SPE 26647

Application of Variable Formation Compressibility for ImprovedReservoir Analysisf3.P. Yale, G.W. Nabor,* and J.A. Russell, Mobil R&D Corp., and H.D. Pham’ ● andMohamed Yousef,~ Mobil E&P US. inc.

SPE Members●Now retired

●●NowwithAbu DhablNatl. Oil Co.tNow withSaudiAramco

Copy@ht 1S53, SocMy of Petroleum Engineer% inc.

Sooktv of PetroleumErIQWrs !

This paper wee prepared for praeematlon at the SSlh Annual Tecfmlcal Conference and Exhibllion of the SocIeIy of Pelrofeum Engineers held Irr Houatom Texas, 3-6 October 1993.

This paper wee eelectmf for presenla!ion by an SPE Program Committee followlng review of information contained In an abetract aubmltfed by the author(s). Contents 01the paper,ae presented, have not been reviewed by the Soclaly ot Petroleum Englneare end are aubjecf to correction by the author(e). The material, ae presented, does not necessarily retlwtany poaltlon of the SocIefy of Petroleum Engineers, Ite officers, or mambera. Pa?era presented al SPE meetings are subject to publication ravlew by Edltorlal Commitleea of the Societyof Petroleum Englneeta. Permleelorrto COPYlareefrlafed !0 en abatracf of not more than 300words. Illuatratlonemay not be copied. Tha abatract should contain conspicuous acknowledgmentof where am by whom ma paper fe preeerded. Write Ubrarlan, SPE, P.O. Box S32438, Richardson, TX 76083-3S3S, U.S.A. Telex, 16324S SPEUT.

ABSTRACTFormationcompressibility has long been recognizedasan important factor influencing production behavior fromoverpressuredoil and gas reservoirs. However,forrmtfon compressibii”~ data are not routineiy collectedand the us~ of formation compressibilityin reservoiranalysis and simulation is often oversimp!ifjed.

This paper discusses more accurate methodstodetermine fomation compressibilityand introducesanew method for anafyzing overpressuredoii and gasreservoirswttkh utilizes the variability of formationcompressibilitywith declining resefvoir pressure. Thenewiy deveioped method departs from earlier proposedmethods in the use of _ rather than &@ fom’tationcompresslbiiii by empfoyinga “pore volume formationvoiume factorn,13f,that property integratespore voiumecompressibilityeffects over the fuli pressure range ofinvestigation. Using the new concept of Elf,the materiaibalance equation (MBE) can be modified to inciude theeffects of pressure dependent formation compressibiiiiy.

We find that the formation compressibilityin highiyoverpressured unconsolidated reservoirs can be thesame order of magnitude as gas compressibilityandsignificantlyhigher than oil compressibility. in sometypes of reswvoirs, an order of magnitudechange information compressibilitycan occur during drawdown.We show that in many ove~ressured andlorunconsolidated reservoirs, proper integration of accurateformationcompressibilities is importantfor resetveestimates,determination of drive energies,and overallreservoirdevelopment plans. For exampie,we find thatthe use of compressibilityvaiues in the MBE which aresignificantly fewer than those which exist in the reservoircouid suggest a strong waterdrive where one does notexist.

1. lNTRODUCTIONit is recognizedthat a decrease in pore voiumeaccompaniesa decilne in reservoir pressure. Therelative change in pore volume per unit of pressurechange, i.e., the formation compressibility,depends onthe rock type, its degree of competence,and the tectonicsetting. Laboratorymeasurementsshow a wide range ofcompressibilityievels over the spectrumof rocks fromcompetent carbonates to unconsolidatedsands,Compressibilitydeclines, sometimes drastically, aslaboratorystress is increasedto correspondto reservoirpressure changes from discovery to abandonment.

Formationcompressibilityis a source of drive energy inaddition to that provkfedby expansionof fiuids. its effect,and also that of water, are often ignored in anaiyzingreservoir performancesince the contribution is minorcompared with that of gas or oii plus soiutlongas. Theeffects are usuaily considered, however, whenundersaturatedoii resewoir performance is analyzed andthe contributionsof rock and water expansioncan easilyexceed 10 percent of the totai.

The conditions found in abnormally pressured reservoirsaiso lead to greater significanceof formationcompressibilityas a source of expaneion energy,particularly if the formation is pooriy consolidated.Abnormai pressureat discovery means a lower effectiveresewoir stress condition, and a higher formationcompressibility. Since pressure ievei is often high, gascompressibility[ ( l@)- ( IL?) [ dtidp )] is relatively low,and formationcompressibilitymay in fact be of the sameorder of magnitude; it wlli often exceed oiicompressibiiiiy. Formationcompressibilitycontributionsmay be further magnified if an aquifer--evena smali one--is present since aii of the water-bearingrock present wiiiprovide fomnationcompressibilitydrive,energy.

Referencesand illustrationsat end of paper. 435

? APPLICATIONOFVAFtll$LE FORMATIONCOMpRESSl131Llw SPF 266~

Where resemolr conditions are such that compressibilityis expected to be relativelyhigh, and variable with stresslevel, taboratoly measurementsare definitely indicated.Use of the data In reservoiranalysis is not routine,andapproximationsare often used. in this paper, we addressboth the laboratorymeasurementsand also a method foraccurately incorporatingthat data in resetvoirperformanceanafysis. The resuft is one which is quitegeneral and whkh can be incorporated in existingmateriai balance or resewoir simulation formulations withonly minor modifications. Futier, methods previouslyproposed by otiter investigatorsprove in fact to bespecial cases of the generai approach developed here.

2. FOtlMATiON COMPRESSIBILITYPore compressibility is a laboratory measured rockproperty whkh is defined as the relative change in porevolume cda rock sample divided by the change inlaboratorystress whkh caused the change in porevoiume:

A ~pl Vpcp=—

AIM)”””””””””””””””1

Formationcompressibility,however, is defined in mostreservoir engineering handbooks as the relative changein pore volume divided by the change in reservoirpressure that caused the change in pore voiume:

C,=4.!!M?A/) ““””0”””””’”””” 2

The difference between pore compressibilityandformation compressibifiiytherefore is reiated to thedifference between reservoir pressure and laboratorystress. There are four main stresseswhich act on anyvolumeof reservoir rock. The overburdenstress, az, thehorizontriistresses, ax, Cp and the pore pressureorreservoir pressure,p, wtuch pressesout against theoverburdenand horizontal resemoirstresses. [n thelaboratory,however,most overburden tests are run usinga hydrostatk confining pressure and ambient porepressure. The reservoirstress state and changes in thatstress state mustbe convertedto effectiie hydrostatklaboratorystress to understandthe laboratorydata. Thefoilowing equation has been proposed and derived bymany (Geertsma, 195Z Jaeger and Cook, 1976; Teeuw,1971; Nur and Byeriee, 1971):

where KI, K2, and K are constants dependent on rocktype and pi and p are the reservoirpressureatd=overy and at the present time respective~. ~eb isthe hydrostatic confining pressureappiied to the coresample (minusany pore pressure)to simulate the in-situ

stress conditions. Equation3 is sometimes referredtoas the “effective stress” equation. Tabie 1 gives KI, IQ,KS for various rock types. KI and K,, relate how thethree confining stresses in the reservoir and the reservoirpressure interact. KI can be defined as

