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The results show the VXUSULVLQJSHUIRUPDQFHRIWKHGHYHORSHGJHQHWLFDOJRULWKPWRPDWFKWKHH[SHULPHQWDOGDWDRIWKHVHOHFWHG 7KHÀXLGPDLQVDPSOHVDGYDQWDJHRIWKHXVHGPHWKRG$OVRLV¿QGLQJLWVKLJKVSHHGLQ¿QGLQJDVROXWLRQ PRUHWKDQRQHVROXWLRQZRUNLQJDXWRPDWLFDOO\FRQ¿QLQJWKHUROHRIH[SHUWVWRWKHODVWVWDJHUHGXFLQJ FRVWVDQGKDYLQJWKHSRVVLELOLW\RIHYDOXDWLQJWKHGLIIHUHQWVLWXDWLRQVDUHWKHRWKHUDGYDQWDJHVRIWKLV PHWKRGWRPDWFKRUGLQDU\EODFNRLO397GDWDDQGPDNHVLWDQLGHDOPHWKRGWRLPSOHPHQWDVDQDXWRPDWLF EOS tuning algorithm for black oils. Equations of state (EOS), tuning, genetic algorithms, black oil, chromosome, regression Key words: substances and carbon groups. The carbon groups are not 1 Introduction ,QWKHVWXG\RIUHVHUYRLUVDQDFFXUDWHGHVFULSWLRQRIWKH GLYHUJLQJUHVXOWVDPRQJVWWKHPVHOYHVDUHXVHGWRHVWLPDWH K\GURFDUERQV\VWHPDQGLWVSURSHUWLHVLVLPSRUWDQW,WLVYLWDO the critical properties of the carbon groups required for EOS WKDWZHKDYHDUHDOLVWLFSK\VLFDOPRGHORIRXUUHVHUYRLUÀXLG FDOFXODWLRQV$OOWKHVHIDFWRUVIXUWKHUGHWHULRUDWHSUHGLFWLRQV EHIRUHZHWU\WRXVHLWLQDUHVHUYRLUVLPXODWLRQ2LODQGJDV RI(26IRUUHDOUHVHUYRLUÀXLGV'DQHVK SURSHUWLHVDUHQRUPDOO\REWDLQHGWKURXJKODERUDWRU\WHVWV upon oil and gas samples. Experimental data are obtained deficiencies is to calibrate, or tune the EOS models against XQGHUVRPHVSHFL¿FFRQGLWLRQVDQGXVXDOO\DUHQRWVXI¿FLHQW experimental data. There are no well defined rules for how IRUDUHVHUYRLUVWXG\2QWKHEDVLVRIWKHVHH[SHULPHQWDO to do regression of an equation of state model to match to GDWDDPRGHORIWKHK\GURFDUERQPL[WXUHFDQEHGHYHORSHG ODERUDWRU\PHDVXUHPHQWV7KHSDSHUE\&RDWVDQG6PDUW and used in combination with an equation of state model to (1986) contains an appendix on the choice, selection and FDOFXODWHDGGLWLRQDORLODQGJDVSURSHUWLHVXQGHUQHFHVVDU\ UDQJHOLPLWVRIUHJUHVVLRQYDULDEOHV+RZHYHUWKH&RDWVDQG conditions. 6PDUWPRGHOLVOLPLWHGLQFKRLFHRIUHJUHVVLRQYDULDEOHVDQG 7KHDYDLODEOHHTXDWLRQVRIVWDWH(26 WRPRGHOWKHÀXLG equation of state. SKDVHEHKDYLRUKDYHVRPHLQKHUHQWGH¿FLHQFLHVSDUWLFXODUO\ $OWKRXJKWKHUHLVQRVLQJOHVWDQGDUGPHWKRGIRUWXQLQJ IRUPXOWLFRPSRQHQWPL[WXUHVWKDWXVXDOO\FDXVHWKHPRGHOV WKHYDULRXVDSSURDFKHVDUHEDVLFDOO\VLPLODUDQ\IHUHQFHVGLI WRSUHGLFWHUURQHRXVUHVXOWVHYHQIRUZHOOFKDUDFWHUL]HG EHWZHHQWKHPHDVXUHGDQGFDOFXODWHGGDWDDUHPLQLPL]HG PRGHOIOXLGV5HDOUHVHUYRLUIOXLGVFRPSRVHGRIWKRXVDQGV XVLQJDUHJUHVVLRQIDFLOLW\ZKLFKDGMXVWVYDULRXVHTXDWLRQRI RIFRPSRXQGVDUHGHVFULEHGE\DOLPLWHGQXPEHURISXUH state parameters. This tuned model is then can regarded as DUHSUHVHQWDWLYHRIWKHUHVHUYRLUIOXLG7KHUHIRUHWKHPDLQ HPDLO6HGDHH#XWDFLU &RUUHVSRQGLQJDXWKRU SXUSRVHLVWRPLQLPL]HDQREMHFWLYHIXQFWLRQGH¿QHGDVWKH 5HFHLYHG-XQH VXPRIZHLJKWHGVTXDUHGGHYLDWLRQVDVVKRZQEHORZ 7KHFXUUHQWDSSURDFKLQWKHRLOLQGXVWU\WRRYHUFRPHWKHVH IXOO\GH¿QHG*HQHUDOL]HGFRUUHODWLRQVRIWHQZLWKVLJQL¿FDQWO\ 200 Pet.Sci.(2012)9:199-211 the regression parameters in detail. ªº §· < X< ,QWKHFRPPHUFLDO397VRIWZDUHDQH[SHULHQFHGXVHU ji j ' «» W ¨¸ (1) VSHQGVDORWRIWLPHWRVSHFLI\PDQ\GLIIHUHQWSDUDPHWHUV ¨¸ «» j ¬¼©¹ LQDQLQWHUDFWLYHZD\$VXLWDEOH(26WKDWFDQEHWXQHG against the experimental data, matching parameters of pred exp ZKHUHȌ DQGȌ represent predicted and experimental WKHVHOHFWHG(26ELQDU\LQWHUDFWLRQFRHIILFLHQWV%,& YDOXHVUHVSHFWLYHO\ W is the weighting factor, N expresses weighting factors for different properties of the fluid which data WKHQXPEHURIPHDVXUHGGDWDSRLQWVWREHILWWHGDQG X DUHPHDVXUHGH[SHULPHQWDOO\WKHPHWKRGDQGQXPEHURI GHVLJQDWHVWKHUHJUHVVLRQYDULDEOHV7KHRSWLPXPYDOXHV pseudo-components if splitting of the plus fraction is needed, RIYDULDEOHVDUHREWDLQHGE\PLQLPL]LQJWKHPXOWLYDULDEOH DQGDVXLWDEOHPRGHOWRFDOFXODWHWKHRLOYLVFRVLW\DQGLWV UHJUHVVLRQIXQFWLRQ¨ FRHI¿FLHQWVVKRXOGEHGHWHUPLQHGE\WKH397H[SHUWWR¿QG DJDLQVWH[SHULPHQWDOGDWDWKHRQO\SUDFWLFDOZD\WRSHUIRUP Then the software uses the selected parameters to perform UHJUHVVLRQLVE\WULDODQGHUURU7KHUHIRUHLWLVHIILFLHQWWR DPXOWLYDULDEOHUHJUHVVLRQDQGE\DGMXVWLQJWKHVHOHFWHG XVHRSWLPL]DWLRQWHFKQLTXHVWRVROYHWKHSUREOHP'LIIHUHQW PDWFKLQJYDULDEOHVWULHVWR¿QGWKHEHVWPRGHOIRUWKDWUXQ classes of search techniques like calculus-based techniques, $WWKHHQGRIHDFKUXQWKH397H[SHUWHYDOXDWHVWKHUHVXOWV JXLGHGUDQGRPVHDUFKWHFKQLTXHVDQGHQXPHUDWLYH DQGGHFLGHVWRDFFHSWRUUHMHFWWKHIOXLGPRGHODQGWKHQ WHFKQLTXHVDUHGHYHORSHGWRGHDOZLWKWKHRSWLPL]DWLRQ selects another set of items for the next run. These trial and +RZHYHUWKHVWURQJQRQOLQHDULW\RIWKH(26WXQLQJSURFHVV HUURUSURFHGXUHVFRQWLQXHXQWLODSK\VLFDOO\DFFHSWDEOHPDWFK is found. DQGXQOLNHO\WREHVXFFHVVIXO7KHUHIRUHDQDOWHUQDWLYH ,WLVFOHDUWKDWWXQLQJRI(26LQWKLVZD\LVDWHGLRXVDQG DSSURDFKZRXOGEHWRXVHKHXULVWLFW\SHPHWKRGVOLNHJHQHWLF WLPHFRQVXPLQJWDVN,QWKLVVWXG\DOPRVWDOOWKHSURFHVVHV algorithms. Strong features of genetic algorithm such as its DUHGRQHDXWRPDWLFDOO\LQDSURJUDP7KHSURJUDPVHOHFWV the EOS model, assigns weighting factors, and