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SPE 16959

SPE

Material Balance Calculations for Solution-Gas-Drive ReservoirsWith Gravity Segregationby A.K. Ambastha and K. Aziz, Stanford U.

SPE Members

CopyrIgtN 1987. Semaly 01 Petroleum Engineers

This paper was prepared 101 preaanlatlon at the 62nd Annual Techmcal Conference and ExhMion ot the SOClely of pelfoleum Engineers held mDallas, TX September 27-SO. 1987

This paper was selecled for preeenlalion by an SPE Program Commmee foilowmg review of mtormaoon conlamed m an abstract subm,lted by Iheauthor(s) Conlenls of the paper. as presented. have not been reviewed by the SO@y of pelrOIeum E ngmeefs and are aubfecl to correction by theautnoqsl The mstenel. es presented. does not naceaeenly reftect any POSIIIOI of the Smety ot Petroteum E ogmaers. IIS ofkers. or members Papers z

preeented @ SPE meetmga em subfacs to pubhcetw review by Ec!ttonat CommdSeea et the Scscmty of Peooteum Engmeare Penmesuon to copy ISreetwtad to an ebetrect ot not mcae than W words Iltustmtms may not be COPA T@ ebetmct shoutd contain conspcuoue acknWMgment otsend by whom the paptM IS pmsanted Wrjte Pubkceowa Mawtaf. WE p o SOX ~. R@~rd~. TX ~ Telex. KW6$t SPEDAL

AB!YflMcT e. A numb of studies’-= of solution-gaadrive Ie6ewoirs

2nthiswork, areseMlrai tnu2atoria UsedtttatudyLhe under different dtions have been pubkiahed. Different col@itioM

~ofww~~mviw-- ~=thst arise during* exploitation of @ resemtira are:

effects on reaenfoir performances. Tbs titivity study of tcsenfoi.pcrtbnnance to btockand titnc-stepai=tiwa that fortypieat

1. Mernal g8a drive =chanbL vohJrnc’tiG ~

Solution-gasdrive rWer@ra, large erlurain mwragereaervoirpn?servoim produce by Reservoirfluid expansion. ‘fheptoduutionisatuibuted totiquid e4Mnsion andtorOck e0tnpreSaibU-

aurcmay reauttfrom impmpercOmIol Oftime-atep inthesimukstor. ity, 56 *--flE eunparison of aimtdatittn results with ItE Tfrmer’1 method

reservoir preaaum dmpadown to the bubbkpoint

ahowathat thelatter pmdicta fsatermaervoir pmsaure andoilsatum-P==u= Ask reaWvoirpre$Wre declitl= furtheL oil@=

tion decline, and thus. mderpdets the msmoir producing life andeontracta becauae of themtcaae ofaolution ga%andtircpnJ-duction is due to g~ expansion. As gas saturation rcaehea the

tecovery. critical value, free gas begins to now, tealdting in high gas-oilWe propose a new material balance methd for predkting the ratios and low oil recovetiea.

perfonnanm of thick, homogeneous, depletion-dnve reservoirs. ‘this 2. External gsa drive meehaniarm h many imxancca, reser-method accounts for the vertical pressure and saturation gradknts,and the secondary gas cap, The thickness of the srmndary gas cap

voir pseasure in solution-gasdrive reservoir+ is maintained by

cart be estimated with good accuracy using an ideslizuf saturationgas injection, and oil is displaced by injected gas, This is m-ferred to as external gas drive mechanism.

profile. An iterative prucedure mAues avetage reservoir prrs me towell pmsum. The pnxsure and saturation at the well am then used

3. Gravity aegrtgation (or gravity drsinage). For high relief

to calculate the producing gas-oil ratio. The idetized satutatiorrreservoirs with good along-dip permeab~iy, favorable contk

profile is also used to develop pseudo-functions to simplify he simu-tions exist for gravity segregation of injected gas or gas

lation of solution-gss-dnve reservoir with gravity segreg~tion,released from solution. Gravity segregation is an importantfactor in attaining high oil twtvery from solution-gasdrivereservoirs. 011 re.cmveriesof the enter of 60 to 80% can M

2NTRODUCT20Nobtaiid with effective gravity segregation.*Z Reservoirswith significant gravity segregation also ahow low producing

while it is usually pssible to do a detailed (and expensive!)Uueedlmensional, muhiblock reservoir simulation study to make

gas-oil ratios in structurally lower wells. Crqft and Huwkitt#describe these drive mechanisms in more detail.

predictions for soluiion-gssdnve reservoirs, it is otlen instructive tofirst do some simple material balance ealctdsticmsto make “ball-park”

Tamer’ and Mush? proposed methods to predict the perfor-

projections. This is especially impott.arit because simulation ofmance of depletion (solution-gas)dnve teservoira under internal gas

solution-gasdrive reservoirs involving gas percolation and gravitydrive mechanism, using rock and fluid properties. ‘lM assumptions

ae~gation is a numerically difficult problem that can gobble upof both methods include negligible gravity segregation forces. ‘2hu6.

large amounts of computer time. The evolution of solution gas andthese authors considered only thin, horizontal reservoirs. Both

its rapid movement to the top of the reservoir is both the source ofmethods use the material balance principle (atadc) and a producing

numeriesl dKticulties and the justification for assuming that gasgas-oil ratio equation (dynamic) to Predict reservoir performance at

movement is essentially hawtarw us in performance predction csl-pressutes, where gas saturation excezds the critical value. A mors

cuhtions using material balance.detailed description of both methods appears in Crqft and HMVMIIS.MThe points to note about these prediction techniques are:

‘The performance predktion of a hydnxatbon reservoir underdifferent drive mechanisms is of interest to any practicing restwoir 1.Tme is not a factor in Ike methods because neither watel

influx rmr gravity segregation are considered. Time historyReferencaa and illustrations at eod of psper. must be inferred horn the resenfes and well production rates.a

259

MATEJUALBALANCBCalculations m-$

2 SOL~ONQAS-DRIVE lU3SERVOlRSWIIM ORAVR’Y SEOREOATION S?81- !

Thus.thepredictodperfosmsnceofsnintraid gsa-iivcreaer-Voir is independent of Oifproduction rate.

2. Predicted frsctionaf recovery fiotn intemaf gas drive re9ervoiris independent of sixe. Thus. reservoir performance calctsls-tions can be based on an initisf content of one stock tank bar-~.24.2J

3. Absolute rcsmoir pcrmeabdity is not a factor in USCpredictedperformance of an internal gas drive reservoir.

Both methods assume uniform prcssum? and saturationthroughout the reservoir (tank type model). That is, the whole reser-voir is tr?ated as a singfe block from the resmoir simulation stand-point. We compate the mwsftsof single block simulation with thoseof Tarner’si methed later. To improve the mlitillity of predictions,scveraf investigators have tried to remove the assumption of uniformpnxsure and saturation throughout the reservoir in matenaf bafancecalculations.