KI=(CrX+ ay+IYZ) /(3 CYZ). . . . . . . . . . 3a

oz can be estimated using an overburden gradientof 1psi per foot of depth or from integratinga density log. K2is equivalent to the Biot “alpha” parameter and is definedby Geertsma (1957) and Nur and Byertee (1971)as:

K2 = (f ‘cb/cg~), . . . . . . . . . . . . 3b

& relates how the drawdownof the reservoir pressureincreasesthe stress on the formation. it can be definedas:

K3 = K2[(l+v)/(3 - TV)] . . . . . . . . 3C

Equation3Cis identical to the “uniaxial correctionfactor”deriied by Teeuw (1971) with the exceptionthat heassumes K2to be unity.

From Equation3, we can see that hydrostaticporecompressibilitytests, therefore, can be correctedtofomation compressibility through the foilowing equation:

cf=&cp . . . . . . . . . . . . . . . . . . 4

r TABLE 1

I ‘O”s’’”‘oR“““v’‘THs W.AIl&I Bock WIQ K h &

IConsolidated Sandstones* 0.85 0.80 0.45

IFriabie Sandstones 0.90 0.90 0.60

IUnconsolidated Sands 0.95 0.95 0.75

ICarbonates’ 0.85 0.85 0.55

*These K2constants for are vaiid for many consolidatedsandstones and carbonates. For weii cementedfomations with porosities iower than 15%, the Kz factorcan be between 0.4 and 0.8 due to the formation’s lowbuik compressibility (see Equation 3b).

2.1 Uniaxial CompactionAs fluids are withdrawnfrom the reservoir, it is assumedto compact only in the verticai direction (uniaxiaicompaction)because the verticai extent of the reservoir isso smali comparedto its iateral exient (Cieertsma,1957;Teeuw, 1971; de Waal, 198$). This leads to a decreasein the horizontal stressesand therefore to a decrease inthe average confiningstress. This has the effectof

436

ABOR.RUSS~. PHAM.ANDYOUSAF 3

fessenfngthe increase in effective stress as the fluidpressure in the resetvoir is decreased. The fi constantin equation 4 accounts for the changes in horizontalstresses (see Equatfon3c). The variation in Poisson’sratio, v, between consolidated and unconsolidatedclasticwdments leads to a variation In Kj of 0.45 forconsolidated sandstones to 0.75 for completelyunconsolidated sediments. Therefore, for a consolidatedsand, a drawdownof 2000 psi Is simulated in the labor-atory by an increase in effective stressof only $00 psi.

This lmlaxfal compaction of the reservoirduringdrawdown has led some to suggest that thecompressibility shoukf be measured unkodaliy,mimkking the “no fateral deformation” bounda~condition and allowing the sample to deform only in thevertkal direction &achance and Andersen, 1987;Andersen, 1985 de Waal, 1986). Theoretkaliy, however,(G$ertarna,1957 Jaeger and Cook, 1976) the volumetricchange in pore volume Is due only to the change In theJWQ5WQvolumetrk stresses on the sample,thereforepropertycorrected hydrostatic tests should be equivalentto uniaxial tests.

We argue that the diffkutties in maintainingthe “no fateraldeformation” boundary condfiion along the entire lengthof a sample durfng a trkudaitest as well as the cost anddiffkufty of the tests make uniaxial tests unfavorable.Published data on unkaxialcompaction (Lachance andAndersen, 198* Andersen, 1985) show data whkh areboth signifkantly 1sssand signfficantfymore than aspredicted by theoretically corrected hydrostaticcornpressibiiff tests. We suggest, therefore, thatformation cornjmssibility be calculated by performinghydrostatic pore compressibilii tests and correctingtoformation compressibility using Equation4.

2.2 Laboratory Methods forPcw Compremdbility

Laboratory pore compressibility measurementsare doneby determiningthe pore volume of a core sampleas afunctbn of effective laboratorystress. The pore volume isusualfydetermined efther by measuringthe total fluidsqueezed out of a Ifqdd saturated sample andsubtracting it from the pore volume at ambient condfiionsor by measuring the pore volume directly of a driedsample at each pressure level using the Soyle’s faw gasexpansion technfque.

Since pore compressibNtyis related to the derivative ofthe pore volume versus stress curve, the accuracyofcompressfbijitydata is dependent on the ability of theapparatus to measure very small changes in porevolume. For thfs reason, Ikjukf squeezeout on sampleswith more than 10CCpm volume gfvesbettercompressibility results than Boyle’s faw measurementsortests on small samples.

We have found that on samplesfrom friable orunconsolidatedformations, sampte integtity as well assample volume is a concern. Pore compressibilityis verysensftiveto the degree of damage or disturbanceof the

437

sample in weak sediments. ASshown in Figure 1, fujldiameter samples from the same unconsolidatedformationas a set of plug sampleshave significantlylowercompressibilities. We suggest that core damageduring plugging and cleaning disturbed the samplesenough to csuse this difference. The authors have foundthat ambient pressure porositiesof the plug sampleswere 2 to 8 porosity units higher than the full diametercore samples,

To maintain sample integrii to insure valid porecompressibilitymeasurements,the authors recommendthat unconsolidatedcore samples be frozen on well siteto prevent sample disturbance and desiccation duringshipping; that fulj diameter samples be used to preventdisturbance from plugging and to maximize accuracyand that the frozen samplesbe placed in the pressurevessel before cleaning and allowed to thaw under someminimumstress (100 to 300 psi, generally). Brfnesqueeze-out pore volume testing can be done before anycleaning provided care b taken to fully liquid saturatethesample and that ambient pore volume is measuredafterthe test is complete.

We have also found that the creep associatedwith thedeformation of unconsolidated rocks can causecompressibilitytests run at high rates of pressureincreaseto be invalid. One of the authors and others (deWaal, 1985) have observed creep in unconsolidatedcoresamplesto be logarithmicwith time. The magnitudeofthe creep being the most signffkant in poorly sorted,clayrich unconsolidatedcore samples. It is unfeasibleto runtests at reservoirdrawdown rates of 100 psi per monthbut standard laboratory rates of 1000 to 2000 psi perhour do not allowthe creep to occur, We suggest thatcompressibilitytests on core samples run at ratesbetween 50 and 5 psi per hour for unconsolidatedsamplesand 500 to 50 psi per hour for weaklyconsolidatedformations allow a significant portion of fhecreep to occur thus improvingthe accuracyof thecompressibilitydata.

