Three Dimensional Finite Volume Solutions of Seepage and Uplift in Homogonous and Isotropic Foundations of Gravity Dams

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1 t WEA Iteratoal Coferece o TEM CIENCE a IMULATION ENGINEERING, Vece, Italy, Noveber -, 7 9 Tree Deoal Fte Volue oluto of eepage a Uplft Hoogoou a Iotropc Fouato of Gravty Da AEED-REA ABBAGH-ADI *, BABAK BAAT ** *Aocate Profeor, ** M.c. Grauate Cvl Egeerg Departet, KN Too Uverty of Tecology, No. Valar treet, 997- Tera-IRAN a@tu.ac.r a NIKO E. MATORAKI Mltary Itute of Uverty Eucato (AEI) Hellec Naval Acaey Tera Catyraou 59, Praue, GREECE ator@wea.org Abtract: I t paper, a tree-eoal vero of NAIR fte volue eepage olver evelope for tetraeral e trouce. Te uercal aalyer utle for olvg te eepage oogeeou a otropc porou ea a uplft uer gravty a wt uptrea cut off wall. Te reult of uercal olver ter of uplft preure uereat of te a bae wt uptrea cut off wall are copare wt aalytcal oluto obtae by applcato of fro coforal appg tecque for a cotat ut rato of fouato ept over a bae ( T b= ). Te accuracy of te reult copute uplft preure preet acceptable agreeet wt te aalytcal oluto for varou rato of cut off wall over te a bae legt (/b). Key-Wor: - Nuercal Aaly, Da Bae eepage a Uplft, NAIR oftware, Fte Volue Meto Itroucto Te proble of eepage flow uereat of gravty a ca be forulate ter of a olear partal fferetal equato. Te equato ecrbe a cotat ety flu flow a eterogeeou a otropc porou ea []. Altoug eprcal forulato are uggete for ple cae, ue to te eretly cople bouary coto a trcate pycal geoetre ay practcal proble, a aalytcal oluto ot poble for coplcate a fouato []. T paper preet a fte volue e eto for oelg water flow a aturate eterogeeou porou ea wt cople bouary yte. Te oluto oa crete wt tetraeral cell a te cotrol volue are cotructe arou te tetraeral vertce. Ug t trategy te partal fferetal of flu volue coervato equato are crete to a yte of fferetal/algebrac equato. Tee equato are te reolve te. Tee eto are utable for trcate pycal geoetre a flow troug tree eoal aturate porou ea wt cotat volue. ulato reult for two cae of oogeeou a eterogeeou a otropc porou ea uereat of a gravty a wt uptrea cut off are preete a copare wt aalytcal oluto obtae by applcato of fro coforal appg tecque for a cotat ut rato of fouato ept over a bae ( T b= ). Te accuracy of te reult copute uplft preure are aee by coparo of copute reult for varou rato of cut off wall over te a bae legt (/b) wt te aalytcal oluto obtae ug coforal appg tecque by Pavlovy,95 []. Proble Forulato Te proble of eawater eepage govere by a partal fferetal equato for te grou water flow tat ecrbe te ea trbuto te eterogeeou oe of teret uereat of a gravty a. Te flow equato for a cofe aturate porou ea ca be wrtte a []: ( ) = ( =,, ) t () Nuercal Aalyer for cetfc a Iutral Requreet

2 t WEA Iteratoal Coferece o TEM CIENCE a IMULATION ENGINEERING, Vece, Italy, Noveber -, 7 9 Were te referece yraulc ea referre to a te frewater ea; a copoet of te yraulc couctvty teor; te pecfc torage; t te. Te paraeter p te flu preure; g te gravtatoal accelerato a elevato. If ea graet flu recto (ecoary varable) efe a, F = ( =,,) () A ece, te equato tae te for: ( F ) = ( =,,) () t Te bouary coto for t equato ay be tate a follow []: -Drclet bouary coto: ( b, b; t) = (, ; t) I B () -Neua bouary coto: V. = V ( b, b; t) I B (5) were te outwar ut vector oral to te bouary;, ) a patal coorate o te ( b b bouary; a V are te Drclet fuctoal value a Neua flu, repectvely. It oul be ote tat, for te oogeeou a otropc porou ea te followg relato are val. = = = y y =, =, = y () FVUM eto ca be categore to two approace, aely, verte-cetere or cellcetere. Te clafcato of te approac bae o te relatop betwee te cotrol volue 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 Cotrol Volue fore by gaterg all te eleet arg a oal pot. T approac very lar to te Galer Fte Eleet Meto wt lear eleet [5,]. 