Loper and Cafho@ relaxed ths aasusnption of uniform pms-sum and satumtion in the reservoir for the Tarrier’s*method by usingtwo-phase steady state. radiaf flow theoty. E1-Khatib’4presented amodified Tamer method to predkt the performance of depletiondriveoil ~rvoirs by using the average pressure and saturation to estima~well pmsure and saturation, assuming the rescwoir to be in pacudoa-teady state. He then used rock and fluid pmpetties comspmdhg toWelf pteaaure and saturation to C.slcldae * pmducii gas-oil rario.Ef-KhaiiP also pmsesttsd the effects of dndnsge srca and productionrate on the msesvoir performance using his pmdktirst method. Rfilferet al.” employed tluid flow and material balance analysis to a seriesof succusive atedy state conditions in the resewoir. They dividedthe reservoir into a nurnhr of annular rings awmunding a welf. Theeffects of production rate, well spacing, and sock and fluid pmptieson the pcrfom2smccof solution-gasdrive reservoirs have also beenconsidered by aeveraf invcatigstors.*M AUof these studka deal withthin horizontal reservoirs whcsc gravity effects are negligibk. OurmmMkation of the Turner’s’ method, to lx discussed later, is similarto thatof E1-Khurib/4but we account for the effect of gravity aegte-gation.

Scvetaf invcsr gatomllsl-m have propmed methods for pcrfor-marw? prediction under extemaf gas drive mechanism with gravitysegregation. Tetwilliger er aL” mcrstion that recovery by gravitydrainage is rate sensitive, and that rather sham dccrcasc in recovetywould occur at production rates above the “maximum rate of gravitydrainage”. Hence, this maximum rate should not be exceeded. Theydefined the “maximum rate of gravity drainage” as the production ratefrom a ltKs% liquid satumted system under a flow gradient equal tothe gravity gradient or static pressure gradient diffcrcnsiaf bctwccn oifand gas due to dcnsi[y difference. ‘fWs “maximum rate of gravitydminagc”, assuming negligible resistance to flow of gas and negligi-ble capillary effects, is given by:n.3$

7.83x10-6 k krOA Ap sin (a)f?o =

lb

(1)

In Eq. (l), qOis oil production rate in m. bbl/day, k is absolute per-meability in md, A is cross-scctionaf area open to flow ‘n ft2, Ap isthe density difference in lb/f[3 bctw~en Ihe oil and p“= phases atrcsmoir conditions, a is the dip angle in dcgrccs, and BOis the oilviscosity in cp, Afcf%rd’2,and Shreve and Welch’ used Darcy’s lawwith gravity component and materiaf balance concept to predict hperformance of solution-gas-drive rcscmoits incorporating gravitydrainage and gas cap pressure maintenance. In both studies, thereservoir was divided into scvcmf blocks in the vertical direction.Martin” prcsemed a method, based on the relations governing theaverage velocity of flow of each phase and the equations of continui-ty, to describe the memoir performance under prcsssuwmaintcnamx

2

~He=-dcumpkkawgstimofmobik MDybmi’ analyxed free-fail gravity dninage system where tb pso-duction rstewas soblydetemtincd by gravity effects. HeemasdadSnapproximste theory of free-fall gmfky dmbssgepmesxedbyCardwefl andhrsons.mllw-m preaemdfora~-presaute system, where gasiainjezre4fatt& mpofthestructmtnoffset the void that occurs because of oil production,

S0s22Sinvestigators have also focuawd on pert&ttsance predic-tion of intemaf gas drive tcservoirs with gravity segtcgathBzwrchueft”conaidescd the effect of grsvity upas the reservoir per-fonname of a high mliif pool, He outlined a pmcedttre tbr strucax-sUy weighting rock and fluid properties for u in the msterisd bsl-ance equation. He used actuaf swervoir performance data to ptedictthe future pe.rfonnsncc of a gravity five rcsctvoir. He stated thatwhen controlled, produced gas-oil ratios wilf be low and wilf ds-crcase rather than incm.se with pressure declii in reservoirs withgravity wgmgation. His statement was based on past field perfbr-msnce of high relief maervoirs.

Cookts analytically studied the effkcta of depletion rate, reser-voir geometry and withdrawal distribution on natural depletion. Heconsidered distributed and segregated flow aystema. Irr the distributedflow study, thm was no vertical penneabilii and gas could onlynow along h dip d-on. Wlthdrawsb wem uniform SloQgthemetvoirauchthsttbmwsasmtotal thddsstigratkxt atanypczIt&Distrihlted ffow SyateS2sshowed two astWadmftonts snddscperi&-rnsnce was rate sensitive. However, for segregated Ilow with vcsticdPm@@, Wf~ wssmuch ksarstesemdtive. CooPrnthatweUs in Wcstem Vemstsdaand O@f COsthsve@muntb~ducizsg mmtbbehsvior, andevenabw vesticalpam@sB&(CooPsused 10md)is aufEdem fhrgsssegregadm Maahswamd-’~ak=——= Pf@=m=Umdfor de@atbmdlivese.servoir inthestripper stage using stesdysWeanalysk Intheatripper stage, tbreservoir prcmre Isso Iowt!sst tlwgravityisthesobdrivii fome+snd ttsus, thepr&cdon rateisqtdtebw. H*?Ouuineaamethodt opredictt hcperfomssnce of gravity drains&RSSZVOimwith tklhbtg prssaum. He mpmsems ths ~tvti by aserieaof blocka, writrxmatcnal balsrsx eqwions forgsssnd oilineach block, and uses Darcy’s law to dtibe fluid flow in and out ofblocks. He calculates rate of advance of gas-off contact using Shrewand We&h’ method. and describes an iterative procedure to predictthe reservoir pcrfomsance. With the easy availab~ty of modemreservoir simulators, there appears to be no justification for the use ofsuch comp~catcd and yet restrictive techniques.

in tfds study, a nusnencaf simulator is used to study gas perco-lation and gravity segregation, and their effects on reservoir perfor-mances. Figwe 1 shows the grid arrangement for one- and two- di-mensional simulations. For onedimensional verticaf simulation, thememoir is divided into NZnumber of blocks in the verticaf (z) direc-tion (Fig. la). For twodimcnsionaf simulation, the nxervoir is divid-ed into NZ and NX number of blocks in the verricaf (z) and the hor-izontal (x) directions, nxpcctively, as shown in Fig, lb. The sensi-tivity of memoir performance to model, mervoir and operationalparameters is repotted. A simple materiaf balance method forpredicting the performance of thick, homogeneous depletiondsivereservoirs is proposed. A by-product of this work is a set ofpseudo-functions that can be used in reswvoir simulation to studytfueedimensionaf flow problems with two-dimensional sreaf grids,

Since gravity segregation or gas percolation is the dominantmechanism in the type of reservoirs bdng considered, a brief reviewof the literature dealing with the simulation of gas percolation alongwith a sensitivity study is pRsentcd next.

SIMULATIONOF GAS PERCOLATION

Highfy nonfincar finite-differenceequations arise in a simulationof counter-cument flow due to gravity segregation. TM problem kespcciafly pronounced in thick pinnacle secfs or biohenns,s

SPE 16939 ANILK. AMBA!WHAANDKHALIDA22Z 3

Gaaprcdatitascanscvemlylimi tthatime-atepsim thatcssrbeuaedtoairn@atc suchprobkma, ifthettWsmkaRWti asetseatedexplicitly. Tlsiaproblem was recognized by Coars,w and McCreary.*‘fWy suggested approximate methods so that airmdatiorrcould cOntin-ue with large time-step size and without any numerical instabilityproblems. Later, Send and Aziz41showed that U2ebes way to solvegas percolation problem is by implicit treatment of transrnissibiitiea.‘lWr Teat Pmbiem No. 2 is the same as that reported by McCfeaty.~Azu and Setkv~2 discuss the comparison of restths from differentmethods for this problem. In this study. wc usc fUUYimPUcit~-ment of transmissibflitim.

