A SIMULTANEOUS SOLUTION APPROACH FOR COUPLED SURFACE AND SUBSURFACE FLOW MODELING

Transcription

1 IMULTEOU OLUTIO PPROCH FOR COUPLED URFCE D UURFCE FLOW MODELIG b Oa Guuz a Musafa M. al Mula Evoal ulaos Laboao cool of Cvl a Evoal E Goa Isu of Tcolo laa G 0 MEL-0-0 pl 00

2 IMULTEOU OLUTIO PPROCH FOR COUPLED URFCE D UURFCE FLOW MODELIG b Oa Guuz a Musafa M. al Mula Evoal ulaos Laboao cool of Cvl a Evoal E Goa Isu of Tcolo laa G 0 T o o c s po s bas as suppo b cool of Cvl a Evoal E Goa Isu of Tcolo a b Goa a Ga aoal a Ga Coll Poa. T poc as as ou Mula Evoal ulaos Laboao of cool of Cvl a Evoal E Goa Isu of Tcolo. MEL-0-0 pl 00

3 uos M. Oa Guuz s a aua su Mula Evoal ulaos Laboao cool of Cvl a Evoal E Goa Isu of Tcolo. D. Musafa M. al s a Pofsso a Dco of Mula Evoal ulaos Laboao cool of Cvl a Evoal E Goa Isu of Tcolo. cols Ts po s bas o o couc Mula Evoal ulaos Laboao a cool of Cvl a Evoal E Goa Isu of Tcolo. Paal facal suppo fo s o as pov b Goa a Ga of aoal a Ga Coll Poa a Goa Isu of Tcolo. uos a aful fo cou suppo of o suos aa of voal ol sus. Copuaoal svcs of s o pov b Mula Evoal ulaos Laboao facls. Oa Guuz a Musafa M. al sulaous soluo appoac fo coupl sufac a subsufac flo ol Rpo o.: MEL-0-0; Poc o.: E-0-G09; pl 00 90p. os Coupl oloc ol sulaous soluo ac av cal flo vcall-ava oua flo

4 Tabl of Cos bsac.... Iouco.... Maacal Mol..... Cal Flo Mol Gov Equaos Ial Coos oua Coos Goua Flo Mol Gov Equaos Ial Coos oua Coos Coupl of Mols...0. ucal oluo..... Cal Flo Mol..... Goua Flo Mol..... ulaous oluo of Coupl Mol...8. pplcao..... Hpocal Cas: l Cal..... Hpocal Cas: Cal o Lo laaa Was Coclusos...7 Rfcs...8 v

5 pp I. Dvao of F Dffc Equaos of Cal Flo...50 pp II. Dvao of Paal Dvavs of F Dffc Equaos of Cal Flo...59 pp III. Ipola Fucos Gal Mo...66 pp IV. Dvao of Gal Fo of Goua Flo Equao...70 pp V. Dvao of El Ials...8 pp VI. ucal Iao of El Ials...97 v

6 Ls of Fus. Coupl ol oa.... Cal flo / oua flo aco (Guuz a al 00) Dsczao of oa Global a quao a copo blocs Pscal sup of pocal oa sl cal Goua a a v sa a po a v sca a upsa boua (P-) Goua a pofl a vaous s alo po (P-) Goua a a v sa a po a v sca a upsa boua (P-) Goua a pofl a vaous s alo po (P-) Pscal sup of pocal oa cal o Goua a a v sa a vaous pos oa a v sca a upsa boua (P-).... Goua a pofl a vaous s alo asc- (P-).... Goua a pofl a vaous s alo asc- (P-).... Goua a pofl a vaous s alo asc- (P-) Goua a a v sa a vaous pos oa a v sca a upsa boua (P-) Goua a pofl a vaous s alo asc- (P-) Goua a pofl a vaous s alo asc- (P-)... 7 v

7 8. Goua a pofl a vaous s alo asc- (P-) Dscz oa of Lo laaa Rv as Dsca oaps obsv al a Rsvll a saos.... Docoo a sao a cuv.... ula vs. asu sa valus Docoo a sao.... ula vs. asu sca valus Docoo a sao Goua a sbuo poc aa a 09// apl cal o a ub sc T sac- us o foula plc f ffc sc fo cal (f Fa 97) Global a local cooa sss a as l cocp oua cooas o as a acual l Rlaosp b boua cooa lobal cooas... 9 Ls of Tabls Tabl. Iao Pos a W Coffcs Gauss uaau Foula (af Zcz a Talo 989) v

