NUMERICAL SIMULATION OF OIL SPILL TRAJECTORIES IN THE SEA

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1 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, loranopols C BRAZIL on: AX: Absrac Ts papr prsns a mamacal modl and s nmrcal ramn for forcasng ol splls rajcors n sa. In ordr o sma s slcks rajcors a n modl, basd on mass and momnm consraon qaons s prsnd. T modl consdrs spradng n rgms n nral and scos forcs conrbalanc gra and aks no accon ffcs of nds and ar crrns. T nmrcal modl s dlopd n gnralzd coordnas, makng modl asl applcabl o compl coasal gograps. D o smlar of rslng qaons os sd n allo Wars modls, an adapaon o n Volm od and gnralzd coordnas of m-implc n Dffrnc mod prsnd b [Casll and Cng 199] s sd n s modl. ITRODUCTIO T sa ransporaon of crd ol b ankrs or offsor pplns as a sgnfcan assocad rsk of an accdnal spll. Wn sc splls occr nar ors or fsng rgons, damag s sll ors. 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 knoldg of splld ol poson and ara cord b slck s of fndamnal mporanc o ak appropra acons agans pollon, lk s of floang 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. As an fld mcancs problm, o approacs for compng ol slcks rajcors ar commonl nconrd n lrar; Lagrangan and lran modls. Lagrangan modls [n Yapa 1988] conss bascall n rprsnng ol slck b an nsmbl of a larg nmbr of small parcls, c ar adcd b a loc rslng from combnaon of acon of nds and crrns. Tn, slck s ddd no p sapd sgmns or srps, dpndng f form of slck s narl crclar or longad and [a 1971] spradng formlas ar appld o ac sgmn o comp spradng of slck. a s formlas consdr spradng of an ol slck n calm ars, r a slck, nall crclar, ll rmans crclar, js ncrasng s damr. or lran approac, o modl ar sall nconrd, os basd n mass and momnm qaons appld o ol slck [Hss and Krr 1979], [Bnqé. al.198], and os basd on a concon-dffson qaon [Vnkas 1988] (among ors), n c dffs par of qaon rprsns spradng of ol b slf and conc rms rprsns adcon of ol b crrns and nds. T modl prsnd n s papr blongs o frs cagor of lran 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 nd and ol aporaon. ATHATICAL ODL T gornng qaons for slck rajcor, ar oband b ngrang ar-oks qaons along cknss of slck. gr 1: Varabls consdrd n rcal ngraon of gornng qaons. gr 1 sos scmacall an ol slck bng ranspord b nsons rd b ar crrns and nds. T ol flo s gornd b mass and momnm qaons for ncomprssbl flos. Ts qaons ar: ( ) ( 1 ) j j p ollong [Hol 197], can consdr a ol scos s mc larg an ar scos. Ts, rcal loc gradns n ol ar mc lss an s gradns n ar or n nd. I ( )

2 s, rfor, a good appromaon o consdr a flo paramrs (loc and prssr) do no ar across cknss of slck. Afr ngraon, gornng qaons oband ar, ( 3 ) B T j j j g ( 4 ) r bar arabls rprsn rcal ngral arags, s ol slck cknss and s a paramr c rlas ol and ar dnss o /. T rms τ rprsn sar nsons on op and boom of slck rd b nds and ar crrns, rspcl. Ts nsons r calclad as, nd nd f T C V ( 5 ) ar ol ar f B C V ( 6 ) URICAL OLUTIO As can b sn gornng qaons ar smlar o allo Wars qaons. Tn sm-mplc mod prsnd b [Casll and Cng 199] s sd for ramn of coplng bn cknss and loc. In s cas, n Volm mod co-locad arabls and gnralzd coordnas r sd. Tn modl s applcabl o arbrar gomrs sc as compl coasal gograps. T ransformd qaons n rms of compaonal doman coordnas, and ar, V U ( 7 ) g V U B T γ ( 8 ) g V U B T γ ( 9 ) T arabls and ar coordnas n gnralzd coordna ssm, and, and γ ar componns of coaran mrc nsor, s acoban of ransformaon and U and V ar conraaran locs dfnd as V U ( 1 ) Ingrang s qaons n olm sod n gr and sng WUD [Rab & Torranc 1979] as nrpolaon fncon 1 1 ( 11 ) ( 1 ) and cnral dffrncs for cross dras, 4. ( 13 ) on obans for olm, [] W g ( 14 ) [] W g ( 15 ) r [ ] s an s an plc conc-dffs fn olm opraor 1, gn, for a gnrc scalar b [ ] V A A B nb ˆ ( 16 ) and rprsns conc-dffs balanc of scalar arabl a olm, n gnralzd coordnas. In s cas, rprsns loc componns and. gr : Conrol olm on compaonal doman. 1 rr dals cold b sn n aladno ()

