AN HYDRODYNAMIC MODEL FOR THE CALCULATION OF OIL SPILLS TRAJECTORIES

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1 AN HYDRODYNAMIC MODEL FOR THE CALCULATION OF OIL SILLS TRAECTORIES Emlo Ernso aladno Clos Ramndo Malska Compaonal Fld Dnamcs Laboraor SINMEC, Mcancal Engnrng, Fdral Unrs of Sana Caarna, Floranopols SC BRAZIL on: FAX: Absrac. T am of s papr s o prsn a mamacal modl and s nmrcal ramn o forcas ol splls rajcors n sa. T knowldg of rajcor followd b an ol slck splld on sa s of fndamnal mporanc n smaon of ponal rsks for ppln and ankrs ro slcon, and n combang pollon sng floang barrrs, drgns, c. In ordr o sma s slcks rajcors a nw modl, basd on mass and momnm consraon qaons s prsnd. T modl consdrs spradng n rgms wn nral and scos forcs conrbalanc gra and aks no accon ffcs of wnds and war crrns. T nral forcs ar consdrd for spradng and dsplacmn of ol slck,.., s consdrd s ffcs on momn of mass cnr of slck. T mass loss casd b ol aporaon s also akn no accon. T nmrcal modl s dlopd n gnralzd coordnas, makng modl asl applcabl o compl coasal gograps. Kwords: Enronmnal Flows - Ol Spll - Nmrcal Smlaon Gnralzd Coordnas 1. INTRODUCTION In rcn ars, proccpaon w nronmn prsraon b ndsrs, gornmn aors and soc n gnral as ncrasd consdrabl. Ts s parclarl r n prolm branc bcas of sa ransporaon of crd ol b ankrs or pplns bcas of sgnfcan rsk of an accdnal spll. T problm s a dsp low frqnc of sc accdns, consqncs ar g. Ts splls ar mc mor damagng wn occr nar sorlns bcas, bsds nronmnal mpacs, conomcal damags rangs from fsng o orsm. T rcn ol spll n Ganabara Ba, Ro d anro, Brazl, casd b a ppln rpr s a srong ampl of s broad mpac. T dald knowldg of splld ol poson and ara cord b slck s of fndamnal mporanc o ak appropra acons agans pollon, lk s of floang

