An Analytical Solution of Effect of Water Content on Solute Transport in Saturated and Unsaturated Porous Media

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1 A Aalyical Solio o Ec o War o o Sol Trasor i Sara a Usara Poros Mia Prav Kmar M arm o Mahmaics KS School o Egirig & Maagm BagalorKaraaka ia Shobakmar M arm o Mahmaics Maharai s Scic ollg or Wom Bagalorkaraaka ia aarya Hg 3 3 arm o Mahmaics Gov Firs Gra ollg Sirsi Karaaka ia S R Shra arm o Mahmaics T Joh si o Tchology Bagalor Karaaka ia Absrac:- Mos o h rsarchrs s h cooria rasormaio - i orr o vala h qaio or avcio-isrsio o a movig li i oros mia Frhr h boary coiios = a = a = a = or ar s which rsls i a symmrical cocraio isribio Th objciv o his ar is o i h solio o irial qaio i logiial ircio ha avois his rasormaio hs givig ris o a asymmrical cocraio isribio is h show ha h solio aroachs ha giv by symmrical boary coiios rovi h isrsio coici is small a h rgio ar h sorc will o b cosir Th solio has b obai or h isrsio mol o logiial miig wih variabl coicis i a ii lgh iiially sol r omai h bgiig homogos omai is si or avcio-isrsio alog wih iorm low Th solio has b obai or h iorm vlociy by cosirig saially o h hrogiy o h omai a h isrsio roorioal o h sqar o h vlociy Th vlociy is liarly irola a small icrm alog h ii omai Th i coiio has b cosir coios o iorm a o icrasig ar boh Th solios ar obai or boh h omais by sig haml s horm a igral solio chiq Th w i sac a im variabls rocsss has b iroc Th cs o h cy o isrsio wih im a h hrogiy o h omai o h sol rasor ar si saraly wih h hl o grahs Ky wors: Avcio isrsio asorio gral rasorms Fick s lawhaml s horm NTROUTON rc yars cosirabl irs a aio hav b irc o isrsio homa i low hrogh oros mia Schiggr 95 Jog 958 a ay 956 hav rs saisical mas o sablish h cocraio isribio a h isrsio coici Avcio isrsio qaio lais h sol rasor o combi c o covcio a isrsio i a mim is a arial irial qaio o arabolic y riv o h ricil o cosrvaio o mass sig isio qaio o h growig srac a sbsrac hyro virom graaio h avcio isio qaio has raw sigiica aio o hyrologiss civil girs a mahmaical molrs s aalyicalmrical solios alog wih a iiial coiio a wo boary coiios hl o rsa h coamia or olla cocraio isribio bhavior hrogh a o mim lik air rivrs laks a oros mim lik aqir o h basis o which rmial rocsss o rc or limia h amags may b orc has wi alicaios i ohr iscilis oo lik soil hysics rolm girig chmical girig a bioscics h iiial works whil obaiig h aalyical solios o isrsio roblms i ial coiios h basic aroach was o rc h avcio isrsio qaio io a isio qaio by limiaig h covciv rms was o ihr by irocig movig coorias Ogaa a Baks 96; Harlma a Rmr 963; Bar 97; Gvaas a Volkr 983; Aral a Liao 996; Marshal al 996 or by irocig aohr variabl Baks a Ali 96 Ogaa 97; Lai a Jriak 97; Mario 97 a Al-Niami a Rsho 977 Th Lalac rasormaio chiq has b s o g sir solios Som o h o-imsioal solios hav b giv Tracy 995 Shra by rasormig h oliar avcio isio qaio io a liar o or sciic orms o h moisr co vs rssr ha a rlaiv hyralic cociviy vs rssr ha crvs which allow boh wo-imsioal a hr-imsioal solios has b riv A mho has b giv o solv h rasor qaios or a kiically asorbig sol i a oros mim wih saially varyig vlociy il a isrsio coicis Va Koo 996 Shra