determines LQDSDUDOOHOPRGH)RUUHVWHWDO LWVDELOLW\WRFKDQJH WKHOLPLWDWLRQVRIPDWFKLQJSDUDPHWHUVLQDVPDUWZD\DQG VHYHUDOYDULDEOHVVLPXOWDQHRXVO\DQGZRUNZLWKIHUHQWGLIGDWD WKHQXVHVWKH*$WR¿QGWKHSURSHUYLVFRVLW\PRGHOELQDU\ VWUXFWXUHVDWWKHVDPHWLPHVXJJHVWWKLVRSWLPL]DWLRQPHWKRG interaction coefficients and matching parameters of the WREHDEHWWHUFKRLFHWRVROYHWKHSUREOHPVRIPDWFKLQJ397 VHOHFWHG(26DQGYLVFRVLW\PRGHO$OVRWKHSURJUDPFKHFNV GDWDDXWRPDWLFDOO\ the different cases for the number of pseudo-components *HQHWLFDOJRULWKPV*$V DUHVWRFKDVWLFWHFKQLTXHVZKRVH LIVSOLWWLQJRIKHDY\FRPSRQHQWVLVRFFXUULQJR7VLPSOLI\ VHDUFKPHWKRGVPRGHODQDWXUDOHYROXWLRQ7KHLUDSSURDFKLV WKHSUREOHPLQRXUSURJUDPWZRGLIIHUHQW(26QDPHO\ based on a stochastic-directed trend with roots on ideas from SDUDPHWHUV3HQJ5RELQVRQ35 (26DQGSDUDPHWHUV QDWXUDOHYROXWLRQDQGIXQGDPHQWDOLGHDVRI'DUZLQ2YHU 6RDYH5HGOLFK.ZRQJ65. (26ZHUHWHVWHGDQGWKH WKHODVW\HDUVPDQ\HQJLQHHULQJSUREOHPVDUHVROYHGE\ SURJUDPZDVUXQIRUHDFKFDVHWKRURXJKO\$OVRIRUHDFK VHOHFWHG(26¿YHIHUHQWGLIYDOXHVRIZHLJKWLQJIDFWRUVDQG VDOHVPDQSUREOHP*UHWHQVWHWWHHWDO QXFOHDUUHDFWRU three different situations regarding the number of pseudo- components were tested. These were: no splitting, 2 pseudo- et al, 1995), and aircraft design (Parnee and Watson, 1999). FRPSRQHQWVDQGSVHXGRFRPSRQHQWV,QDVVLJQLQJWKH *$VZHUH¿UVWXVHGLQSHWUROHXPHQJLQHHULQJLQIRUJDV weighting factors, the general rules were applied. For SLSHOLQHRSHUDWLRQ*ROGEHUJ 1LNUDYHVKHWDO example, the highest weighting factor was assigned to the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¿HGE\*$7KHQDOOWKHVHOHFWHGLWHPV XVHGD*$DSSURDFKWRLPSOHPHQWUHVHUYRLUFKDUDFWHUL]DWLRQ are fed into the commercial PVT software, which has been E\FRQGLWLRQLQJWKHUHVHUYRLUVLPXODWLRQPRGHOWRWKH coupled with the program, as the input data. The commercial production data on a structural model. VRIWZDUHSHUIRUPVDPXOWLYDULDEOHUHJUHVVLRQDQGFKDQJHV ,QWKLVVWXG\DSSOLFDWLRQRID*$WRWKHSUREOHPRI WKHYDOXHVRIVHOHFWHGSDUDPHWHUVLQWKHLUSUHGHWHUPLQHG matching PVT data for three real black oil fluid samples is OLPLWUDQJHVLQDQDWWHPSWWRPLQLPL]HWKHREMHFWLYHIXQFWLRQ GHVFULEHG7KHUHVHUYRLUÀXLGVDUHVDPSOHGIURP,UDQLDQRLO and presents the best solution. The regression method used UHVHUYRLUV in the PVT software uses the Newton numerical method to find the minimum of a residual function that is defined 2 Description of the method DVWKHGLIIHUHQFHEHWZHHQWKHREVHUYHGGDWDPDWUL[DQGWKH calculated data matrix. The residual function depends on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¿FLHQW WKHPRGHOGHVFULELQJWKHSKDVHEHKDYLRURIWKHUHVHUYRLUÀXLG 6LQFHWKHUHDUHPDQ\PDWFKLQJSDUDPHWHUVWRWXQHWKH(26 Pet.Sci.(2012)9:199-211 201 E\WKH*$WRPRGLI\WKHUHJUHVVLRQYDULDEOHVIRUWKHQH[W a predetermined number of iterations for some special cases UXQ7KLV*$SURFHVVLVLWHUDWHGIRUWKDWVSHFLDOFDVHXQWLOD HJWKHVHOHFWHG(26FRXOGQRWPRGHOWKHÀXLGEHKDYLRURU PDWKHPDWLFDOO\DFFHSWHGVROXWLRQLVREWDLQHGDFFRUGLQJWR VSOLWWLQJWKHSVHXGRFRPSRQHQWFDXVHVWKHPRGHOWRJH GLYHU LWVURRWPHDQVTXDUH506 YDOXH7KHVROXWLRQVDYHGE\ WKHUHIRUHLWVHOHFWVWKHEHVWVROXWLRQWKDWPD\QRWEHWXQHG the program and the new fluid model replaces the original against experimental data. Furthermore, sometimes the tuning (26DQGWKH*$SURFHVVLWHUDWHGIRUWKLVQHZ(267KLV RIWKHPRGHOLVSK\VLFDOO\XQDFFHSWDEOHDQGLWFDQQRWSUHGLFW F\FOHLVUHSHDWHGWKUHHWLPHVDQGDWWKHHQGRIWKHVHF\FOHV WKHUHDOIOXLGEHKDYLRUDWRWKHUFRQGLWLRQVFRUUHFWO\VRWKH the program compares these 3 solutions and selects the best PRGHOFDQQRWEHXVHGLQDUHVHUYRLUVLPXODWLRQ RQHZLWKWKHOHDVW506IRUWKHVSHFLDOFDVH7KHIXQFWLRQRI 7KHUHIRUHLQWKHODVWVWDJHD397H[SHUWVKRXOGHYDOXDWH *$ZLOOEHGHVFULEHGLQWKHQH[WVHFWLRQLQGHWDLOV1RWHWKDW WKHVHPDWKHPDWLFDOO\DFFHSWHGVROXWLRQVDQGE\FRQVLGHULQJ WKH506YDOXHGRHVQRWDOZD\VGHFUHDVHWKURXJKWKHWKUHH WKHPHQWLRQHGDVSHFWVDQGXVLQJHQJLQHHULQJMXGJPHQW iterations since the weighting factors do not change through VHOHFWVWKHSK\VLFDOO\DFFHSWHGRQHV6RE\XVHRIWKLV WKHVHLWHUDWLRQVLQ$OVRWKHPRGLI\LQJSURJUDPWKHPDWFKLQJ SURJUDPZHFRQ¿QHWKHUROHRIDQH[SHUWWRWKHODVWVWHS The complete process is shown in Fig. 2. PDWKHPDWLFDOO\DQGSK\VLFDOO\XQDFFHSWDEOHVRLWLVSRVVLEOH 3 Genetic algorithms WKDWWKHILUVWPRGHOLVWKHEHVWDQGPRGLI\LQJWKHPRGHO degrades it. Fig. 1 shows three different cases extracted from *HQHWLFDOJRULWKPVDUHKHXULVWLFW\SHPHWKRGVWKDW WKHSURJUDP,QFDVHDWKH506YDOXHGHFUHDVHVWKURXJKWKH ZHUHILUVWSURSRVHGE\+ROODQG DVDQDEVWUDFWLRQRI LWHUDWLRQVVORZO\ZKLOHLQFDVHEWKH506YDOXHLQFUHDVHV ELRORJLFDOHYROXWLRQGUDZLQJRQLGHDVIURPQDWXUDOHYROXWLRQ WKURXJKWKHLWHUDWLRQV&DVHFVKRZVWKDWWKH506YDOXH and genetic for the design and implementation of robust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ÀLFWVDPRQJ 0.15 REMHFWLYHV7KHUHIRUHLWLVUREXVWZKHUHPXOWLSOHVROXWLRQV 0.14 exist. The genetic algorithms start with an initial population 0.13 RIUDQGRPO\FKRVHQIHDVLEOHVROXWLRQVWRWKHSUREOHPEHLQJ 