Finite Element Simulation of Viscous Fingering in Miscible Displacements at High Mobility-Ratios

Transcription

1 Paula A. Ssini t al. Paula A. Ssini pssini@coc.ufrj.br Dnis A. F. d Souza dnis@lamc.copp.ufrj.br Alvaro L. G. A. Coutino alvaro@nacad.ufrj.br Fdral Univrsity of Rio d Janiro UFRJ COPPE, Dpt. of Civil Enginring PO Box 68506, Rio d Janiro, RJ, Brazil Finit Elmnt Simulation of Viscous Fingring in Miscibl Displacmnts at Hig Mobility-Ratios Numrical simulations of viscous fingring instabilitis in miscibl displacmnts at ig mobility-ratios ar prsntd. Anisotropic disprsion and monotonic viscosity profils ar considrd. T coupld st of partial diffrntial quations is approximatd by t smi-discrt SUPG stabilizd finit lmnt formulation plus a discontinuity capturing tcniqu to improv stability around t moving sarp fronts. T prssur quation is discrtizd by t standard Galrkin mtod, and a postprocssing scm is usd to improv t numrical valuation of Darcy s vlocity. In t rsulting scm all variabls (concntration, prssur and vlocity) ar approximatd by qual ordr linar triangular lmnts. A omognous cannl and a radial systm wr studid. Complx nonlinar viscous fingring mcanisms for ig mobility-ratio miscibl displacmnts wr obsrvd. Kywords: viscous fingring, miscibl displacmnt, stabilizd finit lmnts Introduction 1 Simulation of miscibl displacmnt flows as bn widly studid in t past yars du to its spcial importanc for t ptrolum industry. Enancd oil rcovry, solut transport in aquifrs, packd bd rgnration and rcovry of avy oil and bitumn ar xampls of som applications (Homsy, 1987). Suc procsss involv t injction of a displacing matrial tat is miscibl wit t rsidnt fluid. T displacmnt of a mor viscous fluid by a lss viscous on lads to a mcanical instability known as viscous fingring. Tis instability is ssntially govrnd by t mobility-ratio MR, wic is dfind as t ratio btwn t viscosity of t displacd fluid and tat of t injctd. Howvr, otr important factors also influnc t svrity of viscous fingring: trognous prmability filds, gravity, anisotropic disprsion, vlocity dpndnc of disprsion, nonmonotonic viscosity profils, tc. Hig mobility-ratio displacmnt procsss prsnt vry complx fingring pattrns. Bsids fingr intraction mcanisms, alrady known for low valus of mobility-ratio as silding, sprading, tip splitting, coalscnc and fading (Manickam and Homsy, 1993, 1994, 1995), otr mcanisms bcom dominant at ig mobility-ratios. Ts ar namd doubl coalscnc, sid-brancing, gradual coalscnc, singl-sidd tipsplitting, strtcd coalscnc, trailing lob dtacmnt, altrnating sid-brancing, skwring and dns brancing. As a non- Nwtonian fluid is mor unstabl tan its Nwtonian countrpart, for t sam low valus of mobility-ratio, in miscibl displacmnts wit non-nwtonian fluids wr also obsrvd som of tos complx fingring mcanisms (Azaiz and Moamad, 004). All of ts ar dscribd by Islam and Azaiz (005). Ty study ig mobility-ratio miscibl displacmnts of Nwtonian fluids involving isotropic disprsion. T miscibl displacmnt involving incomprssibl fluids in a rigid porous mdium can b dscribd by diffrnt systms of quivalnt quations. Many autors (Manickam and Homsy, 1993, 1995; Moissis t al., 1987; Sing and Azaiz, 001; Tan and Homsy, 1986, 1988) considr mor convnint and fficint to work in trms of vorticity and stramfunction. Coutino and Alvs (1996, 1999), Loula t al. (1999), Dias and Coutino (004), Coutino t al. (004), Juans and Patzk (00, 004) mploy primitiv variabls: prssur, vlocity and concntration. Papr accptd Novmbr, 009. Tcnical Editor: Francisco Ricardo Cuna Wn simulating miscibl displacmnts a varity of mtods was mployd, suc as finit diffrncs, rportd by Cristi t al. (1991) and Waggonr t al. (1991). Otr altrnativ as bn t psudo-spctral mtods as, for xampl, a Hartly transform-basd scm usd by Zimmrman and Homsy (1991) to simulat unstabl miscibl displacmnt. Ty xamind t ffct of anisotropic disprsion on nonlinar viscous fingring. Latr Zimmrman and Homsy (1991, 199a) xtndd ts mtods to tr dimnsions and prformd two dimnsional isotropic simulations in muc broadr and longr domains. Manickam and Homsy (1994, 1995) also usd a Hartly transform basd spctral mtod to invstigat t nonlinar volution of viscous fingring instabilitis in miscibl displacmnt flows wit nonmonotonic viscosity profils and latr ty studid vrtical miscibl displacmnt flows drivn by bot viscosity and dnsity contrasts. Otr works tat mploy psudo-spctral mtods ar Ruit and Miburg (000) and Sing and Azaiz (001). T modifid mtod of caractristics combind wit mixd finit lmnts is otr intrsting mtod (Moissis t al., 1987, 1988, and 1993). Otr mtod was prsntd by Fast and Slly (004). Ty dvlopd a moving ovrst grid scm for simulating t dynamics of fluids intrfacs. Ty usd a tin body fittd grid tat conforms to t dforming tim dpndnt boundary and is coupld to fixd Cartsian grids. In suc a way ty obtaind ig accuracy in t intrfac position. Rcntly, Wang and Zabaras (006) us a pairwis Markov Random Fild (MRF) to modl sourc idntification in miscibl displacmnts, obtaining accurat solutions. Finit diffrncs ar usd by most of commrcial rsrvoir simulators; owvr, tis mtod prsnts difficultis to andl complx gomtris. Psudo-spctral mtods ar igly accurat and ar vry fficint in t simulation of t growt of viscous fingring in porous mdia on paralll macins (Mangiavacci t al., 1997). T combination of a finit lmnt modifid mtod of caractristics and a mixd finit lmnt mtod gnrats a mtod wit vry littl numrical disprsion, but tis combind mtod involvs diffrnt intrpolation scms for prssur, vlocity and concntration. Coutino and Alvs (1996) mployd in t simulation of miscibl displacmnts in random trognous mdia a paralll finit lmnt mtod wr all variabls wr approximatd by qual ordr intrpolations. In tis mtod, prssur is computd by t standard Galrkin mtod, and a global post-procssing tcniqu (Malta t al., 1995) is usd to valuat vlocitis. A paralll implmntation on a distributd mmory macin using global and also local post-procssing tcniqus to comput igrordr vlocity approximations is dscribd by Loula t al. (1999). 9 / Vol. XXXII, No. 3, July-Sptmbr 010 ABCM

2 Finit Elmnt Simulation of Viscous Fingring in Miscibl Displacmnts at Hig Mobility-Ratios Wn solving t advction dominatd concntration quation, t intrst is in stability and accuracy. Stabilizd finit lmnt mtods ar particularly intrsting for tos cass. Coutino and Alvs (1996) adoptd t Stramlin Upwind Ptrov-Galrkin (SUPG) formulation, dvlopd by Brooks and Hugs (198) to control spurious numrical oscillations plus a discontinuity capturing tcniqu known as Consistnt Approximatd Upwind (CAU), dvlopd by Galão and do Carmo (1988). T rsulting smi-discrt quations ar approximatd in tim by a prdictormulticorrctor algoritm wit variabl tim stpping. Coutino t al. (004) prsntd a rviw of t main matmatical rsults for t stabilizd solution in bot spac-tim and smi-discrt framworks. Coutino and Alvs (1999) applid t formulation dscribd abov to simulat viscous fingring in miscibl displacmnts. Ty studid t volution of viscous fingring in a omognous mdia wit anisotropic disprsion and also otr numrical problm wit a nonmonotonic viscosity profil. Ty usd in bot of cass a rctilinar gomtry and t sam valus for t aspct ratio, A = 8, t mobility-ratio, MR = 0, and t global Pclt numbr, P G = Ty sowd tat t formulation is abl to simulat t volution of viscous fingring in diffrnt pysical situations aciving good paralll prformanc. In tis work, w study t ability of t formulation (Coutino and Alvs, 1996; 1999; Dias and Coutino, 004; Coutino t al., 004) dscribd abov to simulat som vry ig mobility-ratio displacmnt procsss (MR = xp(6)). At tis tim, w us two diffrnt gomtris: a rctilinar Hl-Saw cll and a radial cas, wr w us unstructurd grids. W considrd only anisotropic disprsion for bot numrical xampls. W obsrvd wn comparing our rsults wit otr works, Azaiz and Moamad (004), Islam and Azaiz (005) and Saron t al. (003) tat t stabilizd finit lmnt mtod applid r is abl to rprsnt corrctly t mcanisms of viscous fingring rlatd to t cas of ig mobility-ratio miscibl displacmnts. T rmaindr of tis papr is organizd as follows. In t nxt sction, w brifly prsnt t matmatical formulation adoptd. Sction 3 sows t smi-discrt stabilizd finit lmnt formulation for t spatial discrtization of t govrning quations. Also, in tis sction, w brifly dscrib t vlocity postprocssing tcniqu and t tim intgration scm. Two numrical xampls ar analyzd in t nxt sction. Bot ar miscibl displacmnt flows wit monotonic viscosity profils at ig mobility-ratios. T first on is a rctilinar Hl-Saw cll, and t otr a radial