Simulation of Gas Condensate Reservoir Performance

Transcription

1 Simulati f Gas Cdesate Reservir Perfrmace Keith H. Cats, if: SPE, iercmp Resurce Develpmet aa Egieerig, c. Summary This paper presets a geeralized equati f state (EOS) that represets several widely used cubic EOS's. The geeralized frm is btaied by maipulati f Marti's EOS 1 ad is applied i this smdy. A cmpet pseudizati prcedure that preserves desities ad viscsities f the pseudcmpets ad the rigial mixture as fuctis f pressure ad temperature is described. This prcedure is appiied with material balace requiremets i geerati f tw-cmpet, blackil prperties fr gas cdesates. Agreemet betwee resultig black-il ad fully cmpsitial simulatis f gas cdesate reservir depleti is demstrated fr a very rich, ear-critical cdesate. Als, agreemet betwee EOS cmpsitial results ad labratry expasi data is shw. The fully cmpsitial simulati ecessary fr belwdewpit cyclig is perfrmed fr the eat-critical cdesate with a wide rage f cmpet pseudizatis. Results shw the well-kw ecessity f splittig the C 7+ fracti ad idicate a miimal set f abut six ttal cmpets ecessary fr acceptable accuracy. L,t;::ducti Gas cdesate reservirs are simulated frequetly with fully cmpsitial mdels. This paper presets a pseudizati prcedure that reduces the multicmpet cdesate fluid t a pseud tw-cmpet mixture f surface gas ad il. T4is aliws the use f a simpier, less expesive, mdified black-il mdel that accuts fr bth gas disslved i il ad il vapr i the gas. A majr questi i the use f the black-il mdel is whether the tw-cmpet descripti ca represet adequately the cmpsitial phemea active durig the depleti r the cyclig f gas cdesate reservirs. This questi is especially pertiet t ear-critical r very rich gas cdesates. This paper, therefre, icludes a cmparis f black -il ad cmpsitial simulatis fr depleti ad belw-dewpit cyclig f a aturally c-. currig, rich cdesate ly 15 F [8.3 0c] abve its critical temperature. Like a umber Qf ureprted cases fr leaer cdesates, the tw mdels give very similar results fr depleti. additi, the tw mdels give idetical results fr cyclig abve dewpit prvided that certal cditis are satisfied. Hwever, the black-il mdel is t applicable t cyclig belw dewpit, s results f the cmpsitial mdel are cmpared fr differet multi- cmpet descriptis t estimate the miimal umber ad idetity f cmpets ecessary fr acceptable accuracy. The cmpsitial calculatis reprted here use variats f the Redlich-Kwg 2-5 ad Peg-Rbis6 EOS's. This paper discusses a geeral cubic EOS frm based wrk by Marti 1 that ecmpasses all these EOS's. A geeral-cmpet pseudizati prcedure is preseted, filwed by its applicati t gas cdesates, The black-il PVT prperties btaied ad the agreemet betwee labratry test data ad EOS calculated results are give fr the rich cdesate. Black-il ad cmpsitial simulati results are the cmpared fr depleti ad belw-dewpit cyclig f the cdesate. Fially, the cmpsitial-mdel cyclig results are cmpared fr differet degrees f pseudizati (lumpig) f cmpets. A Geerai Frm fr Cubic EOS;s Use f a EOS i cmpsitial simulati f reservir perface ad labratry tests requires tw basic equatis that give the cmpressibility factr z ad the (ugacity f each cmpet fr a hmgeeus mixture (phase). The tw equatis, ad Z=Z(p, T, x)... (a)!;=!;(p, T, x), i=l, 2...,... (1b) give these quatities as fuctis f pressure, temperamre, ad phase cmpsiti x= (x;}. A umber f EOS' s have bee develped ad are i wide use. These are the Redlich ad Kwg Z (RK'), mdificatis by Zudkevitch ad Jffee 3 ad Jffee et al. 4 (ZJRK) ad by Save 5 (SRK), ad the Peg ad Rbis 6 (PR) EOS.!v1ai! shws that all cubic EOS's ca be represeted by a sigle geeral frm. Use f Marti's wrk ad basic thermdyamic relatiships yields geeralized frms fr Eqs. a ad b as fllws: -em] +mz)b(b+ )]z-[ab+m]mzbz(b+l)]=o, "Nw with Scietific Sftware-tercmp. Cpyright 1985 Sciety f Petrleum Egieers '... (2a) "itials withi paretheses dete the varjus EOS's. JOURNAL OF PETROLEUM TECHNOLOGY

2 ad i "'i=l-=-l(z-b) PXi ad PR flbi = flg ad flai =f~[1+( wi -O.26992wr)(- T~iS )]2. The ~ ad g values fllw: _ BBi) _z_+_m.::..zb_ +-(z-), B i z+mlb B where... (2b) A= L: 'L: x jx ka jk... (3a) j~1 k~1 B= L: xjbj,...,... (3b) j=l ad B j =flbjpr/trj,... (3d) Aj =flajp r/th..... (3e) The Ojk are biary iteracti cefficiets, symmetric i j ad k with Ojj =0. Fr the RK, SRK, ad ZJRK equatis, m =0 ad m2 = 1. Fr the PR equati, ml =1+v'2 ad m2=1-v'2. Egs. 2a ad 2b stem frm the maipulati f Marti's results. Their geeral frm is useful i miimizig the vlume f cde ecessary (i cmpsitial mdels r PVT prgrams) t represet differet EOS's. Further details f the derivati f Eqs. 2a ad 2b are give i Appedix A. As discussed by the authrs f the varius equatis, the fla ad fib appearig i Eqs. 2a thrugh 3e are theretically uiversal cstats, g ad (lb. determied whe the EOS is frced t satisfy the va der Waals cditis: (dpfdv)r ad (d 2 p/dv 2 )r=0 at the critical pit. practice, hwever, the fla ad fib values are treated geerally as cmpet-depedet fuctis f temperature, flai(t) ad flbi(t), as fllws: SRK: flbi =, f~ OCTOBER 1985 ad flai=f~[+( wi wr)(1- T~i5 )]2, RK, SRK, ad ZJRK: PR The ZJRK equati determies ai(t) ad b;(t) fr a give cmpet, i, at a give temperature, T, belw critical s that cmpet vapr pressure ad saturated liquid desity at T are matched exactly. 3,4 Eqs. 2a ad 2b are used i cjucti with the Newt Raphs techiques described by Fussell ad Yasik 7 i the PVT ad cmpsitial mdels (described belw) t perfrm saturati pressure ad flash calculatis. Pseudizati The term pseudizati detes the reducti i the umber f cmpets used i BOS calculatis fr reservir fluids. Pseudizati is imprtat i reservir calculatis because f the large umber f real cmpets (e.g., i C7+ fracti) i reservir fluids. Cmpsitial mdel cmputig times ca icrease sigificatly with the umber f cmpets used. We ca thik f pseudizati i terms f either lumpig cmpets r cmbiig streams. Csider a hmgeeus mixture f cmpets f cmpsiti {zj. deted simply as mixture z. Mixture z ca be divided it m (m < ) mixtures r streams xl, x 2... Xm SO that Z=Z"'lxl-i.e., Zi = L: "'exl.... (4) 1=1 where ad L: Xl = 1.0,... (5) j=l L: "'1= (6) f=! The mixtures xl are pseudcmpets, rmalized by Eq. 5, ad "' is the ml fracti f Pseudcmpet Xl i Mixture z. The Mixture z might be flashed at a lw pressure ad temperature with liquid ad gas separatr prducts resultig. Therefre, the tw cmpets, x = separatr gas ad x 2 == separatr liquid, represet tw pseudcmpets satisfyig Eq. 4, btaied by a cmbiati f streams. With sme versimplificati, this is the basis f the black-il treatmet that has bee used 1871

