Mathematical Solution of Transport of Pollutant in Unsaturated Porous Media with Retardation Factor

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1 nrnionl Journl of Ali Enginring Rsrch SSN Volum 3 Numbr (8). - Rsrch ni Publicions. h:.riublicion.com Mhmicl Soluion of Trnsor of Pollun in Unsur Porous Mi ih Rrion Fcor Prvn Kumr M rmn of Mhmics Achry nsiu of Tchnology Bngluru ni rinm@gmil.com r. S R Suhnr rmn of Mhmics Prsincy Univrsiy Bngluru ni srsuhnr@rsincyunivrsiy.in Absrc An nlyicl firs orr soluion o h on-imnsionl vcion-isrsion quion ih silly vribl rrion fcor is riv using gnrli ingrl rnsform mho o invsig h rnsor of sorbing bu ohris non-rcing solus in hyrulic homognous bu gochmiclly hrognous orous formions. Th soluion is riv unr coniions of sy-s flo n rbirry iniil n inl bounry coniions. Th rsuls obin by his soluion gr ll ih h rsuls obin by numriclly invring Llc rnsform-gnr soluions rviously ublish in h lirur. Th soluion is vlo for hir or flux y inl bounry coniion hich is licbl hn consiring rsin solu concnrions n smi-infini orous mium. For mhmicl simliciy i is hyohsi h h sorion rocsss r bs on linr quilibrium isohrms n h h locl chmicl quilibrium ssumion is vli. Th rsul from svrl simulions comr ih ricions bs on h clssicl vcion-isrsion quion ih consn cofficins inic h rly ims silly vribl rrion ffcs h rnsor bhvior of sorbing solus. Th roh momns corrsoning o consn n vribl rrion r no ncssrily qul. Th cnr of mss rs o mov mor sloly n solu sring is nhnc in h vribl rrion cs. A l im hn h rvl isnc is much lrgr hn h corrlion scl of h rrion fcor h roh momn for h vribl rrion cs is inicl o h cs of invrin rrion. Th nlyicl soluion rsn in his r rovis mor flxibiliy ih rgr o h inl coniions. Kyors :Avcion-isrsion Equion issrsion officin uhml s Thorm Rrion Fcor. NTROUTON Th imc of silly vribl hyrulic rmrs on h rnsor n sring of consrviv non-rcing solus in nurl subsurfc hs bn h focus of mny rcn suis. Glhr. l. (979) Suhnr ( ). Arl.l (996) n ohrs hv rovi mhoologis for imroving h scriion n ricion of non-rcing solu rnsor in comlx srucur formions comr ih h ricion bs on h clssicl vcion-isrsion quion ih consn cofficins. On h ohr hn h rnsor of sorbing solus in gochmiclly s ll s hyruliclly hrognous orous mi hs rciv lil nion. For h imornc cs of rnsor of sorbing solus in gochmiclly homognous orous mi h ffcs of sorion r commonly ccoun for by imnsionlss rrion fcor hich my b fin s h rion of h vrg inrsiil flui vlociy o h rogion vlociy of h solu. Excluing h ossibiliis of mss rnsor limiions n solu rnsformion or cy ny obsrv flucuions on h rrion fcor r ribu solly o h vribiliy of h isribuion cofficin hich is n xrimnlly obin msur of sorion or solu rnion by h soli formion. Sorion rocsss cn b comlx n n on mny vribls incluing mrur rssur soluion H n ionic srngh sorbn surfc chrg sorbn soriv cciy n h rsnc of scis h coml for sorion sis. Sil or morl flucuions in ny of hs vribls ccoringly ffc h isribuion cofficin n consqunly h movmn of sorbing solus in subsurfc orous mi. For xml h isribuion cofficin of nonolr orgnic solus (synhic orgnic chmicls mjor consiuns of grounr oxic olluns) is corrl ih h orgnic crbon conn of h sorbn (Krckhoff (98) n Suhnr ()). Alhough such corrlion is no fully rlibl for vry solu-sorbn sysm (uris n Robrs 985) i cn xlin o som xn h vribl rrion obsrv in fil xrimns (Robrs.l 986). Grbin (987) & Suhnr () mloy scrl mhos o nly rciv solu mcro-isrsion unr h ssumion h h log-hyrulic conuciviy is linrly rl o boh h orosiy n h isribuion cofficin. His rsul inic h solu sring is nhnc hn hr is ngiv corrlion bn h log-hyrulic conuciviy n h isribuion cofficin. Th rsn ork is focus on h rnsor of olluns bu ohris non-rcing solus unr locl quilibrium coniions in on-imnsionl unsur orous mium. Anlyicl soluions r mloy o solv h on-imnsionl vcion-isrsion quion ih uniform sy flui flo coniions n silly vribl rrion fcor for smi-infini mium n fluxy inl bounry coniion.