1 2 3 $VWKHDOJRULWKPSURJUHVVHVDGGUHVVHGIURPWKHVHDUFKVSDFH Iteration the current population of the solution is known as the parent SRSXODWLRQDQGE\DSSO\LQJJHQHWLFRSHUDWRUVWKDWPRGHO Fig. 1506YDULDWLRQVWKURXJKWKUHHLWHUDWLRQV the genetic processes occurring in the nature, i.e. selection, parameters in this case. FURVVRYHUDQGPXWDWLRQWKHILWWHULQGLYLGXDOVDUHVHOHFWHG This process is iterated for the 30 different cases and at and their genetic information is recombined and modified WKHHQGRIHDFKFDVHWKHEHVWVROXWLRQORZHVW506 ZLOOEH to generate the new population known as offspring. The $OOVDYHGWKHVHSURFHVVHVDUHSHUIRUPHGDXWRPDWLFDOO\LQWKH offspring are inserted into the population, replacing the parent SURJUDP$WWKHHQGPDWKHPDWLFDOO\DFFHSWHGVROXWLRQV population and producing a new generation. The genetic DUHREWDLQHGE\WKHSURJUDP DOJRULWKPFRYHUVDOPRVWDOOSDUWVRIWKHVROXWLRQVSDFHLQHDFK 7KHOHDVWVTXDUHVILWWRWKHREVHUYDWLRQGDWDLVQRW JHQHUDWLRQRIVHDUFKDQGE\HYROXWLRQRIWKHDOJRULWKPWKH QHFHVVDULO\WKHJRDORIHTXDWLRQRRIEHVWDWHSUHFLVH¿WWLQJ7 UHVSRQVHSRLQWLVGLUHFWHGWRWKHRSWLPXPYDOXH7KLVSURFHVV WKHJRDOLVWRFUHDWHDIOXLGPRGHOWKDWEHKDYHVOLNHWKH LVUHSHDWHGXQWLOWKHRSWLPL]DWLRQFULWHULRQLVPHW 7KHUHUHVHUYRLUDUHÀXLGDVSHFWVRIWKHUHVHUYRLUÀXLGWKDWWKH $FFRUGLQJWRWKHGHVFULEHGVWUXFWXUHWKHIRUPXODWLRQ PRGHOPXVWFDSWXUHDFFXUDWHO\ RID*$IRUDVSHFLILFSUREOHPUHTXLUHVWKHGHILQLWLRQRI The phase diagram of the final model should represent WKUHHPDLQLVVXHVWKHLQLWLDOL]DWLRQRISRVVLEOHVROXWLRQV WKHUHDOIOXLGEHKDYLRU6RPHWLPHVPRGLI\LQJWKHPDWFKLQJ and representing them in a genetic format, the selection of LQGLYLGXDOVDFFRUGLQJWRWKHLUILWQHVVYDOXHDQGWKHJHQHWLF DUHXQUHDOLVWLFSK\VLFDOPRGHOVRIRXUUHVHUYRLUÀXLGDQGWKH RSHUDWRUVVXFKDVFURVVRYHUDQGPXWDWLRQXVHGWRJHQHUDWH SKDVHGLDJUDPFRXOGQRWEHFUHDWHG$OVRLWLVSRVVLEOHWKDW new solutions. WKHSURJUDPFRXOGQRWFRQYHUJHWRDIDYRULWH506YDOXHLQ ,QWKHIROORZLQJVHFWLRQWKHVHHOHPHQWVZLOOEHGLVFXVVHG RMS SDUDPHWHUVLQGLVFULPLQDWHO\UHVXOWVLQEDGO\WXQHGPRGHOVWKDW SDUDPHWHUVLQGLVFULPLQDWHO\PD\UHVXOWVLQWKHPRGHOVWKDWDUH 202 Pet.Sci.(2012)9:199-211 3-PR EOS 2 Types 3-SRK No splitting No of pseudo-components 2 pseudo-component 3 Options 3 pseudo-component 5 Options Weighting factors 2Ø3Ø5>30 Iterations EOS matching parameters (ȍ , ȍ , P , T ...) A B C C Produce the first BIC :Genome population (275 genomes) Viscosity model and its parameters (LBC...) 3 Iterations Produce the children population PVT software Mutation Output Produce the next population (Offspring) Crossover Calculate RMS for each genome Eq. (2) Selection: Select the good parents NO A certain number of population Calculate fitness Save the result value Condition for each genome A suitable individual found Eq. (3) Result 1 Result 2 YES Result 3 Result Solution 1 Solution 2 Solution 3 Compare the results Save this solution for the special case Solution & select the best one Solution 30 Selection of the good models by user Fig. 26LPSOL¿HGGLDJUDPRIWKHSURFHVV WKDWZHKDYHXVHGLQWKLVVWXG\IRUWKH*$ chromosome for the EOS parameters is a two-dimensional DUUD\DQGIRUPVDQ 1&2036DUUD\RIELQDU\QXPEHUV 3.1 Initialization 1&2036VKRZVWKHQXPEHURIFRPSRQHQWVRIWKHIOXLG VDPSOH$OVRWKHFKURPRVRPHIRU%,&LVDWZRGLPHQVLRQDO The information that is to be held within the genome is DUUD\DQGIRUPVD1&2036 1&2036DUUD\RIELQDU\ WKH(26DQGYLVFRVLW\PRGHOPDWFKLQJSDUDPHWHUV%,&DQG QXPEHUVDQGWKHFKURPRVRPHRIWKHPRGHOVRIYLVFRVLW\LV 6LQFHWKHVHOHFWHG(26LVHLWKHU35WKHPRGHOVRIYLVFRVLW\ DRQHGLPHQVLRQDODUUD\RIELQDU\QXPEHUVZLWKWKUHHELWV RU65.WKHPDWFKLQJSDUDPHWHUVDUH ȍ , ȍ , P , T , acentric a b c c )LJWKURXJK)LJVKRZW\SLFDOFKURPRVRPHVWKDWDUH IDFWRUDQGYROXPHVKLIW$OVRWKHPRVWZLGHO\XVHGYLVFRVLW\ XVHGLQWKLVVWXG\1RWHWKDWELQDU\LQWHUDFWLRQFRHIILFLHQW PRGHOVQDPHO\/RKUHQ]%UD\&ODUN/%& 3HGHUVHQDQG PDWUL[LVV\PPHWULFDQGDFKURPRVRPHOLNH)LJLVDJRRG J3HWHUVHQDUH$DVEHUXVHGLQWKLVZRUN6R V and Z are also c c UHSUHVHQWDWLYHRILW DGGHGWRWKHPDWFKLQJSDUDPHWHUVIRUWKH/%&FRUUHODWLRQLI ,QWKHVHFKURPRVRPHVPHDQVWKDWWKHSDUDPHWHULV WKLVPRGHOLVVHOHFWHGIRUPRGHOLQJYLVFRVLW\ DUHJUHVVLRQYDULDEOHLQWXQLQJRI(26DQGPHDQVWKDW 1RZZHQHHGWRVSHFLI\WKHVWUXFWXUHRIWKHJHQRPH,Q WKHSDUDPHWHULVQRWDYDULDEOH1RWHWKDWXVLQJSURSHUWLHV spite of the fact that the general structure for genome is a RIWKHOLJKWHUFRPSRQHQWVDVUHJUHVVLRQYDULDEOHVLVQRW RQHGLPHQVLRQDODUUD\ZLWKHDFKQXPEHUVSHFL¿HGE\ELQDU\ ELWVZHKDYHFKRVHQWRXVHDQRQVWDQGDUGVWUXFWXUHIRUWKH DUHFRPPHQGHGPHWKRGDVWKHVHYDOXHVDUHZHOOGHILQHG JHQRPH7KHYDULDEOHVKDYHEHHQVSOLWLQWRWKUHHJURXSVZLWK DOVRFULWLFDOYROXPHVDQG ZIDFWRUVDUHRQO\QHHGHGIRU each group being allocated to a separate chromosome. The WKH/%&YLVFRVLW\FRUUHODWLRQ7KHUHIRUHLQVHOHFWLQJWKH Pet.Sci.