cll. T papr nds wit t summary of our obsrvations and main conclusions, in sction 5. Nomnclatur A = aspct ratio, dimnsionlss D // = longitudinal disprsion cofficint, dimnsionlss D = transvrsal disprsion cofficint, dimnsionlss H = caractristic widt of t domain, dimnsionlss L = caractristic lngt of t domain, dimnsionlss MR = mobility-ratio, dimnsionlss P = Pclt numbr, dimnsionlss R = constant usd in t dfinition of t MR, dimnsionlss U = caractristic vlocity of t fluid, dimnsionlss Grk Symbols = domain Γ = boundary φ = porosity of t porous mdium, dimnsionlss σ = stability paramtr to t post-procssd vlocity, dimnsionlss δ = nonlinar diffusion paramtr, dimnsionlss τ = SUPG stability paramtr, dimnsionlss η = pntration disturbanc, dimnsionlss ζ = magnitud of t concntration disturbanc, dimnsionlss Subscripts // rlativ to t longitudinal dirction or t paralll gradint dirction of t solution rlativ to t transvrs dirction 0 rlativ to initial tim rlativ to t lmnt c rlativ to t concntration magnitud t rlativ to post-procssing G rlativ to global 1 rlativ to t Cartsian componnt x rlativ to t Cartsian componnt y Govrning Equations T matmatical modl for t miscibl displacmnt of on incomprssibl fluid by anotr, in a rigid porous mdium, can b dscribd by a systm of partial diffrntial quations (Pacman, 1977). Ts quations in a domain R wit boundary Γ at a tim intrval [0, T] can b writtn as: v = q on [0, T] (1) v = A ( c). p on [0, T] () c φ + ( cv D( v ) c) = cq ˆ on [0, T] (3) t wr v is t total Darcy vlocity of t fluid mixtur, p its prssur, c its concntration and φ t porosity of t porous mdium. T tnsor A(c) is givn as A () c = K (4) μ() c wr K is t position-dpndnt absolut prmability tnsor and μ () c is t viscosity of t fluid mixtur, wic dpnds on its concntration c. T anisotropic diffusion-disprsion tnsor Dv ( ) is 1 v1 v D// 0 v1 v Dv ( ) = v v v1 0 D v v1 wr v 1, v ar t Cartsian vlocity componnts, and D // and D ar rspctivly t longitudinal and transvrs disprsion cofficints. T wlls ar rprsntd by sourc and sink trms dnotd by q. T function ĉ is spcifid at t sourcs and is qual to t rsidnt concntration at t sinks. W assum t usual no-flow boundary conditions v n = 0 in Γ, t [0, T] (6) wr n is t unit outward normal. T initial and boundary concntration conditions ar known, c(,0) = c0( ) on x x (7) (5) J. of t Braz. Soc. of Mc. Sci. & Eng. Copyrigt 010 by ABCM July-Sptmbr 010, Vol. XXXII, No. 3 / 93

3 Paula A. Ssini t al. Dv ( ) c. n = 0 on Γ, t 0, T (8) Uniqunss for prssur solution is nsurd imposing t normalization condition, p( x, t) d= 0, t 0, T (9) T viscosity µ(c) in Eq. (4) is assumd, as in Tan and Homsy (199), to vary xponntially wit t concntration, tat is, μ() c = Rc (10) wr R is a constant suc tat MR R = (11) bing MR t mobility-ratio, or t ratio btwn t viscositis of t rsidnt and t displacing fluids. It is important to not tat wn MR > 1, nonlinar ffcts associatd to t coupling of t quations and t convctiv trm strongly influnc stability and accuracy of numrical approximations. Finit Elmnt Formulation Finit Elmnt Discrtization In tis sction, w prsnt t smi-discrt stabilizd finit lmnt formulations applid to t govrning quations for miscibl displacmnts. T smi-discrt formulation is caractrizd by a finit lmnt discrtization in spac followd by a finit diffrnc discrtization in tim. T Galrkin formulation is applid to t prssur quation and t SUPG formulation (Brooks and Hugs, 198) plus t discontinuity-capturing oprator CAU (Consistnt Approximatd Upwind) dvlopd by Galão and do Carmo (1988) is mployd in t concntration quation. W considr t spac domain dividd in nl lmnts,, nl = 1, nl, wr = U = 1 and i I j = ø. W associat to tis discrtization t standard conforming st of picwis trial and wigting finit lmnts spacs. Tn, by substituting () in (1) and introducing t finit lmnt approximation, t classical Galrkin formulation for t prssur quation is writtn as wr w A ( c ) p d= w qd (1) w is t discrt wigting function for prssur and p is t discrt prssur. T wak variational approximation for t concntration quation is writtn as wlc c ( ) d+ nl τ v wr( ) c c d+ = 1 nl ( ) ˆ δ c w c c d= wcc qd = 1 (13) wr w c is t wigting function for concntration and c is t discrt concntration. T diffrntial oprator Lc ( ) is givn by c L( c ) = φ + ( c v D( v ) c ) (14) t and t lmnt lvl discrt rsidual is dfind as, R ( c ) = L( c ) cˆ q. (15) T first intgral in Eq. (13) is t Galrkin trm, t first summation of lmnt-lvl intgrals is t SUPG advction stabilization trm and t scond is t discontinuity-capturing trm, ndd to add stability around t moving sarp concntration fronts. In t cas of advction-dominatd flows wit no gravity ffcts and using linar lmnts, t particular form of t SUPG paramtr τ (Brooks and Hugs, 198) is dfind blow: wr bing wr 1 P τ = min,1.0 v 3 P is t local Pclt numbr dfind as 1 P = ( v) 3 v T Dv an stimat of t lmnt siz (16) (17) = A (18) A is t lmnt ara, and v t vlocity in t lmnt. T form of t nonlinar diffusion paramtr δ (Coutino and Alvs, 1999) in t CAU oprator is as follows: δ 1 ( ) P R c = min //,0.7 3 c 1 P // = 3 v // T // Dv// ( v ) v c // = c c (19) (0) v (1) wr P // is t local Pclt numbr corrspondnt to v // wic is t vlocity projctd in t paralll gradint dirction of t solution, c. Not tat δ = 0 wn c is zro. 94 / Vol. XXXII, No. 3, July-Sptmbr 010 ABCM

4 Finit Elmnt Simulation of Viscous Fingring in Miscibl Displacmnts at Hig Mobility-Ratios Vlocity Post-Procssing Wn simulating miscibl displacmnts t vlocity computd dirctly from Darcy s law is lss accurat tan t otr variabls. Post-procssing scms may b usd to obtain bttr approximations. Hr, w adoptd t global post-procssing scm of Malta t al. (1995). Tis scm is basd on t combination of Darcy s law variational formulation and t rsidu of t mass consrvation quation. Givn t prssur p and t concntration c and dfining 1 1 { w ( ) ( ), w n = 0 in Γ } () U H H wr R is a domain wit boundary Γ, t vlocity postprocssing consists in finding v% t U suc as w U w av nl 1% t pn d σ % t q d = 1 w ( A v + ) + w ( v ) = 0 (3) wr H 1 () is t usual finit lmnts spac of finit dimnsion functions, t magnitud v% t is t post-procssd vlocity and 1 σ = MR τ (4) Tis tcniqu givs igr ordr rats of convrgnc for t rcovrd vlocity, wic ar vn bttr tan tos obtaind wit mixd formulations (Arbogast and Wlr, 1995), Douglas t al. (1983). Using tis approac, t problm variabls, prssur, vlocity and concntration ar approximatd by standard qual ordr intrpolations. Stabl and accurat finit lmnt approximations ar obtaind for t concntration combining tis post-procssing tcniqu to comput vlocity wit t SUPG formulation for t transport quation. It is also important to not tat otr altrnativ tcniqus can also b mployd. Loula t al. (1999) mployd bot global and local post-procssing tcniqus to comput igr-ordr approximations. Local post-procssing is a gnralization of t global post-procssing basd on last-squard rsidual of t balanc quation, irrotationality condition and Darcy s law at som spcial points of suprconvrgnc of t gradint. Otr possibilitis ar, for xampl, a mixd stabilizd finit lmnt mtod wit two stabilizd variational formulations (Masud and Hugs, 00). T first accommodats continuous vlocity and prssur intrpolations and t scond accommodats continuous vlocity and discontinuous prssur. Rcntly, Bocv t al. (006) compar four diffrnt finit lmnt mtods for t Darcy quations and ac on uss vlocity and prssur approximations of t sam intrpolation ordr. Tim-Marcing Algoritm T concntration tim-drivativs ar approximatd by t gnralizd trapzoidal rul (Hugs, 1987). Tus, w obtain t following block-itrativ prdictor-multicorrctor algoritm to advanc t solution in tim: Block 1: Solv Prssur Equation Block : Comput Post-Procssd Vlocity Fild Block 3: Solv Concntration Equation T itrativ procss continus up to som convrgnc critria is mt. In tis algoritm t linar systm of quations corrsponding to t prssur quation is solvd by prconditiond conjugat gradints wil tat corrsponding to t concntration quation is solvd by t prconditiond GMRES algoritm. An lmnt-by-lmnt Gauss Sidl prconditionr is usd in bot cass. T systms of quations corrsponding to t vlocity postprocssing ar solvd using simpl Jacobi itrations. Hr w us variabl tim stps du to t strong nonlinar coupling btwn t prssur and concntration quations. W us an automatic timstp slction stratgy basd on t fdback control tory as prsntd in Coutino and Alvs (1996, 1999) and Valli t al (005). T most tim consuming stp in t block-itrativ scm is solving for prssur. Tis usually accounts for mor tan alf t CPU tim. Numrical Exprimnt Numrical simulations