TWO-DIMENSIONAL ANALYTICAL SOLUTIONS FOR POINT SOURCE CONTAMINANTS TRANSPORT IN SEMI-INFINITE HOMOGENEOUS POROUS MEDIUM

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1 Jurnal f Engineering Siene and Tehnlgy Vl. 6, N. 4 (011) Shl f Engineering, Taylr s University TWO-IMENSIONAL ANALYTICAL SOLUTIONS FOR POINT SOURCE CONTAMINANTS TRANSPORT IN SEMI-INFINITE HOMOGENEOUS POROUS MEIUM R. R.YAAV*, ILIP KUMAR JAISWAL epartment f Mathematis and Astrnmy Luknw University, Luknw-6007, Uttar Pradesh, India *Crrespnding Authr: yadav_rr@yah..in Abstrat An analytial slutin is btained fr the advetin-dispersin f a nstant input nentratin alng unsteady hrizntal flw in semiinfinite shallw aquifer. Expnential frm f time-dependent seepage velity and first rder deay are nsidered. Pint sure pllutant reahes the grundwater level frm a surfae and mves vertially dwn-wards, frm its eah pint and starts spreading in the hrizntal plane. Tw separates transfrmatins are intrdued. Laplae tehnique is used t btain the analytial slutin f the present prblem. Keywrds: Advetin, ispersin, Grundwater, Shallw aquifer, Seepage velity. 1. Intrdutin The intensive size f natural resures and the large prdutin f wastes in mdern siety ften pse a threat t the grundwater quality and already have resultant in many inidents f grundwater ntaminatin. egradatin f grundwater quality an take plae ver large areas frm plane r diffuse sures like deep perlatin frm intensively farmed fields, r it an be aused by pint sures suh as septi tank, garbage dispsal sites, emeteries, mine spils and il spils r ther aidental entry f pllutants int the undergrund envirnment. Anther pssibility is ntaminatin by line sures f pr quality water, like seepage frm plluted streams r intrusin f salt water, frm eans [1, ]. Analytial slutins in ne-dimensinal prblems thrugh semi-infinite r finite prus media have been presented by several researhers. A nn-exhaustive 459

2 460 R. R. Yadav and. K. Jaiswal list f referenes must inlude at least the wrks f Bastian and Lapidus [3], Banks and Ali [4], Ogata [5], Marin [6] and van Genuheten [7]. Mst f these papers reveal idealisti assumptins, suh as prus medium f nstant prsity, seepage flw and dispersin. In deviating frm the abve ideal nditins, Shamir and Harleman [] presented an analytial slutin in tw layered prus medium. Banks and Jerasate [8], Rumer [9] and Yadav et al. [10] nsidered dispersin alng unsteady flw. Al-Niami and Rushtn [1] nsidered unifrm flw where as Kumar [11] tk unsteady flw, against the dispersin in finite prus media. Mst f these wrks have inluded the attenuatin effet due t adsrptin, first rder radi-ative deay and/r hemial reatins. Brue and Street [1] nsidered bth lngitudinal and lateral dispersin in semi-infinite nn-adsrbing prus media in steady flw field fr a nstant input nentratin. Shen [13] presented a generalized lsed frm slutin fr threedimensinal dispersin in saturated ambient prus media resulting frm sures f finite extent with time-dependent input nentratin. Hunt [14] has given ne-, tw- and three-dimensinal slutin fr instantaneus, ntinuus and steady pllutin sures in unifrm grundwater. Al-Niami and Rushtn [15] btained the analytial slutins nsidering lngitudinal and lateral dispersin in a tw-layered prus medium. Güven et al. [16] used the Aris mment methd t analysis the dispersin f nseutive slute in a hrizntal stratified aquifer. Prakash [17] presented analytial slutins t predit tempral and spatial distributin f nentratin in ne-, tw-, and three-dimensinally fully saturated unifrm prus media flw fr a pint, line r parallelpiped sure in an istrpi prus medium. Latinpuls et al. [18] studied the hemial transprt in tw-dimensinal aquifer. Ellswrth and Butters [19] disussed threedimensinal slutins used fr transprt prblems invlving arbitrary Cartesian rdinate systems. Lgan and Zltnik [0] btained slutins f the nvetin-diffusin equatin with deay fr peridi bundary nditins n a semi-infinite dmain. Aral and Lia [1] examined slutins t tw-dimensinal advetin-dispersin equatins with time- dependent dispersin effiients. In partiular, they develped instantaneus and ntinuus pint sure slutins fr nstant, linear, asymptti, and expnentially varying dispersin effiients. Wrtmanna et al. [] presented an analytial slutin fr advetin-diffusin equatin t simulate the pllutant dispersin in the planetary bundary layer. Sirin [3] assumed pre flw velity t be a nn-divergene free, unsteady and nnstatinary randm funtin f spae and time fr grundwater ntaminant transprt in a hetergeneus medium. The signifiane f the new velity rretin term is investigated n a tw dimensinal transprt prblem driven by a density dependent flw. Smedt [4] presented a mdel fr slute transprt in rivers inluding transient strage in hyprhei znes. The mdel nsists f an advetin-dispersin equatin fr transprt in the main hannel with a sink term desribing diffusive slute transfer t the hyprhei zne. The system f equatins is slved analytially fr instantaneus injetin f a nservative traer in an infinite unifrm river reah with steady flw [1]. But all suh tw r three-dimensinal dispersin studies have been dne alng steady and unidiretinal r lngitudinal (perpendiular t the vertial) flw field. In almst all the slutins derived in tw-dimensin s fr nly lngitudinal mpnent f velity where nsidered, negleting vertial (transverse) Jurnal f Engineering Siene and Tehnlgy August 011, Vl. 6(4)

