Fluctuating Groundwater Flow in Homogeneous Finite Porous Domain

Transcription

1 The SIJ Trasactis Cmputer Sciece Egieerig & its Applicatis (CSEA), Vl. 4, N. 3, July 6 Fluctuatig rudwater Flw i Hmgeeus Fiite Prus Dmai Nyambae Zachary Nyakudi*, Mathew Kiyaui** & Jhaa K Sigey*** *Departmet f Pure ad Applied Mathematics, Jm Keyatta Uiversity f Agriculture ad Techlgy, Nairbi, KENYA. zayaya{at}gmail{dt}cm **Departmet f Pure ad Applied Mathematics, Jm Keyatta Uiversity f Agriculture ad Techlgy, Nairbi, KENYA. mathewkiy{at}yah{dt}cm ***Departmet f Pure ad Applied Mathematics, Jm Keyatta Uiversity f Agriculture ad Techlgy, Nairbi, KENYA. ksigey{at}yah{dt}cm Abstract The equatis gverig the flw csidered i the study are -liear ad therefre t btai their slutis, a efficiet fiite differece scheme has bee develped as utlied i chapter three. The mesh used i the prblem csidered i this wrk is divided uifrmly. The varius flw parameters were varied, e at a time while hldig the ther parameters cstat. This was repeated fr all the flw parameters ad the results preseted graphically. A theretical mdel was develped fr the advecti-dispersi prblem i e-dimesial prus media with tw csideratis: e, the dispersi cefficiet was directly prprtial t the seepage velcity ad the secd the flw was peridic. The prus dmai was hmgeeus, istrpic ad f adsrbig ature. A time depedet peridic pit surce was csidered at the surce budary. Differet budary cditis were csidered at utlet f the dmai. The resultig systems f liear algebraic equatis were slved usig Matlab sftware. The implicit Crak-Nicls scheme was used with the ccepts f stability tested ad the scheme was fud t be stable. raphical illustratis f ccetrati prfiles versus time ad psiti were preseted fr differet set f data ad the results discussed. The results preseted shw that flw ccurs frm regis f high hydraulic head t regis f lw hydraulic head util a steady head value is achieved. This agrees with Darcy s cclusi that hydraulic head decreases i the directi f flw. Keywrds Crak Nicls Numerical Scheme; Fiite Differece Methd; Fluctuatig rudwater; Hmgeeus; Istrpic; Prus Dmai; Seepage Velcity. Abbreviatis Crak Nicls Scheme (CNS); Fiite Differece Techique (FDT); eeralized Itegral Trasfrm Techique (ITT); Laplace Trasfrm Techique (LTT); Partial Differetial Equati (PDE); Particle Swam Optimizati (PSO); Represetative Elemetal Vlume (REV); Successive Over Relaxati (SOR); Three Dimesial (3D). I. INTRODUCTION IN this chapter, a backgrud f grudwater fluctuatis, defiitis f the mai terms used i the study, ad literature review related t the preset wrk is give. Obectives ad ustificatis f the study are als utlied... Backgrud f rudwater Fluctuatis rudwater cmprises 97 percet f the wrld s readily accessible freshwater ad prvides the rural, urba, idustrial ad irrigati water supply eeds f billi peple arud the wrld. Pressure grudwater is cstatly grwig sice the mre easily accessed surface water resurces are already beig used ad gettig exceeded by the demad. I the last few decades, this pressure has bee evidet thrugh rapidly icreasig pumpig f grudwater, accelerated by the availability f cheap drillig ad pumpig techlgies. [Albert et al., ]. The availability f grudwater varies ad fluctuates widely due t shrt-term ad lg-term chages i climate, withdrawal ad lad use. Precipitati ad evaprati have a peridic impact grudwater level fluctuati whereas sme atural factrs such as crustal mvemet ad tide ad huma factrs prvide sme radm chages. Systematic ad lg-term measuremets f water-level are essetial fr the evaluati f grudwater cditi fr sustaiable develpmet f the resurces ad fr the develpmet f grudwater mdels fr plicy aalyses as well as frecast treds [Ma K. Jha, ]. I grudwater flw mdelig, several schlars have de studies i gelgic settigs that vary i terms f rck frmatis, ladfrms, ad altitude, which have resulted i widely varyig grudwater bearig strata situatis i the wrld. I recet years therefre, there have bee several mdelig develpmet ad applicati t simulate the chages i grudwater depth uder bth ctiuus ad discrete cditis. Prblems which ivlve ISSN: Published by The Stadard Iteratial Jurals (The SIJ)