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 oly te bac tree eoal eleet, wc tetraero wt four oe are coere [7]. Terefore, eac oe te oluto oa aocate wt oe cotrol volue. Coequetly, eac cotrol volue cot of oe tetraeral eleet, a llutrate Fgure. Te CV ca be aeble a tragtforwar a effcet aer at te eleet level. Te flow acro eac cotrol urface ut be etere by a tegral. Nuercal Forulato Durg te lat twety year 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 geerally ow, proble Coputatoal Flu Dyac (CFD). T ucce otly ue to te coervatve ature of te cee a te fact tat te ter appearg te reultg algebrac equato ave a pecfc pycal 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 year, everal cotrol volue bae-utructure e (FVUM) eto ave ay way overcoe te tructure ature of te orgal cotrol volue eto. I geeral, te Fgure - ub-oa aocate wt oe of te coputatoal fel Te FVUM cretato proce tate by utlg te tegrate for of te equato (). By applcato of te Varatoal Meto, after ultplyg te reual of te above equato by te tet fucto φ a tegratg over a uboa, we ave, F φ + = t φ ( =,, ) (7) Te ter cotag patal ervatve ca be tegrate by part over te ub-oa a te equato (5) ay be wrtte a,

3 t WEA Iteratoal Coferece o TEM CIENCE a IMULATION ENGINEERING, Vece, Italy, Noveber -, 7 95 φ + t F F φ = ( =,,) () Ug gau vergece teore te equato tae te for: φ + t F Γ φ (. Γ ) = ( =,, ) (9) Were Γ te bouary of oa. Followg te cocept of wegte reual eto, by coerg te tet fucto equal to te wegtg fucto, te epeet varable e te oa ca be approate by applcato of a lear cobato, uc a N = oe ϕ []. = Accorg to te Galer eto, te wegtg fucto φ ca be coe equal to te terpolato fucto ϕ. I fte eleet eto t fucto yteatcally copute for ere eleet type a calle te ape fucto. For a tetraeral type eleet (wt four oe), te lear ape fucto, ϕ, tae te value of uty at ere oe, a ero at oter egborg oe of eac tragular eleet ( ) []. Eteg te cocept to a ub-oa to te cotrol volue fore by te eleet eetg oe (Fgure ), te terpolato fucto ϕ tae te value of uty at te ceter oe of cotrol volue a ero at oter egborg oe (at te bouary of te cotrol volue Γ ). Noteworty tat, t a eetal property of wegt fucto, ϕ, wc oul atfy oogeeou bouary coto o T at bouary of ub-oa []. Tat wy te tegrato of te N lear cobato = oe = ϕ (a approato) over eleet of ub-oa tae te value of (te value of te epeet varable cetral oe ). By t property of te ape fucto ϕ ( ϕ = o bouary Γ of te uboa ), te bouary tegral ter equato (7) tae ero value for a cotrol volue wc te value of T aue ow at bouary oe. After ottg ero ter, te equato (7) tae te for, ϕ ϕ F = (,,) = () t I orer to rve te algebrac forulato, every gle ter of te above equato frt apulate for eac eleet te te tegrato over te cotrol volue perfore. Te reultg forulato val for te cetral oe of te cotrol volue. For te ter wt o ervatve of te ape fucto ϕ, a eact tegrato forula ue a, a b c ϕϕ ϕϕ = (a!b!c!! ) (a+ b+ c+ + ) = / (for a= a b=c==), were te volue of te tetraeral eleet []. T volue ca be copute by te tegrato forula a, = ( ) [ δ l ] () were a δ l are te average recto coorate a projecte area (oral to recto) for every e face oppote to oe of te eleet. Terefore, te traet ter φ for t eac tetraeral eleet (e te ub-oa) ca be wrtte a, φ = ( ) t t Coequetly, te traet ter of equato () for te ub-oa (wt cetral oe ) tae te for, t ϕ = (.a) t Now we try to crete te ter cotag patal ervatve, F ( ) equato (). ce te oly uow epeet varable = ϕ a te ape fucto, ϕ, are coe pece we lear every tetraeral eleet, te eat graet flu ( F fore by 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 bouary tegral ug Gau vergece teore [9], a ece, ϕ = (. ) ϕ. Here copoet of te e face eleet oral to te recto. Te crete for of te le tegral ca be wrtte a, ϕ.( ) [ ϕδl ], were [ ϕ δl ] fore by coerg te e of te eleet oppote to te oe, a te, ultplcato of t copoet perpecular to te recto by ϕ te average ape fucto value of t tree e oe. Hece, te ter