2 nrnionl Journl of Ali Enginring Rsrch SSN Volum 3 Numbr (8). - Rsrch ni Publicions. h:.riublicion.com Th min objciv of h suy is o rovi mhmicl mol for br unrsning of rnsor of ollun hrough unsur orous mi. A mhmicl mol is n imorn ool n cn ly crucil rol in unrsning h mchnism of grounr olluion roblms. is simlifi scriion of hysicl rliy xrss in mhmicl rms. Mhmicl mols h m o simul moshric rocsss involv in grounr olluion r bs in gnrl on h quion of mss consrvion for iniviul ollun scis. Such mols rl in on quion h ffcs of ll h hysicl scs n ynmic rocsss h influnc h mss blnc on grounr hich inclu rnsor iffusion rmovl of olluns n loss or rnsformion hrough chmicl rcions.. MATHEMATAL MOEL Th Avcion-isrsion quion long ih iniil coniion n bounry coniions cn b rin s ( n) S n Th quilibrium isohrm bn soluion n sorb hs S is givn by K K is h isribuion cofficin. ( n) K n ( n) K n R L us k = R = R = R. niilly sur flo of flui of concnrion = ks lc in h orous mi. () Thus h rori bounry coniions for h givn mol ( ) ( ) () ( ) Th roblm hn is o chrcri h concnrion s funcion of n hr h inu coniion is ssum h origin n scon y or flux y homognous coniion is ssum. is iniil concnrion. To ruc quion (3) o mor fmilir form k (3) Subsiuing quion (3) ino quion () givs Th iniil n bounry coniions () rnsform o () (5) Equion () my b solv for im nn influx of h flui =. Th soluion of quion () my b obin rily by us of uhml s horm (rsl n Jgr 97). f F x y is h soluion of h iffusion quion for smi-infini mi in hich h iniil concnrion is ro n is surfc is minin concnrion uniy hn h soluion of h roblm in hich h surfc is minin is mrur Fx y This horm is us rincilly for h conucion roblms bu h bov hs bn scili o fi his scific cs of inrs. onsir no h roblm in hich iniil concnrion is ro n h bounry is minin concnrion uniy. Th bounry coniions r Th Llc rnsform of quion () is L Hnc i is ruc o n orinry iffrnil quion Th soluion of h quion is q. q A B q hr Th bounry coniion s rquirs h B = n bounry coniion = rquirs h A hus h riculr soluion of h Llc rnsform quion is q Th invrsion of h bov funcion is givn in ny bl of Llc rnsforms. Th rsul is rf ( ) Using uhml s horm h soluion of h roblm ih iniil concnrion ro n h im nn surfc coniion = is (6)

3 nrnionl Journl of Ali Enginring Rsrch SSN Volum 3 Numbr (8). - Rsrch ni Publicions. h:.riublicion.com ) ( Sinc is coninuous funcion i is ossibl o iffrni unr h ingrl hich givs 3 Th soluion o h roblm is 3 (7) Puing hn h quion (7) cn b rin s (8) Sinc h riculr soluion of h roblm b rin s (9) hr n.. EVALUATON OF THE NTEGRAL SOLUTON Th ingrion of h firs rm of quion (9) givs. () For convninc h scon ingrl my b xrss on rms of rror funcion (Hornsin 95) bcus his funcion is ll bul. Noing h. Th scon ingrl of quion (9) my b rin s () Sinc h mho of rucing ingrl o bul funcion is h sm for boh ingrls in h righ si of quion () only h firs rm is consir. L n h ingrl my b xrss s. () Furhr l in h firs rm of h bov quion hn. (3) Similr vluion of h scon ingrl of quion () givs.agin subsiuing ino h firs rm h rsul. Noing h Subsiuion ino quion () givs. () Thus quion (9) my b xrss s (5) Hovr by finiion

4 nrnionl Journl of Ali Enginring Rsrch SSN Volum 3 Numbr (8). - Rsrch ni Publicions. h:.riublicion.com 3 rfc lso rf. Wriing quion (5) in rms of rror funcions g rfc rfc (6) Thus Subsiuion ino quion (3) h soluion is rfc rfc R-subsiuing for n givs rfc rfc (7) hr bounris r symmricl h soluion of h roblm is givn by h firs rm h quion (7). Th scon rm is quion (7) is hus u o h symmric bounry imos in h mor gnrl roblm. Hovr i shoul b no lso h if oin gr isnc y from h sourc is consir hn i is ossibl o roxim h bounry coniion by hich ls o symmricl soluion. V RESULTS & SUSSON This suy rsns nlyicl soluions for on-imnsionl vcion isrsion quions in unsur orous mium in fini omin. Th rnsform mho coul ih h gnrli ingrl rnsform chniqu is us o obin h nlyicl soluions. Soluions r obin for boh firs- n hir-y inl bounry coniions. Th vlo nlyicl soluions for fini omin r comr ih soluions for h smi-infini omin o clrify ho h xi bounry influncs h on-imnsionl rnsor in orous mium sysm. Fig. : Brk-hrough-curv for vs im for =m R=. =.5 & = Fig. : Brk-hrough-curv for vs im for =m R=. =.5 & =.5 Fig. 3: Brk-hrough-curv for vs im for =m R=. =.5 & =.5 Fig. : Brk-hrough-curv for vs im for =m R=. =.5 =.75 &.