of ig mobility-ratio Nwtonian miscibl displacmnts in omognous mdia wit anisotropic disprsion and monotonic viscosity profils ar prsntd. A rctilinar and a radial Hl-Saw cll ar studid. Rctilinar Hl-Saw Cll W simulat a miscibl displacmnt flow wit anisotropic disprsion as invstigatd by Zimmrman and Homsy (1991, 199a, 199b) and also by Coutino and Alvs (1999). T rsulting govrning quations wr scald in t sam mannr as in t prvious works. T disprsion cofficints and dimnsionlss paramtrs ar also dfind by Zimmrman and Homsy (006). T computational domain is a rctilinar Hl-Saw cll, wit an aspct ratio A = 4 bing A = L/H, wr L and H ar rspctivly t caractristic lngt and widt of t computational domain. T mobility-ratio was st wit a ig valu: MR = xp(6.0), and global Pclt numbr, P G = 1000, bing P G = UL/ D// (5) wr U is t caractristic vlocity of t fluid. T initial condition for concntration is % ζ η (6) c 0 ( x) = c+ f( x)xp( x / ) wr f(x) is a random function, ζ is a magnitud of t concntration disturbanc, and η is t pntration of t disturbanc. W adoptd ζ = 0.01, η = L and, 1, x = 0 c% = (7) 0, 0 < x L T computational domain dimnsions ar: caractristic lngt L = 500 and caractristic widt H = 15. Tis domain was discrtizd in 56 x 104 clls, ac cll subdividd into 4 triangls in a diamond pattrn, gnrating a structurd ms wit 55,569 nods and 1,048,576 lmnts. W study t viscous fingring bavior at t cannl inflow. T simulation tim was 860 tim units. At tis tim, t fingrs ad not racd t nd of t cannl, but tis priod was noug for obsrving t dvlopmnt of viscous fingring mcanisms. W can obsrv r classical nonlinar intraction mcanisms as rportd by Zimmrman and Homsy (1991, 199a, 199b) and by Coutino and Alvs (1999). Ts mcanisms ar: silding, wr a fingr tat noss aad silds t growt of t nigboring fingrs; fingr fading, wr a fingr fads in J. of t Braz. Soc. of Mc. Sci. & Eng. Copyrigt 010 by ABCM July-Sptmbr 010, Vol. XXXII, No. 3 / 95

5 Paula A. Ssini t al. concntration; coalscnc, wr t tip of t coalscing fingr bnds and mrgs into t narst silding fingr; pairing, producd by t action of a pairwis mcanism upon t microscopic fingrs causing tm to mrg and tip splitting, wr t stramwis dirctd silding fingr sprads at t tip and splits into two vn fingrlts; t widr part sprads and gos troug anotr tip-splitting mcanism, and t procss continus. W may also find otr mor complx mcanisms obsrvd for miscibl displacmnts involving two non-nwtonian fluids, as dscribd by Azaiz and Moamad (004), and all of tm ar idntifid by Islam and Azaiz (005), vn toug ty simulat ig mobilityratio miscibl displacmnt flow wit isotropic disprsion. W can clarly obsrv in Figs. 1, and 3 two diffrnt typs of mcanisms: coalscnc and splitting of brancs. Dpnding on tir spcific faturs, ac mcanism, witin ts two wid catgoris, adopts a suitabl nam. Exampls of first group ar: doubl coalscnc, wr two fingrs adjacnt to a longr fingr approximat into its bas and mrg into it slowly; gradual coalscnc, wr a sligtly inclind fingr gradually mrgs into t closst on and it continus dvlopmnt insid of it and strtcd coalscnc, wr a silding fingr is bordrd of bot sids and mixturs wit t adjacnt fingrs. T scond group contains t following mcanisms: sid brancing, wr a silding fingr dg dvlops; it mrgs in t closst fingr continuing dvloping insid of it; singl-sidd tip-splitting, wr a silding fingr splits in two brancs always for t sam sid; altrnating sid-brancing, wr a branc sparats in otr brancs altrnatly for ac sid; skwring, wr t dg of a silding fingr dvlops; and dns brancing, wr a branc sparats in svral brancs simultanously. Numrical solutions for concntration ar sown in Figs. 1 and rspctivly for two diffrnt tim squncs, at t simulation bginning and at an intrmdiat tim. Figur 3 sows a configuration at t nd of simulation. All figurs wr scald up for bttr apprciating t viscous fingring mcanisms. In all Figurs w provid concntration contours in t rang [0, 1]. W obsrvd r, bsids classical viscous fingring instabilitis, som of t nonlinar intraction mcanisms tat appar in miscibl displacmnts involving two non-nwtonian fluids, as rportd by Azaiz and Moamad (004) and also in Islam and Azaiz (005), in tis cas for miscibl displacmnts at ig mobility-ratio wit isotropic disprsion. Tus, in Fig. 1 w can obsrv mcanisms as doubl coalscnc (DC) and singl-sidd tip-splitting (STS). In Fig., at a latr tim, w obsrv sidbrancing (SB) and gradual coalscnc (GC) and, in Fig. 3, w can apprciat trailing lob dtacmnt mcanism (TLD). W also obsrvd in Fig. 3 tat tr ar som brancs significantly mor dvlopd tan t rst. Tis typ of bavior was also obsrvd in non-nwtonian fluids wit low valus of mobility-ratio (Azaiz and Moamad, 004). Ts obsrvations suggst tat Nwtonian ig and non-nwtonian low mobility-ratio flow displacmnts dvlop similar instability pattrns. Hr, w also sow som computational data to illustrat t difficultis ncountrd in simulations of tis typ. Figurs 4 and 5 prsnt rspctivly t numbr of block itrations witin ac tim stp and t squnc of tim stps producd by t automatic stpsiz control algoritm. Not in Fig. 4 tat 4 to 6 block itrations ar ndd. In t simulation bginning t tim stp is small, but as t solution progrsss, t tim stps vary ordrs of magnitud bcaus of t fast dvlopmnt of complx fingring mcanisms, as sown in Fig. 5. STS DC Figur 1. Concntration contours for a tim squnc at t simulation bginning sowing doubl coalscnc (DC) and singl-sidd tip-splitting (STS) fingring mcanisms MR = xp(6.0), P G = 1000 and A = 4. SB S Figur. Concntration contours for a tim squnc at an intrmdiat simulation tim sowing sid-brancing (SB) and gradual coalscnc (GC) fingring mcanisms MR = xp(6.0), P G = 1000 and A = 4. TLD G GC Figur 3. Concntration contours at t nd of simulation sowing trailing lob dtacmnt (TB) fingring mcanism MR = xp(6.0), P G = 1000 and A = 4. DC STS GC DC STS SB 96 / Vol. XXXII, No. 3, July-Sptmbr 010 ABCM

6 Finit Elmnt Simulation of Viscous Fingring in Miscibl Displacmnts at Hig Mobility-Ratios c 0 ( r) = c + ζ f ( r)xp( r / η ) % (8) wr r is t position vctor in polar coordinats, f (r) is a random function, ζ and η ar rspctivly t disturbanc magnitud and pntration. W adoptd ζ = 0.01, η = R and 1, r = Ri 0 θ < π c% = (9) 0, Ri < r R 0 θ < π Figur 4. Non linar itrations witin ac tim stp. T simulation tim was 97 tim units. W sow rsults for rspctivly two diffrnt tim intrvals in Figs. 7 and 8: on at t simulation bginning and t otr at t simulation nd. Eac tim intrval contains a squnc of tr diffrnt tim instants in ac figur tat wr scald up for bttr apprciating t viscous fingring faturs; in bot figurs w prsnt concntration contours in t rang [0, 1]. T numrical rsults obtaind prsnt similar bavior pattrns wn compard wit xprimntal and otr numrical rsults (Saron t al., 003). W also obsrvd, as for t rctilinar cas, similar viscous fingring mcanisms obsrvd for miscibl flow displacmnts involving two non-nwtonian fluids, as dscribd by Azaiz and Moamad (004) and also in Islam and Azaiz (005), in tis cas for miscibl displacmnts at ig mobility-ratio wit isotropic disprsion. W can obsrv mcanisms as altrnating sid-brancing (ASB) and trailing lob dtacmnt (TLD) in Fig. 7. W also obsrvd sid-brancing and gradual coalscnc in Fig. 8. Figur 5. Tim stp siz variation. ASB ASB TLD ASB Radial Hl-Saw Cll W also analyz otr miscibl displacmnt flow wit anisotropic disprsion following again Zimmrman and Homsy (1991, 199a, 199b). Hr, t diffrnc is tat w study viscous fingring mcanisms in a radial gomtry. As in t rctilinar cas, P G = 1000 and MR = xp(6.0). It was also mployd a random initial condition. T computational domain is a radial configuration wit intrnal radius R i = 5.0 and xtrnal radius R = W mploy an unstructurd ms wit 99,17 nods and 197,634 lmnts. A ms dtail around t intrnal radius is sown in Fig. 6. Figur 7. Concntration contours for a tim squnc at t simulation bginning sowing altrnat sid-brancing (ASB) and trailing lob dtacmnt (TB) fingring mcanisms MR = xp(6.0) and P G = SB SB GC Figur 8. Concntration contours for a tim squnc at t simulation nd sowing sid-brancing (SB) and gradual coalscnc (GC) fingring mcanisms MR = xp(6.0) and P G = Figur 6. Finit lmnt ms dtail around intrnal radius. T initial random condition for concntration is Hr, w also sow som computational data. Figur 9 prsnts t squnc of tim stps producd by t automatic stp siz control algoritm. Diffrntly from t rctangular Hl-Saw cll, r w not a stp siz incras at arly simulation tims and as t solution progrsss and t fingr mcanisms volv, t stp siz diminiss. T numbr of block-itrations also incrass in t initial stps until 6 block-itrations pr stp, and rmains fixd until t nd of t simulation. J. of t Braz. Soc. of Mc. Sci. & Eng. Copyrigt 010 by ABCM July-Sptmbr 010, Vol. XXXII, No. 3 / 97

