Assessment and Application of 3D Galerkin Finite Volume Explicit Solver for Computation of Seepage and Uplift in Dam Foundation

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1 INTERNATIONAL JOURNAL OF MATHEMATICAL MODEL AND METHOD IN APPLIED CIENCE Aeet a Applcato of D Galer Fte Volue Eplct olver for Coputato of eepage a Uplft Da Fouato.R. abbag-a, N.E. Matora, a B. Baat Abtract I t paper, evelopet of a Galer fte volue tree-eoal eepage olver o e of tetraeral ecrbe. Te uercal aaler utle for olvg te eepage porou ea a uplft uer gravt a wt uptrea cut off wall. Te reult of uercal olver ter of uplft preure atural fouato of a gravt a wt uptrea cut off wall are copare wt aaltcal oluto obtae b applcato of coforal appg tecque for a cotat ut rato of fouato ept over alf of a bae ( T b. Te accurac of te reult copute uplft preure for oogeeou a otropc coto preet acceptable agreeet wt te aaltcal oluto for varou rato of cut off wall ept over alf of a bae (/b. Havg aee te accurac of te oel, t apple to evaluate te qualt of te reult of te coo eprcal relato for uplft preure etato. I orer to preet te ablt of te verfe oel to cop wt real worl proble, t apple to olve eepage troug a atural porou fouato of a gravt a wt tree cle laer wt fferet coeffcet of pereablt. Kewor Galer Fte Volue Meto, eepage a Uplft, Multlaer Da Fouato T I. INTRODUCTION HE proble of eepage flow uereat of gravt a ca be forulate ter of a o-lear partal fferetal equato. Te equato ecrbe a cotat et flu flow a eterogeeou a otropc porou ea [. Altoug eprcal forulato are uggete for ple cae, ue to te eretl cople bouar coto a trcate pcal geoetre a practcal proble, a aaltcal oluto ot poble for coplcate a fouato [. Ma reearcer ave wore o eepage proble aal a olvg t goverg equato. Tee wor Maucrpt receve Marc, 7: Reve receve October, 7 aee-rea abbag-a Aocate Profeor Cvl Egeerg Departet of K.N. Too Uvert of Tecolog, Valar t. Tera, IRAN (poe: +9-5-; fa: ; e-al: a@tu.ac.r. No E. Matora, Profeor of Mltar Ittute of Uvert Eucato (AEI Hellec Naval Acae, Tera Catraou 59,Praue, GREECE (e-al: ator@wea.org. Baba Baat PD Caate of Cvl Egeerg Departet of Ar Kabr Uvert of Tecolog, Hafe t. Tera, IRAN (e-al: bbaat@aut.ac.r. Iue, Volue, 7 5 ffer b te tecque ue to olve te proble. Je et al. ( preete a fte fferece eto (FDM bae o bouar-ftte coorate (BFC traforato. Te curvlear gr te, wt coputatoal bouar beg cocet wt te pcal bouar, uercall obtae b olvg te Poo equato. eepage aal ca te be oe b FDM a ufor trafore ortogoal coorate te [. eraf et al. (95 propoe a fte fferece eto to tu treeeoal eepage aotropc eterogeeou fouato epecall for eart a [. Kou ( preete a leat-quare pleetato of te fte-eleet eto to evaluate trea fucto te oluto of fel proble. Te eto prograatcall lar to te oluto of te Laplace equato, a bae o te evelopet of a trea fel tat ortogoal to a alrea calculate potetal fel [5. Grfft a Feto (997 brougt togeter rao fel geerato a fte eleet tecque to oel tea eepage troug a treeeoal ol oa wc te pereablt rao trbute pace [. Grfft a Feto (99 alo cobe rao fel teor a fte eleet tecque wt Mote-Carlo ulato to tu te tattc of et graet precto a a fucto of ol pereablt varace a patal correlato [7. Boufael et al. (999 vetgate tea eepage fro two-eoal oa ug a eole forulato for varabl aturate ea [. L et al. ( preete a Eleet-free eto (EFM for eepage aal wt a free urface bae o te ovg leat quare eto wc ee ol te forato at oe. It avo troubleoe ofcato of te e a te fte eleet eto [9. Plar (99 tue uplft preure effect crace cocrete gravt a. A paraetrc tu o te fluece of uplft preure o tre tet factor a cracpropagato agle perfore. Dewe et al. (99 revewe a copare te proceure for clug uplft preure te a-calculato eto. Te propoe tree eparate oel for clug uplft preure a fte eleet aal [. T paper preet a Galer fte volue eto for oelg water flow a aturate oogeeou porou ea wt cople bouar te. Te oluto oa

