Green functions of mass diffusion waves in porous media

Transcription

1 Jurnal f Physics Cmmunicatins LETTER OPEN ACCESS Grn functins f mass diffusin wavs in prus mdia T cit this articl: Duxing Yang 8 J. Phys. Cmmun. Viw th articl nlin fr updats and nhancmnts. Rlatd cntnt - Claking and anamrphism fr light and mass diffusin Sébastin Gunnau, André Diatta, Tania M Puvirajsingh t al. - Numrical mthds fr finding statinary gravitatinal slutins Óscar J C Dias, Jrg E Sants and Bnsn Way - Extndd Plfka xpansin fr stchastic dynamics B Bravi, P Sllich and M Oppr This cntnt was dwnladd frm IP addrss n 5//8 at 7:6

2 J. Phys. Cmmun. (8) OPEN ACCESS RECEIVED Fbruary 8 REVISED 6 March 8 ACCEPTED FOR PUBLICATION March 8 LETTER Grn functins f mass diffusin wavs in prus mdia Duxing Yang Institut f Crustal Dynamics, China Earthquak Administratin, Bijing, 85, Ppl s Rpublic f China yangdx@mail.iggcas.ac.cn Kywrds: mass wav, diffusin, grn functin, prus mdia PUBLISHED March 8 Original cntnt frm this wrk may b usd undr th trms f th Crativ Cmmns Attributin. licnc. Any furthr distributin f this wrk must maintain attributin t th authr(s) and th titl f th wrk, jurnal citatin and DOI. Abstract A frmulism f frquncy-dmain mass diffusin-wavs in prus mdia is drivd by mans f Furir transfrm. In analgy t cnvntinal thrmal-wav filds, intrnally cnsistnt Grn functins in Cartsian crdinats fr linar mass diffusin-wavs is als prsntd fr infinit, smiinfinit and finit-siz dmains in thr-dimnsinal spacs. Th Grn functins ar utilizd t analyz th rspns f a particular typ f mass diffusin physical systm t any arbitrary tracr surc distributins. This mthd allws th intrductin f frquncy-dpndnt physically intrinsic prprtis f prus mdia. Th Grn functins prsntd in this lttr may significantly advanc undrstanding in linar mass diffusin-wav physics in prus mdia, and can b applid t rtriv spatial-tmpral diffusiv-wav filds frm ambint mass fluctuatins in glgical rsrvirs.. Intrductin Grn functins ar usd t dscrib th rspns f a particular typ f physical systm t a pint surc in spatial-tmpral dmains []. Thrfr, th physical fild rspns t any arbitrary surc distributin can b fund by a cnvlutin intgral f th distributin with th Grn functin vr th surc vlum []. Grn functins in diffusiv-wav physics hav bn prpsd by Mandlis []. Grn functin tchniqus hav bn widly applid in thrmal-wav physics [4 7]. Mandlis [8] has prsntd mthd and Grn functins f diffusin-wav filds. Diffusin wavs aris frm th classical parablic diffusin quatin with an scillatry frc functin in hmgnus mdia [9]. Diffusin-wav mthds hav bn dvlpd in th study f hat transfr [], diffusiv nutrn wavs [], diffusiv viscsity wavs [], prssur diffusin in prus mdia [ 6] and mass transprt [7]. Diffusin wavs ar havily dampd with rlativly slw vlcity and shrt wavlngth. Th pntratin dpth and cmplx wavnumbr dscrib diffusin-wav bhavir. Diffusin wavs in prus mdia by an accumulatin-dpltin law [4]. Rcntly diffusin-wavs f mass transprt hav bn rciving much attntin in gchmistry, and fund applicatins in vlcanic ruptin linkd by radiactiv lmnt transfr [8] and rganic cmpund migratin [9] in sil. Linar mass diffusin-wav mathmatical frmalisms ar basd n harmnic wav slutins, r Laplac transfrm mthds []. Ths thris hav intrducd thrtical tratmnts f mass diffusin-wav filds. Grn functins can b usd t mdl th rspns f a systm t a prscribd xcitatin with knwing th intrnal prprtis f th systm. Tmpral-spatial Grn functins fr mass diffusin-wavs hav bn prsntd by Patrsn []. Mass fild rspns t any arbitrary mass surc distributin can b rprsntd as a cnvlutin intgral f th distributin with a Grn functin vr surc crdinats. An imprtant qustin ariss vr th xistnc f th spctrum f chmical tracr cncntratins bsrvd in a glgical rsrvir, which ncmpasss th ntir spctral bandwidth. T analyz th cnvntinal mass diffusin-wav bhavir, th widband spctral cncntratin masurmnts must b rducd t a singl spctral cmpnnt. Th Grn functins f frquncy dmain must b usd t calculat mass-diffusin wav filds. In this lttr, w prsnt a frmulism f frquncy-spatial mass diffusin in prus mdia, and prsnt Cartsian-crdinat mass diffusin-wav Grn functins fr infinit, smi-infinit, and finit-siz dmains in thr dimnsinal spacs. Th linar mass diffusin-wav filds ar mathmatically analyzd with Grn 8 Th Authr(s). Publishd by IOP Publishing Ltd