7 Paula A. Ssini t al. Conclusions Figur 9. Tim stp volution. In tis papr, it was prsntd numrical simulations to obsrv viscous fingring pattrns in miscibl displacmnts at ig mobility-ratios. W studid numrically two diffrnt gomtris: a rctilinar and a radial Hl-Saw cll. T numrical rsults for bot cass point out tat, bsids classical mcanisms of viscous fingring, otr typs of mcanisms bcom dominant at larg valus of mobility-ratios. Ty ar doubl coalscnc, sidbrancing, gradual coalscnc, singl-sidd tip-splitting, trailing lob dtacmnt and altrnating sid-brancing. Som of ts mcanisms, alrady idntifid in non-nwtonian fluids wit low valus of mobility-ratio, ar rportd by Azaiz and Moamad (004). All of tm ar idntifid by Islam and Azaiz (005) tat studid ig mobility-ratio miscibl displacmnt of Nwtonian fluids involving isotropic disprsion, using a diffrnt numrical approac. Trfor, t stabilizd finit lmnt formulation applid r was abl to rprsnt corrctly t svral complx mcanisms of viscous fingring rlatd to ig mobility-ratio as alrady obsrvd by otrs in miscibl displacmnt flows. Nw stabilizd formulations av bn dvlopd for Darcy flow in rcnt yars. Som xampls ar spac-tim formulations, as in Coutino t al. (004), a mixd stabilizd finit lmnt mtod prsntd by Masud and Hugs (00), four diffrnt finit lmnt mtods using vlocity and prssur approximations of t sam ordr dscribd by Bocv t al. (006) and multi-scal finit lmnt formulations rportd by Juans and Patzk (00). W want to point out tat ts stabilizd formulations applid to complx problms, as t cas of tis papr, would b a grat advanc. In suc a way w xpct tat t rang of t mobilityratio valu will b xtndd vn mor. Acknowldgmnts Tis work is partly supportd by CNPq grant /0-4. Ms Ssini is supportd by CNPq grant /0-0, wil Mr. Souza is supportd by ANP grant Rfrncs Arbogast T., Wlr M.F., 1995, A caractristic-mixd finit lmnt mtod for advction-dominatd transport problm, SIAM J. Numr. Anal., Vol. 3, pp Azaiz, J. and Moamad, A.A., 004, Fingring instabilitis in miscibl displacmnt flows of non-nwtonian fluids, Journal of Porous Mdia, Vol. 7, pp Bocv, P.B., Dormann, C.R., 006, A computational study of stabilizd, low-ordr C0 finit lmnt approximations of Darcy quations, Comput. Mc., Vol. 38, pp Brooks, A.N. and Hugs, T.J.R., 198, Stramlin upwind/ptrov- Galrkin formulation for convction dominatd flows wit particular mpasis on t incomprssibl Navir-Stoks quations, Computr Mtods in Applid Mcanics and Enginring, Vol. 3, pp Cristi, M.A., Muggridg, A.H., Barly J.J., 1991, 3D simulation of viscous fingring and WAG scms, SPE 3138, Proc. 11t SPE Symposium on Rsrvoir Simulation, Anaim, CA. Coutino, A.L.G.A., Alvs, J.L.D., 1996, Paralll finit lmnt simulation of miscibl displacmnt in porous mdia, SPE Journal, Vol. 4, No. 1, pp Coutino, A.L.G.A., Alvs, J.L.D., 1999, Finit lmnt simulation of nonlinar viscous fingring in miscibl displacmnts wit anisotropic disprsion and nonmonotonic viscosity profils, Computational Mcanics, Vol. 3, pp Dias, C.M., Coutino, A.L.G.A., 004, Stabilizd finit lmnt mtods wit rducd intgration tcniqus for miscibl displacmnts in porous mdia, Int. J. Numr. Mt. Engrg.., Vol. 59, pp Coutino, A.L.G.A., Dias, C.M., Alvs, J.L.D., Landau, L., Loula, A.F.D., Malta, S.M.C., Castro, R.G.S., Garcia, E.L.M., 004, Stabilizd mtods and post-procssing tcniqus for miscibl displacmnts, Comput. Mtods Appl. Mc. Engrg., Vol. 193, pp Douglas, J.Jr., Ewing, R.E., Wlr, M.F., 1983, T approximation of t prssur by a mixd-mtod in t simulation of miscibl displacmnt, R.A.I.R.O. Analys Numr., Vol. 17, pp Fast, P. and Slly, M.J., 004, A moving ovrst grid mtod for intrfac dynamics applid to non-nwtonian Hl-Saw flow, Journal of Computational Pysics, Vol. 195, pp Galão, A.C., Carmo, E.G.D., 1988, A consistnt approximat upwind Ptrov-Galrkin mtod for convction-dominatd problms, Computr Mtods in Applid Mcanics and Enginring, Vol. 68, pp Homsy, G.M., 1987, Viscous fingring in porous mdia. Ann. Rv. Fluid Mc., Vol. 19, pp Hugs, T.J.R., 1987, T Finit Elmnt Mtod, Prntic-Hall, Englwood Cliffs. Islam, N., Azaiz, J., 005, Fully implicit finit diffrnc-psudo spctral mtod