2 INTERNATIONAL JOURNAL OF MATHEMATICAL MODEL AND METHOD IN APPLIED CIENCE crete wt tetraeral cell a eac Cotrol Volue (CV cotructe arou te tetraeral vertce. Ug t trateg te partal fferetal of flu volue coervato equato are crete to a te of fferetal/algebrac equato. Tee equato are te reolve te. Tee eto are utable for trcate pcal geoetre a flow troug tree eoal aturate porou ea wt cotat volue. ulato reult for te cae of oogeeou a otropc porou ea uereat of a gravt a wt uptrea cut off preete a copare wt aaltcal oluto obtae b applcato of coforal appg tecque for a cotat ut rato of fouato ept over alf of a bae ( T b. Te accurac of te reult copute uplft preure are aee b coparo of copute reult for varou rato of cut off wall ept over alf of a bae legt (/b wt te aaltcal oluto obtae ug coforal appg tecque b Pavlov,95 [. Te, te verfe oel apple to evaluate te qualt of te reult of te coo eprcal relato for uplft preure etato a ter error are vetgate. Fall, te ablt of te evelope Galer fte volue olver to cop wt real worl eterogeeou proble eotrate b t applcato to olve eepage troug a atural porou fouato of a gravt a wt tree otropc cle laer. I orer to prove better uertag fro te oluto reult color coe urface of te copute preure ea, veloct vector a flow et are preete a te copute uplft preure uereat of te a copare wt te reult of coo eprcal relato. II. MODELING ALGORITHM A. Mateatcal Moel Te proble of eawater eepage govere b a partal fferetal equato for te grou water flow tat ecrbe te ea trbuto te eterogeeou oe of teret uereat of a gravt a. Te geeral for of flow equato for a cofe aturate eterogeeou a aotropc porou ea ca be wrtte a [: ( (,, ( t Were, te referece raulc ea referre to a te frewater ea, a copoet of te raulc couctvt teor, te pecfc torage a t te. If ea graet flu recto (ecoar varable efe a, F (,, ( a ece, te equato tae te for: ( F (,, ( t Te bouar coto for t equato a be tate a Iue, Volue, 7 follow [: - Drclet bouar coto: ( b, b; t (, ; t B ( - Neua bouar coto: V. V ( b, b; t B (5 were te outwar ut vector oral to te bouar, (, b b a patal coorate o te bouar, a V are te Drclet fuctoal value a Neua flu, repectvel. It oul be ote tat, for a oogeeou a otropc porou ea te followg relato are val. (,, Wle, for a oogeeou a aotropc porou ea te followg relato are val. (7,, Furterore, for a eterogeeou a aotropc porou ea te followg relato are val. (,, B. Nuercal Moel Durg te lat twet ear tere a bee a trog focu upo te utlato of te Fte Volue eto for olvg flu flow a eat trafer proble or, a t ore geerall ow, proble Coputatoal Flu Dac (CFD. T ucce otl ue to te coervatve ature of te cee a te fact tat te ter appearg te reultg algebrac equato ave a pecfc pcal terpretato. I fact, te tragtforwar forulato a low coputatoal cot copare wt oter eto ave ae Fte Volue Meto te preferre coce for ot CFD practtoer [. Over te lat te ear, everal Fte Volue eto bae o Utructure Me (FVUM ave a wa overcoe te tructure ature of te orgal CV eto. I geeral, te FVUM eto ca be categore to two approace, ael, verte-cetere or cell-cetere. Te clafcato of te approac bae o te relatop betwee te CV a te fte eleet le utructure e. Te approac ecrbe ere te verte-cetere, wc ue lear ape fucto of tetraeral eleet a te terpolato fucto wt te CV fore b gaterg all te eleet arg a oal pot. T approac ver lar to te Galer Fte Eleet Meto wt lear eleet [,5.

