DEVELOPMENT OF TECHNIQUES FOR GENERAL PURPOSE SIMULATORS

Transcription

1 DEVELOPMENT OF TECHNIQUES FOR GENERAL PURPOSE SIMULATORS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF PETROLEUM ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Hu Ca June 00

2 @ Cyrht by Hu Ca 00 All Rhts Reserved

3 I ertfy that I have read ths dssertatn and that, n my nn, t s fully adequate n se and qualty as a dssertatn fr the deree f Dtr f Phlshy. Dr. Khald Azz (Prnle Advsr I ertfy that I have read ths dssertatn and that, n my nn, t s fully adequate n se and qualty as a dssertatn fr the deree f Dtr f Phlshy. Dr. Lus Durlfsky I ertfy that I have read ths dssertatn and that, n my nn, t s fully adequate n se and qualty as a dssertatn fr the deree f Dtr f Phlshy. Dr. Mart Gerrtsen Arved fr the Unversty Cmmttee n Graduate Studes:

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5 Abstrat Wth the exlsn f mutatnal wer and develment f new and advaned smulatn tehnques, reservr smulatn has evlved nt a mature tehnly. Tday s smulatr s mre rbust, mre mlex and mre useful than what was avalable twenty years a. Mdern smulatrs nrrate a lare varety f tns and they an be used t answer mrtant questns abut reservr manaement. Under these rumstanes, a d desn fr the smulatr s essental. S the frst bjetve f ths researh s t devel a new General Purse Researh Smulatr (GPRS, whh an be used later by a lare researh ru t reserve and enhane the wrk f several nvestatrs ver a erd f tme. Cmstnal smulatn s the majr fus f ths researh. It s mre mlex than blak-l smulatn due t the lare number f mnents and varus ssues related t flash alulatns. Beause f ths mlexty, a lare number f mstnal mdels have been rsed. They use dfferent equatns, varables and mlt levels. The send bjetve f ths researh s t evaluate the erfrmane f dfferent mstnal mdels. T d ths n a nsstent manner, a new General Frmulatn Arah s develed t derve any knd f mdel, and ths arah s used n GPRS t mlement varus mstnal mdels. The IMPSAT (mlt ressure and saturatns and exlt mle fratns mdel has been dsussed n the lterature. Here we nvestated ths methd n detal, nludn ts haratersts, stablty, nstrutn and erfrmane. The IMPSAT mdel s atually ne f the mdels that the General Frmulatn Arah an enerate. Besdes ths, a seres f new IMPSAT based AIM (adatve mlt mdels are als rsed, and ther erfrmane s mared wth the erfrmane f the tradtnal AIM mdel whh uses IMPES (mlt ressure and exlt saturatns and mle fratns. Varus mdels and tehnques f GPRS have been tested, nludn the blak-l mdel, mstnal mdel, unstrutured rd handln and mult-nt flux arxmatn, and ther results and erfrmane are mared wth exstn smulatrs. The erfrmane f dfferent mstnal mdels has been evaluated usn a wde rane f rblems. We fund that the natural varables (ressure, saturatns and mnent mle fratns are d fr the fully mlt (FIM mdel, whle the verall v

6 varables (ressure, verall mnent mle fratns are d fr the IMPES mdel. We have als demnstrated that the IMPSAT mdel and the IMPSAT based AIM mdels have very d erfrmane fr a wde rane f rblems, and they are mre effent than the tradtnal mdels. The IMPSAT mdel s enerally ver 50% faster than the IMPES mdel due t ts mrved stablty. Fr rblems that are nt very hard, the IMPSAT mdel s heaer than the FIM mdel sne t redues the number f unknwns that have t be slved mltly. The IMPSAT based AIM mdels are mre flexble, mre stable and less exensve than the tradtnal AIM mdel. Wth rerly tuned erentaes f rdblks fr eah frmulatn (IMPES, IMPSAT and FIM, the IMPESIMPSATFIM mdel wll always uterfrm the tradtnal IMPESFIM mdel, and be the best mdel t use fr all rblems. v

7 Aknwledements I wuld lke t thank my advser Dr. Khald Azz fr hs surt, enuraement and udane durn ths wrk. I wuld als lke t thank Dr. Lus Durlfsky and Dr. Mart Gerrtsen fr readn the dssertatn and rvdn valuable mments. I als wsh t thank the rest f the Petrleum Enneern Deartment fr ther surts and ntrbutns t my aadem ahevements. Seal thanks are due t my wfe, Je L, fr her surt and fr vn me the jy f lfe, ur newbrn sn. I wuld als lke t thank my mther n law fr heln wth the are f ur baby. My arents als deserve my rattude fr surtn and enuran my eduatn frm the very bennn. I als wsh t thank ExxnMble, ChevrnTexa and Shlumberer fr summer jbs that rvded valuable exerene and mtvatn fr my researh. Fnally, I lke t thank the manes surtn the Reservr Smulatn Industral Afflate Prram (SUPRI-B at Stanfrd Unversty fr ther fnanal ntrbutn and nterest n my researh. Ths researh was artally surted by the Untes State Deartment f Enery thruh ntrat number DE-AC6-99BC53. Ths surt s reatly areated. v

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9 Cntents Abstrat...v Aknwledments...v Table f Cntents...x Lst f Tables...x Lst f Fures...xv Intrdutn.... Develment f Smulatn Tehnly..... Exlsn n Cmutatnal Pwer..... Develment f Smulatn Tehnques...4. Cmstnal Smulatn Frmulatn and Slutn Methd Mass Balane Tye Vlume Balane Tye Flash Calulatn Equlbrum Rats Flash Equalty f Fuates Flash Treatment f Phase Dsaearane and Reaearane Other Numeral Tehnques Outlne f Researh Stes...5 x

10 Bas Asets f GPRS...7. Grddn...7. Cntrl Vlume Frmulatn and Struture f Jaban Matrx Netwrk Mdeln Mult-Pnt Flux Reresentatn and Imlementatn Well Treatment Flash Calulatn and Treatment f Phase Dsaearane and Reaearane Lnear Slver Tmeste Cntrl and Reservr Intalzatn General Frmulatn Arah General Nnlnear Equatn Set General Frmulatn Arah Stes Swth Frm the Natural Varables t Other Varables Redue Full Set t Prmary Set Redue the Imlt Level f the Fully Imlt System Stablty Analyss Influene f Equatn and Varable Seletns Seletn f Prmary Equatns Dfferent Varable Seletns IMPSAT and IMPSAT Based AIM mdel IMPSAT Mdel Why Des IMPSAT Mdel Wrk? IMPSAT Stablty Crtern? Cst f IMPSAT Mdel Hw t Buld IMPSAT Effently? IMPSAT Based AIM Mdel...03 x

11 5 Mdel Valdatn and Perfrmane Valdatn f GPRS Blak-Ol Mdel Cmstnal Mdel Unstrutured Grd and Mult-Pnt Flux General Frmulatn Arah Nn-nventnal Well Perfrmane f Cmstnal Mdels n GPRS Fully Imlt Mdels IMPES Mdels IMPSAT Mdel AIM Mdels Cnlusns and Future Wrk Cnlusns Future Wrk...50 Nmenlature...5 Referenes...55 Aendx A: Overvew f GPRS...65 Aendx B: Blak-Ol Smulatn Usn Cmstnal Frmulatn...79 Aendx C: Flash Calulatn...8 x

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13 Lst f Tables 5. Iteratn and tmn marsns fr FIM mdels and Case rblem Iteratn and tmn marsns fr FIM mdels and Case rblem Iteratn and tmn marsns fr FIM mdels and Case 3 rblem Iteratn and tmn marsns fr FIM mdels and Case 4 rblem Iteratn and tmn marsns fr FIM mdels and Case 5 rblem Iteratn and tmn marsns fr IMPES mdels and Case rblem Iteratn and tmn marsns fr IMPES mdels and Case rblem Iteratn and tmn marsns fr IMPES mdels and Case 3 rblem Iteratn and tmn marsns fr IMPES mdels and Case 4 rblem Iteratn and tmn marsns fr the IMPSAT mdel and Case rblem Iteratn and tmn marsns fr the IMPSAT mdel and Case rblem Iteratn and tmn marsns fr the IMPSAT mdel and Case 3 rblem Iteratn and tmn marsns fr the IMPSAT mdel and Case 4 rblem Iteratn and tmn marsns fr AIM mdels and Case rblem Iteratn and tmn marsns fr AIM mdels and Case rblem Iteratn and tmn marsns fr AIM mdels and Case 3 rblem Iteratn and tmn marsns fr AIM mdels and Case 4 rblem Iteratn and tmn marsns fr AIM mdels and Case 5 rblem...45 x

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15 Lst f Fures. State f the art CPU erfrmane.... Maxmum ratal mdel sze...3. Effet f rdblk rdern n struture f Jaban matrx An unstrutured rd wth faults and multlateral wells A eneral lyhedrn General struture f the Jaban matrx A smle unstrutured rd wth wells Jaban struture fr the system n Fure Grdblk nnetn array when the blk-based arah s used Cnnetn array when the nnetn-based arah s used Jaban struture fr tw-nt flux dervatves Jaban struture fr mult-nt flux dervatves Wellbre flw Perfrmane marsn between GPRS slver and BltzPak Redutn ress frm full set t mlt rmary set Struture f the full set Jaban matrx and RHS Struture f the nverse f the transfrmatn matrx fr a Tye B mdel Redue full set t rmary set by Gaussan elmnatn Struture f the rmary set Jaban matrx and RHS Fully mlt Jaban matrx f ne rdblk Jaban matrx f ne rdblk after exlt treatment f flux term Husehlder refletn deuln Lal nversn deuln Illustratn fr lal nversn df / ds as a funtn f S FS as a funtn f S fr dfferene K values Ol saturatn at the well blk Maxmum stable tmeste sze fr the IMPES mdel Maxmum stable tmeste sze fr the IMPSAT mdel Blak-l reservr...06 xv

16 5. Ol rdutn rate at the rduer frm GPRS and Else Gas l rat at the rduer frm GPRS and Else Cmstnal reservr Well blk and blk (,,5 ressures frm GPRS and Else 300 wth t max 30 days Well blk and blk (,,5 l saturatns frm GPRS and Else 300 wth t max 30 days Well blk and blk (,,5 l saturatns frm GPRS and Else 300 wth t max 30 days fr GPRS and t max 0 days fr Else Unstrutured rd Water ut results frm GPRS and FLE fr unstrutured rd Well blk ressure fr the FIM mdel wth Tye A and Tye B varables Well blk l saturatn fr the FIM mdel wth Tye A and Tye B varables Well blk ressure fr mdels wth dfferent mlt levels Well blk l saturatn fr mdels wth dfferent mlt levels Blak-l reservr wth a dual-lateral rduer Ol rdutn rate at the rduer frm GPRS and Else 00 fr a dual-lateral well rblem Gas l rat at the rduer frm GPRS and Else 00 fr a dual-lateral well rblem Hetereneus ermeablty feld (lnk fr Case Well blk l saturatn marsns fr FIM mdels and Case rblem Well blk l saturatn marsns fr FIM mdels and Case rblem Well blk l saturatn marsns fr FIM mdels and Case 3 rblem Well blk l saturatn marsns fr FIM mdels and Case 4 rblem Well blk l saturatn marsns fr FIM mdels and Case 5 rblem Stable tmeste sze fr the IMPES mdel and Case rblem Stable tmeste sze fr the IMPES mdel and Case rblem Stable tmeste sze fr the IMPES mdel and Case 3 rblem Stable tmeste sze fr the IMPES mdel and Case 4 rblem...7 xv

17 5.7 Well blk l saturatn marsns fr IMPES mdels and Case rblem Well blk l saturatn marsns fr IMPES mdels and Case rblem Well blk l saturatn marsns fr IMPES mdels and Case 3 rblem Well blk l saturatn marsns fr IMPES mdels and Case 4 rblem Stable tmeste sze fr the IMPSAT mdel and Case rblem Stable tmeste sze fr the IMPSAT mdel and Case rblem Stable tmeste sze fr the IMPSAT mdel and Case 3 rblem Stable tmeste sze fr the IMPSAT mdel and Case 4 rblem Well blk l saturatn marsns fr the IMPSAT mdel and Case rblem Well blk l saturatn marsns fr the IMPSAT mdel and Case rblem Well blk l saturatn marsns fr the IMPSAT mdel and Case 3 rblem Well blk l saturatn marsns fr the IMPSAT mdel and Case 4 rblem Well blk l saturatn marsns fr AIM mdels and Case rblem Well blk l saturatn marsns fr AIM mdels and Case rblem Well blk l saturatn marsns fr AIM mdels and Case 3 rblem Well blk l saturatn marsns fr AIM mdels and Case 4 rblem Well blk l saturatn marsns fr AIM mdels and Case 5 rblem...4 A. Struture f an l feld...66 A. Feld level system mdel...66 A.3 Dman level system mdel...67 A.4 Reservr level system mdel...68 A.5 Well level system mdel...69 A.6 Frmulatn level system mdel...70 xv

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19 Chater Intrdutn Petrleum Reservr Smulatn has me a ln way sne ts brth n the 950s. It s wrthwhle t take a bref lk at the develment f smulatn tehnly befre desrbn new develments resultn frm the researh rerted here. Sne mstnal smulatn s the majr nterest f ths researh, dfferent asets f mstnal mdeln are summarzed, nludn frmulatn, hase behavr alulatns and varus numeral tehnques used n the smulatr. Fllwn ths, an vervew f the urrent researh rjet s ven, nludn the desn and mlementatn f a new General Purse Researh Smulatr (GPRS, the develment f a new flexble General Frmulatn Arah, the develment f new mstnal mdels and evaluatn f dfferent mstnal mdels.. Develment f Smulatn Tehnly Reservr smulatn s the art and sene f usn numeral tehnques t slve the equatns fr heat and mass flw n rus meda, nsdern the arrate ntal and bundary ndtns (Azz and Settar, 979. Sne the avalablty f dtal muters t the etrleum ndustry n the 950s, mre and mre f the reservrs have been studed wth the ad f smulatrs (muter rrams. Reservr smulatn has evlved substantally sne ts brth. Tday, reservr smulatn s a mature tehnly, and t s wdely used n reservr manaement. Nearly all majr reservr develment desns are based at least artally n smulatn results (Watts, 997. The revlutn n reservr smulatn ver the last 50 years has been fueled by tw majr fatrs, the exlsn n mutatnal wer and the develment f new and advaned smulatn tehnques.

20 .. Exlsn n Cmutatnal Pwer The earlest muters were lttle mre than addn mahnes by tday s standards. Wthut substantal rress n mutn, rress n reservr smulatn wuld have been mssble. Develments n mutn wer are nentrated n fur areas, CPU seed, memry, arallel mutn and rrammn lanuaes. Eah f them has ntrbuted t the develment f smulatn n ts wn way Mfl/s Year Fure. State f the art CPU erfrmane (Dnarra, 00 CPU seed Fure. ves the mutn seed n mllns f flatn nt eratns er send (Mfl/s fr the fur fastest CPUs f ther tme. CPU seed nreased by nearly 000 tmes frm 970 t 996, just as redted by Mre s Law (the number f transstrs er square nh n nterated ruts dubles every 8 mnths. In the last several years, the rate f nrease f CPU seed aears t be slwn dwn. Memry A steady nrease f avalable memry enables us t run larer and mre mlex rrams. In the 970s, the best muters avalable had nly several Meabyte

21 f memry. Tday, a d PC uld have Gabyte (Gabyte 000Meabyte f memry, and a arallel suermuter uld have ver 50 Gabyte f memry (Dru et al., 00. Parallel mutn Fure. shws snle ressr erfrmane. Tday, hherfrmane mutn s aheved by usn multle ressrs n arallel. The resultn erfrmane (arallel seedu deends wdely n rblem tye, sze, and number f ressrs. Generally 3 ressrs lead t a seedu f 5-5 tmes fr lare (hundreds f thusands f rdblks reservr mdels (Watts, 997, and hher seedu fatrs are exeted fr even larer mdels. Prrammn lanuaes Frm the early versn f Frtran t tday s Frtran90 and C, rrammn lanuaes have develed thruh several staes. Tday s bjetrented lanuaes (suh as C faltate the desn and develment f lare and mlex muter rrams. Mdular arah and all knds f sure de ntrl sftware (suh as SureSafe frm Mrsft make ru rrammn easy and feasble, the st f develment s redued, and rrams develed usn mdern tls are easer t mantan..e06.e05 Number f rdblks.e04.e03.e Year Fure. Maxmum ratal mdel sze 3

22 A steady derease n the re f muters has made smulatn mre and mre affrdable. A dret nsequene f nreasn mutn seed and memry s the rwth f mdel sze. In 960, the larest mdel sze was arund 00 rdblks. By 970, t had rwn t abut 000 rdblks. In 997, t was ruhly 500,000 rdblks fr smulatns n a snle ressr (Watts, 997. Tday wth arallel mutn, the mdel szes are bemn even larer. In a reent meetn Dru (00 rerted smulatns wth u t 6 mlln rdblks fr three-hase blak-l rblems, and rutne smulatn wth ne mlln rdblks. Fure. summarzes these results, t shws a nstant rate f rwth n mdel sze frm 960 t 997, whh s ruhly rrtnal t the rwth n mutn seed. The fast rwth n mdel sze n the last several years s rmarly due t the use f arallel muters. The rwth n mutatnal wer enables enneers t smulate lare and mlex mstnal rblems, and t devel and aly new smulatn tehnques. As a result, t s bemn ssble t mre aurately aunt fr reservr heterenetes and ress haratersts. At the same tme, enneers and sentsts are lkn fr ways t redue the st f reservr smulatn by redun the number f unknwns er rdblk and by develn mre rbust lnear slvers and rendtners... Develment f Smulatn Tehnques In reent years, new tehnques have aeared n all areas f smulatn, nludn rddn, flud mdeln, numeral arxmatns, lnear slvers, reservr and elal mdeln, et. All f these tehnal advanes are makn smulatn mre aurate and realst, and the smulatr mre rbust, fast, stable, and user-frendly. Grddn In the early staes f ths tehnly, all reservr smulatns were erfrmed n retanular Cartesan rd, radal rd was develed later t smulate near well flw. Lal rd refnement was develed t aheve better auray n hh flw rens (Cment and Sweet, 973; Naul, 99; Pedrsa and Azz, 985. Develment f rner-nt emetry rd (Pntn, 989; Dn and Lemnner, 995; Peaeman, 996 made t ssble t use nn-retanular rdblks, rvdn the aablty t mdel faults and ther mlex elal features. U t ths nt, all rds were 4

23 strutured, where the nehbrs f a rdblk uld be easly dentfed frm ther, j, k ndes. In the last deade, unstrutured rds (Henemann and Brand, 989; Pala and Azz, 994; Gunasekera et al., 997; Kberber, 997; Aavatsmark et al., 997; Verma and Azz, were ntrdued, whh an easly fre rdblks t nfrm t majr elal features. In unstrutured rds, the nnetns between rdblks are flexble, and a nnetn lst s used t kee trak f the nneted rdblks. Mst unstrutured rds are nn-rthnal rds, whh ften requre the use f mult-nt flux arxmatns. Tday, users an defne majr reservr unts t whh the smulatn rd must nfrm, wthn eah unt, the rd an be enerated autmatally wth sme udn nut frm the user (Gequest, 000. Autmat rddn sftware akaes are bemn avalable frm mmeral vendrs, suh as, FlGrd frm Gequest, GOCAD frm the GOCAD ru and GrdPr frm Prram Develment Crratn (PDC. In the last several years, flw based unstrutured rds (Aut et al., 998; Edwards et al., 998; Castelln et al., 000 have been rsed, whh are nrmally frmed by eneratn streamlnes and s-tental lnes frm a fne rd snle-hase smulatn. Cnsderable smthn s requred t make the rd sutable fr smulatn. These rds ruhly fllw majr elal features, suh as faults, and they are nentrated n hh flw rate areas, suh as wells r hh ermeablty rens. Edwards et al. (998 shwed that, f the rds fllw the streamlnes exatly, the mult-nt flux alulatns an be redued t tw-nt flux alulatns. Besde develments n rddn tehnques, Lm et al. (994 rsed a new arah t the reresentatn f rd nfrmatn. In the nventnal arah r the blk-based arah, rdblks and wells are traked when mutn flux terms. Whle n the new nnetn based arah, the netwrk f nnetns are nsdered. A nnetn always lnks tw ndes, eah nde an be a rdblk nde, a well nde r a surfae falty nde. A ell (rdblk lst s used fr the alulatn f aumulatn terms, and a nnetn lst s used fr the alulatn f flux terms. In Chater, we wll dsuss the nstrutn f the Jaban matrx and the resdual usn ths arah. Ths nnetn-based arah s extremely nvenent fr unstrutured rds, sne the 5

24 nnetn lst tself has n struture at all. Besdes ths, t s als sutable fr dman demstn, surfae falty mdeln (Lm et al., 994 and mult-nt flux alulatns. Flud mdeln Intally, all smulatns were based n the blak-l flud mdel, where the hydrarbn system s reresented by tw seud-mnents, l and as, ardn t ther status at standard ndtns. In the early 980s, mstnal smulatn, where the hydrarbn system s reresented by an arbtrary number f mnents and seud-mnents (e.. Azz, 996, beame mre mature and ready t use. The develment f mstnal smulatn makes t ssble t smulate vlatle l reservrs, CO fldn and ther EOR resses. Hwever, mstnal smulatn s muh mre exensve than blak-l smulatn, due t the larer number f unknwns er rdblk and mlex flash behavr. Numeral arxmatns Cnventnal smulatrs use fnte-dfferene methds wth tw-nt flux alulatns. Mre reently, mult-nt flux alulatns (Verma and Azz, 996; Gunasekera et al., 998 are bemn mre and mre mmn due t the use f full tensr ermeablty (Lee et al., 994; Lee et al., 997 and nn-rthnal rds. At the same tme, ntrl vlume methds (Aavatsmark et al., 997; Verma and Azz, 997 have beme the methd f he fr tday s smulatr, beause f ther easy handln f unstrutured rds. Lmted wrk has als been dne n hher-rder shemes (Sammn, 99; Chen et al., 99 and fnte element methds (Yun, 978; Fun et al., 99; Sukrman and Lews, 994 t aheve hher auray. Lnear Slver Slvn the lnear system s the snle mst stly art fr a smulatn. S t s extremely mrtant t have a d lnear slver. Intally, dret slvers were used, but as the rblems have beme larer and larer (mre and mre rdblks, teratve slvers have beme mre and mre mmn. The erfrmane f teratve slvers deends n the qualty f rendtners. In the etrleum ndustry, mst f the effrt n ths area has been n buldn better rendtners. Any teratve slver an be used as a rendtner fr ther teratve slvers, and dfferent rendtners an be mbned tether t frm multstae rendtners. Tradtnal rendtners nlude, Inmlete LU demstn (ILU (e.. Behe and 6

25 Frsyth, 983, Gauss-Sedel (GS (e.. Azz and Settar, 979, Alebra Mult-Grd (AMG (Stueben, 983, et. Ther erfrmane deends strnly n the nature f the lnear system. Fr near-ellt system, AMG wrks well, fr near-hyerbl system, bth ILU and GS wrk well. In reservr smulatn, we have a near-ellt ressure equatn and near-hyerbl saturatn equatns. Fr ths knd f mxed system, nne f the snle stae rendtners (ILU, GS, AMG, et wrk well, we need a smarter rendtner. Walls et al. (985 ntrdued the Cnstraned Pressure Resdual (CPR rendtner, whh s seally desned fr reservr smulatn equatns. It uses a tw-stae arah t slve the ressure art and the saturatn art f the reservr equatns searately, whh make t the mst rmsn rendtner fr fully mlt smulatns. Unfrtunately, a d rendtner s stll mssn fr the Adatve Imlt (AIM Methd (Frsyth and Sammn, 986. Mst f the avalable mmeral lnear slvers have been desned fr strutured rds, whh result n Jaban matres wth a banded struture. Fr unstrutured rds, the Jaban matrx s a eneral sarse matrx, and fr suh systems the exstn slvers lse ther effeny. Fr rblems wth strutured rds, the best teratve slvers an slve a rblem wth n unknwns n O(n.~. tme (see tmn results n Setn.7. Hwever a lt f wrk needs t be dne fr unstrutured rd slvers and AIM system slvers. Fr unstrutured rds, the rdern f rdblks and nnetns are qute mrtant fr the effeny f the lnear slver, and these ssues wll be dsussed n Setn.7. Reservr and Gelal Mdeln Reservr smulatns have als exanded n terms f tns and features. Mdern reservr smulatrs an smultaneusly handle multle reservrs, surfae faltes (Shzer and Azz, 994; Byer, 000 and rk mehans. Furthermre, there s mre and mre eratn amn reservr enneer, elst and ehysst t aheve mre realst mdeln f reservr ely fr smulatn rjets (e.. Jurnel, 990; Balln et al., 993. Wth all f these develments n smulatn tehnly and muters, the smulatr has als reahed a new stae. Tday s smulatr s mre rbust, mre mlex, and easer t use than what was avalable 0 years a. Beause f all the features that are 7

26 requred n a mdern smulatr, t takes mre tme t devel a smulatr and mre effrt t mantan t. In the future, smulatrs wll be requred t slve even mre mlex rblems than they an handle tday. Under these rumstanes, a d desn fr the smulatr s vtal. Frtunately, tday we have all knds f desn tls and sutable muter lanuaes t devel tehnly t meet the future needs f the ndustry. Researh students wrkn n ths area need a lt f tme t rram the bas smulatr befre they an even start t exlre researh ts f nterest t them. Als, after a student leaves, mst f hs/her develment bemes unusable wthn a shrt tme, due t lak f d desn and dumentatn. Hene develn a d envrnment fr smulatn researh has beme essental fr a researh ru lke the ne at Stanfrd Unversty. The frst bjetve f ths researh s t devel a General Purse Researh Smulatr (GPRS wth a hhly mdular struture. Ths wll enable eah future develer t wrk n extendn mdules f nterest t hm/her, wthut havn an ntmate knwlede f every aset f ther mdules. A new researher wll be requred t understand nly the mst bas struture f the smulatr. Hefully, ths arah wll save tme n develn new tehnques, and hel reserve the wrk f new students. After the bas desn and mnents have been ut n lae, the entre researh ru wll be able t ntrbute t the rjet. An vervew f the GPRS rram, nludn ts desn, s nluded n Aendx A. Insde ths new General Purse Researh Smulatr, we wll nrrate many f the avalable new smulatn tehnques that were develed n the last deade, a number f whh are unavalable n mmeral smulatrs. Sme f the features f artular nterest t ur researh ru are, unstrutured rd (strutured rd s handled as a seal ase f unstrutured rd, netwrk mdeln (nnetn based arah, bth tw-nt flux and mult-nt flux alulatn fr netwrk mdeln, advaned lnear slver tehnly (need further develment, fully uled surfae faltes mdel (future develment, and fully uled e-mehanal mdel (future develment. 8

27 Currently, the frst fur features have been nluded n GPRS, and the furth feature needs further develment. We lan t add n the last tw features n the near future. GPRS s usn a flexble mstnal frmulatn, whh s the majr fus f ths researh, s t s wrthwhle t take a further lk at varus asets f mstnal smulatn.. Cmstnal Smulatn Petrleum reservr fluds ntan thusands f hemal mnents that affet ther hysal rertes and hase behavr durn rdutn (e.. Azz, 996. It s nt ratal t desrbe etrleum fluds n terms f ndvdual mnents. Instead, seudmnents (rus f mleules wth averae hysal rertes are used t desrbe the reservr flud. As mentned earler, we have tw mdels fr the flud, blak-l and mstnal. Ths seletn s usually based n the vlatlty f the l (e.. Azz, 996. An l, wth as slublty rat (R s less than 750sf/stb, l frmatn vlume fatr (B less than.4bbl/stb and API ravty less than 30, s tyally referred t as blak-l. In ths ase, hase behavr s smly reresented by B and R s, whh are nly funtns f ressure, and flash alulatns are nt needed. The blak-l mdel s just a seal ase f the mstnal mdel, where the hase equlbrum relatns an be redued t lnear relatns between mnent mle fratns and ressure. Beause f ths, we an erfrm blak-l smulatn usn mstnal frmulatn (refer t Aendx B fr detals, ths s the ase n GPRS. Whenever t s narrate t use a blak-l mdel, the l and as must be desrbed by mre than tw seud-mnents. In ths ase, hase behavr s reresented by an Equatn f State (EOS and hase equlbrum relatns, and ths requres flash alulatns. Smetmes, even when the reservr flud satsfes the blak l ndtn, we may stll need t use a mstnal mdel fr smulatn suh resses as CO njetn, N njetn, et. As we frm blak-l t mstnal mdels, the number f equatns and mnents nreases, and the he f equatns and varables ets mre mlex. The 9

28 erfrmane f the smulatr (nverene f Newtn teratns and nverene f lnear slver teratns an be strnly nfluened by these fatrs. The he f the mlt level (number f mlt varables fr eah rdblk s anther mrtant fatr here. Deendn n the varable tye, we may als need t swth varables when a hydrarbn hase dsaears r reaears. The essental asets f mstnal smulatn are dsussed n detal n the next several subsetns, nludn frmulatn, hase behavr and varus numeral tehnques. Here, we nly nsder sthermal mstnal mdels... Frmulatn and Slutn Methd We an searate dfferent mstnal mdels nt tw bas tyes, mass balane tye and vlume balane tye, ardn t the bas equatns they slve. All f the fllwn equatns are fr three-hase systems, and water and hydrarbn hases are ttally searated. In the fllwn dsussn, s the number f hydrarbn mnents, and n s the number f mnents, n s the number f hases. n h n -... Mass Balane Tye Just as the name mles, equatns fr ths tye f mdel are the mass balane equatn fr eah hydrarbn mnent (r seud-mnent and fr water (e.. Cats, 980; Yun and Stehen, 983: F, [ Vφ( Sx S y ] [ T ( λ x Φ λ y Φ ] l, t l W W ( x q y q 0 W,, nh (. F W ( Vφ Sw w ( Tλw w Φ w l, ( wqw 0 (. t w, l W where, x and resetvely, and y are the hydrarbn mnent mle fratns n the l and as hases q W, W q and q W w are the hase vlumetr flw rates f well W. 0

29 Transmssblty s alulated by ka T α L (.3 Phase mblty s determned by kr λ µ (.4 Phase tental dfferene between rdblk and j s defned as Φ,, j Φ, j Φ, (, j, β,, j ( D j D (.5 where α and β are unt nversn nstants (Fr feld unts, α E-3 and β.0/44. Smetmes t s nvenent t relae ne f the hydrarbn mnent mass balane equatns by a ttal hydrarbn mass balane equatn btaned by summn Eqn.. fr all hydrarbn mnents. Fh, [ Vφ( S S ] [ T ( λ Φ λ Φ ] l, t W ( q W q W 0 l (.6 In addtn t mass balane equatns, we need a hase equlbrum relatn fr eah hydrarbn mnent. Sne water and hydrarbn mnents are ttally searated, we nly need tw-hase equlbrum relatns between the l and as hases: F f e,, f 0,,, n h (.7 where f, and f, are the fuates f hydrarbn mnent n the l and as hases, resetvely. Aendx C rvdes detals f flash and fuaty alulatns.