2 Lar i has b show ha som larg sbsrac ormaios hibi variabl isrsiviy roris ihr as a cio o im or as a cio o isac Mahro a Marsily 98; Sosio al 986; Glhar al 99 Aalyical solios wr vlo or scribig h rasor o issolv sbsacs i hrogos smi iii oros mia wih a isac isrsio o oial ar alog h iorm low Yas Th moral mom solio or o imsioal avciv-isrsiv sol rasor wih liar qilibrim sorio a irs orr graaio or im ls sorcs has b ali o aaly soil colm rimal aa Pag al 3 A aalyical aroach was vlo or o-qilibrim rasor o raciv sols i h sara o rig a iilraio risribio cycl Svrio a lma Shra Th sol is rasor by avcio a obys liar kiics Aalyical solios wr rs or sol rasor i rivrs iclig h cs o rasi sorag a irs orr cay Sm 6 Shra Por low vlociy was assm o b a o-ivrgc r say a o-saioary raom cio o sac a im or gro war coamia rasor i a hrogos mia Siri 6 A wo-imsioal smi-aalyical solio was rs o aaly sram aqir iracios i a coasal aqir whr growar lvl rsos o ial cs Kim al 7 A mor irc mho is rs hr or solvig h irial qaio govrig h rocss o isrsio is assm ha h oros mim is homogos a isoroic a ha o mass rasr occrs bw h soli a liqi hass is assm also ha h sol rasor across ay i la o microscoic vlociy variaios i h low bs may b qaiaivly rss as h roc o a isrsio coici a h cocraio grai Th low i h mim is assm o b iircioal a h avrag vlociy is ak o b cosa hrogho h lgh o h low il his ar h solios ar obai or wo sol isrsio roblms i a logiial ii lgh rscivly h irs roblm im sol isrsio o icrasig or crasig ar alog a iorm low hrogh a homogos omai is si h sco roblm h mim is cosir hrogos hc h vlociy is cosir o osiio variabl Th vlociy is liarly irola i osiio variabl which rrss a small icrm i h vlociy rom o o h ohr o h omai This rssio coais a aramr o rrs a chag i hrogos rom o mim o ohr mim isrsio is assm roorioal o sqar o vlociy ach roblm h omai is iiially sol r Th i coiio is o iorm a varyig ar rscivly Nmrical solio has also b obai or h cas i which isrsio varis liarly wih vlociy a has b comar wih h aalyical solio obai i h rvios cass TEMPORALLY EPENENT SPERSON ALONG UNFORM FLOW Bcas mass is cosrv h govrig irial qaio is rmi o b whr is sol cocraio a osiio alog h logiial ircio a im is isrsio coici a is h avrag vlociy o li or sricial vlociy To sy h morally sol isrsio o a iorm i cocraio o coios ar i a iiially sol r ii omai w cosir m a Wh m is a coici whos imsio is ivrs o h im variabl Ths m is a rssio i oimsioal variabl m Th rssio o m = or m = or = Th ormr cas rrss h iorm sol isrsio a h lar cas rrss h iiial isrsio Th coicis a i qaio may b i as iiial isrsio coici a iorm low vlociy rscivly Ths h arial irial qaio alog wih iiial coiio a boary coiios may b wri as: m 3 iially sara low o li o cocraio = aks lac i h mim A = h cocraio o h la sorc is isaaosly chag o = Ths h aroria boary coiios ar Th roblm h is o characri h cocraio as a cio o a whr h i coiio is assm a h origi a a sco y or l y homogos coiio is assm is iiial cocraio To rc qaio 3 o a mor amiliar orm w ak m k m Sbsiig qaio 5 io qaio 3 givs m Th iiial a boary coiios 3 rasorm o 5 6

