The solution of unconfined water seepage problem in saturated-unsaturated soil using Bathe algorithm and Signorini condition

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1 IOP Cofeece Seie: Eath ad Eviometal Sciece PAPER OPEN ACCESS The olutio of ucofied wate eepage poblem i atuated-uatuated oil uig Bathe algoithm ad Sigoii coditio To cite thi aticle: Zhogi Dou et al 7 IOP Cof. Se.: Eath Evio. Sci Related cotet - A modified iteative eemble Kalma filte data aimilatio method Baoxiog Xu, Yulog Bai, Yizhao Wag et al. - Effect of hyteei o the tability of a embakmet ude taiet eepage K Liu, P J Vado, P Aold et al. - NONLINEARITIES IN THE ITT FW-3 PHOTOMULTIPLIER David L. DuPuy View the aticle olie fo update ad ehacemet. Thi cotet wa dowloaded fom IP adde o 4/5/8 at 9:36

2 3d Iteatioal Cofeece o Advace i Eegy, Eviomet ad Chemical Egieeig IOP Publihig IOP Cof. Seie: Eath ad Eviometal Sciece 69 (7) 7 doi:.88/755-35/69//7 The olutio of ucofied wate eepage poblem i atuated-uatuated oil uig Bathe algoithm ad Sigoii coditio Zhogi Dou,, Jiwe Wu, Hogmei Zhag * ad Kai Huag School of Eath ad Eviomet, Ahui Uiveity of Sciece ad Techology, Huaia, Ahui 3, Chia Huaia Vocatioal Techical College, Huaia, Ahui 3, Chia hzyxydz@63.com Abtact: Bathe algoithm covet poblem of detemiig ie ifiltatio face ito a mateial olieaity poblem, ad Sigoii coditio covet outflow bouday ito covetioal wate bouday. Becaue a uatuated eepage poblem i alo a oliea flow poblem, though the oit applicatio of thi method, olvig vaiatioal iequalitie i avoided. I covetioal mateial olieaity fiite elemet famewok with mall modificatio, uig NR algoithm, the olvig of ucofied wate eepage poblem icludig uteady ad uatuated wate eepage poblem i ealized. The followig ae dicued: () the impovemet of covegece of Bathe method ad it geealized fom i a thee-dimeioal model; () ue Switchig Algoithm of Sigoii coditio to olve wate eepage poblem; (3) the ude-elaxatio method of impovig the covegece ad ma coevatio whe olvig uatuated ad uteady wate eepage poblem. At lat, though claic calculatio example, the calculatig accuacy ad applicatio field of the popoed method ae evaluated.. Itoductio Cuetly, ucofied wate eepage poblem ca be olved uig eithe atuated oil wate eepage theoy o uatuated oil wate eepage theoy. Satuated oil wate eepage theoy i cocluded i uatuated oil wate eepage theoy, ad the goveig equatio of them ae the ame [-]. Howeve, uatuated oil wate eepage theoy thik that the pemeability coefficiet of oil deped o ot oly oil popetie, but alo peue head. Whe uatuated oil i ued to olve wate eepage poblem, uatuated oil zoe ad atuated oil zoe ae ot eceay to be claified, ad thi itegate the olvig poce of atuated ad uatuated wate eepage poblem, ad thi method ha wide applicatio popect. But the elatio betwee pemeability coefficiet ad peue head of uatuated oil i difficult to acquied, ad geeally i egieeig, the focu i the flow patte ad peue ditibutio i atuated oil zoe. So i egieeig, the elatio betwee pemeability coefficiet ad peue head i uually implified ito Heaviide Step Fuctio. Thi implifyig method eue the olutio accuacy of atuated oil zoe, at the expee of the olutio accuacy of uatuated oil zoe. Bathe method ue thi kid of implificatio [3]. Bathe method i a kid of fixed-gid method, ad the fiite-elemet mehe emai uchaged duig olvig poce. May othe method of olvig ucofied atuated wate eepage poblem ae belog to fixed-gid method, uch a Deai eidual flow method [4], Yutia Zhag iitial flow Cotet fom thi wok may be ued ude the tem of the Ceative Commo Attibutio 3. licece. Ay futhe ditibutio of thi wok mut maitai attibutio to the autho() ad the title of the wok, oual citatio ad DOI. Publihed ude licece by IOP Publihig Ltd