3 ORGNAL wet GAS GAS G2 i----gas G3 -cmpet calculati. A pwer law ca be substituted fr viscsity, ad dispersi ctrl r extesi, similar t the wrk f Kval, 8 ca be itrduced. Equivalece f this tw-cmpet ad full -cmpet simulatis requires that rigial reservir il cmpsiti be uifrm ad that ijected slvet cmpsiti be cstat fr all time. A secd case is cyclig abve dewpit r depleti f a gas cdesate reservir. This case is described i detail i the fllwig sectis. fig. 1-Examples f gas cdesate separati yieldig tw cmpets-il ad gas. fr decades. Each f these tw pseudcmpets icludes sme f each f the cmpets preset i the rigial mixture. lumpig cmpets, each pseudcmpet csists f a subset f the rigial cmpets, ad e f the members f this subset are preset i ay f the ther pseudcmpets. Fr example, a Mixture z f =8cmpbets, CO 2, C> C 2, C3, C 4, C s, C6, ad C7+, might be pseudized t m=5 pseudcmpets, xl=c r, {x'=c0 2, C 2 }, {X 3 =C 3, C 4 }, {x 4 =C S, C6}, ad x S =C7+' This pseudizati prcedure des t demstrate hw the pseudcmpets are defied r btaied, r des it relate i ay way t the tw-phase behavir f the Mixture z r ay f the pseudcmpets. The prcedure assumes that m pseudcmpet defiitis r cmpsitis, xl, are give ad are btaied frm sme rigial -cmpet Mixture z. The prcedure deteies pseudcmpet prperties (e.g., Pc> T e, a, ad b ) ad pseudcmpet biary iteracti cefficiets s that tw cditis are satisfied fr all pressures ad temperatures.. EOS calculatis will yield idetical desity (zfactr) ad viscsity fr each pseudcmpet whether perfrmed i a sigle-cmpet. mde r i a cmpet mde. 2. Fr all mixtures f the m pseudcmpets (icludig the rigial mixture, z), the EOS calculatis will yield idetical mixture desity ad viscsity whether perfrmed i a m-pseudcmpet mde r i a -cffipofiefit mde. Appedix B gives the equatis defiig pseudcmpet critical prperties, {2 a, {2 b, ad pseudbiary iteracti cefficiets that satisfy these cditis. The usefuless f t1.is pseudizati prcedure is shw i tw cases. First, i CO 2 r slvet fldig calculatis where direct ctact miscibility is assumed, the ijected CO 2 mixture r slvet ad the rigial reservir il cmpsiti ca be pseudized t pseudcmpets 1 ad 2. The cmpsitial mdel ca be ru tl-te i a tw-cmpet mde with the EOS (ad viscsity crrelati), givig desity ad viscsity variati vs. pressure ad cmpsiti idetical t that btaied i a full 1872 Pseudizati f a Gas Cudesate Fluid A gas cdesate reservir rigially abve dewpit pressure is discussed ad prcedural steps are utlied briefly. The calculatis are described i detail i Appedix C ad are illustrated i cecti with real fluids i sectis belw. First a stad-ale, EOS PVT prgram is used t match labratry PVT data, usually icludig dewpit pressure, expasi tests at reservir temperature, ad surface separati,data. The EOS is the used t flash the rigial reservir fluid thrugh desired sigler multistage surface separati (Fig. ). The gas ad liquid -cmpet Cmpsitis G3 ad L3 are selected as pseudcmpets ad 2-gas ad liquid r il. This EOS flash gives -cmpet cmpsitis, mlecular weights, ad desities at fial-stage -separatr cditis f the tw pseudcmpets, gas ad il, which are used i the mdified black-il mdel. Thus the black -il mdel prducti expressed as stck-tak barrels f il ad stadard cubic fcet f gas ca be cverted t mls r mass f each f the cmpets i the rigial reservir fluid. The tw pseudcmpet prperties are calculated as described i Appedix B. Appedix C shws that the PVT prgram perfrms a cstat-cmpsiti r cstat-vlume expasi t calculate a tw-cmpet r black-il PVT table f R ' R s ' 1'", c, ad rs as sigle-valued fuctis f pressure at reservir temperature. The table mits B g ad ' g because they are btaied frm the EOS i pseud twcmpet mde i the black-il simulatr. The reas tr missi is thate g ad 'g are t sigle-valued fuctis f pressure i cyclig calculatis; rather, they deped cmpsiti ad pressure. Pseud tw-cmpet prperties required by the EOS ad viscsity-crrelati calculatis i the black-il simulatr are geerated by the PVT prgram ad are read as iput data t the simulatr. This prcedure (Appedix C) differs i several respects frm a calculati f black-il prperties fr vlatile ils r cdesates give recetly by Whits ad Trp. 9 We have t calculated differeces i black-il PVT curves yielded by the tw appraches r determied the effect f ay such differeces black:il mdel results. Fr cyclig abve dewpit, the black-il simulatr reprduces gas desity ad viscsity variati with pressure ad cmpsiti idetical t that btaied i a -cmpet cmpsitial mdel simulati, subject t tw cditis. The reservir rigially must ctai a udersaturated (abve dewpit) gas cdesate f uifrm cmpsiti, ad ijected (r cyclig lea) gas must hp. th". e.,.. f.",.", e... "...,.ti,...,., "''''C'...i"'f":i... "".-l ",.c...,,-=> OJ.....,u...., uj.""....,... t'uj.q~vh 5"", U....Llll... U. a.., p;::.vuuv... VlUpVet. f ijected-gas cmpsiti des t equal that f the surface-separati gas, results f the black-il ad fullcmpsitial mdels will differ. JOURNAL OF PETROLEUM TECHNOLOGY

4 Fr depleti belw dewpit pressure, we have fud clse agreemet betwee gas deliverability ad istataeus ill gas prducig ratis calculated by the twcmpet black-il ad -cmpet cmpsitial simulatis. This has ccurred fr a umber f cdesates ragig frm very lea t ear-critical ad extremely rich. Essetially, this agreemet reflects the fact, ted by Jacby ad Yarbrugh t ad Fussell, that cmpsiti has a egligible effect K values fr depleti f gas cdesates. Jacby ad Yarbrugh used labratry test data, while Fussell used labratry data i fieldscale, sigle-well, cmpsitial simulati f depleti. Fr cyclig belw dewpit, the tw-cmpet simulati gives results (e.g., Cs+ recvery vs. time) that ca be quite iaccurate, especially fr rich cdesates as illustrated belw. The iapplicability ftw-cmpet calculatis t belw-dewpiut cyclig reflects the fidigs f Ck et al. 12 They shwed that accuracy f calculated vaprizati durig gas cyclig f vlatile ils requires that the C H fracti be split it a umber ffractis. additi, Fussell ad Yarbrugh 13 shwed that cyclig results i a sigificat cmpsiti depedece f K values ad that cmpsiti depedece cat be btaied frm vlumetric (expasi) test data ale. Descripti f Mdels This PVT prgram is a geeral-purpse, stad-ale prgram cded t use ay f the RK, SRK, ZJRK, r PR EOS's. Hwever, we will describe ly thse features pertiet t its use i this paper. The prgram icludes a liear regressi calculliti that perfrms a autmatic adjustmet f EOS parameters t match a variety f labratry PVT measuremets. The regressi variables are specified by the user ad may be ay subset f the EOS parameters. These parameters are ~i ad az; fr each f the cmpets ad the ( - 1)/2 biary iteracti cefficiets. We will dete the regressi variable set selected i a give case by {vi}' i = 1, The data t be matcbed i a sigle regressi may idude ay umber f sample cmpsitis at the same r differet specified temperatures. Fr each sample, data etered may iclude (1) saturati pressure, (2) desities f equilibrium gas ad il at saturati pressure, (3) K values at that pressure, (4) cstat cmpsiti, cstat vlume, adlr differetial expasi data, icludig vlume percet liquid, gas ad il desities, ad Kvalues at each expasi pressure, ad (5) multistage separati data, icludig GOR, gas ad il desities, ad K values fr each stage. The set f all zer data etered is deted by {dj},j=l, The regressi is a liear prgrammig calculati that places default r user-specified upper ad lwer limits each regressi variable Vi' Subject t these limits, the regressi determies values f v i that miimize the bjective fucti F. J F= :z:; Wj drdj' +d j,..... (7) j=l where dj ad dj are calculated ad bserved values f bservati j. The term Wj is a weight factr, iteraliy set r specified by the user. Default values are 1.0 fr mst data but are 40 fr saturati pressure ad 20 fr OCTOBER 1985 primary-phase (il fr a il sample, gas fr a cdesate) desity at saturati pressure. The prgram regresses cdesate data, determies tw pseudcmpets frm specified surface separati cditis, ad calculates pseudcmpet EOS parameters ad lbe tw-cmpet black-il PVT table (Appedix C). The results are stred i a data file i a frmat acceptable t the black-il simulatr. The prgram allws a splittig f a sample plus fracti (e.g., C H ) it a umber f exteded fractis. This calculati is a slight mdificati f a prbabilistic mdel preseted by Whits. 14 He demstrates excellet agreemet betwee data ad his mdel's calculati f the exteded aalysis give by Hffma et al. 15 Withi a sigle executi, the prgram ca perfrm, i sequece, splittig f the plus fracti; regressi with the resultig -cmpet represetati; user-specified pseudizati-grupig r lumpig-t 1 cmpets ( 1 <); repeat f regressi with cmpets; pseudizati t z ( < 1) cmpets; repeat f regressi, etc. Calculatis f expasis r ther tests, with prited cmpariss f calculated vs. bserved data, ca be iterspersed i this sequece alg with strage i a data file f EOS parameters i a frmat acceptable t the cmpsitial simulatr. Exteded fracti (C 7 thrugh e,) prperties (Pc, T" M, T B, etc.) are stred iterally, as shw by Whits. 14 Whits's values are mdified smewhat frm values give by Katz ad Firzabadi.16 The black-il mdel used here is a fully implicit, threedimesial, three-phase mdel described previusly 17 that accuts bth fr il i the gas phase ad fr gas disslved i the il phase as fuctis f pressure, thereby allwig gas cdesate r il reservir simulati. the cdesate case, the mdel calculates B g ad J1. g frm a EOS t allw depedece cmpsiti ad pressure i cyclig calculatis. The cmpsitial mdel used i this wrk is. a altered versi f a implicit mdel described previusly. 18 That mdel has bee exteded t use ay f the fur EOS's metied i this wrk ad a MPES frmulati. Cdesate Reservir Depleti Applicatis Rich-Gas Cdesate A. Table 1 lists data fr the aturally ccurrig rich-gas Cdesate A with bttmhle ad recmbied sample dewpits f 3,025 ad 3,115 psia [20857 ad kpa] at the 325 F [163 C] reservir temperature. Sigle-stage separati data at psia [4307 kpa] ad loo F [38 C] give a liquid ctet f586 bbllmmscf [3.29x 10-3 m 3 /std m 3 ]. This fluid is clse t critical because the critical temperature was 310 F [154 C]. Liquid yield fr the three-stage separati f Table 1 is STB/MMscf [46.2 dm 3 /kml] at 14.7 psia [101.4 kpa] ad 60 F [15.6 C]. Cmplete defiiti f a set f Cdesate A cmpsitial results requires a descripti f the EOS used, the regressi data set, the regressi variable set, ifrmati whether the bttmhle r recmbied sample cmpsiti was used, ad the umber ad defiitis f cmpets used. The results (Figs. 2 thrugh 5) were btaied frm the ZJRK EOS fr the bttrhle sample. The regressi data set ifrmati is icluded i Table