3 k m 7 is hs rqir ha qaio 6 may b solv or a im il o h li a = Th solio o qaio 5 may b obai raily by s o haml s horm arslaw a Jagr 97 is h solio o h isio qaio or smi-iii mia i which h iiial cocraio is ro a is srac is maiai a cocraio iy h h solio o h roblm i F y which h srac is maiai a mrar F y This horm is s ricially or ha cocio roblms b h abov has b sciali o i his sciic cas o irs osir ow h roblm i which iiial cocraio is ro a h boary is maiai a cocraio iy Th boary coiios ar 8 Th roblm is raily solv by alicaio o h Lalac rasorm which is i as L Hc i qaio 6 is mlili by a igra rm by rm i is rc o a oriary irial qaio m Th solio o h abov qaio is q q Th boary coiio as whr a boary coiio a q is 9 rqirs ha = rqirs ha hs h ariclar solio o h Lalac rasorm qaio is q Th ivrsio o h abov cio is giv i ay abl o Lalac rasorms Th rsl is r m Uiliig haml s horm h solio o h roblm wih iiial cocraio ro a h im srac coiio a = is m Sic is a coios cio i is ossibl o iria r h igral which givs m m m 3 Th solio o h roblm is 3 m m Pig ca b wri as m h h qaio m m 3 Sic k h m ariclar solio o h roblm may b wri as k whr a m m m

4 Evalaio o h igral solio Th igraio o h irs rm o qaio givs 5 For covic h sco igral may b rss o rms o rror cio Horsi 95 bcas his cio is wll abla Noig ha Th sco igral o qaio may b wri as 6 Sic h mho o rcig igral o a abla cio is h sam or boh igrals i h righ si o qaio 6 oly h irs rm is cosir L a aig a sbracig Th igral may b rss as 7 Frhr l i h irs rm o h abov qaio h 8 Similar valaio o h sco igral o qaio 6 givs Agai sbsiig io h irs rm h rsl is Noig ha Sbsiio io qaio givs 9 Ths qaio may b rss as k m Howvr by iiio rc Also rc Wriig qaio i rms o rror cios w g rc rc k m Ths Sbsiio io qaio 5 h solio is rc rc R-sbsiig or a givs m rc m m rc R-sbsi h val o h i rms o w g m rc m m rc

5 whr boaris ar symmrical h solio o h roblm is giv by h irs rm h qaio Th sco rm is qaio is hs o h asymmric boary imos i h mor gral roblm Howvr i shol b o also ha i a oi a gra isac away rom h sorc is cosir h i is ossibl o aroima h boary coiio by which las o a symmrical solio 3 SPATALLY EPENENT SPERSON ALONG NON-UNFORM FLOW Th hrogiy o oros omai was i by scal isrsio a low hrogh h mim has b cosir iorm Yas 99 b h low vlociy may also o osiio variabl i cas h omai is hrogos Zoo a Kigh 997 hav cosir h vlociy as a h sol isrsio roorioal o sqar o vlociy i as ; i a smi-iii omai B hs rssios o o rlc ral variaios o hrogiy o h mim bcas as isrsio a vlociy also bcom oo larg ac h variaio i vlociy o hrogiy shol b small so ha h vlociy a ach osiio saisis h arcy s law i cas h mim is oros or saisis h lamiar coiio o h low i a o-oros mim a ssial coiios or h vlociy aramr i h avcioisio qaio This acor is ak car o i h rs work a vlociy is liarly irola i osiio variabl sch ha i icrass rom a val a b a = L whr b may b a ral = o a val cosa Ths a b L a 3a Whr is h aramr accoig or h hrogiy o h mim shol b small so ha h icras i vlociy is o small orr Sol isrsio is assm roorioal o sqar o h vlociy so w cosir a 3b As a is a o-imsioal rm hc a ar isrsio coici a vlociy rscivly a h origi = o h mim Th omai is assm iiially sol r A i cocraio is assm a h origi a a l y homogos coiio is assm a h ohr o h omai Th avcioisio qaio assms h orm a a is rhr rc io a arial irial qaio wih cosa coicis by sig a rasormaio Ulimaly w s h sam iiial a boary coiios o solv h abov isrsio roblm or isrsio o-iorm Th rocr is sam as solv i h arlir cas Th h sir solio may b wri as a rc 5 a a a rc a a A lo o logarihmic robabiliy grah o h abov solio is giv or varios vals o h imsiolss gro Th igr shows ha as bcoms small h cocraio isribio bcoms arly symmrical abo h val = i Howvr or larg vals o asymmrical cocraio isribios bcom oicabl This iicas ha or larg val o or