3 3d Iteatioal Cofeece o Advace i Eegy, Eviomet ad Chemical Egieeig IOP Publihig IOP Cof. Seie: Eath ad Eviometal Sciece 69 (7) 7 doi:.88/755-35/69//7 method [5], a well a famou Lacy method [6], ad o o. Though the deivatio bai of thee method i ot the ame, they evetually eed olve a poblem caued by Heaviide Fuctio ad imila to the igid-platic mateial olieaity poblem, ad ue the method of Newto-Rapho(NR) of claic mateial olieaity fiite elemet famewok to olve. Of coue, becaue of the popetie of Heaviide Fuctio, it ca meet covegece poblem whe thee method ae ued to olve ucofied atuated wate eepage poblem. Yua Wag [7] itoduce elemet adutmet paamete ad ito Heaviide Fuctio to modify it, ad impove the covegece of thee method. Coepod with the fixed-gid method, aothe kid of method, that ca be ued to olve ucofied atuated wate eepage poblem, i meh(ode) adutmet method. Thi method i alway ued. But whe olvig o-homogeeou oil wate eepage poblem, thi method ofte fail to olve becaue of difficult to adut the ode at boudaie. Beide, becaue of the chage of mehe, it i icoveiece to olve ock-oil couplig poblem. Baed o thi, thee i a ted of eplacig thi method with fixed-gid method. No matte what the method i, aothe poblem will be ievitably ivolved: the detemiatio of outflow bouday. Whe modelig, the exact age of outflow bouday i ukow ad eed to be detemied by iteative calculatio. Thee ae may diffeet olutio o thi poblem. Fo example, A. Laabi ad F. D. E. Smedt [8] et the outflow bouday a cotat wate bouday, if calculated flow of a ode i poitive (wate oaked ito the oil), the et thi ode a impeviou bouday at the begiig of ext-tep calculatio, util covegece. But pactice how that the covegece of thi method i ot good, ad ocillatio i eay to appea. The method of C. S. Deai ad G. C. Li [4] i that a laye of high-pemeable elemet i attached o the poible outflow bouday. But Hog Zheg [9] thik that thi method will be icoveiet whe multi-outflow poit. H. Zheg [] poit out that poible outflow bouday hould atify the Sigoii coditio, ad he etablihe vaiatioal iequality ad ue o-fully complemetay algoithm to olve. Whe uig Sigoii coditio to olve ucofied atuated wate eepage poblem, the outflow bouday ad impeviou bouday ae uified teated, ad thi ca eaily olve the poblem of detemiig the outflow bouday. But H. Zheg method will evetually olve a et of dicete vaiatioal iequalitie, ad thi i quite difficult to olve. Actually, Sigoii coditio i fequetly ecouteed bouday coditio i cotact poblem, ad thee ae elatively matue iteative algoithm i covetioal fiite elemet olvig method. Fo example, M. Aitchio ad M. W. Poole [] popoed Switchig Algoithm, ad thi algoithm coveietly ealized the olutio of Sigoii coditio, with almot o chage i the covetioal fiite elemet olutio famewok. Baed o Bathe algoithm, thi pape popoe a exchage algoithm fo wate eepage poblem, coduct umeical imulatio with Sigoii coditio, ad detemie wate eepage outflow