2.3 Variability of Formation CompressibilityOne of the reasons why formation compressibilityhasbeen left out or underestimatedin reservoir analysis isthat it has been assumedthat pore compressibilitykfairly constant with stressand of the same order ofmagnitudeas the compressibilityof water. EvenHammerlindl(1972) who recognized the importanceofcompressibilityin resetvolr analysis, used a constanthigh formation compressibility value. Figures 3 through5 showthe varjabitityof pore compressibilitywithpressureand rock type. The figures representcompilationsof data for consolidated, friable, andunconsolidatedciastic sediments.

Definftkmsof the degree of consolidation are vague. Forthe pwpose of our compilationsthe following generalguidelines apply. Consolidated sandstones haveundergone sigrdfkant diagenesis and have thek grainswell cemented and dropping a core sample on the floordoes not cause ft to disintegrate. In the consolidated

4 FORMATIONCOMPRESSIBILITY SPI?P664~

sandstonestested, porosity ranged from less than 1Ye to25% with a mean porosity of 15%.

We define “friable” samplesas having Iit!leor no cementbetween the grains but holding together evsn aftercleaning and drying. Friablecores, howe~er, willgenerally break or disintegrate if dropped onto the floor.Porosityof the samples tested ranged between 20% and33Y0,with the mean porosityfor our data set at 23.1?4.We have found that the compressibilityof very clean, wellsorted unconsolidatedsands generally fall into this“friablencategory even if they have no cement.

We define “unconsolidated”samplesas those which fallapart completely after drying ancVorcleaning withporositlesbetween 27% and 40Y0. They generally haveno cement between tho grains and are poorly sortedan~or have large clay iractbns. Our data set ofunconsolidated samples was populated primarily withturbidite-typeGulf Coast sands with a mean porosityof32.5?40.

Figure2a and 2b show the differences in grain sizedistributionsbetween a clean, well sorted sand (whosecompressibilityfalls into our “friible” category) and a clayrich, poorly sorted sand (whichfalls into our“unconsolidated”category). Both sands are uncon-solidatedfrom the point of view of having no cementbetween their grains, but they have widely dflerentformationcompressibilities. We have found this strongcorrelation between degree of sorting and compressibilityin a number of unconsolidatedformations.

Figure3 shows formation compressibility versuspressureon a toglog plot for a collection of 121consolidated sandstones from over 45 formations fromaround the workf reported in the published literature(Chieriii et. al. 1967, Dobrynin 1963, Fatt 1958a, 19513b,Wyble 1958, Yale 1984) and measured by the authors.Note the general downward trend versus pressure withan order of magnitude change in compressibilityover thepressure range. Note the order of magnitudevariationofcompressibilitywithin rocks which are all conskfered“consolidated sandstones”.

Figures4 and 5 show the formation compressibilityoffriable to unconsolidatedrocks which make up asurprislngtylarge number of resemoirs. These ranges offormation compressibilitiesare large enough to figureprominently into the total compressibilityequation forboth oil and gas reservoirs, especiallythose which areoverpressured. The data in Figure4 are from 140 coresamplesfrom 7 reservoirsin the North Sea, Afriia, andthe U.S. Gulf Coast which we consider %iable”. The datain Figure5 are from 14 full diametercore samplesfrom 4reservoirsin the CM of Mexko and Afrka which areunconsolidatedand poorly sorted. Note from Figures4and 5 that neady all the samples have compressibilitiesgreaterthan that of water at stressesup to 10000psi.Comparingall three figures, we see over 2 orders ofmagnitude variation in compressibilii at any givenpressuredepending on rock type. Also note that theslopesof the three data sets are dflerent.

These three figures show the importanceof includingvariable fonmationcompressibility in reservoir analysis.Gas compressibilityat 8000 to 15000 PSIcan be in therange of 200 to 20 microsips. In overpressuredreservoirs,where the “effective stress” (see Equation3)can be 3000 to 1000 PSI,formation compressibilitycanbe 1 to 50 mlcrosips.

We find that it is the change in gas and formationcompressibilitywith pressure which causes the familiarchange in slope of the p/z versus cumulative productionplots in overpressuredreservoirs. As reservoirpressuredecreases, gas compressibility increases and formationcompressibilitydecreases. The change in slope of plzversus production plots for overpressured reservoirscanbe due to a change from a formation compressibilityinfluencedsrdem to a gas compressibility dominatedsystem,

2.4 Type Curves for Formation CompressibilityPore compressibilitymeasurementsare not performedroutinetyfor all reservoirsand data are especiallysparsefor those formationswhere it is most important(i.e. friableand unconsolidatedformations). Figure 6 and Table 2give “Type Curves’ which can be used to estimateformation compressibility in clastic formations if cwe dataare not available. The three type curves (and theequationsgiven in Table 2) are least square fits throughthe data compiled in Figures3,4, and 5.

W= CURVES FWMAIW COwFtE~CL~STIC RESERVOIRS

Cf = A(cr-B)~+D

he typecurvesin Figure6 aredefinedbythe abovequationwhenx

r = KI* (overburdenstress) - K2● ~ + /(3 ● (pi -p) (psi)mdII,B, C, D are constants depending on rock type asIescribed below.

Unconsol Fri$bk(poorly s;~ed)

ate~(&well sortedunconsol.)

A -2.805X 10s 1.054X 10~ -2,399X105

E 300 500 300

c 0.1395 -0.2250 0.06230

D 1.183X 104 -1.1(X4Xlo~ 4.308X 105

We caution against the use of type curves unlesscoredata is not available. Many times in unconsolidatedorfriable reservoirs,very little if any core is availableso thatestimatesfrom type curves are necessary. We remind

438

YALE.NABOR.RUSSELI PHAM.ANDYCXJSAF 5

the reader that the “unconsolidated”and “friable”datasets do not cover a wide variety of reswvoirs and therewill be formatkms which can be considered‘unconsolidated” or “friable” which have compressibilitiessignifkantly different from those presented in the typecurves. We do belleve, however,that the quality of thedata In the formationstested is very good due to themeasurement procedures followed.

3. THE PORE VOLUME FVF - A NEW CONCEPTIn order to easily incorporatevariable formationcompressibility into reservoiranalysis we define a “porevolume FVF” (formationvolume factor) as:

~,sv’/v’ ., OO. O., . . . .. O.. O 5

It is convenient, though not strictly necessary,to chooseone atmosphere and reservoir temperature as thestandard or reference condition, where Br= 1.0. Thepore volume FVF is easify related to formationcompressibilii. In differential form the formationcompressibilityequation (Equation 2) can be written as

Cfdp=dVp/Vp= d~lnVP~ . . . . . . . 6

which can be integrated between limitspscand p to give

JP

h (b”/ VPSC)= cfo’p=/(P) . . . . . . 7Psc

or equivalently

8f=el@~ . . . . . . . . . . . . . . . . . 8

The laboratorytest from which CPis determineddoes, infact, give a nearly direct determinationof Bt. The ratio ofsamplepore volume at any stress level to pore volume ata stress tevel correspondingto that reached in theresewolr when pressure declines to standard pressuregivesthe pore volume formation volume factoq the dataneeded are an inftfalpore volume and fluid volumeexpelled as a function of stress applied to the sampleand, of course, a refationsuch as Equation 3 which tiesreservoirpressureto laboratorystress. The laboratorymeasurementdoes not even have to be carried to the“standardcondtiion” stress level; it need onfy cover astress range which encompassesthe expected range ofreservoirpressure. This amountsto defining a referenceconditfon tied to the hjghest stress level reached (i.e.,reservoir pressure below the lowest expected operationalpressure).