2 The SIJ Trasactis Cmputer Sciece Egieerig & its Applicatis (CSEA), Vl. 4, N. 3, July 6 the flw f water ad slutes separately r simultaeusly thrugh saturated/usaturated prus media have received much atteti i differet fields f sciece ad techlgy. I this regard, the advecti-dispersi equati which is applicable i may disciplies like chemical egieerig, biscieces, evirmetal scieces ad petrleum egieerig is a key equati i this study. The flw f water thrugh the prus media is a prblem which ca be described thrugh the use f the differetial equati kw as advecti-dispersi equati. The advecti-dispersi phemea ccur i may physical situatis icludig the trasfer f heat i fluids, flw thrugh prus media, the spread f ctamiats i fluids ad i chemical separati prcesses [Naafi & Haeezhad, ]. I this case study therefre, a theretical mdel has bee develped fr the advecti-dispersi prblem i e-dimesial hmgeeus fiite prus media with a peridic flw ad with the dispersi cefficiet beig directly prprtial t the seepage velcity. The results btaied frm the cstructed mdel will be helpful i the aalysis ad predicti f grudwater distributi i terms f recharge ad discharge, bth i explitati ad explrati reservirs... Literature Review rudwater mdels play a imprtat rle i the develpmet ad maagemet f grudwater resurces, ad i predictig effects f maagemet measures. Decisimakers therefre require adequate ifrmati these iteractis i rder t frmulate sustaiable water resurces develpmet strategies. This has see several mdelig develpmet ad applicati t simulate the chages i grudwater depth uder bth ctiuus ad discrete cditis. Ma K. Jha [] i his study predictig grudwater level usig Furier series itegrated with least Square estimati methd., demstrated that the methd culd be used t predict the grudwater level i the case f seasal-sesitive grudwater fluctuatis. It was bserved that the desiged methd was able t mdel the grudwatertable data, cllected at the Haga Ste Park stati i reesbr, Nrth Carlia, with a fair degree f accuracy with a testig mea square errr f.735. Tth [4] fr the first time used aalytical slutis t ivestigate grudwater flw i hypthetical small draiage basis. He fud theretically the existece f hierarchically ested grudwater flw systems: lcal, itermediate (subregial) ad regial. Tpgraphy, gelgy ad climate tured ut t be mar factrs fr the frmati f three subflw systems f gravity-drive flw i a hmgeus ad istrpic grudwater basi. Freeze & Withersp [8] were the first t use umerical mdels t simulate steady state regial flw patters i hypthetical layered aquifer systems. Their mdels were used t aalyze the effects f water table cfigurati ad hydraulic cductivity regial flw patters ad t quatify basi yields. A trasiet saturated umerical mdel was later develped by Freeze & Cherry [7] t ivestigate the relati betwee ifiltrati rates, water table rise ad base flw hydrgraph. The mdel was further used t predict maximum basi yield as a fucti f pumpig patter ad recharge ad discharge characteristics f a hypthetical basi. Ferris [6] derived a sluti fr the steady peridic respse f the piezmetric heads i a semi-ifiite aquifer subect t a siusidal variati f the stream water level. Field studies f the grudwater recharge/ discharge prcess i aquifers (especially i castal area) shwed that the tide culd imprtatly regulate the tempral ad spatial blueprit f grudwater recharge/discharge ad the salt ccetrati i the ear-shre grudwater. Kumar [8] i his study dispersi f pllutats i semi-ifiite prus media with usteady velcity distributi, csidered the flw velcity usteady/-uifrm i hmgeeus prus dmai. Aalytical slutis were develped fr the dispersi prblem i -adsrbig ad adsrbig, semi-ifiite prus media i which the flw was csidered e dimesial ad the average flw velcity usteady. Hamdy et al., [] derived a steady state aalytical sluti fr the grudwater flw equati i a hmgeeus ucfied aquifer whse sluti culd be used fr calculatig the hydraulic heads i a cmplex flw field caused by multi iecti-pumpig wells havig differet rates i a dmai subected t a uifrm recharge. The aalytical methd, used fr flw simulati, was cupled with the Particle Swarm Optimizati (PSO) methd, used fr ptimizati, t slve the grudwater maagemet prblem. Ra & Sarma [3] develped a aalytical sluti fr determiig a grudwater prfile resultig frm lcalized recharge t a fiite ucfied aquifer with mixed budary cditis. I their study, they used the exteded fiite Furier trasfrms ad the methd f images t arrive at the aalytical sluti. Their aalytical results were validated by experimetal results. Kizelbach & Ackerer [6] studied that i steady flw cditi, tempral variatis yield a larger trasverse dispersivity, i cmparis t lgitudial dispersivity. A tempral fluctuati i the recharge r i the budary cditis causes variatis i the grudwater velcity field that lead t dispersive mixig. de & Kikw [9], fud that hydraulic trasiets ca lead t ehaced lateral dispersi ad demstrated that directial variati i hydraulic