small vals o isac h coribio o h sco rm i qaio 5 bcoms sigiica as aroachs iy RESULTS AN SUSSONS ocraio vals ar vala rom h or aalyical solios iscss i a ii omai a ims yars = 3 a or i vals = = kmyar = 5 km yar Figrs rrss moral cocraio isrsio ar o iorm i a i o icrasig ar rscivly alog a iorm low hrogh a homogos mim scrib by h aalyical solios qaio rscivly igr h iorm i cocraio val is a all ims a h cocraio val a = icrass wih im Ths h rsciv i boary coiios ar saisi h igr h o crvs rrss h solios or a rssio m = m which is o crasig ar h igrs h soli crv rrss h rsciv solios a = yar or aohr rssio m = m which is o icrasig ar may b obsrv ha i cas o iorm i h cocraio val a a ariclar osiio is highr or h lar rssio o m ha ha or h ormr rssio o m Th irc icrass wih h isac alog h omai B i cas o a i cocraio o icrasig ar is val is lss or icrasig ar o m ha ha or crasig ar o m This r is o imiishig ar o = byo which h r rvrss For all h crvs raw i igr a val myar = is chos Boh h aalyical solios o scio may b solv sig ohr rssios o m which saisy h coiios sa a h os o h scio

6 Figr : Tmoral sol isrsio alog iorm low o iorm i scrib by solio qaio Figr : Brak hrogh crv or isrsio alog wih iorm low Th isribio is symmrical or vals o chos som isac rom h sorc A aml o brak hrogh crvs obai or isrsio i a cylirical vrical colm is show as Figr Th horical crv was obai by glcig h sco rm o qaio Figr 3 givs h cocraio vals vala rom aalyical solios qaios 5 or saially isrsio o iorm i a i o icrasig ar rscivly alog o-iorm low hrogh a hrogos omai Th soli crvs i igr 3 rrs h solio qaio 3 i which a val a = km is ak Usig rssios i may b vala ha o h hrogiy o h mim h vlociy varis rom a val o kmyar o a val o kmyar a isrsio varis rom a val o kmyar o a val o kmyar alog h omai km This igr also shows h c o hrogiy o h isrsio ar A o crv is raw or h val a = km may b obsrv ha h cocraio vals vala rom h solio qaio 5 alog a mim o lssr hrogiy which irocs lssr variaio i vlociy a isrsio alog h colm ar slighly highr ha hos a h rsciv osiios o a mim o highr hrogiy ar h origi b cras a asr ra as h ohr o h mim is aroach This comariso is o a = yar This val is chos o sr ha h acor a i coiio rmais osiiv or h vals chos or a Th isribio is symmrical or vals o chos som isac rom h sorc A brak hrogh crv is obai or isrsio i or ir h as show i Figr Th horical crv was obai by glcig h sco rm o qaio 5 5 ONLUSONS Figr 3: Saially sol isrsio alog wih o-iorm low i scrib by solio qaio 5 osiraio o h govrig irial qaio or isrsio i low hrogh oros mia giv ris o a solio ha is o symmrical abo = or larg vals o Erimal vic howvr rvals ha is small This iicas ha lss h rgio clos o h sorc is cosir h cocraio isribio is aroimaly symmrical Thorically oly as ; howvr oly rrors o h orr o magi o rimal rrors ar iroc i h oriary rims i a symmrical solio is assm Th solio is obai or o imsioal avcio isio qaio wih variabl coicis alog wih wo s o boary coiios i a iiially sol r ii omai hav b obai i wo cass: Figr : Brak hrogh crv or isrsio alog wih o-iorm low

