SIMULATION OF MULTI-PHASE FLOW THROUGH POROUS MEDIA USING CVFE (CONTROL VOLUME FINITE ELEMENT)

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1 Sciniic ullin h Plihnica Univrsiy Timisara Transacins n Mchanics Spcial issu Th 6 h Inrnainal Cnrnc n Hydraulic Machinry and Hydrdynamics Timisara, Rmania, Ocbr -, 4 SIMULTION OF MULTI-PHSE FLOW THROUGH POROUS MEDI USING CFE (CONTROL OLUME FINITE ELEMENT Gabril Irinl MRCU, Lcurr Cmpur Scinc Dparmn Prlum- Gas Univrsiy Plisi In CREŢU, Pr. Dparmn Hydraulics Prlum- Gas Univrsiy Plisi *Crrspnding auhr: v ucursi 39,, Plisi, Rmania Tl.: ( , Fax: ( , g_i_marcu@yah.cm STRCT Th papr prsns h applicain Cnrl-lum Fini Elmn (CFE mhd in simulain muli-phas (ar, il, gas l hrugh prus mdia. Th numrical mdl dvlpd using h CFE mhd prsns advanag in simulain auld rsrvirs r simulain cmplx gmry rsrvir. ls, CFE mhd can accura rprsns h lcain lls insid rsrvir, ac hich cnvninally carzian grid can d. Th numrical mdl uss bh IMPES (Implici Prssur- Explici Saurain and ully- implici chniqus. On s cas r validain mdls is dn by cmparisn ih h simpl cas n quasi-linar rsrvir ih lls: n r ar incin and h hr r prducin. ls, his s cas can b slvd using h racinal l hry (ucly-lvr. Th cmparisn givs h saisacry rsuls, vn i h ar saurain disribuin baind by CFE mhd prsns a crain lvl numrical disprsin. nhr s uss a quarr rm h classical iv pin parn r ar incin. This mdl cnain lls lcad n ppsi crnr n rcangl, n ll is r ar incin ih cnsan ra, h hr ll is r luid (il, gas and ar prducin. Th rsuls ar cmpard ih hs prvidd by hr numrical rsrvir simular, giving als h saisacry rsuls. KEYWORDS CFE mhd, IMPES mhd, ully implici mhd, muliphas l simulain. NOMENCLTURE [m ] ra riangl lmn a,b,c, riangl cicins [-] luid vlum acr F [m 3 /day] luid lux g [m/s ] graviy [md] abslu prmabiliy r [-] rlaiv prmabiliy h [m] rsrvir hicnss m [-] prsiy N linar basis uncin p [bar] prssur p c [bar] capillary prssur p s [bar] llbr prssur q l ra pr uni vlum Q [m 3 /day] luid ra (prd/in r s [m] ll radius r c [m] drainag radius S [-] phas saurain [day] im T ransmisibiliy µ [cp] luid viscsiy [bar] ndal l pnial Subscrips and Suprscrips riangl lmn luid g gas phas i,, indics riangls n, im lvl il phas ar phas REITIONS CFE Cnrl lum Fini Elmn IMPES Implici Prssur Explici Saurain 635

2 . INTRODUCTION Numrical simulain muliphas and mulidimnsinal l hrugh prus mdia is accmplishd by using numrical mhd r ging an apprxima sluin r l quains sysm. Ths quains ar highly nn-linar bcaus nn-linar dpndnc rlaiv prmabiliy, capillary prssur upn saurain and vlum acr and viscsiy upn prssur. Th classical sluin mhd uss ini-dirncs prvid a s algbraic quains ha culd b asy slvd. Hvr h hydrcarbn rsrvirs hav irrgular bundaris and cnain nnisrpic and hrgnus rcs, ih varius disribuin luid hrugh rsrvir. S ha, uss classical mhds ih rcangular carsian grid has many disadvanags spcially in auls and