8 oao T follo sbols a us s po: a : Locao of sapl po Gaussa quaau (-) : cv coss-scoal aa of cal flo (L ) o : Iacv coss-scoal aa (off-cal soa) (L ) : cv coss-scoal op (L) o : Iacv coss-scoal op (L) c : U ss p cosa Ma s quao (-) E : Du paa (-) f : Fuco (-) : Gavaoal acclao (LT - ) : T -cooa of paac quao f v oa (-) : T -cooa of paac quao f v oa (-) : Dosa boua coo a sufac lvao (L) : Goua aulc a abov a au (L) 0 : Ial oua a (L) ĥ : ppoa oua aulc a abov a au (L) ( ĥ ) av : El ava appoa oua a abov a au (L) : Cal flo a sufac lvao (sa) abov a au (L) 0 : Ial a sufac lvao cal (L) T : Toal aloss a uco (L) u : Upsa boua coo a sufac lvao (L) H D : pcf a boua coo valu (L) I : Iflao a (LT - ) v

9 : Cal covac (L T - ) c : Coaco/paso coffc (-) : Haulc coucv of cal boo s (LT - ) : Haulc coucv loual co (LT - ) : Haulc coucv asvs co (LT - ) L : Mou flu u o laal spa flo/ouflo (L T - ) L : Toal cal flo oa l (L) : Cal b css (L) c : Ma s ouss coffc cal (L -/ T) : ub of lls (-) : Toal ub of os cal o o oua flo oa (-) : fuco (-) : sap fuco (-) : ub of os cal (-) q L : Laal flo/ouflo p cal l (L T - ) q C : Ha-p boua coo flu valu (L T - ) q : pcf flu boua coo valu (L T - ) : Cal sca (L T - ) 0 : Ial cal sca (L T - ) : Dosa boua coo sca (L T - ) u : Upsa boua coo sca (L T - ) : Wll sca (LT - ) R : Rsual Gal o (-) R : Haulc aus (L) s : cal paa ulaao coffc (-) s c : uos faco fo cou quao (-) s : uos faco fo ou quao (-) : Coaco/paso slop (-) f : Cal/floopla boua fco slop (-)

10 : pcfc l of ucof aquf (-) : T cooa (T) a paa (-) V : Flo vloc (LT - ) : W cosa Gaussa quaau (-) : W p of cal b (L) : Global loual cooa (L) : Global asvs cooa (L) z b : Top lvao of boo pvous la abov a au (L) z : oo lvao of cal b abov a au (L) α : T paa Gal o (-) α c : paa o-rapso o us o valua fs sa of uo vaabls (-) β : Mou coco coffc fo o-ufo vloc sbuo (-) δ : Dac la fuco (au - ) γ : Iao-p ulaao coffc (-) ε : Mau ca aulc a fo all os b o aos (L) : Local asvs cooa (L) θ : W faco fo spac vav fou-po sc (-) : Local loual cooa (L) : Global oua flo oa boua (L) : pcf a boua oua flo ol (L) : pcf flu boua oua flo ol (L) : Ha-p boua oua flo ol (L) ψ : W faco fo vav fou-po sc (-) : Global oua flo oa (L ) : Local oua flo oa (L ) : Global coffc a : Global loa vco f : Vco of aba fucos

11 F : Loa vco ĥ : Global appoa aulc a vco J : Jacoba a o-rapso sc M : Mass a : U oal vco o boua : ffss a : Vco of uo vaabls l o : Ics

12 ulaous oluo ppoac fo Coupl ufac a ubsufac Flo Mol Oa Guuz a Musafa M. al Mula Evoal ulaos Laboao cool of Cvl a Evoal E Goa Isu of Tcolo laa G 0 bsac Coupl oloc ol of sufac a subsufac sss as a a lo of popula las o cas. Cal flo a oua flo ols a coupl o oba a b usa of s oloc paas a as. I al s coupl pocss s o a av faso ul suffc covc s acv fo coo paas l s ff oas. I s su popos a soluo oolo bas o sulaous soluo of cal a oua flo. T o s bas o a of solv a sl lobal a a oc a a solv spaa acs fo ac flo oa a avl pov soluo. Ts soluo cqu s s b coupl a o-soal sa flo ol a uss copl fo of. Va quao a o-soal vcallava oua flo ol. T popos sulaous soluo appoac povs a o ffc soluo fo s coupl flo pobl a s supo o appoa soluo oba ou a av appoac.. Iouco T ov of a oloc ccl s a copl poo bcaus of uous flo paas a acos b s paas. Dsc ol of s sufac a subsufac paas as b su svl a vaous ols of ff copls av b foula pas (McDoal a Habau 988; al 990; Fa 99). I all of s sus s l o o aco b ff paas. Ev ou s ols pov oo suls sula olo of oas sa o so vaos fo obsv aa s acos bco cosabl poa. Ts sc a sola au of s ols ova sacs o focus o coupl ols. P a au (97) a Wools (97) a Fz (97) ao als o o coupl ols. Mo cl Cua a cla (979) a a Y (98a) a a Wl (99) Vaaa a

