by gas-lift systems, especially by gas-lift systems with leaky gas-lift valves.7. Wellbore storage may decrease markedly whenthe fluid in the wellbore reaches the bubble-pointpressure during a buildup test. At this point, free gasgoes into solution and the compressibility of the wellbore system decreases, thus decreasing the wellborestorage coefficient.

ConclusionsI f wellbore storage changes during a test, falloff andinjectivity tests generally appear to be very different.I f they are very different, the injectivity test is thepreferred test. I t is easier to analyze and more reliablethan the falloff test, and is more likely to give usableresults. We recommend that injectivity tests be run ifchanging wellbore storage is a problem in injectionwell testing.The material presented here also applies to production well testing.Nomenclature

Bw = water formation volume factor, RB/STBCt = total system compressibility, psi-1C = wellbore storage coefficient, RB/psiCD = dimensionless wellbore storage

ffi . C 5.6146 Ccoe clent, D = 2 h 27repCt rwh = formation thickness, ftk = formation permeability, mdm = slope of the correct semilog straight line,psi/cyclep = pressure, psi

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Pw. = bottom-hole shut-in pressure, psiPWf = bottom-hole flowing pressure, psi

Pl hI' = pressure 1 hour after beginning of test,taken from the correct semilog straightline (extrapolated i f necessary), psiAp = pressure change, psiq = surface flow rate, STB/D

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

At = change in time, hoursp'w = water viscosity, cp

cf> = formation porosity, fractionReferences1. Matthews, C. S. and Russell, D. G.: Pressure Buildupand Flow Tests in Wells, Monograph Series, Society ofPetroleum Engineers of AIME, Dal las (1967) 1.2. Kazemi, Hossein, Merrill, L. S. and Jargon, J. R.: "Problems in Interpretation of Pressure Falloff Tests in Reservoirs With and Without Fluid Banks," J. Pet. Tech. (Sept.1972) 1147-1156.

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3. Ramey, H. J., J r. : "Non-Darcy Flow and Wellbore Storage Effects in Pressure Build-Up and Drawdown of GasWells," J. Pet. Tech. (Feb. 1965) 2 2 3 ~ 2 3 3 ; Trans., AIME,234.4. Agarwal, Ram G. and Ramey, H .. J., J r. : "Annulus U!1-loading Rates as Influenced by Wellbore Storage and SkmEffect," Soc. Pet. Eng. J. (Oct. 1972) 453-507; Trans.,AIME,253.5. van Everdingen, A. F. and Hurst, W.: "The Applicationof the Laplace Transformation to Flow Problems in Reservoirs," Trans., AIME (1949) 186, 305-324.6. Agarwal, Ram G., AI-Hussainy, Raft and Ramey, H . J.,Jr.: "An Investigation of Wellbore Storage and SkinEffect in Unsteady Liquid Flow: I. Analytical Treatment,"Soc. Pet. Eng. J. (Sept. 1970) 279-290; Trans., AIME,249.7. Earlougher, Robert C., Jr. , and Kersch, Keith M.: "FieldExamples of Automatic Transient Test Analysis," I . Pet.Tech. (Oct. 1972) 1271-1277.8. Hazebroek, P., Rainbow, H. and Matthews, C. S.: "Pressure Falloff in Water Injection Wells," Trans., AIME(1958) 213, 250-260.9. Ramey, H. J., Jr.: "Short-Time Well Test Data Interpretation in the Presence of Skin Effect and Wellbore Storage," J. Pet. Tech. ( Jan. 1970) 97-104; Trans., AIME,249. JPT

Paper (SPE 4371) was presented at SPE-AIME 48th Annua l FallMeeting, held in Las Vegas, Nev. , Sept. 30-0ct. 3, 1973. Copyright 1973 American Institute of Mining, Metal lurgical, andPetroleum Engineers, Inc.

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