5 Dewpit (psia) Ml Fracti CO 2 N, C, C, C, C 4 C, C, C 7 + TABLE l-rich-gas CONDENSATE A DATA AT 325'F Bttmhie Sample 3, Recmbied Sample 3, Three-Stage Separati (Recmbied Sample) c 7 + Prperties Specific gravity = Mlecular weight = 148 P Stage (psia) T GOR Gas Liquid Liquid (OF) (scf/bbl) Gravity Gravity Mlecular Weight Cstat Cmpsiti Expasis at 325 F (Bttm hle Sample) Liquid p Relative Vlume Gas (psia) Vlume ~ z Factr 9, , , , , , , , , , , , ,900 ' , , , , , Recmbied Sample Liquid Shrikage liquid p Relative Vlume Pit (psia) Vlume , ~ 2 8, , , , , ,115" , , , , , , , , , , , , , Gas z Factr Gas gravitles relative t air= 1.0. Liquid gravities relative t water => 1.0. Gas stadard vlumes at 14.7 psia. eo F. *Oewpit JOURNAL OF PETROLEUM TECHNOLOGY

6 The fur regressi variables were ~ ad g f methae ad C 7 +. Seve cmpets were used-c thrugh C 6 ad C 7+ -with N 2 lumped with methae ad CO 2 lumped with Cz. This seve-cmpet regressi is referred t as Regressi 1. With regressi, the ZJRK EOS predicted bubblepit pressures f2,934 ad 2,915 psia [ ad kpa] fr the rigial ie ad lumped seve cmpets, respectively. Fig. 2 cmpares calculated ad bserved expasi data. The pr agreemet f vlume percet liquid values i the 1,600- t 2,600-psia [ t kPaJ pressure rage reflects, i part, missi f data i this rage frm the regressi data set. (Fig. 9, discussed later, idicates the better agreemet btaied whe mre pits f the expasi test are iciuded i the regressi.) Fig. 3 shws the black-il PVT prperties calculated by the material balace methd described i Appedix C. The prperties are virtually idepedet f whether they are calculated frm the cstat-cmpsiti expasi r frm a cstat-vlume expasi. The peculiar curve shapes, icludig the icrease f B g with pressure ear dewpit, are t a csequece f the particular EOS used. The PR ad SRK equatis, subjected t the same regressi, give curves virtually idetical t thse shw i Fig. 3. The B g behavir ear dewpit is a simple csequece f the mass cservati priciple. The same behavir is ted. regardless f whether B g is calculated frm -cmpet r frm pseud tw-cmpet EOS r material-balace (Appedix C) calculatis. Table 3 gives data fr a sigle-well, e-dimesial (10) radial, depleti simulati. Frmati thickess, permeability, ad prsity are 200 ft [60.96 m], 5 md, ad 0.25, respectively. itial reservir pressure is 4,016 psia [ kpal with water ad gas saturatis f 0.2 ad 0.8, respectively. Water remais immbile thrughut the depleti. The well is flwed deliverability agaist a cstat bttmhle fiwig.pressure (BHFP) f 1,250 psia [8618 kpa]. Eight radial gridblcks were used. Results were isesitive t use f mre blcks P PSA TABLE 2-REGRESSON DATA SET DeW-pOit pressure, psia Gas z-factr at dewpit VN s at 1,490 psia VN s at 7,000 psia Vlume fracti f liquid at 2,970 psia Sigl~-stage GOR, (scf/bbl). Sigle-stage gas gravity " '00 0 Data 3,025 0: , RCH GAS CONDENSATE A-, BDTTOMHOLE SAMP!..E, 325 OF CONSTANT COMPOSTON EXPANSON DATA 600 '---.""--~"~-~'''--' VOLUME % LQUD 000 CALCULATED FROM ZJRK EOS 9 COMPONENTS Calculated After Regressi 3, , ","---"T-'--'-,,. " ",..., P, PS1A X O-s Fig. 2-Vlume percet liquid ad relative vlume vs. pressure RCH GAS CONDENSATE A, ~25~F BOTTOMHOLE SAMPLE - CONSTANT COMPOSTON EXPANSON 000 CONSTANT VOLUME EXPANSON GOO, '. ZJRK COS USED , Rs, Be, '5, '0. SCF RB STS 'S "" "f6 Si'B ~ MSCF BOO 1,6 300, "0 '", '" '" ~OOO ~ P,PS1A Fig. 3-Calculated black-il PVT prperties. OCTOBER