13 Loau (00) a Moa a Y (00) av all foula coupl ols of vaous copls o sula acos of sufac a subsufac sss. T lvl of copl s all b ub of sos clu ol as ll as cluso of all possbl pscal poa. I s a os avac ol oul volv a -soal sufac flo copo bas o copl av-os quaos a a -soal vaabl saua subsufac flo copo. Cul o suc ol ss oloc lau u o vabl abacs of () suffc copuaoal po (.. soa capac a pocss sp); a () suffc calbao a vfcao aa. caus of s laos s sll v p fo oloc ol o us appoa ols ov la scal applcaos. I s sac aa os of Vaaa a Loau (00) a Moa a Y (00) bo clu a soal vaabl saua subsufac flo copo a o-soal oa av sufac flo copo. I us b o a v s fal copl appoa foulao s ols ca ol b opa o s o sall scal applcaos. I o o pfo paccal sulaos of la scal oas s cssa o a fu splfcaos foulao of pscal ss. O suc appoa foulao of coupl sufac/subsufac flos volvs us of a o-soal appoac fo sufac flo sas a vs a a o-soal vcall ava appoac fo subsufac flo saua zo. Ralss of lvl of copl coupl pocss all ols s bas o a of solv fo coo paas l sufac a subsufac copos. Ts soluo ca b o a av o o-av faso. T o-av cqu volvs a o- soluo of sufac a subsufac flo quaos a s also o as al coupl. O o a av cqu volvs uous soluos of bo sufac a subsufac quaos ul ouco of subsqu soluos bco lss a a olac co. Iav coupl s sos o as al coupl. caus of fac a av coupl alos volv uous soluos of sufac a subsufac flo quaos a sp qu o copuaoal as oppos o al coupl alos. O o a s av pov of soluo all povs o accua suls a us s all pov o b supo ov al coupl. I s su a soluo cqu s popos fo coupl soluo of sufac a subsufac flos. I s appoac sufac a subsufac flo quaos a solv sulaousl sa lobal a sucu a c s o s call sulaous soluo appoac o ffa fo al a al coupl os. c s appoac las fo av soluo of ov quaos s copaabl fas a av coupl alo. Fuo as s o css o cc fo covc of a av soluo a us ls a o accua soluo a av coupl appoac. To s applcabl of s appoac a o-soal sa flo ol bas o ac av fo of Va quaos s coupl a o-soal vcall

14 ava saua oua flo ol o sula sufac/subsufac acos alo v b. T aacal bacou of coupl ol s v sco sco of s po ov paal ffal quaos of cal flo a saua oua flo ols a ps o accopa al a boua coos. I sco ucal soluo oolo fo coupl ol s ps spaal a poal sczao of quaos s v a sulaous soluo oolo s pla. T appoac s fs appl o o pocal oas o s s capabls a laos a fall o a as sou Goa fo al- ol.. Maacal Mol T coupl ol s a cobao of a o-soal cal flo ol a a osoal oua flo ol (Fu ). T cal flo ol s bas o ac av fo of. Va quaos a oua flo ol s bas o vcall-ava ass cosvao quao of oua flo. T l b o ols s pov va laal flo a v-boo fac us lav valus of v a sufac lvao a oua a. Local cooa ss Global cooa ss To-soal oua flo O-soal oa cal flo oa Fu. Coupl ol oa

15 .. Cal Flo Mol... Gov Equaos T ov quaos of o-soal cal flo ol a v b cou a ou quaos of o clu ffcs of aual cal o a caacscs. T ou quao s bas o copl ac av fo of usa o-ufo. Va quaos (Fa 99): sc ( o ) ql 0 () s ( β / ) f L 0 () s c a s a suos facos fo cou a ou quaos s acv coss-scoal aa of flo o s acv (off-cal soa) coss-scoal aa s sca s s loual sac alo cal/floo pla q L s laal flo p cal l a povs l oua flo ol (posv fo flo a av fo ouflo) β s ou coffc fo vloc sbuo s avaoal acclao s a sufac lvao sa (.. sa) L s ou flu u o laal spa flo/ouflo f a a cal/floo pla boua fco slop a coaco/paso slop spcvl. T ou flu u o spa flo/ouflo cal/floo pla boua fco slop a coaco/paso slop a valua as: 0 L ql fo flo fo ouflo () c Δ( / ) Δ () (5) c f / c R c s paso/coaco coffc Δ s ac l c s u ss p cosa c s Ma s ouss coffc s flo covac faco a R s aulc aus. T aulc aus s f as