rnirs dscripin, ramn lls, rinain cs. Whil muliphas l nar h ll culd b prprly mdlld hrugh a cnning simular [], i is diicul d his in larg simulain sudis ha invlv many lls [9]. Mhds r rducing ngaiv inluncs inadqua ramn lls r aul ar uss lcal grid rinmn (LGR prpsd by Hinmann hrs [4], Quandall ss[], Frssyh and Sammn [], r uss hybrid grid prpsd by Pdrsa and ziz[9]. Fr rducing rinain c Yansi and McCran [] prpsd a nin-pin ini dirnc mhd hich is usd als in ECLIPSE simular prgram. Th Cnrl lum Fini Elmn (CFE mhd as prpsd as a chniqu r alling lxibl grids in Navir-Ss luid l simulain and hn i has dvlpd in l simulain luids hrugh prus mdia. Impran cnribuin a mhd dvlping in prlum ara has brugh by Hinrichs, Hinmann and hrs [5], using a msh calld PEI (Prpndicular-iscrs. Thy mphasisd h main advanags mhd spcially in rducin rinain c and lls ramn. Palagi [8], Nacul and ziz [6] uss h rni grids, a lcal rhgnal grids, r hrgnus rsrvir simulain and r rducing rinain c. Fung and Nghim[7] and Eymard Snir[] prsnd h gnral mdl r numrical slving using cnrl vlum ini lmn. In his papr a cnrl vlum ini lmn chniqus cupld ih implici prssur xplici saurain (IMPES mhd is usd r dvlping a phas il-ar simular. Fr mdl validain a simpl linar l xampl is prsnd. Th rsuls ar asy cmpard ih classical ucly-lvr mdl.. MTHEMTICL MODEL Th l gvrning quains ar hs basd n vlumric (blac il mdl rin in ingral rm. This implis a ishrmal l and validiy Darcy s la. In a blac il simular mass is nihr crad r dsryd, and hrr culd ri h marial balanc quain, and can b xprssd as: chang mass mass in - mass u Th l mass ccurs du l and rm h blc (F, l hrugh lls (q. Th chang mass (M ccurs du chang in luid dnsiis, chang in phas saurain, r chang in prssur. dm d F q ( Thrr r ach phas h quains can b rin as: ms ms d d r µ r µ q d q d r nd r nd ( Th srag rm ds n invlv prssur xplicily, bu h prsiy m, h vlum acr and viscsiy µ ar assumd b uncins nly prssur. Ths prpris ar baind rm PT analysis. 3. NUMERICL MODEL Rrncs [6][7][][3] giv a daild dscripin r building a numrical mdl r Cnrl lum Fini Elmn Numrical Mdl. Firs h rsrvir in dividd in vlum ini lmns, ach lmn having hmgnus prpris. Cnsidr n his lmn P P P P 3 P 4 P 5 and n his assciad riangl Figur. rni plygn ih assciad riangls 636

3 L N usual linar basis uncins dind n riangl. Cnsqunly: N N N N (3 hr N ( a b x c y and is ini x y lmn ara x y, a, b, c ar cnsan x y cicins rm riangl gmry. Th rlain ( is valid r ach riangl in discrisain, hnc applying h Gauss divrgnc hrm bain h ls bn nd and hr nds in riangl : yr xr F h dx dy (4 µ y µ Th surac ingral in rlain (4 is vr all h dgs cnrl vlum, s h l rm phas r subcnrl vlum rin F, in asscia riangl invlvs h lux acrss h vlum bundaris and. Similar r vrx and in riangl culd ri F, F. T calcula ( F, i mus b calcula: y y N N N N y (5 Th drivaiv N can b rin as (6: N b ( a b x c y (6 N c ( a b x c y y y r in marix rm (7: [ b ] b b Φ (7 [ c ] c c C Φ y In rlain (7 can bsrv ha h drivaivs marix has cnsan cicin r h linar inrplain uncins. Hnc r h nds riangl can ri h l rm as: T Φ (8 F hr: F F F ; Φ F ( and: ( T T T T T T ( T T T T T ( T T ( ih T is ransmissibiliy bn nds and : yr xr T h c c b b (9 4 µ µ Th rlain (8 is vry usul hn h glbal quains sysm is baind by assmblanc riangl ini lmns asscia ih vlum ini lmns. Fr nd in riangl can ri ( T ( F ( T nalgus r nd and ( T ( ( T ( F ( F T T Th abv rlains (, dmnsrad ha h inrblc l culd b rin in a similar rm ih hs baind by classical mhd ih ini dirnc, bu ransmissibiliis ar baind in dirn ay. Th accumulain rm in quain ( is simply discrisd using a rgrssiv ini dirnc mhd, r n subvlum bain: ms d h ms ( ms n Wll rprsnain is basd n h mdl prpsd by Hinmman and hr[4]. Thy cnsidrd ha h prssur numrically calculad is qual ih h avrag prssur vlum cnrl. Hnc p s p Q πh r (3 rc µ ln r s 637

4 hr: r c π ih v al ara vlum cnrl. Thus h numrical mdl r l hrugh prus mdia r cnrl vlum hich surrunds nd is: N ( F ( Q h N ms n ms (4 hr N is numbr nds cnncd ih nd and is ara subvlum. Th numrical mdl -phas l prsns srng nn-liniariis prducd by prssur and saurain dpndncis quain cicins. Thrr, us bh ully-implici schm and IMPES schm. Th ully-implici schm cnsidrs all h prpris a nx im lvl (. Thrr r ach blc can ri h rsidual quain r phas : N ( R ( Q ( F h N ms n n ( P ( n S n ms (5 X (6 Cnsidring h unnns r cnrl vlum, ar prssur il phas and ar saurain hn l quain hrugh prus mdia can b rin as in marix rm: ( X X, X..., R (7, 3, X Np caus srng nliniariis us h Nn mhd r slving h sysm quains. Cnsqunly linariz h sysm quains: J X R (8 Whr J is h Jacbian marix, and X is h dirncs unnns a succssiv irains. Th IMPES( IMplici Prssur Explici Saurain mhd invlvs h assumpin n changs in capillary prssur vr h im sp, namly: P p p (9 c Th numrical mdl r nd in riangl is: ( T ( ( T ( ( T ( ( T ( m ms ( S q q ( Cnsidring ngligibl h graviainal rm i.. (z z z in h ar quain g h S rm as: S ( T T ( p P ( T ( p P ( T ( p P S C n hc m [ c n ( p p hr C is a cicin givn by: C m dp ds c ms c c Q hc p ] ( ( Rplacing h S in il phas quain bain an quain nly in P unnns. Thus riing h quains r hr nds riangular lmn ( can bain h lmnal quains in marix rm: F X M (3 hr M is rms vcr, F is lmnal marix unnns cicin (sinss marix, X is ndal unnns vcr cnaining nly h prssur valus. Th lmnal quains has 3 unnns r prssur ndal valu in riangl. Th lmnal quains assmblag in glbal marix can b maing in ays: rm cnrl vlum ini lmn and rm riangl. ssmblag rm CFE has advanag simplr calculain pr vlum and rapid implmnain bundary cndiin, bu i has disadvanag supplmnary spac r srag mr inrmain abu cnsiun riangls. ssmblag rm riangls is mr icinly in inr-nd luid ls dscripins, bu has disadvanags diicul calculain prs vlum. Glbal marix has many lmns quals ih zr; i is a spars marix. Thrr h srag in cmpur mmry is mad in rm band marix r using h spcial mhds r spars marix srag. I us a band marix hn mus adp an rdring mhd nds in grid, hich assur a minimum band idh. Th auhrs hav usd r numrical mdl dvlpmn n classical mhd spars marix srag, h randm cmpac mhd. This mhd allcas hr vcrs in mmry: n ral valus vcr r nn-zr marix lmns and ingr vcrs r 638