7 TABLE 3-RESERVOR AND FLUD DATA FOR RADAL SNGLE-WELL SMULATONS' CW psi- 1 4x10-6 e" psi x10-6 Surface gas gravity, air = 1.0 0_7856 Surface ii desity, lbm/cu ft Pi' psia 4,016 PS psia 3,025 P, Sw, Si Sgl Slw S~ S~g k~ krwc k" kr h,ft k, id '" r W' ft GridbJck ceter radii, ft r e' ft Well prductivity idex, RB-cp/D-psi BHFP, psia 'Rich-gas Cdesate A at 325 F [S 1(1 - Swc)~2 [(SL -Sw, -S~g)/(1 -?w, -S~g)l, where SL =Sw +8 0 = 1 - Si\ ,8.01,16.03,32.1, 64.3, 128.7,257.7, ad ,250 adlr a smalier first-blck radius. The exterir radius crrespds t a 40-acre [16.l9-hal well spacig_ Fig. 4 shws calculated surface gas prducti rate (deliverability) ad istataeus prducig GOR vs. time. Fig. 5 shws calculated fractial recveries f gas ad il vs. time. The black-il (fixed) -cmpet, SUfface gas, ad il cmpsitis were read as iput data ad used t calculate C5+ recveries frm the vlumetric surface il ad gas prducti. Fig. 5 cmpares the calculated CS+ recvery vs. time with results by the cmpsitial simulati. Figs. 4 ad 5 idic~te very clse agreemet betwee black-il. ad cmpsitiill simulati f rich Cdesate A. Hwever, sme differeces t reflected i thse figures are as fllws. The surface gas ad liquid gravities f the cumulative black-il prducti are cstat at (air=lo) ad , respectively. The crrespdig prperties f cumulative gas ad il prducti i the cmpsiiial simulati raged frm ad , iithily, t ad , respectively, after 8 years. While the C 5 + recveries f tie tw simulatis agree well, the distributis withi that cut d L The fractial recveries f C s, C 6, ad C7+ at 8 years were , , ad ad , , ad fr the black-il ad cmpsitial simulatis, respectively. The differece i cmpet distributi is caused primarily by a sigificat ctributi f mbile liquid flw t ttal prducti. Oil-phase relative permeability is zer fr il saturati belw 0.3, Calculated il saturati i the first (ear-wellbre) gridblck reached i maximum f at 195 days ad subsequetly declied t 0.36, 0.34, ad at 700, 1,160, ad 2,920 days, respectively. Calculated pressure i the first gridblck was 1,355 psia [9342 kpal at 2,920 days. Calculated pressure ad liquid saturati prfiles vs. time are t shw but are almst idetical fr the tw simulatis. The iss i deliverabllity caused by. liquid drput ca be characterized rughly by examiati f gas relative permeability. At a ear-well il saturati qf 0.32 (Sw =0.204), the gas relative permeability is abut 0.108, cmpared t a value f 0.3 at zer liquid saturati. Thus, this simple basis, a deliverability factr is 0.108/0.3 (r 0.36), which traslates t a reducti i deliverability f 64 %. These rich-gas Cdesate A results are preseted as the mst severe test (f the black-il apprximati) that we have ecutered t date. a umber f simulati cmpariss fr leaer cdesates, we have ted similar r better agreemet i regard t gas rate ad GOR fr sigle-well depleti. Hwever, we did t calculate the C 5+ r idividual cmpet recveries i thse cases. 6~---r----r----r----r-~-r , 750, RCH GAS CONDENSATE A,325 F 1-0, RADAL $NGLC-WELL DEPLETON -- COMPOSTONAL SMULATON " 6 BLACK OL SMULATON qg. 3 MMSCF/D 450 OL/GAS RATO, 8BLS/MMSCF /), OLGAS RATO 350 as 0~--~----~2----~'~--~4~--~5----~----~--~ 6 7 a l50 TlME,YEARS Fig. 4-Cmparis f black-il ad cmpsitial simulati results JOURNAL OF PETROLEUM TECHNOLOGY

8 TABLE 4-CALCULATED RESULTS FOR CONSTANT"COMPOSTON EXPANSON Rlch.Gas Cdesate A~ 325 F (Recmbied Sample) p (1) (2) (3) (4) (5) (6) (7) (psia) ~ ~ ~ ~ ~ P, ~L 9, , ,115 0, " " 3, , , , , ; , , (1) Vlume fracti liquid data. (2) Calculated frm PA EOS with ie cmpets. (3) Calculated frm black-il PVT lable, usig tw pseudcmpet B g values frm PR EOS. (4) RS wet gas/msc! wet gas, calculated frm PR EOS with ie cmpets, (5) AS ~et gas/msc! wet gas, calculated frm PR EOS with tw pseudcmpets. (6) Desity f gas frm flash separati f cell gas (frm desgated pressure) at psia, 10QF, bm/cu ft, calculated frm PR EOS with ie cmpets. (7) Specific gravity f separatr liquid at psia, 100 F. These are fixed desities f (surface) gas ad il used i the black-il simulatr, RCH GAS CONDENSATE A,325 F '"0, RAO,AL SNGLE"WELL DEPLETON -- COMPOSTONAL SMULATON a BLACK OL SMULATON F= RECOVERY lor GNAL-N- PLACE ~~~~~~~~ ~r-~~~~-'-', C5+ 0 GAS 0.20 F F.15.2, ~~---±2---3~~4~-±5---6~~7~~8 TME,YEARS 0~~---t2--~3~-±4---5f-~6~-±7~~8 TME, YEARS Fig_. 5-Cmparls f recveries frm black-il ad cmpsitial simulati. OCTOBER

9 TABLE 5-WELLSTREAM COMPOSTONS CALCULATED FROM PR EOS AND BLACK-OL PVT TABLE" Cmpet CO, N, C, C, C, C, C s C, C 7 + C s Origial Reservir Fluid , Black-Oil Gas Oil Wellstream at 1,792 psia Black-Oil Nie-Cmpet EOS "Rich-gas Cdesate A at 325 F, recmbied sample RCH GAS CONDENSATE A, 325 f, BOTTOM HOLE SAMPLE ONE-DMENSONAL SMULATONS =-- COM-POSTONAC MODEL /' YEAR (7 COMPONENTS) r BLACK OL MODEL / 12Y7S j, L.L ,5,6 7.9,9 t X/L Fig. 6-Calculated il saturati prfiles after 11 years f cyclig. We ited, therefre, geeralizati f the Cdesate A simulati results related t Cs + r cmpet recveries. The black-ijii PVT repre;etati is based fixed desities f surface gas ad il. Hwever, the -cmpet cmpsitial calculatis fr a expasi yield varyig reservir gas,cmpsitis that, flashig at fixed surface Separati cditis. yield varyig cmpsitis ad desities f surface gas ad liquid. We might expect tw-cmpet black-il.ad -cmpet cmpsitial simulati results fr depleti t differ i sme prprti t this variati i surface cmpsitis ad desities. Tables 4 ad 5 idicate the differeces i reservir liquid drput ad surface desities ad cmpsitis resultig frm black-il vs. cmpsitial depleti calculatis. Of curse, flw effects are icluded i these differeces. The ie-cmpet EOS results listed i Tables 4 ad 5 were btaied frm the PR equati with the rigial ie cmpets (C7+ was t split it fractis). Regressi was perfrmed the recmbied sample expasi data. Table 4 shws that the black-il gas desity ad the specific gravity f the il are 2.73 bm/cu ft [43.73 kg/m3] ad (water = 1.0), respectively. The ie-cmpet cmpsitial surface flashig f cell gas frm the decliig expasi pressures gave gas de sity ad liquid gravity varyig frm the latter values at the 3,115-psia [ kPa] dewpit pressure t bm/cu ft [44.71 kg/m3] ad respectively, at 1,792 psili [ kpa]. additi, TableA shws the effect f the.use f the tw-pseudcmpet Bg calculated liquid drput. The ie-cmpet ad twpseudcmpet EOS values f Bg are cmpared directly als.. Table 4 gives black-il (fixed) surface gas ad il cmpsitis calculated by the PR equati with ie cmpets. These cmpsitis result frm flashig the rigial reservir fluid at psia [4307 kpa] adloo'f [38'C]. Assumig wellstream cmpsiti equal t reservir gas cmpsiti allws calculati f the black-il wellstream cmpsiti at ay pressure frm values f r, (STB/scf), frm surface il desity ad mlecular weight, ad frm surface gas ad il (fixed) cmpsitis. Table 5 cmpares this calculated black-il wellstream cmpsiti at 1,792 psia [ kpa] with the iecmpet PR EOS calculated gas cmpsiti at the same expasi pressure. Cyclig Calculatis This cyclig calculati uses the reservir ad,fluid prp..: erties give i Tables 1 ad 3. The rigial reservir fluid is rich-gas Cdesate A (bttmhle sample) described with seve cmpets, C +N 2, C2+C0 2, C 3 thrugh C 6, ad C7+. The hriztal reservir legth, width, ad thickess are 933.4, ad 200 ft [284.5, 71.1, ad m], respectively. The calculatis are D ad use 20 gridblcks with gas ijecti it Gridblck. Prducti is frm Gridblck 20 deliverability, agaist a bttmhle pressure (BHP) f 1,250 psia [ kpa] with a well prductivity idex f 0.37 RB-cpD-psi [8.53XO- 6 res m 3 Pa s/d kpa]. itial pressure is psia [ kpa] cmpared t the dewpit pressure f 3,025 psia [ kpa]. The reservir is prduced fr 1- year with ijecti. Frm Years thrugh 12, gas is ijected at a cstat rate f 400 Mscf/D [ std m 3 /d]. This gas is the primary separatr (624.7 psia [4307 kpa], 100'F [38'C]) gas f cmpsiti (0.7567, , , , , , ad ), as calculated frm the ZJRK EOS after Regressi (partially displayed i Fig. 2). The black -il simulati was perfrmed with the PVT prperties shw i Fig. 3. The cmpsitial mdel JOURNAL OF PETROLEUM TECHNOLOGY