bouday. Though oit applicatio of Bathe algoithm ad Sigoii coditio, the ucofied wate eepage poblem i atuated oil will be coveted ito covetioal oliea cotitutive poblem. Ad ue covetioal NR algoithm to olve ucofied wate eepage poblem, icludig uteady wate eepage flow ad uatuated wate eepage flow poblem.. Goveig equatio of wate eepage poblem ad it fiite elemet dicete fom Duig the aalyi poce, aume that the total te ad poe peue keep cotat, o the goveig equatio of wate eepage poblem [] i: h h h h kx k y kz wmw () x x y y z z t whee: h i watehead; k x, k y ad k z i epectively x, y ad z diectio compoet of pemeability coefficiet; i pecific weight of wate; m w i the lope of the cuve of volumetic oil wate cotet alog with the chage of poe peue u, d du; t i time. Afte backwad diffeece i time, the fiite elemet dicete fom of Equatio ()baed o the watehead (heeiafte efeed to h- type) i: m w

4 3d Iteatioal Cofeece o Advace i Eegy, Eviomet ad Chemical Egieeig IOP Publihig IOP Cof. Seie: Eath ad Eviometal Sciece 69 (7) 7 doi:.88/755-35/69//7 whee, T C C h t t h t Q t t (a) t t e e g i T i i i i T i i T B K B dv B K B V i i i J W e e g i T i i i i T i i C N N dv N N J W V i i i whee, T, C i epectively hydaulic coductivity matix ad wate eleae matix; K i pemeability coefficiet matix of oil; B i i watehead gadiet itepolatio matix of elemet; h, h N i itepolatio fuctio matix of elemet; t t ; Q ukow ode watehead at the time of t ad time of tt (b) t t i epectively colum matix of i colum matix of ode flow at the t t ; J i om of Jacobi matix; W i itegal poit weight; m ; w w e i the total umbe of elemet i model; g i the umbe of itegatio poit of a elemet; i ad i epectively coute of elemet ad itegatio poit. C ad ae all cotat. Whe a o-atuated wate Whe wate i cofied, T, eepage flow, they ae ot cotat, Equatio (a) will be a et of oliea equatio ad eed iteatio to olve it. Fo ucofied uatuated wate eepage poblem, afte tep oliea pemeability coefficiet matix K i itoduced ito Equatio (a), Bathe algoithm i acquied. 3. Bathe algoithm ad Sigoii coditio 3.. Bathe algoithm 3.. Satuated wate eepage flow. A typical ucofied atuated wate eepage poblem i how i Fig.. Heaviide tep fuctio H h z i itoduced ito Bathe algoithm a elative pemeability coefficiet k. Accodig to Equatio (3a), the atuated pemeability coefficiet k i aduted ad the get pemeability coefficiet k. Solvig domai i expaded to the whole domai ABCDA. Pemeability coefficiet of atuatio zoe i k, ad pemeability coefficiet of uatuated zoe i. The aumptio of uatuated zoe d beig ahydou dy i: Fig. Sketch of wate eepage i eath dam 3

5 3d Iteatioal Cofeece o Advace i Eegy, Eviomet ad Chemical Egieeig IOP Publihig IOP Cof. Seie: Eath ad Eviometal Sciece 69 (7) 7 doi:.88/755-35/69//7 k k h zk Hh zk h h z Hh z h (3a) k z (3b) z whee: z i tatical watehead; i a mall umbe, theoetically, coideig the tability of the umeical calculatio, take value a.; h i the height. H, thi will eult to ocillatio duig olvig poce ad difficult to covege. I ode to impove it covegece ad accuacy of umeical itegatio, Yua Wag [7] itoduce elemet adutmet paamete ad ito Heaviide Fuctio to modify it, ad covet Equatio (3), which i imila to igid-platic poblem, to the poblem which i imila to ideal elatoplatic poblem (a how i Fig.). Becaue Equatio (3a) ad (3b) cotai Heaviide tep fuctio h z