16 ao of coss-sco aa o p bu ca b appoa as ao of cosssco aa o op fo la vs. I s poa o o a ou flu u o spa flo s assu o b lbl a s o cos ol. T laal flo a povs l b cal flo ol a oua flo ol s f as: > ( z ) q L (6) ( z ) ( z ) s v boo s coucv s v b css z s v boo lvao a s p of v b as so Fu. I ca b s fo psso v Equao (6) a laal flo s a fuco of a sufac lvao a oua a a povs coupl cas b sufac a subsufac flo sss. T coupl s plcl b ov quao fo cal flo ol. RIVER z UIFER Ipvous La z b Dau Fu. Cal flo / oua flo aco (Guuz a al 00) 5

17 ... Ial Coos I o o sa as soluo al valus of ol uos (.. sca a a sufac lvao) a o b spcf alo o-soal sa oa. T al coos ca b oba fo: () fl aa; () a pvous usa ol soluo; o () soluo of sa o-ufo flo quao. I a cas al coos a v as: ( 0) 0 (7) ( 0) (8) 0 0 a 0 ps sca a a sufac lvao cal a b of sulao spcvl.... oua Coos I o-soal cal flo ol a o ff ps of boua coos spcf a () al; a () al bouas of oa. T al boua coos a v a os upsa a osa pos of cal o as al boua coos a spcf a al uco pos of cal o.... Eal oua Coos T popos ol s capabl of ol a o of v cals. T o s all copos of sval upsa a al cals a a sl osa cal. Tfo ol ca accooa sval upsa boua coos a a sl osa boua coo. a upsa boua a sca o a sa oap ca b us as boua coo. Ts coos a pss as sca a sa ss a a v as: ( ) ( ) 0 (9) u ( ) ( ) 0 (0) u a u ps upsa boua sca a a sufac lvao valus spcvl. lal boua coo a osa boua ca also b f as a sca o a sa oap a spcf as: u ( L ) ( ) () 6

18 ( L ) ( ) () a ps osa boua sca a a sufac lvao valus a L s oal oa l. Moov s also possbl o f osa boua coo as a sl-valu a cuv a loop a cuv o a ccal p sco. T sl-valu a cuv aps a pacula sa valu o a cospo sca valu a ca b pss b us la polao a abl of sa-sca aa: ( L ) ( ) () a a coscuv abula aa ss of a cuv a s sa a osa boua. loop a cuv o o a aps a sa valu o sval possbl sca valus p o aulc coos of cal a ca b pss us Ma s quao: c () / ( L ) R / f c f s v b of ou quao as: f ( / ) (5) Fall s also possbl o us a ccal p sco as osa boua coo os osa po of ol oa s a cooll sucu suc as a. I s pacula cas ccal p s app o ccal sca va follo quao: / ( L ) (6) s coss-scoal op of cal. 7

19 ... Ial oua Coos o o o cals sc a cal o fos a uco al boua coos a spcf o sasf ass a balac. T popos ol os o allo fo loop os a qu a s alas a sl ouflo cal fo a uco. T ass balac quao a a uco ca b spcf as: o (7) s oal ub of flo cals o uco s sca a of flo cal o uco o pss sca a b of ouflo cal fo uco a / cospos o ca soa uco. Fo a ol applcaos s a coo pacc o assu a ca soa a uco s lbl copa o ca soa a cal (a a Y 98b; Fa 99; Ja al. 000). Cosqul ass balac quao ucs o a spl cou quao. O o a quao s as: V V o T (8) o ( ) ( ) ( ) a V a sa a flo vloc a of flo cal o uco ( ) o a V o a sa a flo vloc a b of ouflo cal fo uco a T s oal aloss uco. W all flos all bacs o a uco a subccal a a los uco s lbl quao splfs o: ( ) ( ) (9) o.. Goua Flo Mol... Gov Equaos T ov quao of o-soal oua flo a asoopc oooous ucof aquf pcpl pabl cos o ac cooa cos s v b: 8

20 ( z ) ( z ) b ( zb ) ( zb ) [ ( ) δ ( ) δ ( )] 0 q L b ( ) δ ( () ) δ () I (0) s vcall ava aulc a z b s op lvao of boo pvous la a a asoopc saua aulc coucvs s flo a of ll loca a po ( ) oa (.. posv fo a c ll a av fo a sca ll) s ub of lls oa δ s Dac la fuco q L s laal flo a v-boo fac f b Equao (6) (.. posv fo laal flo a av fo laal ouflo) a a Casa cooa copos of paac quao of v oa s ub of vs oa s paa I s flao a a s spcfc l of aquf.... Ial Coos T al valus of aulc a o a spcf as al coos of oua flo ol:... oua Coos ( 0) ( ) 0 () T ff ps of boua coos ca b spcf alo ff al bouas of oua flo oa. Tp- o spcf a boua coos a us o ol bouas o aulc a valus. I s also o as a Dcl boua coo a s v as: ( ) H ( ) () D H D s o aulc a valu. Tp- o spcf flu boua coos a us o ol bouas o flu valus. I s also o as a ua boua coo a s v as: q ( z ) ) () b 9