3 Tw-dimensinal Analytial Slutins fr Pint Sure Cntaminants Transprt 461 mpnent, while in present study bth the dispersin mpnents alng lngitudinal and lateral diretins and velity alng these tw diretins are als nsidered. The seepage velity is expnentially dereasing funtin f time. Prus medium is nsidered hmgeneus, istrpi saturated and f semiinfinite in hrizntal plane. The seepage velity is expnentially dereasing funtin f time. The first rder deay term whih is prprtinal t velity is als nsidered. Analytial slutin is btained fr unifrm input sure nentratin with the help f Laplae transfrmatin tehnique.. Mathematial Frmulatin f the Prblem Let the pllutant invades the grundwater level frm pint sure. The pllutant being f a signifiantly higher density than the grundwater mves twards the bttm f the shallw aquifer alng vertially dwnward, frm its eah pint the pllutant is bund t spread in the hrizntal plane alng the unsteady grundwater flw. Let at ne suh pint is the sure nentratin. The lngitudinal and lateral (transverse) diretins in the hrizntal plane extend up t infinity where n nentratin at any time. Let u [LT -1 ] and v [LT -1 ] be mpnent f hrizntal and lateral flw velity and x [L T -1 ] and y [L T -1 ] be the dispersin effiients alng lngitudinal and lateral diretins respetively. The general partial differential equatin desribing hydrdynami dispersin in hmgenus, istrpi prus media an be written as = x + y u t ( t) v( t) γ ( t) (1) u where γ [T -1 ] is the first rder deay. Let u e mt =, v v e mt mt =, and γ = γ e () where u and v are initial velity mpnents alng x and y axes respetively, (x, y, t) is the nentratin at any time t in hrizntal plane and γ is the first rder deay nstant. Rumer [9] established a relatinship fr steady and unsteady flw with an expnentially r sinusidal varying flw velity. S let x = α u and y = α v (3) where α is the effiient having the dimensin [L] and depends upn pre gemetry and average pre size diameter f the prus medium. These tw an be written by using f Eq. () as mt mt x = e and x y y e where α = (4) x = u and y αv = are initial dispersin effiient mpnents alng the tw respetive diretins. Initial and bundary nditins fr present prblem are as fllws, = 0, t = 0, x 0, and y 0 (5) i.e., initially the grundwater is slute free, =, t > 0, x = 0, and y = 0 (6) Jurnal f Engineering Siene and Tehnlgy August 011, Vl. 6(4)