30 Fnally sme lnear nstrants have t be satsfed: Callary ressure nstrants: F F ( w, w w (, 0 0 (.8a (.8b Saturatn r vlume nstrant: n F S S 0 (.8 r F V V 0 V φ T (.8d where V φ s the re vlume, and V T s the ttal flud vlume. Cmnent mle fratn nstrants: n h F x 0 n h (.8e F y 0 (.8f Fr ths system, the equatns are: Tye F Fh Fw Fe F F w, FS F F, Number nh - Ttal n h 6 nh

31 Same number f varables need t be seleted, ne ssble he s: Tye Number 3 S 3 x y nh nh Ttal n h 6 There are a ttal f n h 6 equatns and n h 6 varables. It s nether ratal nr neessary t slve all f these equatns tether. Ardn t Gbbs hase rule, the thermdynam derees f freedm fr ths system are f therm n n. They fx the ntensve state f the system. Besdes ths, the vlume fratn f eah hase ( n - saturatns must als be determned t fx the extensve state. Assumn sthermal ndtns, we have ne less deree f freedm. The fnal derees f freedm (Azz, 997 fr an sthermal mstnal system are f ( n n ( n n S we nly need t slve n extensve state f the system. These f the n h 6 ttal equatns t fx bth the ntensve and n equatns are ur rmary equatns. They are als the equatns slved smultaneusly fr fully mlt (FIM mdels. There shuld be the same number f rmary varables (varables fr the FIM mdel. After slvn fr the rmary varables, ther varables (sendary varables an be slved fr frm the remann equatns (sendary equatns. We an treat all f the rmary varables mltly, lke n the FIM mdel. In rder t further seedu the smulatn ress, we an als treat nly sme f the rmary varables mltly, frst slve fr them, then slve fr the remann rmary varables by exlt r sequental mlt methds (Azz and Settar, 979; Watts, 986. Nte, nly the frst n h equatns ( F, F h, number f ndeendent mass balane equatns ( F, F w, F h, F e are nn-lnear, and the ttal F w s equal t the number f 3

32 rmary equatns and varables. We an use the lnear nstrant equatns t remve tw ressures, ne saturatn and tw mnent mle fratns. Fnally, nly n h number f nn-lnear equatns and varables s left, whh s the full set f equatns and varables. Out f ths full set, n rmary equatns and varables are seleted. Sne mass balane equatns reresent flw n the reservr, and ther number always equals the number f rmary equatns, they are the natural he fr rmary equatns. But the seletn f rmary varables s wdely vared amn mdels. After the rmary set s seleted, the equatns and varables left ver n the full set frm the sendary equatns and varables. There are a lt f mdels that beln t ths lass, and they are dfferent n the way that rmary equatns and varables are hsen. Sme f the bas mdels are revewed next. nh The Fussell and Fussell Mdel (979 Fr ths mdel the rmary equatns are the hase equlbrum relatns and the saturatn nstrant equatn. The rmary varables are, l (hydrarbn lqud mle fratn, and y r x,,, n h, deendn n whether the system n a rdblk s redmnantly var r lqud. The nh hase equlbrum relatns are used t remve all f the rmary varables exet ressure frm the saturatn nstrant equatn. The saturatn nstrant equatn s the fnal IMPES (Only ressure mlt frmulatn ressure equatn. Ths s the nly mdel that uses hase equlbrum relatns as rmary equatns, and t s rarely used. The Cats Mdel (980 Fr ths mdel the rmary equatns are the n mass balane equatns fr eah hydrarbn mnent and fr water. The rmary varables are,, S,S, y,3,, n h (when bth the l and as hases are resent,, S, x,,, nh -(when n free as s resent, and, S, y,,, nh -(when the l hase dsaears. In eneral, adjaent rdblks may have dfferent sets f rmary varables, and we need t swth varables when a hydrarbn hase dsaears r reaears. The varable seletn (ressure, saturatns and mnent mle fratns n ths mdel s nrmally alled the natural varable seletn, sne the mass balane equatns an be dretly 4

33 exressed n terms f these varables and alulatn f dervatves f the Jaban matrx s very easy. Ths mdel was ntally rsed as a FIM mdel, but t an als be redued t an IMPES mdel by varable elmnatn. the The Yun and Stehensn Mdel (983 Fr ths mdel the rmary equatns are n mass balane equatns and the saturatn nstrant equatn, and the last hydrarbn mnent mass balane equatn s relaed by the ttal hydrarbn mass balane equatn. The rmary varables are, hydrarbn n ne unt vlume f flud, vlume f flud and z F S S (ttal mles f W w S w (ttal mles f water n ne unt (verall mle fratn f mnent,,, nh -.The mass balane equatns are used t elmnate all f the rmary varables exet ressure frm the saturatn nstrant equatn. The saturatn nstrant equatn s the fnal IMPES ressure equatn. Ths arah des nt deend n the resene f ndvdual hases, sne all f ts rmary varables are verall quanttes. On the ther hand, saturatn s nt ne f the rmary varables, s we have t use the han rule t evaluate sme f the dervatves f the flw term wth reset t saturatn, suh as relatve ermeablty. Fr ths mdel, there are n rmary equatns and varables, ne mre than neessary. Smetmes, ths s dne t faltate the Jaban matrx alulatn, eseally when ertan nstrant equatn s nnlnear wth reset t the seleted rmary varables (here, the saturatn nstrant equatn s a nnlnear equatn n terms f the rmary varables. Atually eah small ( n x( n blk system an be redued t a n x n equatn as a sendary equatn, then there wll be nly the blk system. An alternatve arah s t treat ths extra nnlnear nstrant n rmary equatns. The Chen, Lee, and Chen Mdel (985 Fr ths mdel the rmary equatns are n mass balane equatns fr eah hydrarbn mnent and fr water. The rmary varables are, W and r z F (ttal mles f mnent n ne unt vlume f flud,,, nh max. (r max s defned as the r wth reset t whh the dervatve f the saturatn nstrant s larest.. Ths mdel als des nt deend n the resene f ndvdual hases. Als, ths mdel needs t use the han rule t evaluate sme f the flw term dervatves. The equatns f ths mdel are nrmally slved fully mltly. n 5

34 Eah f the mdels desrbed abve uld be ether fully mlt r nly mlt n ressure. In the rnal Yun and Stehensn mdel (Yun and Stehensn, 983, nly ressure s treated mltly, whle n the rnal Cats mdel (Cats, 980, all rmary varables are treated mltly. There are a lt f ther mdels, and eah f them s atually a varatn f ne f these bas mdels. The varatns are btaned by hann the level f mltness, hann ntensve and extensve varables, and s n. In summary, all f these sthermal mstnal mdels (exet the Fussel and Fussel mdel selet mass balane equatns as the rmary equatns, and there are basally tw tyes f rmary varables (Azz, 996. Tye A varables: ne ressure n - saturatns n- n mnent mle fratns Tye B varables: ne ressure n-verall quanttes, suh as verall mnent mle fratns The Cats mdel (Cats, 980 belns t Tye A. The Yun and Stehensn mdel (Yun and Stehensn, 983 and the Chen, Lee and Chen mdel (Chen et al., 985 beln t Tye B. Eah tye f varables has ts wn advantaes and dsadvantaes. The mmn nts and dfferenes between these tw tyes f varables are ven belw. Cmmn Pnts: Mass balane equatns are always the rmary equatns. Phase equlbrum relatns are always used t elmnate the sendary varables frm the rmary equatns. After slvn fr the rmary varables, the sendary varables are udated exltly va the hase equlbrum relatns rdblk by rdblk Bth tyes f mdels an be ether fully mlt (FIM r nly mlt n ressure (IMPES. 6

35 Dfferenes: Tye A varables are the natural varables, equatns an be exltly exressed n them, and the Jaban matrx s easy t alulate. Fr Tye B varables, we have t use the han rule t alulate the dervatves fr the Jaban matrx. Fr examle, the han rule s requred t alulate dervatves f hase relatve ermeablty wth reset t verall mnent mle fratns. In Tye A, hase varables (saturatn and mnent mle fratns n eah hase are used, s they deend n the aearane and dsaearane f hydrarbn hases. Fr eah rdblk, f a hydrarbn hase hanes status, we need t swth the rmary varables (relan the nn-exstn hase varables wth the rresndn varables fr the exstn hase. Ths enerally leads t dfferent rmary varables n dfferent rdblks. Addtnal wrk s requred t trak the varables used n eah rdblk, hek the exstene f ndvdual hydrarbn hases and erfrm the swth. Tye B uses verall mstns as the rmary varables, and the same rmary varables an be used fr all rdblks.... Vlume Balane Tye Ths tye f mdels (As et al., 98; Kendall et al., 983; Watts, 986; Azz and Wn, are based n the vlume balane net, and use the vlume balane equatn as the ressure equatn t frm IMPES tye frmulatns. Sne the re sae f rus medum must be fully flled wth fluds resent, the re vlume must be equal t the ttal flud vlume at any tme and lae. Ths an be exressed as V V V V V φ T w (.9 The re vlume V V s slely a funtn f ressure: φ n n n n φ Vφ Vφ r ( (.0 where r s the rk mressblty. 7

36 8 The ttal flud vlume T V s a funtn f bth ressure and verall mstn f eah mnent: ( ( (,,,,, n n n j M M n T n w n w M n w T n n M M n T n T n T M M M V M M M V V V V h j w w (. where, M and w M are the verall mle amunts f hydrarbn mnents and water. n n M M and n w n w M M are atually the aumulatn terms n the mass balane equatns, whh an be wrtten as, n n n n U t M M,,, h n (., n n w n w n w U t M M (.3 where: [ ],,,, ( ( Φ Φ n W n n n n n n n n Q y T x T U λ λ (.4 [ ],,, ( Φ n W w n w n n w w n n w Q T U λ (.5 Nte that, fr IMPES, the transmssblty terms ( λ, λ, w λ,,, w, x, y are fxed at the ld tme level. The tme level fr terms wth suersrt n, n deends n the mltness f the mdel. The fnal ressure equatn s btaned by substtutn Eqn..0, Eqn.. and Eqn..3 nt Eqn... After rearrann, t an be wrtten as [ ],,,,,,, ( n n w M n w T n n n j M M n T n T n n M M n T r U M V U M V t V V V V h j w w φ φ (.6

37 After slvn fr ressure frm Eqn..6, ther varables are udated exltly rdblk by rdblk. The key art f ths methd s t alulate the dervatves f ndvdual hase vlumes V V V wth reset t bth ressure and verall mles f eah mnent (,,. M M In rder t alulate these dervatves analytally, we need t make full use f the hase equlbrum relatns. Dfferent authrs have rsed dfferent methds t alulate these dervatves. Mdels f ths tye were rsed by As et al., 98; Kendall et al. 983; Watts, 986; Azz and Wn, Cmarn the mass balane tye mdels wth the vlume balane tye mdels, the vlume balane tye mdels are nly mlt n ressure, basally t s an IMPES arah. Cats (999 has nted ut that the IMPES ressure equatn s unque, ndeendent f the manner f dervatn, he and rdern f varables and equatns. S n matter whh tye f mdel we use, as ln as t fnally redues t the IMPES ressure equatn, numerally they are equvalent t eah ther. They are nly dfferent n tw asets: The number f eratns t enerate the fnal IMPES ressure system s dfferent. Frm ths nt f vew, the Yun and Stehensn mdel s mre effent than the Cats mdel (Cats, 999. The Jaban matrx s saled dfferently frm rdblk t rdblk, whh uld nfluene the nverene rate f teratve lnear slvers (Cats, 999. w In tw aers, Azz and Wn (989 and Wn et al. (990 shwed the nnetn between these tw tyes f mdels, and hw t derve the vlume balane mdel startn frm the Jaban matrx f a mass balane tye mdel. In ths researh, we wll just treat the vlume balane tye mdel as a seal IMPES ase f the mass balane tye mdel. Mdels that treat nly ressure mltly are referred t n the lterature by dfferent names: IMPES (mlt ressure and exlt saturatns, IMPEM (mlt ressure and exlt verall mnent mass/mle. As stated earler, reardless f the varables seleted, the ressure equatns are always equvalent t eah ther. In ths wrk, we d nt dstnush between them, and refer t them as IMPES mdels. 9

38 .. Flash Calulatn Cmared t blak-l mdels, mstnal mdels are nt nly mre mlated n ther frmulatns, they als requre addtnal flash alulatns. In blak-l mdels, the hase equlbrum relatns an be redued t a seres f lnear relatns between mnent mle fratns and ressure, s there s n need fr flash alulatns. In mstnal mdels, we need t d flash alulatn fr eah rdblk at eah teratn f eah tmeste. There are tw tyes f flash alulatns:... Equlbrum Rats (K Flash Ths s the smlest arah (Yun and Stehensn, 983 t flash alulatns. The equlbrum rats are funtns f bth ressure and mle fratns, n ths arah we assume that these funtns are knwn. We have the fllwn equlbrum relatns: y K x z lx ( l y (.7 (.8 where l s the hydrarbn lqud (l mle fratn n hydrarbn flud. Frm Eqn..7 and Eqn..8, we an further exress x and z x l ( l K K z y l ( l K y n terms f l, z and K : (.9 (.0 Frm Eqn..8e, Eqn..8f, Eqn..9 and Eqn..0, we an derve the Rahfrd-Re equatn (Rahfrd and Re, 95, whh s slely a funtn f l fr a ven K : ( K z 0 y x ( y x (. l ( l K z and The slutn redure redues t the fllwn tw stes:. Slve fr l frm Eqn... Slve fr x and y frm Eqn..9 and Eqn..0 0

39 ... Equalty f Fuates Flash At equlbrum, F f f 0 e,,,,, n h, whh are a seres f nnlnear equatns. They an be slved by ether the suessve substtutn methd r the Newtn-Rahsn methd (Fussel and Yansk, 978. The slutn redure fr the suessve substtutn methd s ven belw:. Slve fr l frm Eqn... Slve fr x and y frm Eqn..9 and Eqn Calulate the hase mressblty fatr Z fr eah hase frm an Equatn f State (EOS 4. Calulate the fuaty fr eah mnent n eah hase 5. Chek nverene ( f / f.0 < ε,, new ld 6. If nt nvered, udate equlbrum rats by K K f / f, and reeat stet6 (,, Aendx C derves all f the analytal exressns f dervatves needed fr the Newtn-Rahsn methd. The suessve substtutn methd nveres slwly, but nverene s uaranteed, the Newtn-Rahsn methd nveres qukly, but t nly nveres near the true slutn. S, a tyal arah s t use the suessve substtutn methd frst, after the errr s redued t a ertan level, the Newtn-Rahsn methd s used t fnsh t qukly (Nhem et al., 983. Fr dfferent Equatn f State (EOS, we have dfferent equatns t alulate the mnent fuates. Cub Equatns f State are mmnly used n the etrleum ndustry, and amn them the Pen-Rbnsn EOS (Pen and Rbnsn, 976 s the mst ular ne. A rerly tuned Cub EOS s enerally adequate fr mstnal smulatn. Besdes hase equlbrum alulatns, ths redure als yelds hase rertes. The l hase mleular weht s alulated as MW n h MW x and the as hase mleular weht s alulated as MW n h MW (. y. (.3

40 The hydrarbn hase denstes are alulated as ZRT (.4 where mressblty fatr Z s ne f the three rts f the Cub EOS, fr the l hase the smallest real rt s used and fr the as hase the larest real rt s used. In the early days, mst smulatrs used the suessve substtutn methd, tday mst f them refer the faster Newtn-Rahsn methd r a mbnatn f bth methds. Sme smulatrs searate flash alulatn frm flw alulatn, eseally fr smulatrs usn Tye B varables...3 Treatment f Phase Dsaearane and Reaearane In a rdblk, there may be u t tw hydrarbn hases (l and as. Whether bth hydrarbn hases exst r nt, deends n ressure, temerature and verall mstn. Durn smulatn, we need t establsh the exstene f hydrarbn hases n all rdblks. If nly a snle hydrarbn hase exsts, equatns and varables fr sme mdels may have t be haned. Phase Dsaearane When a rdblk has tw hydrarbn hases, we need t mntr ts ndtn t fnd when t wll hane t a snle hydrarbn hase. Basally, there are three ways t d t: Fr eah Newtn teratn, we slve the flw equatns, and et the saturatn slutns. If ether S r S s neatve, t s set t zer befre ntalzn fr the next Newtn teratn (Cats, 980. In ths ase, the rresndn hydrarbn hase dsaears. A flash alulatn s erfrmed. If the lqud mle fratn, l, <0, then the l hase des nt exst anymre, else f l >, then the as hase dsaears (Nhem et al., 983. Ths knd f flash s alled neatve flash (Whtsn and Mhelsn, 986. A new set f seud-equlbrum rats (PER s s used, and nly ne set f equatns and nstrants are used at all tmes t smulate the ress reardless f the hase ndtn (Abu-Kassem and Azz, 985. Ths methd s sutable fr bth mstnal and thermal mdels.

41 Phase Reaearane When a rdblk has nly a snle hydrarbn hase, we als need t mntr ts ndtn. Beause whenever t hanes t tw hases, we may need t hane equatns and varables aan. The fllwn three arahes an be used fr ths: Fr eah Newtn teratn, f ether S r S s zer n a rdblk, then a saturatn ressure s alulated fr the snle hydrarbn hase flud resent n the blk. If the alulated saturatn ressure s less than the rdblk ressure, the rdblk remans n snle hydrarbn hase mde, therwse, bth hydrarbn hases exst and the absent-hase saturatn s set t 0.00 (Cats, 980. A flash s erfrmed, f l s between 0 and, then bth hydrarbn hases exst (Nhem et al., 983. We an aelerate the nverene f flash alulatns by ettn the ntal values frm the nehbrn blks (Yun and Stehensn, 983. The thrd treatment (Abu-Kassem and Azz, 985 f hase dsaearane an be aled wthut any hane, sne the same set f equatns and nstrants s used at all tmes reardless f the hases resent...4 Other Numeral Tehnques Partal Jaban Udate (Yun and Stehensn, 983 If a rdblk satsfes the nverene rtera, ts ld Jaban values are retaned. In ther wrds, the Jaban terms are udated nly fr un-nvered rdblks. Ths tehnque an save sme tme n Jaban alulatns, and t des nt have a lare nfluene n the nverene behavr f Newtn teratns. At nverene, rret results are uaranteed. Relaxed Vlume Balane Cnet (Cats et al., 998 Ths tehnque an be used n bth the FIM and the IMPES mdels, tyally t s used n IMPES mdels. Fr the lnear nstrant equatns, suh as Sw S S 0, we use S Sw S where e S w S δs e δsw δs t dretly substtute fr S S, δs nt the nnlnear equatns n rder t remve them. When e 0, vlume s nserved, when e s nt zer, vlume s nt nserved, t s alled the relaxed vlume balane arah beause sme vlume balane errr (hase 3

42 saturatns d nt exatly sum t.0 s aeted. Hwever, exat mass balane at eah Newtn teratn s aheved. In effet, ths redure amunts t teratn ut vlume balane errr whle keen exat mass balane. Ths relaxed balane net an be used fr any lnear nstrant equatn. If exat balane s nt aheved at the last tmeste, then ths balane errr an be rreted n the urrent tmeste. One Ste Flash Calulatn Theretally, we shuld d exat flash alulatn fr eah Newtn teratn. But at mst tmes, t s wrthwhle t d nly a few flash teratns wthut full nverene. Ths s arrate beause the result f eah Newtn teratn s nly an arxmatn t the fnal slutn, and exat flash s nt requred. On the ther hand, If the flash results are far frm nverene, t may wrsen the nverene rate f Newtn teratns. Frm the abve dsussn, we an see that mstnal smulatn s muh mre mlex than blak-l smulatn. There are a varety f mstnal mdels, and they are qute dfferent frm eah ther. Sme analyss and evaluatn f the erfrmane f ndvdual mstnal mdels has been ublshed (Cats, 999, but t s far frm mlete. There are tw majr reasns fr the lak f arrate marsns: It s hard t mlement dfferent mstnal mdels n ne smulatr. They just have t many dfferenes. There s a lt f wrk nvlved n rrammn, befre we an even start t evaluate dfferent mdels. Cmstnal smulatrs are muh harder t devel than blak-l smulatrs. Ths knd f develment s tme nsumn, requres ne t deal wth mlex ssues assated wth flash alulatns and rrammn. Sne the fully mlt (FIM mdels are t exensve fr lare rblems and the tmeste lmts n the IMPES mdel are t strt fr hard rblems, we need a new mdel that s heaer than the FIM mdel and muh mre stable than the IMPES mdel. Anther bjetve f ths researh, n addtn t develn a General Purse Researh Smulatr (GPRS, s t evaluate the erfrmane f dfferent sthermal mstnal mdels, and where arrate, devel new mdels. Befre the wrk an be started, we need t devel a General Frmulatn Arah whh an be used t 4

43 derve any knd f mdel. Then, durn rrammn, dfferenes between dfferent mdels an be mnmzed, and we an qukly hane frm ne mdel t anther mdel. Ths shuld lead t a nsstent evaluatn f the erfrmane f dfferent mdels..3 Outlne Of Researh Stes The majr fus f ths researh s t devel a eneral-urse smulatn framewrk wth emhass n smulatr desn and mstnal smulatn. Ths wll be aheved n fur majr stes: Desn and Imlementatn f GPRS As mentned earler, a well-desned smulatr an mrve ru rdutvty and hel reserve wrk f dfferent nvestatrs. A d desn usn a mdular bjet-rented arah s requred. A lt f advaned tehnques wll be mlemented and tested n ths smulatr, nludn unstrutured rd handln, netwrk mdeln, mult-nt flux alulatn, lnear equatn slutn tehnques, et (Chater. General Frmulatn Arah and Its Imlementatn n GPRS A General Frmulatn Arah wll be develed, and ths eneral arah shuld be aable f yeldn any knd f mdel. Bth varables f Tye A and Tye B shuld be avalable and the level f mltness fr eah rdblk shuld be flexble (frm t number f mnents. The detals f ths General Frmulatn Arah are n Chater 3. Develment f New Cmstnal Mdels Anther urse f ths researh s t devel new mstnal mdels. A new mdel usn mlt ressure, saturatns and exlt mnent mle fratns (IMPSAT s dsussed n Chater 4. Evaluatn f Dfferent Cmstnal Mdels The fnal ste f ths researh s t evaluate the erfrmane f dfferent mstnal mdels under dfferent ndtns, whh nlude lare number f mnents, lare systems, hhly hetereneus systems, et. These asets are dsussed n Chater 5. Our fnal al s t fnd a way t mnmze the mutatnal tme wthut sarfn auray. 5

44 6

45 Chater Bas Asets f GPRS One f the bjetves f GPRS s t buld a eneral framewrk fr researh smulatrs, and allw a ru f students t wrk wth t and reserve the wrk f dfferent nvestatrs. In rder t make ths ress easer, t s wrthwhle t dsuss the bas asets f GPRS n detal, nludn rddn, frmulatn, flux alulatn, well treatment, flash alulatn, lnear slvers, tmeste ntrl and reservr ntalzatn. Wthn GPRS, we ntend t nrrate the mst advaned smulatn tehnques develed n the last deade. S far, we have nluded unstrutured rd, ntrl vlume frmulatn, netwrk mdeln, tw-nt and mult-nt flux fr unstrutured rd, neatve flash alulatn, mstnal well ntrl, and advaned lnear slver tehnly. In ths hater, we wll dsuss the bas asets f GPRS, revew the advaned tehnques used n eah art, and exlan hw these tehnques are nrrated.. Grddn Tday, the trend n smulatr develment s t searate the rddn art frm the flw alulatn (smulatr art, and use the utut frm the rddn art as nut fr the smulatr. The rddn art s nrmally dne by a rddn sftware. Thruh ths arah, ne smulatr shuld be able t ule wth dfferent rddn sftware, and eah rddn sftware shuld be able t rvde serve fr multle smulatrs. Als, ths arah an lwer the develment st and nrease flexblty fr bth the rddn art and the smulatr art. Hwever t s nt very sutable fr dynam rddn, where the rd hanes wth tme. Frtunately, dynam rddn s rarely used n tday s reservr smulatrs. Ths searated arah s artally adted n GPRS, and GPRS an et ts rd nfrmatn n tw ways: 7

46 Internal eneratn GPRS has a Cartesan rd mdule, whh an enerate all f the neessary nfrmatn fr a Cartesan rd. A smlar mdel fr Radal rd may be added later. External read n Fr mre advaned rds, suh as unstrutured rds, the rd nfrmatn s read frm the utut f a rddn sftware. Atually, ths tn s sutable fr all knds f rds, nludn Cartesan rd. Basally, GPRS s a reservr smulatr wthut muh rddn aablty, and unstrutured rd smulatns are surted by nterfan wth ther rddn sftware. In the etrleum ndustry, the avalable mmeral rddn sftware nludes FlGrd frm Gequest, GOCAD frm the GOCAD ru and GrdPr frm Prram Develment Crratn (PDC, et. N matter whh rddn sftware, they always share the fllwn mmn funtns: Read nfrmatn (latn, sze and hysal rertes frm a elal fne rd, read majr elal features, suh as faults (latn, lenth, dretn and read well sefatn (trajetry, dameter and skn Perfrm autmat rddn ardn t user nut (rd tye, dmensn and ther tns, and enerate smulatn rd Perfrm autmat usaln frm the elal fne rd t the smulatn rd, alulate the smulatn rd rertes, buld nnetns between rdblks and alulate well ndes (the nnetn between wellbre and well blk. Nte that sme rddn akaes d nt erfrm usaln, suh as GrdPr. Outut all f the smulatn rd rertes and well nfrmatn n a sef frmat, whh an be read later by a smulatr Nrmally, eah rddn sftware s seally desned t serve several smulatrs, fr examle, FlGrd was ntally desned t rvde nut fr bth ECLIPSE (Gequest smulatr and FIRST! (MOBIL smulatr. In rder t d that, rddn sftware nrmally defnes a sef utut frmat fr eah served smulatr. If the utut fle s a text fle, t s ssble t wrte an external rram t read n and refrmat the nfrmatn and make t sutable fr ther smulatrs, suh as GPRS. 8

47 In rder t make the rddn sftware and the smulatr wrk tether, we have t defne whh nfrmatn shuld be assed n frm the rddn sftware t the smulatr. The neessary nfrmatn nrmally nludes a lst f ell rertes, a lst f nnetn rertes, and a lst f well data. On the ther hand, the smulatr shuld be able t handle the assed rd nfrmatn, suh as unstrutured rd, mult-nt flux. GPRS s desned t handle the mst eneral rd tye and rd nnetn data, and t shuld be able t nterfae wth any rddn sftware. The rddn nfrmatn that GPRS needs nludes the fllwn: Lst f ell rertes. Fr eah ell, we need, dmensns (D, DY, DZ (nt neessary, deth, rsty, ermeablty (nt neessary, nluded n the transmssblty and WI data, and vlume. Lst f nnetn rertes. Fr eah nnetn, we need, number f rdblks fr eah nnetn (fr tw-nt flux, t s always, rdblk ndex fr eah rdblk, and transmssblty nstant fr eah rdblk. Lst f well data. Fr eah well, we need, number f rdblks that ths well enetrates, ndex fr eah rdblk enetrated by ths well (well blk, and well ndex (WI fr eah well blk. The rdern f the ell lst and nnetn lst s qute mrtant t the effeny f a smulatr, eseally n the lnear slver art. Cnsdern a smle 4 ase, Fure. shws tw dfferent ell rderns and the rresndn Jaban matrx strutures. In Fure.a, rdern s frm left t rht, and the Jaban Matrx has a trdanal struture. In Fure.b, rdern s randm, and the rresndn Jaban matrx has a randm struture, whh makes t muh mre dffult t slve than the trdanal matrx. Fr strutured rds, the best way t rder s t rder n the smallest dmensn 9

48 frst and rder n the larest dmensn last. But fr unstrutured rds, suh as the ne shwn n Fure., t s nt s easy t dede whh knd f rdern s the best. In eneral, when we rder the ell lst f an unstrutured rd, we wsh t reserve as muh struture nfrmatn as ssble, and enerate a Jaban matrx wth the smallest ssble bandwdth. The rdern f the nnetn lst des nt nfluene the struture f the Jaban matrx, but stll, n rder t make the best ahe utlzatn n flux alulatns, t s best t ruhly fllw the rdern f the ell lst a left t rht rder b randm rder Fure. Effet f rdblk rdern n struture f Jaban matrx Fure. An unstrutured rd wth faults and multlateral wells (FlGrd,

49 . Cntrl Vlume Frmulatn and Struture f Jaban Matrx There are dfferent tyes f numeral methds fr the dsretzatn f reservr flw equatns. Tradtnally, fnte dfferene methds have been used, n reent years, wth the develment f unstrutured rds, mre and mre use s ben made f ntrl vlume methds (Aavatsmark et al., 997; Verma and Azz, 997, whh make the handln f rreular shaes muh easer. In ase f Cartesan rds, ntrl vlume methds and fnte dfferene methds rdue the same dsretzed equatns. Fnte element methds (Yun, 978; Fun et al., 99; Sukrman and Lews, 994 are anther tn, but they are nt wdely used n the etrleum ndustry. The ntrl vlume frmulatn s used n GPRS. Reservr flw equatns are a seres f nnlnear equatns, whh are nrmally slved by the Newtn-Rahsn methd, where a Jaban matrx and a rht hand sde (RHS are alulated at eah Newtn teratn. In ths setn we wll frst ntrdue the ntrl vlume mstnal frmulatn (sne mstnal smulatn s ne f the majr fuses f ths researh, then we wll shw the eneral struture f the Jaban matrx. In the fllwn several setns, we wll dsuss the alulatn f eah ndvdual art f the Jaban matrx. The ntrl vlume methd s based n mass balane ver a ntrl vlume. The mass aumulatn n a ntrl vlume s equal t the flux (mass flwn n frm ts bundary mnus the rdutn (mass rdued frm a well that maybe n the blk, njetn s neatve. Cntrl vlume uld be f any shae, fr mst unstrutured rds, t s a lyhedrn, lke the ne shwn n Fure.3. Fure.3 A eneral lyhedrn 3

50 The vlume f ths lyhedrn s V, and t has ns surfaes. The rate f mass aumulatn f mnent anbeexressedas ACC V φ n ( S n t n φ ( S n (. where, s the mle fratn f mnent n hase. The ttal flux s the sum f the fluxes arss eah f ts ns surfaes, whh an be wrtten as ns Flux Flux s, (. s Flux arss surfae s s alulated as Flux s, ( v s (.3 where, v, s t s equal t s the hase vlumetr flw rate arss surfae s. Ardn t Dary s law, v, s ( Tλ Φ s (.4 where, Ts the transmssblty nstant arss surfae s, and Φ, s s the hase tental dfferene arss surfae s. By substtutn Eqn..4 and Eqn..3 nt Eqn.., we et the exressn fr the ttal flux: Flux ns s [ T ( λ Φ ] (.5 s s The mnent mass rdued frm a well an be exressed as W W Q ( q (.6 where, q W W q s the hase vlumetr rdutn rate frm the well, whh an be wrtten as W W WI λ ( (.7 3