Fig. Relatio betwee elative pemeability ad peue watehead If peue watehead of a itegatio poit of a elemet i le tha, the thi itegatio poit make a miimum cotibutio to the flow of the ode ( k ). If peue watehead of a itegatio poit of a elemet i geate tha, the thi itegatio poit make a maximum cotibutio to the flow of the ode ( k ). Ad if peue watehead i betwee ad, the liea itepolatio. Afte adutmet accodig to thi method, Equatio (3b) become: h z h z k h z (4) h z The key of Equatio (4) i the value of ad. S. J. Lacy ad H. Pevot [6] povide the method of takig the value of fo ectagula mehe of plae poblem, that i, the value of i the the maximum value of peue watehead diffeece betwee the lowet ode ad the lowet itegatio poit of a ectagula elemet. The method of takig the value of i imila to [7], that i, the value of i the the maximum value of peue watehead diffeece betwee the highet ode ad the highet itegatio poit of a ectagula elemet. Exted thi method to othe elemet, icludig 3D elemet, ad the eult ae how i Fig.3. Notig that: the value of ad i popotioal to l (whe D poblem, l i the poitio alog the aothe diectio; whe 3D 4

6 3d Iteatioal Cofeece o Advace i Eegy, Eviomet ad Chemical Egieeig IOP Publihig IOP Cof. Seie: Eath ad Eviometal Sciece 69 (7) 7 doi:.88/755-35/69//7 poblem, l i vecto um of two poitio vecto alog the aothe two diectio), o fo 3D elemet. (a) (b) Fig.3 Value of coectio coefficiet ad 3.. Uatuated wate eepage flow. Pemeability coefficiet of uatuated wate eepage flow deped o effective oil wate atuatio, ad i a highly oliea poblem. Fequetly-ued model of pemeability coefficiet fo uatuated oil ae Va Geutchte model [], Fedlud model [3], Gade model [4], ad o o. Though Gade model i ot eceaily uitable fo the actual oil, it make Equatio () eay to lieaize ad eay to calculate aalytic olutio. Becaue Bathe algoithm i ealize though oliea cotitutive of pemeability coefficiet (Equatio (3) o (4)), covetig Equatio (3) o (4) ito Va Geutchte model, Fedlud model o othe oliea cotitutive model of pemeability coefficiet ca coveietly ealize the imulatio of uatuated wate eepage flow. Futhemoe, becaue the olieaity of cotitutive model of uatuated wate eepage flow i lowe tha Equatio (3), it covegece i geeally bette tha Equatio (3) o (4). 3.. Sigoii coditio of outflow bouday ad it ealizatio Bathe algoithm effectively covet poblem of detemiig ie ifiltatio face ito mateial olieaity poblem. Howeve, thee i a poblem to be olved, that i, the detemiatio of outflow poit G (a how i Fig.). GF i outflow bouday, ad GCBE i impeviou bouday. I Bathe algoithm, outflow poit G ca be obtaied uig extapolatio method, but thi i quite difficult [], epecially o 3D poblem. Takig advatage of Sigoii coditio, outflow bouday GF ad impeviou bouday GCBE ae teated togethe i thi pape. Ad it i t difficult to fid that, bouday FGCBE atifie the followig Sigoii bouday bouday : h y, q (5) h y q 3.3. Switchig Algoithm of Sigoii coditio I thi pape, Switchig Algoithm (SA) popoed by J. M. Aitchio ad M. W. Poole []. The advatage of SA i that Sigoii bouday bouday ca be ued to umeical imulatio at the expee of ut mall chage, ad covetioal fiite elemet olutio famewok doe ot have to chage. The detail of SA ae the followig: ) Iitialize poible outflow bouday a impeviou bouday, ad tat iteatio calculatio. ) Duig iteatio calculatio, accodig to the eult of the peviou iteatio, poible outflow 5