(2012)9:199-211 203 regression parameters in the chromosome of the EOS and YLVFRVLW\PRGHOVSDUDPHWHUV)LJ WKHSURJUDPDVVLJQHGD (3) SUREDELOLW\QXPEHUWRHDFKEORFNIROORZLQJJHQHUDOUXOHV)RU (1 ) example, the plus fraction row in the matrix has the highest SUREDELOLW\WKHFULWLFDOYROXPHDQG ZIDFWRUFROXPQVKDYH ]HURSUREDELOLW\ZKHQWKHVHOHFWHGYLVFRVLW\PRGHOLVQRW where fUHSUHVHQWVWKHILWQHVVYDOXHRIWKHLQGLYLGXDO i M /%&RUWKHSUREDELOLW\RIWKHOLJKWHUFRPSRQHQWVURZVDUH UHSUHVHQWVWKHQXPEHURILQGLYLGXDOVLQWKHSRSXODWLRQDQG ]HUR RMSUHSUHVHQWVWKHRXWSXWRIWKHREMHFWLYHIXQFWLRQIRUWKH LQGLYLGXDO i. S shift Omega-A Omega-B P T V Zcrit Acentric-factor crit crit crit $V\RXFDQVHHWKH¿WQHVVIXQFWLRQLVDUHODWLYHIXQFWLRQ H S 0 0 0 1 0 0 1 0 DQGWKHUHIRUHWKHILWQHVVYDOXHVDUHEHWZHHQDQG$OVR 0 0 0 C 0 0 0 0 0 WKHJUHDWHUWKH¿WQHVVYDOXHWKHEHWWHUWKHFKURPRVRPHLV C 0 0 0 0 0 0 0 0 :KHQWKHILWQHVVRIHDFKLQGLYLGXDOLVVSHFLILHGWKH C 0 0 0 0 0 0 0 0 VHOHFWLRQRSHUDWRUFKRRVHVEHWWHULQGLYLGXDOVDFFRUGLQJWR C 0 0 0 0 1 0 1 0 4 WKHLU¿WQHVVYDOXHV7KHVHOHFWLRQPHWKRGXVHGLQWKLVVWXG\ C 0 1 0 1 1 0 1 0 LVDVWRFKDVWLFVDPSOLQJPHWKRGFDOOHG³URXOHWWHZKHHO´ DNOLHYD%DQFKHYD 96KRSRYDDQG C 1 1 1 1 1 0 1 0 7+ This method is the simplest proportionate selection Fig. 3 Example of a two-dimensional chromosome for the EOS parameters ¿WQHVV7KHEHWWHUWKHFKURPRVRPHVDUHWKHPRUHFKDQFHVWR H S C C C C C C EHVHOHFWHGWKH\KDYH,QWKLVPHWKRGWKHLQGLYLGXDOVRIWKH 2 1 2 3 4 6 7+ H S SRSXODWLRQDVVXPHDVVORWVRIWKH5RXOHWWHZKHHO(DFKVORW LVDVZLGHDVWKHSUREDELOLW\IRUVHOHFWLRQRIWKHFRUUHVSRQGHG C 1 1 C 1 0 C 0 0 0 0 LVSURSRUWLRQDOWRWKHYDOXHRIWKHILWQHVVIXQFWLRQRIHYHU\ C 0 1 0 0 1 C 1 0 1 1 0 0 7+ chromosome. Fig. 6 shows the principle of this selection method. Fig. 4([DPSOHRIDWZRGLPHQVLRQDOFKURPRVRPHIRUWKH%,& Lohrenz-Bray-Clark Pedersen Aasberg-petersen 1 0 0 Fig. 5([DPSOHRIDRQHGLPHQVLRQDOFKURPRVRPHIRUWKHYLVFRVLW\PRGHOV &RQVLGHULQJWKHVHOLPLWDWLRQVWKHSURJUDPZLOOSURGXFH the parent population. Fix pointer 3.2 Selection :KHQWKHSDUHQWSRSXODWLRQLVLQLWLDOL]HGVROXWLRQVIURP this population are taken and used to form a new population RIIVSULQJ 7KLVLVPRWLYDWHGE\DKRSHWKDWWKHQHZ population will be better than the old one. Solutions which are then selected to form new solutions are selected according WRWKHLU¿WQHVVWKHPRUHVXLWDEOHWKH\DUHWKHPRUHFKDQFHV Fig. 65RXOHWWHZKHHOVHOHFWLRQ WKH\KDYHWRUHSURGXFH6RWKHILUVWVWHSLVWRFDOFXODWHWKH The wheel is rotated and the chromosome that stops in ¿WQHVVYDOXHRIHDFKLQGLYLGXDO IURQWRID¿[HGSRLQWHULVVHOHFWHGWKH&OHDUO\FKURPRVRPHV ,QWKLVVWXG\WKHIROORZLQJURRWPHDQVTXDUH506 ZLWKDODUJHUILWQHVVYDOXHZLOOEHVHOHFWHGPRUHRIWHQ HTXDWLRQLVXVHGDVDQREMHFWLYHIXQFWLRQ 5RXOHWWHZKHHOVHOHFWLRQJLYHVSUHIHUHQFHWRWKHEHWWHU 2 LQGLYLGXDOVLQWKHSRSXODWLRQDQGH[HUWVDODUJHSUHVVXUHRQ ªº 1 §· the search process. (2) «»¨¸ 7KLVSURFHVVFDQEHGHVFULEHGE\WKHIROORZLQJDOJRULWKP 1«» ©¹ ¬¼ >6XP@&DOFXODWHWKHVXPRIDOOFKURPRVRPH¿WQHVVLQ population (S). where NUHSUHVHQWVWKHWRWDOQXPEHURIH[SHULPHQWDOSRLQWV W X represents i i S). WKHFDOFXODWHGYDOXHDQG xUHSUHVHQWVWKHH[SHULPHQWDOYDOXH >/RRS@*RWKURXJKWKHSRSXODWLRQDQGVXPWKH¿WQHVV 1RWHWKDWWKHVPDOOHUWKH506YDOXHWKHEHWWHUWKH YDOXHRIHDFKFKURPRVRPHLQDVHTXHQWLDOSURFHVV:KHQWKH FKURPRVRPHLV6RWRVLPSOLI\WKHSUREOHPWKHRXWSXWRI sum is greater than the random number, stop the summation WKLVREMHFWLYHIXQFWLRQLVXVHGWRFDOFXODWH¿WQHVVYDOXHE\WKH SURFHVVDQGVHOHFWWKHSUHYLRXVFKURPRVRPH IROORZLQJ¿WQHVVIXQFWLRQ UHSUHVHQWVWKHZHLJKWLQJIDFWRUIRUHDFKSRLQW >6HOHFW@*HQHUDWHDUDQGRPQXPEHUIURPWKHLQWHUYDO FKURPRVRPHLVJUHDW7KHVL]HRIWKHVORWLQWKHURXOHWWHZKHHO VFKHPH,QWKLVPHWKRGSDUHQWVDUHVHOHFWHGDFFRUGLQJWRWKHLU 204 Pet.Sci.(2012)9:199-211 This loop is iterated while the required numbers of )RUWKHRQHGLPHQVLRQDOFKURPRVRPHVRIYLVFRVLW\ chromosomes are selected. PRGHOVWKHFURVVRYHULVPHDQLQJOHVVVLQFHMXVWRQHEORFNRI this chromosome should be selected that shows the method of 3.3 Crossover and mutation operators H: nor select none. So, this operator will not perform on these The selected parents should produce the next population E\FURVVRYHUDQGPXWDWLRQRSHUDWRUV%HFDXVHRIWKHVSHFLDO ZLWKQRFURVVRYHU JHQRPHVWUXFWXUHLQWKLVVWXG\ZHXVHVSHFLDOO\GHVLJQHG 7KHPXWDWLRQPHWKRGXVHGLQWKLVVWXG\LVDNLQGRIRQH FURVVRYHUDQGPXWDWLRQRSHUDWRUV 7KHFURVVRYHUDGRSWHGKHUHLVDNLQGRISRLQWFURVVRYHU RIWKHUDQGRPQXPEHUDUHQRUPDOL]HG7KHVHFRRUGLQDWHV 7KHQIRUFURVVRYHULQWKHWZRGLPHQVLRQDOFKURPRVRPHV VSHFLI\WKHPXWDWLRQSRLQWOLNHWKHFURVVRYHURSHUDWLRQ RI(26DQGYLVFRVLW\PRGHOSDUDPHWHUVWKH x-coordinates and flip (0 to 1, or 1 to 0) the gene in the two-dimensional RIWKHVHQXPEHUVDUHQRUPDOL]HGEHWZHHQDQGDQGWKH chromosomes. yFRRUGLQDWHVRIWKHVHQXPEHUVDUHQRUPDOL]HGEHWZHHQDQG )RUWKHYLVFRVLW\PRGHOFKURPRVRPHWKHSURJUDP WKHQXPEHURIFRPSRQHQWVRIWKHIOXLGVDPSOH1&2036 JHQHUDWHVDUDQGRPQXPEHUEHWZHHQDQG,IWKHUDQGRP $OVRIRUFURVVRYHULQWKHWZRGLPHQVLRQDOFKURPRVRPHV number is less than 0.33, the program will flip the first RI%,&WKH x and y-coordinates of these numbers are block and if the number is between 0.33 and 0.67, flip the QRUPDOL]HGEHWZHHQDQG1&20367KHFRRUGLQDWHV VHFRQGEORFNHOVHWKHWKLUGEORFNZLOOEHIOLSSHG,QWKHVH