Introduction of a Mathematical Storage Function Model Based on lumping Process of Infiltration Theory

Transcription

1 Manucipt eceive Oct. 8, 7; evie Ma., 8 Siamak Boaghpou, See Ahma Mibaghei an See Aman Hahemi Monfae Intouction of a Mathematical Stoage Function Moel Bae on lumping Poce of Infiltation Theo SIAMAK BODAGHPOUR, Aitant Pofeo, Civil Engineeing Depatment, K.N.Tooi Univeit of Technolog, TEHRAN, IRAN, boaghpou@kntu.ac.i SEYYED AHMAD MIRBAGHERI, Pofeo, Civil Engineeing Depatment, K.N.Tooi Univeit of Technolog,TEHRAN, IRAN, mibaghei@kntu.ac.i SEYYED ARMAN HASHEMI MONFARED, Gauate Stuent of Civil & Wate Reouce Banch, K.N.Tooi Univeit of Technolog, TEHRAN, IRAN, a_hahemi_m@ina.kntu.ac.i Abtact: - Etimation an wate peiction of the infiltation of ainfall ha been alwa an impotant tak fo hologit. Richa equation i aopte a a funamental equation to evelop a toage function moel. At fit tage, a new bouna conition ha been intouce an it accuac ha been co checke with epeiment. In the econ tage, on the lumping poce of non-imenional fom of Richa equation, a compenation facto ha been popoe to equalize the emi-lumpe equation. Finall the elation between toage an ichage ha been achieve b full lumpe equation. Ke-wo: - Richa equation, lumping of unatuate flow equation, emi lumpe equation, toage function moel Intouction A huge volume of eal pecipitation infiltate in to the goun uface of vaiou catchment aea in the wol an pouce etenive wate eouce une the goun uface. Suitable etimation an peiction of goun wate can be ueful to manage the ometic, agicultual an inutial eman in iffeent aea. Mathematical moel i one of the impotant toll to povie valuable infomation about goun wate at minimum time an cot. Baicall in the natue, infiltation of wate into the oil epen on the popetie of oil uch a haulic conuctivit, wate etention, pooit, oil lae thickne, lope length, lope epth an alo ainfall enit. Of coue, thee ae man othe paamete in the fiel which ae ifficult to obtain. On the othe han, the computation bae on Richa equation take much time. Theefoe, to eceae the computation time, a toage function moel nee to be eive. Fujita,m.( 98) an takaao,t.(985) eive a toage function moel uing kinematic wave theo an matubaahi,u.(994) in t eplain the lumping poce in etail. One of the aim of thi pape i a etail elaboation of lumping poce uing Richa equation. Reeach patten Infiltation of ain into the oil can be epee b Richa equation in it two imenional unatuate flow fom a the following: θ V V () t Whee (θ ) i the moitue content in the oil,(t ) i time, an ( V ) an ( V ) ae flue in ( ) an ( ) iection. Equation () applie to a oil column an it chematic conition i hown in figue (). Whee (l ) i length, ( ) i epth an (α ) i lope length of oil column. A ainfall enit a ( ) i applie to the uface of oil column. The flue in ( ) an ( ) iection can be eive b Dac equation. Φ V k () Φ V k (3) ( Φ ) i wate potential in the oil an ( k ) i haulic conuctivit of oil which can be epee a: θ θ k k θ θ (4) Whee ( k ) i atuate conuctivit of oil, ( ) epen on oil popet, ( θ ) an ( θ ) ae maimum (atuate) an minimum wate content of the oil. The equation of wate content can be eive b Havekamp equation. a θ ( θ θ + θ a + Ψ ) (5) ISSN: Iue 6, Volume 4, June 8