3 J. Phys. Cmmun. (8) functins, which can b usd fr prcisly prdicting th radiactiv lmnt transfr (disprsin-dcay) in th larg-scal nuclar glgical rsrvirs. This mthd allws th intrductin f frquncy-dpndnt matrial prprtis f glgical systms. By virtu f th uniqunss, rapidity f cnvrgnc, and clsd-frm rprsntatins f Grn functins, mass diffusin will b prcisly prdictd fr all physically accptabl bundary cnditins. It is hpd that mthdlgis f Grn functins will frm a mathmatically rigrus and usful rfrnc fr nhancmnt f shal gas, il rcvry, and prdictin f undrgrund radiactiv lmnt lakag. Furthrmr, Grn functins can b usd t invrsly dtrmin th mass diffusivity, mass surc and dcay cnstant f radiactiv lmnts in th Earth.. Mathmatical mthds and grn functins.. Grn functin f mass diffusin wav Basd n th wll-knwn diffusin thrm [9], a mass diffusin (disprsin-dcay) wav [] gnratd by a surc functin q( r, t) in prus mdia is givn by D J( r, t) - lj( r, t) - J( r, t) -q( r, t) ( ) t whr D is th mass ctiv diffusivity in a prus mdium and th functin f prsity, and l prsnts a dcay cnstant f a radiactiv tracr. J( r, t)dnts mass cncntratin. Th mass diffusin quatin () is ssntially a mdifid hat diffusin quatin [9] with th lambda trm, lj( r, t), crrspnding t th dpltin. Th undrlining physical phnmnn f th dpltin trm includs th glgical dcay f radiactiv tracrs, and adsrptin f th mbil tracr r slut t th surfac f prus mdia. Using Furir transfrm [], th mass diffusin-wav bys, in th frquncy dmain, th fllwing quatin D F ( r, -( l j F ( r, -Q( r, ( ) It nts that th drivatin f quatin () frm quatin () rquirs that th mass cncntratin is bundd. Th slutin t th hmgnus (disprsin-dcay) diffusin wav quatin () is xprssd as ò F( r, w ) J( r, t ) -tdt ( ) - Q ( r, is a spctrum f th surc distributin q( r, t). F( r, prsnts th mass cncntratin at lcatin r du t th randm frcing Q ( r,. Th Grn functin (s appndix) fr th linar mass (disprsin-dcay) diffusin wavs in thr-dimnsins is givn by quatin (4): g( r, t r, t ) 8[ pd ( t - t )] - l( t- t) ( r-r ) 4D ( t-t) H( t - t ) ( 4) Th Grn functin (4) is th slutin t th mass diffusin quatin () whn th surc is a dlta functin. Whr rr ( )is th crdinat f th bsrvatin (surc) pint with rspct t th rigin, t ( t )prsnts th, t t bsrvatin (surc apparanc) tim, and H dnts th Havisid functin Ht ( - t). Th, t < t Grn functin f quatin (4) satisfis D g( r, t r, t) - l g( r, t r, t) - r r r r t g (, t, t ) - d ( - ) d ( t - t ) ( 5 ) hr th Grn functin fr parablic quatins in thr variabls g ( r, t r, t)dscribs th rspns f th systm at th pint r t a pint surc lcatd at r. Th pint surc is givn by d ( r - r ) d ( t - t ), th prduct f Dirac dlta functins. is th Laplac pratr. Taking th tmpral Furir transfrm f quatin (4) G ( r r ; w ; t ) g( r, t r, t ) -j wtdt ( 6) ò - and using mthd prsntd by Mandlis [8, 6], wfind D G( r r ; w; t ) - ( l j G( r r ; w; t ) -d( r - r ) -j wt ( 7) Us has bn mad f th ntatin [8] G( r r, -j wt G( r r ; w; t ) ( 8)