3.1 Modified Fluid Formation Volume FactorsBased on the above formulations we defi”” a modifiedgae/oil/waterFVFas

l&fj/& . . . . . . . . . . . . . . . . . . 9

wherej refersto gas, oil, or water. With this definition,we have the advantage of simultaneouslyconsideringthe changes, with pressure, of both fluid and the porespace associated with that fluid. in material balancework, use of these factors allows us to center attentiononfluid volume changes, knowing that pore space changesare being carried along automatkalty, The resuft, as weshall see, is a compact form of equation which accuratelyconsidersall facets of the formation and fluid expansionprocesseswhite retaining an appearance similar to thatwith which reservoir engineers have long been familiar.

4. MATERIAL BALANCE EQUATIONWe will derive the materfal balance equation (MBE) for ablack oil system, using the modified formation volumefactors just introduced. The systemmay be comprisedofthree zones gas cap, oii zone, and pot aquifer. Phasespresent consist of hydrocarbon vapor, hydrocarbonliquid, and brine which are more commonly called freegas, oil, and water. Gas is also looked upon as acomponent,and may be present either in free form ordissoived in oii and water. Oil and water are not solublein gas or in each other. A common (average)pressurecharacterizesail zones and phases.

Since the contribution of water-saturatedformation todrive energy may be considerable, the distributionofwater in the system is of Importance. First, averageconnate water saturation may be different in the gas capand oii zone. Second, we allow for the presenceof a potor “steady state” aquifer which is in immediatepressurecommunicationwith the hydrocarbonzones. This couldbe underlyingwater or simply a small aquifer. In theusual analysis, the energy contribution from a smallaquifer might be neglected, but the possibility of high andvariable formation compressibility enhances theimportanceof such a contribution, especially inoverpressuredsystems. Fina!ly,we will allow for watarand gas influx from a “transient”aquifar. Precisetreatment of such influx requires separate analysiswhichis beyond the scope of this paper, but the overall effectsare easily included in the general formulation.

The analysis begins by relating the pore volumes of theoil, water, and free gas phases to the total pore volumeofthe system.

Nt30/+ WBW~+GF/Bgj=VPSCB/j. . . . . . . 10

from which

VP3C=N&j+ W~Wi+@j&i . . . . . . . . 11

After some depletion, influx of water and gas, andshrinkageof pore volume, the ‘followingwill apply:

(N- Np)13. +(w-wp+w..)B.+(GFI+G~I-G~-GP )13a= VXcBj . . 12

I

I

I

439

$ APPI l~F VARlw FORh&TiONCOMEJ3EWlRlL11’Y SP-

The term (GsI- G9)represents the difference in solutiongas content between initkd and current conditions andcan be written after combining like terms tw

(3s/- (3,= N(F/s/-Rs)+Np F?s. . . . . . s 13

We now go through the algebraic steps of solvingEquation 12 for VP8C,equating the result to Equation 11,and then gathering all terms dealing with production orinflux on the right hand side of the equation while allothers are gathered on the left we get:

(A/ {[ E.+(/?#?.)Bg] -E.,}) +

W {~W-~Wl} + GFI@a-~~l)=

Np(Bo-Ffs E#) +

(WP-We)EW+Gp E9 . . . . . 14

we can define a modified two-phase formation volumefactor by dividing the standard two phase factor by Bfi

Bt=@(R#?s)Eg . . . . . , , . . , 15

Notethat ~ti = ~01

A final step to reachthe form desired requires relating Wand GFito N. We define two quantities

Fe =&

pore volume ratio, gas cap/oil zone= pore volume ratio, pot aquifer/oil zone

Then

*[ I+ FW+F’] . .0 16B~iVP% = -h{

and the pore volume of water can be found by multiplyingeach of the terms within brackets by the appropriatewater saturation for each zone:

~ [ SWI+ FgcSwgi+ FXJ ..17Elwjw = _

After dwision by f3~j,substitutions and rearrangement:

w=[ 1

I@ Swi+ F“cSvgi+F/)ii . . . . 18

Ewj 1- sw~

For free gas,

@, = [

@ Fw(l-swi

Bgi 1l_sw,~)““o””” 19

When the appropriate substitutions are made in Equation14, the final result is:

N (g.- I@g)+(W” - We)& + G#gN= P*

(( )Br/ f-l +

[ 1( )

Swi+ FgcSwg\+ FW - 20aY_l +

I-swi -

[%$Q$J][*-l))

While the preceding equation is a very general form, itdoes require a calculationof Weby other means, Inaddition, using the produced _ ratio:

Rp = Gp/Np, . . . . . . . . . . . . . 21

we can rearrangeterms to yield:

The numerator is sometimesreferred to as the “expandednet=production-plus-excess-gas”formulation.

Forgas reservoirswith associatedaquifers, the sameapproachmay be used to derive the analog of Eq. 20:

:zi!i%-23The terms appearing In the denominatorof the Equations20,22, and 23 are worthy of examination. Eachof theterms[( ~1 E1’ji)- 1] representsthe expansionof a unttvolume of initial fluid, including its dissotved gas, and thecontractionof its associatedpore space. The factorswhich multiply [(@ Ej] -1] are volume ratiosat initialconditionsfor (water/oil), (free gas/oil) or (watdfree gas);the multiplierfor the first term is unityof coursesincetheamatysisis based on a unit of either oil or of free gas.

The water term is often neglected in material balanceformulations,but It shoukt not be. In the general formshown here, its significancebecomes more obvious,especially in overpressured reservoirswhere formationand gas or oil compressibilitiescan be comparable inmagnitude. The water term may in fact be dominantforquite modestvalues of Fp

This can be demonstratedby noting that

In EW = lnBw-ln~

Y~R. RWWHUkiAM. ANDYOUSAF 7

and taking the derivative and rearranging:

(%/BW,) = @ - (Pi-p)

The exponent is srnaii, since compressibilitiesaretypically 106 in order of magnitude while pressurechanges are 10+3in magnitude, so:

Bw/E#l+ m (lx-p)or

()*-1 & m (Pi-P) . . . . . 24B~j

Similar expressionsmay be developed for oii and itsdissotvedgas, and also for free gas, and the pore spaceassociated with each.