3 INTERNATIONAL JOURNAL OF MATHEMATICAL MODEL AND METHOD IN APPLIED CIENCE I a fte eleet e, te ub-rego are calle eleet, wt te vertce of te eleet beg te oal locato. For te verte-cetere approac ol te bac tree eoal eleet, tetraero wt four oe are coere [. Terefore, eac oe te oluto oa aocate wt oe CV. Coequetl, eac CV cot of oe tetraeral eleet, a llutrate Fg.. Te CV ca be aeble a tragtforwar a effcet aer at te eleet level. Te flow acro eac cotrol urface ut be etere b a tegral. Fg.: ub-oa aocate wt oe of te coputatoal fel Te FVUM cretato proce tate b utlg te tegrate for of te equato (. B applcato of te Varatoal Meto, after ultplg te reual of te above equato b te tet fucto φ a tegratg over a ub-oa, we ave: F φ + φ (,, (9 t Te ter cotag patal ervatve ca be tegrate b part over te ub-oa a te equato (5 a be wrtte a: φ φ + F φ F t ( (,, Ug gau vergece teore te equato tae te for: φ φ + φ (. F t Γ ( Γ (,, were Γ te bouar of oa. Followg te cocept of wegte reual eto, b coerg te tet fucto equal to te wegtg fucto, te epeet varable e te oa ca be approate b applcato of a lear cobato, uc a oe N ϕ [7. Accorg to te Galer eto, te wegtg fucto φ ca be coe equal to te terpolato fucto ϕ. I fte eleet eto t fucto teatcall copute for ere eleet tpe a calle te ape fucto. For a tetraeral tpe eleet (wt four oe, te lear ape fucto, ϕ, tae te value of ut at ere oe, a ero at oter egborg oe of eac tragular eleet ( [7. Eteg te cocept to a ub-oa to te CV fore b te eleet eetg oe (Fg., te terpolato fucto ϕ tae te value of ut at te ceter oe of CV a ero at oter egborg oe (at te bouar of te CV Γ. Notewort tat, t a eetal propert of wegt fucto, ϕ, wc oul atf oogeeou bouar coto o T at bouar of uboa [. Tat w te tegrato of te lear oe cobato N ϕ (a approato over eleet of ub-oa tae te value of (te value of te epeet varable cetral oe. B t propert of te ape fucto ϕ ( ϕ o bouar Γ of te uboa, te bouar tegral ter equato (9 tae ero value for a CV wc te value of T aue ow at bouar oe. After ottg ero ter, te equato (9 tae te followg for: ϕ ϕ F (,, ( t I orer to rve te algebrac forulato, ever gle ter of te above equato frt apulate for eac eleet te te tegrato over te CV perfore. Te reultg forulato val for te cetral oe of te CV. For te ter wt o ervatve of te ape fucto ϕ, a eact tegrato forula ue a, a b c ϕϕ ϕϕ Λ(a!b!c!! (a + b + c + + Λ/ (for a Λ a bc, were Λ te volue of te tetraeral eleet [5. T volue ca be copute b te tegrato forula a, Λ ( Λ [ δ ( Λ were a δ are te average recto coorate a projecte area (oral to recto for ever e face oppote to oe of te eleet. Terefore, te traet ter φ for eac t tetraeral eleet Λ (e te ub-oa ca be wrtte a: φ Λ ( Λ t ( Λ t Coequetl, te traet ter of equato ( for te ub-oa (wt cetral oe tae te followg for: t ϕ (5 t Iue, Volue, 7 7