To understand the mechanisms to be incorporated in the materi-al balarxx csfcrdations. wc simulate the solution-gasdrive rmwvoirdeacnbcd by AfcCreay.m Saturation and pmsurc dependent proper-ties given in Tables 1 and 2 of Ref. 40 were used along with the ad-ditional data in Table 1 of this paper. We usc a block centered gridscheme and the closed outer boundaries for both one- and two- dl-mensionaf simulations. it4cCreary’@ problem is referred to as Ex-ample 1 in the following.

Simulation Approach

AS’Ioncdimensiorral simulation model was devefoped based onthe simrdtaneoua solution technique with fully impticit treatment oftzansmissibilkis. In this model, gas pressure. p,, and oil saturation.S., am the primary variables, and the noniinear equations art solvedby Newton’s method. An upstm.amweighting for relative pcsmeabiti-dea and an arithmetic averaging for pressure depmdem terms areused. The derivatives of dtfferent terms with respect to saturation orpressure are calculated numericaf.fyover saturation or pressure changeof 0.0005 or 10 psi, respectively.

Free gas production starts when gas saturation in tk WCUblockexceeds critical gas saturation. Free gas flow rate is calculated by:

(2)

The variables T, and TOin Eq. (2) are the gas and oil transmissibili-ties. rcspec[ively, in the well block at the currwrt time.

The simulation model can be run using a constant time-step sizeor a simplified version of automatic time-step control method used byGrabowski e! al 43 in their fully implicit generaJ purpose thermalmodel (lSCONl). The time-step sequence is cafculatcd by

(3)

The variable to is an empirical paranrctcr, and 5, and q, arc the actualand spcciticd changes in the primary variable i, respectively, at USCcurrent lime step. Two-dimensional simulations were carried out us-ing the ECLIPSE simulator.~ Results of one-dimensional runs onECLIPSE matched the results fmm our one.dimcnsionid simulator.

Sensitivity of Simulation Results

Figures 2 and 3 show the caicufatcd pressure and saturationprofiles at 900 days for runs with constant time-step sin of 10 daysfor Example 1. The results arc shown for different biock sizes. Thepnxsurc profiles am identical for block sizes of 9 and 4.5 ft. Thecorresponding saturaticm profiles are identical, except for slightdifferences for a fcw blocks close to the top of the memoir. Blocksixes larger than 9 ft also produce comparable pressure pmtilcs.However, saturation pmfrles for block sizes larger than 9 ft differ

fromdm$e for bktck*of9m 4.sft. Tkrefos&8bloc& sizCof9ftisc4maidered adequam tostudythc effecta oftime-atepaizeontheaimulation reaulta (Fig& 4x2ds). ‘ttEpmasalm pmtUescakulat-oduaing diffetcntt irste-atepsizcsazaquitediffe~ whereas the aa-turation protile aarecloaei ygrouped. -fbcpmaausc pso61eain Rg.4atmwthat thcuseofa small consmnt dn2e-stsp aiz#detaya thealtaisFmentofbubble poimpreaauminali Mockrhamlwouldrcault inahigher average memoir pmsauze at any time compared to largetime-step size rum. Also, the saturation pcdiiea am mom sensitive toblock size, and pressure profiles to time-step size. These resultsshow that while the model is stable for large time steps, relhbleresults can onfy be obtained by careful choke of time-step and blocksize.

The solid lii~s on Figs. 2 and 3 show idea&cd pressure and sa-turation protiles. ‘tWse idealized profdes arc used to morMy theTarner’sl method in the next section. l12e idctdii satmstion profileis also used in the A-lx to develop a set of pseudo-fimciions tosimplify the simulation of solution-gas-drive reae2voim with gravitysegregation. ‘lWae pseudo-functions can be used to nhrce tkdimensiotilty of the simulation problem for sohstion-gaa-dsivereser-voirs by one.

Figures 6 and 7 show a comparison of computed resetvoir per-formances for several constant time-step sise nma with block aiae of9fL Average zwservoirfmaureand oilsaturadon arocomputed aatiSCV@ltSS2CtSiC~ Figs2re6ahow8that theaimrdation uPtothetime, whenthe bubbtepointp scaaumiacmaaed inaUb&&setarlretmndfoz thcfhture behavior intkscconamst time-s&paizcrtms.Even though average oil saturatha arecloaely grouped fordifkzttruns @g. 7) as eFpecaed fmm Fig. 4, the average ,eoir presama(Fig. 6)azeser&ive totime-step aiae. Tlse ZCS&SWith8@O12@C

time-step control arealao shovmon Fl@.6arsd7. Fortleautomdcrime-step contmlrrm, t%oh~ar xi% arcsetto lday, l,30paiarxf0.1. reapeaively, in Eq. (3). Aa expcctul, conatam time-amp runwithlOdaya time-step sizeappmc&a thezestdta from automatictime-step Contmf run. ‘Ilmugh M aimwll graphically, chan@W ~between o.5and2, ~bctwcCSs20and50 pai, mbetweeno.loand0.20. and &} between 1 and 10 days does not change the simulationresults. These restdts imply that the simulation should be started witha small time-step aim, and increased later, when the trends for aver-age memoir pressure and oif saturation have been established. Wehave used automatic time-step control parameters within the indcatedrange for all subsequent oncdimensionai nsns reponed in this study.

Solid lines on Figs. 6 and 7 are the predictions from tkTarrier’s’ mcrhod using the material balance fores proposed by Tra-~ 2$ F@.f.s (j ~d 7 dso ShOWfht b Si2nUlSdon~.dh wi(h au-tomatic time-step control match the calculations fmm the Tarner’slmethod to a time by which average oil saturation has dipped slightlybelow O-SW-SJ. Mobile 8ss. that has high mo~fiy tOP~ola@ ispresent in the mervoir beyond this time, causing slower avengereservoir pressure and oil saturation declines compared to the prcdietions from the Tarner’sl method. 011 saturation of approximately(I-sin-s=) in the lower portion of the reservoir @lg. 3) irsdicatesthatmobile gas travels to the top portion of the reservoir instantaneously.Thus, a high gas saturation region develops in the top portion of therescffoir, which may be termed “secondary gas cap”. Figure 2shows that the pnxsure is uniform in the secondary sss cap at agiven time. llw development of a secondary gas cap causes a sloweraverage rcacrvoir pmssssre decline. ‘flsc Tamer’s’ methodsignificantly underptilcts ttactvoir producing life and cumulative oilrecovery because the Tarner’sl method does not consider gravityeffects. Afso, the Turner’s*method considers the whole reservoir asone block. FIgurcs 8 and 9 show the results of one block simulationfor Example 1 with automatic time-step cormol. As expected, there.subs from the Tarner’sl method compare favorab!y with those fromsingle block simulation.