7 3d Iteatioal Cofeece o Advace i Eegy, Eviomet ad Chemical Egieeig IOP Publihig IOP Cof. Seie: Eath ad Eviometal Sciece 69 (7) 7 doi:.88/755-35/69//7 bouday ca be detemied i the ext iteatio a followig: () If the bouday of a ode, which i o the poible outflow bouday, i impeviou bouday i peviou iteatio, but i cuet iteatio, the calculated peue watehead i geate tha (it iitial value i.), the et thi bouday a cotat wate bouday, ad the value of watehead i tatical head. () If the bouday of a ode, which i o the poible outflow bouday, i et a cotat wate bouday i peviou iteatio, but calculated wate eepage flow i geate tha i cuet iteatio, the et thi bouday a outflow bouday. It ca be foud i example behid that the covegece ate of thi algoithm i vey fat, ad thi algoithm ha little impact o the oveall olutio efficiecy. 4. Fiite elemet implemetatio 4.. NR iteatio Subtitutig Equatio (3a) ad (4) ito (a), ad uig Switchig Algoithm (Equatio (5)), a FEM pogam i compiled. NR iteatio method i ued to olve, ad withi a time tep t, iteatio equatio ae how a followig: k k k A h R k k k C A T t k k R Q b F (6) k k k k C k F T h h h t Covegece coditio of thi iteatio i: k k k h h h k k R F TOL Whe uig modified NR method to olve, A k ut eed to be chaged ito A. Thu, A k eed t to be decompoed at each time tep if Sigoii coditio i atified. Geeally, the efficiecy of modified NR i highe tha NR. I Equatio (6) ad (7), k i the cuet iteatio tep, TOL i covegece toleace ad it value i geeally et.. i elaxatio facto, ad accodig to covegece, the age of it value i.. Whe take a mall value, ocillatio of iteatio eult ca be uppeed. But if the value of i quite mall, covegece ate will be lowed. So the value of eed to be aduted accodig to pecific cicumtace. 4.. Ude-elaxatio (UR) poce o k ad Duig uig Equatio (6) to olve with iteatio ad updatig T k ad C k poce o k ad m w (7), ude-elaxatio m w ca effectively uppe the ocillatio of iteatio eult [5]. The the eult k, that of umeical expeimet i thi pape how that, uig UR ude-elaxatio method [5] o i k take the aveage value of peviou two iteatio eult, ad thi ca igificatly uppe the ocillatio of Equatio (6) ad impove the covegece of NR algoithm. Ue UR ude-elaxatio method [5] o m, that i: w 6

8 3d Iteatioal Cofeece o Advace i Eegy, Eviomet ad Chemical Egieeig IOP Publihig IOP Cof. Seie: Eath ad Eviometal Sciece 69 (7) 7 doi:.88/755-35/69//7 k k mw k u u Equatio (8) ca igificatly impove the ma coevatio of dicete-fomed Equatio(6). (8) 5. Applicatio Accodig to above algoithm, fou calculatio ample ae coducted ad aalyzed. Fit two of them ae atuated ad teady wate eepage poblem, ad lat two of them ae uatuated ad uteady wate eepage poblem. Equatio (4) i ued a eepage model to olve atuated ad teady wate eepage poblem. Howeve, uatuated wate eepage model mut to be ued to olve uatuated ad uteady wate eepage poblem, oe i Gade model, ad the othe i Va Geuchte model. Paamete fo olvig uatuated wate eepage model i calculatio ample ae how i Table. Table Paamete fo uatuated wate eepage model Satuatio No-atuatio paamete Name of Type of paamete model oil k /(m. - m w /(kpa ) ) Gade Silt Va Sady Geuchte oil Silt Note: ad ae fittig paamete. Fo each calculatio ample, both D ad 3D model ae calculated, ad compae the calculated eult with the publihed eult o aalytical olutio of ome liteatue. Duig calculatig, ix-ode quadatic tiagula elemet ad te-ode quadatic tetahedal elemet ae epectively ued i D ad 3D model. 