51 where, W WI s the well ndex (Peaeman, 978, a nstant, the well blk and W s the hase ressure f s the wellbre ressure fr the well n the well blk. Substtutn Eqn..7 nt Eqn..6, we et W W W Q WI [ λ ( ] (.8 Cmbnn Eqn.., Eqn..5 and Eqn..8, we et the fnal ntrl vlume mstnal frmulatn: V φ { n ns s [ T s ( S ( λ n n φ Φ t (.9 s ( S ] WI W n [ λ ( W ]} n, n Nte that eah f the terms n the flux art and the well art an be evaluated at ether tmeste n r tmeste n, deendn n the mltness f the mdel. Nrmally there s nly ne well assn thruh a ntrl vlume, f there are mre, all f the well fluxes are summed. Usn the Newtn-Rahsn methd, the Jaban matrx s enerated by alulatn the dervatves f Eqn..9 wth reset t all f the unknwns (varables. In GPRS, the Jaban matrx s alulated and stred searately fr the reservr art (aumulatn and flux terms and fr the well art, and t s later eed tether nly n the lnear slver mdule. The struture f the reservr art f the Jaban s shwn n Fure.4a. Basally t nssts f three arrays, Da, OffD_A and OffD_B. Da s the danal art, eah f ts entres s fr ne rdblk, and the latn f eah entry fllws the ell lst. OffD_A rerds the sarse struture f the uer tranular art, and OffD_B rerds the sarse struture f the lwer tranular art. Eah f ther entres s fr ne nnetn, and the latn f eah entry fllws the nnetn lst. Fr examle, fr a nnetn between rdblk A and B (A<B, OffD_A stres the dervatve f equatns f rdblk A wth reset t varables f rdblk B, and t s lated at (A, B f the Jaban matrx. Smlarly OffD_B at (B, A ntans the dervatves f equatns f rdblk B wth 33

52 reset t varables f rdblk A. Fr mult-nt flux, OffD_A and OffD_B wll nlude mre entres, we wll dsuss ths n Setn.4. Fr Jaban alulatns f the reservr art, the aumulatn art and the flux art are further searated. The aumulatn art ls thruh the ell lst, and t nly adds terms n the Da array; the flux art ls thruh the nnetn lst, and t ntrbutes t all three arrays. Fr mult-mnent system, eah f the entres n these arrays s a small dense matrx. The eneral struture f the well art f the Jaban fr a snle well s shwn n Fure.4b. Addtnal wells ntrdue addtnal rws and lumns. Here besdes the reservr varables, eah well may ntrdue an extra well varable, whh s nrmally the well bttm hle ressure (BHP. Frm Fure.4b, we an see that the well art f the Jaban nssts f fur arrays, RRJ, RWJ, WRJ and WWJ. RRJ reresents the dervatves f well terms n Eqn..9 wth reset t the reservr varables, whh are always lated at the danal, and nly have entres at the well blks. RWJ reresents the dervatves f well terms n Eqn..9 wth reset t the well varable. Ths extra well varable s nly neessary fr sme tyes f wells, suh as wells under nstant rate ntrl, and n that ase, an extra well equatn s needed t mlete the system. WRJ and WWJ reresent the dervatves f ths extra well equatn wth reset t the reservr varables and the well varable resetvely. Eah entry n RRJ s nrmally a small dense matrx, fr RWJ, t s a lumn vetr, fr WRJ, t s a rw vetr and fr WWJ, tsa snle value. OffD_A OffD_B Da RRJ RWJ WRJ WWJ a reservr b well Fure.4 General struture f the Jaban matrx 34

53 In rder t make the struture f the Jaban matrx learer, a smle tw-well examle s nluded here. The rd s shwn n Fure.5. Well s mleted n rdblks and, and well s mleted n rdblks 6, 3 and 0. The rresndn Jaban matrx fr ths system s shwn n Fure.6, where A reresents a term enerated by the aumulatn art, F reresents a term enerated by the flux art, and W reresented a term enerated by the well art. The lr n Fure.6 reresents the strae, and same lred tems are stred n the same array Fure.5 A smle unstrutured rd wth wells 35

54 W W W W AFW F F F W F AFW F F F W F AF F F F AF F F F F F AF F F F F F F AF F F F F F AF F F F AF F F F F F F AF F F F F F F AFW F F W F AF F F F F F AF F F F F F F AFW F F F W F F AF F F F AF F F F F AFW F W F F F AF W W W W W W W Fure.6 Jaban matrx struture fr the system n Fure.5 36

55 .3 Netwrk Mdeln In Chater, we have mentned that there are tw knds f methds fr rd reresentatn, the blk based arah and the nnetn based arah (Lm et al., 994, als alled netwrk mdeln. Tradtnally, the blk based arah has been used, where the nnetns (fluxes are assated wth blks, but the number f nnetns fr a rdblk may vary deendn n the emetry f the rdblk, as n the ase fr unstrutured rds. In addtn, lal rd refnement may nrease the number f nnetns. As a result, eah rdblk may have dfferent number f nehbrs. An array keen trak f the nneted rdblks has t be dynamally dmensned fr eah rdblk, (r dmensned fr the larest ssble ase. Fure.7 shws a smle examle f the nventnal rdblk trakn array. The number f nnetns fr eah rdblk vares between tw and fur. Frm Fure.7, we an see that the blk based nnetn nfrmatn s als redundant, eah f the nnetns atually aears tw tmes. The number f nnetns s even hher f a nne-nt sheme s used (Yansk and MCraken, 979; Sammn, 99. In eneral, the blkbased arah bemes very nnvenent fr these ases Blk Cnnetns,3,4,5 3,4,6 4,3,5,6 5,4,7 6 3,4,7 7 5,6 Fure.7 Grdblk nnetn array when the blk based arah s used The mutatn f flux terms s nly neessary when ars f rdblks are nneted. Furthermre, t s bvus that a stve flux nt a ven rdblk autmatally mles a neatve flux fr the ther nneted blk. Ths means that the flux s nly a rerty f ndvdual nnetns, nt rdblks. S the key dea n the nnetn based arah s t fus n nnetns nstead f rdblks when 37

56 mutn flux terms. The frst majr task n the nnetn-based arah s t establsh an array ma r data struture ntann all ssble ars f nnetns. As an examle, the rdblk nnetns n Fure.7 an be rearraned nt a lst f nnetn ars, shwn n Fure.8. The frst rdblk ndex n a nnetn ar s always smaller than the send rdblk ndex, n rder t avd redundany. One the nnetn array has been establshed, the mutatn alrthm n the flux art s exatly the same fr any dmensnal rblems, sne a nnetn s basally a lne nnetn tw nts, whh s always ne-dmensnal. Ths nnetn-based arah s very nvenent fr unstrutured rds, nstead f trakn the number f nnetns fr eah rdblk, we trak the nnetns fr the whle dman Cnnetn N. Cnneted ars,,3 3,4 4,5 5 3,4 6 3,6 7 4,5 8 4,6 9 5,7 0 6, 7 Fure.8 Cnnetn array when the nnetn based arah s used The nnetn-based arah s used n GPRS, and the fllwn mdfatns are made t make t mre sutable fr unstrutured rd smulatn and mdeln f wells and surfae faltes: AddnelllstTradtnally, rdblks (ells are reresented by three-dmensnal (, j, k ndes, nstead f that, a ne-dmensnal ndex (ell lst s used n GPRS, whh an be alulated as ( j k n y nx fr strutured rds, where nx and n y are number f rdblks n x and y dretns. Dn that, the mutatnal alrthm n the aumulatn art s exatly the same fr any dmensnal rblem, and the whle smulatr bemes ndeendent f the rd tye and dmensns. Ths 38

57 arah als saves sme memry by redun a three-dmensnal ndexn array t a ne-dmensnal ndexn array. Reservr nly nnetn lst In Lm et al. (994, the nnetns are between tw ndes, eah nde an be a reservr nde (rdblk, a well nde (wellbre r a surfae nde (surfae falty, whh s the mst eneral nnetn based arah. But the reservr, the well and the surfae faltes, eah have ther wn haratersts. Ths leads t dfferent knds f alulatns and dfferent knds f strutures n the Jaban matrx fr eah art f the system. Hene the nnetn lst n GPRS s nly fr the reservr art, the well art and the surfae falty art are treated searately. The advantae s that the struture f the Jaban matrx s reserved, and ths struture an be used later by the lnear slver t mrve ts effeny. Ths arah als makes the alrthm learer and easer t understand. The redure fr the reservr art f the Jaban matrx and resdual alulatn usn ths mdfed nnetn based arah s as fllws:. Frst, l thruh the ell lst, and evaluate the aumulatn terms and ther dervatves fr eah rdblk. The aumulatn terms are assned t the resdual, and the dervatves are assned t the danal f the Jaban matrx at the rresndn rdblks.. Next, l thruh the nnetn lst, alulate the ustream dretn fr eah hase and evaluate the flux terms and ther dervatves fr eah nnetn. The flux terms are added t the resdual f the nneted blks, ne s stve and the ther s neatve, the dervatves are added t the rret latns n the Jaban matrx, as shwn n Fure.9 fr a tw-nt flux alulatn. Flux F AB s frm rdblk A t rdblk B, A and B are the varables at rdblk A and B, and F A and F B are the FAB FAB FAB equatns at rdblk A and B. and - are stred n Da, and s F stred n OffD_A,and- AB A A s stred n OffD_B. B B 39

58 Jaban RHS A B A B F A F AB A F AB B F AB F AB F B F AB A F AB B F AB Fure.9 Jaban struture fr tw-nt flux dervatves.4 Mult-Pnt Flux Reresentatn and Imlementatn In the last deade, wth the develment f unstrutured rds and usaln methds, mre and mre smulatns are usn nn-rthnal rds and full tensr ermeabltes. Fr these knds f smulatns, mult-nt flux arxmatns (Gunasekera et al., 998; Verma and Azz, 996 are neessary. At the same tme, muters are bemn mre werful, and new tehnques (e.. Edwards, 998 are ben develed t redue the st f mult-nt flux smulatn, bth f them make mult-nt flux arxmatn mre ratal. The nnetn-based arah was rnally rsed fr tw-nt flux. We eneralze t t mult-nt flux. In ths setn, frst we wll mare the dfferenes between tw-nt flux and mult-nt flux alulatns, then we wll shw the alrthm used fr mult nt flux alulatns. Tw-nt flux and mult-nt flux are dfferent n three areas: reresentatn, ustream dretn alulatn and Jaban matrx struture. Reresentatn In tw-nt flux, a nnetn between tw rdblks nly deends n these tw rdblks, enerally, eah nnetn an be reresented n C by nt A, nt B, duble T, whh means that there s flw (nnetn between rdblks A and B and the transmssblty nstant fr ths nnetn s T. Twnt flux an be alulated as Flux _ P, TAB ( λ A B ( Φ, B Φ, A (.0 40

59 But n mult-nt flux, a nnetn between tw rdblks may deend n n (> rdblks. Fr strutured rds, n s fxed, suh as n6 fr tw-dmensnal Cartesan rd and n8 fr three-dmensnal Cartesan rd (Verma, 996. But fr unstrutured rds, t s nt fxed, and there s n uer lmt t t. A mult-nt unstrutured nnetn s reresented by nt n, nt *A, duble *T n GPRS, whh means that there s a nnetn between rdblks A[0] and A[], and ts flux deends n n rdblks, whh are A[0], A[] A[n-], and the transmssblty nstant assned t eah rdblk s T[0], T[] T[n-]. Mult-nt flux an be alulated as Flux MP n ( λ ( T[ l] Φ (. _, A[0] A[], A[ l] l 0 There s n flw thruh a nnetn when hase tentals are equal, n rder t uarantee that, the transmssblty nstants shuld satsfy n l 0 T[ l] 0 (. Usn Eqn.. t remve T[0] frm Eqn.., we an et Flux n ( λ ( T[ l] ( Φ Φ (.3 _ MP, A[0] A[], A[ l], A[0] l Eah ndvdual term n the summatn f Eqn..3 s alled a sub-nnetn. Cmarn Eqn..3 wth Eqn..0, we nte that they are qute smlar. If n, and we defne A[0] as A, A[] as B, and T[] as T, then Eqn..3 redues t Eqn..0. Ths s beause tw-nt flux s just a seal ase f mult-nt flux, where eah nnetn nly has ne sub-nnetn. Ustream dretn alulatn In Eqn..0 and Eqn..3, λ shuld be evaluated usn ndtns n the ustream blk fr eah nnetn and fr eah hase. In tw-nt flux, the hase ustream dretn s slely determned by the hase tental dfferene, and λ s evaluated as 4

60 ( λ A B ( λ ( λ A B,, f f Φ Φ, B, B Φ Φ, A, A < 0 (.4 0 n In mult-nt flux, the hase ustream dretn s determned by ( T[ l] Φ n r ( T [ l] ( Φ, A[ l] Φ, A[0],and λ s evaluated as l ( λ ( λ ( λ n A[0] l 0 A[ 0] A[] n A[],, f f l 0 ( T[ l] Φ ( T[ l] Φ, A[ l], A[ l] < 0 0 l 0, A[ l] (.5 Nte, λ an nly take the value at rdblk A[0] r A[], nt the ther rdblks. Eqn..5 an be redued t Eqn..4 f there are nly tw nts (T[] s always stve fr flux between A[] and A[0]. Jaban matrx struture If a mult-nt flux arxmatn s used, we wll have many mre ff danal terms n the Jaban matrx, and the mutatnal st wll be muh hher. Fr strutured rds, bands fr the Jaban matrx wll nrease frm 5 t 9 n tw dmensn, and frm 7 t 7 n three dmensn. Fr unstrutured rds, the struture f the Jaban matrx bemes mre mlated wth mre nn-zer terms. We have revewed the dfferenes between tw-nt flux and mult-nt flux, next we wll shw the reservr art f the Jaban matrx and resdual alulatn fr the mult-nt flux arxmatn. Ths s dne by ln thruh the nnetn lst. Fr eah nnetn, we frst alulate the hase ustream dretns, then l thruh ts sub-nnetns and evaluate the flux terms and ther dervatves fr eah subnnetn. The flux terms are added t the resdual f the nneted blks (A[0] r A[], ne s stve and the ther s neatve, the dervatves are added t the rret latns n the Jaban matrx, as shwn n Fure.0 fr a mult-nt flux alulatn. Flux F AB s frm rdblk A t rdblk B, and ne f ts sub-nnetns s related t rdblk C. A, B and C are the varables at rdblk A, B and C. F A and 4

61 F B are the equatns at rdblk A and B. and F AB C F are stred n OffD_A, and- AB A F AB A F and - F and - AB C AB B are stred n Da, F AB are stred n OffD_B. Nte,n the mult-nt flux alulatns, the number f entres n OffD_A and OffD_B equals the ttal number f sub-nnetns, and OffD_A s n lner strtly uer tranular and OffD_B s n lner strtly lwer tranular. Jaban RHS B C A B F AB F A F B A F AB F A AB A B F AB B F AB B C F F AB C AB C F AB F AB Fure.0 Jaban struture fr mult-nt flux dervatves.5 Well Treatment We have dsussed hw the reservr art f the Jaban matrx and RHS are alulated n GPRS, nw we wll dsuss the well art. Frm Setn., we knw that the well art f the Jaban nludes fur arrays, RRJ, RWJ, WRJ and WWJ. RRJ and RWJ are the dervatves f well terms n flw equatns wth reset t the reservr varables and the well varable resetvely. WRJ and WWJ are the dervatves f the extra well equatn wth reset t the reservr varables and the well varable resetvely. RRJ s always needed, whle the exstene f RWJ, WRJ and WWJ deends n well ntrl. In ths setn, we wll frst dsuss the alulatn f RRJ, then we wll dsuss when an extra well equatn and a rresndn well varable are needed. Well equatns used n GPRS fr the blak-l and the mstnal mdels wll be resented next. RRJ always exsts, and t s a blk danal matrx wth nly nn-zer terms at the well blks. After t s alulated, t an be dretly added t the reservr art f the 43

62 44 Jaban wthut ntrdun any extra nn-zer terms. In Setn., the well equatn was wrtten as W W W WI Q ] ( [ λ (.8 where, W WI s the well ndex, s the hase ressure f the well blk and W s the wellbre ressure. If ths well has multle erfratns, we need t relate W t the ressure at a referene deth. We assume varable radent and n frtn n GPRS, and use the fllwn relatn t alulate the wellbre ressure at well blk : l l l W l W l W W W W W W W W W W D D D D D D D D 0 ( ( ( ( ( ( ( ( (.6 Here the referene deth s defned at the frst well blk, and the wellbre ressure at ths deth ( W 0 s nrmally alled the well bttm hle ressure (BHP. W s the ttal flud densty wthn the wellbre at well blk, whh an be alulated as (Nlen, 990 W W W q q, ( (.7 W q, s the hase vlumetr rdutn rate f ths well at well blk, whh s alulated frm Eqn..7. Cmbnn Eqn..8, Eqn..6 and Eqn..7, we an easly see that, f the well bttm hle ressure W 0 s fxed, well term wll nly deend n the reservr varables. If needed, the well bttm hle ressure wll be the extra well varable. RRJ s the dervatves f Eqn..8 wth reset t the reservr varables fr a fxed well bttm hle ressure. The exstene f RWJ, WRJ and WWJ deends n whether we need an extra well equatn and well varable. Fr a BHP ntrlled well, where W 0 s fxed, Eqn..8 nly deends n the reservr varables, RWJ s zer, and there s n need fr WRJ and WWJ.

63 RRJ by tself s already mlete. After we slved fr the reservr varables, the well hase rdutn rates an be exltly alulated frm Eqn..7. Fr ther tyes f well ntrls, suh as nstant hase vlumetr flw rate ntrl (at well head r well bttm, the BHP f a well s nt fxed, we have the extra well varable, and we need an extra well equatn t mlete the system. The well equatn deends n bth the desred ntrl at the well and the flud mdel (blak-l r mstnal. Here we nly revew the well equatns used n GPRS. Currently, nly tw tyes f well ntrls are mlemented n GPRS, BHP ntrl and nstant hase vlumetr flw rate (at standard ndtns ntrl. An extra well equatn and varable are needed fr nstant hase flw rate ntrlled wells. The extra well varable s always the well bttm hle ressure, the extra well equatn s dfferent fr blak-l and mstnal mdels, and bth f them are based n mnent mass balane wthn the well bre. As shwn n Fure., there s flw frm the well bttm t the well head, and the well uld have multle erfratns. Well head Well bttm Fure. Wellbre flw 45

64 Assumn steady state flw n the well, the mnent mass flw rate at the well bttm (summatn fr all well blks shuld be equal t the mnent mass flw rate at the well head, whh an be exressed as W WH WI λ Q (.8 l l, l, l,, l (, l l where, WH Q s the mass flw rate f mnent at the well head. Sne the extra well equatn s dfferent fr blak-l mdel and mstnal mdel, they are dsussed searately, usn a nstant l hase flw rate ntrlled well as an examle. Blak-l mdel Cmnents n blak-l systems are defned as hases at standard ndtns (well head. In rder t avd nfusn between mnents and hases, we wll use a suersrt - t ndate mnents whenever neessary. At the well head, blak-l mnents and hases are equvalent t eah ther, and the mnent mass flw rate shuld equal the hase mass flw rate, whh an be exressed as Q WH WH WH WH Q q, SC q (.9 WH Substtutn Eqn..8 nt Eqn..9 and rearrann, we et l WI λ l, l, l, SC,, l (, l W l q WH 0 (.0 WH where, q s the fxed l hase vlumetr flw rate at the well head, and, SC s the l mnent densty at the standard ndtn, whh s an nut rerty. Fr blak-l mdels, Eqn..0 s the extra well equatn used n GPRS. Cmstnal mdel Thns beme a bt mre mlated fr the mstnal mdel. Here, well head s atually a searatr whh erates at standard ndtns. The nut t ths searatr s the ttal mass flw frm the well (summatn f WH Q ver all, the utut s the ndvdual hase flw at standard ndtns. Fr nstant l hase flw rate, we have the fllwn relatn fr the l hase mass flw rate: SP WH SP SP l Q q (. 46

65 Substtutn Eqn..8 nt Eqn.. and rearrann, we et l SP W l SP WI lλ, l, l (, l l ( q 0 (. SP where, SP q s the fxed l hase vlumetr flw rate (utut f the searatr, and SP s the l hase densty at the searatr, and l SP mle fratn at the searatr. T et SP and mnent mle fratns at the searatr are frst alulated as z Q WH n h WH s the hydrarbn lqud (l SP l, the verall hydrarbn / Q (.3 then a flash alulatn s erfrmed at standard ndtn and fxed mstnal mdels, Eqn.. s the extra well equatn used n GPRS. z. Fr Water mnent s fully searated frm the hydrarbn mnents n GPRS. S, fr fxed water flw rate ntrlled wells, the extra well equatn s always l WI λ l w, l w, l ( w, SC, l W l q WH W 0 (.4 Fr njetrs, eah hase mblty s relaed by the ttal mblty r the njeted hase vlume fratn wehted hase mblty (Almehadeb and Azz, 989; Nlen, 990; Cats et al., 998. In GPRS, fr multhase njetn the mblty f the njeted flud s alulated as the vlume fratn ( E f that flud multlyn the ttal mblty f flud n the njetr blk (Almehadeb and Azz, 989, whh an be wrtten as nj λ E λ (.5 The mnent mass balane wthn the wellbre s nly fr the njeted mnents, and the ttal mass flw rate s nly the summatn ver the njeted mnents, rresndnly RRJ and RWJ are nly alulated fr the flw equatns f the njeted mnents. 47

66 Smetmes, after the reservr art f the Jaban and the well art f the Jaban have been mbned, RWJ an be elmnated wthut ntrdun any extra terms, suh as fr snle blk wells wth nstant rate ntrl. Wth ths, we an slve fr the reservr varables frst and exltly udate the well varable later. The detals f ths eratn are shwn n Setn 3.. Fr mult-blk wells wth nstant rate ntrl, RWJ an als be elmnated, but t wll ntrdue sme extra nn-zer terms n RRJ. Fr ths reasn, t s referable nt t erfrm elmnatn n RWJ fr ths knd f wells..6 Flash Calulatn and Treatment f Phase Dsaearane and Reaearane Flash alulatns are neessary n mstnal smulatn, and fr ths we need t selet an equatn f state and hase equlbrum relatns. In ths setn, frst we wll dsuss the equatn f state and hase equlbrum relatns used n GPRS, then we wll dsuss the tw dfferent rles that flash alulatns lay n GPRS, fnally we wll shw the treatment f hase dsaearane and reaearane n GPRS. The Cub equatn f state used n GPRS an be wrtten as (e.., Nhem et al., 983; Walas, 985 Z 3 sz qz r 0 (.6 s ( u B where, q A ( w u B ub 3 r AB wb wb Parameters u and w deend n the ub equatn f state seleted. Fr the Pen- Rbnsn EOS (Pen and Rbnsn, 976, u and w-. Z s the hase mressblty fatr, whh s ne f the three rts f Eqn..6. Fr the l hase, the mnmum real rt s used, and fr the as hase, the maxmum real rt s used. At hase equlbrum, we have the fllwn relatn: f f,,, n h (.7,, where, f, and f, are the fuates f mnent n the l and as hases. Fr the Cub equatn f state they are alulated as (Walas,

67 f Φ,, e, r, (.8 where, Φ a b b ( Z ln( Z B b n A b ( B u 4w b h h j n h n,, ( k,, b j a a j, a a j Z B( u ln[ Z B( u and u u a R T b B RT A 4w ] 4w (.9 (.30 Aendx C defnes all arameters and rvdes the frmulas and dervatves used fr flash alulatns. Eqn..7 s used n tw ways n GPRS, as a nstrant equatn t relate the sendary varables t the rmary varables and as a searate flash rutne t hek the state f hydrarbn hase n a blk. These tw tasks are dsussed next. As a nstrant equatn Flw equatns are the rmary equatns, they deend n bth the rmary varables and the sendary varables. T frm a mlete system, we need sme nstrant equatns t remve the sendary varables frm the rmary equatns. Phase equlbrum relatns (Eqn..7 are ur sendary equatns, whh als deend n bth the rmary varables and the sendary varables. They an be used as the nstrant equatns t buld relatns between the sendary varables and the rmary varables. Ths rle fr flash alulatns s nly neessary when bth hydrarbn hases exst n a rdblk. Fr rdblks wth nly ne hydrarbn hase, half f the hase varables dsaear, and the rmary equatns deend nly n the rmary varables. Fr examle, fr a rdblk wth nly as and water hases, the mass balane equatns wll nly deend n, and y,,, n. When ths s the ase, there s n need fr nstrant equatns. h As a searate flash rutne Fr rdblks wth nly ne hydrarbn hase, the rmary equatns deend nly n the rmary varables, and there are n need f S 49

68 sendary equatns. But smetmes, a snle hydrarbn hase an hane bak t bth hydrarbn hases, and the sendary equatns are needed aan. S we need t mntr the state f hydrarbn hases when a rdblk has nly ne hydrarbn hase. Ths s dne by a searate flash rutne. A smle suessve substtutn flash rutne s used n GPRS, whh s smlar t the ne shwn n Subsetn... T seed u the nverene, the values f Newtn teratn are used fr the ntal uess. K and l at the ld Frm the abve dsussn, we an see that flash alulatns lay dfferent rles under dfferent stuatns. In rder t dstnush these stuatns, we need t detet the dsaearane and reaearane f hydrarbn hases. Fr mdels usn the natural varables (varable Tye A, treatment f hase dsaearane and reaearane s als needed fr hsn the rret hase varables. Phase dsaearane and reaearane are deteted n GPRS and treated as desrbed belw: Phase dsaearane When a rdblk has tw hydrarbn hases, fr eah Newtn teratn, we hek the saturatn slutns. If ether S r S s neatve, the rresndn hydrarbn hase has dsaeared. In ths ase, befre ntaln the next Newtn teratn, we frst set the neatve saturatn t zer, then assn the verall hydrarbn mnent mle fratns ( z t mnent mle fratns n the l r as hases ( x and hase rdblk. y, fnally mark ths rdblk as a snle hydrarbn Phase reaearane When a rdblk has nly ne hydrarbn hase, at the end f eah Newtn teratn, we d a neatve flash (Whtsn and Mhelsn, 986, ressure and verall hydrarbn mnent mle fratns are the nuts. If l s between 0 and, then bth hydrarbn hases exst, and befre ntalzn the next Newtn teratn, we frst assn the hydrarbn hase saturatns as Sw S ( l S w (.3 S S (.3 50

69 then set mnent mle fratns n the l and as hases ( x and y frm the results f flash alulatn, fnally mark ths rdblk as a bth hydrarbn hase rdblk. In mstnal smulatns, flash alulatn s a very mrtant art, and t uld aunt fr u t 30% f the ttal runnn tme. The flash rutne used n GPRS s qute smle, and a mre rbust and effent flash rutne s needed..7 Lnear Slver In GPRS, the Jaban matrx s alulated and stred searately fr the reservr art and fr the well art, and t s later eed tether nly n the lnear slver art. Ths arah basally searates the slvn f the lnear system frm the buldn f the lnear system, and reates a searate lnear slver mdule. All we need t d s t transfrm the Jaban struture f GPRS nt the sef frmat that a lnear slver requres. In ths way we an easly nterfae wth avalable lnear slvers, suh as BltzPak (Landmark, 998, we an als devel and use ur wn slvers. Bth dret and teratve slvers are used n GPRS. The dret slvers nluded are a full matrx slver and a band matrx slver, bth are qute slw and memry nsumn, and an nly be used fr small systems. Fr teratve slvers, we use a Generalzed Mnmum Resdual (GMRES (Saad and Shultz, 986 slver, whh s sutable fr any sarse matrx system. Fr lare rblems, teratve slvers are memry effent and muh faster than dret slvers. Fr teratve slvers, rendtners are used t seed u the nverene. The erfrmane f teratve slvers deends n the rbustness f the rendtners. The fllwn rendtners are nluded n GPRS, rdered ardn t ther rbustness: a danal saln; b blk danal saln; Inmlete LU demstn (ILU0 (e.. Behe and Frsyth, 983; d Alebra Mult-Grd (AMG (Stueben, 983, whh s nly sutable fr the ressure system; and e Cnstraned Pressure Resdual (CPR (Walls et al., 985, whh s ur best rendtner. GPRS als has an nterfae t BltzPak, whh s a mmeral lnear slver akae frm Landmark Grahs Crratn. BltzPak s nly sutable fr strutured rds, where the 5

70 Jaban matrx has u t 7 bands, whle GMRES (Saad and Shultz, 986 n GPRS s sutable fr any rd and any sarse matrx system. Fr GPRS, BltzPak s the fastest fr strutured rds, and CPR rendtned GMRES s the best fr unstrutured rds. Fr the rendtners used n GPRS, nly CPR s seally desned fr reservr smulatn equatns, and t s als the best rendtner n GPRS. CPR s a tw stae rendtner. In the frst stae we deule the full system nt a ressure art and a saturatn art usn an IMPES-lke deuln ress, and slve fr ressure frm the ressure art (AMG rendtned GMRES s used fr the ressure slve, then we use the ressure slutns t redue the resdual f the full system. In the send stae we start frm the redued resdual and wrk n the remann deuled system, and use a lal rendtner, suh as ILU0, t et the full slutns fr the redued resdual. The fnal slutn s the frst stae ressure slutns lus the send stae full slutns. CPR erfrms very well fr all knds f rblems, nludn rblems that are hard fr IMPES. CPR s mre sutable fr fully mlt systems. 00 GPRS BltzPak y3e-05x Trend (BltzPak Trend (GPRS tme (se y 8E-06x E03.E04.E05.E06 number f unknwns Fure. Perfrmane marsn between GPRS slver and BltzPak 5

71 Here we mare the erfrmane f CPR rendtned GMRES (GPRS slver wth the erfrmane f BltzPak n dfferent sze rblems. The results are shwn n Fure.. We use a Cartesan rd, 4 mnent mstnal rblem. The rblem s slved fully mltly. In Fure., we rvde the trendlnes usn wer relatns. Fr the GPRS slver, the wer s abut.08, fr BltzPak, the wer s abut.5. The sts f bth slvers are ruhly lnear wth the number f unknwns, whh s the best we an et fr a lnear slver. The nstant fr the GPRS slver (3E-5 s larer than that fr BltzPak (8E-6. Ths dfferene s due t the fllwn tw reasns: frst GMRES n GPRS des nt make use f the 7 band struture f the Jaban matrx, send the mnents f the GPRS slver are frm ubl dman sftware, whh may nt be very effent. Anyway, fr lare mdels, GPRS slver nly takes abut 50% mre tme than BltzPak. Cnsdern that the GPRS slver s sutable fr any rd system and any sarse matrx, we an nlude that the slver n GPRS s reasnably effent and rbust. In the slver art f GPRS, mst f the mnents are frm dfferent ubl dman sftware, suh as the dret slvers are frm LAPACK (Andersn, 999, GMRES s frm Iteratve Math Lbrary (IML (Dnarra et al., 996, ILU0 s frm SarseLb (Pz et al., 996, a sarse matrx akae, and AMG s frm GMD sftware (Rtzdrf, 99. The slvers and rendtners n GPRS are arrate fr fully mlt and IMPES systems (CPR s used t rendtn the fully mlt system, and AMG s used t rendtn the IMPES system. But, fr an adatve mlt system, suh as AIM, mst f the rdblks are IMPES, nly a small number f rdblk are fully mlt, CPR lses ts effeny (sends t muh tme n the ressure slve at the frst stae, whle the whle system s nly a lttle bt larer than the ressure system, and AMG s nly sutable fr the ressure system. Mre wrks s needed t devel sutable slvers fr AIM methd. Fr unstrutured rds, the rdern f the ell lst s qute mrtant fr the effeny f lnear slvers, as shwn n Setn.. Currently, there s n rerdern dne t the ell lst n GPRS. Addtnal wrk s needed t devel d rdern methds fr the ell lst. 53