261

MA7ZRIAL BALANCS CAIXUIATIONS m4 soLuTloNuASDRIVB REs8Rvuslts Wmi GRAVITY SmR%GAmm SB 16%9

. .*

7.--:MGDIFIEDTARNERMRHGD s!ep I

lheaaaumptions oftbepmposed modified Tamer method are Wkightofseumduy gaacap, ?,is~by

1. An omdimenaional kar resesvoir geomemy with gas per-LWhltingUpwatd Ss%loil moving downward, (l-s=- SF) - (r&_ ~

2. Uniform reservoir porosity, Zc =(l-s=- Ssc) - (QG

(4)

3. Thermodynamic equilibrium between the oil and gas phasesat all time%

4. Gss liberation meehanism in the resmoir the same as thatwhere (r&_ is ti average oil Saturadon calculated from the7brner’sl method. ‘Ilw average oil asturadon in the secdary gas

used to determine the fluid properties, e.ap,(YO),C,is:

5. Uniform absolute penneabilii everywhere in the reservoir toavoid pockets of tzapped gas,

(rJk =(l-sw-sr)+sa

6. A linear saturation protik in secondary gas cap ~tg. 3). 2(5)

7. No water enerosehtnent and negligible water pmductiom

8. Negl@iblecapillary effeets, Equation (5) is a direct mmqumee of a linear saturation pdile as-

9. Uniform cKSSS-seetionalmea open to flow everywhere in thesumption in the aeeondsry gas cap.

memoir,

10. Production wells in the bottom part of the reservoir, and Stap II

11. No gas injection for pmsaurc maintenance operations. Giiaamrasion atthewellboreitc

Bssedon rhesimulation result& three stages ofdepletion ofa Sa= l-SW-SF-esohttion-gasdrive memoir with gravity segregation atw

(6)

l. Freasumeverywhminth resemir iaabovetrubbk pointpesaum. ’’fhereisno&ee gaain*teaemoir and average

llteeatimation of thewenpresaW& ~uaingthe average raar-voir pressure, ~- involves an itemdve

Oilsaturation istksatnea stheinitiai Oilsssumtiom~~~-

following

2. PsWaureintopportionofth esesenmirisbelo wbubbkpoint (i) cakukteoilplwauregradk ntp./144pa i/ftatthepmasurafxeaswe, but pressure in botzotn portion is still above bubble “ h-.point pressure. Atthisstage ofde-thereismobikgssintoppostiom htnotint kbottosnp ortionoftk~-

(ii) CaleulstewelI psesmuc, ~andthepresaue attbreaervoirbottom, ;, ~.

voir. Seeondary gas cap h= started forming. The averagereservoir preaswe can be anywhere between al@sUy lowerand higher than the bubble point pressure. F.

[1pw=~Tamcr+~ ‘-2. m

3. Fressure everywhere in the memoir is below bubble pointpressure. Thus, during this stage, the average rtservoir pres-sure is lower than bubble point pressure. There is free gas PO ~everywhere in the reservoir. ‘i& oil saturation at the well j=fim+~ (8)falls below the initial oil saturation and stabilizes at(l-S#F-). The term e is a small quantity. We usec = ld. During this stage, the average oil saturation is For the first iteration, we use ~= hR or ~ fmm the txdeulation atbelow the oil saturation at the weU. ‘ ~ previous time level, if any.

During the first and seeond stages of depletion, the oil satuta- fiii) Stabilized pressure in the secondary gas cap is

tion at the weU is the same as the initial oil saturation, (l-S=). Dur-ing these stages, the conventional form of Tamer’st method is sde-quate for performance predictions. During the third stage of deple-tion when the average oil saturation first qusls or falls below p,=~w-ao-zc-z’”)

(l-S=-S,.), we propose the use of a modified Tamer method. Thesolid lines of Figs. 2 and 3 show the idealized pressure and saturationprofiles at any time in the thki stage of depiction. Figurt 10 shows = ~Twntr -%[’-’4

(9)a schematic diagmrn illustrating the idealized p~sure and saturationprofiles along with several variables to be used in the development ofs modified Tamer methed. The variable u is the distsnee betweenthe well and the reservoir bottom. The variable F represents the dw-

(iv) The csletdated average reservoir pressure based on tiidealized pressure protile is

tance from the reservoir bottom to the @nt where thz reservoir ptes-sum. equals average reservoir pressure calculated from the Turnsr’s)method, h- ‘llw hdgltt of the aeeondary gas cap is 26. The van- !1

~c+;

sbles ~, pw, and p,. represent the ptesaures as the reservoir txmom, at p,, Zc+ z (h - Z=)

the well, and in the seeondary gas cap, respectively. A stepby-stepj&=

h(lo)

development of a .nodified Tamer methed is now presented.

262

SPE 16939 ANILILAMBAsf’HA AND KHALIDAZZZ s

(v) If&_ -&tiaksarhaa aaped&d tolermminpaLtkn 19*atd20tbm!ugh araapccdvely. Tha ori@na12’#lrne##’mateoda convergence has been reached Ctthewise, assume: performc!d pooriyfor these exm@eaalao. F4puea17 through22

showm$eWoir psrformsIlos bsf&egaa@akdu@L %W. .dmlelmd (s-s) simulation weuaesgrid blocks iatba Amction

&M =%GUAmcticQ ‘fkcme4medor@ verdcal

“1) :2s!! Y&d& ‘The Weu is Compkted it! * 27thblock fiomthetop. Figun?a 17through 22ahowthat fbrbothcxam-*

and re~ steps (ii) thlolrgh (v) Untif convergence. we Weused a tolerance of 20 psi for prcssute convergence in moat 1. TIE computed performances from a twodimmsional (x-x)

cases. Simldationdonotchange aathevmllisahifted hmhntauy.2. m computed perfmmances tlom twodimenskmal *Z) and

Onedimenaioaal vertical (z) Silmdadons ate the same. ‘11111%Step III the Simuladon of atdudon-gaadrive reaenfoirs with gravity

Oil sanmstion at the wellbcxe, S- and weltbwe pressure, k are segregation is easendauy an mdimenaiortal psobkm from

used to evaluate rock and fluid properties to estimate the producing the standpoint of “average” reservoir perfmmance.

gas-oil ratio by: 3. MOditkd Tamer method dSfSCOt’ily fSS!diCtSthe lCSSSVOiSperRnmarrxa with8mdnnane norofabottt3% inthepm-

“’=R-+[4WHW ’12) 4 FGz=?z::lute vertical penneabilky and OUproduction rate., if the oil

~is completes the calculations for a time step using the pmpoaedprdcdon raredoeamc excecd”maxitnm~tyurate” given by Eq. (l).

moWkd Tamr methd. Sii Od~ratsa uaedinthe Exampka 2amt3 arebelow "maximum gravity drabageraw givatby Eq. (lJ&aandon

Vdldakn of Modified Tarrier Method 4iaeaauyjuatilkd obaerv* !and2aIc expMnedby Pig. 23.

Like the original Tarner’d method, modi6ed Tarnsr method Figum23showatheofJ aawadon atsfttblth smaervdrtopfbr

doesnot useabsolute venical permeability andoilpmductio nrate. Exmnple 3atcJamwveoit mcOvcryof14. 72%. ou~

TIIS simulated reservoir performance ploftka an!shown fordifb@ wall bc#iwla for HabmMbU,‘wfdmk~ Figurc23alsD ahowadIeoil aaura&mrn5 fkfFosntbe rueNoirsopOfabsolute vertical permeawty andod producdon rate(Fii.11

through 13) for Exampkl, pmvidedtheoU pmdudonratedocanot tbronedimedod venicaI (z)afmuiadlxL Eventhtmgh thsoil aa-

exc@ “maximum rate of gravity drabge” given by Eq. (1). Fig- turatkm pmfueaam difkanL avemgeo$l Utumdon &sq@w&wdy

urealithrOugh 13 C0r0pamt&pluUcted prlbrmanafmmtk thLssme lkwaucaau. Though notahDwn gr@UUy, tMs~

moditkd Tenmr metbl wkh omdbsuional simulation ~ta for tionistrtle forthepmalra protuaaaiao. Ills satwdonmdprcaawa