Some order~f-magnitude cakulations can now bemade. if we choose a system at 10,000psi and 225°Fastypical ef an overpressuredresetvoir setting with aweakly consolidated or unconsolidated formation, we canestimate:

Cw s 3(10+) pShl (Ostf, 1984)Cg = 37(10~ pshl (Bradley, 1987)C?(frbl) =10(10~) ps~l (friable sand)Cf (Uc) =35(10~ pstil (unconsolidated sand)

it foitowsthat

CW+ C~(frbi) = 13(10~ psklc~+Cf(lJc) = 38(10*) psi-l

compared to

Cu+ Cj(frbi) = 47(104) psi-lCe+ Cf(UC) = 72(10+) pSt’

Tim, the unit expansibilityof water and its pore space isnearty30 percentof that of gas and its pore space for aweakiy consolidated sand and over 50% for anunconsolidatedsand. if SWI= 0.2, the water termappearing in the denominator of Equation 23, for gasresewoirs, wiil dominate if FW >2.7 for a weak sandand for FW >1.3 for an unconsolidated sand. For oilreswvoirs, an estimate of two-phase compressibilitywiilbe system-spectfb, but we can reasonablyargue that itwiii be less than gas compressibility. The water term wiiithen exceedthe oii term at even lowervalues of F’pa.

While the preceding development aimed to illustrate theneed to account for water-bearingformation in materialbalance anatysis,the key issue is actuaily the highformationcornpressibiiity. in the example, formationcompressibiiii contributes over 20 percent of theexpansion energy associated with gas-bearing rock, andover 75 patent of the energy associatedwith water-bearing rock for weak fofrnations. For unconsolidatedformatbn, fonmationcompressibifii contributes nearly

!50%of the energy associated with gas-bearingreservoks. Formationcompressibilityeffects should beincluded, and water-bearing rock shouid not be ignored,even though its total voiume may appear to be quitemodest.

These facts have long been recognized in anaiyzingperformanceof overpressured gas reservoirs(Hammeriindi,1971; Bass, 1972). However,these andother investigators (Ramagostand Farshad, 1981;Bernard, 1987) have suggested onty approximationsfordeaiing with the problem. The fownulationproposed hereexplicitly includes the effects of ali contributing fiulds andtheir assodated pore space, and has the added attractionof allowing variable compressibilities to be included withrelative ease.

5. MBE ANALYSISThe MBE presented in Equations20 and 23 Is morecomprehensivethan those usuaiiy presented, but it hasthe same format except for the use of the modifiedformationvoiurnefactors ~O,W,uin place of the BO,W,g,Themodifiedfiuid formation volume factors can be cafcuiatedindependentlyas a pre-analysis step, and used in placeof the usual fiuid volume factors in MBE’s in current use.it is readityapparent this MBE formulationwiii reducetoconventionalpresentationsof the MBE (see, for example,Dake, 1978; Bradiey, 1987) if appropriate simplifyingassumptionsare made.

As an example, consider the gas materiai baianceEquation23. if we divide both numeratoranddenominatoron the right hand side by Eg,solve theresuftingexpression for ( 1/ ~g) and then substituteBt(p/z) = (constant)s( 1/ ~g),we obtain, after somealgebra:

if we assume Wtt= O,then GF1= G. We also introducethe approximations:

Bf = M 1 -cf(PFP)l

BW = 8~J 1 + Cw(p-p)]

where C~and Cware taken to be smail and constant,The equation which ultimately resuits b:

p , _ Gv(~pt?+%)+cd~w+1)(p;_p)=()[z 1- s~j 1(9,-(*)(9J%+ %%) 26

441

●

lLllY SPF 26647

The preceding equatfon is that developed by f3ass(1972). If, F-= Oand #Yp=O,then:

( )[ ,_& ‘]=(9,-(?)/(%)27g 1 -(Q+ cwav/)(p/-p~

which was proposed by Ramagostand Farshad (1981).

Any one of the Equations 25 through 27 can be plotted as‘corrected” ( ph ) versus “corrected” t3Pand the lineextrapolatedto an intercept to estimate @J or (3,provkfedof course that FWcan be estimatedwithsufficientaccuracy to allow an accurate correction to becalculated. Equation 25 has an advantaga for caseswhere Influxcan reasonablybe taken as zero, and theoverpressuredgas resewoir may well fit this case. Sinceall variable effects are properly allowed for, F maybe

1%determinedby trial and error as the value wh h leads tothe best straight-line fit of the pressureand productiondata. Equations 26 and 27 are not really suitable since cfwill in fact change rather rapidfyas ( pi - p ) increases.

& SIMULATION CONSIDERATIONSVariable compressibility Is easlfy handled at the partialdifferential equation level by substituting OSCEf~forporosity wherever ff appears in the equations.Manipulationof@ as a pressuredependent variableshould be straightforward. it maybe preferabletoreformulatethe equations In terms of the modified fluidvolumefactors B,since these variables can be developedoutsfde the context of the simufatlonequations, therebyreducingthe numerkal cafculatlon required. Since Eltisa continuous, stowtychanging function of resewoirpressure,there is no reason to anticipate that the ~functions will be any more diffkuft to handle numericallythan the B] functlcmsthemselves.

7. CASE HISTORIESTwentyover-pressured gas reservoirs were selected andanalyzed with a computer program developed by usingthe new method and the rock compressibilitycorrelationsdiscussedabove. Followingare two of the case historiesstudied.

One factor needed in the anafysis is a determinationofrocktype so the propers or P relationshipcan be used.If core data are not avalfable,type curves for formationcompressibiiifycan be used although it is aiwayspreferable to use fabomiory compressibilitydata from theformatfcnof interest. If type curve compressibilityis usedyet the degree of consoildatton is not certain or avaiiable,one shoufd conduct sensitivity studies for ail appropriaterocktypes to determinethe best sultabie solution. Forthese case histories, formation compressibilityis takenfrom the type cuwes presented earlier.

7.1 Caso 1The first selectedcase histoty was the Anderson “L”reservoirfrom the Mobii-Davkffieid presented by Duggan

(1972). The Anderson “L” is an wer-pressured gasresewoir having an InftlaipressutQof 9507 psia at11,167feet subsea depth, or a gratiient of 0.843 psilft,Tabie 3 provides other pertinent data on this reservoir.in this case, ft Isassumedthat FW, WO, Geand RSWequalzero, and the “L” sand is weakly consolidated.