72 .8 Tmeste Cntrl and Reservr Intalzatn Fully mlt mdels are unndtnally stable, and we an have a tmeste f any sze. But n rate, n rder t ntrl the number f Newtn teratns fr eah tmeste, a reasnable tmeste sze s requred. Fr mdels that are nt fully mlt, suh as IMPES, ther maxmum tmeste szes are lmted by ther stablty. Setn 3.3 derves all f the stablty rtera used n GPRS. In GPRS, a maxmum tmeste sze tma and a mnmum tmeste sze tmin sefed by the user, and the tmeste szes fr all mdels must satsfy Eqn..33 at all tmes: t t MIN t MA are (.33 The nreasn r dereasn f tmeste sze frm tmeste n t tmeste n s verned by the fllwn relatn (Azz and Settar, 979: η δ ωη n n ( ω t t mn ver (.34 where refers t the rdblk number, η s the sefed desred hane, δ s the n hane ver t and ω s a tunn fatr wth a value between 0 and. Nrmally η and δ have dfferent values fr dfferent varables, s a dfferent varable and the mnmum f all used n GPRS are: t n η 00 Psa, fr ressure η 0., fr saturatns S η 0.0 t n s alulated fr eah s used fr the next tmeste. Tyal values f η fr mnent mle fratns The tmeste szes f fully mlt mdels are deded slely by Eqn..33 and Eqn..34. The tmeste szes f artally mlt mdels are verned nt nly by Eqn..33 and Eqn..34, but als by ther stablty rtera. At the start f smulatn, we need t assn ntal values t reservr varables (ressure, saturatns and mnent mle fratns. Fr blak-l mdels, the 54

73 redure mentned n Azz (996 s used, and t uarantees ntal equlbrum f reservr fluds. Fr mstnal mdels, the fllwn redure s used:. Assn the same ntal ressures (ressure at a referene deth t all rdblks.. Assn ntal water saturatns fr all rdblks. If a rdblk s belw the water-l ntat (WOC, assn, therwse assn nnate water saturatn. 3. Assn ntal verall mstns z fr all rdblks. Intal verall mstns are nutted as an verall mstn vs. deth table, and a table lk u s erfrmed fr eah rdblk t dede ts ntal verall mstns. 4. Assn ntal mnent mle fratns n the l and as hases, x and y fr all rdblks. A flash alulatn s erfrmed at nstant ressure, temerature and verall mstns z fr eah rdblk. The resultn x and y are assned t eah rdblk. At the same tme, the l and as hase denstes, and,andthe hydrarbn lqud mle fratn l are alulated fr eah rdblk. 5. Assn ntal l and as saturatns fr all rdblk as Sw S ( l S w (.35 S S ( Reassn ntal ressures fr all rdblk ardn t deth: 0 0 ( S ( D D (.37 where 0 s the ressure at referene deth 0 D. If neessary, ste 4-6 are reeated t aheve better ntal equlbrum. The effet f allary ressure s nred n ths redure. If the ntal verall mstn vs. deth table reflets real ndtns n the reservr, the alulated ntal equlbrum s d. Otherwse the ntal ndtn wll be far frm equlbrum, suh as fr the ase when the same ntal verall mstn s used fr all deths. 55

74 56

75 Chater 3 General Frmulatn Arah One f the bjetves f ths researh s t evaluate the erfrmane f dfferent sthermal mstnal mdels. In rder t make the mlementatn f these mdels easy and nsstent, a new General Frmulatn Arah s develed, whh an be used t derve any knd f mdel (dfferent equatn and varable seletns and dfferent mlt levels. Wth ths arah, we an mnmze dfferenes amn dfferent mdels, qukly hane frm ne mdel t anther mdel, and evaluate ther erfrmane n a nsstent manner. Ths General Frmulatn Arah als enables us t devel and try ut new mdels, and t s als sutable fr ther tyes f smulatrs, suh as thermal smulatrs. Ths new arah s desned t slve a eneral nnlnear equatn set. In ths study, we have nly exlred ts alatn t sthermal mstnal mdels. T better understand the stes nvlved n ths General Frmulatn Arah, t s wrthwhle t frst revew sme f the mrtant nts abut mstnal smulatn dsussed n Chater : Reservr smulatn nvlves a lare number f uled nnlnear equatns. Cmlexty f the mathematal mdel nreases dramatally frm blak-l t mstnal mdels. Fr mstnal mdels, dret slutn f ths lare system (full set f equatns and varables s almst mssble, due t the hh mutatnal and memry requrements. There s n need t slve all f the uled equatns tether. Fr eah tye f smulatn, there s a mnmum number f ndeendent varables and equatns (rmary varables and equatns that determne the thermdynam state (ntensve and extensve f the system. Fr sthermal mstnal system, ths thermdynam state s determned by n varables, r the number f mnents n the system. By slvn fr ths mnmum number f varables frst (fully mlt 57

76 mdels, we an save a lare amunt f mutatnal tme. Even s, ths arah s stll t exensve fr lare rblems. In fully mlt (FIM mdels all f the rmary varables are treated mltly. T further redue the st f smulatn, we an treat sme rmary varables exltly, and redue the number f mlt unknwns er rdblk. Nrmally, suh mdels are muh faster than FIM mdels, eseally fr rblems wth lare number f mnents. Hwever, exlt treatment f sme rmary varables wll lmt the maxmum tmeste sze, f the maxmum allwable tmeste sze s t small, suh as less than ne day, suh mdels uld be even slwer than FIM mdels. Exlt treatment f sme rmary varables may als ause sme balane errrs (mass r vlume. There are numeral tehnques t redue r swth the balane errrs (Cats et al., 998, but we annt ttally remve them. Full Set f Varables and Equatns Prmary Set Sendary Set Imlt Exlt Fure 3. Redutn ress frm full set t mlt rmary set In eneral, we shuld start wth the full set f equatns and varables, frst redue them t the rmary set (mnmum number, then redue the mlt level f the rmary set t further seed u the smulatn f neessary, as shwn n Fure 3.. Ths tw ste redutn ress s nluded n the General Frmulatn Arah. The full set f equatns and varables are the startn nt fr ths arah. In Chater, we already nted ut that, the full set f equatns are always the mass balane equatns and the hase equlbrum relatns, but the seletn f the full set f varables s nt unque. GPRS uses the natural varables as ts bas full set f varables t 58

77 faltate the alulatn f dervatves. If ther sets f varables are needed, suh as Tye B varables, we frst buld exlt relatns between the new varables and the natural varables, then use the han rule t swth frm the natural varables t the new varables. Ths s atually the frst ste f the General Frmulatn Arah. In summary, ths General Frmulatn Arah nludes three stes. Frst, we defne a base mdel whh uses the natural varables, and buld ther mdels frm ths base mdel by varable swthn. Then, we redue the full set t the rmary set, and buld the FIM mdel. Fnally, f neessary we further redue the mlt level f the rmary set. Thruh these three stes, we ntrl the varables used, and the mltness f eah varable. These three stes are erfrmed rdblk by rdblk, s eah rdblk an have dfferent varables and dfferent mlt levels. Ths rvdes a natural way t frm adatve mlt shemes, suh as the AIM mdel (Frsyth and Sammn, 986. In ths hater, we wll frst take a lk at the eneral nnlnear equatn set that frms the bass f ths arah. Then we wll exlan the eratns nvlved n eah f the abve three stes. Next we wll d sme stablty analyss fr the exlt treatment f sme rmary varables. Fnally we wll dsuss the nsequenes f usn dfferent knds f equatns and varables. 3. The General Nnlnear Equatn Set In rder t make ths arah sutable fr any knds f smulatns, we base t n the mst eneral nnlnear equatn set, whh an be exressed as F ( 0 (3. where F s the full set f equatns, and s the full set f varables. Ths full set f equatns and varables an be slt nt a rmary art and a sendary art: F (, s 0 (3. Fs (, s 0 where, subsrt stands fr rmary, and subsrt s stands fr sendary. 59

78 F 0 s the rmary equatn set, and we always selet the mass balane equatns fr t. Its Jaban matrx struture (dervatves f the mass balane equatns wth reset t the full set f varables lks lke the ne shwn n Fure.4 and Fure.6, here we have a reservr Jaban and a well Jaban. The reservr Jaban has bth danal terms and ff-danal terms. The well Jaban nludes fur arrays, RRJ, RWJ, WRJ and WWJ, andnrmallyrrj s added t the danal art f the reservr Jaban. F s 0 s the sendary equatn set, we selet the hase equlbrum relatns (a seres f nnlnear nstrant equatns fr t. Sne t nly deends n ndvdual rdblks, ts Jaban matrx wll be a blk-danal matrx. Cmbnn the Jaban matrx f F and F s, we have the Jaban matrx fr the full set f equatns and varables, whh s shwn n Fure 3.. danal Jaban RHS A B... A B A RW M C D N A B... A B M 0 0 C D N A WR B WR A WW M WW Fure 3. Struture f the full set f Jaban matrx and RHS Here we defne the fllwn matres: F F A, B, F s C, s D F s, (3.3 s and vetrs: M F, N Fs. (3.4 60

79 6 where A, B and M are enerated by the rmary equatns, and C, D and N are enerated by the sendary equatns. Eah matrx r vetr has ts wn strae array, and the strae fr matres A and B s further searated nt three arts (Da, OffD_A and OffD_B as shwn n Setn.. In ths General Frmulatn Arah, we erfrm eratns n ths full set Jaban matrx, elmnate terms t zer level by level, and fnally extrat a mnmum set fr the lnear slver. All f the eratns are dne lally, n addtnal strae s requred. Ths General Frmulatn Arah s aled t sthermal mstnal mdels n GPRS. We assume that the hydrarbn mnents and water are ttally searated, then 0 F wll be the mass balane equatns fr eah hydrarbn mnent and fr water: [ ] 0 ( ] ( [ (,, Φ Φ W W W l l q y q x y x T y S x S V t F λ λ φ (3.5,, h n 0 ( ( (,, Φ W w W w l l w w w w w w q T S V t F λ φ (3.6 Otnally, we an relae ne f the hydrarbn mnent mass balane equatns by a ttal hydrarbn mass balane equatn: [ ] 0 ( ] ( [ (,, Φ Φ W W W l l h q q T S S V t F λ λ φ (3.7 0 s F wll be the hase equlbrum relatns: 0,, e f f F,,, h n (3.8 Ths full set f equatns s used fr all f the sthermal mstnal mdels.

80 The bas full set f varables,, S, y, S, x,,, n h - base, are hsen t be whh are atually the natural varables, r Tye A varables. Fr ther varables, a swth frm the natural varables t the new varables s requred. Here we have a ttal f n h varables n n h nnlnear equatns. Callary ressure nstrants are used t remve water and l ressures, the saturatn nstrant s used t remve water saturatn, and mnent mle fratn nstrants are used t remve the mle fratns f the last hydrarbn mnent n the l and as hases. Blak-l mdels are treated as seal ases f the eneral mstnal mdel, where sendary equatns and varables are n lner needed and mnent mle fratns are ether nstants r nly funtns f ressure, refer t aendx B fr detal. Fr thermal mdels, F wll nlude the enery balane equatn, and wll nlude temerature r nternal enery. S far, we have shwn the base mdel (the full set f equatns n terms f the base varables, whh s als the startn nt fr the General Frmulatn Arah. In the fllwn setn, we wll exlan the three stes f ths arah. Fr eah ste, the eratns fr the reservr art and fr the well art are dsussed searately. Nte that RRJ f the well Jaban s already added t the danal art f the reservr Jaban, s t wll nt aear n the fllwn dsussn. 3. General Frmulatn Arah Stes The frst ste f the General Frmulatn Arah s t swth frm the base varables t the new varables, and ths ste s nly neessary when varables ther than the natural varables are used. 6

81 3.. Swthn Frm the Natural Varables t Other Varables In GPRS, the base mdel uses the natural varables, and t s strahtfrward t alulate F dervatves wth reset t these varables, and they are. If anther mdel usn a base new set f varables s desred, then we must mute dervatves wth reset t the F new varables. We an d ths by the han rule: F F base (3.9 where base base an be alulated frm the relatn between and base.thewhle swthn ress s summarzed belw:. Buld exlt relatn ( base between and base, and alulate the base transfrmatn matrx TRAN ( fr eah rdblk. base F F. Calulate the new dervatves by TRAN. Ths s dne fr all terms n the reservr Jaban and fr WRJ n the well Jaban, and the results are stred F bak n. base 3. Redue the full set system t the fnal mlt rmary set system (ste and 3 f the General Frmulatn Arah 4. Slve the new lnear system and udate exlt varables, et δ and udate the new υ υ varables by δ 5. Udate the base varables base by slvn the exlt relatn ( base,whh may be nnlnear n sme ases. base Dfferent mdels wll have dfferent transfrmatn matres due t dfferent relatns between the new varables and the base varables, but the abve fve stes wll be exatly the same. Next we wll revew examles f new varables, and exress eah f base 63

82 them n terms f the base varables. Based n a revew f exstn mstnal mdels, the ssble hes fr (besdes the base mdel varables are dentfed as: S w, x n h, nh z, r, r w, F l water saturatn y, mle fratn f the last hydrarbn mnent n the l and as hases verall mle fratn f eah hydrarbn mnent verall densty f eah hydrarbn mnent verall densty f water mnent verall densty f hydrarbn mnents hydrarbn lqud (l mle fratn and ther relatns t the base varables are ven belw: S w S S (3.0 x x n h n h nh y y n z r r w h S x S S S S x S w ( S S F S S l S S S y y (3. (3. (3.3 (3.4 (3.5 (3.6 (3.7 where, w w (,, x,, xn and, y,, yn ( h ( h Nte that ths ress s equvalent t the use f han rule t et the dervatves n Tye B mdels. S t shuld nt ntrdue any extra wrk. Fr Tye A mdels, all f the transfrmatn matres are dentty matres, and ths ste s nt neessary. Besdes the base mdel, a Tye B mdel s mlemented n GPRS, where the full set f varables s 64

83 65, r,,, h n - w r, S,and y,,, h n -. Wthn ths set,, r,,, h n - and w r are the rmary varables. The fllwn relatns are used t alulate the transfrmatn matrx: w w y y S S S S r y S x S r ( (3.8 The dervatve f the verall mnent denstes wth reset t the base varables are alulated as x S x S x r y S y S y r x S r y S r y S x S r δ δ,,,,, (3.9 and 0 ( w w w w w w w x r y r S r S r S S r (3.0 The fnal struture f the nverse f the transfrmatn matrx ( TRAN s shwn n Fure 3.3. The transfrmatn matrx s alulated as the nverse f TRAN n GPRS,

84 reardless f ts nternal struture. But, fr matres wth seal strutures, lke the ne n Fure 3.3, there are mre effent ways t alulate ts nverse. S S y ' x ' r S x S y y x Sδ ' y S, ' y Sδ ' x S, ' x r w S w ( S S w w y I 0 Fure 3.3 Struture f the nverse f the transfrmatn matrx fr a Tye B mdel Nw that we have the full set f equatns and varables, and we als have dfferent hes f varables, the next ste s t redue the number f equatns and varables t a set that s slved tether. Frst we redue t t the mnmum number ( n f equatns and varables requred t establsh the thermdynam state (the FIM mdel, then we an further redue the mlt level. 3.. Redue Full Set F ( 0 t Prmary Set F ( 0 In rder t wrte an teratve ress t slve F ( 0, we uld use the Newtn- Rahsn methd: F υ υ υ υ ( δ F (, and δ (3. r we an exress F ( n Taylr seres: F F n υ υ υ 0 ( F ( F ( ( δ (3. 66

85 67 Eqn. 3. and Eqn. 3. are equvalent. In the fllwn analytal dervatn, Taylr seres are used t derve the rmary set. In addtn, we wll als shw a matrx manulatn methd, whh an aheve the same fnal al as the analytal dervatn methd. The matrx manulatn methd s used n GPRS, and we wll dsuss t n mre detal later. Analytal Dervatn In the last setn, we have defned the fllwn matres: F A, s F B, s F C, s s F D and vetrs: F M, s F N where matres A, B, C and D tether frm the full set Jaban matrx fr Eqn.3., and M, N are the rresndn rht hand sdes (RHS fr the rmary art and fr the sendary art, as shwn n the frst frame f Fure 3.4. The frst ste s t relate the sendary varables t the rmary varables. We start wth the sendary equatn set 0, ( s s F. Use f Taylr seres n ths equatn set yelds s s s s s s s s s D C N F F F F δ δ δ δ υ υ υ υ υ υ υ ( (, (, ( 0 (3.3 Rearrann Eqn. 3.3, we et s C D N D δ δ υ υ ( ( (3.4 The send ste s t wrte the rmary equatns n the rmary varables. Use f Taylr seres n the rmary equatn set 0, ( s F yelds

86 68 s s s s s B A M F F F F δ δ δ δ υ υ υ υ υ υ υ ( (, (, ( 0 (3.5 Substtutn Eqn.3.4 nt the abve equatn, we et ] ( [( 0 C D N D B A M δ δ υ υ υ υ υ (3.6 Rearrann terms, we have υ δ υ ( ( N BD M C BD A (3.7 Ths s the fnal rmary system, whh nly nludes the rmary varables as unknwns. After we slve fr the rmary varables, the sendary varables an be udated exltly frm Eqn. 3.4 rdblk by rdblk. We an aheve the same al by erfrmn matrx manulatn dretly n the full set Jaban matrx and RHS. Matrx Manulatn (Gaussan Elmnatn Here we erfrm Gauss elmnatn n the blks (A, B, C, D and M, N f the full set Jaban matrx and RHS. The al s t elmnate B t zer, then the rmary equatns wll deend nly n the rmary varables. Ths s llustrated n Fure 3.4. Fure 3.4 Redue full set t rmary set by Gaussan elmnatn A B C D M N A B D C I M D N 0 B A- B D C B D C M- B D N B D N (3 ( ( A B B D C B M B D N

87 Frst, we multly the sendary equatn set by D -, then we multly t aan by B, and fnally we subtrat the sendary equatn set frm the rmary equatn set. The fnal rmary equatn set an be extrated and wrtten as ( A BD C δ ( M BD N (3.8 whh s the same equatn as Eqn We an als extrat a sendary equatn set frm the send frame f Fure 3.4, after rearrane t, we have υ υ δ ( D N ( D C δ s (3.9 whh s the same equatn as Eqn The three eratns n Fure 3.4 are erfrmed fr eah rdblk and fr eah art f the Jaban matrx. Frm Fure 3., we knw that matres A and B exst n the danal art, the ff-danal art and the well art f the full set Jaban. Matres C and D nly exst n the danal art. We need t elmnate all arts f B. Nte that matres A, B, C and D are always n the same lumn (they are the dervatves wth reset t the varables n the same rdblk, A, B and M are n the same rw (equatns fr the same rdblk, and C, D and N are n the same rw. Dfferent arts f B are elmnated as dsussed belw: Danal art f B (Da array L thruh the ell lst, selet the danal A, B, C and D fr the same blk and ther rresndn RHS, M and N, and erfrm the three ste elmnatn ress. In addtn, D - s alulated and stred bak n D, D - C and D - N are alulated and stred bak n C and N t avd reeatn alulatns and usn extra memry sae. Off-danal art f B (OffD_A and OffD_B arrays L thruh the nnetn lst, fr eah A and B n the ff-danal art, selet the C and D whh are n the same lumn as A and B. M s rresndn t A and B, and N s rresndn t C and D. Only the last ste f the elmnatn s needed here, sne D - CandD - N have already been alulated. Ths ress needs t be dne fr bth the OffD_A array and fr the OffD_B array. 69

88 70 Well art f B If a well has the WRJ array, fr eah A WR and B WR n the WRJ, selet the C and D whh are n the same lumn as A WR and B WR.M WW s rresndn t A WR and B WR, and N s rresndn t C and D. Only the last ste f the elmnatn s needed here. If a well enetrates multle rdblks, there wll be several A WR and B WR, and the elmnatn ress wll be erfrmed fr eah f them. T make the ress learer, fr the system shwn n Fure 3., the fllwn eratns are erfrmed t redue B t 0: Danal art eratns: D D D D C D C C D C N D N N D N ( C D B A A ( N D B M M ( C D B A A ( N D B M M Off-danal art eratns: ( C D B A A ( N D B M M ( C D B A A ( N D B M M Well art eratns: ( C D B A A WR WR WR ( N D B M M WR WW WW where ( C D B A A means that ( C D B A s alulated and stred bak n A. The udated A and M an later be extrated fr fully mlt slve r fr further redutns. After the abve eratns, the A and M f the full set Jaban and RHS n Fure 3. are redued t the rmary set Jaban and RHS (fully mlt. The struture f the resultn Jaban s shwn n Fure 3.5.

89 danal Jaban RHS A... A A RW M A... A M A WR A WW M WW Fure 3.5 Struture f the rmary set Jaban Matrx and RHS After slvn fr the rmary varables tether, the sendary varables f eah rdblk are udated exltly by δ s, ( D N ( D C δ δ D N ( D C δ s, (,, We have mentned n Chater that, fr snle blk wells wth nstant rate ntrl, we an deule the reservr Jaban frm the well Jaban by elmnatn RWJ, then we an slve fr the reservr varables frst and exltly udate the well varable later. If the well n Fure 3.5 nly has snle blk enetratn and RWJ exsts, the fllwn addtnal eratns are erfrmed n GPRS: A A A RW ( A WW A WR M M A RW ( A WW M WW After the rmary reservr varables have been slved, the well varable s udated by δ A M ( A A δ W ( WW WW WW WR, Here, Newtn teratns are erfrmed n bth the rmary equatns and the sendary equatns. If the sendary equatns are nly used as nstrant equatns, as mentned n Chater, then all f the N's n the abve equatns wll be zer. 7

90 Sme varatns n the eneral eratns are neessary as dsussed belw. If the transmssblty s treated exltly, then there s n need t elmnate ffdanal arts f B, beause flux terms nly have dervatves t ressure, whh s always a rmary varable, and all ff-danal arts f B are zer. Matrx D s atually the Jaban matrx fr flash alulatns. When we udate the sendary varables, t s equvalent t dn ne ste f flash alulatns fr eah Newtn teratn. Fr exat flash alulatns at eah Newtn ste, we an use t as the ntal uess fr flash. Fr the blak-l mdel, there are n sendary varables, and ths ste s nt needed. Fr mstnal mdels, f a rdblk has nly ne hydrarbn hase, then the rmary equatns wll nly deend n the rmary varables as shwn n Setn.6, and ths ste s als nt neessary. S far, we have derved the fully mlt Jaban matrx and RHS. Fr fully mlt smulatns we an just st here. But n a lt f stuatns, we desre t redue the mlt level t seed u the alulatns. The next subsetn wll shw hw ths s dne Redue the Imlt Level f the Fully Imlt System F ( 0 T redue the mlt level f the fully mlt system, we frst slt the rmary varables nt an mlt art and an exlt art, and set the dervatves f flux terms wth reset t the exlt varables t zer, then we deule the mlt varables frm the exlt varables. The send ste s basally a deuln ste. Ths ress needs t be dne fr eah rdblk that s nt fully mlt. After the mlt varables have been slved fr tether, the exlt varables an be udated exltly rdblk by rdblk. In ths art, we wll dsuss the varus deuln methds avalable, ther dfferenes, and whh ne s used n GPRS. Fr eah rdblk, the fully mlt Jaban matrx s shwn n Fure 3.6, whh s ne rw f the matrx equatn n Fure

91 danal V V V 3 P Off-danal V V V 3 P RWJ W RHS Fure 3.6 Fully mlt Jaban matrx f ne rdblk Here the left art s the Jaban matrx A, whh nludes a danal art, several ffdanal arts and several well arts (RWJ. The rht art s the rht hand sde M. P s the ressure r the mlt rmary varables, V,V and V 3 are the exlt rmary varables. We an redue the mlt level f the fully mlt system n Fure 3.6 n tw stes:. Frst, set all flux term dervatves wth reset t the exlt varables t zer, r smly dn t alulate them. Sne the ff-danal arts f A are the flux term dervatves, ths treatment wll leave arts (dervatves wth reset t the exlt varables f the ff-danal A t zer. Danal A wll als have sme hane, but t wn t be zer. Ths s llustrated n Fure 3.7. danal V V V 3 P Off-danal V V V 3 P RWJ W RHS Fure 3.7 Jaban matrx f ne rdblk after exlt treatment f flux terms. Next, deule the mlt varables frm the exlt varables. There are dfferent knds f deuln methds avalable, tw ular hes are Husehlder refletn (QR demstn and lal nversn (Larx et al., 000. Cats 73

92 (999 has nted ut that the IMPES ressure equatn s unque, ndeendent f the manner f dervatn, he and rdern f varables and equatns. Based n t, we an nlude that the he f deuln methds des nt have t muh nfluene n the fnal mlt system. Here we nly revew the tw mst ular deuln methds, Husehlder refletn and lal nversn: Husehlder refletn The ress f usn Husehlder refletn (Larx et al., 999 s summarzed n Fure 3.8. We start frm the system n Fure 3.7, and mve eah art (dark retanles n Fure 3.8 nt a matrx r lumn vetr. Then we erfrm a QR demstn (AQR, where Q s an rthnal matrx, Q - Q T,and R s an uer tranular matrx n the danal matrx, and use the transse f Q t multly all f the matres and vetrs. After that, the danal matrx bemes R, whh nly has values n the uer tranular art, the last equatn wll be nly n terms f the mlt varables, and t s extrated and assed t the lnear slver. After t s slved, the exlt varables an be easly udated by makn use f the uer tranular struture f the danal matrx. The same results an be aheved by erfrmn Gaussan elmnatn n the system n Fure

93 danal V V V 3 P Off-danal V V V 3 P RWJ W RHS A ' A RW A M A QR R ' Q A Q A RW Q M danal V V V 3 P Off-danal P RWJ W RHS Fure 3.8 Husehlder refletn deuln 75

94 Lal nversn The ress f usn lal nversn (Larx et al., 000 s summarzed n Fure 3.9. The ntatn used here fllws the nrmal ntatn used fr blak-l IMPES mdels, subsrt stands fr ressure r mlt varables, and subsrt s stands fr saturatn r exlt varables. Here we als start frm the system n Fure 3.7, but we d an addtnal slttn ste t searate the mlt art frm the exlt art, and mve eah art (dark retanles n Fure 3.9 nt a small matrx r lumn vetr. Then we use the redure n Fure 3.4 t redue As t zer. Fnally the send equatn an be extrated and slved fr mlt varables. After that, the exlt varables are udated frm the frst equatn. Nte that we erate n the matres and vetrs n the same rw (beln t the same rdblk. Lke the redure used n Fure 3.4, t s dne searately fr the danal art, the ff-danal art, and the well art (RWJ. danal V V V 3 P Off-danal V V V 3 P RWJ W RHS Ass As ' A s RW A s M s A s A ' A RW A M As A s A ss A s A s A ss A ' s A s A ss A RW s A s A ss M s 0 A s A A ss A s A s A ' A ss A ' s A s A RW A ss A RW s A s M A ss M s Fure 3.9 Lal nversn deuln 76

95 Fr the IMPES mdel, numeral exerments (Vasslevsk, 00 shw that fr mst ases, the ressure system frm Husehlder refletn s mre symmetr than the ressure system frm lal nversn, whh s better fr the lnear ressure slver. But n GPRS, we use the lal nversn methd anyway, beause ths redure s smlar t the redure used n Fure 3.4, and n addtnal wrk s needed t mlement t. The methd used fr deuln des nt make muh dfferene, sne the fnal mlt systems are always equvalent. danal A ss As Jaban... ' A s RW A s RHS M s As A ' A RW A M Fure 3.0 Illustratn fr lal nversn T make ths ste learer, fr the system shwn n Fure 3.0, the fllwn eratns are erfrmed t redue Danal art eratns: A A M s A A s M A ss ( A ( A s s A A ss ss A s M As s t0ngprs: Off-danal arts eratns: A ' A ' ( As Ass As Well arts (RWJ eratns: A RW A ( A s A ss A RW s A, ' A, RW A and M frm the fnal mlt system, they are extrated and assed t the lnear slver. After ettn the mlt varables, the exlt varables are udated exltly by 77

96 δ ex A ss ( M s A s δ m A ' s δ ' m A RW s δ W Cats et al. (995 rsed usn IMPES redutn fatr t enerate the ressure system. Basally t s the alatn f the redure n Fure 3.9 fr the ase where nly the ressure s treated mltly, suh as n IMPES. In that arah, eah equatn (ne rw n Fure 3.7 s multled by a salar, and the resultn n equatns are added, and the IMPES redutn fatr s a vetr nsstn f these salars. The values f these salars are determned s that the addtn remves all f the varables exet ressure. The IMPES redutn fatr s atually the Frmulatn Arah. Fr IMPES, t s a vetr, sne As Ass As matrx n the General s a rw vetr and matrx. Cats (999 shwed that IMPES redutn fatr leads dretly t the value f ttal mressblty n multhase rdblks, and ths n turn rvdes an errr hek fr blak-l PVT data, suh as f the alulated ttal mressblty s neatve, then there must be sme nnssteny n the blak-l PVT data. Befre any deuln, arts f the transmssblty are treated exltly. Fr eah exlt art, we an ether fx t at the ld tme level r udate t teratn by teratn. The frst arah has less numeral dffusn. The send arah s atually an nexat Newtn methd, whh un nverene ves the same slutn as the FIM methd. Frm Eqn..5, we an see that transmssblty has three nn-nstant arts (hase mblty, hase densty and mnent mle fratn, and eah art an be treated dfferently. Fr eah rdblk, we have the fllwn fur tns fr the treatment f transmssblty n GPRS: Otn All three arts are fxed at the ld tme level, and nly dervatves wth reset t the mlt varables are alulated. Ths tn s used fr the IMPES mdel. Otn Cmnent mle fratns are fxed at the ld tme level, hase mblty and hase densty are udated teratn by teratn, and nly dervatves wth reset t the mlt varables are alulated. Ths tn s used fr the IMPSAT mdel, where bth ressure and saturatn are treated mltly, and mnent mle fratns are treated exltly. Ass s a 78

97 Otn 3 All three arts are udated teratn by teratn, and nly dervatves wth reset t the mlt varables are alulated. Ths tn an be used fr any mdels, but t erfrms rly fr IMPES, whh s atually the IMPES mdel rsed fr an adatve mlt sheme by Thmas and Thurnau (983. Russell (989 shwed that n rder t make ths methd stable, the CFL number must be less than Otn 4 All three arts are udated teratn by teratn, and all dervatves are alulated reardless f the mlt level. Ths tn s used fr the FIM mdel. Fr the IMPES and IMPSAT mdels, ths tn an als be used, the redutns n mlt level are erfrmed fr all nn-zer terms, and fnally sme terms are nred t frm the mlt system. The rresndn mdels are nrmally alled quas IMPES and quas IMPSAT mdels. Generally, quas IMPES s nt used as a frmulatn, and t s nly used as a way t enerate a ressure system, whh an be used later fr rendtnn urse. In summary, the rsed General Frmulatn Arah an be used t btan any desred frmulatn, reardless f the tye f mdel, tye f varables and level f mltness. All f the eratns are erfrmed n ndvdual rdblks, s the extensn t the use f dfferent mlt levels fr dfferent rdblks (Adatve Imlt Methd, AIM s strahtfrward. Ths arah als allws the use f dfferent mdels fr dfferent reservrs (reservrs are nly nneted thruh well rus and bundary nterfaes, and eah reservr s handled by a snle ressr n arallel mutatn. Ths General Frmulatn Arah nly defnes the eneral stes and methds t reah the fnal al, and t s sutable fr any mdel. But n rder t make the smulatr mre effent, we need t nsder the detals f eah sef mdel. Fr examle: In many ases, matres are banded r sarse, we shuld nly erfrm eratns n the nn-zer terms. Fr sme mdels, suh as varable Tye B mdels, whh use z as the rmary varables, we knw the analytal frm f aumulatn term dervatves wth reset 79