Example 1. ~ moditied T-r method aatiafactorUypredkta the pm61eaat diffelerElo@orla andtfmea ahowaimuarbelwior. ‘nlat

perfonnsIIc4 of solution-gas-drive reservoir of Example 1. is why, * reservoir performance from x-z and z aimuhtiom?are ~

MtrdMed Tiuner method is mt ap@cablq if gas bmsks throughsame befors gas bakthmugh. Thiadiscussion alsoappuesto Exam-ple 2.

into the wefl blcck and hence, mducca oif saturation in the weU blocksignificantly below (l-S=-SW). Under rhc condWns of gas bmtc-thmugh, the producing gas-oil ratio will increasedramatically as op- Implications of Idealised Saturation Protileposed to declining produced gas-oil ratio in the absence of gas break- F@we 24 shows how oif saturation at 5 ft from the Icservoirthrough on Fig. 13. The declining produced gas-oil ratio on Fig. 13 top varies with cumulative oil recovety (or time) for the thee exam-is in agreement with Burrclmeff’s’” statement that under controlled pies. For Examples 2 and 3, residual oil saturation is 0.15. For Ex-conditions, produced gas-oif rw:o may be low, and decline with pres- ample 1 residual oil saturation is 0,3895. OU saturation at tfw topsure decline in ~servoirs with gravity segregation. portions of the reservoir approaches residual oil aatmtion faster for

Two more examples are considered for funher validation of Example 1 than Examples 2 and 3 because of b steeper oif relativemodifitxl Tamer method. Example 2 uses saturation and pm.ssure permeability curve mar residual oil saturation for Example 1 (Hg.dependent propemies given in Tables 1 and 2 of Ref. 39. Additional 14). 011 saturation at 5 ft from the top does not reach residuaf oil sa-data for Example 2 is providexl in Table 2. Example 3 uses saturs- tumtion at any time for Examples 2 and 3, But moditkd Tamertion dependent properties as given in Table 2 of Ref. 39 and PVT method satisfactorily predkts the reservoir perfonnana assundng aproperdes as given in Table 1 of Ref. 46, Additional data for Ecam- linear saturation profUe with S. at the top and (l-SW+J at Z. in theple 3 am the same as in Table 2, except the initial pmsutt which is secondary gas cap.given by gravity equilibrium with oil pressure of 4550 psia at 5 ft Since modilled Tamer method satisfactorily predicts the mser-from the top of the column. The two sets of relative permeability voir prfonnance with an inaccurate saturation profUe, it may also becuwes used in this study am presented in Fig. 14. T?Msolution gaa- possible to predict the resefvoir performance just as accurstety, aa-oil ratio (GOR) and oil formation volume factor (FVF) of the three suming complete segregation with S- from the resewoir top to 2. andsets of PVT properties are presented in Figs. 15 and 16. Normalized (1-S6-$=-s) fmm ZCto the reservoir bottom. ModMed Tar&rpreasrne in Figs. 15 and 16 is obmined by dividing the pressure bybubble point pressure for each PVT data set. Cwves A, B, ad C

method was used for the three examples with 13Jq = S- ‘lhe mstdts

Rfer 10 the PVT data sets of McCre~,* Coars,s andA1-Khal@h etfrom assuming ~,),, = Se wete identical with UMmaulta from aaaum-

uf.,a respectively. The bubble point presatm for curves A, B, and ing ~,),, given by Eq. (S), except for aligtu ovew%ietations of pro-

C ~e 1275, 1644, and 4514.7 psi, respectively. I%e simulated per- ducing gas+il ratio late in reservoir tife by assuming CJ,C. Smfonnance afong with the performance pmdictcd from modified Tamer ‘lltus, the assumption of incomplete or complete gravity segregationmethod for the Examples 2 and 3 are presented on Figs. 17 through does not affect calculated “average” reservoir performance

263

MATERIAL BALANCECAlzuLA’f10N8 Rx6 SOLUTIO?WAS—DRIWRESERVOlR8 WITH GRAVHY SEGREGATR3N SPB 169s9

Signfflcsmtly.vertical hrekky-Levereu theory can approximate %e- NOMENCLATUREtual” aaturadon profile most closely. But a signitlcant isnpaet on cal-culated reservoir performance is not expeued. However. the time to A= csoss-s4ctional ams opal to flow

gSS ~Sk-U@ is ti~ by th; UIOVCSIHli Of SCCOndSrygas B. = Gil formation volume factor ‘

cap with time, and may depend snore ssmngly on the shape of m- Bt = Gas formation volume f-r

rmmedaaturadon profile. c, = Rock compmsaibility8 . Gravitational asxeleraion, 32.2 ft/see?

In the simulation suns, gas breaks through etiler for a=z simu- gc= Conversion eonstanL 32.2 lbm-fk/lbf-sa?lation for a welf in the end block (127) or (527) compared to a welf h= Reswoir thicknessin the center block (327), as expected. G= breakthrough for one- k= Pemleabfitydimensional vertical (z) simulation occurs after gas breakthmugfr for a kti = Relative permesbtiy of phase Iwefl in the center block for two-&mensiorsafsimulation This is be- kmt = Relative permesbtii to oilcause at any time, gas is more Unifom’y distributed in the top por- at critical gas asturadontions of the reservoir for an one-dimensionaf vertical simrdstion than K* = Pseudo relative ~rmeabiiity to phase 1a two-dimensional simulation. Producing gas-oil ratio increases m. Number of blocks in z directiondramatically, and a high oil production rate can not be sustaiswd after ~. Number of blocks in x dirwticxsgas bmkthrough. Thus, even though overalf reservoir performances p. Pmsrm

are the same from twodlmensionsl and one-dimensional simsdations, ~= Pressure at the resemoir bottomthere art differences in the time at which gas breakthrough occurs. F= Average reservoir pressure

-=McxMied Tamer method, using @JQ given by Q. (5) and p; .

Pseudo Capillarypreastrse

Q = SW predicts gas breakthkxrgb (2<. 275 R = center of welfProduction rate, volJDay

R= C&oil ratioblock) after a cumrdative oil smovery of about 35% and 75%, rcqtec- S= saturationtively, for Example 2. But twodimensiod (*z) simulation shows $= Depth averaged oil saturation~w_~@kamtiveofl-~oftit~% f. = Average Oti SSWration

for the well in k center block (327). This example cakadadon = Timeshowa that moditied Tarmr method using lii aaturadon pro61e ;=pmdicrs tbetimetogas

TmaaMbWty ofphaaef=&&Jkr,Wakthmugh Snotc accutaely than rnodi6ed x= Horiaorad disunm

Tamer method using the Cotnple&segsBgationasstur@on. lmprov- 2= Vertical distanceing nsoditied Tamer method for mom realistic mpmrmtm “0ssof h Zc =saturadon pmtlle in the aemdary gas cap wifl improve Uwpmdktion

~sas@Pw-

of @ time to gas bmskrbrougts, Further impsovemeats to themodified Tamer method as presented in this study carsbe either based Gmk symbofaon empirical data or on one-dimensional disptacesnent theory,

a= M @eAr= Time-step size4= Density diffemce betweezsoil and gas, W@

CONCLUSIONS s . Change in the Mrinsic variable during a time-step

1. Sensitivity study of simulation results to time-step and block size ‘= Arbitrarily small quantity

shows that even though simultaneous solution technique is rmcon- ~= Arbitrssily specified SKIrm

ditionally stable, it is not sufficient to ensure reliable answers with y= Dens@ in terms of pstssure/distance = Pglge

large time-step and block sizes. The simulation should be stwted p. Viscosity

with a small time-step size, which can he increased later, when co= Empirical parameter

average reservoir pressure and saturation trends have beers esta-+. Porosity

blished.e. = porosity at pressure #p. Fluid density, lbnt/ ft3

2. The Twrcr’si method significantly underpnxhcts the nxervoirproducing life and cumulative oil recovery. The Tarner’slmethod has been modified empirically to predict the performance Svbacriptsof a solution-gasdrive tesmoir with gravity segregation, ‘flrismodified method satisfactorily matches the simulated perfor- g= Gssmsnces for three examples, and is applicable before gas break- gc = Ciiticsf gasthrough. i= ith specified norm or change in ith inthnsic variable