RIWKHVHQXPEHUVVKRZWKHFURVVRYHUUHJLRQLQWKHWZR FKURPRVRPHVLIWKHYDOXHRIWKHVHOHFWHGEORFNIRUPXWDWLRQ dimensional chromosome. Fig. 7 shows different situations LVWKHYDOXHZLOOFKDQJHWRDQGWKHYDOXHVRIWKHRWKHU IRUFURVVRYHULQWZRGLPHQVLRQDOFKURPRVRPHVDFFRUGLQJWR EORFNVFKDQJHWREXWLIWKHYDOXHRIWKHVHOHFWHGEORFNIRU x and y-coordinates of the random numbers. mutation is 1, no change will be occurred. This is because RQO\RQHYLVFRVLW\PRGHOVKRXOGEHVHOHFWHG %RWKWKHFURVVRYHUDQGPXWDWLRQRSHUDWRUVDUH SHUIRUPHGRQWKHHQWLUHSRSXODWLRQH[FHSWYLVFRVLW\PRGHOV FKURPRVRPHVZLWKGLIIHUHQWSUREDELOLWLHV,WPHDQVWKDW A=(a1, a2) DQLQGLYLGXDOPD\IHUVXIFURVVRYHUPXWDWLRQRUERWKRUQR B=(b1, b2) change with respect to its generated random number. For H[DPSOHLIWKHFURVVRYHUSUREDELOLW\LV C and the mutation SUREDELOLW\LV M¿UVWZHJHQHUDWHDUDQGRPQXPEHUIRUHDFK LQGLYLGXDO,IWKHUDQGRPQXPEHULVVPDOOHUWKDQ C and M, FURVVRYHUDQGPXWDWLRQDUHSHUIRUPHGRQWKHLQGLYLGXDO When the offspring population is produced, we compare DOOLQGLYLGXDOVRISDUHQWDQGRIIVSULQJSRSXODWLRQVDQG UHSODFHWKHEHWWHULQGLYLGXDOVLQWKHQHZSRSXODWLRQXQWLOWKH QHZSRSXODWLRQLVFRPSOHWHG$FFRUGLQJWRWKHVWXG\XQGHU A=(a1, a2) FRQVLGHUDWLRQWKLVPHWKRGFDXVHVWKHSURJUDPWRFRQYHUJH B=(a1, a2) IDVW$OVRXVLQJWKLVPHWKRGLQVHUWVDQHOLWLVPRSHUDWRULQWR WKHSURJUDPDXWRPDWLFDOO\ ,QWKLVVWXG\VRPHUXOHVRIWKXPEDUHXVHGWRGHWHUPLQH LVHTXDOWRWKHQXPEHURIYDULDEOHVWKHQXPEHURIJHQHUDWLRQV LVWKHFURVVRYHUSUREDELOLW\LVDQGWKHPXWDWLRQ SUREDELOLW\LVWKDWLVHTXDOWRYDULDEOHQXPEHU'H J -RQJ*ROGEHU A=(a1, a2) 4 Sample data B=(b1, b2) 7KHÀXLGVVWXGLHGZHUHWKUHHUHDOEODFNRLOVDPSOHVIURP ,UDQLDQRLO¿HOGV7KHFRPSRVLWLRQVRIWKHVHÀXLGVDUHOLVWHG LQDEOH71RWHWKDWDVD¿UVWVWHSEHIRUHDQ\UHJUHVVLRQLV FRQVLGHUHGWKHFRQVLVWHQF\DQGTXDOLW\RIWKHPHDVXUHGGDWD are checked. 7KHGDWDWKDWKDYHEHHQXVHGLQWXQLQJRI(26DUH from two experiments performed on the fluids: constant A=(a1, a2) C=(c1, c2) B=(a1, a2) D=(d1, d2) Table 2 to Table 5 show the summaries and results of these Fig. 7 ([DPSOHVRIFURVVRYHULQWZRGLPHQVLRQDOFKURPRVRPHV tests. FRPSRVLWLRQH[SDQVLRQ&&( DQGGLIIHUHQWLDOOLEHUDWLRQ'/ WKH*$SDUDPHWHUV7KHSRSXODWLRQVL]HLVLQGLYLGXDOVWKDW WHFKQLTXH,QWKLVPHWKRGWZRUDQGRPQXPEHUVDUHSURGXFHG LVSURGXFHG7KHQOLNHWKHFURVVRYHURSHUDWRUWKHFRRUGLQDWHV SRLQWPXWDWLRQWHFKQLTXH,QWKLVPHWKRGRQHUDQGRPQXPEHU FKURPRVRPHVDQGWKH\ZLOOEHUHSHDWHGLQWKHQH[WSRSXODWLRQ PRGHOLQJYLVFRVLW\FDQQRWVHOHFWPRGHOVVLPXOWDQHRXVO\ Pet.Sci.(2012)9:199-211 205 Table 1&RPSRVLWLRQRIWKHVDPSOHV WHVWIRUEODFNRLO5HVXOWVRIWKH'/ Table 3 Solution *DV Oil DOXHPRO9 Pressure 2LOGHQVLW\ B *25 FRPSUHVVLELOLW\ YLVFRVLW\ &RPSRQHQWV psia g/cm 5%67% 6&)67% factor cp Black oil-1 Black oil-2 Black oil-3 6044 0.760 1.395 1.335 + S 1.39 0.00 1.45 5041 0.753 1.408 1.286 N 0.88 0.36 0.23 4043 0.746 1.421 1.237 &2 5.18 0.51 3.84 2 3532 0.742 1.428 1.212 3030 0.738 1.436 1.188 & 22.57 25.24 26.03 2523 0.734 1.444 1.163 & 6.94 7.91 7.81 2323 0.732 1.448 1.153 & 5.91 5.48 5.67 2223 0.731 1.450 1.148 i& 0.98 1.07 1.01 2123 0.730 1.452 1.144 2023 0.730 1.453 1.139 n& 2.97 3.39 3.09 1925 0.729 1.455 1.134 i& 0.93 1.42 1.02 1845 0.728 1.457 547.10 1.129 n& 1.03 1.73 1.15 1521 0.734 1.427 478.08 0.875 1.146 & 3.07 4.98 2.73 1223 0.741 1.394 406.76 0.884 1.196 & 4.06 2.60 6.16 921 0.750 1.358 334.26 0.897 1.274 621 0.760 1.320 259.84 0.915 1.370 & 4.14 1.49 3.69 322 0.770 1.274 172.76 0.935 1.500 & 3.69 2.76 2.77 14.7 0.841 1.078 0.00 1.000 2.161 & 3.45 3.10 2.75 WHVWIRUEODFNRLO5HVXOWVRIWKH'/ Table 4 & 2.11 2.44 3.28 Oil Solution *DV Oil & 30.68 35.52 27.30 Pressure B Pressure 12+ GHQVLW\ *25 FRPSUHVVLELOLW\ YLVFRVLW\ psia 5%67% psia g/cm 6&)67% factor cp Total 100.00 100.00 100.00 4992 0.792 1.293 4986 1.313 4493 0.789 1.298 3989 1.232 Table 2WHVWV6XPPDU\RI&&(DQG'/ 3995 0.786 1.303 2993 1.150 Value 3498 0.783 1.309 2493 1.109 Parameter Black oil-1 Black oil-2 Black oil-3 3001 0.779 1.315 1995 1.068 2503 0.776 1.321 1719 1.047 &&(WHVW 2206 0.773 1.325 1473 1.056 Saturation pressure, psia 1852 1404 2014 2107 0.772 1.326 1204 1.090 6ROXWLRQ*256&)67% 552 366.30 498.08 2008 0.772 1.327 905 1.204 2LOJUDYLW\RIUHVLGXDORLO $3, 24.35 24.46 20.71 1909 0.771 1.329 605 1.376 WHVW'/ 1811 0.770 1.330 305 1.570 o 1719 0.769 1.332 420.19 14.7 2.611 Test temperature, F 255 220 255 1513 0.773 1.316 381.66 0.873 Saturation pressure, psia 1845 1719 2013 1263 0.780 1.295 336.43 0.889 6ROXWLRQ*256&)67% 547 420.19 498.03 1013 0.786 1.275 291.02 0.900 )RUPDWLRQYROXPHIDFWRU 1.457 1.332 1.405 763 0.796 1.250 245.40 0.91 #VDWXUDWLRQSUHVVXUH B 5%67% 513 0.802 1.229 196.32 0.928 2LOJUDYLW\RIUHVLGXDORLO $3, 24.46 22.54 20.88 263 0.812 1.198 140.05 0.946 1RWHV*25*DVRLOUDWLR6&)67%6WDQGDUGFXELFIHHWSHUVWRFNWDQN 14.7 0.864 1.062 0.00 1.000 EDUUHO5%6&)5HVHUYRLUEDUUHOSHUVWRFNWDQNEDUUHO 206 Pet.Sci.