4 t WEA Iteratoal Coferece o TEM CIENCE a IMULATION ENGINEERING, Vece, Italy, Noveber -, 7 9 N F F ( for a cotrol ) [ ( ϕδl ) ] volue (cotag N eleet arg t cetral oe). ce te ape fucto ϕ tae te value of uty oly at cetral oe of cotrol volue a ero at te oe locate at te bouary of cotrol volue, ϕ = / for te face coecte to te cetral oe of cotrol volue a ϕ = for te bouary face of te cotrol volue. O te oter a te u of te projecte area (oral to recto) of tree e face of every 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 δl ] (.b) = Were [ δ l ] te copoet of te bouary face (oppote to te cetral oe of te cotrol volue ) perpecular to recto. Note tat, F copute at te ceter of tetraeral eleet of te cotrol volue, wc aocate wt e. Te ea graet flu recto, F =, at eac tetraeral eleet ca be calculate ug Gau vergece teore, F = = T ( ), were ( ) te projecto of e face of te eleet perpecular to recto. By epreg te bouary tegral crete for a, ( ) ( δl ), for eac eleet e te cotrol volue. Terefore, we ave, [ F ] = ( δl ) Were, l = () δ te copoet of t face of a tetraeral eleet (perpecular to te recto) a te average ea of tat face a te volue of te eleet. Note worty tat for cotrol volue at te bouary of te coputatoal oa, cetral oe of te cotrol volue locate at t ow bouary. For te bouary e coecte to te to te oe tere are o egborg eleet to cacel te cotrbuto. Hece, ter cotrbuto rea a tey act a te bouary e of te ub-oa. Terefore, tere o cage to te ecrbe proceure for coputato of te patal ervatve ter F ( ϕ ). Fally, ug epreo (.a) a (.b) te equato () ca be wrtte for a cotrol volue (wt ceter oe ) a: M = [ F δl ] ( =,,) () t = Te volue of cotrol volue, ca be copute by uato of te volue of te eleet aocate wt oe. Te reulte uercal oel, wc lar to No-Overlappg cee of te Cell-Verte Fte Volue Meto o utructure ee, ca eplctly be olve for every oe (te ceter of te ub-oa wc fore by gaterg eleet arg oe ). Te eplct oluto of ea at every oe of te oa of teret ca be oele a, N t+ t t δ t = ( F l ) ( =,) (5) = Now we ee to efe a lt for te eplct te tep, δ t. Coerg teral ffuvty a α= κ ρc wt te ut ( ), te crtero for eaurg te ablty of a ateral for ea cage. Hece te rate of ea cage ca be epree a,. Terefore, te approprate e for local δ t te teppg ca be coere a, δ t= β ( β ) (5) β coere a a proportoalty cotat coeffcet, wc t agtue le ta uty. For te teay tate proble t lt ca be vewe a te lt of local coputatoal tep towar teay tate. However, tere are fferet e of cotrol volue utructure ee. T fact ple tat te u agtue of te above relato be coere. Hece, to ata te tablty of te eplct te teppg te global u te tep of te coputatoal fel oul be coere, o, δ t= β ( ) ( β ) () Noteworty tat for te oluto of teay tate proble o utable fe utructure ee, te ue of local coputatoal tep tea of global u te tep ay coerably reuce te coputatoal effort. I orer to tablg te uercal oluto, te tep retrcte by:

5 t WEA Iteratoal Coferece o TEM CIENCE a IMULATION ENGINEERING, Vece, Italy, Noveber -, 7 97 t = a( ) (7) Were area of eac cotrol volue a (=,,) yraulc couctvty recto. Verfcato Tet Cae To verfy te above ecrbe uercal oel, a tet cae coere, for wc aalytcal oluto avalable. Te aalytcal oluto of te eepage a uplft preure troug te oogeeou a otropc a fouato reult are obtae for a uber of rato of cut off wall over te a bae legt (/b) ug coforal appg tecque. Te paraeter were coe o tat te aalye cae correpo to toe aalytcally olve by Pavlovy, 95 []. Te geoetry of te a fouato wt a uptrea cut off wall at te a bae tet cae ceatcally ecrbe fgure. Te bouary coto eploye preet uercal ulato are alo ow Fgure. Te fouato rego coere to be a oogeeou a otropc a wt 5 5 = y = = 5 a ec = repreete a crete for by a treeeoal tetraeral e for a cubc a fouato ow Fgure. T 7 9 b Da ecto T 5 Fgure ow a typcal copute color coe urface of ea te oogeeou a otropc a fouato of a wt up trea cut off. Fgure 5 a, repectvely, preet typcal copute color coe velocty vector a flow et a oogeeou a otropc a fouato of a wt up trea cut off. H: Fgure : Typcal copute color coe urface of ea te oogeeou a otropc a fouato of a cutoff Da Bae 5 5 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 Fgure 5: Typcal copute velocty vector te oogeeou a otropc a fouato of a Bae Beroc L Da Bae y Fgure: Proble ecrpto of altwater truo a coatal cofe aqufer [] cutoff cutoff Da Bae Fgure : A tree-eoal tetraeral e for a fouato Fgure : Typcal copute flow et te oogeeou a otropc a fouato of a Fgure 7 preet plot of uplft preure trbuto uereat of a wt up trea cut off for varou rato of cut off wall over te a bae legt (/b) for a cotat ut rato of

6 t WEA Iteratoal Coferece o TEM CIENCE a IMULATION ENGINEERING, Vece, Italy, Noveber -, 7 9 fouato ept over a bae ( T b= ) /b P/(Gaa.) /b= /b=. /b=. /b=. /b=. Fgure 7: Uplft preure trbuto uereat of a wt up trea cut off for /b ( T b= ) 9 evelope for coputg te eepage flow a uplft preure uer gravty a wt cut off wall. Te oel eplctly olve te equato of grou water flow o te tree eoal utructure e ug Galer Fte Volue Meto evelope for lear tetraeral eleet. Te oel ca prect preure ea trbuto geoetrcal cople porou ea. I orer to verfy te accuracy of oel reult, te eepage flow troug a oogeeou a otropc a a fouato olve for varou rato of uptrea cut off wall over te a bae legt (/b) for a cotat ut rato of fouato ept over a bae ( T b= ). Te copute reult of uplft preure trbuto are copare wt te aalytcal oluto obtae by applcato of coforal appg tecque by Pavlovy,95. Acceptable agreeet betwee te reult of te preet ulato a aalytcal oluto ecourage applcato of te oel for oelg eepage flow eterogeeou a aotropc porou ea. 7 Dp (%) /b Aalytcal Coputatoal Fgure : Te coparo of te copute reult for varou (/b) wt te aalytcal oluto of Pavlovy,95 [] Fgure preet plot of uplft preure rop D = ( ) / uereat of a wt up p L R trea cut off for varou rato of cut off wall for a rage of /b for T b=. I t relato L a R are preure ea uptrea a ow trea of te cut wall a te fferece of water ea at uptrea a ow trea of a. Te average error betwee uercal reult a aalytcal oluto.5%, wle te au error copute a 7%. A ca be ee te accuracy of te reult copute by preet vero of NAIR utructure fte volue oel for oluto of eepage flow a coputato of uplft preure are qute acceptable. Cocluo I preet paper, a D uercal oel bae o fte volue utructure e (FVUM) eto Referece: [] Bear, J., "Hyraulc of Grouwater", McGraw-Hll, New our, 979. [] U Ary Corp of Egeer, Egeerg a Deg; eepage aaly a cotrol for a, EM --9, capter (Deterato of pereablty of ol a cecal copoto of water), Aprl 99. [] La N. Re,, eepage ol, Jo Wley & o Ic. [] Hoffa K.A a Cag.T, "Coputatoal Flu Dyac for Egeer", Egeerg Eucato yte, [5] egol. G., Per, G. F. a Gray, W. G. (975) A galer fte eleet tecque for calculatg te traet poto of te altwater frot. Water Re., (), 975, -7. [] abbag-a.r. a Matora E.N. (7), "Effcet yetrc Bouary Coto for Galer Fte Volue oluto of D Teperature Fel o Tetraeral Mee ", Te 5t IAME / WEA Iteratoal Coferece o Heat Trafer, Teral Egeerg a Evroet Voulage Beac, Greece [7] Topo J.F., o B.K. a Weaterll N.P., Ha boo of gr geerato, CRC Pre, New or, 999. [] Rey J.N. a Gartlg D.K., Te Fte Eleet Meto Heat Trafer a Flu Dyac, CRC Pre,.