5.. Satuated ad teady wate eepage poblem Two calculatio ample fo atuated ad teady wate eepage poblem ae how i Fig.4, ad thee two ample ae widely ued to Veify othe method [6,6-8]. The fit calculatio ample i the typical Mukat poblem. AD ad BE ae wate boudaie, the height of wate ae epectively m ad m; EC i poible outflow bouday, ad othe boudaie ae impeviou bouday. The ecod calculatio ample i imila to the fit oe, the diffeece i that oil i ot a igle. The pemeability coefficiet of the zoe at the ight ide of lie mm i time of left. Fig.4 how the calculated ifiltatio face uig modified NR algoithm. It ca be foud that: () The calculated eult of thi pape ae coitet with publihed eeach eult of othe eeache, ad i the fit calculatio ample, the calculated locatio of outflow poit i i good ageemet with the aalytical olutio i liteatue [9]; The calculated eult of D i coitet with the eult of 3D, ad thi idicate that the value takig method fo ad (a how i Fig.3) i feaible; (3) Both the calculated eult of D ad 3D idicate that the calculated eult ae ot eitive to the ize of elemet; (4) Equatio (4) ca be effective to be ued to olve atuated wate eepage flow poblem. Compaio of calculatig efficiecy of NR ad modified NR algoithm i how i Table. Fom Table, it ca be foud that modified NR algoithm i moe teady ad high-efficiecy tha NR algoithm to olve atuated wate eepage flow poblem. O the flow of eepage, all calculated eult ae coitet with aalytic olutio. It ca alo be foud fom Table that the efficiecy of detemiig outflow bouday uig Sigoii coditio i vey high, ad umbe of iteatio i mall elative to oveall calculatig umbe, epecially i the example #. I the example #, becaue the oil i ot homogeeou, a lage umbe of calculatio ae ued to detemie the ie ifiltatio face. 7

9 3d Iteatioal Cofeece o Advace i Eegy, Eviomet ad Chemical Egieeig IOP Publihig IOP Cof. Seie: Eath ad Eviometal Sciece 69 (7) 7 doi:.88/755-35/69//7 SN of ample (a) Homogeeou (b) No homogeeou Fig.4 Veificatio fo atuated teady wate eepage poblem Dimeio D 3D D 3D Table Efficiecy compaio of NR ad modified NR method Model Iteatio Solvig Time/ Flow/(m 3.d - ) MS/m mnr NR mnr NR mnr NR. (5) 8(4) (9) (9) (6) 8(6) (3) () (3) 55(45) (3) NC 5.79 NC 7.45 NC. 7(4) 33(63) (46) 785(68) Aalytical Solutio Note: () MS mea meh ize; NR mea Newto-Rapho algoithm; mnr mea modified Newto-Rapho algoithm; NC mea Not Covege. () The umbe iide of backet mea the umbe of iteatio fo detemiig the outflow bouday uig Sigoii coditio ad witchig algoithm. (3) + mea that the olutio ae calculated baed o Dupuit aumptio (accodig to the eeach eult of O. D. L. Stack [], the calculated olutio baed o Dupuit aumptio ae exact olutio. Beide, if Equatio (3) i choe a eepage cotitutive model, oly ample # with big-ize elemet ca covege well, othe ca t each the covegece citeia of Equatio (7), eithe NR algoithm o modified NR algoithm, ad eithe D model o 3D model. Thi idicate that Equatio (4), which i modified with ad, ha a bette covegece. 5.. Uatuated ad uteady wate eepage poblem 8