(2012)9:199-211 WHVWIRUEODFNRLO5HVXOWVRIWKH'/ Table 5 R7DQDO\]HWKHUHVXOWVPRUHHI¿FLHQWO\HUURUYDOXHVZHUH calculated for different properties. Table 7 shows the results Solution *DV Oil Pressure 2LOGHQVLW\ B RIWKLVDQDO\VLV,QFDOFXODWLQJHUURUYDOXHVWKHIROORZLQJ *25 FRPSUHVVLELOLW\ YLVFRVLW\ psia g/cm 5%67% equation was used: 6&)67% factor cp 5016 0.781 1.363 1.722 ªº §· 4021 0.774 1.376 1.634 u 100 (4) «» ¨¸ 1«» ©¹ ¬¼ 3019 0.766 1.390 1.546 2617 0.763 1.395 1.510 where N represents the total number of experimental points IRUWKHSURSHUW\ XLVWKHFDOFXODWHGYDOXHIRUWKHSURSHUW\ 2516 0.762 1.397 1.501 i and xLVWKHPHDVXUHGYDOXHIRUWKHSURSHUW\ 2416 0.761 1.398 1.492 (UURUYDOXHVLQWKHGHWHUPLQDWLRQRIYDULRXVSDUDPHWHUV Table 7 2315 0.760 1.400 1.483 2215 0.760 1.402 1.475 (UURUYDOXH Parameter 2114 0.759 1.403 1.466 Black oil-1 Black oil-2 Black oil-3 2013 0.758 1.405 498.03 1.457 Saturation pressure 0.003 0.014 0.018 1614 0.766 1.373 428.80 0.885 1.484 )RUPDWLRQYROXPHIDFWRU 0.197 0.304 0.206 #VDWXUDWLRQSUHVVXUH 1212 0.776 1.337 350.21 0.897 1.626 5HODWLYHYROXPH 1.447 0.339 0.194 814 0.786 1.300 270.92 0.913 1.876 2LOGHQVLW\ 0.571 0.605 0.422 411 0.797 1.258 183.64 0.937 2.238 2LOIRUPDWLRQYROXPHIDFWRU 0.543 0.842 0.381 14.7 0.863 1.075 0.00 1.000 4.288 2LOYLVFRVLW\ 5.548 4.245 5.832 *25 2.293 3.912 1.105 5 Results and discussion Z factor of gas 0.340 0.368 0.613 %\UXQQLQJWKHSURJUDPIRUWKHEODFNRLOVDPSOHV answers are produced for each sample. For black oil-1, six YHUDJHHUURUYDOXH$ 1.368 1.329 1.096 DQVZHUVIRUEODFNRLOVHYHQDQVZHUVDQGIRUEODFNRLO nine answers were acceptable as engineering aspects. Fig. $OVRDEOH7VKRZVWKHYDOXHVRIPDWFKLQJSDUDPHWHUV 8 to Fig. 10 show the results of the tuning of EOS against DEOHDQG7VKRZVWKHYDOXHVRI%,&VDVDV\PPHWULFPDWUL[ PHDVXUHGGDWD$VWKHILJXUHVVKRZWKHVHOHFWHGPRGHOV IRUÀXLGVDPSOHDIWHUWXQLQJRIWKHPRGHO SUHGLFWWKHIOXLGEHKDYLRUZHOO$OVRD7EOHVKRZVWKH 5HVXOWVVKRZWKDWWKHSUHVHQWHGPHWKRGSHUIRUPHGZHOOLQ WXQLQJWKH(267KHHUURUYDOXHVVKRZWKDWWKHRSHUDWLRQ*$ at saturation pressure. LVUHDOO\VXUSULVLQJLQDOOFDVHV7KHPDLQDGYDQWDJHRIRXU algorithm is its high speed in finding the solution. While Table 66DWXUDWLRQSUHVVXUHDQGRLOIRUPDWLRQYROXPHIDFWRU WXQLQJRI(26LVDWHGLRXVDQGGLIILFXOWZRUNHYHQIRUDQ WHVWDWVDWXUDWLRQSUHVVXUHIRUWKH'/ H[SHULHQFHGUHVHUYRLUHQJLQHHUDQGRIWHQQHHGVDORQJWLPH Parameter 0HDVXUHGYDOXH &DOFXODWHGYDOXH WKHSURSRVHGPHWKRGFDQ¿QGWKHVROXWLRQE\FRQ¿QLQJWKH Black oil-1 VXSHUYLVLRQE\DQH[SHUWWRWKHODVWVWDJHLQDVKRUWWLPHDQG VDYHWLPHDQGH[SHQGLWXUHLQUHVHUYRLUVWXGLHV)XUWKHUPRUH Saturation pressure, psia 1845.00 1844.95 WKLVPHWKRGSUHSDUHVVHYHUDOSRVVLEOHDQVZHUVIRUDSUREOHP VLPXOWDQHRXVO\,QWKHPDQXDOWXQLQJRI(26ZHFRXOG¿QG 2LOIRUPDWLRQYROXPHIDFWRU 1.457 1.460 #VDWXUDWLRQSUHVVXUH5%67% RQO\RQHVROXWLRQDIWHUDORQJWLPHEXWLQWKLVPHWKRGWKH DOJRULWKPWHVWVGLIIHUHQWVLWXDWLRQVDQGGHOLYHUVWKHEHVW Black oil-2 DQVZHUVDWWKHHQGRIHDFKHYDOXDWLRQ7KHUHIRUHRQHPLJKW WHVWVHYHUDODOWHUQDWLYHVLQDVKRUWWLPHDQGFKRRVHWKHGHVLUHG Saturation pressure, psia 1719 1718.752 DQVZHUV$OVRLWLVSRVVLEOHWRXVHWKHVHGLIIHUHQWVROXWLRQV 2LOIRUPDWLRQYROXPHIDFWRU DVDVHQVLWLYLW\DQDO\VLV$OWKRXJKWKHILJXUHVDQGWDEOHV 1.332 1.328 #VDWXUDWLRQSUHVVXUH5%67% VKRZWKHEHVWDQVZHURI*$WKHRWKHUSK\VLFDOO\DFFHSWHG DQVZHUVRIWKHDOJRULWKPDUHVDWLVI\LQJIRUDUHVHUYRLUVWXG\ Black oil-3 $VVWDWHGHDUOLHUDWWKHHQGRIWKHSURJUDPWKHUHZHUH Saturation pressure, psia 2013 2012.641 JRRGDQVZHUVDQGZHVHOHFWHGVRPHRIWKHP'HILFLHQF\ in the other answers could be related to weighting factors. 2LOIRUPDWLRQYROXPHIDFWRU 1.405 1.408 HLJKWLQJIDFWRUVSOD\DQLPSRUWDQW:UROHLQWXQLQJRI(26,Q #VDWXUDWLRQSUHVVXUH5%67% WKLVPHWKRGZHLJKWLQJIDFWRUVDUHVSHFL¿HGDWWKHVWDUWRIWKH UHVXOWVIRUVDWXUDWLRQSUHVVXUHDQGRLOIRUPDWLRQYROXPHIDFWRU Pet.Sci.(2012)9:199-211 207 1.00 3.5 Measured Measured 0.98 Calculated 3.0 Calculated 0.96 2.5 0.94 2.0 0.92 1.5 0.90 1.0 0.88 0.5 0.86 0.0 0 200 400 600 800 1000 1200 1400 1600 1800 0 1000 2000 3000 4000 5000 6000 Pressure, psia Pressure, psia 54 3.0 Measured Measured Calculated Calculated 2.5 2.0 1.5 1.0 0.5 0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000 Pressure, psia Pressure, psia 0.60 1.50 0.55 1.45 1.40 0.50 0.45 1.35 0.40 1.30 0.35 1.25 1.20 0.30 0.25 1.15 1.10 0.20 Measured Measured 0.15 1.05 Calculated Calculated 0.10 1.00 1000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000 0 2000 3000 Pressure, psia Pressure, psia Fig. 8 SURJUDPDQGDUH¿[HGWLOOWKHHQGRIRQHHYDOXDWLRQZKLOHLQ manual tuning these factors can change through the process )XUWKHUPRUHLWLVSRVVLEOHWKDWDWXQHGPRGHOLVDFKLHYHGLQ +HQFHE\E\WKHLPSURYLQJWKHXVHUZHSURJUDPLQWKLVZD\ GLIIHUHQWZD\V6LQFHWKH*$LVDVWRFKDVWLFDOJRULWKPDQG PD\H[SHFWWRKDYHPRVWRIWKHJRRGDQVZHUVGHVLUDEOH WKH¿YHJURXSVRIZHLJKWLQJIDFWRUVDUHSURGXFHGUDQGRPO\ 1RWHWKDWWXQLQJRI(26LVQRWDXQLTXHZRUNDQGDÀXLG for each case, we would expect different results and tuned Oil FVF, RB/STB Oil density, Ib/ft Relative volume Oil viscosity, cp Gas Z factor GOR, MSCF/bbl VDPSOHFRXOGEHPRGHOHGZLWKVHYHUDOGLIIHUHQWWXQHGPRGHOV 5HVXOWVRIWKHWXQLQJRI(26IRUEODFNRLO 208 Pet.Sci.