20th International Congress of Mechanical Engineering THREE-DIMENSIONAL MULTIPHASE BLACK-OIL SIMULATOR BASED ON GLOBAL MASS FRACTION

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1 Prceedins f COBEM 2009 Cpyriht 2009 by ABCM 20th Internatinal Cnress f Mechanical Enineerin Nveber 15-20, 2009, Graad, RS, Brazil THREE-DIMENSIONAL MULTIPHASE BLACK-OIL SIMULATOR BASED ON GLOBAL MASS FRACTION Wil Mreira Nunes, iuslav@ail.c a Duarte da Csta Sarent, duartecsta_sarent@yah.c.id b Federal University f Ceará Mechanical Enineerin Departent a, Cheical Enineerin Departent b André Luiz de Suza Araúj, andre@ifce.edu.br Institut Federal de Educaçã, Ciência e Tecnlia d Ceará Francisc Marcndes, arcndes@ufc.br Federal University f Ceará Departent f Metallurical Enineerin and Material Science Abstract. In eneral, petrleu reservir siulatrs fr the black-il del are based n pressure and saturatin. Hever, t deal ith the as phase appearance/disappearance se nuerical tricks are necessary. In rder t avid this prble, this paper presents the black-il del in ters f pressure and ass lar fractin. The equatins are discretized thruh the finite-vlue ethd usin bundary-fitted crdinates. The results are presented in ters f vluetric rates and saturatin fr il, as, and ater phases. Als, the results are cpared ith the cercial siulatr IMEX fr CMG cpany. Keyrds: Multiphase fl, Glbal ass fractin, Bundary-fitted crdinates, Finite-vlue ethd 1. INTRODUCTION The unknns in st f petrleu reservir siulatrs, based n black-il del, are pressure and saturatins. Hever, t deal ith the as phase appearance/disappearance se nuerical prcedures are necessary. These prcedures include, fr instance, se as residual saturatin r chane the variables durin the siulatin hen the as phase appear r disappear, Maliska et al. (1997). On alternative apprach t by-pass this nuerical prble is t replace the phase saturatin by the lbal ass fractin as suested by Prais and Capanl (1981). In this apprach, even hen all as phase is disslved in il phase the lbal ass fractin f as ill be different fr zer, Maliska et al. (1997). In this paper the abve entined apprach is applied t ultiphase fls invlvin il, as, and ater in cnjunctin ith bundary-fitted crdinates. The ass balance equatins in ters f lbal ass fractin f il, as and ater are discretized thruh the finite-vlue ethd and linearized by the Netn s ethd. The results are presented in ters f vluetric rates f il, as, and ater and phase saturatins. 2. PHYSICAL MODEL The black-il equatins fr the ater, il, and as cpnents in ters f the lbal ass fractins and pressures are iven by t t ( φρ ).(. ) Z = K Φ ( φρ ).(. ) Z = X K Φ X (1) (2) ( ) ( ) ( Z ). ( 1 X ) K. K. 1 X t φρ = Φ + Φ (3) Suin Equatins (1) thruh (3) the fllin lbal ass cnservatin equatin is btained: K K K t φρ = Φ + Φ + Φ (4) ( ).(... ) Equatin (4), in this rk, ill be called pressure equatin. In Eqs. (1) thruh (4), φ dente the prsity, ρ the averae density, Z, Z, Z are respectively the ater, il, and as lbal ass fractins, X the ass fractin f