10 TABLE 6-COMPONENT DEFNTONS Case 13** Case 11 Case 9 Case 7 Case 6 Case 9L Case 5L Case 4L CO, (0.0253) CO, C, +N2 C C, +N2 CO, c, +N z C, +N2 1 +N2 N, (0.0860) N, C, +CO, C, +CO, C 2 -+C0 2 +C 3 +C 4 N, C 2 +C0 2 C 2 +C0 2 +C 3 +C 4 C, (0.6927) C, C, C 3 +C 4 C 5 +Cs C, C 3 +C 4 Cs+Cs C, (0.1348) C, C, C s +Ca F7 C 2. C s +Cs C,. C, (0.0404) C, C, F7 Fa C, C,. C, (0.0208) C, C, Fa F, C 4 C, C, F7 F, C, C, C, Fa C, C7 F7 F, C7+ F7 Fa Fa F, F, FlO F11 Objective Fucti Attaied by Regressi ~Rtch-gas Cdesate A, recmbied sample. '~Cmpsiti f ijected cyclig gas is give i paretheses. simulati was derived with the ZJRK EOS with parameters determied i Regressi 1. The tw mdels gave.calculated il saturatis f abut 0.26, average reservir pressure fabut 2,400 psia [ kpa], ad a pressure differece (Blcks 1 thrugh 20) f 270 psi [1862 kpa] at the ed f year. At the ed f 12 years, the blackil ad cmpsitial mdels shwed average reservir pressures f 1,709 ad 1,636 psia [ ad kpa], respectively, fractial il recveries f ad , respectively, ad fractial C7+ recveries f ad , respectively. Fig. 6 cmpares calculated il saturati prfiles at Years ad 12. These results clearly idicate the iapplicability f the black-il mdel t cyclig belw dewpit, at least (r especially) fr cdesates apprachig the richess f Cdesate A. additi, the cmplete vaprizati regi calculated by the cmpsitial mdel casts dubt abut the accuracy f that simulati. Ck et al. 12 demstrated the erreus vaprizati f cmpsitial cyclig calculatis with lumped C 7 + fractis. Egieers ad researchers have recgized this errr fr may years. The remaiig calculatis described here relate t cmpsitial cyclig calculatis with differet umbers ad grupigs f cmpets fr the recmbied sample f Cdesate A. Fixed sets f regressi data ad regressi variables were used i all cases. Table 6 defmes Cases 13,11,9,7,6, 9L, 5L, ad4l. The case umber reflects the umber f cmpets used, ad "L" detes that C 7 + was retaied as a lumped, sigle cmpet. Case 13 splits C 7 + it five fractis, while Cases 11 thrugh 6 split the C7+ fracti it three cmpets. Pseudizati r lumpig f cmpets (defied i Table 6) was perfrmed i accrdace with the pseudizati prcedure preseted i this paper. Fractis F7, F8, ad F9 are defied idetically fr Cases 11, 9, 7, ad 6 ad differ frm the Case 13 Fractis F7, F8, ad F9. The PR EOS was used i all fllwig calculatis with the biary iteracti cefficiets f Katz ad Firzabadi 16 except fr CO 2 -HC=O. ad where chaged by regressi. The regressi data fr all cases csisted f (1) dewpit pressure, (2) gas desity at dewpit, (3) cstat cmpsiti expasi data (see Ta- OCTOBER 1985 ~ PSA :!.200 :! & '"" "" , " DATA RCH GAS CONOENSATE A, RECOMBNEO SAMPLE, 325-F CONSTANT COMPOSTON EXPANSON 000 CALCULATEO FROM PR EOS 13 COMPONENTS CASE CAL.CUL.ATEO FROM PR EOS 4 COMPONENTS CASE 4L. " " " VOL1JM~ Ok LQUD " ", " 'w's"t " " " 1' : P, PSA Fig. 7-Vlume percet liquid ad relative vlume vs. pressure. ble 1) fr pits 13, 14, 15, 17, ad 20: V/Vs ad vlume fracti liquid, ad (4) GOR ad gas gravity fr siglestage separati at psia [4307 kpa], loo F [38 C]. The regressi variable set fr all cases was 0 ~ ad g f C7+, the methae-c7+ biary iteracti cefficiet, ad Z f methae. cases where C 7 + was split, the first three variables were values fr all split fractis-e.g., fr Case 13 O~ ad g f cmpets 9 thrugh 13 ad biary f C 1 vs. Cmpets 9 thrugh 13. (After these calculatis were perfdled, we fud better results fr this ad ather cdesate with Katz 1879

11 RCH GAS CONDENSATE A, RECOMBNED SAMPLE ONE-DMENSONAL, COMPOSTONAL (PR EOS) SMULATON CASE C7 f. FRACTONS L X/L Fig. 8-Calculated il saturati after 11 years f cyclig. TABLE 7-CALCULATED RESERVOR GAS AND lftjljl.... v...."a... A..,.."... _A r-lllll""'"1 ;;,a:::r-.m.h.um U.H.;:) U:.';:)t:.;:) A'U VSCOSTES FOR CASE 13 AT 325 F Desity (bm/cu ft) Viscsity (cp) p Reservir Separatr Reservir Separatr (psia) Gas Gas Gas Gas 6, , , , ad Firzabadi's 16 biaries fr all C 1 - HC except the regressed C, -Fl [last fracti] biary.) The agreemet betwee labratry data ad the PR EOS calculatis fr Case 13 is shw i Fig. 7. The bjective fucti values (see Eq. 7) attaied fr all cases are listed i Table 6. Calculated results fr all cases, except 6 ad 4L, plt Fig. 7 withi a fracti less tha f symbl size. Case 4L results are shw fr pits where symbls d t itersect (Fig. 7). The D, hriztal cyclig calculatis d t reflect gravitatial frces r the effects f adverse mbility rati cfrmace. These might be imprtat i abvedewpit cyclig. Calculated reservir gas ad primary separatr gas desities ad viscsities fr Case 13 at 325 F [163 C] are shw i Table 7. These values idicate that gravity verride ad adversembility cfrmace effects might be csiderable i abve-dewpit cyclig. The values belw dewpit are ly f academic iterest because f phase equilibrati accmpayig vaprizati durig belw-dewpit cyclig. The ld cyclig simulati was perfrmed with the cmpsitial mdel fr all Cases 13 thrugh 4L. The ijeeted gas cmpsiti used icludes C 5 + (Table 6). Tables 8 ad 9 shw calculated C5+ ad C7+ recveries vs. time fr a cases thrugh 12 years, which crrespds t a ijecti f 1.9 HCPV's f lea gas. All simulatis resulted i early idetical il saturatis ad average reservir pressures f abut 26% ad 2,540 psia [17 513!cPa], respectively, after 1 year f depieti. At the ed f 12 years, all cases resulted i average reservir pressure ad a pressure differece f abut 1,600 psia [ 032 kpa] ad 210 psi [1448!cPa], respectively (Blcks 1 thrugh 20). Fig. 8 cmpares calculated il saturati prfiles at 12 years fr Cases 13, 6, ad 9L. The prfiles fr Cases 11, 9, ad 7 resemble that shw fr Case 6. The prfiles fr Cases 5L ad 4L clsely resemble that shw fr Case 9L. The "duble plateau" character f Case 6 is accmpaied by COTeSpdilg regis f essetially uifrm equilibrium gas ad il cmpsitis. Fr example, calculated TABLE 8-CAl.CULATED C,. AND C 7 CYCLNG RECOVERES' C 5 +- Recvery Time Case (Number f Cmpets) (years) L 5L 4L j"" ~~ "t ~..,,'" u. "... u. e..:> V.l'::;v C 7 + Recvery Case (Number f Cmpets) Q, "Rich-gas Cdesate A at 325 F, recmbied sample. 1D, cmpsitial simulati JOURNAL OF PETROLEUM TECHNOLOGY