4 J. Phys. Cmmun. (8) Th Grn functin can b writtn by D G( rr, - ( l j G( rr, -d( r- r) () 9 By intrchanging th crdinats r «r in quatins () and (9), multiplying quatin () by G ( r r, and quatin (9) by F( r,, thn subtracting th rsulting xprssins, w hav D [ G( r r, F ( r, -F( r, G( r r, ] -Q( r, G( r r, d( r - r) F( r, Applying a rciprcity prprty [] f Grn functins G( r - r, G( r - r, and th shifting prprty f Dlta functin [8], carrying ut an intgratin by parts in th rgin f th surc vlum, and using Grn s thrm [, 8], it givs F ( r, w ) Q ( r, w ) G ( r r ; w ) dv D [ G ( r r, w ) F ( r, w ) s s -F( r s, G( r r s, ] ds ( ) Th tim mdulatin factr t is implicitly cmplid in quatin (9). Hr ( ) dv dnts an intgratin vr a vlum V. Th intgral ( ) ds is vr th surfac S which bunds and nclss th vlum V. r s prsnts a crdinat pint n S. ds n ds, whr n is th utward nrmal t th surfac S. G ( r r, is a widband Furir spctrum f Grn s functin g ( r, t r, t). Th quatins f (), (9) and () ar th fundamntal frmula fr th disprsin-dcay diffusin-wav fild in prus mdia. It pints ut that 8 th final xprssin f mass (disprsin-dcay) diffusin filds must b multiplid by t... Thr dimnsinal infinit Grn functins Appndix dscribs th Grn functin f mass diffusin-wavs (disprsin-dcay) fr infinit thrdimnsinal spacs G( r r, l - r-r D 4pD r - r ( ) Hr r - r ( x - x ) ( y - y ) ( z - z ) is a mdul f th distanc vctr ( r - r ) frm th l pint surc ( x, y, z) t th rspns pint ( x, y, z) in th Cartsian crdinat, whr is th mdifid D mass diffusin-wav fild wavnumbr... Thr dimnsinal smi-infinit grn functins If a linar disprsin-dcay diffusin fild is dividd by a surfac at th bsrvatin crdinat z, a smiinfinit gmtry is cnsidrd. Th Grn functin can b btaind dirctly frm quatin () by th mthd f imags prsntd by Mandlis [6]. In mass diffusin wav filds, th cmplx wavnumbr is l sw ( ) ( ) D If th Grn functins satisfy hmgnus Dirichlt cnditins at z, G ( r r, w )z, z, applying th mthd f imags [, 6], it asily yilds l l r r r D D Grr (, - 4 pd r - r r - r r Hr r - r and r - r ar thus givn by r - r ( x - x ) ( y - y) ( z - z ) r - r ( x - x ) ( y - y) ( z z ) Fr this chic f z, th abv-mntind xprssins rduc t r - r r - r ( x - x ) ( y - y) ( z) If th Grn functin satisfis Numann cnditins at z, G ( r r, z, z. Th impulsiv imag surc argumnt rquirs that th mass-wav fluxs cancl ut at th intrfac. Th Grn functin is givn by ( )

5 J. Phys. Cmmun. (8) G ( r r, l l r r r D D 4 pd r - r r - r r ( 4).4. Thr dimnsinal grn functins fr finit gmtry Mrs and Fshbach [] has prsntd mthd f imags, whr imag surcs must b lcatd at surc crdinats as shwn in figur f rfrnc [6]. Grn functins satisfy ithr Dirichlt r Numann bundary cnditins at z, L. Fr this cas, Grn functin is knwn fr an infinit gmtry and it is rquird fr a gmtry with finit bundaris. Using th mthd similar t drivatin f quatins () and (4), th Grn functin is givn by G ( r r, 4pD l l n - r-rn - r- rn å n- D D r - r r - r Hr ± crrspnds t Numann r Dirichlt bundary cnditins. With th array f imag surcs fund in Figur.7 f rfrnc [8], th fllwing xprssins ar givn by r - r ( x - x ) ( y - y) [ z - ( nl z )] n n r - r ( x - x ) ( y - y) [ z - ( nl - z )] Th xact bhavir f mass diffusin wavs dpnds n th bundary cnditins. Th influnc f diffrnt bundaris is clarly shwn frm quatin (5). n n ( 5). Applicatin f grn functins W cnsidr th smi-infinit rgin with mass cncntratin spcifid by F ( r s,t) Fvr a intrfac plan z. Th Grn functin satisfis a hmgnus Dirichlt cnditin n th surc plan z, and bys quatin (). Withut bulk surcs cnsidrd in half-spac ( x, y, z), and using G( r r s, w )z, th mass-wav fild at a givn psitin dirctly fllws frm quatin (): F ( r, -D F( r, G( r r, ds ( 6) s s whr th bundary surfac intgral vr ds is d S nidxdy. n i prsnts th inward nrmal vctr t th surfac ds. By mans f n z - n i z dnting th utward nrmal vctr by n, quatin (6) can b xprssd as Frm quatin (), it fllws that ( 7) r, D r s, F ( F( r r z G ( s, z dxdy 8 ( ) l l - r - r z Gr rr r z D l D ( - -, z pd r D r r ( x - x ) ( y - y ) z ( 9) Cmbing quatins (8) and (9), and multiplying by t yilds l z r, j t j D l w - r F w F w D dxdy p òò ( ) ( ) - r r 4