7 moral isrsio alog wih iorm low hrogh homogos mim a saially isrsio alog o-iorm low hrogh hrogos mim which sol isrsio is assm roorioal o h sqar o vlociy Th alicaio o a w rasormaio which irocs aohr sac variabl o h avcio-isio qaio maks i ossibl o s Lalac rasormaio chiq i gig h solio Nmrical solio has b obai sig a wo-lvl lici ii irc schm Th rsciv aalyical a mrical solios hav also b comar a vry goo agrm bw h wo has b o Th aalyical solio o h sco roblm i cas o iorm i has b comar wih h mrical solio o sam roblm b assmig isrsio varyig wih vlociy Sch aalyical solios may srv as ools i valiaig mrical solios i mor ralisic isrsio roblms aciliaig o assss h rasor o ollas sol cocraio away rom is sorc alog a low hrogh soil mim hrogh aqirs a hrogh oil rsrvoirs 6 REFERENES [] Al-Niami A N S a Rsho K R Aalysis o low agais isrsio i oros mia; J Hyrol [] Al Kmar ili Kmar Jaiswal a Nav Kmar Aalyical solios o o-imsioal avcio-isio qaio wih variabl coicis i a ii omai; J Earh Sys Sci [3] Baks R B a Ali J isrsio a asorio i oros mia low; J Hyral iv Vol [] Baks R B a Jrasa S R isrsio i say oros mia low; J Hyral iv Vol [5] Bar J yamics o lis i oros mia Nw York:Amr Elsv o 97 [6] rak J Th Mahmaics o isio Loo: Oor Uiv Prss 975 [7] hrysikoolos V a Sim Y O imsioal virs rasor i homogos oros mia wih im- isribio coici; J Hyrol [8] Ebach E H a Whi R Th miig o lis lowig hrogh ack solis; J Am s hm Egg [9] Glhar L W Wly a Rhl K R A criical rviw o aa o il-scal isrsio i aqirs; War Rsor Rs [ ]Gvaas V a Volkr R E Erimal ivsigaios o coi aqir ollio rom rcharg basis; War Rsor Rs [ ]Harlma R F a Rmr R R Logiial a laral isrsio i a isoroic oros mim; J Fli Mch [ ]Lij F J Tori N a va Gch M Th Aalyical solios or o-qilibrim sol rasor i hr-imsioal oros mia; J Hyrol [3]Li S H Noliar asorio i layr oros mia low; J Hyral iv []Lisrom F T a Borsma L Aalyical solios or covciv isrsiv rasor i coi aqirs wih ir iiial a boary coiios; War Rsor Rs [5]Loga J a Zloik V Th covcio isio qaio wih rioic boary coiios; Al Mah L [6 ]Loga J Sol rasor i oros mia wih scal isrsio a rioic boary coiios; J Hyrol [7]Ogaa A Thory o isrsio i gralar mia; US Gol Sr Pro Par [8]Ogaa A a Baks R B A solio o h irial qaio o logiial isrsio i oros mia; US Gol Srv Pro Par -A 9 96 [9]Siri H Gro war coamia rasor by o ivrgc-r say a o-saioary vlociy ils; J Hyrol [ ]Shra SRA solio o h irial qaio o logiial isrsio wih variabl coicis i a ii omai J o Ali Mahmaics & Physics Vol No 93- []Shra SR A solio o h irial qaio o isrsio alog iorm a o-iorm low wih variabl coicis i a ii omai J o Mahmaical Aalysis Vol3 No 89-5 []Shra SR A aalyical solio o o-imsioal avcio-isio qaio i a oros mia i rsc o raioaciv cay Global Joral o Pr a Ali Mahmaics Vol8 No 3- [3]Shra SR Raji J & Niraja M Mahmaical Solios o rasor o ollas hrogh sara oros mia wih asorio i a ii omai J o ombi Rsarch & vlom Vol No 3- []Shra SR Prav Kmar M & Ramsh T Mahmaical Aalysis o rasor o ollas hrogh sara oros mia wih asorio a raioaciv cay J o ombi Rsarch & vlom Vol No -8 [5 ]Shra SR Raji J & Niraja M Mahmaical mollig o rasor o ollas i sara oros mia wih raioaciv cay a comariso wih soil colm rim Sciiic J o Egirig & Tchology Vol 7 No 5 [6]Tarakovsky M a Frico V i A aalyical solio or coamia rasor i o iorm low; Trasor i Poros Mia [7]va Gch M Th a Alvs W J Aalyical solios o h oimsioal covciv-isrsiv sol rasor qaio; US Agriclr Tch Bll No [8]Yas S R A aalyical solio or o-imsioal rasor i hrogos oros mia; War Rsor Rs [9]Yas S R A aalyical solio or o-imsioal rasor i oros mia wih a oial isrsio cio; War Rsor Rs [3]Zoo a Kigh J H Aalyical solios or avcio a avcio isio qaio wih saially variabl coicis; J Hyral Egg