for t simulation of ig mobility-ratio miscibl displacmnts, Int. J. Num. Mt. Fluids, Vol. 47, pp Juans, R., Patzk, T.W., 00, Multipl-scal stabilizd finit lmnts for t simulation of tracr injctions and watrflood, In: SPE/DOE 13 t Symposium on Improvd Oil Rcovry, Tulsa, Oklaoma, SPE Juans, R., Patzk, T.W., 004, Multiscal-stabilizd finit lmnt mtods for miscibl and immiscibl flow in porous mdia, Journal of Hydraulic Rsarc, Vol. 4, pp Sp. SI. Loula, A.F.D., Garcia, A.L.M., Coutino, A.L.G.A., 1999, Miscibl displacmnt simulation by finit lmnt mtods in distributd mmory macins, Comput. Mtods Appl. Engrg., Vol. 174, pp Malta, S.M.C., Loula, A.F.D., Garcia, A.L.M, 1995, A post-procssing tcniqu to approximat t vlocity fild in miscibl displacmnt simulations, Matmática Contmporâna, Vol. 8, pp Mangiavacci, N., Coutino, A.L.G.A. and Ebckn, N.F.F., 1997, Paralll psudo-spctral simulations of nonlinar viscous fingring in miscibl displacmnts, In: Offsor Enginring, ditd by FLLB Carniro t al., Computational Mcanics Publications, Soutampton, UK, pp Manickam, O., Homsy, G.M., 1993, Stability of miscibl displacmnts in porous mdia wit nonmonotonic viscosity profils, Pys. Fluids A, Vol. 5, pp Manickam, O., Homsy, G.M., 1994, Simulation of viscous fingring in miscibl displacmnts wit nonmonotonic viscosity profils, Pys. Fluids, Vol. 6, pp Manickam, O., Homsy, G.M., 1995, Fingring instabilitis in vrtical miscibl displacmnts flows in porous mdia, J. Fluid Mc., Vol. 88, pp Masud, A., Hugs, T.J.R., 00, A stabilizd mixd finit lmnt mtod for Darcy flow, Comput. Mtods Appl. Mc. Engrg., Vol. 191, pp Moissis, D.E., Millr, C.A., Wlr, M.F., 1987, A paramtric study of viscous fingring. In: Numrical Simulation in Oil Rcovry, ditd by M.F. Wlr, Springr-Vrlag, Nw York, pp Moissis, D.E., Millr, C.A., Wlr, M.F., 1988, A paramtric study of viscous fingring in miscibl displacmnt by numrical simulation, Numrical Simulation in Oil Rcovry, Vol. 11, pp / Vol. XXXII, No. 3, July-Sptmbr 010 ABCM

8 Finit Elmnt Simulation of Viscous Fingring in Miscibl Displacmnts at Hig Mobility-Ratios Moissis, D.E., Wlr, M.F., Millr, C.A., 1993, Simulation of miscibl displacmnts viscous fingring using a modifid mtod of caractristics: ffcts of gravity and trognity, SPE Rsrvoir Advancd Tcnology Sris, Vol. 1, pp Pacman, D., 1977, Fundamntals of Numrical Rsrvoir Simulation, Elsvir, Amstrdam. Ruit, M., Miburg, E., 000, Miscibl rctilinar displacmnts wit gravity ovrrid, part 1, omognous porous mdium, J. Fluid Mc., Vol. 40, pp Saron, E., Moor, M.G., McCormick, W.D. and Swinny, H.L., 003, Coarsning of fractal viscous fingring pattrns, Pys Rv Ltt, Vol. 91, pp Sing, B., Azaiz, J., 001, Numrical simulation of viscous fingring of sar-tinning fluids, Can. J. Cm. Engg., Vol. 79, pp Tan, C.T., Homsy, G.M., 1986, Stability of miscibl displacmnts in porous mdia: rctilinar flow, Pys. Fluids, Vol. 9, pp Tan, C.T., Homsy, G.M., 1988, Simulation of nonlinar viscous fingring in miscibl displacmnt, Pys. Fluids, Vol. 31, No. 6, pp Tan, C.T., Homsy, G.M., 199, Viscous fingring wit prmability trognity, Pys. Fluids A, Vol. 4, No. 6, pp Valli, A.M.P, Cary, G.F., Coutino, A.L.G.A., 005, Control stratgis for timstp slction in finit lmnt simulation of incomprssibl flows and coupld raction-convction-diffusion procsss, Intrnational Journal for Numrical Mtods in Fluids, Vol. 47, No. 3, pp Waggonr, J.R., Castillo, J.L., Lak, L.W., 1991, Simulation of EOR procsss in stocastically gnratd prmabl mdia, Proc. 11 t SPE Symposium on Rsrvoir Simulation, Anaim, CA, Wang, J., Zabaras, N., 006, A Markov random fild modl of contamination sourc idntification in porous mdia flow, Intrnational Journal of Hat and Mass Transfr, Vol. 49, No. 5-6, pp Zimmrman, W.B., Homsy, G.M., 1991, Nonlinar viscous fingring in miscibl displacmnts wit anisotropic disprsion, Pys. Fluids A, Vol. 3, pp Zimmrman, W.B., Homsy, G.M., 199a, Tr-dimnsional viscous fingring: a numrical study, Pys. Fluids A, Vol. 4, pp Zimmrman, W.B., Homsy, G.M., 199b, Viscous fingring in miscibl displacmnts: unification of ffcts of viscosity contrast, anisotropic disprsion, and vlocity dpndnc of disprsion on nonlinar fingr propagation, Pys. Fluids A, Vol. 4, pp J. of t Braz. Soc. of Mc. Sci. & Eng. Copyrigt 010 by ABCM July-Sptmbr 010, Vol. XXXII, No. 3 / 99