gradiet is mre imprtat tha the magitude variati i the hydraulic gradiet. Webster & Taylr [5] bserved that grudwater flw directi rtates with time due t variati i pressure gradiet. They ivestigated the efficiecy ad character f rtatial dispersi usig Mte Carl simulatis f the dispersal f cluds f particles thrugh a highly idealized prus medium ad demstrated that rtatial dispersi behaves as a diffusive prcess ad that it culd be may times mre effective tha mlecular diffusi r shear dispersi as a trasprt mechaism. They further demstrated that passig waves ca greatly ehace slute trasfer betwee the bed ad the verlyig water. ISSN: Published by The Stadard Iteratial Jurals (The SIJ)

3 The SIJ Trasactis Cmputer Sciece Egieerig & its Applicatis (CSEA), Vl. 4, N. 3, July 6 Jeg et al., [4] derived aalytical slutis t describe the peridic grudwater flw ad demstrated that usteady seepage velcity which iflueces the raifall ifiltrati ad water level variati with seepage flw is based tw-phase flw ccept. Jaiswal et al., [], Kumar et al., [7] ad Yadav et al., [7], btaied aalytical slutis fr advecti diffusi equati with variable cefficiets, fr temprally ad spatially depedet dispersi prblems. Pérez et al., [], preseted a exact sluti f advecti diffusi equati with cstat cefficiets fr bth trasiet ad steady-state regimes ad slved aalytically usig the eeralized Itegral Trasfrm Techique (ITT). Che & Liu [] derived aalytical slutis fr edimesial advective-dispersive trasprt i fiite spatial dmai with three simple time-depedet ilet cditis icludig cstat, expetially decayig ad siusidally peridic iput fuctis ad demstrated the applicability f sluti. Parametric aalysis was perfrmed t illustrate the saliet behavir f slute trasprt resultig frm a peridic iput fucti. The geeralized sluti was evaluated by meas f the umerical itegrati ad the develped geeralized aalytical sluti was useful fr the develpmet f the aalytical slutis fr sme specified time-depedet iput fuctis r umerical evaluati f ccetrati distributi fr arbitrary time-depedet iput fuctis. Jaiswal et al., [3] develped a theretical mdel fr the advecti-dispersi prblem i e-dimesial prus media with the dispersi cefficiet directly prprtial t the seepage velcity. They studied the ifluece ccetrati prfiles due t differet budary cditis i the dmai. The derived sluti was als exteded t a semi-ifiite dmai. The Laplace Trasfrmati Techique (LTT) was used t get a umerical sluti where ew time variables were itrduced. The cause f the variatis i flw velcity is rmally assiged t variatis i hydraulic cductivity. T crrectly represet a dispersi mdel several key factrs shuld iclude cause variatis i velcity ad thse caused by a fluctuatig budary cditi. The purpse f this study is t derive umerical slutis t e-dimesial advectidispersi equati with peridic flw alg tw set f budary cditis resultig frm time depedet peridic iecti. I this study, therefre, a mathematical mdel is develped fr the advecti-dispersi prblem i edimesial prus media with peridic flw ad with the dispersi cefficiet beig directly prprtial t the seepage velcity usig the Fiite Differece Techique t btai a umerical sluti..3. Statemet f the Prblem The availability f grudwater varies ad fluctuates widely due t shrt-term ad lg-term chages i climate, withdrawal ad lad use. Water-level measuremet frm bservati wells prvides critical ifrmati regardig hydrlgic stresses actig aquifers ad hw these affect grudwater dyamics such as recharge, strage ad discharge. rudwater level fluctuati is ctiuus i ature but chages frequetly t discrete the iterrupti f huma activity. A ctiuus sequece ca be simulated by discrete data. There have bee several mdelig develpmet ad applicati t simulate the chages i grudwater depth uder bth ctiuus ad discrete cditis. I this study, a theretical mdel is develped that mdels fluctuatig grudwater flw i hmgeeus fiite prus dmai fr the advecti-dispersi prblem i edimesial prus media csiderig a flw that is peridic ad the dispersi cefficiet directly prprtial t the seepage velcity usig the Fiite Differece Techique..4. Justificati The results btaied frm the cstructed mdel will be useful i the aalysis ad predicti f grudwater distributi i terms f recharge ad discharge, bth i explitati ad explrati reservirs. Petrleum egieers will be able t use the results frm the mdel t aalyze ad predict the prblems f grudwater utflw it il-bearig strata. The mdel culd be used t mitr ad advise the pssibility f ptimal use f wells. I the cstructi f dams, determiati f flw patters f grudwater is vital i hydraulics. Such