6 J. Phys. Cmmun. (8) With th chang f variabls x - x h and y - y x, quatin () bcms l z r, j t D F ( F w p òò - h x z ( h x z ) h x z l - h x z D dhx d ( ) By changing Cartsian crdinats t plar crdinat systms [9] as r h x and dd hx rrj dd dnting th radial crdinat by r, and th angular crdinat by j, this givs l r, j t D F ( F w ò rz r z ( r z ) r z l - r z D Th mass diffusin-wav fild rprsntd by quatin () is in a cmpact frm, which can b sparatd ut l tw cmpnnts including th ral and imaginary parts. Th ral k r and imaginary k im squar rt f is k r jk With rspct t Eulr s frmula [], it rsults in with REAL and IMAG dfind in th frm im ò dr D ( ) l l w l l w j - ( ) D D r, z REAL j IMAG k z F ( F ( ) - r r rdr ( 4) kr REAL r z ( r z ) r z sin ( wt - k r z) im kim cs wt - k im r z ( ) r z kim kr IMAG cs ( wt k z z - im r ) - r r z ( r z ) r z sin ( wt - k r z) im Th physical mass-wav fild is th ral part f quatin (4). Th disprsin-dcay f radiactiv lmnts (rubidium) is simulatd in a nuclar glgical rsrvir [8]. Figur dpicts th dpth-dpndnt amplitud f th mass-wav fild () with tim-mdulatin angular frquncy as a paramtr. It shws that th slp f th spatial dcay rmarkably incrass with incrasing w. Th D mass diffusin lngth,, knwn as th pntratin dpth, dscribs th rt man squar dpth [4] f th w diffusin-wav pntratin int th dmain z >. As a frquncy incrass, a mass pntratin dpth dcrass, and th diffusin-wav attnuatin acclrats. Th imaginary part f th wavnumbr (s quatin ()) can attain nugh nrgy t appar, as rlativly high frquncis ar rachd [5]. It prvs that th mass diffusin wavs ar disprsiv, and intrnally dampd. Figur illustrats similar dpth dpndnc at a cnstant mdulatin frquncy f Hz, with dmain mass diffusivity as a paramtr. Th ct f incrasing mass diffusivity n th slp f th dcay curvs is sn t b similar t that f dcrasing frquncy, as xpctd frm th structur f quatin (). Such phnmna can b xplaind n th physical grunds. As mass diffusivity incrass, th pntratin dpth accrdingly incrass, a fact that crrspnds t an incras in th dgr f mass transfr. Th diffusivity is a transprt prprty f th prus mdia, which is linarly prprtinal t th prsity []. It indicats that cts f prsity n bhavirs f mass diffusin-wav filds bcm similar t that f th crrspnding diffusivity. Figur shws th cts f varius valus f th paramtr λ n th amplitud f th mass diffusin-wav dpth prfil. It is sn that with incrasing λ th slp f th spatial dlay incrass. Th diffrncs in amplitud ar snsitiv t th valu f λ.asλ dcrass, th dgr f mass dcay dcrass, and th mass diffusin-wav attnuats slwly, lading t mass accumulatin within a prus mdium. As shwn in figurs, amplitud prfils clarly xhibit th pridical bhavirs. Th wav-frnt is wll bhavd and capturd. Th pridically 5