2 Prceedins f COBEM 2009 Cpyriht 2009 by ABCM 20th Internatinal Cnress f Mechanical Enineerin Nveber 15-20, 2009, Graad, RS, Brazil cpnent il in il phase,,, are respectively, the ater, il, and as phases ultiplied by the density f the each phase,,, are the ass fl rate f ater, il and as phases, respectively per unit f bulk vlue f the reservir. These ters represent the sink r surce ters fr the cntrl vlues hich culd cntain a ell. K is the abslute pereability tensr hich is assued, in this rk, as a dianal tensr. The ptential f each phase (Φ) is iven by Φ= P ρ z p p p (5) here p dentes the ater, il, r as phase, P is the pressure, ρ p is the density f phase p, is ravity, and z is the depth, hich is psitive in the upard directin. Inspectin Eqs. (1) thruh (5), e can seen that there are 6 unknns fr (Z, Z, Z, P, P, and P ) and nly three equatins, since Eq. (4) is just a cbinatin f Eqs. (1) thruh (3). The clsin equatins ces fr capillarity pressure relatins and the ass cnservatin equatin hich requires Z+ Z+ Z= 1 (6) 3. EVALUATION OF THE PHYSICAL PROPERTIES In rder t evaluate the physical prperties a flash calculatin prcedure as suested by Prais and Capanl (1981) and Cunha (1996) ill be used. In this prcedure, e first evaluate the as-il slubility rati by R s ( 1 Z Z ) ρ STC, = in, Rs( P ) ρ, STC Z (7) here STC eans standard cnditins and Rs(P ) dentes the as-il slubility rati in PVT data f Black-il del. Then, e evaluate the ass fractin f each phase (α p ). The ass fractin f ater, il, and as phase are iven by α α ρ α STC, = Z ; = Z 1 + Rs ; = 1 α ρ, STC α (8) Further details f the ass fractin f each phase can be fund in Cunha (1996). The next step is t evaluate the density f each phase, and finally the saturatins. The saturatins are evaluated by α p ρp αp = αp np ρ p (9) here n p dentes the nuber f phases. 4. TRANSFORMATION OF THE EQUATIONS In this rk Eqs. (1), (2), and (4) are used t evaluate Z, Z, and P. In rder t deal ith cplex features f the reservirs such as irreular bundaries and faults, bundary-fitted crdinates are eplyed. The equatins are ritten in cputatinal plane usin the fllin transfratin: ξ = ξ( x, yz, ) ; η = η( xyz,, ) ; γ = γ( xyz,, ) (10) Usin the relatins iven by Eq. (10), Eqs. (1), (2), (4) can be ritten in a reular cputatinal plane, Thpsn et al. (1985) and Maliska (2004). Therefre, the fllin equatins fr the cpnents ater, il and the pressure equatins are btained in the cputatinal plane:

3 Prceedins f COBEM 2009 Cpyriht 2009 by ABCM 20th Internatinal Cnress f Mechanical Enineerin Nveber 15-20, 2009, Graad, RS, Brazil Water equatin: φρz = D11 + D12 + D13 + D21 + D22 + D23 t ξ ξ η γ η ξ η γ D31 + D32 + D33 (11) γ ξ η γ Oil equatin: φρ Z = X D + D + D + X D + D t J t ξ ξ η γ η ξ η + X D + D + D γ ξ η γ J t (12) Pressure equatin: φρ = D11 + D12 + D13 + D21 + D22 + D 23 + t ξ ξ η γ η ξ η γ D31 + D32 + D33 + D11 + D12 + D13 + γ ξ η γ ξ ξ η γ D21 + D22 + D23 + D31 + D32 + D33 + η ξ η γ γ ξ η γ D11 + D12 + D13 + D21 + D22 + D23 + ξ ξ η γ η ξ η γ + + D31 + D32 + D33 γ ξ η γ (13) In the transfred equatins J t is the Jacbian f the transfratin and D tensr invlves fluid, reservir and eetric infratin. The expressin fr the Jacbian and D tensr fr the ater cpnent are iven by D = K + K + K D = K + K + K = D ( ξ ξ ξ ) ; ( ξ η ξ η ξ η ) 11 x xx y yy z zz 12 x x xx y y yy z z zz 21 D = K + K + K = D D = K + K + K ( ξγ ξγ ξγ ) ; ( η η η ) 13 x x xx y y yy z z zz x xx y yy z zz D η γ K η γ K η γ K D D γ K γ K γ ( ) ( ) = x x xx+ y y yy + z z zz = 32 ; 33 = x xx+ y yy + z J K z z t t ( ( ) ( ) ( )) 1 ξ η γ γ η η ξ γ γ ξ γ ξ η η ξ J= x yz yz x yz yz + x yz yz (14) here ξ x, ξ y,.., η x, η y,..,γ x, γ y, are the direct etrics f the transfratin, hich are evaluated nuerically as a functin f the inverse etrics (x ξ, x η,..y ξ, y η,..z ξ, z η,..). Expressins fr direct etric as a functin f the inverse etric can be fund in Maliska (2004).