a mdel may be used t prvide the ecessary ifrmati. These results may be sigificat fr grudwater resurce maagemet ad ca be used t bechmark umerical mdels. The mdel will therefre make a sigificat ctributi t the existig kwledge f mathematical mdelig f grudwater flw..5. Obectives a) T determie the ifluece f varius flw parameters ccetrati prfiles f grudwater due t differet budary cditis i the dmai. b) T develp a exteded mathematical mdel fr the advecti-dispersi prblem i e-dimesial prus media with a peridic flw ad with the dispersi cefficiet beig directly prprtial t the seepage velcity. II. MATERIAL AND METHODS.. Assumptis The fllwig assumptis were made i rder t simplify the equatis gverig fluid flw i this study; a) The flw is peridic b) The dispersi cefficiet is directly prprtial t the seepage velcity c) The prus dmai is hmgeeus, istrpic ad f adsrbig ature d) N slip cditi hlds betwee fluid particles ad surfaces.. Mathematical Mdel f the Prblem Mathematical mdels are used i simulatig the cmpets f the cceptual mdel ad cmprise a equati r a set f ISSN: Published by The Stadard Iteratial Jurals (The SIJ)

4 The SIJ Trasactis Cmputer Sciece Egieerig & its Applicatis (CSEA), Vl. 4, N. 3, July 6 gverig equatis represetig the prcesses that ccur, fr example grudwater flw, slute trasprt etc. The differetial equatis are develped frm aalyzig grudwater flw (ad trasprt) ad are kw t gver the physics f flw (ad trasprt). The reliability f mdel predictis depeds hw well the mdel apprximates the actual atural situati. Assumptis are made i rder t cstruct a mdel, because the field situatis are usually t cmplicated t be simulated exactly. Mathematical mdels f grudwater flw ad slute trasprt ca be slved geerally with tw brad appraches: a) Aalytical sluti f the mathematical equati gives exact sluti t the prblem, i.e. the ukw variable is slved ctiuusly fr every pit i space (steadystate flw) ad time (trasiet flw).aalytical mdels are exact sluti t a specified, well simplified grudwater flw r trasprt equati. Because f the cmplexity f the 3D grudwater flw ad trasprt equatis, the simplicity iheret i aalytical mdel makes it impssible t accut fr variatis i field cditis that ccur with time ad space. Fr these prblems, (variatis i field cditis) such as chages i the rate/directi f grudwater flw, stresses, chages i hydraulic, chemical ad cmplex hydrgelgic budary cditis, the assumptis t be made t btai a aalytical sluti will t be realistic. Numerical sluti techiques are therefre used t slve mathematical mdels f this ature. b) Numerical sluti f the mathematical equati gives apprximate sluti t the prblem, i.e., the ukw variable is slved at discrete pits i space (steady-state flw) ad time (trasiet flw). Numerical mdels are able t slve the mre cmplex equatis f multidimesial grudwater flw ad slute trasprt. May umerical slutis t the advecti-dispersi equati have bee reprted. The mst ppular techiques are as the Fiite Differece Techique (FDT) ad Fiite Elemet Methd (FEM). I this study, the Fiite Differece Techique is used t slve the grudwater flw mdel..3. verig Equatis The equatis gverig grudwater flw is discussed i the sectis that fllws.3.. Advecti-Dispersi Equati Fr purpses f this study, the gverig equati fr a grudwater flw mdel represetig a mathematical descripti f the assumed trasprt mechaisms ad prcesses i ideal case which iclude the effect f adsrpti, i e dimesi may be writte as [Jaiswal et al., 3], c p F c D x, t u x, t c t t x t p (..) Where c is the slute ccetrati i the liquid phase ad F is the ccetrati i the slid phase. As is geerally kw, the mass trasprt equati uses hydrdyamic dispersi, which is the cmbiati f mechaical dispersi ad diffusi, hwever mlecular diffusi is egligible due t very lw seepage velcity. The advecti-diffusi Equati (..) has served as the mai theretical framewrk fr mdelig ad trasprt f slute i prus media ad fr addressig critical evirmetal issues r waste dispsal peratis durig the last few decades, Jury & Fluhler [5]. I equati (..), D ad u may be cstats r fuctis f time r space. Lapidus & Amuds [9], csidered tw cases, amely; F k c k ad (..) F k c k (..3) t respectively, equilibrium ad -equilibrium istherm betwee the ccetratis i the tw phases, where ad are empirical cstats f the medium. The istherm is liear if, ad is -liear if. Fr simplicity, the frmer relatiship is adpted i the preset aalysis. This assumpti is geerally valid whe the adsrpti prcess is fast i relati t the grud-water velcity, Cherry et al., [3]. Usig equati (..) i equati (..) fr =, we get liear advecti-diffusi equati [Yim & Mhse, 8], c k c k p c D x, t u x, t c t t x x Where p c c R D x, t u x, t c t x x R p p k (..4a) (..4b) (..4c) The term the left side f equati (..4b) idicates the retardati factr (R) ad chage f ccetrati i time, whereas the tw terms the right side describe hydrdyamic diffusi ad flw velcity respectively. If bth the parameters are idepedet t space variable ad time variable, the these are called cstat diffusi ad uifrm flw velcity respectively. Let us write, ad, [Jaiswal & Yadav, 5]. The the geeral liear advecti-diffusi partial differetial equati i e dimesi becmes c, c D f x t u f x, t c t x x Itrducig a ew idepedet variable X f dx x t, r dx (..5) (..6), d x f x t which is a case f temprally depedet dispersi alg a uifrm flw, i.e., fr ad f x t (..7) f x t f m t,, Where is a usteady cefficiet whse dimesi is iverse f the dimesi f, i.e. f dimesi. Thus is expressed as a -dimesial variable. The ISSN: Published by The Stadard Iteratial Jurals (The SIJ) 3

5 The SIJ Trasactis Cmputer Sciece Egieerig & its Applicatis (CSEA), Vl. 4, N. 3, July 6 fucti is chse such that, fr r. Thus equati (..5) will assume the frm c c c f ( m t ) D u t X X (..8) where, =iitial dispersi cefficiet, = seepage velcity cefficiet Itrducig ather ew time variable T helps t get rid f the time depedet cefficiet the left had side, hece [Crak, 4]; T t dt (..9) f ( m t ) Equati..8 reduces t c c c D u T X X Further usig trasfrmatis (..) Z (, ) (, ) exp( X 4 ad u u K Z T Z D D T ), (..) The advecti-diffusi equati (..) reduces t a diffusi equati i terms f a ew depedet variable, that is; III. K T K D Z (..) NUMERICAL METHODS AND STABILITY 3.. Fiite Differece Techique ANALYSIS The umerical methds ca be categrized as Fiite Differece, Fiite Elemet, Fiite Vlume ad Budary Elemet. At the preset time, i practice, the methd f Fiite Differece is e f the mst valuable methds f apprximatig (umerical) sluti f PDE. I this study, we use Fiite Differeces Techique (FDT) t slve a tw dimesi Advecti-Dispersi equati (..). Fr grudwater flw prblems, the FDT is the mst ppular apprach i slvig differetial equatis umerically sice it has firm theretical fudatis. Als, wig t the frm ad algebraic simplicity f the equatis resultig frm fiite differece apprximatis, develpmet f sluti algrithms is relatively easier. The FDT is based the apprximate substituti f derivatives by differece qutiets rmally btaied by Taylr s series. The methd btais a fiite system f liear r liear algebraic equatis frm the PDE by discretizig a give PDE ad cmig up with a umerical scheme aalgue t the equati, i this case the Advecti-Dispersi equati. Accrdig t Jai [], the resultig differece equati, that is, the umerical schemes, is the slved by iterative prcedures such as auss-jacbi, auss- Seidel, ad SOR. Fr the sluti f the Fiite Differece Equatis t be a reasably accurate apprximati t the sluti f the crrespdig PDE, the fllwig cditis must be satisfied: a) Cvergece: A fiite differece equati is cverget if the sluti f fiite differece equati appraches the exact sluti f the partial differetial equati as the mesh sizes appraches zer. b) Csistecy: Whe a trucati errr ges t zer, a fiite differece equati is said t be csistet r cmpatible with a partial differetial equati. c) Stability: A umerical prcess is said t be stable if it limits amplificati f all cmpets f the iitial cditis. I this study, the ccepts f stability ad cvergece are tested fr the Crak-Nicls umerical scheme develped. Matlab sftware is used t geerate the umerical sluti values. 3.. Discretizati f Advecti-Dispersi Equati I this secti, equati (..8), that is,, is discretized by replacig partial derivatives with their differece aalgues as fllws: 3... Numerical Scheme fr the Advecti-Dispersi Equati c x c t x c i, i, c c c i, i, x c i, i, t c c c ( x) i, i, i, Accrdig t Jaiswal & Yadav [5], we take; f m t D.7 5 u (3..) (3..) (3..3) (3..4) } Substitutig the equatis; (3..), (3..), (3..3) ad (3..4) it equati (..8) gives a implicit Crak Nicls scheme; [ ] (3..5) [ ] Multiplyig equati (3..5) by ad lettig, we get a umerical scheme; (3..6) 3... Stability Aalysis f the Advecti-Dispersi Equati Csiderig the Advecti-dispersi equati: ISSN: Published by The Stadard Iteratial Jurals (The SIJ) 4

6 The SIJ Trasactis Cmputer Sciece Egieerig & its Applicatis (CSEA), Vl. 4, N. 3, July 6 c c c f m t D U t x x (3..7) ad substitutig f(mt) =; D = D =.534 ad U = u =.864 where D(x,t) = D ad U(x,t) = u Let t = 6; m =.5 ad r =.5; we get the Crak Nicls scheme (3..8) Takig, ad varyig as i =,, 3, 4...N with N=8, we get a set f liear algebraic equatis; Writig the system i matrix frm we get, r c, r c, r r c N, r c N, r c c c,,, r c, r r c N, r c c c N,,, N N The abve system ca cmpactly be writte as; ( r ) I A c ( r ) I B E c d w h e r e;, A, B, E Where is the idetity matrix f rder (N-) (N-) ad is a cstat vectr c c d c c,, N, N, c r I A r I B E c d c c d r I B E r I A where is the amplificati matrix. Fr stability, the mdulus f the eigevalues f the amplificati matrix shuld be less tha r equal t uity. Accrdig t We-chyua Yueh [6], we use the frmula; r a a m c s N The eigevalue f A is give by:, where m=,, 3, (N-), N. m m m A () c s c s 4 si N N ( N ) m N The eigevalue f is ( ) c s The eigevalue f E is m m m () c s c s 4 si N N ( N ) The eigevalue f is =. Eigevalue f amplificati matrix is give by [( r ) B E ][( r ) A ] But r =.5 Therefre, m m ( r ) () ( ) [ 4 si ] ( r ) () [ 4 si ] ( N ) ( N ) m ( r ) () ( ) [ 4 si ] ( N ) m ( r. 534) () [ 4 si ] ( N ) a) Fr b) Fr m r s i ( N ) m r s i ( N ) m si, ( N ) is satisfied. m si, ( N ) which is always satisfied. r. 6 r. 6 r r which Hece it ca be see frm the results that the Crak- Nicls scheme is ucditially stable fr the cases (a) ad (b) abve. IV. 4.. Ccetrati Prfiles RESULTS AND DISCUSSION 4... Dispersi Cefficiet Directly Prprtial t Seepage Velcity The dispersi cefficiet is directly prprtial t seepage velcity ccept is used. The direct relatiship betwee dispersi cefficiet ad seepage velcity helps t cvert the time depedet cefficiets i the gverig equatis it cstat cefficiets. If we take r =.5, = ad vary i as i =,, 3, 4 ad substitute it the umerical scheme i equati (3..8) we get systems f algebraic equatis; ISSN: Published by The Stadard Iteratial Jurals (The SIJ) 5

7 Ccetrati C (x,t) Ccetrati C (x,t) The SIJ Trasactis Cmputer Sciece Egieerig & its Applicatis (CSEA), Vl. 4, N. 3, July C C.7 6 C C C,,,,,, C C.7 6 C C C, 3,,, 3,, C C.7 6 C C C 3, 4,, 3, 4,, C C.7 6 C C C 3, 4, 5, 3, 4, 5, C C.7 6 C C C 5, 6, 4, 5, 6, 4, C C.7 6 C C C 6, 7, 5, 6, 7, 5, C C.7 6 C C C 8, 6, 7, 8, 6, 7, C C.7 6 C C C 8, 9, 7, 8, 9, 7, C C.7 6 C C C 9,, 8, 9,, 8, C C.7 6 C C C,, 9,,, 9, Usig the iitial budary cditis C ( x, ) m g / L, ad x L, t C (, t ).5 m g / L, x, t, systems f algebraic equatis i matrix frm as; we write the abve C, C, C 3, C 4, C 5, C 6, C 7, C 8, C 9, C, Table 4.: Ccetrati Values fr Varyig Time (t) at Cstat ad Distace Time t=6 Time t=7 Time t=8 x= x= x= x= x= x= x= x= x= x= The data i the table 4. abve ca be preseted graphically i figure 4. belw; t=6 t=7 t=8 With, ad m.9, we slve the abve matrix equati usig matlab t get the ccetrati sluti values i table 4. belw at cstat ad. Table 4.: Ccetrati Values fr Varyig Usteady Parameter (m) at Cstat ad Usteady Usteady Usteady Distace parameter m=.5 parameter m=.7 parameter m=.9 x= x= x= x= x= x= x= x= x= x= The data i table 4. abve ca be preseted graphically i figure 4. belw; Distace x Figure 4.: raph f Ccetrati agaist Distace at Varyig Usteady Parameter (m) With, ad ad usig matlab sftware, the ccetrati sluti values are btaied i the table belw; m=.5 m=.7 m= Distace x Figure 4.: raph f Ccetrati agaist Distace at Varyig Time 4... Flw Csidered t be Peridic with Peridic Budary Cditi Frm the peridic iitial ad budary cditis:, [Jaiswal & Yadav, 5]: We get systems f algebraic equatis; C C.7 6 C. 9 C C,,,,,, C C.7 6 C. 9 C C, 3,,, 3,, C C.7 6 C. 9 C C 3, 4,, 3, 4,, C C.7 6 C. 9 C C 5, 3, 4, 5, 3, 4, C C.7 6 C. 9 C C 5, 6, 4, 5, 6, 4, C C.7 6 C. 9 C C 6, 7, 5, 6, 7, 5, C C.7 6 C. 9 C C 7, 8, 6, 7, 8, 6, C C.7 6 C. 9 C C 7, 8, 9, 7, 8, 9, C C.7 6 C. 9 C C 9,, 8, 9,, 8, C C.7 6 C. 9 C C,, 9,,, 9, This ca be writte i matrix frm as; C, C, C 3, C 4, C 5, C 6, C 7, C 8, C 9, C, With, ad, we slve the abve matrix equati usig matlab t get the ccetrati sluti values i table 4.3 belw at cstat ad. ISSN: Published by The Stadard Iteratial Jurals (The SIJ) 6

8 The SIJ Trasactis Cmputer Sciece Egieerig & its Applicatis (CSEA), Vl. 4, N. 3, July 6 Table 4.3: Ccetrati Values fr Varyig Usteady Parameter (m) agaist Time (t) Time m =.5 m =.7 m =.9 t= t= t= t= t= t= t= t= t= t= The data i table 4.3 abve ca be preseted graphically i figure 4.3 belw; Figure 4.3: raph f Ccetrati agaist Time at Varyig Usteady Parameters Frm table 3, if the usteady parameter is held cstat agaist varyig time, the graph btaied is as belw: 4.. Discussi Ccetrati distributi alg the flw was csidered. raphical illustratis f ccetrati prfiles versus time ad psiti were preseted fr differet set f data ad the results discussed. The figure 4. shws the effect f usteady parameter ccetrati prfile. At the iput budary,, the usteady parameter has a sigificat effect the ccetrati level. Iitially, as the usteady parameter m icreases frm thrugh t, the ccetrati level rapidly icreases i value frm.95, thrugh t respectively ad twards the utlet budary, ccetrati level decreases tedig t zer. The table 4. demstrates the ccetrati levels at varyig times t = 6, 7 ad 8. At the iput budary, fr the time t (day)=6, 7 ad 8, the ccetrati values are.95, ad respectively. Iitially, the ccetrati levels are varyig peridically with time ad ted t dimiish at the utput budary. This situati i figures 4. ad 4. is realistic i the sese that the retardati f ctamiat trasprt with respect t grudwater is caused by desrpti ad adsrpti f ctamiat cmpets the surface f slid particles. Ctamiat ccetrati decreases i time with trasprt distace due t ctrl f suspeded particles thrugh the filtrati prcess. The table 4.3 illustrates the peridic behaviur f ccetrati prfile i the time dmai at varyig usteady parameter m =.5,.7 ad.9. It is bserved that as time (t) icreases, ccetrati fluctuates siusidally. It is bserved frm figure 4.3 that grud water fluctuatis rise ad fall with differet amplitudes. rudwater ccetrati levels rise ear stream chaels due t bak strage durig fld perids, rise ad fall f sea level due t chagig barmetric pressure, chages i recharge, wids, pumpig iflueces ad earth tides. The areas uder the psitive ad egative sub-cycles are equal t each sub-cycle s mea grudwater levels ver the tidal perid. Durig the psitive sub-cycle, grudwater rises whereas durig the egative sub-cycle, the grudwater falls. Therefre, grudwater flw ad level shw a seasal variati f siusidal ature ver a give perid f time. I the ext chapter, the prcedure fllwed i btaiig slutis t the gverig equatis is reviewed. Cclusis frm the results btaied are als preseted ad recmmedatis fr further wrk are give. V. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS The bective f this study was t study the effect f varius flw parameters i fluctuatig grudwater flw i a hmgeeus fiite prus media. A aalysis f the effects f the varius parameters i fluctuatig grudwater flw has bee carried ut. The equatis gverig the flw csidered i the prblem are -liear ad therefre t btai their slutis, a efficiet fiite differece scheme has bee develped as utlied i chapter three. The varius flw parameters were varied, e at a time while hldig the ther parameters cstat. This was repeated fr all the flw parameters ad the results preseted graphically. It was bserved that grudwater flw ad level shw a seasal variati f siusidal ature ver a give perid f time due t bak strage durig fld perids, rise ad fall f sea level due t chagig barmetric pressure, chages i recharge, wids, pumpig iflueces ad earth tides. It was als bserved that flw ccurs frm regis f high hydraulic head t regis f lw hydraulic head dw the hydraulic gradiet util a steady head value is achieved which agrees with Darcy s cclusi that hydraulic head always decreases i the directi f flw. Further wrk culd iclude: T csider a peridic flw with the dispersi cefficiet beig iversely prprtial t the seepage velcity T use aalytical methds t btai slutis f a tw dimesial advecti-dispersi prblem fr grudwater flw. T determie fluctuatig grud water flw i a hetergeeus fiite prus dmai. REFERENCES [] M. Albert, L. Michael, K. Edward, T. Albert & H. Rafik (), Keya rudwater verace Case Study, Stakehlder Wrkshp, September,, Nairbi, Keya. ISSN: Published by The Stadard Iteratial Jurals (The SIJ) 7

9 The SIJ Trasactis Cmputer Sciece Egieerig & its Applicatis (CSEA), Vl. 4, N. 3, July 6 [] J.S. Che & C.W. Liu (), eeralized Aalytical Sluti fr Advecti-Dispersi Equati i Fiite Spatial Dmai with Arbitrary Time-Depedet Ilet Budary Cditi, Jural f Hydrlgy ad Earth System Scieces, Vl. 8, Pp [3] J.A. Cherry, R.W. illham & J.F. Barker (984), Ctamiats i rudwater-chemical Prcesses i rudwater Ctamiati, Washigt, D. C., Natial Academy Press, Pp [4] J. Crak (975), The Mathematics f Diffusi, Oxfrd Uiversity Press, Ld. [5] D.K. Jaiswal & R.R. Yadav (4), Ctamiat Diffusi alg Uifrm Flw Velcity with Pulse Type Iput Surces i Fiite Prus Medium, Iteratial Jural f Applied Mathematics, Electrics ad Cmputers (IJAMEC), Vl., N. 4, Pp. 9 5 [6] J.. Ferris (963), Cyclic Water-Level Fluctuatis as a Basis fr Determiig Aquifer Trasmissibility, I: Methds f Determiig Permeability, Trasmissibility ad Drawdw, Cmpiled by Betall Ray, U.S. elgical Survey, Water- Supply Paper, Pp [7] R.A. Freeze & J.A. Cherry (979), rudwater, Pretice Hall, Ic. [8] R.A. Freeze & P.A. Withersp (966), Theretical Aalysis f Regial rudwater Flw: Aalytical ad Numerical Slutis t the Mathematical Mdel, Water Resurces Research, Vl., N. 4, Pp [9] D.J. de & L.F. Kikw (99), Apparet Dispersi i Trasiet rudwater Flw Water Resurces Research, Vl. 6, Pp [] A. Hamdy, El-hadur & Ahmed Elsaid (3), rudwater Maagemet usig a New Cupled Mdel f Flw Aalytical Sluti ad Particle Swarm Optimizati, Iteratial Jural f Water Resurces ad Evirmetal Egieerig, Vl. 5, N., Pp.. [] M.K. Jai (984), Numerical Methds fr Scietists ad Egieerig Cmputati, New Delhi, Wiley Easter Ltd. [] D.K. Jaiswal, A. Kumar, N. Kumar & M.K. Sigh (), Slute Trasprt alg Temprally ad Spatially Depedet Flws thrugh Hriztal Semi-Ifiite Media: Dispersi Prprtial t Square f Velcity, Jural f Hydrlgic Egieerig (ASCE), Vl. 6, Pp [3] D.K. Jaiswal, R.R. Yadav & ulraa (3), Slute-Trasprt uder Fluctuatig rudwater Flw i Hmgeeus Fiite Prus Dmai, Jural f Hydrgelgy & Hydrlgic Egieerig, Vl., N., Pp. 7. [4] D.S. Jeg, L. Li & D.A. Barry (), Aalytical Sluti fr Tidal Prpagati i a Cupled Semi-Cfied/Phreatic Castal Aquifer, Advaces i Water Resurces, Vl. 5, Pp [5] W.A. Jury & H. Fluhler (99), Trasprt f Chemicals thrugh Sils: Mechaisms, Mdels, ad Field Applicatis, Advaces i Agrmy, Vl. 47, Pp. 4. [6] W. Kizelbach & P. Ackerer (986), Mdélisati du Trasprt de Ctamiat das u champ d éculemet N- Permaet, Hydrgelgy, Vl., Pp [7] A. Kumar, D.K. Jaiswal & N. Kumar (9), Aalytical Slutis f Oe-Dimesial Advecti-Diffusi Equati with Variable Cefficiets i a Fiite Dmai, Jural f Earth System Sciece, Vl. 8, Pp [8] N. Kumar (983), Dispersi f Pllutats i Semi-Ifiite Prus Media with Usteady Velcity Distributi, A Iteratial Jural f Hydrlgy Research, Vl. 4, N. 3, Pp [9] L. Lapidus & N.R. Amuds (95), Mathematics f Adsrpti i Beds, VI. The Effects f Lgitudial Diffusi i I-Exchage ad Chrmatgraphic Clums, Jural f Physical Chemistry, Vl. 56, Pp [] Ma K. Jha (4), Predictig rudwater Level usig Furier Series Itegrated with Least Square Estimati Methd, America Jural f Egieerig ad Applied Scieces, Vl. 7, N., Pp [] H.S. Naafi & H. Haiezhad (8), Slvig Oe Dimesial Advecti-Dispersi with Reacti usig sme Fiite Differece Methds, Applied Mathematics Scieces, Vl. 58, N., Pp [].J.S. Pérez, L.C.. Pimetel, T.H. Skaggs &.M. Va (9), Aalytical Sluti f the Advecti Diffusi Trasprt Equati usig a Chage-f-Variable ad Itegral Trasfrm Techique, Iteratial Jural f Heat ad Mass Trasfer, Vl. 5, Pp [3] N.H. Ra & P.B.S. Sarma (984), Recharge t Aquifers with Mixed Budaries, Jural f Hydrlgy, Vl. 74, Pp [4] J. Tth (963), A Theretical Aalysis f rudwater Flw i Small Draiage Basis, Jural f ephysical Research, Vl. 68, N. 6, Pp [5] I.T. Webster & J.H. Taylr (99), Rtatial Dispersi i Prus Media due t Fluctuatig Flws, Water Resurces Research, Vl. 8, Pp [6] We-chyua Yueh (5), Eigevalues f Several Triadiagal Matrices, Applied Mathematics E-tes, Pp , Mathematics Subect Classificati: Chia. [7] R.R. Yadav, D.K. Jaiswal, H.K. Yadav & ulraa (), Oe-Dimesial Temprally Depedet Advecti Dispersi Equati i Prus Media: Aalytical Sluti, Natural Resurce Mdelig, Vl. 3, Pp [8] C.S. Yim & M.F.N. Mhse (99), Simulati f Tidal Effects Ctamiat Trasprt i Prus Media, rudwater, Vl. 3, Pp Nyambae Zachary Nyakudi. Mr. Nyambae hlds a Bachelr f Educati degree i Mathematics ad Ecmics frm Mi Uiversity i 999, Keya. He is udertakig his fial study fr the requiremet f master f Sciece i Applied Mathematics frm Jm Keyatta Uiversity f Agriculture ad Techlgy, Keya. He is curretly teachig at Kisii Natial Plytechic i Kisii Cuty. He has kee iterest i udergrud fluid flw ad fluid dyamics. Mathew Ngugi Kiyaui. Prfessr Kiyaui Obtaied his MSc. i Applied Mathematics frm Keyatta Uiversity, Keya i 989 ad a PhD i Applied Mathematics frm Jm Keyatta Uiversity f Agriculture ad Techlgy (JKUAT), Keya i 998. Presetly he is wrkig as a prfessr f Mathematics at JKUAT. He has published ver fifty papers i iteratial Jurals. He has als guided may studets i Masters ad PhD curses. His Research area is i MHD ad Fluid Dyamics. Jhaa K. Sigey. Prf. Prfessr Jhaa Kibet Sigey Obtaied his MSc. I Applied Mathematics frm Keyatta Uiversity, Keya i 999 ad a PhD i Applied Mathematics frm Jm Keyatta Uiversity f Agriculture ad Techlgy (JKUAT), Keya i 5. Presetly he is wrkig as a directr f academic prgrammes at JKUAT Kisii Campus. He has published ver te papers i iteratial Jurals. He has als guided may studets i Masters ad PhD curses. His Research area is i Cmputatial Dyamics. ISSN: Published by The Stadard Iteratial Jurals (The SIJ) 8