7 J. Phys. Cmmun. (8) Figur. Amplitud (a) and phas (b) dpth prfils f th mass diffusin-wav fild in a smi-infinit dmain, quatin (), with angular mdulatin frquncy w, as a paramtr. Th paramtrs usd fr th calculatins wr D.5-9ms - and l.5-9s -. Figur. Amplitud (a) and phas (b) dpth prfils f th mass diffusin-wav fild in a smi-infinit dmain, quatin (), with diffusivity D, as a paramtr. Th paramtrs usd fr th calculatins wr w - Hz and l.5-9 s -. frcd functin with intrnal damping has rmarkabl influncs n mass diffusin-wav mtins. Th mass diffusin-wav filds by th fild-gradint-drivn accumulatin-dpltin ruls [9]. Th applicatin xampl is dirctd t gphysical phnmna. Hwvr th chmical ractins in glgical rsrvirs ar nnlinar. Futur wrks culd b ndd t xtnd th applicatin f Grn functins t nnlinar prblms [8]. 4. Cnclusins W prps a frmulism f frquncy-dmain mass diffusin-wavs in prus mdia, and driv intrnally cnsistnt Cartsian-crdinat mass-wav Grn functins fr infinit, smi-infinit, and finit-siz dmains in thr-dimnsinal spacs. It nts that th thr-dimnsinal smi-infinit Grn functin is f practical imprtanc, bcaus it dscribs th xact bhavir f diffusiv mass xcitd by an arbitrary surc in glgical rsrvirs. Th intgral xprssins fr prpagating mass diffusin-wav filds ar prsntd in hmgnus systms and in practically glgical gmtris. A spcific applicatin is xplicitly implmntd in trms f Dirichlt bundary cnditins in a cas f smi-infinit rgin with mass impuls functin prscribd vr th intrfac plan. Th pridically frcd functin with intrnal damping has rmarkabl influncs n mass diffusin-wav mtins. Th mass diffusin-wav filds by th fild-gradint-drivn accumulatin-dpltin ruls. Grn functins btaind fr mass diffusin-wavs can b usd fr all physically accptabl bundary 6

8 J. Phys. Cmmun. (8) Figur. Amplitud (a) and phas (b) dpth prfil f th mass diffusin-wav fild in a smi-infinit dmain, quatin (), with dcay cnstant l, as a paramtr. Th paramtrs usd fr th calculatins wr w - Hz and D.5-9 ms -. cnditins undr fixd gmtris. Th xact bhavir f mass diffusin wavs dpnds n th bundary cnditins. It is hpd that Grn functins prsntd in this articl will frm a mathmatically rigrus and usful rfrnc fr nhancmnt f shal gas r il rcvry, and migratin f radiactivity in nuclar glgy. Furthrmr, Grn functins can b usd t dtrmin th mass diffusivity and fluid prprtis f glgical rsrvirs. Acknwldgmnts Th supprt f th Natinal Natural Scinc Fundatin f China undr Grant N. 447 is acknwldgd. Th annymus rviwrs hav givn cmmnts and suggstins, which imprvd this lttr. Appndix Th Grn functin frmulism f th linar mass (disprsin-dcay) wav fild is givn by D G( r - r, - ( l j G( r - r, -d( r - r ) ( A) Using Furir transfrm in spatial-frquncy G( r - r, js R G s, w ds, p ( ) whr r - r, and applying th Dirac dlta functin [7] as ( ) th Furir intgral d ( R) js Rds, p and quating intgrands, w btain ( ) G( s, ( A) sd l js R G( r - r, ds p ( ) sd l By intgrating vr th sphrical shll f radius s, with ds s sin qdsdqdj, it yilds p p jsr cs q G( R, sds dj sin qq d ( p) D s s jr( p) D j sr - -j sr jsrsds ds s jr( p) D ò- s s Nw lt ò ò ò ò s s s ( A) ( A4) j Rzzdz f () z º ( A5) z s It is bsrvd that th intgrand f f ()has z simpl pls vr th cmplx cntur at 7

