Adomian Decomposition Method for Dispersion. Phenomena Arising in Longitudinal Dispersion of. Miscible Fluid Flow through Porous Media

Transcription

1 dv. Thor. ppl. Mch. Vol. 3 o. 5 - domia Dcomposiio Mhod for Disprsio Phoma risig i ogiudial Disprsio of Miscibl Fluid Flow hrough Porous Mdia Ramakaa Mhr ad M.N. Mha Dparm of Mahmaics S.V. Naioal Isiu of Tchology Sura-3957 Idia mhr_ramakaa@yahoo.com mm@ashd.svi.ac.i S.K. Mhr Dparm of polychiqu Egirig ad griculur Uivrsiy Dahod-3895 Idia srika_mah@rdiffmail.com bsrac I his papr a horical modl is dvlopd for h disprsio problm i porous mdia i which h flow is o dimsioal ad h avrag flow is usady. Th Numrical Soluio ad graphical illusraio of h disprsio problm is prsd by mas of domia Dcomposiio mhod for oliar parial diffrial quaios ad Mahmaica 7.. Mahmaics Subjc lassificaio: 8U3 76S5 Kywords: domia Dcomposiio mhod Miscibl fluid Disprsio Flows i porous mdia.. Iroducio I a miscibl displacm procss a fluid is displacd i a porous mdium by aohr fluid ha is miscibl wih h firs fluid. Miscibl

2 R. Mhr M.N. Mha ad S.K. Mhr displacm i porous mdia plays a promi rol i may girig ad scic filds such as oil rcovry i prolum girig coamiaio of groud war by was produc disposd udr groud movm of miral i h soil ad rcovry of sp liquors i pulpig procss. mog May flow problms i porous mdia o ivolvs fluid miurs calld miscibl fluids. miscibl fluid is a sigl phas fluid cosisig of svral complly dissolvd homogous fluid spcis a disic fluid-fluid irfac dos is i a miscibl fluid. Th flow of miscibl fluid is a impora opic i prolum idusry; a hacd rcovry chiqu i oil rsrvoir ivolvs ijcig a fluid solv ha will dissolv h rsrvoir s oil. Ths problms of disprsio hav b rcivig cosidrabl aio from chmical viromal ad prolum girs hydrologiss mahmaicias ad soil sciiss. Mos of h works rval commo assumpio of homogous porous mdia wih cosa porosiy sady spag flow vlociy ad cosa disprsio coffici. For such assumpio Ebach ad Whi [4] sudid h logiudial disprsio problm for a ipu cocraio ha varis priodically wih im. l-niami ad Rusho [] sudid h aalysis of flow agais disprsio i porous mdia. Hu [6] applid h prurbaio mhod o logiudial ad laral disprsio i o uiform spag flow hrough hrogous aquifrs. Mhr ad Mha [8 9] sudid h Disprsio of Miscibl fluid i smi ifii porous mdia wih usady vlociy disribuio from diffr poi of viw. Th prs papr discusss h approima soluio of h oliar diffrial quaio for logiudial disprsio phoma which aks placs wh miscibl fluids mi i h dircio of flow. Th mahmaical formulaio of h problm yilds a o liar parial diffrial quaio. Soluio has b obaid by usig domia dcomposiio mhod.. Mahmaical Formulaio ad Soluio of h Problm Th problm is o fid h cocraio as a fucio of im ad posiio as h wo miscibl fluid flow hrough porous mdia o ihr sids of h mid rgio. Th sigl fluid quaio dscribs h moio of fluid [4]. Hr h miig aks plac logiudially as wll as rasvrsly a ad a do of fluid havig [ ] cocraio is ijcd ovr h phas. Th do movs i h dircio of flow as wll as prpdicular o h flow. Fially i aks h shap of h llips wih a diffr cocraio [ ].

3 domia dcomposiio mhod for disprsio phoma 3 Fig:. Disprsio of a isaaous poi sourc i a uiform flow fild ccordig o Darcy s law h quaio of coiuiy for h miur i ρ cas of icomprssibl fluids is giv by. ρv Whr ρ is h dsiy for miur ad v is h por spag vlociy. Th quaio of diffusio for a fluid flow hrough a homogous porous mdium wih ou icrasig or dcrasig h disprsig marial Polubariova [] is giv by. v. ρd ρ Whr is h cocraio of a fluid i a porous mdia. D is h offici of disprsio wih i compos D ij. I a lamiar flow for a Icomprssibl fluid hrough homogous porous mdium dsiy ρ is cosa. Th quaio bcoms v.. D 3 us assum ha h spag vlociy v is alog h - ais h v u ad h o zro compos will b D D coffici of logiudial disprsio ad ohr ompos will b zro Polubariova []. Equaio 3 bcoms u 4 Whr u is h compo vlociy alog -ais which is im dpd as wll as cocraio alog ais i dircio ad D > ad i is h cross scioal flow vlociy i porous mdia. u Whr > ad for by Mha[9].Equaio 4 bcoms 5 This is h o liar Burgr s quaio for logiudial disprsio of miscibl fluid flow hrough porous mdia.