12 TABLE 9-CALCULATED K VALUES AT 1,792 psia FROM CYCLNG SMULATON-CASE 13" Psiti, x/l 0.2 D.5 D Time (years) HCPV jected 1.1 D S Cstat- Cmpsiti Cmpet Expasi r ,57 2, , C C, C, D C, C, F, Fa F, F" F" "Aich-gas Cdesate A at 325 P, recmbied sample. Case 6 cmpsitis at 12 years, expressed apprximately withut rmalizati, are fud i Table 10. Table 8 ad Fig. 8 idicate that pseudizati frm 13 t 6 cmpets, retaiig a splittig f the C7+ fracti, results i a mderate t small lss i accuracy f calculated CS+ ad C7+ recveries vs. time. The tabular results (fr Cases 9L, 5L, ad 4L) may seem t idicate that use f the sigle C 7 + fracti ad pseudizati t five r eve fur cmpets yields larger but acceptable errrs i the recveries. Fig. 8, hwever, shws that these cases will yield strgly erreus vaprizati ad crrespdig saturati prfles ad will predict erreusly 100% C 5 + ad C 7 + recveries at lger simulati time. The calculated K values fr Cases 13 thrugh 4L exhibit a surprisig isesitivity t pseudizati r lumpig. Pseudcmpet C 1 +N 2, Cs +C 6, ad C7+ K values fr Cases 13, 9L, ad 4L were calculated as 2:y/2:xi frm the PR EOS cstat cmpsiti expasi calculatis (after regressi). The summati fr each case is ver all cmpets icluded i the pseudcmpet. Fr example, Kc,. fr Case 13 is 2:y/2:xi with summati ver Fractis F7 thrugh F. Fig. 9 shws that these pseudcmpet K values are early idetical ver the etire pressure rage (3,115 t 1,792 psia [ t kpa]) i spite f pseudizati frm 13 cmpets with five split C 7 + fractis t ly fur cmpets. Fig. 9 als illustrates the ear-criticality f Cdesate A. The expasi calculatis shw almst depedece f K values up cmpsiti (that is, calculated equilibrium phase cmpsitis at a give pressure are early the same whether btaied by a cstat cmpsiti r by cstat-vlume expasi). Hwever, the cmpsitial simulati results shw a pruced depedece f K values cmpsiti. Table 9 lists Kvalues at 1,792 psia [ kpal calculated frm the simulati fr Case 13. The K values f prpae ad all heavier cmpets decrease with icreased ctact by ijected gas. Calculated K values fr the heaviest cmpets, Fractis FlO ad F, decrease by factrs f 6 ad 14, respectively. OCTOBER " =t,~.9 CONDENSATE At: RECOMBNED SAMPLE PR EOS USED ', ~1+N2 j \ j A -- CASE 9L 0 CASE 13.8 " " CASE 4L.7.6 C57 K J,] R PSiA Fig. 9-Lumped fracti K values calculated frm cstat cmpsiti expasi. The pseudizati prcedure give i this paper des t relate t r preserve ay tw-phase r saturated mixture prperties. t is f sme iterest, the, t examie the effect f pseudizati ale-with regressi EOS-predicted values f saturati pressure ad assciated prperties fr Cases 13, 11, 9, 7, 6, 5*, ad 4*. Cases 5* ad 4* are the same as Case 6 (C7+ split it 1881

13 TABLE 10-CALCULATED COMPOSTONS FOR CASE 6 AT 12 YEARS O.2<JdL< <JdL<0.85 Cmpet Oil Gas Oil Gas,.. ~ ~ ""'1 T''''2. " v., V./O v., V. C 2 +C0 2 +Ca +C C s +C s F F F Fractis F7, FS, ad F9) except that Fractis FS ad F9 are lumped tgether i Case 5*, ad Fractis F7, FS, a.t'ld F9 are lumped tgether i Case 4*. Table 11 s~ws a surprisigly mild variati f saturati prperties ver this rather pruced rage f pseudizati. Cclusis A sigle frm fr several widely used cubic EOS's is preseted as desity (z-factr) arid cmpet fugacity equatis. This geeral frm stems frm maipulati f results give by Marti. A prpsed pseudizati prcedure preserves siglephase pseudcmpet ad mixture desities ad viscsities as fuctis f pressure ad temperature as the umber f cmpets describig a mixture is reduced. This pseudizati prcedure ad material balace csideratis are used t geerate tw-cmpet, blackil represetatis f gas cdesates. Black-il simulatis f reservir depleti fr a umber f cdesates have shw clse agreemet with results btaied by full cmpsitial mdclig. Qualifyig this agreemet is the fact that we have cmpared the calculated C 5 + depleti recveries ly fr the ear-critical Cdesate A discussed i this paper. Gd agreemet, i tur, betwee cmpsitial (EOS) results ad experimetal labratry expasi data is shw fr Cdesate A, prvided regressi is used. Full c-ompsitial mdelig is ecessary fr accuracy i cyclig f cdesates belw dewpit. Cmpsitial cyclig simulatis were perfrmed fr the ear-critical Cdesate A with the umber f cmpets ragig frm 13 t 4. As ted by ther authrs, accuracy f calculated vaprizati requires a splittig f the C7+ it a umber ffractis. Fr this cdesate, acceptable accuracy f cyclig calculatis required a miimal set f abut six cmpets, icludig methae ad three t fur C 7 + fractis, with itermediates pseudized t tw r three cmpets. Nmeclature b g = B g b = B a Bg = gas frmati vlume factr, RBiscf [res m 3 1std m 3 ] B = il frmati vlume factr, RBSTB [res m 3 lstck-tak m 3 ] c = cmpressibility, vlumelvlume-pressure d j = regressi bservati (data) j f = fugacity, uits f pressure F = bjective fucti value used i regressi h = reservir thickess, ft [m] J = ttal umber f regressi variables J = ttal umber f bserved data values matched by regressis k = abslute permeability k, = relative permeability k mwe = relative permeability t il at cate water ad zer gas saturatis K = K value, y/x L = reservir legth M = mlecular weight = umber f cmpets i a mixture N; = umber f mls f cmpet i i a mixture N m = ttal mls f a mixture p ~ pressure P e = capillary pressure q = flw rate rs = il vapr i gas, STBlscf [stck-tak m 3 1std m 3 ] R = uiversal gas cstat Rs = sluti gas, scf/stb [std m 3 lstck-tak m 3 ] S = saturati, fracti S iw = irreducible water saturati TABLE ll-effect OF PSEUDOZATON ON PREDCTED SATURATON PRESSURE AND ASSOCATED PROPERTES Rich-Gas Cdesate A, Recmbied Sample, 325 F PR EOS Used, N Regressi, All Ps Are Bubblepits p, Case* (psia) KC 1 +N~ KC~+C6 13 3, " ' 3, Case umber= umber f cmpets. *;Same as Case 6 except FB ad F9 are lumped tgether. Same as Case 6 except F7, Fa, ad F9 are lumped tgether Zg Mg "g x ", JOURNAL OF PETROLEUM TECHNOLOGY