2 barrrs, drgns, dsprsans, c. I s also mporan smaon of ponal rsks n slcng ppln ros, locang sorln anks and procmcal ndsrs. Trfor, a modl o forcas m-spac olon of ol slck sold mak par of an nronmnal program a as prpos of ol pollon comba. T frs sds ampng o modl momn of ol slcks (Fa (1969,1971), Fannlop and Waldmann (1971), Hol (1972), Bckmasr (1973), c.) consdr spradng as on-dmnsonal or a-smmrc. Ts modls consdr spradng of ol n calm wars, wr a slck, nall crclar, wll rmans crclar, js ncrasng s damr. Consdrng forcs a gorns spradng procss, Fa (1969), caracrzd spradng b ddng n r pass: Inall, wn cknss of slck s larg and so ar nral forcs, gra acs as ac forc conrbalancd b nral forcs; s s calld gra-nral spradng rgm. Wn man cknss of slck bgns o dcras, and scos forcs rd b war bondar lar wll nall owg nra as rardng forc, conss gra-scos spradng. In fnal nsancs, slck wll b so n a mbalancs of srfac srss bwn ar-war, ar-ol and war-ol wll sbs gra as ac forc, mananng nson rd b war as rardng forc. Ts las rgm s calld scos-srfac nson spradng. For larg splls (>1 4 m 3 ), s rgms las for 1 o 4 ors, for o n das and sral mons, rspcl. Frr modls as rd o smla mor ralscall rajcors b ncldng or procss lk dsprson casd b wnds and war crrns, and os procss wc rprsns mass cangs bwn dffrn nronmnal comparmns (calld fa procsss) lk aporaon, dssolon, mlsfcaon, c. Two approacs for compng ol splls rajcors ar commonl nconrd n lrar; Lagrangan and Elran modls. T Lagrangan modls (Sn Yapa (1988)) conss bascall n rprsnng ol slck b an nsmbl of a larg nmbr of small parcls, wc ar adcd b a loc rslng from combnaon of acon of wnds and crrns. Tn, slck s ddd no p sapd sgmns or srps, dpndng f form of slck s narl crclar or longad. Fa spradng formlas ar n appld o ac sgmn. For Elran approac, wo modl ar sall nconrd, os basd n mass and momnm qaons appld o ol slck (Hss and Krr (1979), Bnqé al. (1982)), and os basd on a concon-dffson qaon (Vnkas (1988), among ors), n wc dffs par of qaon rprsns d spradng of ol b slf and conc rms rprsns adcon of ol b crrns and wnds. T modl prsnd n s papr blongs o frs cagor of Elran modls and s basd on ngraon of mass and momnm qaon or cknss of ol slck. I consdrs spradng n nral-gra and scos-gra rgms, slck ranspor b crrns and wnd and ol aporaon. On mporan qson, wc arss b consdraon of nral forcs, s acclraon of slck as a wol,.., acclraon of slck mass cnr. Ts fac, no consdrd n Lagrangan and Elran dsprson-qaon-basd modls, cold cas mporan dffrncs n smad poson of slck, as wll b sn lar. 2. MATHEMATICAL MODEL Ts modl s basd on ngraon of mass and momnm qaons along cknss of slck. Trfor, aks no accon spradng of ol b slf and ranspor casd b wnds and war crrns. As srfac nson s nglcd and, rfor, onl frs and scond spradng rgms,.. gra-nral and grascos spradng ar consdrd, modl s applcabl p o abo n das afr spll,

3 dpndng on s magnd. T aporaon s consdrd rog a logarmc dca modl prsnd b Sr and Macka (1984). T mamacal modl for ol moon s oband as follows, Fgr 1: Varabls consdrd n rcal ngraon of gornng qaons Fgr 1 sows scmacall an ol slck bng ranspord b sar srss rd b war crrns and wnds. T ol flow s gornd b mass and momnm qaons for ncomprssbl flows. Ts qaons ar ( ) ( τ ) j j p ( 1 ) ( 2 ) Followng Hol (1972), w can consdr a ol scos s mc larg an war scos. Ts, rcal loc gradns wn ol ar mc lss an s gradns n war or n wnd. I s, rfor, a good appromaon o consdr a flow paramrs (loc and prssr) do no ar across cknss of slck. Ingrang gornng qaons, Eqs. ( 1 ) and ( 2 ), across slck cknss as sown n Fg. 1, consdrng drosac prssr dsrbon wn ol, w oban ( ) ( ) ( j) j j τ T τ B g ( 3 ) ( 4 ) wr bar arabls rprsn rcal ngral arags 1, s ol slck cknss and s a paramr wc rlas ol and war dnss ( o w )/. T rms τ w rprsn sar srss on op and boom of slck rd b wnds and war crrns, rspcl. Ts srss wr calclad as, τ ( 5 ) τ T wnd wnd C f ol war ( ) ( 6 ) B war C f V 1 T sffs and j ars from 1 o 2, as, afr ngraon, modl bcoms wo-dmnsonal.

4 Wr wnd f C war f C wr mad and rspcl. Tos als ar commonl sd n s modls (Idlfonso Csa, prsonal commncaon). T war f C al s an mprcall adjsd al, wl wnd f C al s calclad n sc wa a fnal loc of slck mass cnr b abo 3 % of wnd loc (3% rl). 3. NUMERICAL SOLUTION D smlar of gornng qaons w os sd n Sallow Wars Flows, an adapaon of sm-mplc mod prsnd b Casll and Cng (1992) s sd r for a fn olm procdr n gnralzd coordnas and co-locad arabls. Transformng Eqs. (3) and (4) o gnralzd coordnas followng procdr dscrbd n dals, for nsanc, n Malska (1995), w oban ( ) ( ) V U ( 7 ) ( ) ( ) g V U B T β γ β α τ τ ( 8 ) ( ) ( ) g V U B T β γ β α τ τ ( 9 ) T arabls and ar coordnas n gnralzd coordna ssm, α, β and γ ar componns of coaran mrc nsor, s acoban of ransformaon and U and V ar conraaran loc componns dfnd as ( ) ( ) V U ( 1 ) Ts qaons wr dscrzd sng a fn olm approac, m araon was consdrd plcl n momnm qaons and mplcl for mass consraon qaon sd o calcla ol cknss dsrbon. Fg. 2 sows a conrol olm n compaonal doman sd for qaons dscrzaon.