3 B ngraon of mass qaon, n spac and m, a U U ( 17 ) V V n s Hr, nd conraaran locs a olm facs. Usng qs. ( 1 ), and locs a conrol olm nrfacs gn b, g ( ) [] ( 18 ) ( ) 4 [] g ( ) 4 a for as fac, * g U U r * U ( ) ( ) ( ) 4 [] [] and smlarl for or facs of conrol olm. o a for alaon of loc componns a nrfac nd conc-dffs opraor alad a olm nrfacs. As arabls arrangmn sd s co-locad, locs a nrfacs ar no aalabl. T proposon r s o ala s opraor b an arag of opraors calclad a cnrs of adjacn olms,.., [ ] [ ] ( 19 ) ( ) ( 1 ) [ ] ( ) o a no rprsns an armcal arag of loc componns, c s kno o gnra srong nsabls n solon procdr, b s an arag of qaons of moon as sggsd b [arc and alska 1994]. bsng conraaran locs n q.( 17 ) and oprang, fnd for calclang slck cknss fld lk, A A s A A AW An As A B n W s W A o a for comp cknss fld nd o sol a lnar ssm of qaons l loc fld s calclad plcl. n ( 3 ) BOUDARY CODITIO To p of bondar condons r sd n s modl, r doman lms concds sorlns no mass fl as prscrbd and a opn sa locall parabolc condons r assmd. Ts las condon p allos slck o la compaonal doman o affcng cknss dsrbon of slck nsd doman. Tn s possbl o dfn doman js for rgon of nrs bcas prsnc of bondars ll no affc rsls nsd doman. or no mass fl condon, a fro loc componns a condon of prscrbd arabl,.., a Drcl condon prscrbd al qal o zro. If rprsns an componn of loc cor, a, f ( 4 ) ( f ) ( 5 ) T cross dras ar zro bcas prscrbd al s consan along fronr. gr 3: as bondar a compaonal doman. or mass consraon qaon, ll b sd a modolog proposd b [alska 1981] and [Van Doormaal and Rab 1984]. To aod ncss of prscrb cknss als a bondar, s proposd o sbs conraaran loc a fronr n mass consraon qaon, n mass balanc for olm of gr 3 rsls, } U U f ( 6 ) V V n s T or locs ar alad n sam a of nrnal olms. Tn qaon for as bondar olm s, A A W An As ( 7 ) An W As W B o a n s cas conraaran loc a fronr s zro and n sorc rm for q. ( 7 ) s, * * * ( U V V ) n n s s B ( 8 )

4 or opn sa bondars (Locall parabolc condon), sa a r s no araon n an arabl normall o fronr. or olm of gr 3, mans a, U U ( 9 ) and zro dffs fl normal o fronr, Γ Γ ( 3 ) Consdrng qaon ( 9 ) a, for mass balanc a fronr olm, U U ( 31 ) V V n s T prm ndcas arabl s calclad n fncon of aalabl als and s rm ll ak par of ndpndn rm B. In s cas s gn b, * * * B U U V V ( 3 ) T solon procdr for copld ssm s: Inalz all arabls a. T cknss of ol for ol doman s nalzd a small al (sa ) o aod dson b zro. Dfn rgon and cknss of nal ol slck, f an nsananos spll s consdrd. Calcla coffcns 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. RULT To alda modl, frs sp ll b comparson aalabl analcal solons. or s problm sm-analcal solon of [a 1971] ar adqa. scal aldaon rqrs fld masrmns. As as alrad mnond, a s rsls dscrb spradng of an nsananos spll n calm ars. T rsls for gra-nral and grascos spradng rgms ar, rspcl R K g ( gv ) 1/ 4 n 1/ 6 n s s ( 33 ) 3/ gv R K g 1/ ν In abo qaons R s slck rado (n calm ars spradng s a-smmrc) as a fncon of lapsd m afr spll and K s an mprcal ( 34 ) proporonal facor dpndng on spradng rgm. T follong fgrs sos rsls for o spradng rgms consdrd b modl, for dffrn ol dnss and dffrn nal splls. lck Rads (m) lck Rads (m) Inal Volm m 3 Inal Volm 1 4 m 3 Ts odl a (1971) - Inral a (1971) - Vscos m () (a) 7 kg/m 3 9 kg/m m () (b) Ts odl a (1971) - Inral a (1971) - Vscos gr 4: Comparson of orcal [a 1971] and nmrcal solons for a-smmrc spradng n calm ar, for (a) dffrn olms splld and (b) dffrn ol dnss. In frs problm, ar bod as consdrd nall qscn, ar momn ndcd b ol momn. gr 5 sos ffc of bondar condon p as slck appromas o fronr. In cas of mprmabl fronr ol accmlas agans all ncrasng slck cknss and bgns o sprad n ransrsal drcon. or locall parabolc condon, cold b sn a slck las doman o cangs n par rmanng nsd doman.