98 t them, and we an just byass the varable swthn art, and dretly alulate them frm the start. Water equatn nly has dervatves wth reset t ressure and saturatns, whh are bth rmary varables fr the base mdel. Ths means that water equatn s n terms f the rmary varables nly frm the start, and we need nt d any redutn fr t, as ln as bth ressure and saturatns are mlt. The matrx C s enerated frm the hase equlbrum relatns, and t des nt have any dervatve wth reset t saturatns, s sme lumns f C are always zer. We an take advantae f ths, and save sme alulatns. These are sme eneral nsderatns that redue alulatns, and they are far frm mlete. Furthermre, eah sef mdel has ts wn haratersts, and we shuld use them t further smlfy alulatns. Hwever, all f the eratns f ths arah are f frst rder t the number f rdblks, s the st shuld be small mared t the st f lnear slver, eseally fr lare mdels. Wth rer tunn, the seed f ths General Frmulatn Arah shuld be marable wth that f mmeral smulatrs. 3.3 Stablty Analyss After the sendary varables are elmnated frm the rmary equatns, we et the FIM mdel, whh s unndtnally stable. In rate, n rder t ntrl the number f Newtn teratns fr eah tmeste, a reasnable tmeste sze s requred, nrmally t s between 30 days and 00 days. If we further redue the mlt level f the rmary set, the exlt treatment f rmary varables wll make the mdel ndtnally stable, and the tmeste sze wll have t satsfy sme stablty rtera. In real smulatns, we desre t use the maxmum allwable (stable tmeste sze, and stablty analyss s needed t rretly dentfy t. The natural varables are used n GPRS, wthn them, ressure s always treated mltly, and saturatns and mnent mle fratns an be treated ether mltly r exltly. Next, we wll derve the stablty rtera due t the exlt treatment f saturatns and mnent mle fratns. Assumn ne-dmensnal tw- 80

99 8 hase (as/l flw, n sure and snk, the hydrarbn mnent mass balane equatns an be wrtten as ( ( y u x u x y S x S t φ φ (3.30,, h n where u and u are the hase veltes fr the l and as hases. Summn Eqn ver all mnents, we et the ttal hydrarbn mass balane equatn: ( ( u u x S S t φ φ (3.3 By assumn nmressble flw, n ravty and nstant vssty, the ttal flw velty T u bemes a nstant, and Eqn. 3.3 an be smlfed as x f u x f u u f u f x t S t S T T T T φ φ ( (3.3 x S ds df u x f u t S T T ( ( ( φ (3.33 where f and f are the fratnal flws f the l and as hases, and they are nly funtns f saturatn. After rearrann Eqn. 3.33, we et x S ds df u t S T ( φ (3.34 Eqn s the Bukley-Leverett wave equatn, ts stablty s determned by the CFL number, whh an be wrtten as, T x S ds df x t u CFL φ (3.35 Fr three-dmensnal flw, the ttal CFL number s the summatn f CFL numbers n all three dretns (Russell, 989:

100 8,,,,,, T z z T y y T x x T z S y S x S S ds df V t q ds df z A t q ds df y A t q ds df x A t q CFL CFL CFL CFL φ φ φ φ (3.36 where T q s the ttal vlumetr flw rate, x T q, y T q, and z T q, are the vlumetr flw rates aln x, y and z dretns, and z T y T x T T q q q q,,,. V s the vlume f a rdblk, and z A y A x A V z y x. The hysal meann f Eqn s that a hase frnt an nt mve mre than ne rdblk n a stable tmeste. Cats (00 derved the same stablty rtera fr the eneral mult-dmensnal three-hase flw, nludn bth ravty and allary ressure. Fr tw-hase (as/l flw wthut allary ressure, t an be redued t S q ds d q ds d V t CFL λ λ λ λ λ λ λ λ φ (3.37 By nrn ravty, we have T T T T q q f q q q f q λ λ λ λ λ λ (3.38 Substtutn Eqn nt Eqn and rearrann, we et / ( ( ( ( ( T T T T S ds df V t q ds d V t q ds d ds d ds d V t q ds d ds d V t q CFL φ λ λ λ φ λ λ λ λ λ λ λ λ φ λ λ λ λ λ λ φ (3.39

101 83 whh s the same equatn as Eqn Eqn s used as the stablty rtern fr the exlt treatment f saturatns n GPRS. If any mnent mle fratn s treated exltly, we an et ts stablty rtern frm Eqn Assumn nmressble flw and nstant equlbrum rat K,we have x S ds df x y u x y u K u t S x y t y S K S T ( / ( ( / ( φ φ φ (3.40 Substtutn Eqn nt Eqn. 3.40, we an elmnate the send terms n bth sdes, and after rearrann, we et x y y S x S y u x u t y φ φ (3.4 Dvdn by K n bth sdes f Eqn. 3.4, we et x x y S x S y u x u t x φ φ (3.4 The stablty rtern fr bth Eqn. 3.4 and Eqn. 3.4 s, x y S x S y u x u x t CFL φ (3.43 Smlarly, fr three-dmensnal flw, the stablty rtern s y S x S y q x q V t CFL φ (3.44 Cats (999 mentned that the stablty rtern due t the exlt treatment f mnent mle fratns s w T S q V t CFL φ, whh s nly a smlfed versn f Eqn fr S 0 r S 0. In Cats new aer (Cats, 00, the same stablty rtern as develed here (Eqn was rsed, but he t t by frst assumn

102 q 0 r q 0, and ettn the stablty rtern fr the ase where nly snle hydrarbn hase an flw, then fnally eneralzn t t the ase where bth hydrarbn hases flw. Our dervatn s mre eneral. The hysal meann f Eqn s, fr a stable tmeste, the amunt f mnent flwn ut f a rdblk an nt exeed the amunt f mnent n lae n that rdblk. In ther wrds, a mnent an nt mve mre than ne rdblk n a stable tmeste. Eqn s fr eah ndvdual mnent, and t nly needs t be satsfed when ether y r x s treated exltly. Eqn s used as the stablty rtern fr the exlt treatment f mnent mle fratns n GPRS. We wll further analyze and mare these tw stablty rtera (Eqn and Eqn n Chater Influene f Equatn and Varable Seletns Cmstnal mdels are dfferent n bth ther mlt levels and ther equatn and varable seletns. In the last setn we have dsussed the stablty rtera due t the exlt treatment f varables. In ths art, we nsder the nfluene f equatn and varable seletns n the effeny f mstnal smulatns. Frst we dsuss the effet f seletn rmary equatns, then we mare the dfferenes between dfferent varable seletns. Fr eah we wll dsuss ther nfluene n Newtn teratns and lnear slver teratns Seletn f Prmary Equatns In reservr smulatn, we an selet dfferent equatns, but n matter what equatns we selet, eah equatn s basally a lnear mbnatn (frst assn a effent t eah equatn, then sum them u f all f the equatns shwn n Subsetn... Fr examle, the ttal mass balane equatn s the summatn f all f the mnent mass balane equatns, n ths ase the vetr f effents s (,,,. If we nsder the rdern f equatns, a new rdern an be vewed as a seal lnear mbnatn f the ld rdern. Suh as, effents (0, 0, 0,, 0 mve the furth equatn t the urrent latn. Equatn seletn s dne rdblk by rdblk, the Jaban matrx and the RHS fr ne rdblk an be wrtten as 84

103 A δ B (3.45 where A s ne rw f the full Jaban matrx A, B s ne rw f the RHS B, and s all f the unknwns. Any mbnatn f equatns f rdblk an be reresented by re-multlyn Eqn by a matrx n bth sdes: M ( A δ M B δ (3.46 Eqn s nly fr ne rdblk. Equatns fr ndvdual blks an be mbned t btan the fnal full system: M ( A δ M B (3.47 where, M M M M n, A A A A n,and B B B B n Eqn an be reast as ( M A δ ( M B (3.48 Here, the Jaban matrx hanes t MA, the RHS hanes t MB, but the slutn vetr s stll δ. The slutn vetr s always δ n matter what the M s, the same slutn s uaranteed as ln as Eqn 3.48 s slved aurately. Same slutns fr lead t the same resduals at eah Newtn teratn. Sne the nverene f Newtn teratns deends n the redutn f resduals, the number f Newtn teratns wll stay the same. In ther wrds, equatn seletn wll nt nfluene Newtn teratns. On the ther hand, the hane f the Jaban matrx wll nfluene lnear slver teratns, beause A and MA have dfferent ndtn numbers. Alternatvely, we an vew Eqn as a left rendtnn f the ld lnear system A δ B,where M s the rendtnn matrx. In ths ase M s a blk danal matrx, s t s sme knd f blk danal saln left rendtnn. In ths sense, the best equatn seletn s 85

104 the ne that makes the blk danals f MA as lse t a danal matrx as ssble, just as the blk danal saln rendtner des. In ther wrds, we lke t selet and aln the equatns n suh a way that equatn s mst nfluened by varable, then the blk danals f MA wll be as near t danal as ssble (danal dmnane. Based n these bservatns, we an nlude that, at the Jaban level, dfferent equatn seletns are equvalent t erfrmn dfferent blk danal left rendtnns t the ld system, whh nly nfluene lnear slver teratns and has n nfluene n Newtn teratns. A d equatn seletn shuld ruhly aln the equatns n the way that varable has the best nfluene n equatn (danal dmnane. Alyn a blk danal saln left rendtnn t the Jaban matrx uld aheve the same effet. Of urse, dfferent equatn seletns wll nfluene the amunt f wrk n alulatn the Jaban matrx and the RHS Dfferent Varable Seletns Dfferent mdels an als use dfferent varables. In ths art, we wll frst nsder the nfluene f dfferent varable seletns n Newtn teratns and lnear slver teratns, then we wll dsuss the dfferenes between usn ntensve varables and extensve varables, fnally we wll mare the erfrmane f mdels wth Tye A and Tye B varables n terms f ther mass and vlume balane errrs. In Subsetn 3.., we have mentned that we an hane the varable seletn by multlyn the equatn by a transfrmatn matrx. Ths transfrmatn matrx s fr eah rdblk, and ths ress s dne rdblk by rdblk. We an mbne all f them tether, then at the full Jaban level, ths ress lks lke ( AM ( M δ B (3.49 where, M M M M n. 86

105 Here, the Jaban matrx hanes t AM, the slutn vetr hanes t but the rht hand sde s stll B. δy M δ, The hane f the Jaban matrx wll nfluene lnear slver teratns, beause A and AM have dfferent ndtn numbers. Alternatvely we an vew Eqn as a rht rendtnn f the ld system A δ B,whereM s the rendtnn matrx. In ths ase M s als a blk danal matrx, s t s sme knd f blk danal saln rht rendtnn. Fr the same reasn as fr the seletn f equatns, t s best t aln the varables n the way that makes varable t have the best nfluene n equatn (danally dmnant. By rendtnn, we slve fr by the same slutn as fr the ld system, frst slve fr Y by υ υ Y Y δy υ υ υ υ υ δ MδY υ and we et A δ B. But fr the new varables, we must, then f eah rdblk an be udated exltly by the relatn f ( Y that was bult between and Y fr the alulatn f transfrmatn matrx. In ths ase, we wll et a dfferent slutn f the relatn f ( Y s nnlnear. Suh as fr F 0 wth ntal uess 0, after the frst Newtn teratn, we et δ 0. 5 and δ. 5,f we hane t Y, then the equatn bemes F Y 0 wth ntal uess Y 0, after the frst Newtn teratn, we have Y δ, Y Y δy and Y, and these tw slutns fr are dfferent. Dfferent slutns fr lead t dfferent resduals at eah Newtn teratn. Sne the nverene f Newtn teratns deends n the redutn f resduals, varable seletn wll nfluene Newtn teratns. If the equatns are mre lnear wth reset t the new varables, then the number f Newtn teratns wll derease (fr lnear system, nly ne Newtn teratn s needed t nvere. Varable seletn wll als nfluene the amunt f wrk n alulatn the Jaban matrx and the RHS. The natural varables are the heaest nes fr buldn a fully mlt system. Fr the same tye f varable, t s als ssble t use an extensve varable, suh as the verall mnent mlar densty r vs. the verall mnent mles m.inrdert understand the dfferene between these tw tyes f varables, we need t reall that, n matter whh knds f varables are used, the number f ndeendent varables s always 87

106 the same. Cnsder a mstnal mdel usn the natural varables, whh are all ntensve varables, after the natural varables have been alulated and fxed, all f the extensve varables f ths system wll als be fxed. Fr examle the verall mnent mle m an be alulated as m Vφ S x S ( y (3.50 But smetmes, mdels usn extensve varables use ne mre equatn and ne mre varable than neessary. Suh as fr a three-hase blak-l mdel, f we use ressure and saturatns as varables, ne f the saturatns an be easly remved by usn the saturatn nstrant equatn, sne t s a lnear equatn wth reset t saturatns. But f we use ressure and verall mnent mles m as varables, we an nt easly remve ne f the verall mnent mles, sne the relatn between them s nnlnear. Fr examle n the ase f blak-l mdels, we have B B B R w, B Vφ mw m m w (3.5 Eqn. 3.5 s atually the vlume balane equatn fr blak-l mdels. Fr nvenene ths vlume balane equatn s treated as art f the nnlnear equatn set t mlete the system when alulatn the Jaban matrx and the RHS. Ths des nt mean that the derees f freedm f ths system s 4, n fat, t s nly 3. The vlume balane equatn s nly a nstrant equatn (nly has dervatves n the danal art f the Jaban matrx, and eah f the 4 4 small matres an be redued t a smaller 3 3 matrx. Is there any beneft n usn extensve varables? Yes, smetmes, but nt t muh. If the verall mnent mles m are used, then the aumulatn terms, ( m n n m / t wll be lnear wth reset t these varables, and the mass balane s exat at eah Newtn teratn (Farkas, 997. If the verall mnent mlar denstes n n r are used, then the aumulatn term, [( φr ( φr ] V / t s nearly lnear wth reset t these varables f we nsder that rsty hanes are small wth ressure, and the mass balane wll be nearly exat at eah Newtn teratn (Farkas, 997. But f the verall mnent mle fratns z are used, then the aumulatn term, 88

107 n n [( φ z ( φ z ] V / t wll be nnlnear, sne the ttal flud densty T T T S S hanes a lt wth bth ressure and verall mnent mle fratns, ths wll ause mass balane errrs at eah Newtn teratn. In summary, f the rk mressblty s muh smaller than the flud mressblty, suh as fr as flw, a mdel usn the verall mnent mlar denstes r shuld have smlar erfrmane as a mdel usn the verall mnent mles m. If the rk s nmressble, m Vφr and the nly dfferene between r and m s a nstant salar ( V φ, and bth mdels shuld have the same erfrmane. In Chater, we mentned that there are tw tyes f varables, Tye A and Tye B. Next we wll mare the mdels usn these tw tyes f varables n terms f mass balane and vlume (saturatn balane errrs. Tye A mdels There s mass balane errr at eah Newtn teratn, due t the nnlnearty f the aumulatn terms wth reset t these varables. The vlume balane s exat at all tmes, sne the saturatn nstrant equatn s a lnear equatn. Bth balanes wll be exat when the mdel nveres at the end f eah tmeste (Farkas, 997. Tye B mdels If r and m are used, there s vrtually n mass balane errr at eah Newtn teratn, but there s a vlume balane errr, nw the saturatn nstrant equatn s a nnlnear equatn wth reset t these varables. Aan, bth balanes wll be exat when the mdel nveres at the end f eah tmeste (Farkas, 997. In reservr smulatn, mass balane s a muh mre desrable rerty than vlume balane. Fr Tye A varables, Cats (999 rsed a new arah t aheve exat mass balane at eah Newtn teratn. In eah teratn ste, the Jaban varables are used t alulate well rate and nter-blk fluxes, and the mnent mass hanes are alulated frm the mass balane equatns. As a result, there s n mass balane errr. But the mass balane errrs are transfrmed nt vlume balane errr n eah teratn ste, and the relaxed vlume balane methd s used t ntrl the vlume balane errr. Is there a strn nnetn between mass balane and Newtn teratn numbers? N. Mass balane at eah Newtn teratn deends n the lnearty f the aumulatn term. The Newtn teratn number deends n the lnearty f the flw equatns, whh 89

108 nlude three arts, the aumulatn art, the flux art and the well art. Fr exat mass balane, the aumulatn art must be lnear wth reset t the seleted varables, but the deree f nnlnearty f the flux and well arts wth reset t the seleted varables are stll unknwn. In mstnal smulatns, we are slvn the mass balane equatns and the hase equlbrum relatns tether, and bth f them are nnlnear. The Newtn teratn number deends n the nnlnearty f bth sets f equatns, the nnlnearty f hase equlbrum relatns wth reset t dfferent varables s even mre unredtable. Due t these nsderatns, we nlude that fr mstnal smulatns, the Newtn teratn number s nt strnly nfluened by the mass balane. The results n Chater 5 further valdate ths nt. Based n the abve dsussn, we an draw the fllwn nlusns fr dfferent equatn and varable seletns: Dfferent equatn and varable seletns wll hane lnear slver teratns, and a d equatn and varable seletn shuld aln the equatns and varables n suh a way that varable has the best nfluene n equatn (danal dmnane. The same effet an be aheved by alyn blk danal rendtners fr the Jaban matrx. Dfferent equatn seletns wll nt hane the behavr f Newtn teratns, whle dfferent varable seletns an hane the behavr f Newtn teratns, whh deend n the nnlnearty f the equatns wth reset t dfferent varables. N matter what tyes f varables are used, ntensve r extensve, the derees f freedm f the system are always the same. Mass balane s a ne rerty t have, but t des nt have a dret nnetn wth the nverene f the nn-lnear system (Newtn teratn numbers. Of urse, when seletn equatns and varables, the st f alulatn the Jaban matrx and the RHS shuld als be nsdered. 90

109 Chater 4 IMPSAT Mdel and IMPSAT Based AIM Mdel The General Frmulatn Arah has the advantae f allwn seletn f any level f mltness. Fr sthermal mstnal mdels, the mlt level (number f mlt varables er rdblk ranes frm n IMPES mdels t n n fully mlt (FIM mdels. Wth the nrease f mlt level, stablty nreases, and we an use larer tmeste sze wth fewer Newtn teratns. Hwever wth the nrease n number f mlt varables, the mutatnal st er Newtn teratn als nreases dramatally. In IMPES mdels, the st f eah lnear slve s fxed relatve t n, beause we always slve fr ne varable er rdblk, but n FIM mdels, the st f.~. eah lnear slve nreases as O (. The questn here s hw t balane these tw n fatrs, and selet a sutable mlt level that s fast enuh and stable enuh. T make a smulatn run fast, we need t redue the number f unknwns er rdblk frm n. Ths number s mnmum fr IMPES and maxmum fr FIM. We als lke a mdel t nly slve fr a fxed number (relatve t n f unknwns er rdblk as s the ase fr the IMPES mdel. Under these nstrants, mlt ressure and mlt saturatns aear t be desrable, beause bth varables are ndeendent f the number f mnents. Here we rse the IMPSAT mdel, where nly ressure and saturatns are treated mltly, and all f the mnent mle fratns are treated exltly. Sne ne f the saturatns an be remved usn the saturatn nstrant equatn, we nly have n unknwns er rdblk. In reservr smulatn, tyally we have tw r three hases, s the number f unknwns er rdblk fr the IMPSAT mdel s three r less. Cmared t n unknwns er rdblk fr the FIM mdel. Thus the IMPSAT mdel has the tental t yeld b savns n mutatnal st er lnear slve, eseally fr rblems wth lare number f mnents. In fat, the IMPSAT mdel s nt new, Quandalle and Savary (989 and Bran and Rdruez (995 als rsed smlar deas. In Quandalle and Savary s aer, t was 9

110 alled IMPIMS (mlt ressure and mlt saturatns mdel, and they used t t smulate a labratry exerment. In Bran and Rdruez s aer, t was alled semmlt mdel, and several small test rblems were used fr valdatn. Nether aer rvded a detaled analyss f the mdel, n stablty rtera were derved and n marsns wth the IMPES and FIM mdels were rvded. Furthermre all f the tests were n small smle rblems. In rder t rve that IMPSAT wrks n rate, mre wrk was needed n all f these areas. As a matter f fat IMPSAT s an tn n at least ne f the mmeral smulatrs, but t des nt wrk rerly. In ths study we wll shw that when rerly mlemented, IMPSAT wrks very well fr mst ases. Hefully ths wll lead t reater use f ths methd. In ths hater, we wll frst analyze varus asets f the IMPSAT mdel, then we wll ntrdue a new IMPSAT based AIM mdel, where IMPSAT frmulatn s added nt the tradtnal AIM mdel, IMPESFIM (Frsyth and Sammn, 986. We beleve that ths new AIM mdel an relae the tradtnal AIM mdel, eseally fr lare feld ase studes. 4. IMPSAT Mdel In ths setn, we wll answer the fllwn fur questns abut the IMPSAT mdel: Why des t wrk, what s ts stablty rtern, what s ts st and hw t buld t mst effently? In answern eah questn, we wll dsuss advantaes and dsadvantaes f IMPSAT mared t the IMPES and FIM mdels. 4.. Why Des IMPSAT Wrk? In rder t exlan why IMPSAT wrks, we need t take a lk at the varable uln between rdblks. Fr the natural varable seletn, we have three tyes f varables, ressure, saturatn and mle fratns. Eah f them s uled frm rdblk t rdblk due t the flux terms n the mass balane equatns. Fr ne-dmensnal flw wthut ravty and allary ressure, eah mnent flux wthn a hase an be exressed as the rdut f a transmssblty term and a ressure dfferene between tw nehbrn rdblks: 9

111 Flux, T, and T, C λ x (4. The transmssblty term has fur arts, a nstant C, a hase mblty λ whh nly deends n ressure and saturatns, a hase densty whh nly deends n ressure and mnent mle fratns, and a mnent mle fratn x. In reservr smulatn, ths transmssblty term s always evaluated at ustream ndtns. Deendn n the flw dretns, the flux term f rdblk uld deend n the varables n rdblk, and -: Flux,, Flux,, (,,, S,, S,, S,, x,,, x,,, x,, (4. Due t ths, all f the varables are uled frm rdblk t rdblk, and we need t slve fr all f them tether, as s the ase fr the FIM mdel. The deree f uln s dfferent fr eah varable. The uln between ressures exsts n bth the transmssblty art and the ressure dfferene art. Whle the uln between saturatns and between mle fratns nly exsts n the transmssblty art. If we fx the transmssblty term, the flux term wll beme a lnear funtn f rdblk ressures: Flux,, Flux,, (,, (4.3 Frm Eqn. 4.3, we an see that, ressure s always strnly uled frm rdblk t rdblk, whle saturatns and mle fratns are lsely uled frm rdblk t rdblk. The IMPES mdel s based n ths bservatn, t nres the uln f saturatns and mle fratns between rdblks by treatn the transmssbltes exltly. One ths s dne, ressures are deuled and an be muted mltly fllwed by exlt udatn f saturatns and mle fratns, rdblk by rdblk. Here, we make a new lam: the uln between mle fratns s even weaker than the uln between saturatns. Fr saturatns and mle fratns, the rats f the flux term dervatves t the aumulatn term dervatves are mared, whh are als the rat f the ff-danal term t the danal term n the Jaban matrx, and ruhly 93

112 reresent the deree f uln between rdblks fr eah varable. Fr saturatns, ths rat an be exressed as n ns x n ( Flux µ S RatS n ( (4.4 ( ACC φ x φµ Fr mle fratns, ths rat an be exressed as n S n ( Flux µ S Rat ( ( ACC φ S φµ (4.5 n Here we assume a wer relatn ( k S, n fr hase relatve ermeabltes and r nre the deendeny f hase denstes n mle fratns. Cmarn Eqn. 4.4 wth Eqn. 4.5, we an see that the uln between saturatns s arxmately n tmes strner than the uln between mle fratns. Fr three-dmensnal rblems, due t the ravty effet, the lare densty and vssty dfferene between the as and the l hases wll enhane the mvement f hase frnt, and ths uln dfferene uld be even larer. The exstene f allary ressure wll als nrease ths dfferene, sne allary ressure s assumed t be nly a funtn f saturatns. Based n these bservatns, we rse the IMPSAT mdel, where the hase densty and the mnent mle fratn n the transmssblty art are treated exltly, and the flux terms are nly funtns f ressures and saturatns: Flux,, Flux,, (,,, S,, S,, S, (4.6 In the IMPSAT mdel, bth ressure and saturatns are smultaneusly slved fr frst, then mnent mle fratns are udated exltly rdblk by rdblk. Sne we nly nre the uln between mle fratns, whh s enerally weak, the IMPSAT mdel shuld be mre stable than the IMPES mdel. Hefully ths mrved stablty wll allw arxmately the same sze tmestes as n the FIM mdel. Next we wll derve the stablty rtern fr the IMPSAT mdel and mare t wth the stablty rtern f the IMPES mdel. 94

113 4.. IMPSAT Stablty Crtern In rder t redue the number f tmestes and Newtn teratns, a mre stable mdel than IMPES s desrable. In the IMPSAT mdel, nly the mle fratns are treated exltly, s ts stablty rtern nly needs t satsfy Eqn. 3.44, whh s t q x q y CFLIMPSAT (4.7 Vφ S x S y In the IMPES mdel, bth the saturatns and the mnent mle fratns are treated exltly, ts stablty rtern shuld satsfy bth Eqn and Eqn. 3.44, whh an be wrtten as λ dλ λ dλ q q t λ ds λ ds q x q y CFLIMPES MA (, Vφ λ λ S x S y (4.8 Frm these tw equatns, we an see that, the IMPSAT mdel s always mre stable than the IMPES mdel, but by hw muh? Fr a ne-dmensnal rblem wthut ravty, Eqn. 4.7 and Eqn. 4.8 an be smlfed as qt t CFLIMPSAT V (4.9 Vφ qt t df CFLIMPES MA (, V (4.0 Vφ ds where V f ( K S ( K f, f / S, / S, when K when K when K 0 / (4. The dfferene between these tw CFL numbers deends dretly n the relatve sze f df / ds and V. We an lt these tw funtns fr a tyal fratnal flw urve 3 ( k r S, k r S, µ / µ 0 and / 0. df / ds s shwn as a funtn f S n Fure 4.. Nte that ds funtn f df / has a maxmum value f.8. V S fr dfferent K values n Fure 4., dfferent K s shwn as a values rresnd t 95

114 dfferent mnents, and lhter mnents have larer K values sne lhter mnents are mre lkely t stay n the as hase. The maxmum value n Fure 4. s arund.4. Fr ths examle, we an say that the IMPSAT mdel s abut tw tmes mre stable than the IMPES mdel. Frm Fure 4., we nte that lhter mnents have larer K and V values, s they ause mre nstablty mared t the heaver mnents. Ths s beause lhter mnents are mre lkely t stay n the as hase, whh has a muh hher flw rate than the l hase due t ts small vssty. If we nlude tw extra varables x l, y l (mnent l s the lhtest ne n the mlt varables, ths mdel (IMPSAT wll beme even mre stable. 3.5 df_ds S Fure 4. df / ds as a funtn f S 96

115 .6 V K40 K0 K0 K K S Fure 4. V as a funtn f S fr dfferene K values Fr three-dmensnal rblems, where ravty s mrtant, the lare densty and vssty dfferene between the as and the l hases wll enhane the mvement f hase frnt, and the stablty dfferene between the IMPSAT and the IMPES mdels wll be muh larer than tw tmes. Here we mare ther stable tmeste szes fr a threedmensnal (5x5x5 fur-mnent (C, C, C4 and C7 mstnal smulatn wth ne rner rduer mleted nly n the t layer (Ca, 00. Fure 4.3 shws the l saturatn at the well blk. Intally the reservr was nt at equlbrum, sne the same ntal verall mstn was used fr all deths. U t 300 days ravty dmnated the ress, ausn hase sereatn. After abut 400 days ravty sereatn s mlete and the l hase starts t ne arund the rduer. 97

116 S t(days Fure 4.3 Ol saturatn at the well blk Fure 4.4 rerts the maxmum stable tmeste sze f the IMPES mdel, whh was alulated frm Eqn Here the maxmum stable tmeste sze was arund 3 days, and durn the transtn erd between 300 and 400 days, t was smaller than days. Fure 4.5 rerts the maxmum stable tmeste sze f the IMPSAT mdel, whh was alulated frm Eqn Here, the maxmum stable tmeste sze raned frm 30 days t ver 90 days, als n rblems were bserved at the transtn erd. The rat between these tw stable tmeste szes s frm t 7 tmes. 98

117 6 4 0 dt (days t(days Fure 4.4 Maxmum stable tmeste sze fr the IMPES mdel dt (days t(days Fure 4.5 Maxmum stable tmeste sze fr the IMPSAT mdel 99

118 The results n Fure 4.4 and Fure 4.5 are fr CFL. In rate, we an use a larer CFL number, fr IMPES mdels, nrmally CFL s fne (Cats, 00. Fr the IMPSAT mdel, we have fund that CFL3 r 4 s enerally a safe lmt. Ths means that n rate, the IMPSAT mdel has even reater advantae than that demnstrated by the examle. The marsn between the IMPES and the IMPSAT mdels fr larer CFL numbers are nluded n Chater 5. In eneral, fr mst three-dmensnal rblems, the stable tmeste sze f the IMPSAT mdel an be ne rder f mantude larer than the stable tmeste sze f the IMPES mdel. We have demnstrated that the IMPSAT mdel s muh mre stable than the IMPES mdel, hw abut the st f the IMPSAT mdel mared t the sts f the FIM and IMPES mdels? In the next subsetn, we wll analyze the st f dfferent mstnal mdels Cst f IMPSAT Mdel In rder t mare the st (CPU tme f the IMPSAT mdel wth the sts f the FIM and IMPES mdels, t s neessary t analyze the st f smulatn fr mdels wth dfferent mlt levels. In mstnal smulatns, the mst tme nsumn arts fr eah Newtn teratn are the Jaban nstrutn, the flash alulatn and the lnear slve. The st f eah art deends n mlementatn detals f the ndvdual smulatrs, here we rvde sme ruh estmates fr eah mdel. The st f the Jaban nstrutn deends n bth the seletn f rmary varables and the mlt level, and t hanes snfantly frm mdel t mdel. Fr FIM mdels, t an be exressed as FIM T N [ C N N C FIM Ja, FIM Newtn Ja, blk blk Ja, nn nn N N ] (4. where N blk s the ttal number f rdblks, N nn s the ttal number f nnetns FIM FIM between rdblks, and C, and C, are the effents (nstants. The frst Ja blk Ja nn term n Eqn. 4. s the st f alulatn eah f the N N danal entres f the Jaban matrx, and the send term n Eqn. 4. s the st f alulatn eah f the N N ff-danal entres f the Jaban matrx. C, and FIM Ja blk C, are dfferent FIM Ja nn fr dfferent smulatrs and dfferent varable seletns, and enerally C, and FIM Ja blk 00