5 W N NE NW E SW S SE n w s Fgr 2: Conrol Volm on Compaonal Doman Usng WUDS (Rab & Torranc (1979)) as nrpolaon fncon and alang plcl m dra, w a, akng as fac as ampl, locs a s fac ar gn b [] ( ) ( ) 4 S SE N NE E M g F ( 11 ) [] ( ) ( ) 4 S SE N NE E M g F ( 12 ) Wr [] F s an plc conc-dffs fn olm opraor 2 and rprsns plc concon-dffson balanc of arabl for a conrol olm. I s prssd for a gnrc arabl φ as, [ ] V S A A M M F NB nb ˆ φ φ φ ( 13 ) wr M s mass n conrol olm and A and A nb ar cnral coffcns for momnm qaon a olm and s ngbor olms, rspcl. T sprscrp dnos qan alad a pros m ll. T mass balanc n olm wc s oband b dscrzaon of Eq. ( 7 ), s gn b, ( ) [ ] ( ) [ ] s n w V V U U ( 14 ) Smlarl Carsan locs no prssons for conraaran locs a as fac of conrol olm, on gs ( ) ( ) β α 4 * SE S NE N E M g U U ( 15 ) 2 Frr dals cold b sn n aladno (2)

6 In sam wa, w can oban conraaran locs a or facs of conrol olm. Tn, sbsng s locs no mass qaon, on obans an qaon for ol cknss as 2 A A A n E A NE w A W s A SE n N A A nw s NW S A sw SW B ( 16 ) Ts qaon s sold sng Gass-Sdl mod. No a for momnm qaons no lnar ssm of qaons as o b sold. T solon procdr for copld ssm s: Inalz all arabls a. T cknss of ol for wol doman s nalzd w a small al (sa ) o aod dson b zro. Dfn rgon and cknss of nal ol slck, f an nsananos spll s consdrd. Calcla coffcn of momnm qaons. Drmn loc fld plcl,.. no lnar ssm as o b sold r. W mos rcn locs, calcla coffcns of momnm qaon. Comp conc-dffs opraor o nr alaon of sorc rm of mass qaon. Calclas coffcns and sorc rm of mass qaon and sol ol cknss. Rcalcla ol cknss fld akng no accon mass ransfr procsss lk aporaon, snkng, c. Adanc a m sp, pda all flds and ccl back o sp on. Two ps of bondar condons wr sd. Wr doman concds w sorlns no mass fl was prscrbd and a opn sa locall parabolc condons wr assmd. Ts allows slck o la compaonal doman wo affcng cknss dsrbon of slck nsd doman. 4. MODEL VALIDATION AND RESULTS T frs sp n aldang a nmrcal modl s o compar w aalabl analcal solons. For s problm sm-analcal solon of Fa (1971) ar adqa. scal aldaon rqrs fld masrmns. As was alrad mnond, Fa s rsls dscrb spradng of an nsananos spll n calm wars. T rsls for gra-nral and gra-scos spradng rgms ar, rspcl 2 ( gv ) 1/ 4 R K ( 17 ) R g 1/ 6 2 3/ 2 gv K g 1/ 2 ( 18 ) ν In abo qaons R s slck rado (n calm wars spradng s asmmrc) as a fncon of lapsd m afr spll and K s an mprcal proporonal facor dpndng on spradng rgm. T followng fgrs sows rsls for wo spradng rgms consdrd b modl, for dffrn ol dnss and dffrn nal splls.