4 4 R. Mhr M.N. Mha ad S.K. Mhr Th hory ha follows is cofid o disprsio i uidircioal spag flow hrough smi-ifii homogous porous mdia. Th spag flow vlociy is assumd usady. Th disprsio sysms o b cosidrd ar subjc o a ipu cocraio of coamias. Th porous mdium is cosidrd as oadsorbig. osidr h ipu cocraio is.th govrig parial diffrial quaio 5 for logiudial hydrodyamic disprsio wih i a smiifii oadsorbig porous mdium i a uidircioal flow fild i which is h logiudial disprsio coffici is h avrag cross-scioal cocraio u is h usady spag vlociy is a coordia paralll o flow ad is im. Th iiial ad boudary codiios ar 6 say Sic ocraio is dcrasig as wih disac. Thrfor for h sak of covic f is cosidrd as gaiv poial fucio Mha [9]. 3. domia Dcomposiio Mhod Thoric pproach W solv quaio 4 for ad sparaly ad w g N 7 ad N 8 b h ivrs opraors of ad rspcivly giv by h form:. d ad. dd 9 Th opraig boh sids of quaio 7 ad 8 wih h ivrs opraors 9 w obai B Whr ad B ca b drmid subjcd o h corrspodig iiial ad boudary codiio 6 ad w obai: B Now addig ad ad dividig by w g h followig form B 3 W wri h paramrizd form of 3

5 domia dcomposiio mhod for disprsio phoma 5 4 ad h paramrizd dcomposiio forms of ad N as 5 ad N 6 Whr h domia s spcial polyomials ar o b drmid. Hr h paramr looks lik a prurbaio paramr; bu acually is o a prurbaio paramr; i is usd oly for groupig h rms. Now subsiuio of 5 ad 6 io 4 givs 7 If w compar lik powr rms of from boh sids of quaio 7 ad akig udr cosidraio ha paramr is big provd ha has h uiqu valu hrruaul[3] w g 3 8 N w drmi domia s spcial Polyomials s '. 4. Drmiaio of domia s Spcial Polyomials Th polyomials ar drmid i such a way ha ach dpd oly o... for 3. i.. c. I ordr o do his w subsiu 5 i o 6 ad w hav

6 6 R. Mhr M.N. Mha ad S.K. Mhr N 9 From 6 w coclud ha h domia polyomials hav h followig form:.. Hc h polyomial has h followig form: [ 4 ] If w suggs as a soluio of as a approimaio of oly wo rms of h form 4 6 [ 4 ] W us Mahmaica 7. i ordr o g umrical rsuls.. /{Ep[-]} D[] D[] dd[] aigra[-{ep[-]ep[-]}/4] bigra[/{ep[-]}{ep[-]ep[-]}/4] uepad[/b^-a] uepad [] pepad[/.{. }] Plo[{p}{.9}]

7 domia dcomposiio mhod for disprsio phoma 7 TBE- / TBE- /

8 8 R. Mhr M.N. Mha ad S.K. Mhr Graph- << for fid Graph- <<. for fid Disac Tim Graph-3 << for fid Graph-4 <<. for fid Disac Tim Graph-5 for << <<. Graph-6 for << <<. Plo3D[{/-Ep[-]/33Ep[-]/8/8Ep[-]*3Ep[-]*/8}{}{.}]

9 domia dcomposiio mhod for disprsio phoma 9 Eac Soluio Graph-7 for << <<. Graph-8 for << <<. oclusio Rmark Numrical ad Graphical soluios hav b dvlopd for prdicig h possibl cocraio of a giv dissolvd subsac i usady uidircioal spag flows hrough smi-ifii homogous isoropic porous mdia subjc o h sourc cocraios ha vary gaiv poially wih disac. Fially hr w mak a compariso of umrical soluio wih h approima soluio which shows ocraio dcrass wih disac ad slighly icrass wih im. Th soluio prss by quaio show ha approima valu of cocraio up o wo rms a ay for >. Th aalyical prssios obaid hr ar usful o h sudy of saliiy irusio i groudwar hlpful i makig quaiaiv prdicios o h possibl coamiaio of groudwar supplis rsulig from groudwar movm hrough burid wass. Rfrcs. domia G. pplicaio of h Dcomposiio Mhod o h Navir-Soks Equaios J. Mah. al. ppl pp l-niami. N. S. ad Rusho K. R. alysis of Flow agais disprsio i porous mdia J. Hydrol. Vol pp hrruaul Y. J. Mah. omp. Modlig 6 No

10 R. Mhr M.N. Mha ad S.K. Mhr 4. Ebach E. H. ad Whi R. Miig of fluids flowig hrough bds of packd solids. J. h. E. Vol p Fr.W. oamia hydrology Scod diio Pric hall Hu B. Disprsio calculaios i ouiform spag J. Hydrol. Vol Mario M.. Flow agais disprsio i oadsorbig porous mdia J. Hydrol. Vol Mhr R.K Disprsio of Miscibl fluid i smi ifii porous mdia wih usady vlociy disribuio Iraioal Joural of mahmaical Scics ad Egirig applicaiosijmsevol.3no.iv9. 9. Mha M.N. soluio of Burgr s quaio yp o dimsioal Groud war Rcharg by spradig i Porous Mdia. Joural of Idia cad.mah.vol.8 No. 6Pag Polubariova-Kochia P. Ya Thory of Groud war Movm Prico Ui. Prss 96. Rcivd: Novmbr 9