5 lcain hs lmns in spars marix. Th glbal quains sysm sluin mhd a accun his paricular yp marix srag, h auhrs using h bicnuga gradin mhd hich all a asymmric marix sysm and i is rlaiv quicly cnvrgn mhd. 4. TESTING THE MODEL 4.. Linar cas W cnsidr a linar mdl r sing h mdl, shard in riangular lmn li in igur. Th rni diagram assciad ih his riangular msh is prsnd in igur 3. Figur. Triangular grid r rsrvir. Figur 3. Triangular grid r rsrvir. Th rsrvir has h prpris prsnd in abl and rlaiv prmabiliis vrsus saurain ar prsnd in igur 4 Tabl. Rsrvir prpris Prpry alu Prpry alu Lngh. Widh Prsiy. Oil dnsiy 88 Thicnss War dnsiy Prmabiliy X Prmabiliy Y Oil viscsiy, Iniial cp War viscsiy cp Saurain iniial. Iniial prssur Th il prducr is placd in nds 3 ih ra m3/days and h incr is placd in nds and incin ra is m3/days. Th prssur disribuin and h saurain disribuin ar shing in igur 5 and 6. Kr; Kr Rlaiv Prmabiliis vs Saurain SW [-] Figur 4. Rlaiv prmabiliis curvs Kr Kr x [mrs] Prssur a days Figur 5. Prssur disribuin a days I cnsidr nly cnral nds h ar saurain disribuin a dirn im (,, 5,, 5, days rm saring ar incin is shd in igur 7. W hav cmpard h ar saurain disribuin ih h saurain disribuin baind using ucly Lvr mhd r n dimnsinal ar incin prblm. This cmparisn is prsnd in igur 8. W can s a crain lvl numrical disprsin, bu h rsuls ar similarly. X[mrs] Saurain [-] W ar saurain a days Figur 6. War Saurain a days Saurain Disribuin. CFEMdl War Saurain [-] X[mrs] Prssur[bar] 5 5 Figur 7. Saurain disribuin in cnral nds. 639

6 Figur 8. Cmparisn bn CFE mdl and ucly Lvr mhd. 4.. T-dimnsinal iv pin parn Th scnd cas prsnd hr is a quarr classical iv pin parn usd in ar incin. In his cas hav als lls placd in psi crnrs, Th incin ll has m 3 incin ra and is lcad in bm l crnr, Th prducin ll is lcad in p righ crnr and has a prducin ra m 3. Th hr prpris ar simillary ih hs an rm prvius xampl. W cnsidr hr ind riangular msh. Firs grid calld Msh, has 4 nds and 64 riangls similar ih a carzian grid (igur 9.a, scnd msh calld Msh, has 88 nds and 4 riangls (igur 9.b, h hird calld Msh3, has 7 nds and 4 riangls (igur 9.c. Th prssur disribuin r ach msh a 5 days ar h sar incin is prsnd in igur. Th ar saurain disribuins a sam im, 5 days, ar prsnd in igur. W can bsrv h dirncs inducd by h siz riangls and hir rinain. Fr Msh, h rlaiv larg siz riangls givs a high lvl numrical disprsin. Thrr h ar saurain rn is sprad vr nir diagnal squar, and prssur disribuin prsns variain vn n midl rgin squar. Th scnd msh, Msh, bcaus smallr dimnsin riangls, prsns a smallr numrical disprsin and h prssur n middl rgin has small variains. Th hird msh, Msh3, bcaus larg riangls siz in cnral rgin squar prsns h ndncy laral sprading saurain rn. Th prssur is alms cnsan in middl par squar, prsning larg gradins nly nar lls. Dspi hs dirncs h valus baind ar cmparabl (prssur is bn r Msh, r Msh, bars r Msh3 hy ar rlaiv small rm pin vi prlum nginrs. Figur 9. Triangl mshs r s cas 64