2 Siamak Boaghpou, See Ahma Mibaghei an See Aman Hahemi Monfae ( Ψ ) i uction in the oil an ( a ) i epen on oil popet. Equation () an (3) can be ewitten a follow: V k a a + Ψ Ψ Sinα a Ψ V k Coα a + Ψ (6) (7) The olution of Richa equation ha been coniee b the following bouna conition: an V. Coα (8) an V (9) an V () At ( X l ) whee ichage of toe wate in oil poou meia appea ue to the lope length of oil column, iffeent bouna conition have been popoe b eeache. Fo eample: Ψ () Ψ () Thee equation ae phicall eplaine that the uction Ψ, oen t change abuptl nea the bouna an theefoe the have an effect to epe the ichage. Matubaahi,u. (994) inceae the hoizontal haulic conuctivit to avoi uch a phenomena. In the peent tu, we intouce a bouna conition (equation (3)) an it valiit ha been co checke with epeiment a hown in figue (3), (4) an (5). l an V (3) An epeimental ectangula oil column (5 3 4 cm) with atificial ain on the top intalle in an with teniomete a it i hown in figue (). Wate flow potential though oil poou meia an oil uction ae meaue b teniomete at iffeent epth an itance in oil column. A changing of wate flow potential i moe abuptl at outlet of oil column ( l ) theefoe mot of the teniomete concentate in uch a place. Two cae of lope length (. an. n) ae coniee fo oil column which ae almot imila to lope length of catchment aea in natue. B epaate oil mechanic epeiment on an following popetie ae achieve ( θ. 5, θ.4,, a 65cm, Cm k. 9 ). Uing Sec epeimental conition an bouna conition in equation (3), the Richa equation ha been olve an the eult ae hown in figue (3). Futhe the ( q ) elation an toage i compute in two wa. The elation ( q ) i eive accoing to the equation (4) an the eult i hown in figue (4). Beie, the elation ( q ) i eive b uing equation (5) an the eult i hown in figue (5). Both elation how goo accuac with epeimental eult ( S i the initial toage) t t (4) S S t qt l D θ (, t) S (5), Non imenional fom of the baic equation. Phical paamete of baic equation can be euce b conveting it into the non imenional fom. Fo thi pupoe the following elationhip have been coniee: V V, Y,, Ψ Ψ Φ, t T t S SS, q ~ qq, θ θ θ, R, V V (6) Following aumption have been mae fo non imenionalizing poce. θ θ θ, k, V k,, l S θ l, q k, Ψ, θ t (7) The non imenional fom of equation (),(6),(7),(4) an (5) can be witten a the following: ( ) t l (8) a A (9) A A + Φ A A + Φ A Φ ( Sinα ) l Φ ( Coα ) A + Φ k () () k () A θ A + Φ θ + θ θ (3) The non imenional initial an bouna conition can be witten a: Φ l c ( ) Sinα + ( ) Coα + (4) ISSN: Iue 6, Volume 4, June 8

3 Siamak Boaghpou, See Ahma Mibaghei an See Aman Hahemi Monfae, (5), (6) RCoα, (7), (8) The initial conition in equation (4) mean initiall, wate i available in the oil but thee i no ichage. Thi initial conition ha been atifie the epeiment initial conition. 3 Semi lumpe unatuate flow equation B integation of the equation (8), (), (5) an (6) along we can obtain emi lumpe equation. θ Q + ( ) RCoα (9) t l A Φ Q ( Sin ) A α + Φ l Q Q (3) (3) (3) (θ ) in equation (9) an ( Φ ) in equation (3) ae aveage wate content an uction along epth. Howeve, when the elation in equation (3) eplace in equation (9),it ha been ceate itubance in calculation an it gave efaulte eult. To avoi uch a conition, following elation ha been coniee: an θ o θ θ T cont (33) to equalize the appoimation of integation, following elation ha been coniee. A A + Φ (34) f A A + Φ When elation of equation (34) eplace in equation (3), it ha been ealize that it eto the ituation of bouna conition at ( ). Theefoe to ealize the elation in equation (3), equation (35) ha been intouce. A Φ Q γ Sinα (35) A + Φ l Whee (γ ) i obtaine though vaiou numeical calculation. γ Ep ln A (36) [ ( ) ( )] Fo iffeent ange of paamete the valiit of (γ ) i hown in figue (6). (γ ) i obtaine une unifom ain, theefoe to check it accuac, we applie iffeent tpe of ain a it i hown in figue (7). 4 Mathematical toage function moel The elation between toage an ichage can be eive une the tea tate conition. l Q RCoα (37) Though futhe integation of equation (9) an (35) une the conition ( Φ Sinα >> ), equation l (38) an (39) ae obtaine. S + Q RCoα (38) T l S + γ Q Sin α θ + θ θ (39) ( S ) i toage an figue (8) how the eult obtaine b emi lumpe equation. In pinciple, the equation (9) an (35) have to be olve theoeticall to fin ( S ~ Q ) elation at untea tate. Howeve it i ifficult to olve them theoeticall. Theefoe, a convenient metho ha been peente b autho. Let aume: A εφ e (4) A + Φ B the ubtituting the elation in equation (4) into equation (35) the following elation can be achieve. ε Φ ε Φ γ e Q γ Sin α e (4) lε Conieing the elationhip between an t.with the epect to iffeent given time, mean the tea tate conition of ichage along ( ) whee the elation in equation (43) mean the untea tate conition. Q( ) Q (4) Q ( ) ( ) Q (43) Let aume a new conieate tem whee accoing to bouna conition at : Φ e ε Φ e ε (44) B conieing the new conieate tem an ubtituting the elation in equation (4) into ISSN: Iue 6, Volume 4, June 8