119 C, f the FIM mdel wth Tye B varables are larer than thse f the FIM mdel FIM Ja nn wth Tye A varables, beause we need t use the han rule t alulate the dervatves n the Tye B FIM mdel. Fr the IMPES mdel wth Tye A varables, the st f the Jaban nstrutn s arxmately ven by T [( C C IMPES _ A IMPES _ A Ja, IMPES _ A N Newtn Ja, blk N blk N CJa, nn N nnn GFAN blk ( N (4.3 3 ] where the frst tw terms are the sts f alulatn the full Jaban matrx, where all f the ff-danal entres are N matres, the last term s the st f redun the mlt level frm N t (nvertn eah small matrx D, whse sze s ( N - ( N -, and CGFA s the rresndn effent. Fr the IMPES mdel wth Tye B varables, sne the aumulatn term f the mass balane equatns an be smly exressed as [( φ r n ( φr n ] V / t, all danal entres f the full Jaban wll nly have nn-zer terms at the danal and ne lumn, and the st f the Jaban nstrutn s nly T Ja IMPES B Newtn Ja blk blk Ja nn nn GFA blk IMPES _ B IMPES _ B, _ N [( C, N N C, N N C N ( N ] (4.4 Fr the IMPSAT mdel (Tye A varables, the st f the Jaban nstrutn an be exressed as T N [( C N N C C IMPESAT IMPSAT Ja, IMPSAT Newtn Ja, blk blk Ja, nn nn GFA blk N N N N ( N N 3 (4.5 ] Smlarly, the frst tw terms are the sts f alulatn the full Jaban matrx, where all f the ff-danal entres are the mlt level frm N ( N - N, and ( N N eneralzed as T N N matres, the last term s the st f redun N t N (nvertn eah small matrx D, whse sze s ( N - CGFA s the rresndn effent. Fr mdels wth N mlt varables, Eqn. 4.5 (st f the Jaban nstrutn an be Ja N Newtn [( C N N C N N N C ( N N Ja, blk blk Ja, nn nn GFA blk N 3 ] (4.6 0

120 The st f the flash alulatn s ndeendent f the mlt level, and t an be exressed as flash T N [ C N N C flash Newtn Ja blk flash slve n flash N blk N ] (4.7 where the frst term s the st f alulatn the N N rdblk and the send term s the st f slvn t, and slver used. flash Jaban matrx fr eah flash Cslve and n flash deend n the The st f eah lnear slve deends n the ttal number f mlt unknwns ( N blk N, and t an be wrtten as n T N [ C ( N N slver ] (4.8 lnear _ slve Newtn slver blk where Cslver and n slver deend n the lnear slver used. Tyally nslver. ~. d teratve lnear slvers. The ttal st f smulatn s the summatn f the sts f these three arts. T ttal T Ja T flash T lnear _ slve (4.9 fr Frm Eqn. 4.3 t Eqn. 4.9, we an see that f we knw the ttal number f Newtn teratns, we an ruhly mare the sts f dfferent mdels. We have demnstrated that the IMPSAT mdel s muh mre stable than the IMPES mdel n Subsetn 4.. and mared ts st wth the sts f the FIM and IMPES mdels n Subsetn Next we wll ntrdue an effent way t nstrut the IMPSAT mdel Hw t Buld IMPSAT Effently? There s a very smle way t buld the IMPSAT mdel. In bth the water mass balane equatn (Eqn.. and the ttal hydrarbn mass balane equatn (Eqn..6, nly ressure and saturatns aear exltly. Sne the water hase densty deends nly n ressure, the water mass balane equatn nly nludes ressure and saturatn deendent terms, and t an be dretly used as an IMPSAT equatn wthut any redutn. Fr the ttal hydrarbn mass balane equatn, f we nre the deendeny f hydrarbn hase denstes n mnent mle fratns, then t wll nly nlude 0

121 ressure and saturatn deendent terms and an be dretly used as anther IMPSAT equatn wthut any redutn. IMPSAT needs three equatns fr three-hase flw, and we already have tw. S we nly need t use the redutn stes n Chater 3 n the remann n hydrarbn mnent mass balane equatns t et ne mre IMPSAT equatn, and the st f ths eratn s abut 3 tmes the st f buldn an IMPES mdel. If we nsder the deendeny f hydrarbn hase denstes n mnent mle fratns, the redutn stes n Chater 3 need t be erfrmed n the remann n hydrarbn mnent mass balane equatns t et tw mre IMPSAT equatns, and the st s dubled. 4. IMPSAT Based AIM Mdel Fr lare and dffult rblems, IMPES tmeste sze s t small t be ratal, and even IMPSAT s nt stable enuh. Whle FIM s always stable, t s just t exensve, eseally fr rblems wth a lare number f mnents. In suh ases, we rse addn the IMPSAT mdel nt the tradtnal AIM mdel (Frsyth and Sammn, 986, we all the new AIM mdel the IMPSAT based AIM mdel. Cmared t the tradtnal AIM (IMPESFIM mdel, t s mre flexble, mre stable and less exensve. There are three varatns f the new IMPSAT based AIM mdel: IMPESIMPSAT Here we use the IMPSAT mdel t relae the FIM mdel, and ths mdel s the least exensve ne n terms f st er Newtn teratn. IMPSATFIM Here we use the IMPSAT mdel t relae the IMPES mdel, and ths mdel s the mst stable ne. IMPESIMPSATFIM Here we use the IMPSAT mdel as the thrd frmulatn n the new AIM mdel, and ths mdel s the mst eneral ne. Atually, the frst tw AIM mdels and the tradtnal AIM mdel are all seal ases f ths eneral AIM mdel. All f these AIM mdels are very easy t buld usn the General Frmulatn Arah rsed n Chater 3, where all f the eratns are dne rdblk by rdblk, and we an ntrl the mlt level at eah rdblk searately. 03

122 Deendn n dfferent tyes f flw rblems, ne f these new AIM mdels wll uterfrm the tradtnal AIM mdel. Fr easy rblems, IMPESIMPSAT s the best t use, fr hard rblems, IMPSATFIM s a better he. Wth rerly tuned erentaes f IMPES, IMPSAT and FIM rdblks, IMPESIMPSATFIM shuld be the mst effent mdel t use, eseally fr lare feld ase studes. 04

123 Chater 5 Mdel Valdatn and Perfrmane In ths hater, we wll resent tw sets f results, the frst set s used t valdate ndvdual mdels and tehnques mlemented n GPRS. The results frm ths set are mared wth the results frm exstn smulatrs. The send set s used t evaluate the erfrmane f dfferent mstnal mdels. Here we mare the erfrmane f IMPSAT and IMPSAT based AIM mdels wth the tradtnal mdels (IMPES, FIM and AIM. Befre we start shwn the results frm GPRS and marn them wth the results frm ther smulatrs, t s wrthwhle t dsuss the nverene ntrls used n GPRS. In GPRS, the nverene f Newtn teratns s verned by bth the maxmum resdual and the maxmum varable hane. At nverene, the fllwn ndtn shuld be satsfed: Saled mass balane equatn resduals are less than 0.. Equatns are saled by the aumulatn terms (mnent mass dvdn tmeste sze. Saled hase equlbrum relatn resduals are less than 0.0. These are saled by the mnent fuates n the as hase. The maxmum saled ressure hane (abslute ressure hane dvded by the averae reservr ressure s less than The maxmum abslute saturatn hane s less than The maxmum abslute mnent mle fratn hane s less than The nverene f teratve lnear slvers s verned by the nrmalzed lnear equatn resdual ( B A / B. Ths lmt s set t Valdatn f GPRS Here we shw valdatn results fr the blak-l and the mstnal mdels n GPRS The blak-l mdel results are mared wth the results frm Else 00 (Gequest, 000A, and the mstnal mdel results are mared wth the results frm Else 300 (Gequest, 000A. We have als tested the unstrutured rd handln and mult- 05

124 nt flux arxmatns n GPRS, and mared ur results wth the results frm FLE (Verma, 996. Besdes that, we wll als shw valdatn results fr the General Frmulatn Arah (dfferent varable seletns and mlt levels. Fnally, we nlude a nn-nventnal well rblem, whse results are mared wth the results frm Else Blak-l Mdel Ths s the frst SPE maratve rjet (Odeh, 98, a three-hase blak-l smulatn f a quarter f a 5-st. Ths rd s Eah layer has dfferent rertes. One as njetr s lated at (,,, and ne l rduer s lated at (0, 0, 3, and bth wells are under nstant rate ntrl, shwn n Fure 5.. Intally, the reservr s full f undersaturated l and nnate water. The smulatn was run t 0 years wth a maxmum tmeste sze f 80 days, and the results frm GPRS are mared wth the results frm Else 00. Cmstnal frmulatn was used t smulate ths blak-l rblem n GPRS, and bth smulatrs were run n fully mlt mde. The as was nt allwed t re-dsslve. as Fure 5. Blak-l reservr WOC Bth smulatrs used the same number f tmestes (3 and Newtn teratns (4. Fure 5. and Fure 5.3 shw the results frm GPRS and Else 00. GPRS results are shwn n dts, and Else 00 results are shwn n sld lnes. Fure 5. shws the l rdutn rate, lateau rdutn u untl arund 000 days, then the rduer hanes t bttm hle ressure (BHP ntrl. Fure 5.3 shws the as l rat (GOR at the 06

125 rduer, ntally, there s n free as, GOR s nstant, after arund 660 days, free as aears, and GOR nreases. Basally, the results frm these tw smulatrs are the same, whh means that we have exellent areement between GPRS and Else l rdutn rate (STB/day Else GPRS tme (days Fure 5. Ol rdutn rate at the rduer frm GPRS and Else Else GOR (MSCF/STB 5 0 GPRS tme (days Fure 5.3 Gas l rat at the rduer frm GPRS and Else 00 07

126 5.. Cmstnal Mdel Ths s a fur-mnent, tw-hase (as-l mstnal smulatn (Ca, 00. The rd s One rduer s lated at (5, 5,, whh s under bttm hle ressure (BHP ntrl (shwn n Fure 5.4. The Pen-Rbnsn Equatn f State (Pen and Rbnsn, 976 was used fr flash alulatns. Intally the reservr has bth the l and as hases. The smulatns were run t 000 days wth a maxmum tmeste sze f 30 days, and the results frm GPRS are mared wth the results frm Else 300 (Gequest, 000A. Fure 5.4 Cmstnal reservr GPRS uses Tye A varables, whle Else 300 uses Tye B varables (ressure and verall mlar densty f eah mnent. Bth smulatrs were run n fully mlt mde. Bth smulatrs qukly reahed the maxmum tmeste sze f 30 days, and used ruhly the same number f tmestes (GPRS used 39 tmestes, whle Else 300 used 38 tmestes. GPRS tk an averae f.5 Newtn teratns er tmeste, whle Else 300 tk an averae f. Newtn teratns er tmeste. The slht nrease f Newtn teratn number n GPRS was aused by the dfferene n Newtn teratn nverene ntrl n the tw smulatrs. In Else 300, nly the maxmum hane f ressure and effetve saturatns (the effetve saturatn hane fr a ven mlar densty hane are used fr nverene ntrl (Gequest, 000A. Als ther lmts are larer than the lmts used n GPRS, Else 300 uses 0. atm fr abslute ressure hane and 0.0 fr effetve saturatn hane, whle GPRS uses fr maxmum saled ressure 08

127 hane and fr abslute saturatn hane. Fure 5.5 and Fure 5.6 mare the results frm GPRS wth the results frm Else 300. ressure (sa (,,5_el (,,5_rs rd_el rd_rs tme (days Fure 5.5 Well blk and blk (,,5 ressures frm GPRS and Else 300 wth t max 30 days 0.9 l saturatn (,,5_el (,,5_rs rd_el rd_rs tme (days Fure 5.6 Well blk and blk (,,5 l saturatns frm GPRS and Else 300 wth t max 30 days 09

128 Smlar t the results n Fure 5. and Fure 5.3, GPRS results are shwn n dts, and Else 300 results are shwn n sld lnes. The blak lnes are fr rduer (5,5,, and the blue lnes are fr blk (,,5. Fure 5.5 shws the ressure at blk (,,5 and at the rduer (5,5,. Fure 5.6 shws the l saturatn at these tw blks. The l saturatn at the rduer blk frm the tw smulatrs des nt math very well, the reasn s that Else 300 s nverene rtera (ressure and effetve saturatn hanes are nt as strt as thse used n GPRS. Atually Else 300 es t the next tmeste wthut full nverene n the urrent tmeste. Frm Fure 5.6 we an nte a lth (arund 90 days n Else 300 results. By redun the maxmum tmeste sze frm 30 days t 0 days n Else 300, the results mrve as shwn n Fure 5.7. Else 300 used a maxmum tmeste sze f 0 days, and GPRS stll used 30 days. Nw these tw results are almst n t f eah ther. Hene we an nlude that the mstnal mdel n GPRS wrks well, and at tmes t erfrms even better than the mdel n Else 300. Frm Fure 5.7, we nte that, even fr 0 days, there s stll a small lth (arund 330 days n Else 300 results, rbably stll due t the lse nverene ntrl. 0.9 l saturatn (,,5_el (,,5_rs rd_el rd_rs tme (days Fure 5.7 Well blk and blk (,,5 l saturatns frm GPRS and Else 300 wth t max 30 days fr GPRS and t max 0 days fr Else 300 0

129 5..3 Unstrutured Grd and Mult-Pnt Flux A test rblem eed tether frm Verma (996 was used t valdate the mlementatn f unstrutured rds and mult-nt flux n GPRS. Ths s a quarter f a fve st smulatn f a hmeneus blak-l (water-l reservr (Ca, 00, ne water njetr s lated at the lwer-left rner and ne rduer s lated at the uerrht rner. Bth wells are at nstant rate ntrl. The rd s a tw-dmensnal (x-y tranular rd wth lal rd refnements arund the wells, Fure 5.8a shws the tranular elements, and Fure 5.8b shws the ntrl vlumes (rdblks. Mst f the mult-nt fluxes are arund the wells, where the rdblks are rreular. Ths rblem has 30 rdblks, 838 nnetns, and abut 30% f the nnetns use mult-nt flux alulatns. a tranular elements b ntrl vlumes Fure 5.8 Unstrutured rd (Verma, 996 We ran ths rblem usn GPRS and mared ts results wth the results frm FLE (Prevst, 00. Fure 5.9 shws the water ut at the rduer. Cnsdern the dfferenes n these tw smulatrs, the results mare well. There s nly a small dsreany at the water breakthruh tme. We an nlude that the unstrutured rd handln and mult-nt flux arxmatns n GPRS wrk well.

130 WCT FLE GPRS tme (days Fure 5.9 Water ut results frm GPRS and FLE fr unstrutured rd 5..4 General Frmulatn Arah Ths s the same hmeneus fur mnent (C, C, C4, C7 tw hydrarbn hase rblem that was used t valdate the mstnal mdeln n Subsetn 5... Here we mare the results f mdels wth dfferent varable seletns and mlt levels. The maxmum tmeste sze f 30 days was used fr all runs. Fure 5.0 and Fure 5. mare the results f the FIM mdel wth Tye A and Tye B varables. Tye A results are shwn n sld lnes, and Tye B results are shwn n dts. Fure 5.0 shws the rdblk ressure at the rduer and Fure 5. shws the l saturatn at the rduer. The results frm bth FIM mdels math exatly, ths s beause they are slvn the same equatns and at nverene they shuld have the same slutns. Fure 5. and Fure 5.3 mare the results f mdels f Tye A varables wth dfferent mlt levels, and they are the FIM mdel (all fur rmary varables are mlt, the IMPSAT mdel (the frst three rmary varables are mlt, the IMPSAT mdel (the frst tw rmary varables are mlt and the IMPES mdel (nly ressure are mlt. Fure 5. shws the rdblk ressure at the rduer and Fure 5.3 shws the l saturatn at the rduer. Frm these results, we an see that

131 the IMPES results have less numeral dffusn, and the results frm the ther three mdels are basally the same. 550 ressure at the rduer (sa Tye A Tye B tme (days Fure 5.0 Well blk ressure fr the FIM mdel wth Tye A and Tye B varables 0.7 l saturatn at the rduer (sa Tye A Tye B tme (days Fure 5. Well blk l saturatn fr the FIM mdel wth Tye A and Tye B varables 3

132 550 ressure at the rduer (sa FIM IMPSAT IMPSAT IMPES tme (days Fure 5. Well blk ressure fr mdels wth dfferent mlt levels 0.7 l saturatn at the rduer (sa FIM IMPSAT IMPSAT IMPES tme (days Fure 5.3 Well blk l saturatn fr mdels wth dfferent mlt levels 4

133 5..5 Nn-nventnal Well Ths s the same three-hase blak-l rblem that was used t valdate the blak-l mdeln n Subsetn 5... Here, the rduer s relaed by a BHP ntrlled dual-lateral well, whh s lated at (6-0, 0, 3 and (0, 6-0, 3, shwn n Fure 5.4. Intally, the reservr s full f undersaturated l and nnate water. The smulatn was run t 0 years wth a maxmum tmeste sze f 80 days, and the results frm GPRS are mared wth the results frm Else 00. Cmstnal frmulatn was used t smulate ths blak-l rblem n GPRS, and bth smulatrs were run n fully mlt mde. The as was nt allwed t re-dsslve. as WOC Fure 5.4 Blak-l reservr wth a dual-lateral rduer Fure 5.5 and Fure 5.6 shw the results frm GPRS and Else 00. GPRS results are shwn n dts, and Else 00 results are shwn n sld lnes. Fure 5.5 shws the l rdutn rate and Fure 5.6 shws the as l rat (GOR at the rduer. The results frm these tw smulatrs are almst the same. 5

134 50000 l rdutn rate (STB/day Else GPRS tme (days Fure 5.5 Ol rdutn rate at the rduer frm GPRS and Else 00 fr a dual-lateral well rblem 30 5 GOR (MSCF/STB Else 5 GPRS tme (days Fure 5.6 Gas l rat at the rduer frm GPRS and Else 00 fr a dual-lateral well rblem 6

135 S far, we have shwn that the bas mdules f GPRS wrk well, we an nw start t evaluate the erfrmane f dfferent mstnal mdels usn varus mstnal rblems. The erfrmane f ndvdual mstnal mdels s ndeendent f the rd tye and the methd f flux arxmatn, s n ths art we wll use Cartesan rds and tw-nt flux arxmatns fr all rblems. 5. Perfrmane f Cmstnal Mdels n GPRS A seres f mstnal rblems were used t evaluate the erfrmane f dfferent mstnal mdels. The sze f these rblems ranes frm 5 rdblks t 00,000 rdblks, number f mnents ranes frm 4 t 9, wth bth hmeneus and hhly hetereneus dmans. The mstnal mdels evaluated here nlude fully mlt (FIM mdels wth bth tyes f varables (Tye A and Tye B, IMPES mdels wth bth tyes f varables, the IMPSAT mdel wth Tye A varables and the tradtnal and new AIM mdels. Besdes the smulatn results, we als rert the teratn unts (number f tmestes, number f ttal Newtn teratns and number f ttal lnear slver teratns and the tmn results (slver tme and ttal runnn tme fr eah rblem and eah mdel. All f the smulatns were erfrmed n a SGI rn 00 wrkstatn. Here bth Tye A and Tye B varables were used. Fr a fur hydrarbn mnent and tw hydrarbn hase rblem, the full set f equatns (ttal 8 nlude the mass balane equatns and the hase equlbrum relatns fr eah hydrarbn mnent, wth the fur mass balane equatns as the rmary equatns. Fr Tye A varables, the full set f varables s, S, y, y, y 3, x, x, x 3,where y and x are the mnent mle fratns n the as and l hases, and, S, y, y are the rmary varables. Fr Tye B varables, the full set f varables s, r, r, r 3, S, y, y, y 3,where r are the verall denstes f hydrarbn mnents, and, r, r, r3 are the rmary varables. In ths setn, we wll frst ntrdue mstnal rblems used, then evaluate the erfrmane f ndvdual mstnal mdels. The erfrmane f mstnal mdels s evaluated under dfferent ndtns. Eah rblem s desned fr ne sef ndtn. These ndtns nlude njetn f mnents, heterenety, lare number 7

136 f mnents and lare systems. The fve mstnal rblems used here are dsussed belw (Ca, 00: Case Ths s the same hmeneus fur mnent (C, C, C4, C7 tw hydrarbn hase rblem that was used t valdate the mstnal mdeln n GPRS. One bttm hle ressure (BHP ntrlled rduer s lated at (5, 5,. Ths s a relatvely easy rblem, and bth IMPES and IMPSAT shuld erfrm well. Case Ths s the same rblem as n Case, but here an njetr s added t nrease the dffulty. The njetr s lated at (,,. It s under bttm hle ressure ntrl and C s njeted. Ths rblem s dffult fr the IMPSAT mdel, beause a mnent s njeted when the mnent mle fratns are treated exltly. Case 3 Ths s a hetereneus rblem, wth fur mnents (C, C, C4, C7 and tw hydrarbn hases. The rd s 4 5. One bttm hle ressure ntrlled rduer s lated at (,, -3, and t enetrates nly the t three layers. The ermeablty feld s frm the 9 th SPE maratve rjet (Klluh, 995. Only the t layer, shwn n Fure 5.7, s used. The ermeablty hanes ver fur rders f mantude and t s rrelated aln the x dretn. The rmary bjetve here s t test the erfrmane f dfferent mdels n a dffult hetereneus rblem. Due t hh flw rates n the hh ermeablty rens, ths rblem s hard fr bth IMPES and IMPSAT. Case 4 Ths s a nne-mnent rblem. The flud desrtn s frm the 3 rd SPE maratve rjet (Kenyn and Behe, 987, and the nne mnents are CO, N,C,C,C3,C4-6,C7,C7 and C7 3. The rd s wth hmeneus ermeablty, and ne bttm hle ressure ntrlled rduer s lated at (5, 5,. Due t the lare number f mnents, ths rblem s hard fr bth IMPES and IMPSAT. Case 5 Ths s a lare hmeneus rblem wth 00,000 rdblks ( and fur mnents (C, C, C4, C7. One bttm hle ressure ntrlled rduer s lated at (00, 00, -, and t enetrates nly the t tw layers. Cmared t ther rblems, ths rblem has muh smaller rdblks (00ft 00ft 0ft mared t 000ft 000ft 0ft fr ther rblems. Ths small rdblk sze lmts the maxmum stable tmeste sze f IMPSAT t arund 0.5 days, and t s vrtually 8

137 mssble t run the rblem wth ths mdel. The stablty restrtn s even wrse fr IMPES mdels. S fr ths rblem, nly the FIM and AIM mdels were used. The stablty f the IMPESIMPSAT mdel s even wrse than the stablty f the IMPSAT mdel, s t was als nt used. All f the smulatns were run t 000 days, and the Pen-Rbnsn EOS (Pen and Rbnsn, 976 was used fr flash alulatns. Nw that we have ntrdued all f the rblems, next we wll shw the erfrmane f GPRS fr eah tye f mstnal mdel, startn frm the FIM mdel. Fure 5.7 Hetereneus ermeablty feld (lnk fr Case Fully Imlt Mdels The FIM mdel wth bth Tye A and Tye B varables was used. Mdel erfrmane s mared fr dfferent rblems and dfferent tmeste szes. The maxmum tmeste szes tested were 5, 30 and 00 days (Fr Case 5, nly tmeste szes f 30 and 00 days were used. Fure 5.8 t Fure 5. shw the well blk l saturatn fr eah 9

138 rblem, and the leend n eah fure marks the mdel used, suh as B_30days stands fr Tye B FIM mdel wth a maxmum tmeste sze f 30 days. Tye A varable results are drawn n lnes, and Tye B varable results are drawn n dts. Frm these results, we an draw the fllwn nlusns: Fr the same maxmum tmeste sze, bth FIM mdels math exatly, ths s beause they are slvn the same equatns and at nverene they shuld have the same slutns. Wth the nrease f tmeste sze, there s mre numeral dffusn fr bth FIM mdels and the results are mre smthed ut. Bth FIM mdels are always stable. These three bservatns are true fr all mstnal rblems. 0.7 l saturatn at the rduer A_5days B_5days A_30days B_30days A_00days B_00days tme (days Fure 5.8 Well blk l saturatn marsns fr FIM mdels and Case rblem 0

139 l saturatn at the rduer A_5days B_5days A_30days B_30days A_00days B_00days tme (days Fure 5.9 Well blk l saturatn marsns fr FIM mdels and Case rblem 0.6 l saturatn at the rduer A_5days B_5days A_30days B_30days A_00days B_00days tme (days Fure 5.0 Well blk l saturatn marsns fr FIM mdels and Case 3 rblem

140 0.4 l saturatn at the rduer A_5days B_5days A_30days B_30days A_00days B_00days tme (days Fure 5. Well blk l saturatn marsns fr FIM mdels and Case 4 rblem l saturatn at the rduer A_30days B_30days A_00days B_00days tme (days Fure 5. Well blk l saturatn marsns fr FIM mdels and Case 5 rblem

141 Iteratn unts and tmn results fr rblems dsussed abve are resented n Table 5. t Table 5.5. Fr all f the fully mlt runs, the CPR rendtned GMRES methd was used t slve the lnear system. We an draw the fllwn nlusns based n ur analyss: Fr bth FIM mdels, wth the nrease f tmeste sze, mre Newtn teratns are needed at eah tmeste and mre lnear slver teratns are needed t slve eah lnear system. Ths s beause the system bemes less danally dmnant as the tmeste sze nreases. There s n snfant dfferene n the number f Newtn teratns between these tw FIM mdels. Tye B FIM mdel enerally takes mre lnear slver teratns than Tye A FIM mdel. As a result, Tye B FIM mdel takes mre tme n the lnear slver, and n sme ases, the nrease n CPU tme an be as muh as 30%. We als tred ther slvers and rendtners, suh as ILU0 and BltzPak, and we bserve the same trend fr all f them. Tye B FIM mdel s als mre stly n buldn the Jaban matrx, sne t needs t use the han rule t alulate the dervatves. Wth the hher slver st, the ttal st s als substantally hher. In summary, Tye A FIM mdel sts less than Tye B FIM mdel, at least fr the rblems tested here. The mlementatns f Tye A mdels are mre effent than the mlementatns f Tye B mdels n GPRS beause f the varable swthn, s we an nly mare the relatve st f these tw tyes f mdels. Ths s true fr all f the ther marsns between mdels usn these tw tyes f varables. A_5days B_5days A_30days B_30days A_00days B_00days N tmestes N Newtn_ ters N _ slver ters Tslver(se Tttal (se Table 5. Iteratn and tmn marsns fr FIM mdels and Case rblem 3

142 A_5days B_5days A_30days B_30days A_00days B_00days N tmestes N Newtn_ ters N _ slver ters Tslver(se Tttal (se Table 5. Iteratn and tmn marsns fr FIM mdels and Case rblem A_5days B_5days A_30days B_30days A_00days B_00days N tmestes N Newtn_ ters N _ slver ters Tslver(se Tttal (se Table 5.3 Iteratn and tmn marsns fr FIM mdels and Case 3 rblem A_5days B_5days A_30days B_30days A_00days B_00days N tmestes N Newtn_ ters N _ slver ters Tslver(se Tttal (se Table 5.4 Iteratn and tmn marsns fr FIM mdels and Case 4 rblem 4

143 A_5days B_5days A_30days B_30days A_00days B_00days N tmestes N Newtn_ ters N _ slver ters Tslver(se Tttal (se Table 5.5 Iteratn and tmn marsns fr FIM mdels and Case 5 rblem 5.. IMPES mdels IMPES mdels wth bth Tye A and Tye B varables were used. Mdel erfrmane s mared fr dfferent rblems and dfferent CFL numbers (larer CFL numbers lead t larer tmeste szes. Bth CFL and CFL were used, CFL s always stable, but CFL may nt be always stable. Fure 5.3 t Fure 5.6 shw the stable tmeste sze (CFL fr eah rblem. Fure 5.7 t Fure 5.30 shw the well blk l saturatn fr eah rblem, and the leend n eah fure marks the mdel used, suh as B_CFL stands fr Tye B IMPES mdel wth a maxmum CFL number f. Tye A varable results are drawn n lnes, and Tye B varable results are drawn n dts. The fully mlt result (the dark sld lne fr maxmum tmeste sze f 30 days s nluded fr marsn. Frm these results, we an draw the fllwn nlusns: Fr all mstnal rblems tested here, the stable tme ste szes f the IMPES mdel are small, rann frm.5 days t 6 days. Ths s beause n mstnal smulatns, as hase s always resent, and as flw s hard t handle fr the IMPES mdel due t ts hh mblty. The stable tmeste szes f the IMPES mdel have sudden jums. Ths s beause the CFL lmt f IMPES deends n the fratnal flw dervatve wth reset t the saturatn (saturatn velty, and at the saturatn frnt, t s dsntnuus, whh auses the sudden jum. Fr the same tmeste sze, bth IMPES mdels math exatly, ths s beause they are slvn the same IMPES equatns and at nverene they shuld have the same slutn. Cmared t the FIM mdel, the IMPES mdel has less numeral dffusn (shaer frnts, due t the exlt treatment f transmssbltes. 5

144 Fr Case, and 3, IMPES wth CFL s stable r nearly stable. But wth the nrease f number f mnents, suh as n Case 4, ths lmt s redued, and CFL r.5 s a safe lmt. 6 4 maxmum stable tmeste sze tme (days Fure 5.3 Stable tmeste sze fr the IMPES mdel and Case rblem 8 6 maxmum stable tmeste sze tme (days Fure 5.4 Stable tmeste sze fr the IMPES mdel and Case rblem 6

145 4 maxmum stable tmeste sze tme (days Fure 5.5 Stable tmeste sze fr the IMPES mdel and Case 3 rblem 6 maxmum stable tmeste sze tme (days Fure 5.6 Stable tmeste sze fr the IMPES mdel and Case 4 rblem 7

146 0.7 l saturatn at the rduer FIM A_CFL B_CFL A_CFL B_CFL tme (days Fure 5.7 Well blk l saturatn marsns fr IMPES mdels and Case rblem l saturatn at the rduer FIM A_CFL B_CFL A_CFL B_CFL tme (days Fure 5.8 Well blk l saturatn marsns fr IMPES mdels and Case rblem 8

147 l saturatn at the rduer FIM A_CFL B_CFL A_CFL B_CFL tme (days Fure 5.9 Well blk l saturatn marsns fr IMPES mdels and Case 3 rblem 0.4 l saturatn at the rduer FIM A_CFL B_CFL A_CFL B_CFL tme (days Fure 5.30 Well blk l saturatn marsns fr IMPES mdels and Case 4 rblem 9

148 Iteratn unts and tmn results fr rblems dsussed abve are resented n Table 5.6 t Table 5.9. Fr all f the IMPES runs, the AMG rendtned GMRES methd was used t slve the lnear system. We an draw the fllwn nlusns based n ur analyss: Fr bth IMPES mdels, wth the nrease f CFL number (tmeste sze, mre Newtn teratns are needed at eah tmeste and mre lnear slver teratns are needed t slve eah lnear system. Ths s beause the ressure system bemes less danally dmnant as the tmeste sze nreases. There s n snfant dfferene n Newtn teratn numbers between these tw IMPES mdels. Number f lnear slver teratns s als abut the same fr these tw IMPES mdels. Ths s beause bth IMPES mdels enerate the same ressure system, and the salar dfferene between them s nt enuh t make any dfferene n lnear slver erfrmane. Tye B mdels are bult frm Tye A mdels by varable swthn n GPRS. Ths nreases the st fr Jaban alulatns n Tye B mdels, as shwn n the tables. But n rate, Tye B IMPES mdel an be bult dretly frm the mass balane equatns. In ths ase Tye B mdel wll requre fewer eratns than Tye A mdel t frm the ressure system (Cats, 999. Hene Tye B IMPES mdel shuld be mre effent frm the nt f vew f Jaban matrx nstrutn. In summary, whle ur results dn't shw ths, Tye B IMPES mdel uses less tme than Tye A IMPES mdel, due t lwer st n Jaban matrx nstrutn. A_CFL B_CFL A_CFL B_CFL N tmestes N _ Newtn ters N _ slver ters Tslver (se Tttal (se Table 5.6 Iteratn and tmn marsns fr IMPES mdels and Case rblem 30