4 46 R. R. Yadav and. K. Jaiswal and = 0, = 0, t 0, x and y (7) The nditin (6) indiates that the nentratin is ntinuus arss the inlet bundary and (7) indiates that there is n slute flux at end f bth bundaries. Using Eqs. () and (4) the differential equatin (1) an be written as, e T mt = t x + y u v γ Intrduing the new time variable T by fllwing transfrmatin (Crank [5]) t = 0 v ( t) dt, where v(t) = e mt (9) t r mt 1 mt T = e dt = ( 1 e ) 0 m mt e mt Fr an expressin e whih is taken suh that =1 fr m = 0 r t = 0, the new time variable btained frm Eq. (9) satisfies the nditins T = 0 fr t = 0 and T = fr m = 0. The first nditin ensures that the nature f the initial nditin des nt hange in the new time variable dmain. Thus Eq. (8) an be written as, (8) = T x + y u v γ Let a new spae variable is intrdued as fllws: (10) y = y (11) X x + x Therefre differential equatin (10) redues int T = U X X γ (1) where y = + 1 and x U = u u + v x After taking are f the transfrmatins (9) and (11) the initial and bundary nditins (5)-(7) an be put as fllws = 0, T = 0, X 0 (13) =, T > 0, X = 0 (14) X = 0, T 0, X (15) Intrduing a new dependent variable K(X, T) by fllwing transfrmatin, y x Jurnal f Engineering Siene and Tehnlgy August 011, Vl. 6(4)

5 Tw-dimensinal Analytial Slutins fr Pint Sure Cntaminants Transprt 463 U U = exp X + γ T 4 ( X, T ) K( X, T ) (16) and applying Laplae transfrmatin n Eqs. (1), (14) and (15) and using initial nditin (13), we an get fllwing rdinary bundary value prblem, d K = dx pk (17) K =, X (18) U p 4 and dg = 0, X (19) dx K 0 where K ( X p) = K( X, T ), e dt pt and p is the Laplae parameter. The slutin f Eq. (17) by using the nditins (18) and (19), bemes U p 4 X p / ( X, p) = e (0) Applying inverse Laplae transfrmatin n (0) and then using Eq. (16), the final slutin f the present prblem may be written as, ( ) ( ) 1 β β + γ X, T = exp ( + β γ ) X X U + 4γ T erf T 1 β + X X + U + 4γ T + exp erf (1) T where U β =, 4 1 T = 1 m mt ( e ). y = + 1, y x U = u + v, x x y = y and X x + x 3. Partiular Cases (i) When m = 0 then T = t in the slutin (1), the btained slutin alng steady hrizntal flw may be written as, ( ) ( ) 1 β β + γ X, t = exp X X U + 4γ t erf t Jurnal f Engineering Siene and Tehnlgy August 011, Vl. 6(4)

6 464 R. R. Yadav and. K. Jaiswal ( + β γ ) 1 β + X X + U + 4γ t + exp erf () t where all the variables defined abve are same. (ii) The slutin fr ne-dimensinal dispersin alng unsteady flw thrugh semi-infinite prus media an be btained by substituting v = 0 and = 0 in Eq. (1), i.e. 1 ( ) ( ) β β + γ X, T = exp x ( + β γ ) X X u + 4 erf x T γ x T 1 β + X X + u + 4γ x T + exp erf (3) x T x where u β =, X = x and u,, and T represent initial velity, dispersin x 4 x effiient alng lngitudinal diretin (x-axis) and time variable respetively. y (iii) Cnentratin distributin due t lngitudinal and lateral dispersin but alng lngitudinal unsteady prus medium flw (as nsidered by the previus wrkers) an be btained by substituting v = 0 in the slutin (1), i.e., ( ) ( ) 1 β β + γ + X X u 4γ T X, T = exp erf T ( + β γ ) 1 β + X X + u + 4γ x T + exp erf (4) T u where β =, and T are same as abve Numerial Examples and isussin T illustrate the nentratin distributin f the btained analytial slutin (1) in tw-dimensinal hmgeneus prus medium in semi-infinite dmain, an example has been hsen in whih the different variables are assigned numerial values, where lngitudinal and lateral seepage velity and dispersin effiients are u = 0.95 (m/day), v = (m/day), =1. 05 (m /day) and = (m /day) respetively. The flw resistane effiient m = 0.1 (1/day) and first rder deay term γ = 0.04 (1/day) have been hsen. Figure 1 is drawn fr slutin (1) with different time t = 1.5,.0 and.5 (days) fr unsteady and Fig. fr steady flw (m = 0) respetively. x y Jurnal f Engineering Siene and Tehnlgy August 011, Vl. 6(4)