1= Phase, oil or gas3. Simulated examples show that befote gas hduhrough, the per- n= nth time interval

fonnsnce of a solution-gas-drive reservoir with gravity ae-grega- *= Oiition is independent of absolute vestical Penrseatility and oil pro- of = Residuaf oilductiors rate. if the oil production rate is below the “maximum p. Producinggravity drainage rate”. s= Solution

w= water or well4. New pseudo cspilla~ and relative permeability functions for a Wc = Connate water

solution-gas-drive resemir with gravity segregation have beendeveloped using the idealizd aaturstiors pmtile from this studyand the vertical equilibrium approach proposed by Coats et uf!s

These pseudo-functions should improve perfotmaru predictions ACKNOWLEDGMENTS

from threedlmensionsf simrdations using the two-dimensional Finznciaf support was provided by the Depsnment of Energyamaf grids. Contract No. AC03-glSFl 1564, SUPRI A Industsiaf AffWea,

---264

SF%169S9 ANILK. AMBASTHAAND KHAL3DAZEZ 7

WPfU B Irrdtumial Affiliates, and Stanford Univcraity. We thank5xpioration Constdttts Lmited for the permission to use U@ re$er-Ioir simulator ECIJPSE for this project.

?EPERENCES

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22. HagoortoJ.: ‘Oil Remvery by Gravity Dminage; Sot. Per. Eng.J. (June 1980) 139-150.

23. YpmL J.G.J.: “Analytical and Numerical Modeliig of Immiaei-ble Gravity-Stable @rs Injection Into Stmti6ed Reamoira,” Sot.Pet. Eng. J. (Aug. 1985) 5S4-564.

24. Crafti B.C. and Hawkins, M. F.: Applkd PerrofeunrReservoirEn@eering, mntice Hall IrE., Ertglewood Cliffs, NJ. (1%9)375.

).. Buckley, S.E. and Leverett. M.C.: “Mechanism of ~uid Dis- 25, Wilson, W.W.: “Engineering Study of the Cook Ranch Field,placement in %ndsy Tram., AIME (1942) 146, 107. Shackelford Counry, Texas: Trurrs., AIME (1952) 195.77.

). Welge, H. J.:“Dkplacement of Od from Porous Me@a by Water 26. Katz, D.L.: “Possibilities of Secondary Recovery for the Ok-or Gas,” Trons., AIME (1949) 179, 133. lahoma County Wilcox Sand.” ‘huts., AIME (1942)

L Welge, H.J.: “A Simplified Method for Computing 011Recoveries by Gas or Water Drive,” Trans., MIME(1952) 19S

27. Lewis, LO.: “Gravity Drainage in 011 Fmlds,” Tram., AIME

A, (1942)Y1.

;. Muskat, M.: ‘The Production Hktories of 011 Producing Gas-drive Reservoirs,” Journol of Applied Physics, (1945) 16, 147.

5. Kern,, L.R.: “Displacement Mechanism in Multi-Well $KWIS.”

Tram.. AIME (1952) 195,39.

1. Weve, D.R. and Welch, L.W., Jr.: “Gas Drive and GravityDrainage Analysis for Pressure Maintenance Gpmdons.”Trans., AIME (1956) 207,136.

B. Smith, R.H. reprted by Klot% LA.: “’l%e GravityDminagcMechankam”Journaf c# Per. Tech.. Vol. V, No. 4 (April 1953)19.

9, Woody, L.D., Jr. and Moscrip HI. R.: “performance Cabtla-tions for Combination Drive Reservoir%” Trans.. AfME (1956)207, 129.

10. Burtehaell, E.P.: -Weaemoir Performance of a High ReliefPool,” Trons.. AfME (1949) 171.

11. Terwilliger, P.L., W~y, L.E., HalL H.N., Btidge$. P.M. mMorse, R.A.: “An Experimental ard Tkoriticai hlVeSti@tiOtl ofGravity Drainage Performance Trotrs.,AME (1951) 19% 285.

12. McCord, D.R.: “performance Pmdkxions incorporating GravityDrainage and Gas Cap Pressure Maintenance - LL-370 Are%Bolivar Coastal Field; Truns., AIME (1953) 198,231.

13. Martin, J.C.: “Reservoir Analysis for Pressure MaintenanceOperations Based on Complete segregation of Mobde Fluids,”Trons., AIME (1958) 213,220.

14. E1-Khatib, N.A.F.: “A Modified Methcd for Performance Pre-dictionof Depletion Drive Oil Reservoirs: Paper No. 82-33-04presented at the 33rd Annual Mtg. of the Pctmleurn Society ofCIM held in Calgary (June 6-9, 1982).

15. Cook, R.E.: “Analysis of Gravity Segregation Performance Dur-ing Natural Depletion; Sot. Pet. Eng. J. (Sept. 1962) 261-274.

16. Matthews, C.S. and Lefkovits, H.C.: “Gravity Drainage Pcrfor-msnce of Depletion-type Reservoirs in the Stripper Stage,”Tronr., AIME (1956) 207, 265.

17. Hall, H.N.: “Analysis of Gravity Drainage,” J. Pet. Tech. (Sept.1961) 927-936.

18. Dykstra, H.: “The Prediction of Od Recovery by GravityDrtinage,” J. Pet. Tech. (May 1978) 818-830.

19. Templeton, E.E., Nielsen, R.F., and Stahl, C.D.: “A Study ofGravity Counterblow Segregation.” Sot. Pet. Eng. J. (June1%2) 185-193.

20, Ridings, R,L., Dalton, R,L., Greene, H,W,, Kyte, J.R., and Nau-mann, V.O.: “Experimental and Calculated Behavior ofDissolved-Gas-Drive Systems.” Sot. Per. Eng. J. (March 1%3)4148.

21. Joslii, W.J.: “Applying the Frontal Advance Equation to Veni-cal Segregation Reservoirs,” J. Pet. Tech. (Jan. 1964) 87-94.

28. Tracy, G.W.: “Simplified Form of the Material Balance Eqtra-tion; Trons., AIME (1955) X)4, 243.

29. Loper, R.G. and Calhoun, J.C,, Jr.: “’flte Effect of Well Spacingand Draw&wn on Recovery from fntemal Gas Drive Reser-voirs,” Truns., AIME (1949) 1* 211.

30. E1-Khatib, N.A.F.: Whe Effeet of Drainage Area & Pmduc-tionllarconrhe Rrhman@ of Dcplction Dtiveoil Re.ser-Voirs,” PapersPE llo19pmae@ed at the 57th Amttlal Mtg. ofSFE of AIME. New Gtleam, LA (SepL 26-2!%1982).

31. Miller, C.C., Btownacom~ E.R. and Kie@miclL W.F., Jr.: “ACat@ation oftlKEffect of Producdon Rate uptm UkimateRecovery by solution Gaa Drive,” Trum., AIME (1949) 1=235.