5 nrnionl Journl of Ali Enginring Rsrch SSN Volum 3 Numbr (8). - Rsrch ni Publicions. h:.riublicion.com Th min limiions of h nlyicl mhos r h h licbiliy is for rlivly siml roblms. Th gomry of h roblm shoul b rgulr. Th roris of h soil in h rgion consir mus b homognous in h sub rgion. Th nlyicl mho is somh mor flxibl hn h snr form of ohr mhos for on-imnsionl rnsor mol. Figurs o rrsns h concnrion rofils vrss im in h sorbing mi for h = m n Rrion fcor R=. is sn h for fix vlociy isrsion cofficin n isribuion cofficin K crss ih h s orosiy n crss u o h isribuiv cofficin K n if im incrss h concnrion crss for iffrn im n cy chin. Th nlyicl soluions riv for h fini omin ill hus b riculrly usful for nlying h on-imnsionl rnsor in unsur orous mium ih lrg isrsion cofficin hrs h nlyicl soluion for smi-infini omin is rcommn o b li for mium sysm ih smll isrsion cofficin. Th givn soluion is usful for vliing numricl mol simul soluion bcus rlisic roblms gnrlly hv fini omin. n his r hv conclu h h mhmicl soluions hv bn vlo for ricing h ossibl concnrion of givn issolv subsnc in sy uniircionl sg flos hrough smi-infini homognous n isoroic orous mi subjc o sourc concnrion for silly vribl rrion fcor using chng of vribl n ingrl rnsform chniqu. Th xrssions k ino ccoun h conminns s ll s mss rnsfr from h liqui o h soli hs u o sorion. For simulnous isrsion n sorion of solu h isrsion sysm is consir o b sorbing r roorionl o is concnrion. V REFERENES [] Arl M.M. & Lio B. Anlyicl soluions for oimnsionl rnsor quion ih im-nn isrsion cofficins Journl of Hyrologic Enginring Vol [] Brry.A. n Sorio G. Anlyicl soluion of convcion-isrsion mol ih im-nn rnsor cofficins Wr Rsourc Rsrch [3] Bu V. A gnrli o-imnsionl nlyicl solu rnsor mol in boun mi for flux-y mulil sourcs. Wr Rsourc Rsrch [] BrJ. n A. Vrruij Molling Grounr flo n olluion Ril Publishing o.tokyo 99. [5] Ermk.L. An Anlyicl Mol for Air Pollun rnsor n osiion from oin sourc Amos Environ [6] J.S. hn. W. Liu n.m.lio To-imnsionl Llc-Trnsform Por Sris Soluion for Solu Trnsor in Rilly onvrgn Flo Fil Avnc Wr Rsourc [7] Suhnr S.R. A soluion of h iffrnil quion of longiuinl isrsion ih vribl cofficins in fini omin n. J. of Ali Mhmics & Physics Vol. No [8] Suhnr S.R. A soluion of h iffrnil quion of nn isrsion long uniform n non-uniform flo ih vribl cofficins in fini omin n. J. of Mhmicl Anlysis Vol.3 No [9] Suhnr S.R. An nlyicl soluion of onimnsionl vcion-iffusion quion in orous mi in rsnc of riociv cy Globl Journl of Pur n Ali Mhmics Vol.8 No. 3-. [] Suhnr S.R. Rji J & Nirnjn M Mhmicl Soluions of rnsor of olluns hrough unsur orous mi ih sorion in fini omin n. J. of ombin Rsrch & vlomn Vol. No. 3-. [] Suhnr S.R. Prvn Kumr M. & Rmsh T. Mhmicl Anlysis of rnsor of olluns hrough unsur orous mi ih sorion n riociv cy n. J. of ombin Rsrch & vlomn Vol. No. -8. [] Suhnr S.R. Rji J & Nirnjn M Mhmicl molling of rnsor of olluns in unsur orous mi ih riociv cy n comrison ih soil column xrimn n. Scinific J. on Enginring & Tchnology Vol. 7 No. 5. [3] Ys S R An nlyicl soluion for on-imnsionl rnsor in hrognous orous mi; Wr Rsour. Rs [] Zoou. n Knigh J.H. Anlyicl soluion for vcion n vcion-iffusion quion ih silly vribl cofficins Journl of Hyrulic Enginring