7 Tw-dimensinal Analytial Slutins fr Pint Sure Cntaminants Transprt 465 Fig. 1. Cnentratin Prfile fr ereasing Funtin at ifferent Time fr Slutin (1). Fig.. Cnentratin Prfile fr Cnstant Ceffiients at ifferent Time fr Slutin (). Jurnal f Engineering Siene and Tehnlgy August 011, Vl. 6(4)

8 466 R. R. Yadav and. K. Jaiswal Figure 3 shws mparisn between unsteady and steady flw at time t =.0 (days). The behavir f the nentratins in bth diretins inreases with inreasing time, whereas the nentratin values at partiular psitin in unsteady flw are lwer than the steady flw. Fig. 3. Cmparisn f Cnentratin Prfiles between ereasing Funtin and Cnstant Ceffiients fr Slutin (1) and (), t =.0 (days). The lateral velity mpnent is nsidered ne-tenth f the lngitudinal mpnent. Althugh, it signifiant is small but an nt be ignre fr better auray f result. This effet dereases with time. It means as the time inreases dispersin press ges n dminating ver the transprt due t nvetin. In the ase f unsteady flw similar but lesser effet wuld appear t. Thugh expnential frm f seepage velity is nsidered in the present analysis, there is n reasn why ther frm f seepage velity uld nt be used as lng as the bundary nditins (x, y, t) and (x, y, T) are mpatible. A hrizntal twdimensinal transprt mdel is seleted beause the grundwater flw is essentially hrizntal and vertially mixing f the grundwater ntaminant is a gd apprximatin in suh shallw aquifer. The time dependent behavir f pllutants in subsurfae is f interest fr many realisti prblems where the nentratin is bserved r needs t be predited at fixed psitins. Prblems f slute transprt in tw-dimensin invlving sequential first rder deay reatins frequently urs in sil, hemial engineering and grundwater systems, fr example the migratin f simultaneus mvement f interating nitrgen speies, rgani phsphate transprt and the transprt f pestiides and their metablites. The auray f the numerial methd is validated by diret mparisns with the analytial results. Jurnal f Engineering Siene and Tehnlgy August 011, Vl. 6(4)

9 Tw-dimensinal Analytial Slutins fr Pint Sure Cntaminants Transprt 467 Cnlusins This study presents an analytial slutin t slve the advetin dispersin equatin with lngitudinal and transverse dispersin fr desribing the twdimensinal slute transprt in a hmgeneus prus media. In the derived slutin, bth the mpnents (lngitudinal and transverse) f velity are assumed expnentially deeasing funtin f time, dispersin effiient and first rder deay are diretly prprtinal t velity. The hypthetial studies indiate that the effet f pllutant is nt unifrm but derease as we mve away frm rigin alng either diretin r hrizntal plane. The gverning slute transprt equatin is slved analytially by emplying Laplae Transfrmatin Tehnique (LTT). The derived slutin is an effetive and useful fr further appliatin t verify the newly develped numerial transprt mdel fr prediting the twdimensinal time-dependent transprt f ntaminants. The appliatin results reveal that the slute transprt press at the test site beys the linearly timedependent dispersin mdel and that the linearly time-dependent assumptin is valid in this real wrld example. The prpsed slutin an be applied t field prblems where the hydrlgial prperties f the medium and prevailing bundary and initial nditins are the same as, r an be apprximated by, the nes nsidered in this study. Aknwledgements The authr thanks the tw reviewers, whse helpful mments and suggestins greatly imprved the paper. The send authr gratefully aknwledges the finanial assistane in the frm f UGC- r.. S. Kthari Pst tral Fellwship. Referenes 1. Al-Niami, A.N.S.; and Rushtm, K.R. (1977). Analysis f flw against dispersin in prus media. Jurnal f Hydrlgy, 33(1-), Shamir, U.Y.; and Harleman,.R.F. (1967). ispersin in layered prus media. Jurnal f Hydraulis ivisin, 95(7), Bastian, W.C.; and Lapidus, L. (1956). Lngitudinal diffusin in inexhange and hrmatgraphi lumns. The Jurnal f Physial Chemistry, 60(6), Banks, R.B.; and Ali, J. (1964). ispersin and adsrptin in prus media flw. Jurnal f Hydraulis ivisin, 90, Ogata, A. (1970). Thery f dispersin in granular media. U.S. Gelgial Survey, Prfessinal paper N. 411-I, I1-I Marin, M.A. (1974). istributin f ntaminants in prus media flw. Water Resures Researh, 10(5), Van Genuhten, M. Th. (1981). Analytial slutins fr hemial transprt with simultaneus adsrptin, zer-rder prdutin and first rder deay. Jurnal f Hydrlgy, 49(3-4), Jurnal f Engineering Siene and Tehnlgy August 011, Vl. 6(4)