32. West WJ,, Garvim W.W. ad SheldorL J.W.: “solution of theEquations of Unsteady state -fwo-phaae now in oil h-voir%”Tram, AIME (1954) 201,217.

33. ~ JJ. and Roberts, T.G.: “The Effect of the Relative Per-meability Ratio, the Oil Gravity, and the soltrtkm Gaa-Oil Ratioon the f%mary Recovery ticun a Depletion Type Reservoir.”Trum., AME (1955) 204, 120.

34. Heuer, G.J., Jr., Clark, G.C., and Dew, J.N.: “The Influence ofProduction Rate, Permeability Variation and Well Spacing onSolution-Gas-Drive Perfonnanc%” J. Pet. Tech. (May 1%1)469-74.

35. Levine, J.S. and Prats, M.: “l’he Calculated Performance ofSolution-Gas-Drive Reservoim~ Sot. Pet. Eng. J. (Sept. 1%1)142.

36. Morse, R.A. and Whking, R.L.: “A Numerical Model Study ofGravitational Effects and Production Rate on Solution GasDrive Performanceof Oil Reservoirs: J. Pet. Tech. (May 1970)625.

37. Csrdwell, W.T., Jr., and Parsons, R.L.: “Gravity DrainageTheory: Trons., AIME (1949) 179, 199-215.

38. Richardson, J.G. and Blackwell, R.J.: “Use of SimpleMathematical Models for Predicting Resemoir Behavior: J.Pet. Tech. (Sept. 1971) 1145-54.

39. coats, K.H.: “A Treatment of the Gas Rrcolation Problem insimulation of ?luec-DmenaiorraL Three-Phase Flow in Reser-voir.” Sot. Per. Eng. J. (Dec. 1%8) 413-19.

40. McCreary, J.G.: “A Siiple Method for Controlling gas Per@a-tion in Numerical Slmtdadon of Solution Gas Drive Reser-voir” Sot. Pet. Eng. J. (March 1971) 85-91.

41. Settari, A. and Aziz, K.: “Treatment of Nonlinear Terms in U’ENumerical Solution of Partial Differential Equations for Mtdti-phase Flow in Porous Meda? ht. J. Afbdtf@we Ffow, 1, 1975,817-44.

1.-.

MAIS?UAL BALAWB CAUXJLA’fKWS K1R8 SOLUI’ION-GAS-DRIVBRESERVOIRSWmi c2MvrrY sBGREGAnO!! SE 169S9

42. Aziz, K. and settari, A.: Petrofenm Reservoir SbmdUI@ &#Ied science Publii Ltd., London (1979) 169.

43. GrabowskioJ.W., Vie, P.K., L@ RC,, Behie, A. atd Ru-bin, B.: “A F@y Ir@cit @ner@ purpose f%lite ~lffelwlceThennrd Model for In-situ Combustion and Steam,” Paper SPE83% presented at the S4th hnual Mtg. of the SPE of AIME,LiISVegSS(Sept. 23-26, 1979).

44. ECLIPSE Refenmce Manual, Version S4/11, Exploration Con-SUkSIUS Ltd. (NOV. 1984).

45. Coats, K.H., Dempsey, J.R. and Henderson, J.H.: ‘The Use ofVertical Equilibrium in Two-Dmensional simulation of Three-Dimensional Reservoir Perfonnanee: Sot. Pet. Eng. J. (Mach1971) 63-71,

46 A1-Khalifah, A.A., MIZ, K., and Home, R.N.: “A New Ap-proach to Multi-phase Well Test Analysis,” paper SPE 16743pzwented at 62nd Annual Meeting, Dallas, TX (Sept. 27-30,1987).

Ifweaelect areferemeplsm fortheaswlcabdadm aadsetopOftheresenfoir (z.Oplane), the31atz=(k

Pg-PO=FC=ATZC (A-5)

PIwure diffemmceat z = Ois a pseudo capillary pressure deaot-ed by p.. Substitution of Zcfrom Eq. (A-5) in Eq. (A-1) givess rela-tion between $ and PCax

[ 1(Q,e -(1 -S* - S8C) PC

s=h AT

+(1 - SW - ~gc) (A-d)

Pseudo Relative Permeability

APPENDIX - DEVELOPMENT OF PSEUDO-PUNCTIONS IPseudo relative perrneabilities to oil and gas are given by

IrI this Appendix, we develop pseudo-funetions for solution-gas-drive reservoirs with gravity segregation. The development is krom.wZ. + kmcg (h - z.)based on the verdctd quilibrium concept proposed by Coau et uf.a

~.=h

(A-7)

and idealizd satwation profile ahown in Fig. 3.

Pseudo capillary Pressure

I (A-8)

Depth averaged oil saturation at some position and time is given..

byWhere k.and+ale thetelative penneabilitics tooil and gas,

~= (s0)2. % + (1 - & - ~r) (h - 2=)

~CtiVdy, M Oil WtU@OII Of(~~k given by ~. (5). =- G

(A-1)between Eqa. (A-7) and (A-1) reatdta itx

h

(k,om - k,ag) [~-(1 -SW - S-)]where (~o)z,is given by Eq. (S). Equation (A-1) is beed on the SS- g,= + ~~ (A-9)sum@on of idealized saturation pmfi.leof Fig. 3. Equation (A-1) R- (Qc - (1 -SW - Sgc)lales the saturation F to the secondary gas cap-oil mntsct position, ZC.

At the pliMez = z<:A similar elimination between Eqs, (A-8) srd (A-1) results in:

Pm= Pgc (A-2)

because of insignificant rock capiUarypressure. At any position other k68 =

,g,aw [~-(1 - $w - Sgc)lthan z = z., oil and gas pressures are given by: (&)zc -(1 - Sw - Sgc)

(A-1O)

Po (z)= Pm + y. (z - 2.) (A-3)The expressions developed for pseudo capillary pressure and se-

lative permeability are useful only before g~ b~~~ugh, whichP8 (z) = D& + 78 (Z – Zc) (AA) means that Z, c h or quivdently, ~> (s’),..

1

---

.. ..-$

TABLE L ADDITIONALDATA FOR EXAMPLE 1

RsxavohaiacOifpmduaimruc

GaadmaiIyalaldaldcoXKmOMbxilixloila&l#knl

coXmalCwJtaaawatimInitial psaxoc

Tonf aimulxkm*C

~.. dmmwnal column2327 X 2327 X 135 ft tafl~ ~~ay k the ]OWSI bfockmmd6olwlF0.0s ibm’0.990.01Givenby gravityquitibnum wiUxoil pmssumof17S0paiaatthc topof the column13s0 Daya

FwosiIyfouOWSlbc - O=++CJ(P+)]

Whuc d’= 0.04pa= 17s0 @a

c,= 3.0 x K@ psi-’

TABLE2. ADDITIONALDATA FOR EXAMPLE2

RcsuvOuaiaeOifpmduaimmc

VCnicafFtmdmlyOilIlauityz!atmdaxdomiiti-

Gzsfkoaiv alamdmlcmlwmslxitialoil~

c~ - aaaaahlIxilialprualw

ToialaimlmkmthtRYxoaily

Rock~v

2500x 2500x 3mfl1000 SrBmay

2??0.0s OYfP1.00O.mGivenby gmvitycquihbriumwithoil prcasurcof1700paiax5 ftliomtkt0Pof*mI-5400 bays0.103.0 x IF pat’

1

2

●

●

NZ

1 2 ● ● Nx

NX+l NX+2 . ● 2 NX

● ● ● ● ●

● ● ● ●

i●

● ● ● ● NX NZ

a b

Fig. la umqunau for one- ●u! twdbnendonai aimulxtlons.