The pore vo!ume formation voiume factors (B~ arecalculated from Cf vaiues by using Equations7 and 8.Figure7 shows a graphkai presentation of the rockcompressibilityas a function of reservoirpressure. Wecan use the 13fconcept to ‘correct” the p/z versusproduction piot to account for formation and watercompressibility. As shown In the braced term on the Iefiside of Equation25, we can use a factor C:

C = (Bf/Bf~”(Fpa + 1) - (Bw/BW~*(Fpa + Swfl 28

(1-sw/j

as a muftip!lerfor p/z. Figure8 shows the actuai and thecorrected@zdata plotted against the cumulativewet gasproduction. The eariy extrapolationof the actuai p/zcurve Indkates an apparent gas-inepiaceof 112 Scf,whkh Is about 61 percent higher than the estimatedvoiumetrk gas-in-placeof 69,6 Bcf. However,theextrapolationof the corrected p/z curve using iinearregressionon ail data points yields a correetedgas”in-place of 83,6 Bcf. The gas-in-piaceof 83.6 Bcf was theninput into Equation 25 and the estimated gas productionat each time step was calculated and plotted in Figure8.As shown in Figure 8, the calculated gas productionshows an exceiient matchto the actual data.

To determinethe degree of confkfence in predkting theoriginai gas-in-piace early in the productive iife of thereservoirwhen a few data points are availabie,asensitivity study was conducted where oniy the first sixdata points were considered In the evaluation. in thiscase, the origlnai gas-in-piace determined by iinearregressionon the first six correctedp/z data points isestimatedat 76,0 Bcf. Tabie 4 shows the regressionanaiysis results for the six and the ail=data-pointcases.Aithough the six-data-pointcase shows a higherstandard deviation, both cases give an exceiient best fitto the straight iine, This seemsto imply that the gas-in-piace tends to be under-estimatedwhen consideringonlyetwiydata points. To verffy this point, we performedadditional evacuationsbased on data groups from aminimumof three to a maximumof sixteen data points.The results from these evaluations and our experiencewith other case histories indicated that gas=in-piaceestimatestend to increasewhen more data points areinciuded and become stabie as reservoir pressure dropsto about 70 percent of the originai resewoir pressure.Currentiy,we are evacuatingthe possibie causes of theseempirical resuftso

7.2 Case 2The North Ossun “NS2tY’reservoir (1-iarviiieandHawkIns,1969) k an over-pressuredgas reservoirhaving an initlai pressureof 8921 psi at 12,500 feetsubsea depth, or a gradient of 0.725 psMt. Tabie 5

442

providesother pertinent data on this reservoir.Furthermore,gaod geologic data and considerablecomplex fmdffng in the area suggest a closed reservoirwith a limited wateraquifer. in this case, we also assumethat W* Gormd l?W equaizero.

As in Case 1, Bf is calculated from c~ via Equations7and 8 for consolidated and unconsolidatedsandstones.Figure9 shows ct as a function of pressure. (p/2)C iscalculatedfor the two selected cases(a) unconsolidated sandstone with no associated wateraquifer (Fm = 0), and (b) consolidated sandstone witha water aquifer equai five times the pore votumeof thegas reservoir (F@ = 5).

Figure 10 shows the actual and the modified g%!zdata forCase (a) plotted against the cumulative gas production.The early extrapolationof the actuai p/z curve indicatesan apparent gas-fn-pfaceof 210 Bcf. However, theextrapolationof the modifiedp/z curve @/z)C yieids acorrected gas-in-placeof 105 Bcf whkh is close to thevoiumetdcestimate of 114 Bcf. Aiso, as shown onFigure 10, the cakuiated p/z curve, based on the gas-in-piace of 105 Bcf, matchesvety weii with the actuai data.

To study the contribution of formation compactionandwater expansion from a smaii aquifer to the drive energy,a sensitivitystudy of this resewoir was conducted usingdifferent aquifer sizes (F” and rock compressibilities.For each combination of rock type and aqutier size (Fw),the @/zjC data was cakufated and from whkh acorrected gas-in-place can be determined. Table 6summarizesthe results obtained from twelve differentcases analyzed. Comparing the first unconsolidatedcase (Fpa= O)and the iast wnsoii~ted case (~PtJ= 5)1if is seen that both cases give the iowest standarddeviations whkh indicate the correct gas=in-piaceiswithin the range of 104 to 108 Bcf. i30thcases providesimilarcafcuiation results of @/2)C.

7.3 Drhm Energy Partitioning and ReserveEstimationThe results from this sensitivitystudy indicatethat avarying combination of rock compactionand waterexpansion from a small water aquifer couid provide thesameperfownanceeffects to the reservoirsystemas iorigas the total energy contributionfrom these two factors isthe same. This observation is consistentwith thespeculation raised in the MBE Analysis section of thispaper. Therefore, it is important to utilize knowiedgeofthe geofogkai setting as weii as knowledge of reservoirrock properties to evaluate and confickmtiypredict gas-in=piace from pressure performanceof over-pressuredgasreservoirs. Correct partitioningof drive energies,therefore, is dependent in many cases on accuratemeasurementsor estimatesof formation compressibility.Underestimationof formation compressibilitymaysuggest a waterdrive where one does not exist and viceversa.

Profitabledevelopment of overpressuredancUorunconsolidated resewolrs is dependent on an accurateunderstandingof drive mechanismsand totai reserves.

This is especiaifytrue since many if not most of thesetypes of reservoirsare iocatedoffshore. Accurateformation compressibilitydata and appikation of thatdata in MBE anaiysis and reservoir simuiatkmcansignificantly improve reservoir development in thesetypes of fieids.

8. CONCLUSIONS● Incorporationof variable formation compressibility intoreservoir performanceanatysis is important foroverpressured and/or weakly to unconsolidatedreservoirs.

● Accurate laboratorymeasurementsof porecompressibilityare important and standard methods formeasurementof pore compressibilityon friable tounconsolidatedcores are often inadequate. Tests onfuii diameter, fresh core samplesfrom unconsolidatedformations are preferable to plug samplesand slow ratetests are necessary to account for the aneiastic natureof these formations.

. Use of the modified FormationVoiume Factorasdefined in this pape; ailows variable formationcompressibilityto be incorporated into the MBE andother reservoir performance analyses easily andeffectively.

gUse of variable formation compressibility in materiaibaiance analysis for initiai reserves leads to moreaccurate estimatesof resetves. Useof accuratelaboratory pore compressibilitydata can aiiow accuratereserve estimatesfrom earfy time data in overpressuredsystems.

. incorporationof accurate formation compressibilitymeasurements in reservoir performanceanaiysis canaiiow for the correct partitioning of drive energies andestimatesof remaining reserveswhich can aid in themost efficient developmentof the resewoir.

$. ACKNOWLEDGMENTSWe wouid like to thank the managementsof MobiiResearchand Development Corporation and MobiiExplorationand Producing,U.S. inc. for permissiontopubiish this paper. We would a!so like to thank MartyCohen, Ron Moore,J. Michael Rodriguez,and all theothers who helped on this project.

10. NOMENCLATUREA = constantinTable2B = constantinTable2e = porevolumeformation

voiurnefactor(FVF),RWSTB& = iniiiaiporevolumeFVF,FiB/STS

!$!