(2012)9:199-211 1.00 5.0 Measured Measured 4.5 0.98 Calculated Calculated 4.0 0.96 3.5 0.94 3.0 0.92 2.5 2.0 0.90 1.5 0.88 1.0 0.86 0.5 0.84 0.0 600 1000 1200 1400 1600 1800 0 1000 2000 3000 4000 5000 0 200 400 800 Pressure, psia Pressure, psia 58 3.0 Measured 56 Measured Calculated Calculated 2.5 2.0 1.5 1.0 0.5 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 Pressure,psia Pressure,psia 1.35 0.50 0.45 1.30 0.40 1.25 0.35 1.20 0.30 1.15 0.25 1.10 0.20 Measured Measured 1.05 0.15 Calculated Calculated 1.00 0.10 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 Pressure, psia Pressure, psia Fig. 9 models for each case if we re-run the program. R7HYDOXDWH factors change in each run, the following equation is used in WKH YDULDWLRQFDXVHGE\WKHVWRFKDVWLFDOJRULWKPWKHSURJUDP FDOFXODWLQJWKH506506 YDOXHVWRFRPSDUHWKHUHVXOWV is run 10 times for black oil-1. Fig. 11 shows the number of acceptable solutions as engineering aspects for 10 different §· UXQV$OVR)LJVKRZVWKH506DQGDYHUDJHHUURUYDOXHV (5) ¨¸ IRUWKHEHVWVROXWLRQRIHDFKUXQ6LQFHWKH506GHILQHG ©¹ E\(T GHSHQGVRQWKHZHLJKWLQJIDFWRUVDQGWKHVH Relative volume Oil FVF, RB/STB Oil density, Ib/ft Oil viscosity, cp Gas Z factor GOR, MSCF/bbl 5HVXOWVRIWKHWXQLQJRI(26IRUEODFNRLO Pet.Sci.(2012)9:199-211 209 3.0 1.00 Measured Measured 0.98 Calculated Calculated 2.5 0.96 2.0 0.94 1.5 0.92 1.0 0.90 0.5 0.88 0.0 0.86 1000 2000 3000 4000 5000 0 200 400 600 800 1000 1200 1400 1600 1800 Pressure, psia Pressure, psia Measured 5.5 Measured Calculated Calculated 4.5 3.5 2.5 1.5 40 0.5 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 Pressure, psia Pressure, psia 1.45 0.55 1.40 0.50 1.35 0.45 1.30 0.40 1.25 0.35 1.20 0.30 1.15 0.25 1.10 0.20 Measured Measured 1.05 0.15 Calculated Calculated 1.00 0.10 0 1000 2000 3000 5000 0 1000 2000 3000 4000 5000 Pressure, psia Pressure, psia Fig. 10 6LQFHWKH*$LVDVWRFKDVWLFDOJRULWKPGLIIHUHQWUHVXOWV WUHQGVDUHDOPRVWWKH$OVRVDPHWKHDERYHJUDSKVVKRZWKDW DUHDFKLHYHGIHUHQWIRUUXQVGLIRIWKHSURJUDPDQGQRHVSHFLDO WKHSURSRVHGPHWKRGFRXOG¿QGWKHVROXWLRQLQHDFKUXQ 7KHVHDGYDQWDJHVLQWURGXFHWKHSURSRVHGDOJRULWKPDV 6LQFHERWKWKH506 DQGDYHUDJHHUURUYDOXHVGHSHQGRQWKH a suitable method in tuning EOS against PVT experimental IHUHQFHGLIEHWZHHQWKHREVHUYHGDQGFDOFXODWHGYDOXHVWKHLU GDWDIRURUGLQDU\EODFNRLOV Oil density, Ib/ft Oil FVF, RB/STB Relative volume Oil viscosity, cp GOR, MSCF/bbl Gas Z factor UHODWLRQFRXOGEHIRXQGDPRQJWKHUHVXOWVLQWKHDERYHJUDSKV 5HVXOWVRIWKHWXQLQJRI(26IRUEODFNRLO 210 Pet.Sci.(2012)9:199-211 Table 8 Values of matching parameters for black oil-1 p T V (V ) crit crit crit isc &RPSRQHQWV 2PHJD$ Omega-B Z (V ) $FHQWULFIDFWRU S shift 3 crit isc psia °F ft /(lb·mol) + S 0.45724 0.077796 1296.2 212.81 1.5698 0.28195 0.1 -0.1025978 N 0.45724 0.077796 492.31 -232.51 1.4417 0.29115 0.04 -0.1313342 &2 0.45724 0.077796 1071.3 88.79 1.5057 0.27408 0.225 -0.0427303 & 0.45724 0.077796 667.78 -116.59 1.5698 0.28473 0.013 -0.1442656 & 0.45724 0.077796 708.34 90.104 2.3707 0.28463 0.0986 -0.1032684 & 0.45724 0.077796 615.76 205.97 3.2037 0.27616 0.1524 -0.0775014 i& 0.35598 0.060789 264.53 274.91 4.2129 0.14137 0.1848 -0.0619837 n& 0.32196 0.070838 275.33 305.69 4.0847 0.13693 0.201 -0.0542249 i& 0.45724 0.077796 491.58 369.05 4.9337 0.27271 0.227 -0.0417725 n& 0.45724 0.077796 488.79 385.61 4.9817 0.26844 0.251 -0.0302779 & 0.45724 0.077796 436.62 453.83 5.6225 0.25042 0.299 -0.0072888 & 0.45724 0.077796 426.18 526.73 6.2792 0.25281 0.3 0.0575821 & 0.45724 0.077796 417.66 575.33 6.936 0.26082 0.312 0.031934 & 0.45724 0.077796 381.51 625.73 7.7529 0.25394 0.348 0.0594578 & 0.61738 0.054329 175.47 667.13 8.5539 0.12413 0.385 0.0861113 & 0.32196 0.070838 161.73 706.73 9.4028 0.12149 0.419 0.1139716 & 0.42403 0.056428 43.473 1122.7 22.792 0.058352 2.2726 0.766346 12+ DOXHVRI%,&VIRUEODFNRLO9 Table 9 &RPSRQHQW + SN &2 & & & i& n& i& n& & & & & & & & 2 2 2 1 2 3 4 4 5 5 6 7 8 9 10 11 12+ +S0 N 0.176 0 &2 0.096 -0.012 0 & 0.05 0.1 0.1 0 & 0.05 0.1 0.1 0 0 & 0.05 0.1 0.1 0 0 0 i& 0.05 0.1 0.1 0 0 0 0 n& 0.05 0.1 0.1 0 0 0 0 0 i& 0.05 0.1 0.1 0 0 0 0 0 0 n& 0.05 0.1 0.1 0 0 0 0000 & 0.05 0.1 0.1 0.0279 0.01 0.01 0000 0 & 0.05 0.1 0.1 0.03308 0.01 0.01 0000 0 0 & 0.05 0.1 0.1 0.0363 0.01 0.01 0000 0 0 0 & 0.05 0.1 0.1 0.03896 0.01 0.01 0000 0000 & 0.05 0.1 0.1 0.04092 0.01 0.01 0000 0000 0 & 0.05 0.1 0.1 0.04246 0.01 0.01 0000 0000 0 0 & 0.0814079 0.1 0.162816 0.06297 0.01 0.01 0000 0000 0 0 0 12+ 3HW6FL 211 that modifying the matching parameters indiscriminately does not develop the model necessarily and may results in the 12 PRGHOVWKDWDUHPDWKHPDWLFDOO\DQGSK\VLFDOO\XQDFFHSWDEOH Since the GA is a stochastic algorithm, different results are achieved for different runs of the program and the proposed PHWKRGZDVVXFFHVVIXOLQ¿QGLQJWKHVROXWLRQLQHDFKUXQ References QDJ%XQ'DQG6XQ0*HQ HWLFDOJRULWKPIRUFRQVWUDLQHGJOREDO 0 RSWLPL]DWLRQLQFRQWLQXRXVYDULDEOHV$SSOLHG0DWKHPDWLF 12345 6789 10 &RPSXWDWLRQ WV.