4 Prceedins f COBEM 2009 Cpyriht 2009 by ABCM 20th Internatinal Cnress f Mechanical Enineerin Nveber 15-20, 2009, Graad, RS, Brazil 5. APPROXIMATE EQUATIONS Equatins (11) thruh (13) are interated in the cputatinal plane. Perfrin the interatin f Eq. (11) in tie and in the cntrl vlue f Fi. 1 and adptin a fully iplicit ethd t evaluate the physical prperties in each interface f the cntrl vlue, the fllin equatin fr the ater cpnent is btained: N B Δγ n b Δη W f s P e E η F Δξ S γ ξ Fiure 1 Cntrl vlue in the cputatinal plane ΔV ( φρ Z ) ( φρ Z ) = D11 + D12 + D13 ΔηΔγ D11 + D12 + D13 Δ P P t ξ η γ ξ η γ P e D21 + D22 + D23 ΔξΔγ D21 + D22 + D23 ΔξΔ γ + ξ η γ ξ η γ n D D D ξ η + + Δ Δ D + D + D ΔξΔη ξ η γ ξ η γ P f s b Δ (15) In Equatin (15), P is the ass fl rate f the ater phase hich is psitive fr the prducin ells and neative fr injectin ells. This ter is iven by ΔV P = (16) P The ater bility that is included in the D tensr is evaluated in each interface f the cntrl vlue usin an upind schee. Cnsiderin, fr instance, the interface lest f the cntrl vlue P (see Fi. 1) this bility is evaluated by e, P, se G11 + G12 + G13 > 0 ξ η γ e = E, se G11 + G12 + G13 < 0 ξ η γ e (17) The sae prcedure ust e applied fr the ther interfaces f each cntrl vlue. The sae prcure perfred t Eq. (11) ust be realized t Eqs. (12) and (13). Next, the apprxiate equatins need t be linearized. In this rk, the Netn ethd as eplyed t linearize Eq. (15) and siilar nes fr the il and pressure. All the 19 dianals ere kept at the linear syste. As e have 3 unknns per cntrl vlue, each entry in the atrix dentes a 3x3 subatrix. The linear systes ere slved by the Bi-CGSTAB (van der Vrst, 1992) precnditined ith a riht incplete LU factrizatin f level 1, Marcndes et al (1985).

5 Prceedins f COBEM 2009 Cpyriht 2009 by ABCM 20th Internatinal Cnress f Mechanical Enineerin Nveber 15-20, 2009, Graad, RS, Brazil 5. TEST PROBLEMS In rder t sh the applicatin f the ethdly eplyed in the present rk three case studies ere carried ut. The first ne is the siulatin f a saturated quarter f five spt. In this case, e injected ater in rder t increase the reservir pressure and thus disappear the as phase. Fr this case, nly Cartesian eshes ere eplyed. Table 1 presents the fluid and physical prperties. The relative pereability curves fr all study cases shn in this rk are iven in Tabs. 2 and 3. Als, the capillary pressure as set t zer in all siulated cases. The injectin ell is perfrated aln all layers hile the prducin ell nly in the first layer. Fr Table 1, it is pssible t bserve that il and ater phases are presented in the layer ne and as and ater are presented in layer three. Table 1 Input data fr case 1. Reservir data Initial cnditin Physical prperties Well cnditins K = 9.9 x A = x φ = 0.2 c r = 0.0 r = P = 20 x 10 6 Pa S i = 0.12 S i = 0.88 (layer 1) 0.88 (layer 2) 0.00 (layer 3) B = 1 at 0 Pa c = 1.00 x Pa -1 c = x 10-9 Pa -1 q i = ³/d P f = 20x10 6 Pa Table 2 Oil-as relative pereability as a functin f liquid saturatin (S +S ). S lt k r k r Table 3 il-ater relative pereability as a functin f ater saturatin (S ). S K r K r The secnd case study refers t an undersaturated reservir in a quarter-f-five-spt. Fr this case study, Cartesian and bundary fitted crdinates ere eplyed. The reasn fr the nn-rthnal esh be used here is t investiate the bundary fitted ipleentatin. It is rthhile t entin that the reservir prductin des nt depend n the ay the reservir as discretized and the apprxiate equatins ere slved. On the ther hand, the nuerical results are strnly affected by the eplyed eshes and the ay the equatins are slved. Fr this case, nly the priary recvery is investiated thruh the use f t prducin ells perfrated in all three layers f the reservir. The input data fr this case study case is shn in Tab. 4. Fiure 2 shs the bundary fitted esh eplyed t this case study.