4 INTERNATIONAL JOURNAL OF MATHEMATICAL MODEL AND METHOD IN APPLIED CIENCE Now we tr to crete te ter cotag patal ervatve, F ( φ equato (. ce te ol uow epeet varable ϕ a te ape fucto, ϕ, are coe lear pecewe ever tetraeral eleet, te eat graet flu ( F wc fore b frt ervatve cotat over eac eleet a ca be tae out of te tegrato. O te oter a, te tegrato of te ape fucto patal ervato over tetraeral eleet ca be coverte to bouar tegral ug Gau vergece teore [, a ece: ϕ Α (. Α ϕ. Here, Α copoet of te e Α Α face eleet oral to te recto. Te crete for of te le tegral ca be wrtte a: ϕ.( Α [ Λ ϕ δ, were [ ϕ δ Δ fore b coerg te e of te eleet oppote to te oe, a te, ultplcato of t copoet perpecular to te recto b ϕ te average ape fucto value of t tree e oe. Hece, te ter N F F ( φ for a CV [ Λ ( ϕ δ (cotag N eleet arg t cetral oe. ce te ape fucto ϕ tae te value of ut ol at cetral oe of CV a ero at te oe locate at te bouar of CV, ϕ / for te face coecte to te cetral oe of CV a ϕ for te bouar face of te CV. O te oter a te u of te projecte area (oral to recto of tree e face of ever tetraeral eleet equate to te projecte area of te fourt e face; ece te ter cotag patal ervatve recto of te equato (, ca be wrtte a: M ϕ F [ F δ ( were [ δ te copoet of te bouar face (oppote to te cetral oe of te CV perpecular to recto. Note tat, F copute at te ceter of tetraeral eleet of te CV, wc aocate wt e. Te ea graet flu recto, F, at eac tetraeral eleet ca be calculate ug Gau vergece teore, F Λ Λ Δ T (, were Δ ( Δ te projecto of e face of te eleet perpecular to recto. B epreg te bouar tegral crete for a, Δ δ, for Δ ( ( eac eleet e te CV. Terefore, we ave, Iue, Volue, 7 [ F (7 Λ were, δ te copoet of t face of a tetraeral ( δ eleet (perpecular to te recto a te average ea of tat face a Λ te volue of te eleet. Notewort tat for CV at te bouar of te coputatoal oa, cetral oe of te CV locate at t ow bouar. For te bouar e coecte to te oe tere are o egborg eleet to cacel te cotrbuto. Hece, ter cotrbuto rea a te act a te bouar e of te ub-oa. Terefore, tere o cage to ecrbe proceure for coputato of te patal ervatve ter F ( ϕ. Fall, ug epreo (5 a ( te equato ( ca be wrtte for a CV (wt ceter oe a: M [ F δ (,, ( t Te volue of CV ca be copute b uato of te volue of te eleet aocate wt oe. Te reulte uercal oel, wc lar to No- Overlappg cee of te Cell-Verte FVUM, ca eplctl be olve for ever oe (te ceter of te ub-oa wc fore b gaterg eleet arg oe. Te eplct oluto of ea at ever oe of te oa of teret ca be oele a, N t+δt t Δt ( F Δl C. Coputatoal tep (, (9 Now we ee to efe a lt for te eplct te tep, δ t. Coerg teral ffuvt a α κ ρc wt te ut (, te crtero for eaurg te ablt of a ateral for ea cage. Hece te rate of ea cage ca be epree a, / Δt. Terefore, te approprate e for local te teppg ca be coere a, Δt β ( β ( β coere a a proportoalt cotat coeffcet, wc t agtue le ta ut. For te tea tate proble t lt ca be vewe a te lt of local coputatoal tep towar tea tate. However, tere are fferet e of CV utructure ee. T fact ple tat te u agtue of te above relato be coere. Hece, to ata te tablt of te eplct te teppg, te global u te tep of te coputatoal fel oul be coere. o, Δt β ( ( β ( Notewort tat for te oluto of tea tate proble o utable fe utructure ee, te ue of local coputatoal tep tea of global u te tep a