10 3d Iteatioal Cofeece o Advace i Eegy, Eviomet ad Chemical Egieeig IOP Publihig IOP Cof. Seie: Eath ad Eviometal Sciece 69 (7) 7 doi:.88/755-35/69// Wate ifiltatio poblem ito oil pilla.thi poblem i oe-dimeioal uatuated ad uteady ifiltatio ito homogeeou pilla poblem, a how i Fig.5(a). Gade model i ued a ifiltatio model, ad it paamete i how i Table. The bouday coditio ae: () whe t, B i cotat wate-head bouday, ad H B. m ; () whe t, A i ilet bouday, ad it eepage tegth q =.m/(d.m ). The meh ize of both D ad 3D model ae all.5m. The total legth of time i 5 d, ad thee ae 54 time tep. The legth of a tep i fit time tep i.5d, the legth of a tep i ext 5 time tep i.d, ad the legth of a tep i the lat 4 time tep i d. The calculated eult ae how i Fig.5(b). Fom Fig.5(b), it ca be foud that the calculated eult ae coitet with aalytic olutio i liteatue []. (a) Model of umeical calculatio time (b) Peue watehead ditibutio at diffeet Fig.5 Uatuated oil colum ifiltatio poblem I tem of ma coevatio, Q b i defied a the wate flow of eteig the model fom k bouday i each time tep, Qb F t. Q i defied a the amout of wate toed i the model becaue of the chage of volumetic wate cotet of atuated oil, Q e g dv J w. i I thi D ample (3D model ae imila to D model), the cuve of cumulative ma atio Q Q a time i how i Fig.6. I Fig.6, STS mea mall time tep, ad ude thi b coditio, the total calculatig time i.5d/tep tep +.d/tep 5tep + d/tep 4tep; LTS mea mall time tep, ad ude thi coditio, the total calculatig time i.5d/tep tep +.d/tep 5tep + d/tep 4tep; FM mea fie meh, ad it ize i.5m ad equal to the ize of meh i Fig.5(b); CM mea coae meh, ad it ize i.m. Fig.6 how the impact of ix kid of coditio, uch a diffeet ma matix updatig method, the ize of meh, the legth of time tep, ad o o, o ma coevatio. It ca be foud fom Fig.6 that, UR method i good i tem of ma coevatio, ad the legth of time tep ad the ize of meh ha little impact o it. Ude the thee coditio (UR+SLS+FM, UR+LLS+FM, a well a UR+SLS+FM, a how i Fig.6), cumulative ma atio of each time tep ae all, ad the pheomeo of o-coevatio of ma, poited out by M. A. Celia [] whe uig h-type dicete fom (Equatio ()), doe ot appea. The C, the calculatio method of m i diffeet. M. A. eao i that, whe olvig ma matix Celia [] ued UR method, ad a a cotat, both UR ad UR method ae ued i thi pape. Fom w 9

11 Cumulative ma atio(σq/σqb) 3d Iteatioal Cofeece o Advace i Eegy, Eviomet ad Chemical Egieeig IOP Publihig IOP Cof. Seie: Eath ad Eviometal Sciece 69 (7) 7 doi:.88/755-35/69//7 Fig.6, it ca be foud that, whe uig UR method, the calculated eult of cumulative ma atio ae elatively poo, epecially i the begiig of ifiltatio, the coditio of pemeability coefficiet ad wate cotet chage damatically at thi time. Thi idicate that UR method i ot uitable fo h-type dicete fom. Meawhile, the umeical calculatio eult how that, the efficiecy of NR algoithm i bette tha modified NR algoithm if the demad of accuacy i the ame. Whe the bouday coditio ae the ame, if NR algoithm i ued, the calculatig time of D ad 3D model i epectively.3 ad 3.9, ad if modified NR algoithm i ued, the calculatig time of D ad 3D model i epectively 36. ad 3.. So, mnr algoithm i much lowe tha NR algoithm UR+STS+FM UR+LTS+FM UR+LTS+CM UR+STS+FM UR+LTS+FM UR+LTS+CM.5... Time/d Fig.6 Ma coevatio ude diffeet UR method Beide, if UR method i ot choe ad UR i choe to poce covege, ad covegece of the modified NR algoithm alo