6 Prceedins f COBEM 2009 Cpyriht 2009 by ABCM 20th Internatinal Cnress f Mechanical Enineerin Nveber 15-20, 2009, Graad, RS, Brazil Fiure 2 Case study 2 16x16x3 bundary fitted esh. The last case study refers t a reservir ith a re cplex eetry. Fiure 3 presents the reservir. As the IMEX cde des nt include the crss derivatives in the ass fl rate evaluatin, the results fr this case study are nt cpared ith the IMEX siulatr. Table 5 presents the input data used. This case study als refers t undersaturated reservir and nly the priary recvery is investiated. Table 4 Input data fr case 2. Reservir data Initial cnditin Physical prperties Well cnditins k = 9.9 x h = 15 A = x φ=0.2 c r = 0.0 r = P = x 10³ Pa S i = 0.12 S i = 0.88 B = 1 at 0 Pa c = 1.00 x Pa -1 c = x 10-9 Pa -1 P f = 20x10 6 Pa Fiure 3 Case study 3 12x12x3 bundary fitted esh. Table 5 Input data fr case 3. Reservir data Initial cnditin Physical prperties Well cnditins k = 9.9 x h = 30 A = x φ=0.2 c r = 0.0 r = P = x 10³ Pa S i = 0.12 S i = 0.88 B = 1 at 0 Pa c = 1.00 x Pa -1 c = x 10-9 Pa -1 P f = 20x10 6 Pa 6. RESULTS All the results shn in this sectin ere btained ith the fllin criteria fr the cnverence f the slutin in each tie-step, in pressure and in lbal ass fractin f il and ater, respectively: 68, Pa (10 psi) and 5x10-4. Fiure 4 presents the cuulative prductin f il, as, and ater fr case study 1. Fr Fiure 1, it is pssible t bserve that a d areeent beteen the present frulatin and the cercial siulatr IMEX ere btained.

7 Prceedins f COBEM 2009 Cpyriht 2009 by ABCM 20th Internatinal Cnress f Mechanical Enineerin Nveber 15-20, 2009, Graad, RS, Brazil (c) Fiure 4 Cuulative prductin rates case study 1. a) il b) as c) ater The cuulative prductin rate fr case study 2 are shn in Fi. 5, fr Cartesian, bundary-fitted crdinates, and IMEX usin Cartesian esh. As can be seen fr Fi. 5, a d areeent beteen the results btained ith the present frulatin fr all the tested eshes ere btained. Als, the results are aain very clse t the nes btained ith the cercial siulatr IMEX. Fiure 5 Cuulative prductin rates case study 2. a) il b) as Fiure 6 shs the as saturatin field in 43, and 98 days. Fr this fiure, it is pssible t bserve that the as phase appears just arund the prducin ells in the first days f prducin, and later n, due t the lest density f the as phase the hle as phase are cncentrated in the tp f the reservir (third layer). Fiure 7 presents the ttal vluetric rates f il and as fr the third case study. Fr this case, t bundary fitted crdinates eshes are eplyed. Fr the reasn entined previusly, e have nt cpared the results f the present rk ith the cercial siulatr IMEX. As can be seen in Fiure 7, d areeents ere btained fr bth esh cnfiuratins.