21 q s o flu valu a s u oal o boua. Fall p- o a-p boua coos a us o ol bouas o c coos p o ca aulc a suc as sas a vs a al bouas of oa. I s also o as a Cauc boua coo a s v as: q C > ( z ) ( zb ) ) ql () ( z ) ( z ) q C s a-p flu valu a s also qual o laal flo b o sss. I us b o ov a laal flo s o lo a a-p flu sco coo appls Equao () sc (z - ) bcos cosa aulc a falls blo boo lvao of v ss. I s pacula cas a-p flu s a as a cosa flu coo. T coupl s o plcl b boua coo fo oua flo ol... Coupl of Mols T coupl of cal flo a oua flo ols s pov va laal flo/ouflo. T laal flo/ouflo appas as a souc/s bo cal flo ol a oua flo ol. I s cu sa coupl ol os o allo a o laal flo/ouflo o cal flo ol suc as ovla flo pcpao a vapoao. Tfo follo coupl cas s assu o b val cal flo ol: > pa occus fo cal o oua flo oa. Hc bcos a laal ouflo fo cal flo ol a a laal flo fo oua flo ol. o spa occus b o oas. Hc laal flo/ouflo quaos () a (0) bcos zo ps a o flu coo fo bo ols. < pa occus fo oua flo oa o cal. Hc bcos a laal flo fo cal flo ol a a laal ouflo fo oua flo ol. 0

22 . ucal oluo.. Cal Flo Mol I al avalabl ucal cqus fo soluo of pa a-va quaos ca b v as: () o of caacscs; () f ffc os; a () f l os. Of s os f l o s al us flo s appoa as o-soal suc as cas of a-va quaos. T o o os av b cool appl fo ucal soluo of o-soal usa flo sc 960s. T f ffc os ca fu b classf as plc a plc cqus ac of c ols sc ucal caacscs. ao avaa of plc f ffc o ov o of caacsc a plc f ffc cqu s s sabl ou qu o sasf Coua coo c ss ca fo au alloabl sp. Ts qu o sasf Coua coo of as o of caacscs a plc cqus v ffc s of us of copu. Fuo ca plc scs suc as o popos b Pssa (96) allo us of vaabl a spaal sps c a o l cov fo applcaos ou of floo oaps v sss (u 00). Cos s avaas plc f ffc cqu s us o solv cal flo quaos v b Equaos () a (). Of vaous plc scs a av b vlop " fou-po" sc of Pssa s v avaaous sc ca al b us uqual sac sps a bcos paculal poa fo aual aas cal caacscs a l vaabl v so sacs. lal applcabl of uqual sps s ao poa caacsc of s cqu fo oap ou flooas oul all s lavl qucl a css auall. T f ffc coupas of cou ou a boua coo quaos a v pp. I a cal o scz fos of Equaos () a () fo co of cal flo ol. Ts quaos a fo ac cal o a suppl b scz fos of boua coo quaos. Fo ac cal fal fo of cou quao s as:

23 0 / / / / / / / / / / Δ Δ Δ Δ o c o c o c o c s s s s θ θ (5) T fal fo of ou quao s ffl fo laal flo a ouflo. I s poa o o a laal flo as o cobuo o ou balac a fo s assoca laal flo op ou fo ffc quao. Fo cas of laal ouflo f ffc fo of ou quao s as: [ ] [ ] 0 / / / / / / / / / / / / / / / / / / / / / / / / Δ Δ Δ Δ Δ Δ Δ Δ f f s s s s β β θ β β θ (6) I Equaos (5) a (6) subscps () a () ps spaal a poal cs spcvl. T s subscp () a o fo al coos o fo soluo of a-va quaos a pvous l. c coss scoal aa a cal op a fucos of a sufac lvao ol uo s s quaos a sca a a sufac lvao a () l a os () a (). Tfo a ol fou uos s quaos. ll a s