= gasFVF,RWSTB= iniiiaigas FVF,RWST’B= oii FVF,FiWSTB

f% = initiaioii FVF,REVSTB& = two-phaseFVF,RB/STB

10 APPLICATIONOFV~ FORMATQNCOMPRESSWIW

= irtttiaitwo-phase FVF,RB/STB= waterFVF,RB/STB= hitiafwaterFVF,RB/STB

= ~/&

= &/&

=@/@

= &/*= &/@= &/@= &l@: t%:gt ~ ?dle *

= constanth Equation28 and Figures8 and 10= iltlk Comprwhiiityof thefofrrtation,Vowovpsi=formatbn Cornprssstiliii,VoiNOvpsi= gasV-MIM*VWM= fyairl~e66ibiiity Oftiw fofmation,Vowovpsi= w --wt vo~o~m= totatwatercornpmssibitity,vot’vo~psi=conskmt= porevalue ratb, gascap/oilzone= porevalueratio,potaquif~oflzone=Mafhitiaigasklpklce,scf= inftialfmefjas inplace,ecf=totat gasprodhced,Scf=Soiutbngashpiace, scf=Mtiatsotutiongashpface,scf= htegratedfotrnatiortcornpressblihy= oonstmth Equath 3= constantin Equation3=constrmtfnEquatbn3= oii in place,STB= N/@= Poisson’s ratio= total oil produced, STB= porosity at standard condfiions, fraction= reservoirpressue, psi= initial resewoir pressure, psi= gas in soluftion in oil, scf/RB

f?~ = ~itia! gas in solution in oil, scf/Rf3Swgi = initiai water saturation, gas cap, fractions~i = initiai water saturation,oii zone, fraction@ = initai effective Iaboratofy stress,psiau = effective laboratorystress, psi

UJr,y = horizontal stresses,psia- = overburdenstress, psiv1?

=pore volume at resetvoir conddion, RB

dSc= pore voiume at standard condition, STB

= water in place, STBwe = cumulative water infiux, STBWp = cumulative water produced, STi3z = gas deviation factor

11. REFERENCESAnrhreen,M. A; ‘PredictingReaeivolrConditionPore-Volume

CompressibilityfromHydrostatio4XmseLaboratoryDatt%’paperSF%14213 presentedat the 1985 SPE 60th AnnualMeeting,La6Vega%Sept. 22-26.

~as, t), M.: ‘AnSiyd6ofAbilOmld!yPme8uredGas Raserwrh WithPartialWaterInfluxt paperSPE S650presentedat the 1972 3rdSymposiumonAbnormatSubsuflacePore Pressure,LouiaianeStateUniversity,Mayl&16.

Bernard,W. J.: ‘ResenresEstimationandPedormanca Pmdi@n forGeopreawed Gas ReeervoiratJ. Pet. Sci. Errg.(Aug.19S7)1,15-21.

f3radley,H.B.(Editor-fn-OhlafiPetmbum E~ineerirrg Hmrdbook,SPE, Richardson,Texas(1987).

Chlerioi,G.L., Ciuod,G.M., Eva, F., and Long,Gt.(1967) “Effectofoverburdenpreaswuon aornafmtmphy8kelparemeteraofreservoirrocks,’Proo. 7#r WorfdP6froteum Congress,z 309.

Dake,L. P.: Func&nental# of Reservoir Ehgineedng,EleevterScientificPubtishlngCo., Amsterdam(1978).

de Waal, J. A,: Or Rate Tjpe CompactionBehatior of SandstoneReservoir Rock, Ph.D. thesis,TeohnlacheHogeachoot12alft,(19s6).

Dobrynln,V.M. (1663) ●Effectof ovetirden pressureon somepropertleaof sandstones”,SPfZ/, 2,360,

Dug9an,J. 0.: ‘The Andereon‘L* -An AbrronnallyPmssuradGasReswvoirin SouthTexas,’ JPT(Februery1%’2) 132-1SS.

Fam1.(195S8)‘Carrpmseltilltyofarmdstonesat lowto rncderatepressures’,EMi.AAPG, 42,1924.

FattjL (1958b)“Porevotumecompressibilitiesof sandstonereservoirreeks’, Trans.,AIME, 213,362.

Gewtame, J,: “TheEffeotof FluidPmaure Declineon VolunwtrioChenge8of PorousRooks,’Trans., AlME (1957) 210,331-340.

Harrnarlind, D. J.: “I%adiotingGaa ReservesinAbnrmnatlyPnssauradReaervdrsspaperSPE 3479 preeentedat the 1971SPE of AIME 46thAnnualMeeting,New Orleans,Oct.3-S.

Harville,D.W.,andkiawkirra,M. F; “RockCompreseMityandFallumas ReservdrMeohsniarnsinGeopressuredGas Ra.servoire;JPT(December,1969) 1528-1530.

Jaeger,J. C., andCook,N. G. W.: Fundamentalsof Rock Meohanks,Chapmanand Hall,London(1976).

Kaelan,D. K. (19S5) ‘Automatedwe measurement6ystemforenharwcf ooredataatoverburdenconditions’,paperSPE15165.

Koger,K. M., SaomJ. D., and Mogenstem,N. t%: ‘Te8tiWtoDeterminethe GeoteohnioelPropertiesof Oil Sends,’paperPBiCIM 87-38.59 preisentadat tha 1987 PetroleumSodetyofCIM 36thAnnualMeeting,Calgary.

Lschwwe,D. P., andAndarsen,M. A: ‘Cemparfaonof UnlaxlalStrainand HydrestaticrStressPore-Volume(krnpressibilityintheNuggetSandstone,’paper SPE 11971 presentedat the 1$SSSPE 5SthAnnualMeeting,San Fmndsoo,Oct.5-S.

Nur,A. and Byerfee,J.D. (1971) ‘An exactefteotivestresslawforelatitiedeformationof rookwithflulds’,Jour. Geophys. i?es.,76,6414-6419.

. .

Osif,T. L.: “TheEffectsof SaIt Gas,Te$nperatum,and Pressureonthe Comrxessibllitvof Water.’ rmeerSPE 13174 rxeeentedat the19~ &PE 59th A&uel Technics](%nferenceand Exhibition,Houston,Texas, Sept. 16-19.

Ramsgost,B. P., andFarehed,F. F; ‘PiZ AbnormallyPressuredGasReservoiretpaperSPE 10125 presentedat the 1981 SPE ofAIME 66th AnnualTeohnioalConference,SanAntonio,Ootober5.7.

Teeuw,D.: ‘Predictionof ReservdrCompactionfromLakwatory \CornpressibllityData,’ SPEJ, (September,1971) 263-271.

Teeuw,D.: ‘LaboratoryMeasurementsof (%oningenReswvdrRock,’ Trans.,RoyalDutchSo& of (iieologiatsand MiningEng.(1973) 28, 19-32s

Wytie, D.-O. (1958) ‘Effeotof appliedpressureon the conductivity,porosity,and permeabilityof sandstones,’Trans. AiME, 213,430.

Yale,D.P. (1964) NelworkModeilingof Flow, Storage,endDeformationin Porou$ Rooks,Ph,D,theds, StanfordUnlverdty.