1050 7

‘AZ 11I 1 I

1030 - ● 4.5

~ ,*~~ - : :5

-/ -

+ 225

+

— Idcdized Profile

.~o 30 60 90 lm 150

Di2tanwthmt hetop,fi

Fig.2-anDat othJoefcsia90xl P18-uNP’-0(Ex-x@o 1. At=? OdWS, UmO=900dWS).

.. \

,.“

!a9

[

a7

80’

+ 22.5

— ldeahacd Refile

0.31 1 I I I

o 30 a w 120 1s0Distancefromthetop,fl

FIE3-EWU101 Mock ●,z* on wt.raoon PIOflm IE..mw 1. v. w day.. th.. 9w tiy.)

1100

Wlo

I I I IAt.Days

● 10● 30 7A 45+90

,d

.* ***● **9

;. ..9”

:::: 1::: ::::::2

●● .+*

● +*● .*

. ...+

30 en 90 Im 150

Diatanccfmn thetcp. n

F* 4.m9e10fmu.,18@0no ruWwOwlnmta,m Qlol. 81.9 R. Hmo.9006wy8,

I .0 L ● * d .ma6&.9bb

09 -

At. Days

07● 10

h o 30A 45+90

05 -*

●

I Io,~

o 3“ 60 w 120 1$0

Disumccfrom thetop, ft

n~ s. tw,ct et t#u,..*t.* 0“ ,.1”,,110” p.ttle (En*rnpl. I ,r .9 Il. tom,. 900 6,”s,

! “ U201 I l\ I Io 00s 0.1 01$ 02

Cumulative 0:1 Raovery,Fraction

~Ia 8-hwmq9r.wrvw W.*UW* IrOm *.D smwl.tmn I \t. ? Ill Md TWO mothcd 10+E**rn*b !

208

Lo

— &Metftod ‘AutoowicAt Cuatrol

0.93

0.9

. 10

O’s$-: z+90

<0.8 I I I

o O.OJ 0.1 0.1s 0.2

Cumulative Oil Recovery. Fractkon

Fig 74”* *1 sm”lmim tram s.” Unlilbuon (w. * It) MIJ TM!U’* nmwmd 10I E“- ,.

~ ,W — Tamer Metkl

; ● Single Block Simulation

c!

w,

km

6

$m—o 003 0.1 0.1s 02

Cumulative 0)1 Raovay, Fraction

— Tamer Method

u. 095 -

gSingle Block Simulation

s~ 09 ,-

.-6~ oaf

E

4OH I

o 005 01 01$ 02

Cumulative Oil Recovery. Fmc[ion

FIQ 9-Av.rw. .311wuralm. Iron!●qlc.aiorb ofwlallon and lame, s In.thou lot i!.mnpl. 1

-5-!!+P:.—

\

w

l-sgc -&J PA

“E ‘m —Mcdfd Tamer Methcd

!!

Itm

● 500 STB/D&20md,-

i I2(Q&

i!!’

$“m,“..,

o 005 01 015 02

CumulativeOil Recovery. Fraction

P19 I I . h,,,+ ,,”,vM ~,,”u,. I,OM ,.0 gmnJIS,W ,na nwu,,ud law IW,hod 10I E“,m@, 1.

..0 00s 01 015 02

Cumulative Oil Recovery. Fracoon

F* *$-A””- @ —h8ml-o —and — Wnw-ractaalmnl

Cmnul.10\c011P.ccovcr>. I.r.h.la,m

Fq I 3- P,.XI. <8”9g,, , , ,,,,0 ,,mn ,.0 ,,,n”,.,,.n ,M ,nga,,,..,J ,,,”,, .l.,h~ ,., E,*In*,, ,

hk(’rcwy Ilht.4(II—1 —

,. on \ —— CmII\( Rut V))_= \

$( ;J>

\011

~ (Ml

\

‘\y [1..l \

fj ‘\

u~ ().2

11 ,.,0 0.2 0..1 0.(3 ().X

Od Saturahon

Fq Ia-nel,,,,, pg,maabu,,” c“”,,

.. . . .... . .=. ., ,. . ......

WE 1695g

i

6 c;”A . Ref. 40

1.2

8 -

B - Ref. 39.’

,’ C - Ref. 46

~:: - “’”

i ~, “’;/0 ---;----3 “ ,..”>+

. .. I I I

“o 0.5 I 1.5 2

NormalizedPresaw’c

Fig Is.aahltlall ~oll ,“!0 rot trwaoPVT m ml,.

I@ Ic:.”

A - Ref. 401.8

!?

B - Ref. 39.’.’ C - Ref. 46

~ 16 #“

i 14 /“’”

8

,.”B,.f -d---—- __

12 . . . . ~- A. .,.

II I I I

0 0.s I 1.s 2

Nonnaliad Preaaum

‘a“’”I‘ I I I2-D (X-Z) !hd42i041

● Wefl in bfeck (327)

+ Well in block (L27)-

4 Well in block (S.27)

~

t1200 -0 1000 STWDay

A 2(KI0s?8~y

4 — Modified Tarrier Method

km 1 t I I I

o 0.0s 0.I 0.15 01CumulativeOil Recovery,Frwion

F@ 17-A.crqo ,,WIVOI, LU,W, 101E,9w18 2

1.02.D (x.z) Simulation

. W41 in block (3,27)

i -

+ Well in block (1,27)-

+ Well in blak (5,27)

09

8 I-D (z) Simulation

!

0.8s -0 loflommayA 2W0 ~Day

●— Modlfkj Tuner Method

01. .0 0.05 0.1 0.1s 0.2

CumulaIIweOil Recovery.Fraction

● well in blosk (327)

i -

+ Well knblock(1,27).Aw + Wellinblock(547)

!!! 3s - 1-D (a) Sirnuixth ●

i

o Km STwDeyA 2ooo-y— Modified TunerMethod ●+4N

E ~~o O.m 0.1 0.1s 0.2

Cumulative oil Recove$y, Frxction

● well knW (3#7~

~

+ v$llinblock(}~)

1200 - t WxilInblock(S&L

! -lMO

1.D (z) Skmuluion

1 -

9000 3omdA 10nrd

— ModkfWlkIWMdKd7s0

o &w al 0.1$ 0.2

w

c Wetl in W (3,27)Uso - + Wxuinblock(lm)”

● wellh M (s.27)

mOO-

IOD(~S~u2edooMM -

A 10md— ModilbdTanmMedld

o @w al als alCumuldve oil koveey, Prmion

}.0

i

2.D (X.2) Slrnulmkrlls Well in blak (3,27)

0.9 -

I -

+ Well in block (1,27)-

Q Well In block (S,27)

00

3

!

0.7A 1OM(I

s — Modi!led Txmer MeUmd h0.6-

I

o 0.03 0.1 31$ 0.2

CumulativeOil Recovcv, Frection

~ 0.30s ------- .-. —.- -----

m 0.306“m---s 2-2)(xc) Simulxtia

i :

0.304 — wou b200k(3!27)--- w8ublock(l,27)

0.302. . . . . . . . WX21Wak (s27)

8-. _ . \.D(*)skm~~

● ‘A II!D (z) Skmudion [

A Bxxmple1* 4

● Sxxmple24 Eaxmple 3

-*4

j

0,,-0 0.15 0.2

Cumuhfhe Oil Recovety, Fmction