2D MODELLING OF GROUNDWATER FLOW USING FINITE ELEMENT METHOD IN AN OBJECT-ORIENTED APPROACH

Transcription

1 IMW Symposum 7: Water Mg Evromets R. Cdu & F. Frau Eds 7t - 3st May 7 Caglar Italy D MODELLING OF GROUNDWTER FLOW USING FINITE ELEMENT METOD IN N OBJECT-ORIENTED PPROC bolgasem Kamar-Roua Departmet of Mg Geopyscs Sarood Uversty of Tecology P.O. Bo 36 Ira bstract fte elemet umercal soluto s preseted for groudwater models dscretsed by tragular elemet grd. obect-oreted approac s cosdered for mplemetg te models. Tus a Java program s developed for ts purpose to obta prmarly te uow ydraulc eads at te odes of tragular elemets provded boudary codtos are gve. By aalyss of te ydraulc ead soluto results oe ca easly obta te flow drecto vectors flow veloctes flow rates dfferet drectos. Itroducto For mg ecavatos as well as may oter egeerg operatos groudwater stuato te area uder operato sould be caractersed cosdered te desg of te operato. For ts modellg of groudwater flow s ofte ecessary to estmate te groudwater caracterstcs te area. Groudwater flow equato smple form correspods to te Laplace s equato wc represet a potetal problem. I ts paper a umercal soluto for te groudwater flow equato usg fte elemet metod s obtaed for a practcal case by mplemetg te model te obect-oreted programmg laguage Java. Formulato of te Problem ssumg a asotropc o-orotal layered sol represeted two dmesos of a Cartesa coordate system as sow Fgure we ca wrte Darcy s fltrato law te drectos ξ η see Fg. te followg forms: v - v - ξ ξ ξ η η η were v s te flow velocty s egt of water free surface or level peometrc ead s trasmssvty ydraulc coductvty. I matr otato equatos are wrtte as v ξ ξ ξ. v η η η Fgure. asotropc o-orotal layered sol D model. s computato as to be performed wt a global - coordate system a coordate trasformato as to be made from layer coordates to global oe. For ts purpose a rotato matr τ defed te followg s used for ts trasformato: cosθ - sθ τ 3 sθ cosθ were θ s te agle betwee orotal aes of te two coordate systems.e. aes ξ. Fally for te asotropc sol we obta te trasmssvty matr te global coordate system as:

2 . 4 Te matr s symmetrc. Tus t olds. Wrtg te Darcy s law te global - coordate system for ts case.e. v - - v combg tese equatos wt te equato for mass coservato cotuty equato gve by v v 6 we fally get te Posso s equato for groudwater flow as. 7 For a sotropc sol tus equato 7 cages to te Laplace s equato.e. or Δ 8 were Δ deotes te Laplace s operator. Ts formulato s for te stuato we te costat trasmssvty olds for te etre doma of soluto. For fte elemet solutos wc wll be dscussed ts paper t s geerally cosdered tat te trasmssvty may cage from elemet to elemet but t s costat wt eac dvdual elemet. Te more geeral approac s obtaed by assumg te trasmssvty beg a fucto of space. I ts case te Laplace s equato s wrtte te followg form: or Δ. 9 alytcal solutos to te Laplace s equato may be obtaed by troducg a potetal fucto φ te dervatves of wc gve te veloctes v v.e. v φ v φ wc are always ortogoal to te potetal fucto. lteratvely cosderg te mass coservato equato gve by equato 6 we ca use a specal fucto te stream fucto ψ defed te followg for te soluto of te above dfferetal equatos: v ψ ψ v. Streamles a set of stream fuctos ψ cost. gve wt te doma of soluto are also ortogoal to te equ-potetal fucto les. owever at ay pot of te stream fucto ψ cost. te flow velocty s tagetal to te fucto. Cosderg veloctes gve by equato we ca wrte equato 5 te followg matr form: ψ. ψ Ts matr equato sould be verted order to solve for te water ead potetal fucto.e. ψ. 3 ψ For te Laplace s equato case sotropc sol a potetal Π ests t s gve by te followg surface tegral equato:

