Finite Element Discretization and Simulation of Groundwater FlowSystem
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1 OSR Jol of Mtemtics OSR-JM e-ssn: Volme 5 sse 6 M. - Ap. 13 PP iosjols.og Fiite Elemet Discetiztio Simltio of Gote FloSstem K. R. Aeboe 1 M. D. Se A. Ns 3 13 Deptmet of Mtemtics/Sttistics Feel Uiesit of Tecolog Mi Nigei Abstct:Tis ppe cosiee to imesiol flo i cooite sstem eie No- omogeos Lplce Eqtio fo gote flo sig Dc s eqtio ic fom te bsis fo te fiite elemet iscetiztio pocess. B iscetizig te lic Coctiit Piezometic e i te iectio flo iectio ecto of te gote sstem is obtie. Keos Fiite elemet meto lic coctiit Gote flo Piezometic e Lplce eqtio. toctio Gote is te locte beet te go sfce. A it of oc o seimets is clle qife e it c iel sble qtit of te [1]. Te ept t ic soil poe spces ois i oc become completel stte it te is clle te te tble []. Gote is itl esoce i o eiomet.t epleises o stems ies bitts[3]. oee gote is igl leble to cotmitio fom septic ts gicltl off ig pipe les spills istil ctiities impope isposl pctices [4]. [5] Km-Roi stesse tt i miig ectio s ell s m egieeig opetios gote sittio i e e opetio sol be ccteize cosiee i te esig of te opetio. Epeiece s so tt most impott ogs of te m bo e ot see otsie. Fo emple bloo te et tt pmps it e ot isible et te e e impott [6].. Moel Fomltio We emplote meto of spce iscetiztio of te gote flo ee te compttiol omi is iie ito fiite elemet mes coecte t oes i j.o goeig eqtio is te Dc s eqtio gie b [7] ts: Q i i j A j.1 Wee; Q = Flo te A = Flo Coss Sectio Ae = Piezometic e = lic Coctiit. Accoig [8] Q A. Sbstittig eqtio. ito.1 e e A i j A j.3 Wee = Flo elocit Tt is i j j.4 We o epess eqtio.4 i lee cooite sstem i te iectios..iosjols.og 54 Pge
2 Fiite Elemet Discetiztio Simltio of Gote FloSstem.iosjols.og 55 Pge.5.6 Wee; = Flo Velocit =Piezometic e =lic Coctiit. mti ottio eqtios.5.6 c be epesse s:.7 Eqtios.5.6 epesse i te Globl cooite sstem tes te folloig fom:.8.9 Combiig eqtios.8.9 it te folloig eqtio gie b [8].1 eobti te eqtio fo gote flo s.11 Fo te Aqife e cosietio isotopic soil = =.1 = =.13 Eqtio.11 eces to:.14 Eqtio.1 is eqilet to: 15 Wee =Potetil e Vle. Eqtio.15 is No-omogeeos Lplce's eqtio fo gote flo. We o compte te soltio fo te lic es t te oes of te tigl elemet of te omi e cosietio. lie it eqtio.15 e cosie te eqtio; = i.16
3 Fiite Elemet Discetiztio Simltio of Gote FloSstem.iosjols.og 56 Pge R i.17 Wee =Domi e cosietio =Potetil e les t te Noes R = Resil fctio = =Piezometic e. We itoce bit fctio i.18 Wee =eigtig fctio. Eqtio 3.17 tes te fom;.19 Afte itegtio b pts se of iegece teoem eqtio.19 is itte s 1. Te pojectio of te giet of i te iectio of te ot it oml is gie b: 1..1 lie it eqtio.1 eqtio. c be itte s. Fo omogeos bo sstem Domi e cosietio eqtio. c be epesse opetioll s follos: B.3 Wee } { 1.4 } { = Sobole Spce B.6.7 We e tt U.8 W.9 lie it eqtio.8.9 eqtio.3 c be itte s;
4 Fiite Elemet Discetiztio Simltio of Gote FloSstem B.3 Wee ol elemets of U W espectiel so tt;.31 N 1 N 1 i j i j.3 Wee 1... e bsis fctios i te iectio N. Sbstittig s efie i eqtio.31.3 ito eqtio. it e: N N 1 N J [ ] J 1 J 1 J 1.33 Eqtio.33 c be itte s; N N K J J R 1 J 1.34 Wee J 1 K J [ ] J.35 R.36 Fo bit it follos tt N J 1 K J J R 1... N.iosjols.og o e.37 Te folloig fiite elemet mti eqtio gie b eqtio.38 ols fo tigl elemet it oes i j to Globl eqtio sstem;.38 A soltio fo te lic es t te oes is compte b: Wee K R K R.39 K K K.4 1 K R.41 1 K R R = Vecto Mti cotis o lic es t te Noes 57 Pge