7 Figur. Prssur disribuin a 5 days r h hr mshs Figur. War saurain disribuin a 5 days r h hr mshs 64

8 CONCLUSION Th papr prsns a simpl numrical mdl using rni grid and CFE mhd r simulain ar-il ling hrugh prus mdia. Th rni grid alls a gd rprsnain rsrvir gmry and lls lcains. Th IMPES Mhd is usd r numrical ramn nn-liniariis prducd by prssur dpndncis viscsiy and vlum acr, and saurain dpndncis rlaiv prmabiliis n inrblc ls and accumulain rms. Th CFE simular uss a randm cmpac mhd r marix srag in cmpur mmry, a mhd ha p nly a nn-zr valu in mmry, s i is icinly r big marix srag ih many zr valus. ls, h quain sysm is slvd by h bicnuga gradin mhd ihu marix prcndiin. Th phas simular using h CFE mhd in assciain ih IMPES mhds is validad agains h ucly Lvr mdl r n simpl rsrvir ih n incin ll and n prducin ll, and cnsan physical prpris. Th rsuls numrical xampl ar similarly ih hs givn by ucly-lvr mdl. In -dimnsinal l h msh has crain impranc and hrr is br i h riangl msh has n riangl ih small angls and riangls hav rlaiv sam ara. Evn h riangl msh dn rspc h unirmiy rquirmns r grid h rsuls d n dir vry much, hy ar alms similarly. REFERENCES. EYMRD R., SONIER F., Mahmaical and Numrical Prpriis Cnrl-lum-Fini- Elmn Schm r Rsrvir Simulains, SPERE. Nvmbr 994. FORSYTH P..JR., SMMON P.H., Lcal Msh Rinmn and Mdlling Fauls and Pinchus, SPERE, Jun 986 pg: FORSYTH P..JR., Cnrl-lum-Fini- Elmn Mhd r Lcal Msh Rinmn in Thrmal Rsrvir Simulain, SPERE, Nvmbr 99 pg: HEINEMNN, Z.E., Grn, G., Hanlmann, Using Lcal Grid Rinmn in Mulipl- pplicain Rsrvir Simulain, papr SPE 55 prsnd a 983 SPE Rsrvir Simulain Sympsium, San Francisc, Nv Hinmann, Z.E., Mdling Rsrvir Gmry Wih Irrgular Grid, SPERE, May 99 pg: NCUL, E.C., ZIZ, K., Us Irrgular Grid in Rsrvir Simulain, papr SPE 886 prsnd a 99 SPE nnual Tchnical Cnrnc and and Exhibiin, Dallas, Oc NGIEM. L.X., FUNG L.S.K., Rsrvir Simulains. ih a Cnrl-lum-Fini-Elmn Mhd, SPERE ugus 99 pg: PLGI, C.L., Us rni Grid in Rsrvir Simulain, papr SPE 889 prsnd a 99 nnual Tchnical Cnrnc and Exhibiin, Dallas c PEDROS, O..JR., ZIZ, K, (986 Us Hybrid Grid in Rsrvir Simulain SPERE, Nvmbr. 986 pg: QUNDLLE P., ESSET P., Th Us Flxibl Gridding r imprvd Rsrvir Mdlling, papr SPE 39 prsnd a h 983 SPE Sympsium n Rsrvir Simulain, San Francisc Nvmbr 5-8 pg:5-3. SETRI,., ZIZ, K, (974 Cmpur Mdl r T Phas Cnning Simulain SPEJ, Jun 974 pg:-36.. YNOSIK, J.L. MCCRCKEN T.., Nin Pin Fini Dirnc Rsrvir Simular r Ralisic Prdicin dvrs Mbiliy Rai Displacmns, SPEJ, ugus 979 pg: