Comparison of Explicit Finite Difference Model and Galerkin Finite Element Model for Simulation of Groundwater Flow
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- Ralf French
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1 Iraal Jural f Ivav Rsarc Advacd Egrg (IJIRAE Vlum Issu (Arl 4 Cmars f Elc F Dffrc Mdl ad Galr F Elm Mdl fr mula f Grudar Fl.H.Kular * A.K.Rasg Assca Prfssr Prfssr D. f Cvl& ar Maagm Egg.GGIE&add D. f Cvl Egg. II Bmba Absrac s ar dscrbs Galr f lm (FEFO mdls fr smula f grudar fl dmsal ras ucfd grudar fl ssms. s sud vlvs valda f FEFO mdl rrd aalcal slus ad als cmars f rrd Elc F Dffrc Mdl fr grudar fl smula (FDFO. mdl s furr usd ba sac ad m dsrbu f grudar ad fr rrd sc s cas. ffc f m s sz sac dscrzas umg ras s aalzd mdl rsuls. Krds Fl smula mdl Ucfd grudar fl ac - m dsrbu f grudar ad Valda f umrcal grudar fl mdl I. IRODUCIO umrcal mdls f grudar fl ar rrl ualzd vrs f a cml aufr ssm c armas fl m.. armas umrcal mdls ar ffcd rug s f assums rag gmr f dma as rgs ar smd u aur f rus mdum rrs f flud ad f fl rgm. cml aufr ssm s rad as a cuum c mls a flud ad sld mar varabls ar cuusl dfd a vr aufr dma. cuum s vd as a r f svral rrsav lmar vlums ac rrsg a r f r vlum f a aufr avrag flud ad sld rrs a vr ad assgd ds f surmsd grd usd fr dscrza f dma. umrcal mdls gral l aalzg mlcas f rsd cas rss f aufr ssms. ar advaagus a cval scal ad aalg mdls rsc f rssg racs f smulaus scal rcsss ad r flucs ac r b s f csuv rlass. s mdls als l sszg fld frma ad adlg larg amu f u daa rgal scal rblms [ag ad Adrs 98]. umrcal mdls ar ald r a rrv ss ga sg crllg aufr aramrs a s-scfc sg r grc ss frmula rgal rgular gudls ad ac as scrg ls df rgs suabl fr sm rsd ac suc as. arfcal rcarg. csu f a grudar fl mdl csss f gvrg ua f grudar fl c s bad b cmbg fl cu ua basd csrva f mass Darc s la basd csrva f mmum. fl ua s dscrbd usad r sad sa f fl rgm. Pdr ad Brdf (968 usd alrag drc mlc f dffrc scm slv grudar fl ua. s umrcal scm s mlmd fr -dmsal grudar fl rgus ad asrc cfd aufr. rsuls f mlma f s mdl ar cls agrm s aalcal slus. arasma ad rs (976 dvld a grad f dffrc mdl fr aalzg ras grudar fl varabl saurad rgus aufr. A cmur rgram RU s dvld aalz ras grudar m fr varabl saurad dfrmabl rgus ad muldmsal aufr ssm. rgram as b valdad agas aalcal slus rmal daa ad fld rsuls. Pdr ad Frd (97 dvld Galr f lm mdl smula -dmsal grudar fl a la aufr. aurac f mdl rsuls s cmard f dffrc rsuls ad s fud a b rsuls ar cls agrm ac r. ms al. (984 rsd a Galr f lm mdl usg Pcard ad -Ras algrms dsgd fr slu f -lar grudar fl ua. mdl uss b ragular ad rcagular f lms fr aufr dscrza. fluc ara cffc cu s usd sad f cval umrcal gra scm ba lm marcs. I s fud a s cu s sussful rduc f cmuaal cs. rg al. (4 dvld dscuus Galr md c uss a f-lm dscrza f grudar fl dma. r mdl usd a rla fuc f a arbrar rdr s fr ac lm f dma. dd f a rla fucs ac lm rms dscus rasmssv fl dma. s frmula s s b f g rdr aurac ad arcularl suabl fr aural calculag fl fld rus mda. mulas ar rsd rms f sramls a -dmsal aufr ad cmard slus bad frm sadard f-lm md ad a md flm md. umrcal smulas s a dscuus Galr arma s mr ffc a sadard f-lm md cmug flus ad sramls fr a gv aurac. I: 78-3 IJIRAE ://ra.cm 4 IJIRAE - All Rgs Rsrvd Pag - 3
2 Iraal Jural f Ivav Rsarc Advacd Egrg (IJIRAE Vlum Issu (Arl 4 Kular.H. al. ( dscussd valda f umrcal grudar fl smula mdls ad aalzd ffc f vara f aufr ad fl aramrs umrcal mdl slus. II. GROUDAER FO EUAIO gvrg ua f -dmsal rzal ad ras grudar fl mgus src ad ucfd aufr s gv as [ Illagasar ad Dll 989] ( r s scfc ld [dmslss]; s draulc ad avragd vr vrcal [ ]; s m [ ]; ad ar cms f rasmssv sr [ / ] c ar armad as K ad K rvdd cag ad ucfd aufr s glgbl as cmard s saurad css [Illagsar ad Dll 989]; K ad K ar cms f draulc cducv sr [ / ]; ad ar saal crdas [ ]; s 3 umg ra ( ad c ra ( a umg ad / r c ll [ / ]; s umbr f umg ad/r c lls dma; s umbr f ds dma dsrbud dscarg ad/r rcarg; ( s Drac dla fuc; ad ar Carsa crdas f rg [ ]; ad ar crdas f umg ad / r c ll [ ]; s dsrbud dscarg ra ( ad rcarg ra ( a ds dsrbud dscarg ad/r rcarg [ / ]. Eua ( s subc fllg al cd c s gv as ( ( ( r s al ad vr r fl dma [ ] ad s fl dma [ ]. Eua ( s subc Drcl f budar cd c s gv as ( ( ; (3 r s rscrbd ad vr aufr dma budar [ ]. uma budar cd zr grudar flu ca b gv as r ' (. ( ; b ( (4 b s scfd grudar flu acrss budar [ [ / ] ad s rmal u vcr uard drc. / ]; [ ] s grudar flu acrss budar III. FDFO MODE lc f dffrc md s mld FDFO mdl slv Eua (. I s mdl u dal ad a m lvl s lcl cmud frm fur gbrg ds ads a rvus m lvl. lc f dffrc scm s add bcaus s cmuaall ffc a alrag drc mlc f dffrc scm bu as rsrc f sz f m s usd smula. Hvr sabl f slus ca b surd b csrag lg f a m s. I s mdl r aufr dma s dscrzd rcagular cmuaal clls b surmsg ms crd f dffrc grd vr dma. cmuaal clls ar frmd arud rsc s f grd clum ad r ls c ar rfrrd as a ds. us ac d grd clum d ad grd r d rrss a cmuaal cll. sz f ac rcagular cmuaal cll s ad - ad - drcs rscvl. I FDFO mdl saal drvavs ad mral drvav Eua ( ar armad b cral f dffrc ad frard dffrc scms rscvl c ll rsul fllg ua ( ( ( ( ( r m s sz []; ad ar rvus ad m lvls [] ad s dal d. varus rms f Eua (3. ar rarragd suc a a a all dal ads suld aar sd f ua ad us rss fr dal ad a m lvl ca b gv as (5 I: 78-3 IJIRAE ://ra.cm 4 IJIRAE - All Rgs Rsrvd Pag - 4
3 Iraal Jural f Ivav Rsarc Advacd Egrg (IJIRAE Vlum Issu (Arl 4 I: 78-3 IJIRAE ://ra.cm 4 IJIRAE - All Rgs Rsrvd Pag - 5 (6 Eua (5 r fr all ds.. f aufr dma ll csu ssm f lar algbrac uas c s rssd mar frm as.... (7 us u dal ad vcr Eua (7 s slvd usg drc mar vrs cu avalabl MA- AB vrm. IV. FEFO MODE s mdl mls Galr f lm cu fr cmug ad dsrbu aufr. I s md ral slu f ad s subsud Eua ( c rsuls rsdual. B usg gg fucs as sa fucs gd rsdual s grad ad frcd zr ld ssm f lar uas. s f lar uas s slvd g dal ad dsrbu. grudar fl dma s dscrzd f umbr f ds usg a ragular f lm ms. Eac d f dma s dfd b a d ad f lm b a d. r ds f a f lm ar labld as ad r clcs r aclcs mar. m dma s dscrzd f umbr f dscr m ss. sz f ac m s s. ral slu f grudar ad b usd f lm frmula s gv as ( ( ( (8 r s ral slu f grudar ad [ ]; s al umbr f ds fl dma; s dal grudar ad a a m [ ]; s lar sa fuc a a aufr dma. sa fuc s dfd cs bu cuus mar vr r fl dma c rags frm. ral slu s subsud fr u dal ad Eua ( c rsuls rsdual f grudar ad c s rssd as ( (9 r s rsdual f grudar ad a a ad a m ; [ ]. rsdual f ad s gd ad grad vr r fl dma ba dal ad dsrbu. gral f gd ad rsdual s frcd zr ld ssm f algbrac uas ad sam s gv as d d ( r s gg fuc a a d. gral f Eua (7 s valuad usg Gr s fuc c rducs scd rdr drvavs rms frs rdr drvavs ad s rssd as d d d d d d d ( r d s gralzd varabl rrsg dsac alg budar [ ]. Eua (8 ca b rssd fllg frm b subsug fr ad gv as
4 Iraal Jural f Ivav Rsarc Advacd Egrg (IJIRAE Vlum Issu (Arl 4 I: 78-3 IJIRAE ://ra.cm 4 IJIRAE - All Rgs Rsrvd Pag - 6 d dd dd dd ( Eua (9 ca b r rssd fr dvdual lm as M M dd dd M M d dd (3 Alg umrcal gra fr varus rms f Eua ( fllg ssm f lar uas s bad ad sam ca b r as f B P P G (4 r G s glbal cducac mar c s frmd b assmblg lmal cducac marcs G a ca b rssd as A d d G 4 (5 P s glbal srag mar c s assmbld frm lmal srag marcs P a ca b gv as A d d d d P (6 B s glbal lad vcr c s assmbld frm lmal lad vcrs B a ca b gv as 3 3 A d d B (7 f s glbal budar flu vcr c s assmbld frm lmal budar flu vcrs f a ca b gv as d b d b d f b b (8 Frm ad dsrbu a rvus m lvl u ad dsrbu a m lvl s bad b rcursvl slvg s f algbrac uas gv Eua (. V. REU AD DICUIO cs s Cas s amd a valdag FEFO mdl. valda f fl mdls s amlsd b cmarg mdl smulas rrd aalcal slus [Illagasara ad Dll 989]. s s cas abls vsgas f rss f aufr varg dscarg ras frm ar f umg lls. Furr ssv f FDFO ad FEFO slus vara aufr aramrs suc as rasmssv scfc ld ad m s sz s aalzd. Fr s Cas a rcagular mgus ad src ucfd aufr s cs as s Fgur. rcagular aufr s slcd mal sasf sa csras msd fr aalcal slu. aufr ssm s
5 Iraal Jural f Ivav Rsarc Advacd Egrg (IJIRAE Vlum Issu (Arl 4 3 m lg ad 8 m d. ad a ad bm sds f aufr budar s csdrd av csa valu f m rugu smula ad lf ad rg sds f aufr budar s csdrd av zr grudar flu. umg lls ar lacd a a lca f (4 m 4 m ad (8 m 4 m frm rg as s Fgur. Iall grudar ad s assumd sac a valu f m a all ds aufr dma. ar abl drad causd b umg ar bsrvd a a bsrva ll suad a a lca f ( m m frm rg. aufr aramrs usd smula clud aufr css ( b =3 m csa umg ras fr umg lls ( & = 4.85 & m3/d ffcv rs ( =.3 aufr rasmssv ( =885.7 m/d ad scfc ld ( =.5 rscvl. Fr FDFO mdl aufr s dscrzd usg ms crd f dffrc grd c rsuls 55 cmuaal clls ufrm dal sacg f m b - ad - drcs as s Fgur. Fr FEFO mdl aufr s dscrzd usg ragular f lm ms 448 lms. sz f suar f dffrc cll ad sscls ragular lm s m. al ds Drcl budar cd ad uma budar cds ar 3 ad 8 rscvl. 3 s Cas 5 Drcl Budar uma Budar Erac ll Obsrva ll Z Grd s Dsac (m 5 5 Fgur cmac f aufr mdld s Cas fr smula f -dmsal ras grudar fl u cfd aufr udr umg ll cds Valda f FDFO ad FEFO mdls: Dsac (m ar abl drad valus a a bsrva ll du umg b ar f lls fr das ar cmud b FDFO ad FEFO mdls. m- drad curvs bad b FDFO ad FEFO mdls ad rrd aalcal slu [Illagasar ad Dll 989] ar cmard as s Fgur. Fgur Valda f FDFO ad FEFO mdls fr s Cas drad valus ar cmud as.38 ad.39 m b FDFO ad FEFO mdls rscvl c ar cls agrm rrd drad f.4 m b aalcal slu [Illagasar ad Dll 989] mlg vald f dvld fl smula mdls. m s sz s a as 7 das m Cura umbr crr fr FDFO mdl. I s als s frm cmars f drad curvs a al sags f smula b umrcal ad aalcal slus ar alms sam bu lar sags f smula FDFO slus dva arcabl frm I: 78-3 IJIRAE ://ra.cm 4 IJIRAE - All Rgs Rsrvd Pag - 7
6 Iraal Jural f Ivav Rsarc Advacd Egrg (IJIRAE Vlum Issu (Arl 4 aalcal slu. rcags f dffrc b FDFO ad FEFO ad aalcal slus ar 9.5 % ad 7.% rscvl. s dffrc s aabl grudar fl mdlg. usg a ms-crd f dffrc grd. dal sacg - ad - drc s ufrm as 5 m. al ds f dffrc grd ar 55 l al budar ds ar 6. Cmars f ad dsrbu b FDFO ad FEFO mdls: ad dsrbu bad b FDFO mdl cmars favrabl ad dsrbu bad b FEFO mdl. I s bsrvd frm Fgur 3 a curs f ad 99.6 m ad bad b b mdls ar affcd b budar ffcs vr cur f 99.7 m ad as s sm umrcal dsrs du rsc f mrvus budar lf ad rg sds f aufr. FEFO smulas causd grar dva f 99.7 m ad cur bcaus f umrcal rrr arma f budar flu rms b lar rla fucs. 5 Pumg ll FDFO FEFO 99.7 Dsac (m Dsac (m Fgur 3 Cmars f grudar ad dsrbus b FDFO ad FEFO mdls udr umg ll cds fr s Cas Effc f rasmssv ar abl drad: Fgur 4 ss ffc f varg magud f rasmssv ar abl drad cmud b FDFO ad FEFO mdls. drad valus cmud b FDFO mdl ar ad.37 m fr rasmssvs f ad 7.4 m /d rscvl. Fgur 4 Effc f rasmssv ar abl drad fr s Cas crrsdg valus f ar abl drad bad b FEFO mdl ar ad.38 m rscvl. us s frrd frm rsuls a 5% dcras magud f rasmssv causs drad b dubld ad 5 % cras magud f rasmssv causs drad g lrd b 5 %. s rsuls ar cd ls as ll-drad ua rasmssv dmar las mr mra rl a a fuc f lga- I: 78-3 IJIRAE ://ra.cm 4 IJIRAE - All Rgs Rsrvd Pag - 8
7 Iraal Jural f Ivav Rsarc Advacd Egrg (IJIRAE Vlum Issu (Arl 4 rmc rasmssv Jacb s ua s.5 l 4 r s r a ll fuc rasmssv s ua s ( u 4 r r aramr ll fuc s u r (u ll fuc s ad r s radus f fluc. 4 Effc f scfc ld ar abl drad: Fgur 5 ss cmars f ar abl drad curvs bad b FDFO ad FEFO mdls ffc f vara scfc ld magud. scfc ld s vard frm bas valu f.5 4% lss f a valu...9 ad % mr f a valu...3. Crrsdg s valus drad cmud b FDFO ar ad.469 m r as fr sam rag f scfc ld vara drad cmud b FEFO ar ad.48m. I as rfr b frrd frm rsuls a cag scfc ld as sgfca ffc drad. Bu as s vd frm rsul aalss a cras scfc ld rsuls rduc drad as cmard cras drad causd b dcras scfc ld. s s du fac a ul rasmssv scfc ld rm s l a aramr f ll fuc ad ds s drc rral drad a ll rss ua. Fgur 5 Effc f scfc ld ar abl drad fr s Cas Effc f umg ra ar abl drad: fluc f cag ll umg ras vr ar abl drad s s Fgur 6. Fgur 6 Effc f umg ra ar abl drad fr s Cas rag f vara umg ras f ll ad s csdrd as + 5% ad -5% f all cs umg ras. I ca b s frm Fgur 6 a fr 5% dcras gv umg ra as causs dcras drad b 5 % ad 5% b FDFO ad FEFO mdls rscvl ras 5% cras gv umg ra causs cras drad b % ad 49.89% b FDFO ad FEFO mdls rscvl. I s furr cd a ra f I: 78-3 IJIRAE ://ra.cm 4 IJIRAE - All Rgs Rsrvd Pag - 9
8 Iraal Jural f Ivav Rsarc Advacd Egrg (IJIRAE Vlum Issu (Arl 4 cag drad s mamum ll 5 das afr sar f umg r afr drad ar fud b sablzd us grudar fl rgm s aracg sad sa. s rsuls ar cd ls frm raal csdras. Effc f Cura umbr ar abl dra d : fluc f cag ll umg ras vr ar abl drad s s Fgur 7. m s szs ar a as.5 ad da rscvl. crrsdg Cura umbrs ar calculad as.9 ad.5 rscvl c ar ll bl rmssbl valu f u. FDFO mdl cmus drad f.388 ad.378 m fr m s szs f.5 ad da rscvl afr 5 das f umg ras FEFO mdl cmus drad f.39 ad.385 m fr fr crrsdg valus f m s sz. us s frrd frm rsuls a v 5% cras m s sz as glgbl ffc drad. sz f m s grar a. das rsuls usabl slus du vla f sabl crra. Hc fr gv rasmssv scfc ld ad aufr dscrza m s sz f da s rcmmdd ar abl Drad (m FDFO.5.9 FEFO Pumg m (Das Fgur 7 Effc f Cura umbr ar abl drad fr s Cas VI. COCUIO fllg ar luss f rs sud: Valda f FDFO ad FEFO mdl fr cs s Cas ss a r s a cls agrm b cmud ad aalcal slus al sags f umg vr afr das f umg dffrc b drad bad b FDFO ad FEFO mdl ad aalcal slu ar % ad 7% rscvl. us FEFO mdl as rfrmd br a FDFO mdl. dva f FEFO cmud slus frm aalcal slus lar sags f smula ma b arbud lr rdr rla fuc usd FEFO mdl. Fr cs s Cas FDFO ad FEFO mdl slus ar fud b sabl fr Cura umbr f.4 fr cs m s f da.fdfo ad FEFO slus ar mr ssv umg ra ad mdral ssv rasmssv f aufr ras ffc f scfc ld s fud b glgbl vr umrcal slus. VII. REFERECE [] Pdr G. F. ad Brdf J. D. 968 Alca f dgal cmur fr valua f aufr ar Rsurcs Rsarc [] Pdr G. F. ad Frd E. O. 97 Alca f Galr s rcdur aufr aalss. ar Rsurcs Rsarc 8(:8-. [3] arasma.. ad rs P. A. 976 A grad f dffrc md fr aalzg flud fl rus mda ar Rsurcs Rsarc (: [4] ag H. F. ad Adrs M. P. 98 Irduc Grudar Mdlg. H. Frma ad Cma Yr. [5] ms B. M cus fr mag f lms cmv mdlg fl saurad rus mda. ar Rsurcs Rsarc: [6] Illagasar. H. ad Dll P. 989 A dscr rl md f caracrscs mdl f slu rasr ar abl aufrs ar Rsurcs Rsarc 5(5: [7] rg F. al. 4 Alca f dscuus scral Galr md grudar fl Advacs ar Rsurcs 7: 9-4. [8] Kular.H. ad Rasg A.K. umrcal Erms lu rasr Grudar Fl sm Iraal Jural f Emrgg clg ad Advacd Egrg (8: I: 78-3 IJIRAE ://ra.cm 4 IJIRAE - All Rgs Rsrvd Pag - 3
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