5 5 15 f slck passs rog s bondars, s dos no affc slck poson nsd doman. (m) (m) (a) 1 5 pln falr poson TROBRA Carg/Dscarg dc Opn a ronrs (Locall arabolc Condons) (m) (m) (b) gr 5: lck spradng and drfng a consan ar loc of.5 m/s, n drcon, (a) no mass fl as bondar and (b) locall parabolc as bondar. nall, o so modl fars, as appld o smla an nal spll a cn of arbor of ão ranssco do l, ana Caarna, r r s an ol carg/dscarg pon a 9 km off sor. Trfor, s s a local g spll rsk, c cold b casd b ppln rpr or falr n carg/dscarg opraons. gr 6, sos doman dfnon a rgon of por of ão rancsco do l, ol dc brak locals 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, Coas (o ass l ronr) 1 km gr 6: Doman dfnon for smlaons a por of ão rancsco do l. As s smlaon as onl prpos o so gnral of modl appld o a coasal spll, and no o smla accral a ral problm, crrn fld as consdrd spaall consan and arabl as a sn fncon of m, rng o rprsn appromal dal crrns. Rpors of prmnal masrmns a rgon so prdomnanl sosos crrns rsdal crrns of appromal.5 m/s and mamm dal crrns of.16 m/s. T nd as consdrd from sosoas blong a 3 km/. To smla ppln brak, as consdrd a pollan sorc consan mass fl njcng 1 kg/s drng 1. gr 7 sos m spac olon of an ol slck casd b ppln brk. I cold b sn ffcs of bondar condons, a sorln, r no mass fl condon as mposd, ol accmlas, ncrasng slck cknss. In cas of an opn sa bondars, slck las doman o affcng s sap psram. 1 3

6 4 6 gr 7: Tmporal-spaal olon of an ol slck splld a arbor of ão rancsco do l COCLUIO Ts papr prsnd a mamacal and nmrcal modl o prdc ol spll momns n sa. Rsls for spradng n calm ar r compard sm-analcal solons and agrmn as good. Alog r ar no bncmark solons aalabl for cas r ar mos, rsls for a gnral problm, r ar mos prodcall n m, follo pcd pscal rnds and mass cnr of slck mos ar 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. BIBLIOGRAHY Bnqé,.., Hagl, A., and Voll,. L. (198) ngnrng Applcaon of Compaonal Hdralcs, II, man Adancd blsng rogram, London, pp Casll V. and Cng R. T. (199) m-implc n Dffrnc ods for Tr Dmnsonal allo War lo, Inrnaonal ornal for mrcal ods n lds, Vol. 15, pp a,. A. (1971) scal rocsss n prad of Ol on a War rfac, rocdngs of on Confrnc on rnon and Conrol of Ol plls, Amrcan rolm Ins, Wasngon D. C., pp Hss, K. W. and Krr, C. L. (1979) A odl o orcas oon of Ol on a, rocdngs of Ol pll Confrnc, pp Hol, D..(197) Ol pradng on a, Annal R of ld cancs, pp arc alska (1994) A on-orogonal n Volm od for olon of All pd los Usng Co-locad Varabls, mrcal Ha Transfr, Vol. 6, o. 3, pp alska C. R. (1981) A solon od for Tr- Dmnsonal arabolc ld lo roblms n onorogonal Coordnas, D Tss, Unrs of Warloo, Warloo, Canada. aladno,..() odlagm amáca mlação mérca d Drrams d rólo no ar, c. Tss, rograma d ós-gradação m ngnara cânca, UC, ana Caarna, Brazl. (n orgs) Rab & Torranc (1979) Upsram Wgd Dffrncng cms and Tr Applcaons o llpc roblms Inolng ld lo, Comprs and lds, Vol., pp n, H. T. Yapa,. D. (1988) Ol lck Transpor n Rrs, ornal of Hdralc ngnrng, Vol. 114, 5, pp Van Doormaal.. and Rab G. D. (1984) nancmns of IL od for rdcng Imcomprssbl ld lo, mrcal Ha Tranfr, Vol. 7, pp Vnkas (1988) T Ol pll Baor odl of Canadan Amosprc nronmn rc, ar 1, ATOHR OCA, Vol. 6, pp

AN HYDRODYNAMIC MODEL FOR THE CALCULATION OF OIL SPILLS TRAJECTORIES

AN HYDRODYNAMIC MODEL FOR THE CALCULATION OF OIL SILLS TRAECTORIES Emlo Ernso aladno mlo@snmc.fsc.br Clos Ramndo Malska malska@snmc.fsc.br Compaonal Fld Dnamcs Laboraor SINMEC, Mcancal Engnrng, Fdral Unrs