7 Slck Rads (m) 1 1 Inal Volm m 3 Inal Volm m 3 Ts Modl Fa (1971) - Inral Fa (1971) - Vscos Slck Rads (m) kg/m 3 9 kg/m 3 Ts Modl Fa (1971) - Inral Fa (1971) - Vscos m () (a) m () (b) Fgr 3: Comparson of orcal (Fa (1971)) and nmrcal solons for a-smmrc spradng n calm war, for (a) dffrn olms splld and (b) dffrn ol dnss. In frs problm, war bod was consdrd nall qscn, w war momn ndcd b ol momn and rsls ar sowd n Fgr 3. Fgr 4sows on-dmnsonal olon of an ol slck, consdrng an nsananos spll, n cas a war s mong. In s cas, was consdrd a spaall and mporall consan crrn of magnd of.5 m/s n -drcon Inall Slck () Tcknss (m) X (m ) Fgr 4: On-dmnsonal olon of an slck sbjcd o a consan war crrn of.5 m/s. No a scals ar dsord, mamm cknss s 3 mm and wol doman as 25 km. As was pcd, afr a prod of m n wc slck acclras, mass cnr of slck rmans mong w war loc. I was sn from smlaons rsls a man ffc from consdraon ol nra s acclraon of slck mass cnr. Modls a no consdr nral forcs jsf s n fac a nral spradng pas s r sor, wc s acall r as cold b sn n fgrs abo. B, wa w wan o sow r s a, wn slck ranspor s consdrd, nral forcs cold rard slck dsplacmn for a consdrabl prod of m. Fgr 5 sows slck mass cnr poson and loc as arng w m.

8 Mass Cnr oson X (m) Mass Cnr Vloc ( m/s ) m (s) m (s) (a) (b) Fgr 5: Dsplacmn of slck mass cnr, (a) oson, (b) Vloc. As can b sn acclraon of slck mass cnr s ffc nl abo s (25 ) afr spll and s fac cold affc sgnfcanl smaon of slck poson. Fnall, o sow modl fars, was appld o smla an nal spll a cn of arbor a São Franssco do Sl, Sana Caarna, wr r s an ol carg/dscarg pon a 9 km. off sor. Trfor, s s a local w g spll rsk, wc cold b casd b ppln rpr or falr n carg/dscarg opraons. Fgr 6 sows doman dfnon a rgon of por of São Francsco do Sl, ol dc falr local and dfnon of bondar condons for smlaons. T doman as bn ndd no sa js o cor rgon of nrs, rmndng a, d o locall parabolc condon far from sorln, f slck passs rog s bondars, s dos no affc slck poson nsd doman. pln falr poson ETROBRAS Carg/Dscarg dc Opn Sa Fronrs (Locall arabolc Condons) Coas (No Mass Fl Fronr) 1 km Fgr 6: Doman dfnon for smlaon a por of São Francsco do Sl. As s smlaon as onl prpos o sow gnral of modl and s abl for solng a ral problm, crrn fld was consdrd spaall consan and arabl as a sn fncon n m, rng o rprsn appromal dal crrns.

9 Rpors of prmnal masrmns a rgon sow prdomnanl so-sows crrns w rsdal crrns of appromal.5 m/s and mamm dal crrns of.16 m/s. T wnd was consdrd blowng from so-soas a 3 km/. To smla ppln brak, was consdrd a pollan sorc w consan mass fl njcng 1 kg/s drng 1. T modl can also consdr a mass sorc arabl w m, n ordr o consdr an possbl prssr araons n ppln. 5. CONCLUSIONS Ts papr prsnd a mamacal and nmrcal modl o prdc ol spll momns n sa. Rsls for spradng n calm war wr compard w sm Fgr 7: Tmporal-spaal olon of an ol slck splld a arbor of São Franco do Sl (Cas 2) D o prodc baor of dal crrns, momn of slck s casd prmarl b acon of rsdal crrns and soas wnds. B as, rsdal crrns ar small n s cas, slck momn as a wol,.., dsplacmn of s mass cnr s prncpall casd b wnds. T ffcs of bondar condons can also b apprcad. A sorln, wr no mass fl condon was mposd, ol accmlas, ncrasng slck cknss. In cas of an opn sa bondars, slck las doman wo affcng s sap psram

10 analcal solons and agrmn was good. T rsls for on-dmnsonal problm sow a consdraon of ol nra s mporan as affcs slck rajcor for a larg prod of m. Alog r ar no bncmark solons aalabl for cas wr war mos, rsls for a gnral problm, wr war mos prodcall n m, follow pcd pscal rnds and mass cnr of slck mos w war crrn loc. T modl can b sd o smla n s ol splls n ordr o asss pollon comba asks, so s an mporan ool n an ol spll conngnc plan. I can b also sd o sma ponal rsks n dcson sppor for ankrs and ol dcs ro slcon, dsllrs and grond anks locaon, among or ol sorng asks. REFERENCES Bnqé,.., Hagl, A., and Voll,. L. (1982) Engnrng Applcaon of Compaonal Hdralcs, II, man Adancd blsng rogram, London, pp Bckmasr,. (1973), Vscos-Gra spradng of an ol slck, ornal of Fld Mcancs, Vol. 59, No. 3, pp Casll V. and Cng R. T. (1992) Sm-Implc Fn Dffrnc Mods for Tr Dmnsonal Sallow War Flow, Inrnaonal ornal for Nmrcal Mods n Flds, Vol. 15, pp Fanlop, T. K, Waldman, G. D. (1971) Dnamcs of Ol Slcks, AIAA ornal, Vol. 1, No. 4, pp Fa,. A. (1969) T sprad of ol slcks on a calm sa, Ol on sa lnm, Nw York, pp Fa,. A. (1971) scal rocsss n Sprad of Ol on a War Srfac, rocdngs of on Confrnc on rnon and Conrol of Ol Splls, Amrcan rolm Ins, Wasngon D. C., pp Hss, K. W. and Krr, C. L. (1979) A Modl o Forcas Moon of Ol on Sa, rocdngs of Ol Spll Confrnc, pp Hol, D.. (1972) Ol Spradng on Sa, Annal Rw of Fld Mcancs, pp Sr, W. And Macka, D. (1984) Eaporaon Ra of Splls of Hdrocarbons and rolm Mrs, Enr. Sc. Tcnol, Vol. 18, pp Malska, C. R. (1995) Transfrênca d Calor Mcânca dos Fldos Compaconal, Lros Técncos Cnífcos Edora, Ro d anro, R, Brazl. aladno, E. E.(2) Modlagm Mamáca Smlação Nmérca d Drrams d rólo no Mar, MSc. Tss, rograma d ós-gradação m Engnara Mcânca, UFSC, Sana Caarna, Brazl (n orgs). Rab & Torranc (1979) Upsram Wgd Dffrncng Scms and Tr Applcaons o Ellpc roblms Inolng Fld Flow, Comprs and Flds, Vol.2, pp Sn, H. T. Yapa,. D. (1988) Ol Slck Transpor n Rrs, ornal of Hdralc Engnrng, Vol. 114, N 5, pp Vnkas (1988) T Ol Spll Baor Modl of Canadan Amosprc Enronmn Src, ar 1, ATMOSHERE OCEAN, Vol. 26, pp

NUMERICAL SIMULATION OF OIL SPILL TRAJECTORIES IN THE SEA

NUMERICAL SIMULATION OF OIL SPILL TRAJECTORIES IN THE SEA URICAL IULATIO O OIL ILL TRACTORI I TH A mlo rnso aladno mlo@snmc.fsc.br Clos Ramndo alska malska@snmc.fsc.br Compaonal ld Dnamcs Laboraor IC, cancal ngnrng, dral Unrs of ana Caarna, 884-9 loranopols C