+DQG6PDUW&RD$SSOLFDWLRQ*RI7DUHJUHVVLRQEDVHG(26 397 Run number SURJUDPWRODERUDWRU\GDWD63(5HVHUYRLU(QJLQHHULQJ Fig. 11 Number of physically accepted solutions for 10 different runs $HVK397DQG3KDVH%HKDYLRXURI3HWUROHXP5HVHUYRLU)OXLGV'DQOVW 0.17 1.5 (GLWLRQ (OVHYLHU6FLHQWL¿F3XEOLVKLQJ&RPSDQ\ -RQJ.$$QDQDO\VLVRIWKH'HEHKDYLRURIDFODVVRIJHQHWLFDGDSWLYH 0.16 1.4 $EVWUDFWV7KHVLV8QLYHUVLW\RI0LFKLJDQ'LVVHUWDWLRQV\VWHPV3K' 0.15 ,QWHUQDWLRQDO % 0.14 1.3 )RJ DUW\7&DYDN9)DQG&KHQ\38VHRIWKHJHQHWLFDOJRULWKPIRU 0.13 ORDGEDODQFLQJRIVXJDUEHHWSUHVVHV,Q(VKHOPDQ/-(GV 1.2 0.12 3URFHHGLQJVWK,QWHUQDWLRQDO&RQIHUHQFHRQ*HQHWLF$OJRULWKPV JDQ.DXIPDQQ6DQ)UDQFLVFR0RU 0.11 1.1 RMS* value )RU UHVW+%.R]D-56KLSPDQ-HWDO%XLOGLQJDSDUDOOHO 0.10 Average error value FRPSXWHUV\VWHPIRUWKDWSHUIRUPVDKDOISHWDIORSSHU 1.0 0.09 GD\3URFHHGLQJVRIWKH*HQHWLFDQG(YROXWLRQDU\&RPSXWDWLRQ 23 4 56 7 89 10 JDQ.DXIPDQQ6DQ)UDQFLVFR&RQIHUHQFH0RU Run number GEHUJ'*RO(&RPSXWHU DLGHGJDVSLSHOLQHRSHUDWLRQXVLQJJHQHWLF DOJRULWKPVDQGUXOHOHDUQLQJ3K''LVVHUWDWLRQ7KH8QLYHUVLW\RI RMS and average error values for 10 different runs Fig. 12 0,$UERU$QQ0LFKLJDQ3UHVV *RO GEHUJ'(*HQHWLF$OJRULWKPVLQ6HDUFK2SWLPL]DWLRQDQG 0DFKLQH/HDUQLQJ$GGLVRQ:HVOH\3XEOLFDWLRQ&RPSDQ\0$ 6 Conclusions The results of this study show that the developed genetic *UH WHQVWHWWH-*RSDO55RVPDLWD%HWDO*HQHWLFDOJRULWKPVIRUWKH DOJRULWKPFDQEHVXFFHVVIXOO\DSSOLHGWRWKHWHGLRXVGLI¿FXOW WUDYHOOLQJVDOHVPDQSUREOHP,Q*UHWHQVWHWWH-(GV 3URFHHGLQJV and time consuming operations of tuning of EOS against )LUVW,QWHUQDWLRQDO&RQIHUHQFHRQ*HQHWLF$OJRULWKPVDQG7KHLU H[SHULPHQWDOGDWDIRUWKUHHUHDOEODFNRLOÀXLGVDPSOHV7KH $SSOLFDWLRQV/DZUHQFH$VVRFLDWHV(UOEDXP3XEOLVKHU1- property graphs show the successful tuning of EOS against +RO ODQG-+$GDSWDWLRQLQ1DWXUDODQG$UWLILFLDO6\VWHPV$Q measured data; furthermore, the average error values are Introductory Analysis with Applications to Biology, Control, and below 2 percent for all the cases and prove that the GA $UERU$UWL¿FLDO7KH,QWHOOLJHQFH$QQ8QLYHUVLW\RI0LFKLJDQ3UHVV 0, we cannot argue that the method presented in this study is the DN$.RQ &RLW':6PLWK$(0XOWLREMHFWLYHRSWLPL]DWLRQXVLQJ EHVWIRUPRIWKHJHQHWLFDOJRULWKPIRUWKLVSUREOHP JHQHWLFDOJRULWKPV$WXWRULDO5HOLDELOLW\(QJLQHHULQJ 6\VWH P The strong non-linearity of the EOS tuning process makes 6DIHW\ classical deterministic optimization methods inefficient KDOHZLF]=$OJRULWKP'DWD*HQHWLF6WUXFWXUH (YROXWLRQ0LF 3URJUDPV DQGXQOLNHO\WREHVXFFHVVIXO7KHUHIRUHDQDOWHUQDWLYH HUODJ1<9QG(GLWLRQ 6SULQJHU approach would be to use heuristic type methods like 1LN UDYHVK0$PLQ]DGHK)DQG=DGHK/$6RIW&RPSXWLQJDQG JHQHWLFDOJRULWKPV7KHDELOLW\RIJHQHWLFDOJRULWKPWRXVH ,QWHOOLJHQW'DWD$QDO\VLVLQ2LO([SORUDWLRQVW(GLWLRQ (OVHYLHU continuous and discontinuous variables, changing several 6FLHQWL¿F3XEOLVKLQJ&RPSDQ\ variables simultaneously and the ability of this method to QHH,3DU& DQGDWVRQ:$+3UHOLPLQDU\DLUIUDPHGHVLJQXVLQJFR work with different data structures in the same time, cause HYROXWLRQDU\PXOWLREMHFWLYHJHQHWLFDOJRULWKPV,Q%DQ]KDW: this optimization method to be a good choice to solve the 'DLGD-(LEHQ$(HWDO(GV 3URFHHGLQJVRIWKH*HQHWLFDQ G GDWDDXWRPDWLFDOO\SUREOHPRIPDWFKLQJ397 The main advantage of the method is its high speed )UDQFLVFR LQILQGLQJVROXWLRQV:KLOHWXQLQJWKH(26LVWHGLRXVDQG Q3:3RR DQG3DUNV*72SWLPL]LQJ3:5UHORDGFRUHGHVLJQV,Q 0DQQHU5DQG0DQGHULFK%(GV 3DUDOOHO3UREOHP6ROYLQJIURP difficult work even for an experienced reservoir engineer 1DWXUH1RUWK+ROODQG and often needs a long time to find just one tuned model, 5RP HUR&(DQG&DUWHU-18VLQJJHQHWLFDOJRULWKPVIRUUHVHUYRLU the proposed method can find more than one solution in a FKDUDFWHUL]DWLRQ-RXUQDORI3HWUROHXP6FLHQFHDQG(QJLQHHULQJ VKRUWWLPH$OVRZRUNLQJDXWRPDWLFDOO\FRQILQLQJWKHUROH of experts to the last stage, reducing costs and having the SRYD(DNOLHYD%DQFKHYD*1DQG*9JHQHWLF%$6,&$6KRKPDOJRULW possibility of evaluating the different situations are the other IRUHQJLQHHULQJSUREOHPVVROXWLRQ&RPSXWHUVDQG&KHPLFDO advantages of this method to match PVT data and makes it (QJLQHHULQJ an ideal method to implement as an automatic EOS tuning H]/DQJV2*HQHWLF-RXUQDODOJRULWKPV$QLQRLORYHUYLHZLQGXVWU\HO9 DOJRULWKPIRURUGLQDU\EODFNRLOV RI3HWUROHXP6FLHQFHDQG(QJLQHHULQJ Comparing the RMS values for different iterations shows (Edited by Sun Yanhua) RMS* Number of solutions Average error , % (YROXWLRQDU\&RPSXWDWLRQ&RQIHUHQFH0RUJDQ.DXIPDQQ6DQ RSHUDWLRQLVUHDOO\VXUSULVLQJLQDOOFDVHV+RZHYHUDWSUHVHQW
Petroleum Science – Springer Journals
Published: Jul 28, 2012
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