5 Fiite Elemet Discetiztio Simltio of Gote FloSstem R =Vecto Mti cotiig Potetil e les t te Noes K =Vecto Mti cotiig lic Coctiit les t te Noes ee i j i j i j i j i i.43 j i j = lic e t Noe 1 = lic e t Noe = lic e t Noe 3 = lic e t Noe 4 = lic e t Noe 5 = lic e t Noe 6 = lic e t Noe 7 = lic e t Noe 8 = lic e t Noe 9 = lic Coctiit t Noe 1 = lic Coctiit t Noe = lic Coctiit t Noe 3 = lic Coctiit t Noe 4 = lic Coctiit t Noe 5 = lic Coctiit t Noe 6 = lic Coctiit t Noe 7 = lic Coctiit t Noe 8 = lic Coctiit t Noe 9 = Potetil e Vle t Noe 1 = Potetil e Vle t Noe = Potetil e Vle t Noe 3 = Potetil e Vle t Noe 4 = Potetil e Vle t Noe 5 i j i j 1 i j i j j i j i = Potetil e Vle t Noe 6 = Potetil e Vle t Noe 7 = Potetil e Vle t Noe 8 = Potetil e Vle t Noe 9 j i j i j i j i j i j.iosjols.og Pge
6 Fiite Elemet Discetiztio Simltio of Gote FloSstem ee i K 1 j 3.44 i j i = j i j i j i j i i i i j i j j j j i i i i j j j j i j. ResltsA Discssios 3.1 lic Coctiit fo Nol Elemet Tble 3.1lic Coctiit t Noes 1-9 Nol Elemet lic Coctiit m Soce: [9] Tble 3.1 gies te les fo lic Coctiities of te coespoig les fo Nol Elemets fo te clcltio of lic e. 3. Potetil e Vles fo Nol Elemet Tble 3. Potetil e Vles t Noes 1-9 Nol Elemet Potetil e Vlem Soce: [9] Tble 3. gies te les fo Potetil e of te coespoig Nol Elemet fo te clcltio of lic e..iosjols.og 59 Pge
7 3.3 lic e Vles fo Nol Elemet Fiite Elemet Discetiztio Simltio of Gote FloSstem Tble 3.3 lic e Vles t Noes 1-9 Nol Elemet lic em Tble 3.3 gies te eslt obtie fo lic e sig Tble i eqtio Flo Diectio Vecto Fige 4.1: Flo Diectio Vectos fo lic e Vles Fige 3.1 epesets te Flo Diectio Vectos fo lic e Vles obtie i Tble 3.3. Te os gie te Recge Discge Diectios of te Gote Flo. Tble 3.1 gies te les fo lic Coctiities of te coespoig les fo Nol Elemets.Tble 3. gies te les fo Potetil e of te coespoig Nol Elemet fo te clcltio of lic e. Tble 3.3 gies te eslt obtie fo lic e sig Tble it eqtio Te Flo Diectio Vectos fo lic e Vles obtie i Tble 3.3 is pesete i Fige 3.1. Te os gie te Recge Discge Diectios of te Gote Flo t ec poit of te lic e les; 4.m m 7.861m 35.17m 3.138m 6.345m m 6.483m 4.88m..iosjols.og 6 Pge
8 Fiite Elemet Discetiztio Simltio of Gote FloSstem V. Coclsio We obsee tt t lic e les betee 4.m 4.8m simlte b te iectio ectos so te beo of gote flo s so i figes 3.1. Tis mes tt te te flos fom ig es tos te lo es i semi-cicl spe. Refeeces [1] S. Jib R. K. Somse K. L. Ps K. Si estigtio of emetls i cstllie qife gote fom Diffeet lles of Bgloe Kt Jol of Geogp Regiol Plig [] J. Sime M. T. V Gecte Cotmit tspot i te stte zoe: Teo moelig i J. W. Delle E. Teboo of gote egieeig e. Ne Yo: CRC Pess 6. [3] J. Sime Moels of te flo solte tspot i te stte zoe i M. G. Aeso J. J. McDoell Es. Ecclopei of ologicl scieces 78 Ciceste Egl: Jo Wile & Sos Lt [4] J. W. Delle Elemet gote flo tspot pocessesi J. W. Delle E. Te boo of gote egieeig e. 3 Ne Yo: CRC Pess 6. [5] A. Km-Roi D Moellig of gote flo sig fiite elemet meto i object-oiete ppoc i R. Ci& F. F Es. MWA smposim 7: Wte i miig eiomets Cgl tl 7. [6] P.. Olseie Te gotes of Nigei: A soltio to sstible tiol te ees gl Lecte Seies 17 Feel Uiesit of Tecolog Mi Nigei 1. [7] C. P. Km Gote Flo Moels i N. C. Gos& K. D. Sm Es. Gote moelig mgemet Ne Deli: Cpitl Pblisig Comp [8] T. J. Dbi Gote flo tspot moel Repot o Sesie gote bsi Motee Cot Clifoi Timot J. Dbic. Cosltig ologists CA [9] A. is-n ogeopsicl ogeocemicl ccteiztio of te Bi Bsi qife sstem octol tesis FeelUiesit of Tecolog Mi Nigei 1..iosjols.og 61 Pge
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