24 a cosas o a fucos of s uos. T sul o albac quaos oba b applcao of fou-po sc a ola a a av soluo cqu s qu. W f ffc fos of cou a ou quaos a solv fo ac so pp a ss of ( - ) quaos a fo fo o -l b upsa a osa boua of cal pss ub of os cal. T o uos ac of s quaos l a oal of uos fo ac l. T ss of ( -) quaos uos qus o aoal quaos fo closu of ss. Ts o aoal quaos a suppl b upsa a osa boua coos of cal. W s pocu s pa fo ac cal of o a oal of Σ( ) quaos a fo us fo o ub of cals o a pss oal ub of os ss. T sul ss of o-la quaos uos s solv b a suabl a soluo alo suc as o-rapso av cqu. Of all o-la soluo pocus o-rapso o s o of os coo av cqus us fo soluo of a ss of o-la quaos. I povs a ffc as of cov o a oo v a suffcl oo al uss. Fo a cal o applcao ss of quaos ca b o as fucoal laos o b zo a volvs vaabls a ps b fo : f... ) 0 (7) ( If os vco of uo vaabls a f os vco of fucos f ac of fucos f ca b pa as a Talo ss paso booo of : f f ( δ) f δ O( δ ) (8) a of fs paal vavs s call Jacoba a J. T ls of Jacoba a fo uos a valua pp. I a oao o ca Equao (8) as: f ( δ) f J δ O( δ ) (9) lc o s a s lf a-s qual o zo o ca oba a s of la quaos a a solv fo cocos δ: J δ f (0)

25 Ts a quao s solv b a a solv suc as Gaussa lao o LU coposo fo uo δ a a pov sa of soluo s oba b: δ () supscp pss lvl of ao a uo l. T av soluo s ac b f valus of uos a so a suals v Equao () a foc o zo o v clos o zo. I us b o a covc pocss ps o a oo fs sa fo uo vaabls. Fa (985) sas a a asoabl oo sa fo fs sp s o us al coo of sca a a sufac lvao. Fo all o sps fs sas of uo vaabls ca b oba b us lal apola valus fo soluos a pvous sps acco o alo v blo: Δ ( ) α c () Δ s fs sa of uo vaabls a () l s soluo vco of a valus fo pvous sp - s soluo vco of a valus fo o pvous sps α c s a faco fo 0 o a Δ a Δ a o coscuv sp szs... Goua Flo Mol I oua flo ol lau s uous ols pl ff ucal soluo pocus. T os coo of s pocus a f ffc a f l os (McDoal a Habau 988; al 990). T f l o bca a popula o u o flbl offs sula aquf oas ula bouas as ll as oous aquf pops. I s a Gal f l o bas o o of suals s us ucal soluo of oua flo copo of popos coupl ol. T ucal pocu sas alzao of soluo oa b a f ub of sc o-ovlapp os call f ls ov c uo vaabls a o b pola. I a alzao ls a slc suc a aal pops of oa suc as aulc coucv a spcfc l a a vual ls. I o-soal f l aalss fals of aula a/o qualaal ls a all us o scz aalss oa. lou s ls ca b la quaac o cubc us spl la ls povs suffc accuac a a b soluo sa. ualaal ls a supo as oppos o aula ls u o fac a a copuaoall o ffc a splf as of l pobl oa ou ouc

26 a bas a aula ls possss. Fo s asos la ula qualaal ls fou os a slc o scz oa a vlop bass fucos popos ol (Fu ). T als assoca bass fucos a v pp. Follo alzao of oa a slco of pola fucos a appopa a fo of pobl s vlop. Us Gal sual o a a fo of oua flo quao s v pp b us saa sps of sual ao b pas a copoa aual boua coos. T sul f l a quao oba b appl Gal pocu s v as: M F () M a F sa fo lobal sffss a lobal ass a a lobal loa vco spcvl a ĥ s appoa aulc a vco. Ts lobal acs a vcos a oba b l l coupas acco o cocv of ls soluo oa. T plc foulas of l acs a vcos a v pp 5. s po s clal s a s l als a all copl a ca o b a aalcall. Hc a ucal ao sc s qu o valua s l als. I popos ol a o-soal Gaussa quaau cqu s pl o valua s als ucall. T als of s cqu a scuss pp 6. T oa ffal Equao () oba as a sul of f l sczao ca b solv us a ub of cqus clu o-sp f ffc appoaos. c aulc a s a fuco of s possbl o f o posos a ps o a uo ls spcvl. If o fs a a po b o a uo l (.. α 0 α.0) cospo a coul b calcula as a ava: α α ( α ) () suc a f a po s slc as po b o ls (.. α0.5) a bcos a ac ava of o as a o s. Us Talo ss paso of aulc a aou a po us pos a o oul oba: α α Δ α Δ O ( Δ ) (5)!! 5

27 6!! O α α α Δ Δ Δ (6) lc s qual o o a sco o a subac sco quao fo fs ls: Δ Δ!! 0 α α (7) a af aa ucs o: Δ α α (8) c s alas possbl o oa ffal quao fo a pacula l o oul oba follo quaos fo o ls: F M (9) F M (0) W s quaos a ulpl b paas (-α) a α spcvl a a o o oul oba: F F M M α α α α α α () I us b o a saua ucof aquf flo ass a M s a cosa a c s o a fuco of aulc a a as sa valus fo all sps. Tfo abov foulao ca b splf as: F F M α α α α α α () I s o possbl o subsu fo ava vav s v Equao (8) fo pass o oba:

28 α ( α ) M αf ( α ) F Δ () f aa quao as follo fal fo: M Δ M Δ ( α ) F ( α ) α αf () I s o possbl o oba ff ao scs p o valu of paa. Ev ou fl a valus of paa a possbl sval of s a paculal poa a av sfca pops W α0 quao bcos a plc sc a os o qu soluo of a ss of quaos o o avac soluo acoss. Hov plc scs of cou ucal sabls f sp s a oo la. W α0.5 sc bcos so-call Ca-colso o c pls a cal-ffc appoao b o ls. I s o a s coc of faco cospos o opal sapl of fs poal vav ov sp. If aa of pobl av suffc cou s sc bs s opal accuac pops a suls a v ffc o fo al -pc of as pobl. Ufoual psc of a scou aa la o spuous oscllaos of copu soluo. If α sc bcos a full-plc sc a sss vlop of soluo oscllaos b a a o o-sp o. Tfo s cool us fo os ffcul pobls. Hov soul b o a s sc s o full accua o spcall ffc bu ll ap spuous -fquc ffcs o sol a o scs. T popos ol all uss a - paa of 0.5. Du o o-la of ov quao of ucof aquf flo a suabl av sc s pl o solv ss of o-la quaos. Coo ola soluo cqus suc as o-rapso o o succssv subsuo (Pca) ao o ca b appl s soluo. lou o-rapso o s fas covc qus copuao of paal vavs a s a cosl f l fao. Hc lavl spl Pca ao cqu s appl soluo of oua flo ol. T Pca o s a v spl cqu a s bas o succssvl subsu las valus of aulc a o copu valus ul suffc covc s acv. W Pca o s appl scz oua flo quao ca b as: M Δ M Δ ( α ) F ( α ) α αf (5) 7

29 supscps a ps pvous a cu ao valus of aulc a a uo lvl. Fo all aos os c valus of aulc as a us o oba a pov sa of as a uo lvl acco o follo foula: ( γ ) γ (6) γ s a ao-p ulaao coffc (o a ap paa) a valus b 0 a. T lf a-s valu a () ao pss pov sa o b us ao. Fo v o-la pobls a ca aos b la ou o caus soluo o osclla. I suc cass a ap paa ca b us o sc a ca fo o ao ccl o (Huao al 986). I ac ao ccl valu of ap paa s copu acco o follo pocu: s s s γ (7) s < s s s a scal paa valua acco o follo ul: s ε (8) > γ olε ε a ε ps aulc a ca fo ao a spcvl a s las absolu valu a γ ol s valu of ap paa a pvous ao... ulaous oluo of Coupl Mol T al sp fo ucal soluo of coupl ol s scz aalss oa. I s pocu cal o s scz fs cos sabl qus of cal flo ol. T oua flo oa s scz cos o of aquf. Du sczao of oua flo oa ac o of cal flo ol s slc suc a cocs a o oua flo ol as s Fu. Ts o-o-o cospoc of os alo cal o s ssal fo sulaous soluo of coupl ol. If s a qu fo f sczao of oua flo oa a a po alo cal o u o l vaabl aquf pops 8

30 sczao of cal flo ol us also b of o sasf o-o-o cospoc of os. I s a sulaous soluo of coupl ol s alas bas o fs scz oa a ol focs alo cal o. Fo a o po aalss oa sczao s soll bas o qus of oua flo ol. alss oa Cal o Goua flo sczao Cal flo sczao Fu. Dsczao of oa T ol cpo o o-o-o cospoc of os s obsv a cal uco pos. a uco flo cals a o ouflo cal ucal sczao of cal flo ol qu a os a us o popl ps osa boua os of flo cals a o upsa boua o of ouflo cal. c all os pscall ps sa uco po all cospo o a sl po oua flo ol. Tfo a uco pos os of cal flo ol cospos o o o of oua flo ol. T sco sp of sulaous soluo of coupl ol s o scz quaos of cal a oua flo quaos. Ts a v Equaos (0) a (5) spcvl. W s quaos a fo all os of cal a oua flo oas a ss of quaos s oba fo bo cal a oua flo ss. s po s sss of quaos a assbl o 9

31 a sl ss so a ca b solv o a sulaous a. T assbl fal a quao s so Fu. GW GW 0 GW 0 RIVER RIVER RIVER Fu. Global a quao a copo blocs I assbl lobal a quao s lobal coffc a s lobal loa vco a s lobal uo vco. s s fo Fu lobal a a vcos a oba b cob spaa blocs oba fo cal a oua flo ol. Ts spaa blocs a as: GW α ( / Δ)M (9) RIVER J (50) GW ( M) ( α ) F ( α ) ( / Δ) α F (5) RIVER f (5) GW (5) RIVER δ (5) lou lobal a s so full-a foa calculaos a pfo us a ba a sucu o uc copu o qu o so a solv ss. T oal ba of lobal a ps o lav aus of bas of cal flo a oua flo ols. Tfo sz of lobal 0