Y&E. NAKM+RMSELL pHAM.ANDy~ $A~u 11

.-

TABLE 3

ANDER ONs u n RESFRVOIR DATA

Depth 11167 feet

Initial BHP 9507 psia

Pressure Gradient 0,843 psi/foOt

Bottom-hole Temperature 266 “F

Net Gas Pay Thickness 75 ft

Porosity 24 ~0

Water Saturation 35 %

VolumetricGas In Place 69.6 Bof

TABLE 4

EstimatedOGIP (Bcf) 83,6 76

Correlation Coefficient 0.9982 0.9922

Standard Deviation (Yo) 0,91 6,85

TABLE 6

JJ~~T~ OS~UNM S2Bw IERVQIR DATA

Depth 12500 feet

Initial BHP 8921 psla

PressureGradient 0.725 psVfoot

Bottom-holeTemperature 248 ‘F

Net Gas Pay Thickness 100 ft

Porosity 24 ~0

Water Saturation 34 %

Volumelrlc Gas in Place 114 Bcf

TABLE 6

NORTH O!5SUN *NS2B” RESERVOIR ANAL YSIS RESULTS

[O(3IP @cf) / correlation eoeff./ std.dev.(%)]

Fpa = 0 Epa = 1 Fpa = ~ Fpa =5

Consolidated 15810.986J 1.4 143I 0.991 I 1.4 120I 0.995 I 1.2 104/0 .997/1.1

Weakly Consol. 149I 0.990/ 1.4 12910.994 / 1.2 10210.99611.1 8410.99411.7

Unconsolidated 10510.99611.1 7410.99212.3 4610,982113. 3210.975133.

445

*

M APPLICATIC)N0FVAt31W FORMATIONCOMP~m sP-

FULL DIAMETER VERSUSPLUG SAMPLECCIMPFIESSZBILITY

120

80

40

A UNCONSOLIDATED FORMATIONS

AA A

ACLEANED PLUF&l

INDUSTRY STANDARDA AA

AA

A A

“Pti-’uL-’”” “~& ~~’ A

o0 3000 6000 9000

PRESSURE (PSI)

FIGURE 1Comparisonofcompmsibility from cleaned plugs versus fresh, fulldiameter cores

showing effectofplug damage on pore compressifdlitiy.

10, vchm9 *

9

‘3a1

00.2 0.4 ‘1. oa4@loao 100 200 400 100

?artlelo DisMt.r turn)

FIGURE 2AGrain size distribution for clean.we!lsorted unconsolkfated san”d 2.4, vOllJa**

a.a

1 P La.o

1.0

v 1.6 il; 1.4

: l.2-

FIGURE 26Grain size distribution for c!ay rich,mdy sorted UtICOtN301h%kt0Cfsand

● 0.8

0.60.4o.ao

0.2 0.4 1.@a4610a040 100 aoo 400 1000?utielo Di40ct.r Iumi

446

~~R” Rl=~NDyOusAF 13

WELL CONSOLIDATED SANDSTONES

‘.’-’~

1i- 1 .. -1

3IL ‘J.—__---.’ ‘: ‘2;00.-

!500 2000 5000

EFFECTZVELAS STRESS(P5$1

FIGURE 3Log-1ogplot of FormationCompressibilityversus Effective Laboratory Stress

(121 weli consoikfated sandstone samples)

4

FRIA9LE SANDS a WELL SORTED UNCONSQLIDATM

2. E-5

2. E-6

2.E-4 ~”” 1~

A

A1

bA

4AA

i?.E-7~#

500 2000 5000 20000

EFFECTIVE LAB STRESS(fJsi)

FIGURE 4Log-fog piot of FormationCompressibilityvarsus EffectiveLaboratory Stress

(140 friable sandstone and weil sorted unconsoikfatedsand sampies)

447

●

M APPI ICATK)N OF VAFi,@ F FORMATiON C~lRILITY SPF a

UNCONSOLIDATED SANDS (POORLY’ SORTED)2. E-4t I j

zw~. 4

~&

2. E-5 ;

d 4g

~ ~AAa

fo0 2. E-6 :

zgb

i!9k

2. E-7 i!500 2000 5000 20000

EFFECTIVE LAB STRESS (PS5)

FIGURE 5Log40g plot of Formation Compressibilityversus Effective Laboratory Stress

(14 unconsolidated sand samples)

TYPE CLfFiVES FWt CLASTIC RESERV(IIRS50, I. # 1 1

zEf-ii0IL

} ..

4

.

..

40 -“.

“.“..

“.....O-..,

30 -

.............““””.........20 - ..........

10 -\\\

--c_OTx&T~- ‘______ .

0 t 1 --T-

o 2500 5000 7500 10000

EFFECTIVE LAB STRESS(P5$)

FIGURE 6Typecwvesbased on non4inearregression ofdatain Figures3, 4,and5

448

NAE30R.13LlSSWI PHAM.AN~ ‘fOUSAF 15

14

9

ANDERSON “L” RESERVOIR

, I 1 1 1 1 1

~~3000 4000 5000 6000 7000 Booo 9000 10000

RESERVOIRPRESSURE[PSII

FIGURE 7Formationcompressibil”~ as a function of reservoir pressurefor Anderson “L”

ANDERSON ‘L” HESERVOXR

7000 ~’~

6000

z: 5000

9

“ 4000*QQ 3000

100C

c

e ACTUALA P/z*c

— cALGULATEO---- P/z*c

6 tJoints—-- P/z*c

all points—- P/z

~\\

\\

\ \1

20 40 60 80 100 120-.

CUMULATIVE PRODUCTION (Ecf)

FIGURE $P/Zasafunctionof cumulative gaspmduction

(standard and “variable compressibill~” analysis)

449I

~6 APPJJQYllON OF1/AW FORM~ON COMPRl=SSIBILllY SPF 26647

Nf3F+THOSSUN “NS2B” RESERVOIR

UNCONSOLIDATED

CONSOLIDATE

A

I 1 t I

:000 5000 7000 9000

RESERVOIR PRESSURE (Psia)

FIGURE 9Formation compressibility as afunctlon ofrese~oir pressureforNorth Ossun

(from Type Cutves)

NORTH OEWJN “NS2B” RESERVOIR

7000 I 1 1 I 1 I I I I 1 i

I

e ACTUAL ‘J6000 A P/z*c

%j — CALCULATED -

& 5000—- P/z?ic—- ?/2 -1

: i

~ 40(3(J - J

n ALQ 3000 - \

\d

$\ \

\ i

2000 - \\ \

4

\

\

i\

1000 - \\ \

, \,, , -,0 L t I 1 I

o 20 40 60 $0 100 120 140 160 180 200 220 240

CUMULATWE PIWJOUCTION(BCf)

FIGURE 10P/Zasafunctionof cumulativegasproduction

(standard and ‘variable compressibility”analysis)

450