5 INTERNATIONAL JOURNAL OF MATHEMATICAL MODEL AND METHOD IN APPLIED CIENCE coerabl reuce te coputatoal effort.i orer to tablg te uercal oluto, te tep retrcte b: Δt a( ( were area of eac CV a (,,, raulc couctvt recto. Da Bae 9 III. MODEL VERIFICATION I te followg ecto, te accurac of te reult of te evelope oel verfe b coparo of te copute eepage a uplft preure atural fouato of a gravt a wt a cut off wall wt aaltcal oluto obtae b applcato of coforal appg tecque. Te plot of copute preure ea, veloct vector a flow et are ue to prove better uertag of te uercal oluto reult. To verf te above ecrbe uercal oel, a tet cae coere, for wc aaltcal oluto avalable. Te aaltcal oluto of te eepage a uplft preure troug te oogeeou a otropc a fouato are obtae for a uber of rato of cut off wall over alf of a bae legt (/b ug coforal appg tecque. Te paraeter were coe o tat te aale cae correpo to toe aaltcall olve b Pavlov, 95 [. Te geoetr of te a fouato wt a uptrea cut off wall at te a bae tet cae ceatcall ecrbe Fg.. Te bouar coto eploe preet uercal ulato are alo llutrate. Te fouato rego coere to be a oogeeou a otropc a wt 5 a ec. Te oel repreete a crete for b a treeeoal tetraeral e for a cubc a fouato Fg.. Fg. : A tree-eoal tetraeral e for a fouato Fgure ow a tpcal copute color coe urface of ea te oogeeou a otropc a fouato of a wt uptrea. Fgure 5 a, preet tpcal copute color coe veloct vector a flow et, repectvel, a oogeeou a otropc a fouato of a wt uptrea. Fgure 7 preet plot of uplft preure trbuto uereat of a wt uptrea for varou rato of wall ept over alf of a bae legt (/b for a cotat ut rato of fouato ept over alf of a bae ( Tb. H: T 7 b 9 Da ecto T 5 Bae Beroc L Fg. : Decrpto of verfcato cae Fg. : Tpcal copute color coe urface of ea te oogeeou a otropc a fouato of a Fgure preet plot of uplft preure rop D ( / uereat of a wt uptrea p L R for varou rato of wall ept for a rage of /b for Tb. I t relato L a R are preure ea uptrea a owtrea of te wall a te fferece of water ea at uptrea a owtrea of a. Te average error betwee uercal reult a aaltcal oluto.5%, wle te au error copute a 7%. A ca be ee, te accurac of te reult copute b preet atr free Galer fte volue oel for oluto Iue, Volue, 7 9

6 INTERNATIONAL JOURNAL OF MATHEMATICAL MODEL AND METHOD IN APPLIED CIENCE of eepage flow a coputato of uplft preure are qute acceptable. 9 Da Bae Vr.E-.7E- 5.95E- 5.7E- 5.5E- 5.E- 5.E-.9E-.E-.E-.5E-.E-.E-.E-.E-.9E-.9E-.7E-.55E-.E-.E-.9E-.7E-.9E-.E-.E-.5E-7.E-7.5E-7.E-7 Fg.5 : Tpcal copute veloct vector te oogeeou a otropc a fouato of a Da Bae Fg. : Tpcal copute flow et te oogeeou a otropc a fouato of a /b Dp (% /b Aaltcal Coputatoal Fg. : Te coparo of te copute reult for varou (/b wt te aaltcal oluto of Pavlov,95 [ IV. ERROR OF EMPIRICAL RELATION Tere are oe eprcal relato wc globall etate te value of uplft force uereat of te a bae a fucto of pore preure a fouato. Tee relato efe oe