14 S rg = residual il saturati fr gas displacemet S,"",,;, residual il saturati fr water displacemet S we = cate water saturati t = Marti'sl vlume traslati fr Eq. A-3 T = temperature, R uless therwise stated v = specific vlume, vlume/ml Vi = regressi variable i V = ttal vlume VV s = labratry cell vlume/rigial cell vlume at saturati pressure Wj = weight factr regressi bservati j x = ml fracti r liear distace y = ml fracti i a gas phase z = gas cmpressibility factr Zi = ml fracti f cmpet i i Mixture z ' = specific gravity (air = 1.0 fr gas, water = 1.0 fr liquids) ij = biary ite~acti cefficiet betwee Cmpets i ad h 5 ij = biary iteracti cefficiet betwee Pseudcmpels i ad j ' = viscsity, cp [Pa' s) 1'* = viscsity parameter i the Lhrez et al. 2 1 crrelati p = desity, mass/vlume l' = fugacity cefficiet w = acetric factr a' 0 b = EOS cstats Subscripts B = rmal bilig pit c = ~ritical g = gas i = cmpet i j = cmpet j L = liquid = il r = reduced (e.g., p,=plpc) w = water Superscripts e = Pseudcmpet e Refereces 1. Marti,.J.: "Cubic Equatis f State-Which?" d. ad Eg. Cher.- Fud. (May 1979) 8l. 2. Redlich, O. ad Kwg, J.N.S.: "O the Thermdyamics f Slutis. V. A Equati f State. Fugacities f Gaseus Slutis," Cher. Review (1949) 44, Zudkevitch, D. ad Jffe, J.: AChE J. (1970) 16, N., Jffe, J., Scheider, G.M., adzudkevitch, D.: AChEJ. (1970), 16, N. 3, 4~6. 5. Save, G.: Cher. Eg. Sci. (1972) 27, Peg, D.-Y. ad Rbis, D.B.: "A New Tw-Cstat Equati f State," d. ad Eg. Cher. Fud. (1976) 15, Fussell, D.D. ad Yasik, Jh L.: "A terative Sequece fot Phase Equilibrium Calculatis crpratig the Redlich-Kwg Equati f State," Sc. Pet. Eg. J. (Jue 1978) Kval, E.J.: "A Methd fr Predictig the Perfrmace f Ustable Miscible Displacemet i Hetergeeus Media," Sc. Pet. Eg. J. (Jue 1963) ; Tms., AME, 228. OCTOBER Whits, C.H. ad Trp, S.B.: "'Evaluatig Cstat-Vlume Depleti Data," J. Pet. Tech. (March 1983) Jacby, R.H. ad Yarbrugh, L.: "PVT Merumremets Petrleum Reservir Fluids ad Their Uses," d. ad Eg. Chem. (Oct. 1967) 59, Fus~ell, D.O.: "Sigle-Well Perfrmace Predictis fr Gas Cdes.ate Reservirs," J. Pet. Tech. (July 1973) ; Tras., AME, Ck, A.B., Walker, C.J., ad Specer, G.B.: "Realistic K Values fc 7 + Hydrcarbs. fr Calculatig Oil VaflQrizati Durig Gas Cyclig at High Pressures," J. Pet. Tech. (July 969) ; Tras., AME, 246., 13. Fussell, D.O. ad YarbrC!ugh, L.: ''The Effect fphas.e Data Liquid Recvery Durig Cyclig f a Gas CdeSate Reservir," Sc. Pet. Eg. J. (April (972) Whits, C.H.: "Characterizjg -:lydrcarb Plus Fractis," Sc. Per. Eg. J. (Aug. 1983) Hffma, A.E., Crump, l.s., ad Hctt, C.R.: "Equilibrium Cstats fr a Gas-Cdesate System," Tras., AME (1953) 198, kat~, D.L.. ad Firzabadi, A.: 'Predictig Phase Behavir f Cdesate-/Crpde-Oil Systems Usig Methae teracti Cefficiets," J. Pet. Tech. (Nv. 1978) ; Tras., AME, Patt, J.T."Cats, K.H."ad Spece, K.: "Carb Dixide Well Stimulati: Part -A Parametric Study," J. Pet. Tech. (Aug. 1982) Cats, K.H.: "'A Equati-f-State Cmpsitial Mdel," Sc. Per. Eg. J. (Oct. 1980) Reid, R.C. ad Sherwd, T.K.: The Prperties f Gases ad Liquids, third editi, McGraw-Hill Bk C. c., New Yrk City (1977). 20. Abbtt, M.M.: "'Cubic Equatis f State: A terpretive Review," "Equatis f State i Egieerig ad Research," Ad~'aces i Chemistry Series 182, Am. Chem._ Sc. (1979) Washigt, D.C. 21. Lhrez~ J., Bray, B.O., ad Clark, C.R.: "Calculatig Viscsity f Reservir Fluids Frm Their Cmpsiti," J. Pet. Tech. (Oct. 1964) ; Tras., AlME, 231: APPENDX A Derivati f a Geeral Frm Fr Cubic EOS's The well kw thermdyamic equatis defiig fugacity 19 are 1'=l -=--J f 1 r'( p-- RT) dv p RT v ~ +z--l z,... (A-) fr a pure cmpet, ad ' i=l i = PX<, RTJ ~ an, V 1 [V( ap _ RT)dV - z,... (A-2) fr a cmpet i a mixture, where Ni is the mls f cmpet i i the mls f ttal mixture, N M' Theterm 1'i is the fugacity cefficiet, v is specific vlume (vlume/ml), V is ttal vlume, ad Xi is the ml fracti f cmpet i, NilN M. The cmpressibility factr z is, by defiiti, pvlrt. Marti 1 gives the geeralized cubic BOS, RT Cl p=v-t (v-t+!3)(v-t+')... (A-3) 1883

15 Cursry dimesial aalysis shws that t, {3 ad 'Y must have uits f specific vlume ad that", must have uits f (specific vlume)2 times pressure. Abbtt 20 ivkes the priciple f crrespdig states i selectig RT elp c as the uit f specific vlume. Thus we ca express the EOS parameters as ad (3 = h~rtclp ad aticipate that the varius dimesiless values are uiversal cstats idepedet f cmpet idetity ad temperature. practice, the 0 values i published EOS's are treated geerally as fuctis f temperature ad cmpet idetity. Difficulties i prceedig thrugh detailed derivatial steps frm Eqs. A- thrugh A-4 t Egs. 2a ad 2b are tati ad legthy mechaical maipulatis. The "vlume traslati" tused by Marti i Eq. A-3 is custmarily deted "b" by ther authrs, while Marti's '" is usually deted "a." Because f the tatial cfusi, legth f the maipulatis, ad lack f ay sigificat velty f the result, we will skip the derivatial detqils ad simply meti sme items that may be helpful t a studet r egieer begiig "Yrk i this area. The itegrati i Eq. A-2 is simplified csiderably if the _rder f differetiati ad itegrati is reversed. FC'r a ctiuus fucti p (x,y) J ap(x,y) dy= ~ J p(x,y)dy,..... (A-5) ax ax s Eq. A-2 ca be writte a jv q,-.=--- pdv+l V ~ - z....(a-6), RT ani - ~ Whe the EOS A-3 is substituted it Eq. A-6, itegrated, ad the differetiated with respect t N" the term the right side ( V at V=) cacels ut. Eq. 2a is btaied by dividig Eq. A-3 by P ad rearragig it as fllws: 1 A 1=- Z-T (z-t+b)(z-t+c) where... (A-7) T=tpRT=D,p,T,.,... (A-8a) ad C='YpRT=D 7 P,T"... (A-8c) A=",pR 2 T 2 =D"p,T~..... (A-8d) Rearragemet f Eq. A-7 t a cubic (i z) frm, defiitifml adm2 by B=(+ml)7, C=(+m2)T, ad relabellig T as B (t B f Eq. A-8b) gives Eq. 2a. Mixig rules are required i the itegrati ad differetiati steps f Eq. A-6. The mst widely used rules are the fllwig fmis: r {3= ~ Xj{3j ]=1 B= ~ xjbj,... (A-9) ]=1 fr t, {3, ad 'Y ad a quadratic frm r 1/ f 0:= ~ ~ XjXfCljf j=1 f=l A= L; L; xjx/ajl,...,... (A-O) i=1 f=1 fr ex, where Cije is usually expressed as (l-je)(ajcxe).s with the biary iteracti cefficiet 0jl beig symmetric ij ad e ad Ojj=O. The maipulatis ivlved i Eg. A-6, theri, use ad {3i... (A-) 2NM ~ Xi"'ij, j=l... (A-12) Because xi =N;lN M. Als, simplificatis are pssible at varius stages feq. A-6 maipulatis if the EOS itself (Eq. A-7) is used. The result f maipulatis Eg. A-6 is Eq. 2b. The rst step i Eq. A-6 is, f curse, iserti fp frm the EOS A-3, writte i terms ftlal mls: p 1884 B={3pRT=D~p,T"... (A-8b)...,... (A-l3) JOURNAL OF PETROLEUM TECHNOLOGY