9 J. Phys. Cmmun. (8) l D z j j s. Using th Cauchy Rsidu Thrm, it givs j Rzzdz jsrsds jzr l z - R f z lim j j R j D A6 z z s ò () p p -s p ( ) - s s z Insrting quatin (A6) int quatin (A4),it finally lads t th Grn functin in th frquncy dmain: j - l w R z js Applying th Furir invrsin transfrm [8] f quatin (A7), it givs D G( R, ( A7) 4pRD g( R, t) 8[ pd t] t R - l 4D t Subjct t th tim discntinuity at t t and cnsistnt with th Causality prprty f tim-dmain Grn functin, using th Havisid functin Ht ( - t ), w btain th tmpral-spatial Grn functin f th infinit spac dmain g( r, t r, t ) 8[ pd ( t - t )] - l( t- t) ( r-r ) ( A8) 4D ( t-t) H( t - t ) ( A9) With th similar mthdlgis, th n-dimnsinal Grn functin is givn by g( x - x t - t) pd ( t - t ) - l( t- t) ( x-x ) 4D ( t-t) H( t - t ) ( A) Th mthd f imags [] is usd t driv th tmpral-spatial Grn functin fr smi-infinit dmains g( r, t r, r, t ) Ht ( - t) 8[ pd ( t - t )] - r r r r ( - ) ( - ) - - l( t - t) [ 4D ( t-t) 4D ( t-t) ] ( A) By mans f th mthd similar t drivatin f quatin (A),w find th Grn functin fr finit bundaris g( r, t r, r, t ) n n Ht ( - t) 8[ pd ( t - t )] n r r n r r ( - ) ( n ) - å - l( t - t) ( 4D ( t-t) 4D ( t t) ) ( A) n- In quatins (A), (A), th symbls, prsnt Numann and Dirichlt bundary cnditins, rspctivly. Th trms, ( r - r ), ( r - r ), ( r - r n ) and ( r - r n ) ar dfind in sctins.,.4. ORCID ids Duxing Yang https: /rcid.rg/ Rfrncs [] Mrs P M and Fshbach H 95 Mthds f Thrtical Physics (Nw Yrk: McGraw-Hill) Ch 7.4 [] Bayin S S 6 Mathmatical Mthds in Scinc and Enginring (Nw Yrk: Wily) Ch8 and 9 [] Mandlis A 989 J. Opt. Sc. Am. A 6 98 [4] Thmas R L, Puch J J, Wng Y H, Favr L D, Ku R-K and Rsncwaig A 98 J. Appl. Phys. 5 5 [5] Favr L D, Ku R K and Thmas R L 987 Phtacustic and thrmal wav phnmna Smicnductrs d A Mandlis (Nw Yrk: Nrth-Hlland) Ch 4 [6] Mandlis A 99 J. Phys. A: Math. Gn [7] Grigrian S A, Vartanyants I A and Kvalchuk M V J. Phys. D: Appl. Phys. 4 A78 [8] Mandlis A Grn Functins and Mathmatical Mthds f Diffusin-Wav Filds (Nw Yrk: Springr) [9] Mandlis A, Niclaids L and Chn Y Phys. Rv. Ltt [] Hua H T, Mandlis A, Liu L X and Mlnikv A 7 NDT & E Intrnatinal 9 79 [] Osbrn A G and Dinrt M R Ann. Nucl. Enrgy 6 69 [] Hcking W K, Fuka S, Yamamt M, Tsuda T and Kat S 99 Radi Sci. 6 8 [] Barnblatt G I, Zhltv I P and Kchina I N 96 J. Appl. Math [4] Yang D X, Li Q and Zhang L Z 5 J. Appl. Phys [5] Yang D X, Li Q and Zhang L Z 6 J. Appl. Phys [6] Shapir N M, Drznin D V, Ya Drznina S, Snyukv S L, Gusv A A and Grdv E I 7 Nat. Gsci. 44 [7] Yang D X and Zhang D L Prg. Cmput. Fluid Dy. 85 8

10 J. Phys. Cmmun. (8) [8] Faul H 954 Nuclar Glgy (Nw Yrk: Jhn Wily & Sns, Inc.) [9] David Lgan J Transprt Mdling in Hydrglgical Systms (Nw Yrk: Springr) [] Sun N Z 995 Mathmatical Mdling f Grundwatr Pllutin (Nw Yrk: Springr) [] Baddur N and Mandlis A 5 Phtacustics [] Bar J and Chng A H D Mdling Grundwatr Flw and Cntaminant Transprt (Nw Yrk: Springr-Nthrlands) [] Kasana H S 5 Cmplx Variabls: Thry And Applicatins nd dn (Nw Dlhi: PHI Larning Pvt. Ltd) [4] Wang C H, Mandlis A, Qu H and Chn Z Y 8 J. Appl. Phys. 45 [5] Mandlis A and Fng C Phys. Rv. E [6] Mandlis A 995 J. Appl. Phys [7] Jntschura U D and Sapirstin J 8 J. Phys. Cmmun. 56 [8] Valds-Parada F J, Sals-Cru M, Ocha-Tapia J A and Alvarz-Ramirz J 8 Cmput. Chm. Eng. 5 9