crtero for coerg equvalet eepage legt L e. Te, uplft preure at a pot uereat of te a bae ca be etate b coerg lear varato of preure ea fro H up to H ow alog L e. Oe of te popular relato for calculato of equvalet eepage legt L e propoe b Lae [. Lvertcal + ( L ( e Lae LHorotal Aoter popular relato for calculato of equvalet eepage legt L e propoe b Blg [. ( Le Blg Lvertcal + L ( Horotal I t ecto te etate uplft preure fouato of a a (pecfe te prevou ecto ug above etoe relato are copare wt te uplft preure copute b trouce uercal olver (Fg.9. Followg table preet te uplft preure error caue b ug te two eprcal relato for etatg L e are preete P/(Gaa. /b /b. /b. /b. /b..9 Fg.7 : Uplft preure trbuto uereat of a wt uptrea for fferet /b ( T b Iue, Volue, 7 9

7 INTERNATIONAL JOURNAL OF MATHEMATICAL MODEL AND METHOD IN APPLIED CIENCE. P/(Gaa Coputatoal Blg Lae 5 K, K, K, /b Fg.9 : Uplft preure varato uereat of a wt uptrea for fferet /b ( T b Table : Uplft preure error ug eprcal relato for /b P c /(g w Copute Blg Error(% Lae Error(%..... L e V. MODEL APPLICATION I t ecto, te tree laer fouato rego coere to be a eterogeeou fouato cae wt followg coeffcet of pereablt: ec 5 ec ec Te oel repreete a crete for b a treeeoal tetraeral e for a cubc a fouato. Te geoetrcal feature of te cae are ecrbe Fg.. Fg.7 : Decrpto of te applcato proble Fgure ow a tpcal copute color coe urface of ea te otropc a fouato of a wt uptrea. Fgure a Fgure, preet tpcal copute color coe veloct vector a flow et, repectvel, a eterogeeou fouato of a wt uptrea. Fgure preet plot of uplft preure trbuto uereat of a wt uptrea te eterogeeou fouato. H Fg. : Tpcal copute D color coe urface of ea te otropc a fouato of a Vr.9E-.E- Da Bae.E-.97E-.9E-.E-.75E-.E-.E-.5E-.E-.9E-.E-.E-.7E-.9E-.E- 9.E-7.75E-7.E-7 7.9E-7.5E-7 5.E-7 5.E-7.E-7.5E-7.9E-7.9E-7.E-7 7.9E- Iue, Volue, 7 9

8 Fg. : Tpcal copute veloct vector a eterogeeou fouato of a INTERNATIONAL JOURNAL OF MATHEMATICAL MODEL AND METHOD IN APPLIED CIENCE Da Bae Fg. : Tpcal copute flow et a eterogeeou fouato of a /b uplft preure trbuto are copare wt te aaltcal oluto obtae b applcato of coforal appg tecque b Pavlov, 95. Te agreeet betwee te copute reult obtae fro ecrbe oelg algort a aaltcal oluto are prog. Te coparo of te reult of preet uercal olver wt te eprcal relato ow tat te uplft preure calculate ug te equvalet eepage legt uggete b Lae a Blg are cloe to te copute value ol for all /b. For large value of /b te equvalet eepage legt uggete b Blg wll prouce over etate uplft preure value. Te goo perforace a acceptable reult of te trouce oelg tecque ecourage applcato of te oel for olvg eepage flow a real worl ultlaer porou fouato. Hece, te ablt of te evelope oel for olvg eepage flow troug a atural eterogeeou fouato of a gravt a wt tree otropc cle laer eae. I orer to prove better uertag fro te oluto reult color coe urface of te copute preure ea, veloct vector a flow et are preete a te copute uplft preure uereat of te a copare wt te