16 APPENDX B Calculati f Pseudcmpet Prperties The first f th~ tw pseudizati cditis stated i the paper is satisfied by defiig pseudcmpet prperties as ad L; L; x/x/(1-ij)(a i A).5... (B-a) i:=1 j=1 Bf=fJ1pT/Tp/= L; x/bj,... (B-b) j=l where p/= L; x/pcj,... (B-2a) j=l Tc = L; x/tcj,... (B-2b) j=l v/ = L; x/vcj,... (B-2c) j=! ad M'= L, x/m j, )=1... (B-2d) p.*= L; (x/p.;jm;)+ L; ~t/jm;), j~t j=i.. (B-2e) where use f the Lhrez et al. 21 viscsity crrelati is presumed. The fj a ad fj give by Eqs. B-a ad B-b are idepedet f pressure. They are respectively depedet temperature if ad ly if ay f the fj ai ad bi are depedet temperature. Eqs. B-a thrugh B-2e esure that the EOS will give idetical pseudcmpet desity 's. temperature apd pressure, whether calculated i e-cmpet r -crpet mde. Eqs. B-la ad B-lb are btaied directly frm Eqs. 2a thrugh 3e. Eqs. B-2a thrugh'b-2e are btaied by the use f the equatis f the Lhrez et al. 21 viscsity crrelatiihat is, if e-cmpet desity ad -cmpet desity f the pseudcmpet are idetical ad ecmpet prperties are defied by Eqs. B-2a thrugh B-2e, the that viscsity crrelati will give idetical viscsities whether calculated i e- r -cmpet mdes. The secd cditi fpseudizati is satisfied by Eqs. B-a thrugh B-2e, ad by the additial require- OCTOBER 1985 met that fr each pair f pseudcmpets, the pseudbiary iteracti cefficiet is give by 2 2 L; L; "'i"'j(l- i j)(a i Ai)O.5 i=1 )=1 L; L; XiXj(l-ij)(A i A) (B-3) ;=1 )=1 The pair f pseudcmpets are arbitrarily labelled Cmpets ad 2 the left siqe f Eq. B-3. {xil is the -cmpet cmpsiti f a arbitrary mixture f the tw pseudcmpets-le., where "' +"'2 = \.0 ad O<"'j<l fr j=l, 2. The values f a [ ad 0::2 are arbitrary becau~e they cace~ ut whe the left side f Eq. B-3 is expaded, whe Eq. B-4 is substituted it the right side feq. B-3, ad whe Eq. B-a is used.. The ly ukw i Eq. B-3, after use feq. B-la, is 2_Le., the pseudbiary iteracti cefficiet betwee Pseudcmpets ad 2. This cefficiet" 12 is idepedet f tempemture regardless f temperature depedece f ai adlr i, prvided, f curse, that the" i are idepedet f temperature. APPENDX C Calculati f Black-Oil PVT Prperties fr Gas Cdesates This Appedix describes a calculatial prcedure fr gas cdesates that determies a black-il (tw-cmpet, surface gas ad il) PVT table f B", R,<, P.' c, B g, r.p ad p.g vs. pressure, which will reprduce, apprximately, cstat-vlume r csta~-cmpsiti labr4- try expasi data. The EOS PVT prgram is used first with regressi t match dewpit pressure, available expasi data, ad surface separati data. The "calibrated" EOS is used the t perfrm a specified sigle- r multistage surface separati. The fial-stage prducts, as idicated i Fig. 1, are referred t hereafter simply as gas ad il, with fixed gas gravity (M g) ad gas ad il desities at fialstage pressure ad temperature. The EOS flash f ml f rigial reservir wet gas thrugh the specified separati gives the mls f gas ad il btaied tgether with their -cmpet cmpsitis, gas mlecular weight, desity, ad il desity. STB used here detes barrcls f fial-stage il, ad sef 4etes cubic feet f gas at 14.7 psi. [101.4 kpa] ad 60 F [ls.6 C]. The value f r, STBlscf at dewpit pressure is easily calculated frm these results. The EOS calculatis start with 1 ml f rigial wet gas at reservir temperature T ad dewpit pressure ad prceed stepwise t expad the gas thrugh a specified sequece f decreasig pressures. Each expasi' step is cstat-cmpsiti, fllwed by gas withdrawal befre the ext pressure decremet if the verall expasi is cstat-vlume. ' 1885

17 Fr ay expasi step frm pressure P t lwer pressure P 2, mass cservati f gas ad il require~ that ad... (C-a)... (C-lb) A additial cstrait gives the desity f the reservir il r liquid atpz, Tas Pz =bz(psr+cr.a),... (C-2) where C[ = 14.7 M g /(10.73x520x5.6146), M g "" mlecular weight f fial stage gas, ad Pst = fial stage il desity, bm/cu ft. Rearragig Eq. C-2 gives Pst Pa2 (brsh = --bz +-.- =Clb 02 +is... (C-3) C, c2. Substitutig (brsh frm Eq. C-3 it Eq. C-a ad rearragig terms yields ad V Sgbg+SClb= V (bgsg+br.,s)-sis (C-4a) where, fr clarity, subscript 2 has bee drpped S g, S. b g rs, ad b- The EOS is used t perfrm this expasi ad yields values f Sg, S, V, ad rs at pressure Pz. The value f r, is btaied by flashig the equilibrium gas at pz thrugh the specified surface separati cditis. Eqs. C-4a ad C-4b are slved fr the tw ukws b g2 ad b 02, ad R,2 is the calculated frm Eq. C-3. Whits ad Trp9 calculate Rs ad b 0 by flashig the "reservir" liquid frm each expasi pressure at surface separati cditis: The R, ad b calculated here lack that physical meaig; rather, t!iey are simply values that satisfy a mass balace ad yield crrect reservir il desity. Repetiti f this calculati fr each step f the expasi yields the black-il PVT table desired. Reservir il cmpressibility C is easily calculated frm the EOS, ad [he BOS ad viscsity crrelati 21 give reservir il a!=! gas viscsities vs. pressure. At this pit, the black-il PVT table ca be characterized as fllws. Csider tw calculated expasis at reservir temperature, T, each beig the cstatcmpsiti r cstat-vlume pressure sequece used i the abve calculatis. The first calculated expasi uses the EOS i full -cmpet cmpsitial mde, while the secd uses the black-il PVT table. The tw calculatis will yield idetical S Oiquid dfput) ad reservir liquid ad gas qesities ad viscsities vs. pressure. addiii, if the expasi is cstat-vlume, the tw calculatis will yield idetical values f mass f gas remved at each step. The b g ad fl g values i the black-il table culd be retaied as sigle-valued fuctis f pressure as calculated abve. Hwever, cyclig peratis ca result i udersaturated gas with depedece f b g ad /l g up cmpsiti (rs) i additi t pressure. Therefre, e fial apprximati is made. We pseudize the (surface) gas -cmpet cmpsiti t e pseudcmpet ad the (surface) il t a secd with the pseudizati prcedure decribed i this paper. The black-il mdel the igres the PVT table b g ad flg values ad calculates them frm the BOS ad viscsity crrelati 21 i pseud tw-cmpet mde. We have ted a geerally small differece betwee these tw-cmpetb g ad fl g values ad the table r -cmpet EOS values belw dewpit pressure. Abve dewpit pressure, the tw-cmpet ad cmpet b g ad fl g values are idetical fuctis f pressure ad cmpsiti Crs) as a csequece f the pseudizati methd. Eqs. C-2 ad C-3, we use P2 values calculated frm the EOS rather tha experimetal values. Assumig a differece, greater accuracy f the black-il simulati shuld result frm use f experimetal values. Hwever, whe cmpsitial ad black-il mdel results are cmpared, the EOS il desities shuld be used. 81 Metric Cversi Factrs bbl X E-Ol = m 3 cp X 1.0' E-03 = Pa's ft x 3.048* E-Ol m OF (OF-32)/1.8 C bm/cu ft X E+O kg/m3 bm ml x E-Ol = kml psi x E+OO kpa scf x E-02 = std m 3 ~Cversi factr is exact JPT Origial mauscript received i the Sciety f Petrleum Egieer:s ffice March 2, Paper accepteld fr publicati Feb, 3, Revisea mauscript received July 15,1985. Paper (SPE 10512) first preseted at the 1982 SPE Reservir Simulati Sympsium held i New Orleas Ja. 31-Feb JOURNAL OF PETROLEUM TECHNOLOGY