10 468 R. R. Yadav and. K. Jaiswal 8. Banks, R.B.; and Jerasate, S. (196). ispersin in unsteady prus media flw. Jurnal f Hydraulis ivisin, 88, Rumer, R. (196). Lngitudinal dispersin in steady flw. Jurnal f Hydraulis ivisin, 88(4), Yadav, R.R.; Vinda, R.R.; and Kumar, N. (1990). One-dimensinal dispersin in unsteady flw in an adsrbing prus medium: An analytial slutin. Hydrlgial Presses, 4(), Kumar, N. (1983). ispersin f pllutants in semi-infinite prus media with unsteady velity distributin. Nrdi Hydrlgy, 14(3), Brue, J.C.; and Street, R.L. (1966). Studies f free surfae flw and twdimensinal dispersin in prus media. Reprt N. 63, Civil Eng. ept. Stanfrd Univivrsity. 13. Shen, H.T. (1976). Transient dispersin in unifrm prus media flw. Jurnal f Hydraulis ivisin, 10 (6), Hunt, B.W. (1978). ispersin sures in unifrm grundwater flw. Jurnal f Hydraulis ivisin, 104(1), Al-Niami, A.N.S.; and Rushtm, K.R. (1979). ispersin in stratified prus media: analytial slutin. Water Resures Researh, 15(5), Güven, O.; Mlz, M.J.; and Melville, J.G. (1984). Analysis f dispersin in stratified aquifers, Water Resures Researh, 0(10), Prakash, A. (1984). Grundwater ntaminant due t transient sures f pllutin. Jurnal f Hydrauli Engineering, 110(1), Latinpuls, P.; Tlikas,.; and Mylpuls, Y. (1988). Analytial slutin fr tw-dimensinal hemial transprt in aquifer. Jurnal f Hydrlgy, 98(1-), Ellswrth, T.R.; and Butters, G.L. (1993). Three-dimensinal analytial slutin t the advetin-dispersin equatin in arbitrary artesian rdinates. Water Resures Researh, 9(9), Lgan, J..; and Zltnik, V. (1995). The nvetin-diffusin equatin with peridi bundary nditins. Applied Mathemati Letters, 8(3), Aral, M.M.; and Lia, B. (1996). Analytial slutins fr tw dimensinal transprt equatin with time-dependent dispersin effiients. Jurnal f Hydrlgi Engineering, 1(1), Wrtmanna, S.; Vilhena, M.T.; Mreirab,.M.; and Buske,. (005). A new analytial apprah t simulate the pllutant dispersin in the PBL. Atmspheri Envirnment, 39(1), Sirin, H. (006). Grund water ntaminant transprt by nn-divergene free, unsteady and nn-statinary velity fields. Jurnal f Hydrlgy, 330(3-4), Smedt, F.. (007). Analytial slutin and analysis f slute transprt in rivers affeted by diffusive transfer in the hyprhei zne. Jurnal f Hydrlgy, 339(1-), Crank, J. (1980). The mathematis f diffusin. ( nd Ed.), Oxfrd University Press. Jurnal f Engineering Siene and Tehnlgy August 011, Vl. 6(4)