8 Prceedins f COBEM 2009 Cpyriht 2009 by ABCM 20th Internatinal Cnress f Mechanical Enineerin Nveber 15-20, 2009, Graad, RS, Brazil Gas saturatin Gas saturatin Fiure 6 Gas saturatin field case study 2. a) 43 days b) 98 days Fiure 7 Cuulative prductin rates case study 3. a) il b) as Fiure 8 presents the as saturatin field in 41 and 64 days, and Fi. 9 presents the as saturatin field in 97 and 157 days btained fr case study ( / ) Gas saturatin Gas saturatin Fiure 8 Gas saturatin field case study 3. a) 41 days b) 64 days

9 Prceedins f COBEM 2009 Cpyriht 2009 by ABCM 20th Internatinal Cnress f Mechanical Enineerin Nveber 15-20, 2009, Graad, RS, Brazil 3 3 ( / ) Gas saturatin Gas saturatin Fiure 9 Gas saturatin field - case study 3. a) 97 days b) 157 days Aain, fr Fiures 8 and 9 it is pssible t bserve the sae behavir already seen fr the as saturatin fr case study 2. We ean, at the bein, the as phase ce ut f the slutin f il phase and then quickly spreads ut all ver the last layer f the reservir, due t the lest density f the as phase cpared t the il and ater phases. 7. CONCLUSIONS The present rk has ipleented and tested the black-il del based n lbal ass fractin and pressure in cnjunctin ith bundary fitted crdinates. The equatins ere slved by the finite vlue ethd usin a fully iplicit frulatin. The ipleented frulatin as able t treat the as phase appearance and disappearance ithut any additinal nuerical tricks as ccurs ith the traditinal black-il frulatin based n saturatins and pressure fr several saturated and undersaturated reservirs. 8. ACKNOWLEDGEMENTS The authrs uld like t t thank CNPq (The Natinal Cuncil fr Scientific and Technlical Develpent f Brazil) fr their financial supprt. 9. REFERENCES Cunha, A. R. C., 1996 Methdly fr 3D Petrleu Reservir Siulatin Usin the Black-il Mdel and Mass Fractin Frulatin, M.Sc. Thesis (in Prtuuese), Federal University f Santa Catarina, Flrianóplis, Brazil. Maliska, C. R., 2004, Cputatinal Heat Transfer and Fluid Fl, 2nd Editin (in Prtuuese), Livrs Técnics e Científics Editra S.A., Ri de Janeir, Brazil. Maliska, C.R., da Silva, A.F.C., Czesnat, A.O., Lucianetti, R.M., and Maliska Jr., C.R., 1997, Three-Diensinal Multiphase Fl Siulatin in Petrleu Reservirs usin the Mass Fractins as Dependent Variables, SPE MS, Latin Aerican and Caribbean Petrleu Enineerin Cnference, Ri de Janeir, Brazil. Marcndes, F., Zabaldi, M. C., Maliska, C. R., 1985, Cparisn f Nn-Statinary Methds Usin Unstuctured Vrni Mesh in Petrleu Siulatin (in Prtuuese), XII COBEM, Bel Hriznte, Brazil. Prais, F. and Capanl, E. A., 1991, Mdelin f Multiphase Fl in Reservir Siulatin (in Prtuuese), XI COBEM, Sã Paul, Brazil. Thsn, J. F., Warsi Z. U. A., and Mastin, C. W., 1985 Nuerical Grid Generatin Fundatin and Applicatins, Elsevier Sciennce Publishin C., EUA. van der VORST, H. A., 1992, BI-CGSTAB: A fast and sthly cnverin variant f Bi-CG fr the siulatin f nnsyetric linear systes, SIAM J. Sci. Stat. Cpt., 13, pp RESPONSIBILITY NOTICE The authrs are the nly respnsible fr the printed aterial included in this paper.