149 A_CFL B_CFL A_CFL B_CFL N tmestes N _ Newtn slver ters N _ ters Tslver (se Tttal (se Table 5.7 Iteratn and tmn marsns fr IMPES mdels and Case rblem A_CFL B_CFL A_CFL B_CFL N tmestes N _ 8 8 Newtn slver ters N _ ters Tslver (se Tttal (se Table 5.8 Iteratn and tmn marsns fr IMPES mdels and Case 3 rblem A_CFL B_CFL A_CFL B_CFL N tmestes N Newtn _ ters N slver _ ters Tslver (se Tttal (se Table 5.9 Iteratn and tmn marsns fr IMPES mdels and Case 4 rblem 3

150 5..3 IMPSAT mdel The erfrmane f the IMPSAT (mlt ressure and mlt saturatns mdel s evaluated fr dfferent rblems and dfferent CFL numbers (larer CFL numbers lead t larer tmeste szes. CFL, and 4 were used, CFL s always stable, but CFL and CFL4 may nt be always stable. Fure 5.3 t Fure 5.34 shw the stable tmeste sze (CFL fr eah rblem, mared wth the stable tmeste sze f the IMPES mdel. Fure 5.35 t Fure 5.38 shw the well blk l saturatn fr eah rblem, and the leend n eah fure marks the mdel used, suh as CFL stands fr the IMPSAT mdel wth a maxmum CFL number f. The fully mlt result (the dark sld lne fr maxmum tmeste sze f 30 days s nluded fr marsn. Frm these results, we an draw the fllwn nlusns: The stable tmeste szes fr the IMPSAT mdel are qute d, rann frm 6 days t 90 days, whh s t 7 tmes larer than that fr the IMPES mdel. In the IMPSAT mdel, saturatn s treated mltly, s the exstene f as flw des nt reate any rblem. The stable tmeste szes f IMPSAT always hane smthly, whle fr IMPES we bserved sudden jums. The CFL lmt f the IMPSAT mdel deends n the mnent velty. The results f the IMPSAT mdel aree well wth the results f the FIM mdel, and the slht dfferenes between the tw are mstly due t the dfferenes n tmeste sze. Fr sme rblems, suh as Case and 3, the tmeste sze f IMPSAT reahed 00 days, whle fr FIM the maxmum tmeste sze was lmted t 30 days. In thery, the IMPSAT mdel shuld have less numeral dffusn than the FIM mdel, hwever fr the rblems tested here, we bserve the same level f numeral dffusn fr the IMPSAT and FIM mdels. Fr Case, and 3, IMPSAT wth CFL4 s stable. But wth the nrease f number f mnents, suh as n Case 4, ths lmt s als redued, and CFL r.5 s a safe lmt. 3

151 00 maxmum stable tmeste sze IMPSAT IMPES tme (days Fure 5.3 Stable tmeste sze fr the IMPSAT mdel and Case rblem 35 maxmum stable tmeste sze IMPSAT IMPES tme (days Fure 5.3 Stable tmeste sze fr the IMPSAT mdel and Case rblem 33

152 30 maxmum stable tmeste sze IMPSAT IMPES tme (days Fure 5.33 Stable tmeste sze fr the IMPSAT mdel and Case 3 rblem 30 maxmum stable tmeste sze IMPSAT IMPES tme (days Fure 5.34 Stable tmeste sze fr the IMPSAT mdel and Case 4 rblem 34

153 0.7 l saturatn at the rduer FIM CFL CFL CFL tme (days Fure 5.35 Well blk l saturatn marsns fr the IMPSAT mdel and Case rblem 0.7 l saturatn at the rduer FIM CFL CFL CFL tme (days Fure 5.36 Well blk l saturatn marsns fr the IMPSAT mdel and Case rblem 35

154 0.6 l saturatn at the rduer FIM CFL CFL CFL tme (days Fure 5.37 Well blk l saturatn marsns fr the IMPSAT mdel and Case 3 rblem 0.4 l saturatn at the rduer FIM CFL CFL CFL tme (days Fure 5.38 Well blk l saturatn marsns fr the IMPSAT mdel and Case 4 rblem 36

155 Iteratn unts and tmn results fr rblems dsussed abve are resented n Table 5.0 t Table 5.3. Fr all f the IMPSAT runs, the CPR rendtned GMRES methd was used t slve the lnear system. We an draw the fllwn nlusns based n ur analyss: Fr the IMPSAT mdel, wth the nrease f CFL number (tmeste sze, mre Newtn teratns are needed at eah tmeste and mre lnear slver teratns are needed t slve eah lnear system. Ths s beause the system bemes less danally dmnant as the tmeste sze nreases. Cmared t the IMPES mdel, the IMPSAT mdel needs fewer (ver 50% number f tmestes and Newtn teratns due t ts mrved stablty. IMPSAT may st mre n the lnear slver art than IMPES, beause f ts nreased number f unknwns, but t an als save a lt n the Jaban matrx nstrutn and flash alulatns. Bth f these sts are dretly rrtnal t the number f Newtn teratns, and they are tw f the mst tme nsumn arts n mstnal smulatn. Fnally, the ttal st f IMPSAT s enerally muh lwer than the ttal st f IMPES, and mst f tme, IMPSAT an ut the runnn tme by half mared t IMPES. Fr mst rblems, the mrved stablty f the IMPSAT mdel enables t t use the same number f tmestes and Newtn teratns as the FIM mdel (wth maxmum tmeste sze f 00 days. Beause IMPSAT slves fr fewer unknwns n the lnear slver art, t s a heaer alternatve t the FIM mdel. All marsns here between the IMPSAT mdel and the FIM mdel are based n that the FIM mdel uses 00 days as the maxmum tmeste sze, whh s qute lare fr tyal mstnal smulatns (The default maxmum tmeste sze used n Else 300 s nly 50 days. Of urse, the FIM mdel s unndtnally stable, and t an use any tmeste sze. In summary, the IMPSAT mdel s muh better than the IMPES mdel due t ts mrved stablty, and t s als heaer than the FIM. Fr mst rblems, the IMPSAT mdel an uterfrm bth the IMPES and FIM mdels. But f the rblem s very hard, suh as hh flw rate r small rdblk szes, the stable tmeste sze f IMPSAT an als be qute small (less than day, sne t s rrtnal t ell vlume dvded by the 37

156 flw rate, as shwn n Eqn In suh ases, FIM mdels and AIM mdels are arrate. CFL CFL CFL4 N tmestes N _ Newtn slver ters N _ ters Tslver (se Tttal (se Table 5.0 Iteratn and tmn marsns fr the IMPSAT mdel and Case rblem CFL CFL CFL4 N tmestes N Newtn _ ters N slver _ ters Tslver (se Tttal (se Table 5. Iteratn and tmn marsns fr the IMPSAT mdel and Case rblem CFL CFL CFL4 N tmestes N Newtn _ ters N slver _ ters Tslver (se Tttal (se Table 5. Iteratn and tmn marsns fr the IMPSAT mdel and Case 3 rblem 38

157 CFL CFL CFL3 N tmestes N Newtn _ ters N slver _ ters Tslver (se Tttal (se Table 5.3 Iteratn and tmn marsns fr the IMPSAT mdel and Case 4 rblem 5..4 AIM mdels The tradtnal AIM mdel and three new IMPSAT based AIM mdels were used here. Mdel erfrmane s mared fr dfferent rblems. Fr AIM mdels, there are tw ways t dede the tmeste sze and assn mlt level fr eah rdblk, and they are dsussed bellw: Fx tmeste sze Fr a ven tmeste sze, assn the mlt level fr eah rdblk ardn t the stablty rtera (CFL number at that rdblk. Ths methd s strahtfrward, and the erentae f rdblks fr eah frmulatn hanes frm tmeste t tmeste. Fx erentae Fr a fxed erentae f rdblks fr eah frmulatn, frst dede the maxmum stable tmeste sze whh an satsfy the ven erentae, then assn the mlt level fr eah rdblk ardn t the stablty rtera. Ths methd s nt strahtfrward, n rder t dede the maxmum stable tmeste sze fr a ven erentae, we need t rder the rdblks ardn ther CFL number. Fr AIM systems, the erfrmane f lnear slvers and rendtners deends strnly n the erentae f rdblks fr eah frmulatn. Fr examle, CPR s mre effent fr systems where mst rdblks are FIM, whle AMG s mre effent fr systems where mst rdblks are IMPES. Based n these nsderatns, the methd f fxed erentae s used n GPRS. The erentae fr eah AIM mdel was assned as fllwn: IMPESFIM 90% IMPES rdblks and 0% FIM rdblks IMPESIMPSAT 90% IMPES rdblks and 0% IMPSAT rdblks 39

158 IMPSATFIM 90% IMPSAT rdblks and 0% FIM rdblks IMPESIMPSATFIM 90% IMPES rdblks, 9% IMPSAT rdblks and % FIM rdblks The well blks were fred t be ether FIM r IMPSAT. The maxmum tmeste sze was set t be 00 days. Fure 5.39 t Fure 5.43 shw the well blk l saturatn fr eah rblem, and the leend n eah fure marks the mdel used. The fully mlt result (the dark sld lne fr maxmum tmeste sze f 00 days s nluded fr marsn. Frm these results, we an draw the fllwn nlusns: The results f all AIM mdels aree well wth the results f the FIM mdel. The results f AIM mdels wth an IMPES frmulatn are n t f eah ther, whle the results f the FIM and the IMPSATFIM mdels are n t f eah ther. Ths s mstly due t the dfferene n tmeste szes, IMPES tmeste sze an nt reah 00 days, but wthut IMPES, t an reah 00 days. Due t ths, we bserve bvus dfferenes n the results f dfferent AIM mdels fr Case 3 and l saturatn at the rduer FIM IMPESFIM IMPESIMPSAT IMPSATFIM IMPESIMPSATFIM tme (days Fure 5.39 Well blk l saturatn marsns fr AIM mdels and Case rblem 40

159 l saturatn at the rduer FIM IMPESFIM IMPESIMPSAT IMPSATFIM IMPESIMPSATFIM tme (days Fure 5.40 Well blk l saturatn marsns fr AIM mdels and Case rblem 0.6 l saturatn at the rduer FIM IMPESFIM IMPESIMPSAT IMPSATFIM IMPESIMPSATFIM tme (days Fure 5.4 Well blk l saturatn marsns fr AIM mdels and Case 3 rblem 4

160 0.4 l saturatn at the rduer FIM IMPESFIM IMPESIMPSAT IMPSATFIM IMPESIMPSATFIM tme (days Fure 5.4 Well blk l saturatn marsns fr AIM mdels and Case 4 rblem l saturatn at the rduer FIM IMPESFIM IMPSATFIM IMPESIMPSATFIM tme (days Fure 5.43 Well blk l saturatn marsns fr AIM mdels and Case 5 rblem 4

161 Iteratn unts and tmn results fr rblems dsussed abve are resented n Table 5.4 t Table 5.8. Fr all f the AIM runs, the ILU0 rendtned GMRES methd was used t slve the lnear system. We an draw the fllwn nlusns based n ur analyss: Fr rblems that are easy fr IMPSAT, suh as Case and 3, the IMPESIMPSAT mdel s faster than the tradtnal AIM mdel, sne IMPSAT slves fr fewer unknwns than FIM. On the ther hand, the IMPESIMPSAT mdel s less stable than the IMPSAT mdel, s any rblem that an nt be handled by the IMPSAT mdel, an als nt be handled by the IMPESIMPSAT mdel. Fr rblems that are hard fr IMPES, suh as Case,, 4 and 5, the IMPSATFIM mdel needs far fewer tmestes and Newtn teratns than the tradtnal AIM mdel, due t the mrved stablty f IMPSAT. Hene fr hard rblems, the IMPSATFIM mdel s muh faster than the tradtnal AIM mdel. The ILU0 rendtner used here s qute weak mared t the rendtners (CPR, AMG used fr ther mdels, s t wn t make any sense t use the tmn results here t mare the erfrmane f AIM mdels wth ther mdels. But n thery, wth a d rendtner, AIM mdels shuld be muh faster than FIM mdels, beause they slve fr fewer unknwns. In summary, deendn n the nature f the rblem, ne f the three new AIM mdels wll uterfrm the tradtnal AIM mdel. The IMPESIMPSATFIM mdel s the eneral AIM mdel, wth rer tunn f the erentaes ardn t the dffultes fr the IMPES and IMPSAT mdels, t wll always uterfrm the ther three AIM mdels. 43

162 IMPESFIM IMPESIMPSAT IMPSATFIM IMPESIMPSATFIM N tmestes N Newtn_ ters N _ slver ters Tslver(se Tttal(se Table 5.4 Iteratn and tmn marsns fr AIM mdels and Case rblem IMPESFIM IMPESIMPSAT IMPSATFIM IMPESIMPSATFIM N tmestes N Newtn_ ters N slver_ ters Tslver(se Tttal (se Table 5.5 Iteratn and tmn marsns fr AIM mdels and Case rblem IMPESFIM IMPESIMPSAT IMPSATFIM IMPESIMPSATFIM N tmestes N Newtn_ ters N slver_ ters Tslver(se Tttal (se Table 5.6 Iteratn and tmn marsns fr AIM mdels and Case 3 rblem 44

163 IMPESFIM IMPESIMPSAT IMPSATFIM IMPESIMPSATFIM N tmestes N Newtn_ ters N slver_ ters Tslver(se Tttal (se Table 5.7 Iteratn and tmn marsns fr AIM mdels and Case 4 rblem IMPESFIM IMPESIMPSAT IMPSATFIM IMPESIMPSATFIM N tmestes N Newtn_ ters N slver_ ters Tslver(se Tttal (se Table 5.8 Iteratn and tmn marsns fr AIM mdels and Case 5 rblem 45

164 46

165 Chater 6 Cnlusns and Future wrk 6. Cnlusns A new General Purse Researh Smulatr (GPRS has been develed. It nrrates varus smulatn mdels and tehnques s that ther advantaes and dsadvantaes an be nvestated n a nsstent manner. The desn f GPRS s suh that t an be used by multle researhers. The verall desn and bas mlementatns f GPRS have been mleted. Varus mdels and tehnques n GPRS have been tested, nludn a blakl mdel, several mstnal mdels, unstrutured rd handln and mult-nt flux arxmatn. It s nw ssble fr ther researhers t start wrkn n extensns f ndvdual mdules. A bas ntrdutn t the struture f GPRS s nluded n Aendx A. A new General Frmulatn Arah has been develed and mlemented n GPRS t faltate the mlementatn f dfferent mdels. Ths arah an enerate almst any mdel, reardless f the seletn f varables and the level f mltness. All f the eratns are erfrmed n ndvdual rdblks, s dfferent varables and dfferent mlt levels an be used n dfferent rdblks. Ths s neessary fr the buldn f AIM mdels. The arah resented s sutable fr all knds f smulatns. In GPRS, t s nly used t enerate dfferent mstnal mdels. An extensve study has been nduted fr the new IMPSAT (mlt ressure, mlt saturatns and exlt mle fratns mdel, whh s atually a seal ase f the General Frmulatn Arah. We have analyzed the haraterst f ths IMPSAT mdel, derved ts stablty rtern and suested an effent way fr ts nstrutn. We have als mared the erfrmane f the IMPSAT mdel wth the erfrmane f IMPES and FIM mdels usn varus mstnal rblems. In addtn, we have rsed three new IMPSAT based AIM mdels, and ther erfrmane has been mared wth the erfrmane f the tradtnal AIM mdel. Dfferent mstnal mdels, varyn n ther seletn f varables and mlt levels, have been evaluated usn several mstnal rblems. These rblems were 47

166 seleted t be dffult n ne sef area. Frm the results btaned, we an draw the fllwn nlusns fr dfferent varable seletns, fr dfferent mlt levels and fr the new IMPSAT based AIM mdels: Dfferent Varable Seletns The same mdels (FIM r IMPES wth Tye A and Tye B varables have the same slutns, beause they are slvn the same equatns. Fr the same mdels (FIM r IMPES wth Tye A and Tye B varables, there are n majr dfferenes n the Newtn teratn numbers. Fr FIM mdels, Tye B varables need mre lnear slver teratns (and hene mutatnal st than Tye A varables. Tye B varables als nrease the st f Jaban matrx nstrutn, beause f the need t use the han rule t alulate dervatves. Hene Tye A FIM mdel s mre effent than Tye B FIM mdel. Fr IMPES mdels, there are n snfant dfferenes n lnear slver teratns, sne the ressure system s always the same. Hwever, Tye B varables are mre effent than Tye A varables n Jaban matrx alulatn (Cats, 999. Hene the ttal st f Tye B IMPES mdel s lwer than the ttal st f Tye A IMPES mdel. Dfferent Imlt Levels FIM mdels are mst stable, and they an use lare tmeste szes, but the st f eah Newtn teratn s als hh, eseally fr rblems wth lare number f mnents. IMPES mdels are least stable, beause nly ressure s treated mltly, but ther st s the lwest fr eah Newtn teratn. Fr mstnal rblems, the stable tmeste szes f the IMPES mdel are small, due t the hh as mblty. Als t has sudden jums beause f the dsntnuus hase veltes. Fr relatvely easy rblems, the IMPES mdel s faster than the FIM mdel, beause t slves fr fewer unknwns. Hwever ths advantae s lst fr hard rblems, where the tmeste restrtn makes IMPES even slwer than the FIM mdel. The IMPES mdel has less numeral dffusn than the FIM mdel. 48

167 The IMPSAT mdel s muh mre stable than the IMPES mdel, due t the mlt treatment f saturatns, and t an use tmeste szes f u t 0 tmes larer than what s ssble wth IMPES. Als the stable tmeste szes f the IMPSAT mdel hane mre smthly than the stable tmeste szes f the IMPES mdel. The st f the IMPSAT mdel fr eah Newtn teratn s fxed, ndeendent f the number f mnents, just as s the ase fr the IMPES mdel. Cmared t the IMPES mdel, the IMPSAT mdel needs far fewer tmestes and Newtn teratns, and enerally an ut the runnn tme by half. Fr rblems that are nt very hard, the mrved stablty f the IMPSAT mdel enables t t use the same number f tmestes and Newtn teratns as the FIM mdel. Beause the number f unknwns t be slved s redued, IMPSAT s a heaer alternatve t the FIM mdel. The IMPSAT mdel has the same level f numeral dffusn as the FIM mdel. Dfferent AIM Mdels All AIM mdels are mre effent than the FIM mdel, sne hh flw rates are enerally restrted t a small rtn f the reservr and the FIM mdel s nly needed n the rens f hh flw rates. Fr rblems that are hard fr IMPES, the IMPSATFIM mdel s muh faster than the tradtnal IMPESFIM mdel due t the mrved stablty f the IMPSAT mdel. Fr rblems that are easy fr IMPSAT, the IMPESIMPSAT mdel s faster than the tradtnal IMPESFIM mdel, due t the redutn n the number f unknwns that need t be slved mltly. The IMPESIMPSATFIM mdel s the mst eneral AIM mdel, wth rerly tuned erentaes fr eah frmulatn, t wll always uterfrm the ther three AIM mdels. In summary, the key nlusns abut the erfrmane f mstnal mdels are as fllwn: Tye A FIM mdel s better than Tye B FIM mdel, bth n the ndtn f the Jaban matrx and the st f buldn the Jaban matrx. 49

168 Tye B IMPES mdel s mre effent than Tye A IMPES mdel n the st f buldn the ressure mlt Jaban matrx. The IMPSAT mdel s enerally ver 50% faster than the IMPES mdel due t ts mrved stablty. Fr rblems that are nt very hard fr IMPSAT, the IMPSAT mdel s heaer than the FIM mdel sne t redues the number f unknwns that have t be slved mltly. Ths s eseally true fr rblems wth a lare number f mnents. The IMPSAT based AIM mdels are mre flexble, mre stable and less exensve than the tradtnal AIM mdel. Wth rerly tuned erentaes fr eah frmulatn, the IMPESIMPSATFIM mdel wll always uterfrm the tradtnal AIM (IMPESFIM mdel. 6. Future Wrk GPRS s a eneral-urse researh smulatr. It requres further testn, mrvements and enhanements. Wrk s eseally needed n the fllwn areas: Evaluatn f IMPSAT mdels (nludn sme mle fratns n the mlt set Evaluatn f quas-impsat mdel, where the mle fratns n the transmssblty art are udated teratn by teratn Imrvements n rendtns fr AIM mdels Better slver fr unstrutured rd Imrvements n flash alulatns Imlementatn f tehnques fr arallel mutn Surfae falty mdeln Cmlex well mdeln Autmat seletn f the level f mltness n AIM mdels Thermal mdeln Fratured reservr smulatn Enhanements n the treatment f water t nlude as slublty n water and water var n as Other strutured rd (ylndral, urvlnear, rner-nt Culed mdeln f emehans Interfae wth ther tls, and allw use f varus unts 50

169 Nmenlature A Area f surfae between tw rdblks, [ft ] B Frmatn vlume fatr f hase (blak-l, [bbl/stb: l&wat; ft 3 /SCF: as] D Deth, [ft] F Ttal mles f hydrarbn n ne unt vlume f flud, [mle/ft 3 ] f, Fuaty f mnent n hase,[sa] Gravty fre nstant, [3. ft/se ] K k kr l M M w MW MW n nh n,, Equlbrum rat f mnent Abslute ermeablty, [md] Relatve ermeablty f hase Hydrarbn lqud mle fratn n hydrarbn flud Ttal mles f hydrarbn mnent,[lb-mle] Ttal mles f water, [lb-mle] Mleular weht f mnent, [lbm/lb-mle] Mleular weht f hase, [lbm/lb-mle] Number f mnents Number f hydrarbn mnents Number f hases P Callary ressure between hase and hase,[sa] W Blk ressure, [sa] Wellbre ressure f well W,[sa] Mass flw rate f mnent at well W,[lbm/day] W Q W q Vlumetr flw rate f hase at well W,[ft 3 /day] R, Slublty f mnent n hase, [SCF/STB: R, ] r Ttal mles f hydrarbn mnent n ne unt vlume f flud, [lb-mle/ft 3 ] S T Saturatn f hase Transmssblty, [md ft] V Vlume, [ft 3 ] Vφ Pre vlume, [ft 3 ] V Vlume f hase,[ft 3 ] VT Ttal flud vlume, [ft 3 ] 5

170 v Vlumetr flw rate f hase,[ft 3 /day] W Ttal mles f water n ne unt vlume f flud, [lb-mle/ft 3 ] W WI, Well Index f well W,[md ft] Mle fratn f mnent n hase x Mle fratn f mnent n the l hase, same as, y Mle fratn f mnent n the as hase, same as, z Z Overall mle fratn f mnent Cmressblty fatr Arnyms AIM AMG BHP CPR EOS FIM GMRES GPRS ILU IMPES IMPSAT Tye A Tye B Adatve mlt Alebra mult-rd Bttm hle ressure Cnstraned ressure resdual Equatn f state Fully mlt Generalzed mnmum resdual General Purse Researh Smulatr Inmlete LU demstn Imlt ressure and exlt saturatns Imlt ressure and saturatns Tye A varables (ressure, saturatns and mnent mle fratns Tye B varables (ressure, verall mnent denstes Greek Catal Delta (hane δ Delta (hane α, β Unt nversn nstants Mblty f hase,[/] λ Densty f hase,[lbm/ft 3 ] Densty f mnent at standard ndtn, [lbm/ft 3 ] µ Vssty f hase,[] 5

171 Φ φ Ptental f hase,[sa] Prsty Subsrts Phase Cmnent w Water Suersrts n Tme level n n Tme level n υ W Iteratn level Well 53

172 54

173 Referenes. Aavatsmark, I., Barkve, T. and Mannseth, T.: Cntrl Vlume Dsretzatn Methds fr 3D Quadrlateral Grds n Inhmeneus, Anstr Reservrs, aer SPE 38000, reedns f the 4 th SPE Symsum n Reservr Smulatn, Dallas, T, June 8-, 997. Abu-Kassem, J.H. and Azz, K.: Handln f Phase Chane n Thermal Smulatrs, JPT, Setember 985, As, G., Dleshall, S. and Farkas, E.: General Purse Cmstnal Mdel, SPEJ, Auust 985, Aut, R., Edwards, G.E., Verma, S. and Azz, K.: Flexble Streamlne-Ptental Grds Wth Dsretzatn n Hhly Dstrted Cells, reedns f the 6 th Eurean Cnferene n the Mathemats f Ol Revery, Peebles, Stland, Setember 8-, Almehadeb, R.A., Azz, K. and Pedrsa, D.A.: A Reservr Wellbre Mdel fr Multhase Injetn and Prdutn, aer SPE 794, reedns f the SPE Mddle East Ol Tehnal Cnferene and Exhbtn, Manama, Bahran, Marh - 4, Andersn, E., Ba, Z., Bshf, C., Blakfrd, S., Demmel, J., Dnarra, J., Du Crz, J., Greenbaum, A., Hammarln, S., Mkenny, A. and Srensen, D.: LAPACK Users Gude, Thrd Edtn, Sety fr Industral and Aled Mathemats, Azz, K.: Fundamentals f Reservr Smulatn, lass ntes fr PE3 (Reservr Smulatn lass, summer Azz, K. and Settar, A.: Petrleum Reservr Smulatn, Aled Sene Publshers, Lndn, Enland, Azz, K. and Wn, T.W.: Cnsderatns n the Develment f Multurse Reservr Smulatn Mdels, reedns f the st and nd Internatnal Frum n Reservr Smulatn, Albah, Austra, Setember -6, 988 and Setember 4-8,

174 0. Baker, L.E. and Luks, K.D.: Crtal Pnt and Saturatn Pressure Calulatn fr Mult-Cmnent Systems, SPEJ, February 980, 5-4. Balln, P.R., Azz, K., Jurnel, A.G. and Zul, L.: Quantfyn the Imat f Gelal Unertanty n Reservr Freasts, aer SPE 538, reedns f the th SPE Symsum n Reservr Smulatn, New Orleans, CA, February 8- Marh 03, 993. Behe, A. and Frsyth Jr., P.A.: Pratal Cnsderatns fr Inmlete Fatrzatn Methds n Reservr Smulatn, aer SPE 63, reedns f the 7 th SPE Symsum n Reservr Smulatn, San Frans, CA, Nvember 5-8, Bran, C.M. and Rdruez, F.: A Sem-Imlt Frmulatn fr Cmstnal Reservr Smulatn, aer SPE 7053, SPE Advaned Tehnly Seres, Vl. 4, N., Byer, T.J.: Prendtned Newtn Methds fr Reservrs Wth Surfae Faltes, Ph.D. Dssertatn, Stanfrd Unversty, June Ca, H.: GPRS Pakae (de, data fle and user s manual 00, Castelln, A., Edwards, M.G. and Durlfsky, L.J.: "Flw Based Mdules fr Grd Generatn n Tw and Three Dmensns", reedns f the 7 th Eurean Cnferene n the Mathemats f Ol Revery, Baven, Italy, Setember 5-8, Chen, W.H., Durlfsky, L.J., Enqust, B. and Osher, S.: Mnmzatn f Grd Orentatn Effets Thruh Use f Hher-Order Fnte Dfferene Methds, aer SPE 887, reedns f the 66 th SPE Annual Tehnal Cnferene and Exhbtn, Dallas, T, Otber 6-9, Chen, M.C.H., Lee, S.T. and Chen, W.H.: A New Fully Imlt Cmstnal Smulatr, aer SPE 3385, reedns f the 8 th SPE Symsum n Reservr Smulatn, Dallas, T, February 0-3, Cats, K.H.: An Equatn f State Cmstnal Mdel, SPEJ, Otber 980,

175 0. Cats, K.H.: A Nte n IMPES and Sme IMPES Based Smulatn Mdels, aer SPE 49774, reedns f the 5 th SPE Symsum n Reservr Smulatn, Hustn, T, February 4-7, 999. Cats, K.H.: IMPES Stablty: The Stable Ste, aer SPE 695, reedns f the 6 th SPE Symsum n Reservr Smulatn, Hustn, T, February -4, 00. Cats, K.H., Thmas, L.K. and Persn, R.G.: Cmstnal and Blak-Ol Reservr Smulatn, aer SPE 9, reedns f the 3 th SPE Symsum n Reservr Smulatn, San Antn, T, February -5, Cats, K.H., Thmas, L.K. and Persn, R.G.: Cmstnal and Blak-Ol Reservr Smulatn, aer SPE 50990, SPE Reservr Evaluatn & Enneern, Auust 998, Cment, M. and Sweet, R.A.: Mesh Refnement fr Parabl Equatns, J. Cm. Phys., Deember 973, De, M.D., Nutakk, R. and Orr Jr., F.M.: Shmdt-Wenzel and Pen-Rbnsn Equatn f State fr CO -Hydrarbn Mxtures: Bnary Interatn Parameters and Vlume Translatn Fatrs, aer SPE 8796, reedns f the SPE Calfrna Renal Meetn, Bakersfeld, CA, Arl 5-7, Dn, Y. and Lemnner, P.: Use f Crner Pnt Gemetry n Reservr Smulatn, aer SPE 9933, reedns f the Internatnal Meetn n Petrleum Enneern, Bejn, Chna, Nvember 4-7, Dru, A.H., Pta, J.A., Fun, L.S., Al-Zamel, N., Al-Shaalan, T. and Dreman, W.T.: Meaell Reservr Smulatn: the Quest fr Hh Reslutn n Full-Feld Mdeln, reedns f the 6 th Internatnal Frum n Reservr Smulatn, Salzbur, Austra, Setember 3-7, Dnarra, J.: Perfrmane f Varus Cmuters Usn Standard Lnear Equatns Sftware, Marh 9, 00, htt:// 9. Dnarra, J., Lumsdane, A., Pz, R. and Remntn, K.A.: IML V.., Iteratve Methds Lbrary Referene Gude, Natnal Insttute f Standards and Tehnly, Arl

176 30. Edwards, G.M.: Slt Ellt Tensr Oeratrs fr Reservr Smulatn, reedns f the ICFD Cnferene n Numeral Methds fr Flud Dynams, Oxfrd Unversty, UK, Marh 3 Arl 03, Edwards, G.M., Aut, R. and Azz, K.: Quas K-Orthnal Streamlne Grds: Grddn and Dsretzatn, aer SPE 4907, reedns f the 73 rd SPE Annual Tehnal Cnferene and Exhbtn, New Orleans, Lusana, Setember 7-30, Farkas, E.: Cmarsn f Lnearzatn Tehnques f Nnlnear Partal Dfferental Equatns n Numeral Reservr Smulatn, Ph.D. dssertatn, Reservr Enneern Deartment f the Mntanunverstat Leben, Austra, Frsyth, P.A. and Sammn, P.H.: Pratal Cnsderatns fr Adatve Imlt Methds n Reservr Smulatn, J. Cm. Phys., Vl. 6, 986, Fun, L.S., Hebert, A.D. and Nhem, L..: Reservr Smulatn Wth a Cntrl Vlume Fnte Element Methd, aer SPE 4, reedns f the th SPE Symsum n Reservr Smulatn, Anahem, CA, February 7-0, Fussell, D.D. and Yansk, J.L.: An Iteratve Sequene fr Phase-Equlbra Calulatns Inrratn the Redlh-Kwn Equatn f State SPEJ, June 978, Fussell, L.T. and Fussell, D.D.: An Iteratve Tehnque fr Cmstnal Reservr Mdels SPEJ, Auust 979, Gequest, Shlumberer: Else 00 Tehnal Desrtn 000A, 000, htt:// 38. Gequest, Shlumberer: Else 300 Tehnal Desrtn 000A, 000, htt:// 39. Gequest, Shlumberer: FlGrd Referene Manual 000A, 000, htt:// 40. Gunasekera, D., Chlds, P., Herrn, J. and Cx, J.: A Mult-Pnt Flux Dsretzatn Sheme fr General Plyhedral Grds, aer SPE 48855, reedns f the 6 th SPE Internatnal Ol&Gas Cnferene and Exhbtn, Bejn, Chna, Nvember -6,