deteioate. k, NR algoithm will ot 5... Multi-laye oil lope ifiltatio poblem. The ecod uatuated ad uteady wate eepage poblem i 3-laye oil lope ifiltatio poblem. The calculatig model i how i Fig.7, ad i thi model, the oil i middle laye i ilt, ad ady oil i i the top ad bottom laye. Va Gechte model i choe a ifiltatio model, ad the paamete ae how i Table. The bouday coditio ae: () Whe t, AFE i wate bouday, h A =.m, h F =.m, h E =.3m, ED i outflow bouday, CBA i impeviou bouday; () whe t, DC i ifiltatio bouday, ad -4 ifiltatio tegth i q. m/(.m ). Ad the iitial coditio i the teady-tate watehead ude the bouday coditio (). The chaacteitic of thi poblem i multiple outflow boudaie. J. J. Rulo [3] tudied thi poblem uig ad tak expeimet. The elemet ize of D ad 3D model ued to imulate thi poblem ae all.5m, ad the time tep i 5. Fig.7 how the chage of ifiltatio face with time. Fom Fig.7, it ca be foud that, the calculated eult of thi pape ae coitet with the eult of pofeioal oftwae GeoStudio ad expeimetal eult. Thi idicate that the method popoed i thi pape ca be ued to calculate multiple outflow boudaie poblem.

12 3d Iteatioal Cofeece o Advace i Eegy, Eviomet ad Chemical Egieeig IOP Publihig IOP Cof. Seie: Eath ad Eviometal Sciece 69 (7) 7 doi:.88/755-35/69//7 Fig.7 Calculated eult of 3-laye oil lope ifiltatio poblem I tem of ma coevatio ad calculatig efficiecy, they ae imila to the example #3, that i, cumulative ma atio of UR method i much bette tha UR, the calculatig efficiecy of NR algoithm i bette tha modified NR whe olvig uteady wate eepage poblem. 6. Cocluio Though oit applicatio of Bathe algoithm ad Sigoii coditio, the ucofied wate eepage poblem i atuated ad uatuated oil i coveted ito covetioal oliea cotitutive poblem, ad olvig vaiatioal iequalitie i avoided. I covetioal mateial olieaity fiite elemet famewok with mall modificatio,, uig NR algoithm, the olvig of ucofied wate eepage poblem icludig uteady ad uatuated wate eepage poblem i ealized. The followig cocluio ae obtaied: () The Switchig Algoithm of Sigoii coditio ca be effective to ealize the imulatio of outflow bouday, icludig multiple outflow boudaie. () I tem of ucofied atuated wate eepage, Equatio (4), which i modified with ad, ha a bette covegece, whethe D o 3D model. (3) I tem of elatic-pefectly platic cotitutive model (Equatio (4)) of ucofied atuated wate eepage, modified NR algoithm i teady ad high efficiecy. But i tem of uatuated ad uteady wate eepage, NR algoithm i moe efficiet. (4) Fo h-type FEM dicete fom of uatuated ad uteady wate eepage (Equatio (a)), UR method ca eue ma coevatio of calculated eult, but UR method i difficult to each that. (5) I tem of elative pemeability coefficiet k, UR ude-elaxatio method ca igificatly etai the ocillatio duig olvig ad impove the covegece of NR algoithm. Ackowledgemet Thi wok wa fiacially uppoted by the Natual Sciece Key Poect of Uiveitie i Ahui Povice (KJ6A86). Refeece: [] LAM L., FREDLUND D. G., BARBOUR. S. L.. Taiet eepage model fo atuated-uatuated oil ytem: a geotechical egieeig appoach[j]. Caadia Geotechical Joual, 987, 4(4): [] HUYAKORN P. S., THOMAS S. D., THOMPSON B. M.. Techique fo makig fiite elemet competitive i modelig flow i vaiably atuated poou media[j]. Wate Reouce Reeach, 984, (8): 99-5.

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