reult of coo eprcal relato P/(Gaa. Coputatoal Blg Lae Fg. : Uplft preure trbuto uereat of a wt uptrea A HETEROGENEOU fouato VI. CONCLUION I preet paper, a tree-eoal uercal oel evelope for coputg te eepage flow a uplft preure uer gravt a wt wall. T oel eplctl olve te equato of grou water flow o te tree eoal utructure e ug Galer Fte Volue Meto evelope for lear tetraeral eleet. ce tere o ape fucto te fal forulato of te oel, te trouce atr free oal bae eto coue ver lgt coputatoal overea. Te propoe algort ut te utructure ee a terefore, te evelope oel ca prect eepage flow a preure ea trbuto geoetrcal cople porou ea. I orer to verf te accurac of oel reult, te eepage flow troug a oogeeou a otropc a a fouato olve for varou rato of uptrea wall ept over alf of a bae legt (/b for a cotat ut rato of fouato ept over alf of a bae (T/b. Te copute reult of REFERENCE [ J. Bear, Hraulc of Grouwater, McGraw-Hll, New our, 979. [ U Ar Corp of Egeer, Egeerg a Deg; eepage aal a cotrol for a, EM --9, Capter (Deterato of pereablt of ol a cecal copoto of water, Aprl 99. [. Je, G. Je,. Mao a G. L, eepage aal bae o bouarftte coorate traforato eto, Coputer a Geotecc,,, pp.79. [ J.L. eraf, A.P. ato a M.C. Mato, Tree-eoal eepage troug a a fouato, Queto of ICOLD cogree, Q5: Fouato treatet for cotrol of eepage, R., 95, pp [5 P.D. Kou, Leat-quare Fte-Eleet Evaluato of Flow Net, Joural of Geoteccal a Geo-evroetal Egeerg, ACE, Augut, pp [ D.V. Grfft, G.A Feto a A. Goro, Tree-Deoal eepage troug patall rao ol, Joural of Geoteccal a Geoevroetal Egeerg, ACE, Februar 997, pp.5-. [7 D.V. Grfft a G.A. Feto, Probabltc aal of et graet ue to tea eepage, Joural of Geoteccal a Geo-evroetal Egeerg, ACE, ep. 99, pp [ M.C. Boufael, M.T. ua, A.D. Veoa a M.T. Bower tea eepage Trece a Da: Effect of Capllart Flow, Joural of Hraulc Egeerg, ACE, Marc 999, pp.-9. [9 G L., J. Ge a. Je, Free urface eepage aal bae o te eleet-free eto, Mecac Reearc Coucato,,, pp.9 9. [ G.A. Plar, O te fluece of uplft preure cocrete gravt a, Egeerg Fracture Mecac, Vol. 59, No., 99, pp [ R.R. Dewe, R.W. Rec a V.E. aoua, Uplft oelg for fracture ecac aal of cocrete a, Joural of tructural Egeerg, ACE, Vol., No., October 99, pp.5-. [ L. N. Re, eepage ol, Jo Wle & o Ic.,. [ K.A. Hoffa a.t. Cag,, Coputatoal Flu Dac for Egeer, Egeerg Eucato te, 99. [ G. egol, G. F.Per a W.G. Gra, A galer fte eleet tecque for calculatg te traet poto of te altwater frot. Joural of Water Re., (, 975, pp.-7. [5.R. abbag-a a E.N. Matora, Effcet etrc Bouar Coto for Galer Fte Volue oluto of D Iue, Volue, 7 9

9 INTERNATIONAL JOURNAL OF MATHEMATICAL MODEL AND METHOD IN APPLIED CIENCE Teperature Fel o Tetraeral Mee, 5 t IAME / WEA Iteratoal Coferece o Heat Trafer, Teral Egeerg a Evroet, Voulage Beac, Greece, 7. [ J.F. Topo, B.K. o a N.P. Weaterll, Ha boo of gr geerato, CRC Pre, New or, 999. [7 J.N. Re a D.K. Gartlg, Te Fte Eleet Meto Heat Trafer a Flu Dac, CRC Pre,. Frt Autor' bograp a be fou : ttp://aa.tu.ac.r/~a/ Iue, Volue, 7 9