177 4. Gunasekera, D., Cx, J. and Lndsey, P.: The Generatn and Alatn f K- Orthnal Grd Systems, aer SPE 37998, reedns f the 4 th SPE Symsum n Reservr Smulatn, Dallas, T, June 8-, Henemann, Z.E. and Brand, C.W.: Mdeln Reservr Gemetry wth Irreular Grds aer SPE 84, reedns f the 0 th SPE Symsum n Reservr Smulatn, Hustn, T, February 6-8, Jurnel, A.G.: Gestatsts fr Reservr Charaterzatn, aer SPE 0750, reedns f the 65 th SPE Annual Tehnal Cnferene and Exhbtn, New Orleans, CA, Setember 3-6, Kendall, R.P., Mrrell, G.O., Peaeman, D.W., Sllman, W.J. and Watts, J.W.: Develment f a Multle Alatn Reservr Smulatr fr Use n a Vetr Cmuter, aer SPE 483, reedns f the SPE Mddle East Ol Tehnal Cnferene, Manama, Bahran, Marh 4-7, Kenyn, D. and Behe, G.A.: Thrd SPE Cmaratve Slutn Prjet: Gas Cyln f Retrrade Cndensate Reservrs, aer SPE 78, reedns f the 7 th SPE Symsum n Reservr Smulatn, San Frans, CA, Nvember 5-8, Klluh, J.E.: Nnth SPE Cmaratve Slutn Prjet: A Reexamnatn f Blak- Ol Smulatn, aer SPE 90, reedns f the 3 th SPE Symsum n Reservr Smulatn, San Antn, T, February -5, Klluh, J.E. and Cmmander, D.: Salable Parallel Reservr Smulatn n a Wndws NT-Bases Wrkstatn Cluster, aer SPE 5883, reedns f the 5 th SPE Symsum n Reservr Smulatn, Hustn, T, February 4-7, Kberber, S.: An Autmat, Unstrutured Cntrl Vlume Generatn System fr Gelally Cmlex Reservrs, aer SPE 3800, reedns f the 4 th SPE Symsum n Reservr Smulatn, Dallas, T, June 8-, Larx, S., Vasslevsk, Y. and Wheeler, M.: Iteratve Slvers f the Imlt Parallel Aurate Reservr Smulatr (IPARS, I: Snle Pressr Case, TICAM rert 00-8, (000, htt:// 50. Landmark Grahs Crratn: BltzPak User s Gude, 998, htt:// 59

178 5. Lee, J., Kasa, E. and Kelkar, M.G.: Analytal Usaln f Permeablty fr 3D Grdblks, SPE aer 7875, reedns f the SPE Western Renal Meetn, Ln Beah, CA, Marh 3-5, Lee, S.H., Durlfsky, L.J., Luh, M.F. and Chen, W.H. Fnte Dfferene Smulatn f Gelally Cmlex Reservrs Wth Tensr Permeabltes, aer SPE 3800, reedns f the 4 th SPE Symsum n Reservr Smulatn, Dallas, T, June 8-, Lm, K.T., Shzer, D.J. and Azz, K.: A New Arah fr Resdual and Jaban Array Cnstrutn n Reservr Smulatrs, aer SPE 848, reedns f the SPE Petrleum Cmuter Cnferene, Dallas, T, July 3-Auust 03, Naul, E.C.: Use f Dman Demstn and Lal Grd Refnement n Reservr Smulatn, Ph.D. Dssertatn, Stanfrd Unversty, Marh Nhem, L.., Azz, K. and L, Y.K.: A Rbust Iteratve Methd fr Flash Calulatns Usn the Save-Redlh-Kwn r the Pen-Rbnsn Equatn f State, SPEJ, June 983, Nlen, J.S.: Treatment f Wells n Reservr Smulatn, reedns f the 3 rd Internatnal Frum n Reservr Smulatn, Baden, Austra, July 3-7, Odeh, A.S.: Cmarsn f Slutns t a Three-Dmensnal Blak-Ol Smulatn Prblem, aer SPE 973, JPT, January 98, Pala, C.L. and Azz, K.: Use f Vrn Grd n Reservr Smulatn, SPE Advaned Tehnly Seres, Arl 994, Peaeman, D.W.: Calulatn f Transmssbltes f Grdblks Defned by Arbtrary Crner Pnt Gemetry, aer SPE 37306, 996 (unublshed, but avalable frm SPE e-lbrary 60. Peaeman, D.W.: Interretatn f Well-Blk Pressures n Numeral Reservr Smulatn, SPEJ, June 978, Pedrsa, O.A. and Azz, K.: Use f Hybrd Grd n Reservr Smulatn, aer SPE 3507, reedns f the 8 th SPE Symsum n Reservr Smulatn, Dallas, T, February 0-3,

179 6. Pen, D.Y. and Rbnsn, D.B.: A New Tw-Cnstant Equatn f State, Ind. En. Chem. Fundam., Vl. 5, 976, Pntn, D.K.: Crner Pnt Gemetry n Reservr Smulatn, reedns f the st Eurean Cnferene n Mathemats Of Ol Revery, Cambrde, Enland, 989, P.R.Kn(ed., Clarendn Press, Oxfrd 99, Pz, R., Remntn, K.A. and Lumsdane, A.: SarseLb v..5, Sarse Matrx Class Lbrary Referene Gude, Natnal Insttute f Standards and Tehnly, Arl Prevst, M., rvate mmunatn, Prram Develment Crratn (PDC: GrdPr, htt:// 67. Quandalle, P. and Savary, D.: An Imlt n Pressure and Saturatns Arah t Fully Cmstnal Smulatn, aer SPE 843, reedns f the 0 th SPE Symsum n Reservr Smulatn, Hustn, T, February 6-8, Rahfrd, H. and Re, J.: Predure fr Use f Eletrn Dtal Cmuters n Calulatn Flash Varzatn Hydrarbn Equlbrum, JPT, 4, 95, Rtzdrf, H.: Readme t AMG Pakae, GMD Sftware, Russell, T.F.: Stablty Analyss and Swthn Crtera fr Adatve Imlt Methds Based n the CFL Cndtn, aer SPE 846, reedns f the 0 th SPE Symsum n Reservr Smulatn, Hustn, T, February 6-8, Saad, Y. and Shultz, M.: GMRES: A Generalzed Mnmum Resdual Alrthm fr slvn nn-symmetr lnear systems, SIAM J. S. Statst. Cmut., Vl. 7, 986, Sammn, P.H.: A Nne-Pnt Dfferenn Sheme Based n Hh-Order Stream Tube Mdeln, aer SPE 3, reedns f the th SPE Symsum n Reservr Smulatn, Anahem, CA, February 7-0, Shzer, D.J. and Azz, K.: Use f Dman Demstn fr Smultaneus Smulatn f Reservr and Surfae Faltes, aer SPE 7876, reedns f the SPE Western Renal Meetn, Ln Beah, CA, Marh 3-5, Stueben, K.: Alebra Multrd (AMG: Exerenes and Cmarsns, reedns f the Internatnal Multrd Cnferene, Cer Muntan, CO, Arl 6-8, 983 6

180 75. Sukrman, Y.B. and Lews, R.W.: Three-Dmensnal Fully Culed Flw: Cnsldatn Mdeln Usn Fnte Element Methd, aer SPE 8755, reedns f the SPE Asa Paf Ol & Gas Cnferene, Melburne, Australa, Nvember 7-0, The GOCAD ru: GOCAD, htt:// 77. Thmas, G.W. and Thurnau, D.H.: Reservr Smulatn Usn an Adatve Imlt Methd, SPEJ, Otber 983, Tran, T. and Jurnel, A.: Autmat Generatn f Crner-Pnt-Gemetry Flw Smulatn Grds frm Detaled Gestatstal Desrtns, Stanfrd Center fr Reservr Freastn, Vasslevsk, Y.: Tw Marks n the Cmbnatve Tehnque, rvate mmunatn, Verma, S.: Flexble Grds fr Reservr Smulatn Ph.D. Dssertatn, Stanfrd Unversty, June Verma, S. and Azz, K.: A Cntrl Vlume Sheme fr Flexble Grds n Reservr Smulatn, aer SPE 37999, reedns f the 4 th SPE Symsum n Reservr Smulatn, Dallas, T, June 8-, Verma, S. and Azz, K.: Tw- and Three-Dmensnal Flexble Grds fr Reservr Smulatn, reedns f the 5 th Eurean Cnferene n Mathemats f Ol Revery, Leben, Austra, Setember 3-6, Walas, S.M.: Phase Equlbra n Chemal Enneern", Butterwrth Publshers, Bstn, MA, Watts, J.W.: A Cmstnal Frmulatn f the Pressure and Saturatn Equatns aer SPE 44, SPE Reservr Enneern, May 986, Watts, J.W.: Reservr Smulatn: Past, Present, and Future aer SPE 3844, reedns f the 4 th SPE Symsum n Reservr Smulatn, Dallas, T, June 8-, Whtsn, C.H. and Mhelsen, M.L.: The Neatve Flash, reedns f the Internatnal Cnferene n Flud Prertes and Phase Equlbra fr Chemal Press Desn, Banff, Canada, Arl 30-May 05, 986 6

181 87. Wn, T.W., Frzabad, A. and Azz, K.: Relatnsh f the Vlume-Balane Methd f Cmstnal Smulatn t the Newtn-Rahsn Methd, aer SPE 844, SPE Reservr Enneern, Auust 990, Yansk, J.L. and MCraken, T.A.: A Nne-Pnt Fnte-Dfferene Reservr Smulatn fr Realst Predtn f Adverse Mblty Rat Dslaement, SPEJ, Auust 979, Yun, L.C.: An Effent Fnte Element Methd fr Reservr Smulatn, aer SPE 743, reedns f the 53 rd SPE Annual Tehnal Cnferene and Exhbtn, Hustn, T, Otber -3, Yun, L.C. and Stehensn, R.E.: A Generalzed Cmstnal Arah fr Reservr Smulatn SPEJ, Otber 983,

182 64

183 Aendx A Overvew Of GPRS Mdular bjet rented desn s used fr GPRS. All f the de s wrtten n standard C. Whle the desn wll hel new users n ettn started wth GPRS, t wll stll requre sme effrt t beme fully famlar wth ths de. GPRS s a b and mlex rram, urrently t nludes abut 0 fles, whh are searated nt subdretres. In rder t faltate the use f GPRS, d dumentatn s essental (Ca, 00. T fully understand GPRS, users shuld frst learn the bas struture f GPRS, whh wll be ntrdued n ths art. After that, a d understandn f the bas alrthms and tehnques used n GPRS (refer t Chater fr detals s neessary. Fnally GPRS has a lt f nternal mments, whh shuld be very helful n understandn ndvdual lasses and methds. In ths art, we wll frst exlre the system mdels fr GPRS level by level, frm t dwn, after that we wll dsuss ts flw sequene. We wll als lst the dretres, fles and lbrares f GPRS, and exlan ther use. Fnally, fr end users, we wll exlan the struture f nut fles. A. The System Mdel The system mdel shws the bas lasses and ther relatns, and t s very helful fr the understandn f the struture f GPRS. Sne GPRS s a mlex de, t wll be dffult t exlan the system mdel usn nly ne fure. T make t easer t understand, we wll shw the system mdels level by level, and exlan eah level n detal. Fure A. shws the hysal struture f an l feld, whh nssts f several reservrs, lts f wells, and sme surfae faltes. Frm a mutatnal nt f vew, we an reast the system n Fure A. as the ne shwn n Fure A., whh s atually the t level (feld level system mdel fr GPRS. Currently, surfae falty mdeln s nt nluded n GPRS, s t s left ut n Fure A.. 65

184 Feld Reservrs Wells Surfae Faltes Res Res W W W3 W4 W5 Fure A. Struture f an l feld Feld Bundary Udate Dman Dman Res Wells Res Wells W W W4 W3 W5 Fure A. Feld level system mdel 66

185 In Fure A., eah l feld s slt nt several dmans, and eah dman nludes ne reservr and all f the wells that are n that reservr. Multle dmans are best handled by arallel mutn methds. Ths an be dne by frst assnn eah dman t a dfferent CPU, then slvn eah f them searately, after that, the bundares between dmans need t be udated (assn data between dfferent CPUs. Ths ress has t be reeated untl the whle feld has nvered. If there are mre dmans than CPUs, multle dmans an be assned t the same CPU. Currently, GPRS s nly used fr sequental smulatn. In rder t urade t fr arallel mutatn, sme wrk s needed t mlement the bundary udate. The dman level system mdel s shwn n Fure A.3. As we knw frm Fure A., eah dman nludes a reservr and several wells. Besdes that, eah dman als has an equatn seletr and a lnear slver. Basally, n GPRS, the Jaban matres are alulated searately fr the reservr art and fr the well art, the equatn seletr s used t reast the Jaban wth desred varables and mlt levels. After that the Jaban matres are assed t the lnear slver and eed tether there fr the lnear slve. Fnally, the slutn es bak n the ste dretn. Dman EqnSeletr LnearSlver Reservr Wells swth varables full set t rmary set 3 redue mlt level slve AB al rs al Ja and RHS al Ja and RHS Fure A.3 Dman level system mdel 67

186 As mentned abve, the Jaban matres are alulated searately fr the reservr art and fr the well art. Fure A.4 and Fure A.5 shw the system mdel fr eah f them. Frm Fure A.4, we an see that, eah reservr nludes a rd and a frmulatn, rd art enerates the rd nfrmatn and asses t t the frmulatn art. The frmulatn art alulates the rdblk rertes and bulds the reservr art f the Jaban matrx and the RHS. In GPRS, the rd nfrmatn s ether nternally enerated (urrently nly fr Cartesan rd, r read n frm the utut f a rddn sftware. The system mdel fr the well art has a smlar struture (as shwn n Fure A.5. In ths art, well nfrmatn s enerated frm the well mletn data and assed t the well ntrl mdule, whh alulates the well art f the Jaban matrx and the RHS. Currently, fur tyes f well ntrls are mlemented, and they are BHP, l flw rate, as flw rate and water flw rate ntrl. Reservr Grd Grd nfrmatn Frmulatn Cartesan Inut frm rddn sftware Fure A.4 Reservr level system mdel 68

187 Well Well Cmletn Well Infrmatn Well Cntrls BHP Ol Rate Gas Rat Water Rate Fure A.5 Well level system mdel Fure A.6 shws the last system mdel. The entre reservr related alulatns are arred ut here. The alulatns are ranzed usn fve mathematal mdules: ne fr rk, ne fr flud, ne fr rk/flud, ne fr hase mblty and the last ne fr flw equatns. Mst f the rdblk rertes are alulated n the frst fur mdules, and the reservr art f the Jaban matrx and the RHS are alulated and assembled n the last mdule. If neessary, eah mdule an have multle sublasses fr dfferent mdels. Fr examle the flud art has a blak-l and a mstnal mdule. If a mdule has nnetns t the hysal wrld, suh as the rk mdule, t wll have a nter t the rresndn hysal bjet. All f the mathematal mdules are ranzed n a multlevel nhertane struture, they all share the same ubl nterfae (methds, and dynam bndn s used t determne the atual bjets and methds requred durn a run. 69

188 Frmulatn Rk Mdule Flud Mdule RkFlud Mdule Mblty Mdule Flw Eqn Mdule rk m flash alulate and assemble sendary art Ja. and RHS rela erm assemble hase mblty alulate and assemble rmary art Ja. and RHS BO Flud Mdule Cm Flud Mdule BO Mblty Mdule Cm Mblty Mdule BO Flw Eqn Mdule Cm Flw Eqn Mdule R BOFlu CmFlud RkFlud Fure A.6 Frmulatn level system mdel 70

189 A. Flw Sequene Havn shwn the system mdels and lasses used n GPRS, we wll nw revew the flw sequene f GPRS. The essental stes n a smulatr are ven belw (Azz, 996:. Read nut data (rblem defntn. Intalze (allate data and set ntal ndtns 3. Start tme ste alulatns Intalze wth ld tmeste data Start the Newtn teratn Calulate rdblk rertes Lnearze (alulate and assemble Jaban and RHS Slve the lnear system Perfrm Newtn udate Chek nverene, d anther Newtn teratn f neessary 4. Prnt and lt results at arrate tmes 5. Inrement tme and t Ste 3 f endn ndtns are nt reahed 6. End when run s mlete The mst mrtant ste s Ste 3, whh aunts fr ver 95% f the smulatn tme. The mlementatn f Ste 3 n GPRS s as fllwn: alintprs(; whle(true { alprs(; lnearze(; // rerd ld tmeste values // start Newtn teratns // alulate rdblk rertes // alulate and assemble Jaban and RHS nvf nveref(; // hek resdual nverene mlnearslver->slve( // slve the lnear system newtnudate(; // newtn udate nvy nveredy(; // hek varable hane nverene f(nvf && nvy break; // nvered } // end f Newtn teratns 7

190 Wthn ths ste, fur methds handle mst f the alulatns fr the reservr art. They are: alintpr(, alpr(, lnearze( and newtnudate(. They have smlar struture as that f lnearze(: vd Reservr::lnearze( { // --- eah mathematal mdule lnearzes --- lst<mthmdelbase*>::teratr ter mmthmdels.ben(; whle (ter! mmthmdels.end( { (*ter->lnearze(dt; ter; } } // --- eah well lnearzes --- vetr<well*>::teratr ter mwells.ben(; whle (ter! mwells.end( { f((*ter->welloen( (*ter->lnearze(; ter; } Basally, mst f the reservr alulatns are erfrmed by these mathematal mdules. Sne all f these mdules are nherted frm the same base lass, they share the same ubl nterfae, we an just l thruh them, and let eah f them d ts wrk n turn. The nly thn we need t take are f s t rram the rret ubl methds (abve fur methds fr eah mdule. Usn ths arah, eah Newtn teratn bemes smle and flexble. If at any tme, a new mdule s needed, we nly need t reate t, add t t the lst f mdules, and mlement the abve fur ubl methds. If any f the abve fur methds s nt neessary fr a mdule, then t wll be emty fr that mdule. A.3 Sub-dretres and Fles Besdes the man rram (Man. whh s lated n the base dretry, there are sub-dretres. Here we wll ntrdue eah f them and the fles ntaned n them. 7

191 feld Feld.h/: feld lass flud Physal flud Flud.h/: Phase.h/: Cm.h/: BOFlud.h/: CmFlud.h/: BOPhase.h/: HCPhase.h/: WaterPhase.h/: flud base lass hase base lass mnent lass blak-l flud lass, sublass f Flud mstnal flud lass, sublass f Flud blak-l hase lass, sublass f Phase mstnal hydrarbn hase lass, sublass f Phase mstnal water hase lass, sublass f Phase rd Grd.h/: Cartesan rd lass mth Frmulatn mdules mthmdelbase.h/: mthrkcmmdel.h/: mthfludmdelbase.h/: mthbofludmdel.h/: mthcmfludmdel.h/: mthrkfludmdel.h/: mthmbltymdel.h/: mthcmmbltymdel.h/: base lass fr all mdules rk mressblty mdule, sublass f mthmdelbase base flud mdule, sublass f mthmdelbase blak-l flud mdule, sublass f mthfludmdelbase mstnal flud mdule, sublass f mthfludmdelbase rk flud mdule, sublass f mthmdelbase hase mblty mdule, sublass f mthmdelbase mstnal hase mblty mdule, sublass f mthmbltymdel 73

192 mthflweqnmdel.h/: base flw equatn mdule, sublass f mthmdelbase mthboflweqnmdel.h/: blak-l flw equatn mdule, sublass f mthflweqnmdel mthcmflweqnmdel.h/: mstnal flw equatn mdule, sublass f mthflweqnmdel res Reservr Reservr.h/: EqnSeletr.h/: EqnSeletrCats.h/: EqnSeletrYun.h/: reservr lass equatn seletr base lass varable Tye A (Cats mdel equatn seletr lass, sublass f EqnSeletr varable Tye B (Yun and Steensn s mdel equatn seletr lass, sublass f EqnSeletr rk Physal rk Rk.h/: hysal rk lass rkflud Physal rk flud (relatve ermeablty RkFlud.h/: hysal rk flud base lass RkFludP.h/: hysal tw-hase rk flud lass, sublass f RkFlud RkFlud3P.h/: hysal three-hase rk flud lass, sublass f RkFlud sm Smulatn nstant defntn Cmlbs.h: defne mmn nluded fles EnumDef.h: defne strutures, Enumerates and nstants used n GPRS 74

193 slver Lnear slvers LnearSlverBase.h/: LaakFullSlver.h/: LaakBandSlver.h/: ImlGMRESSlver.h/: BltzSlver.h/: PrendtnerBase.h/: DaPre.h/: BlkDaPre.h/: ILUPre.h/: AMGPre.h/: TrueIMPESPre.h/: QuasIMPESPre.h/: lnear slver base lass full matrx slver frm LAPACK, sublass f LnearSlverBase band matrx slver frm LAPACK, sublass f LnearSlverBase GMRES slver frm IML (Iteratve Math Lbrary, sublass f LnearSlverBase slver frm BltzPak, sublass f LnearSlverBase rendtner base lass danal saln rendtner, sublass f PrendtnerBase blk danal saln rendtner, sublass f PrendtnerBase Inmlete LU demstn (ILU0 rendtner frm SarseLb, sublass f PrendtnerBase Alebra Mult-Grd (AMG rendtner frm GMD sftware, sublass f PrendtnerBase Cnstrant Pressure Resdual (CPR rendtner, true IMPES s used t enerate the ressure equatn, sublass f PrendtnerBase Cnstrant Pressure Resdual (CPR rendtner, quas IMPES s used t enerate the ressure equatn, sublass f PrendtnerBase utl Utltes FleIO.h/: Table.h/: Strlb.h/: fle nut and utut lass table read n and lk u lass strn eratn lass 75

194 FCwraer.h/: MyMatrx.h/: DataVetr.h/: IntDataVetr.h/: DFDataVetr.h/: CnnDataVetr.h/: DataPl.h/: Frtran t C wraer lass fr Frtran subrutnes matrx eratns lass, sutable fr small matrx vetr f data base lass vetr f nteer data lass, base lass f DataVetr vetr f duble data lass, base lass f DataVetr vetr f nnetn data lass, base lass f DataVetr data l lass, stre and rvde aess t all DataVetr bjets well Well.h/: WellCml.h.: WellCnt.h/: WellBHPCnt.h/: WellORateCnt.h/: WellGRateCnt.h/: WellWRateCnt.h/: well lass well mletn lass wellntrlbaselass well bttm hle ressure (BHP ntrl lass, sublass f WellCnt well nstant l flw rate ntrl lass, sublass f WellCnt well nstant as flw rate ntrl lass, sublass f WellCnt well nstant water flw rate ntrl lass, sublass f WellCnt A.4 Lbrares GPRS uses several ubl dman lbrares, and mst f them are used n the lnear slver art. They are dsussed belw: 76

195 STL (Standard Temlate Lbrary Ths lbrary s a ular lbrary that s treated by many as a standard C lbrary. In GPRS, we use the Lst and Vetr temlate lasses frm ths lbrary. LAPACK (Lnear Alebra Pakae (Andersn, 999 Ths s a standard lnear alebra akae, whh an be freely dwnladed frm It has a Frtran versn and a C versn. Currently we are usn the Frtran versn, and wraers are used t nterfae t wth the man C de. Fr mst SGI mahnes, LAPACK s already nstalled. Fr PC s, we use the MKL (Math Kernel Lbrary frm INTEL, whh nludes LAPACK. In GPRS, we use the dret slvers (desv and dbsv frm LAPACK. BltzPak (Landmark, 998 Ths s a mmeral slver akae frm Landmark Grah Crratn. GPRS nterfaes wth t, and uses t as a strutured rd slver. AMG Ths s an Alebra Mult-Grd (AMG slver frm GMD sftware (Rtzdrf, 99, whh an be freely dwnladed frm htt:// It s nly used as a rendtner fr the ressure system n GPRS. SarseLb (Pz et al., 996 Ths s a sarse matres lbrary (freely dwnladable frm htt://math.nst.v/sarselb/ whh nludes sme bas reresentatns fr sarse matrx, suh as mressed rw frmat, and sme bas rendtners, suh as ILU0 (Inmlete LU demstn wthut fll-n. In GPRS, we use t fr sarse matrx reresentatn and ILU0 rendtner. IML (Iteratve Math Lbrary (Dnarra et al., 996 Ths s an teratve slver akae (freely dwnladable frm htt://math.nst.v/ml/, whh nludes GMRES, CG (Cnjuate Gradent, et. In GPRS, we nly use ts GMRES slver. A.5 Inut Fles The man nut fle fr GPRS s nrmally alled rs.n. Wthn t, three fles are nluded (usn INCLUDE keywrd, ne fr the reservr data, ne fr the wells data, and ne fr the ntrl data. Cntents f eah f these fles are desrbes belw: 77

196 Reservr nut fle rd data (rd emetry and rertes flud data, blak-l r mstnal (hase and mnent rertes hase mnent relatn data (exstene f eah mnent n eah hase rk flud data (relatve ermeablty and allary ressure rk data (rk mressblty ntal equlbrum data (WOC, GOC, ntal ressure and verall mstn, et Well nut fle well sefatn (well name, well ru name, whh reservr t belns t, rduer r njetr well mletn data (well blk ndexes, and WI fr eah f them well ntrl data (all avalable well ntrls fr ths well Cntrl fle Tmeste ntrl (ntal, mnmum and maxmum tmeste sze, and ttal smulatn tme Newtn teratn number ntrl (mnmum, maxmum and fxed number f Newtn teratns fr eah tmeste Tmeste nrease ntrl (desred varable hanes and nreasn fatr Newtn teratn nverene ntrl (nverene rtera fr varable hanes and resduals Frmulatn ntrl (whh tye f varables t use, hw many mlt levels, and what are they Lnear slver ntrl (whh lnear slver t use, whh rendtner t use, nverene tlerane and maxmum number f teratns Fr nfrmatn abut the keywrds and data frmat used by the nut fles, refer t the GPRS user s manual, whh s rvded as art f the GPRS akae (Ca,

197 Aendx B: Blak-Ol Smulatn Usn Cmstnal Frmulatn The blak-l mdel s a seal ase f the eneral mstnal mdel, where flash alulatns are redued t exlt relatns, and the rmary equatns an be dretly exressed as funtns f nly rmary varables. Hene a eneral mstnal smulatr shuld be able t erfrm blak-l smulatn ven the rret nut, suh as frmatn vlume fatrs and as slublty. Cnsdern a three-hase (l, as and water blak-l system, assumn as mnent an exst n bth the as and the l hases, and l and water mnents an nly exst n ther wn hases. The eneral mstnal frmulatn an be wrtten as kr kr w [ Vφ ( S x ] [ T ( x Φ ] l [ WI ( x( ] t µ µ l W (B. Fr the blak-l mdel, the hase rertes and mnent mle fratns an be alulated by the relatns shwn belw (Azz, 996: (, hase denstes are nly funtns f ressure, and they are alulated by w w (B. Bw (B.3 B B R, (B.4 where, w, and whh are nut nstants. are the mnent denstes (hases at standard ndtns, B w, B and B are the hase frmatn vlume fatrs, whh are funtns f ressure and nrmally ven as B vs. tables. R, 0 s the as mnent slublty rat n the l hase, whh s als a sle funtn f ressure and ven as a R vs. table. 79

198 µ µ (, hase vsstes are als nly funtns f ressure, nrmally a µ vs. table s ven, and table lku s used t et hase vsstes at sef ressure. (, mnent mle fratns are nt ndeendent varables, they are ether nstants r nly funtns f ressure, and an be alulated by the fllwn relatns:, w w,, w w,, w, w,,, R, R R,, 0 (B.5 (B.6 (B.7 (B.8 An even smler methd s t dretly alulate, R, B frm (Azz, 996, (B.9 80

199 Aendx C Flash Calulatn (Walas, 985 As mentned n Setn.6, the Cub equatn f state an be wrtten as Z 3 sz qz r 0 (C. where, s ( u B q A ( w u B ub 3 r AB wb wb (C. Dfferent Cub equatns f state have dfferent u, w values, and they als use dfferent relatns t alulate arameters a and b f eah mnent: RTC, R TC, Redlh-Kwn, u, w0, b, a 0. 5 P P T Save-Redlh-Kwn, u, w0, Pen-Rbnsn, u, w-, where T C, and P C, b C, RT C,, PC, b where RT C,, PC, where a f w a f w C, R T C, [ fω PC, ω T ( ( T ω R T C, [ fω PC, ( ( C, T T C, ω 0.699ω are the rtal temerature and rtal ressure f eah mnent, T s the reservr temerature, and ω s the aentr fatr f eah mnent. Fr the Cub equatn f state, the mnent fuates and the hase rertes an be alulated frm the fllwn 5 stes (Walas, 985:. Calulate hase arameters and ther dervatves: nh nh a j ( k, j aa j j, (C.3 nh b b ] ] 8

200 8 and n j j j j b b a k a a h, ( (C.4 where j k s the bnary nteratn arameter. RT b B T R a A (C.5 and B B A A, b b B B a a A A (C.6. Calulate mressblty fatr Z and ts dervatves: r qz sz Z (C.7 and q sz Z r Z q Z s Z q sz Z r Z q Z s Z 3 3 (C.8 where 3 ( ( wb wb AB r ub B u w A q B u s (C.9 3. If neessary, erfrm vlume translatn (e.. De et al., 989: n h RT Z Z (C.0

201 83 and n RT Z Z RT Z Z h (C. where b S 4. Calulate hase rertes (hase denstes and ther dervatves: ZRT (C. and Z Z Z Z (C.3 5. Calulate mnent fuates and ther dervatves: e f Φ, (C.4 ] 4 ( 4 ( ln[ ( 4 ln( ( w u u B Z w u u B Z a a b b w u B A B Z Z b b Φ (C.5 f f f Φ (C.6 ] 4 ( 4 ( 4 ( 4 ( ( 4 ( 4 ( ln( [( 4 ( w u u B Z w u u B Z w u u B Z w u u B Z B A w u u B Z w u u B Z B B A A B w u a a b b B Z B Z Z b b Φ (C.7

202 84 j j j f f f Φ δ (C.8 ] ( [ 4 ( 4 ( ln( 4 ] 4 ( 4 ( 4 ( 4 ( ( 4 ( 4 ( ln( [( 4 ( ] ( [ j j j j j j j j j j j j j j j a a k a a a a b b b w u u B Z w u u B Z w u B A w u u B Z w u u B Z w u u B Z w u u B Z B A w u u B Z w u u B Z B B A A B w u a a b b B Z B Z b Z b b Z b b Φ (C.9