3 dd Π. 4 Smlarly for te Posso s equato case asotropc sol te potetal s epressed by dd Π. 5 Fte Elemet Sceme for te Numercal Soluto of te Problem We cosder a typcal groudwater problem as llustrated Fgure. For modellg suc a system t sould be dscretsed boudary codtos sould also be prescrbed. For dscretsato of te system te fte elemet sceme we use a tragular grd as sow Fgure. Fgure. Te groudwater model dscretsed by tragular fte elemets. Te ydraulc ead at te odes of te tragular grd wt odes m elemets are to be determed by te fte elemet soluto preseted ere. Te water eads at te odes are represeted by... ;. Tey may be assembled wt te vector matr were [ ]... T. Wt te elemets te potetal fucto ca be epressed by a lear appromato fucto eutral area coordates. Te appromato ead fucto for eac tragular elemet represeted Fgure 3 ca also be defed learly.e. l l l 6 were l l deotes respectvely te eutral coordates ydraulc eads for te 3 odes represeted by estg eac tragular elemet see Fg. 3. s te potetal s based o te frst dervatves of te state ead fucto te dervatves of te appromato ead fucto gve by te followg epresso ave to be calculated: 7 wc ca be rected te followg form: 8 were s te area of te tragle s defed by te followg determat: det or det. 9 Smlarly te frst dervatves of te appromato ead fucto wt respect to s defed as IMW Symposum 7: Water Mg Evromets R. Cdu & F. Frau Eds 7t - 3st May 7 Caglar Italy

4 . Te trasformato from eutral coordates to Cartesa coordates wt a tragular elemet s gve by:. Te data are computed by. Fgure 3. Eac tragular elemet as tree odes arraged couter-clocwse drecto. Te flow veloctes v v or te flow rate wc s velocty multpled by tcess y of vertcal sol slce are computed usg te Darcy s law applyg te epressos for te frst dervatves of te ead fuctos gve by equato 7 or 8. Usg te Petrov-Galer metod for te wegted resdual approac te fte elemet tegral cosderg te trasmssvty beg costat also usg lear appromato ead fuctos eutral area coordates for eac tragular elemet we costruct te followg matr equato for ay tragular elemet wt te odes : or were s te coeffcet or trasmssvty matr tat s geerated o te elemet level. For te global system te trasmssvty matrces from all elemets ave to be assembled as a result te trasmssvty coeffcets for a ode belogg to two or more elemets are summed up to obta oe trasmssvty value for te correspodg etry of te coeffcet matr te global equato system. We ca wrte te global equato system geeral matr form K were te coeffcet or trasmssvty matr K s sparse symmetrc postve defte wll be of se f te dscretsed groudwater modellg doma cossts of odes. Te vector matr cludes uow ydraulc eads at te odes.e. [ ]... T. soluto for te uow ydraulc eads at te odes s obtaed by prescrbg boudary codtos. To obta te soluto te gve ydraulc eads at odes boudary codtos are multpled by te correspodg elemets of te coeffcet matr K te te results of tese multplcato are correspodgly subtracted from moved to te rgt sde of te equato system wc cotas te ow formato about te system. s a result we obta te o-omogeeous equato system R K were R s te vector matr cotag ow values. Fally by vertg ts matr equato system we obta te uow values ydraulc eads of vector matr.e. R K. s te matr K s sparse symmetrc we use a proper effcet fast solver for symmetrc equato systems suc as Precodtoed Cougate Gradet PCG metod or Colesy algortm to solve te equato system. obect-oreted programmg approac s cosdered for mplemetato of a model obtag te model soluto because of ts fleblty less tme demg for future eteso of te model. For ts a Java

5 IMW Symposum 7: Water Mg Evromets R. Cdu & F. Frau Eds 7t - 3st May 7 Caglar Italy program as bee developed to obta te water eads at te odes of tragular elemet dscretsed grd wt te groudwater modellg doma te fte elemet sceme. Numercal Eamples ltoug we ave mplemeted obtaed te results for dfferet models ere oly te results of a smplfed model sow Fgure 4 ave bee preseted. Te boudary codtos 4 m m are gve. lso we ave cosdered te edges of te elemets alog drectos to be equal to 5 m.e. Δ Δ 5 m. s dcated Fgure 5 te followg ydraulc eads at odes to 9 are obtaed by rug te fte elemet based Java program for te above smplfed model: m m m 3.38 m m 7.86 m m 4.88 m lso te flow drecto vectors ydraulc ead cotour les ave bee sow Fgure 5. s ca be see te water flows from g eads toward low eads a sem-crcular sape. Fgure 4. smplfed model wt odes umbered from to te tragular elemet grd. Fgure 5. Te flow drecto vectors ydraulc ead values at odes dcated by small sold tragles ydraulc ead cotour les for te smplfed model sow Fgure 4. Coclusos umercal soluto for te groudwater flow equato usg fte elemet sceme was obtaed. umber of models dscretsed by tragular elemet grd were cosdered ts wor a Java program was developed to obta prmarly uow ydraulc eads at te odes of tragular elemets. Based o te results te flow drecto vectors ydraulc ead cotour les were sow for a model. Refereces Baer.J. Pepper D.W. 99. Fte elemets. McGraw ll. elma R. 5. Effcet umercal metods formato-processg tecques for modellg ydro evrometal systems. Sprger-Verlag. Wag.F. derso M.P Itroducto to groudwater modelg Fte dfferece fte elemet metods. cademc Press.