4 Siamak Boaghpou, See Ahma Mibaghei an See Aman Hahemi Monfae equation (35) the following elation can be achieve. εφ Q Z e * γsinα ( ) c c e c e (45) c c Whee: lεsinα c (46) Similal, b eplacing the equation (43) in equation (35) we can eive. ε Φ Q ( ) c + + c Z e e e c c c c (47) Finall, the ( S ~ Q ) elation fo tea an untea tate can be obtaine b following equation: Q S Z ( ) G ( c ) (48) γsinα Q S Z ( ) G( c ) (49) γsinα Whee G ( c, ) an G ( c, ) ae obtaine b numeical calculation. G ( c, ).5log c +.33log +.4 (5) G ( c, ) (.64log +.) log c +.3log +.3 (5) 5 Concluion At the fit tage, we intouce a new bouna conition at the outlet of oil column an it accuac i eamine with epeimental eult. In the econ tage, we popoe a compenating facto (γ ) to equalize the emi lumpe equation to the Richa equation. Full lumpe toage function moel in two cae of tea an untea tate ae intouce an eamine in iffeent phical conition. In tea tate, with inceaing an lope length, the eult of toage function moel how highe accuac in compaion with epeimental eult. Beie in cae of untea tate, when an.5 D. 8 the mot uitable eult can be achieve. Refeence: []Boaghpou, S.an Fujita, M.(994)"Stoage function moel bae on infiltation theo", Poceeing of JSCE, Hokaio banch vol.5,38-33 []Matubaahi, U.Takagi, F.anFuuta, S.(994)"On the lumping of unoff moel bae on the unatuate flow theo "JSCE NO.497/II- 8,-9 [3]Fujita, M.Yamamoto, T.Takemoto, N.an Boaghpou, S. (99)"Epeimental tu on the unatuate infiltation poce with uface flow" poceeing of JSCE Hokaio Banch, vol.48, [4]Takaao, T. an Shiiba. M (985) "Lumping of kinematic wave moel Jounal of Diate Pevention Reeach Intitute Annual, No 8B-,3- [5] Fujita, M (98) The impact of Vaiation of Slope Length on Stoage Dichage elationhip Poceeing of JSCE No.34,75-86 Figue : pofile of oil column ISSN: Iue 6, Volume 4, June 8

5 Siamak Boaghpou, See Ahma Mibaghei an See Aman Hahemi Monfae ARTIFICIAL RAIN FILTER TENSIOMETER Figue : pofile of epeimental oil Figue 3: Pofile of ichage α. Figue 4: Relationhip of toage an ichage( S ~ q ),α. Figue 5: Relationhip of toage an ichage ( S ~ q ), α. Figue 6: ange of paamete fo γ ISSN: Iue 6, Volume 4, June 8

6 Siamak Boaghpou, See Ahma Mibaghei an See Aman Hahemi Monfae Figue 7: pofile of ichage a.5,.5, α, l Figue 8: elationhip of toage an ichage ISSN: Iue 6, Volume 4, June 8