32 a s b sz of b ba. I al ba of oua flo ol s b a ba of cal flo ol. I s poa o o a sc lobal ss s o-la a p s solv sval s fo ac sp ul suffc covc s acv fo uo paas. Hc lobal a soluo volvs a av poo o al o-la of ov quaos of bo ols. Hov s av o-la soluo os o affc sulaous soluo of ovall coupl ss. T av soluo s ol us soluo of o-la o subsss. T covc of o-la soluo s cc us o spaa ca fo cal flo a oua flo copos. Tfo alou sss a solv o covc of soluo s s ff ca sc of o-la cal flo s all uc a of ola oua flo. Tpcall o aos a fou o b suffc fo covc of o sub-oa ols. Ev ou o oloc sss coupl s ol a av sfcal ff scals sulaous soluo ul a av soluo qus a coo sp ucal sczao. Paculal bavo of a cal flo ol s all o ac a ovall spos of a oua ol. Ts cosa coul occasoall ca lo sulao pos popos alo bu s alas fas a av soluo appoac a ulzs sa sp sz.. pplcao T ss of sulaos a a us popos ol a soluo alo. I fs applcao a pocal s cas s sula o s ol s capabls a laos a caula oua flo oa a a ovl sl cal oa. I sco applcao sa pocal s cas s of o aalz ol s spos psc of a cal o. Fall applcao s bas o sula a al ss fo sou Goa... Hpocal Cas: l Cal T coupl cal/oua flo ol s fs appl o a pocal saaquf ss o osa pfoac of popos sulaous soluo alo. T pscal sup of pocal oa s so Fu 5. I s applcao sa s a 0 0 lo ufo caula cal a cosa slop of 0.000/ a vs aquf o o qual oas of 000 o ac s of cal. T Ma s ouss coffc of sa s ufo ou ou cal a as a valu of sa flo coos cal cas 00 /s sca a ufo flo p of.56. T css of ss a boo of cal s 0. a aulc coucv of

33 poss s.0e-6/s. T cal boo lvao a os upsa po s v as 0 abov a sa lvl. To vsualz suls asl 0 lo a ucof aquf s assu o av a ufo a soopc aulc coucv of.0e-/s a aquf bas s s a a sa lvl. T sa flo ol s scz b 00 lo ls v a oal of 0 os. T oua flo ol oa s scz b squa ls a s l of 00 v a oal of os a 000 squa ls. Fuo a cosa sp of s us sulaos. I cal flo ol upsa boua coo fo cal s v b a apzoal sca oap a bas sca of 00 /s a pa sca of 50 /s a a o pa of 0 as (Fu 6). T osa boua coo s v b a sl-valu a cuv a aps sca o s oal p. I oua flo ol bouas paalll o sa a spcf as cosa a coo a bouas ppcula o sa a spcf as o-flu coo. Moov al boua sa us ou s spcf as a a-p l souc. T al coos sa flo ol s v as ufo flo coos (.. 00 /s of sca a a cospo.56 of p) a all os. I oua flo ol o ff ss of al aulc a sufacs a us. I fs sulao al oua a aquf s cos o b a a all os. Ts sulao llusas a coo laal flo occus fo sa o aquf. I sco sulao oppos scao s sula a al oua a aquf s cos o b 5 llusa a coo laal flo occus fo aquf o sa. Ts o sulaos of Pobl- a f o as cao- a cao- spcvl a a abbva as P- a P- follo scusso. I bo scaos po copasos of oua a a sa sa a ps fus 6 a 8 a po of aalss oa ( fo upp boua of aquf c also cospos o po of sa). spaal sbuo of oua as a also ps fus 7 a 9 alo l ( ; 5000). alss of oua a ss fus 6 a 8 vals a passa of floo av cas a cas oua as b ca a ou a v as lo as sa sas a a oua as fo a suffcl lo po of. Ts ou s sul of laal flo o aquf (Fu 7). I s also s a ou subss a ba soa s a bac o sa sa sa falls blo oua as. I s also s fo fus 6 a 8 a s a la b pa valus of aulc a a sa sa c clal pss ac bavo of sa flo as oppos o oua flo. T spos of coupl ss o a floo av s cl la o al coos sa a aquf. copaso of fus 7 a 9 osa ffc of al oua a aquf a s poso lav o sa sa. W aulc a aquf s a sa sa (Fu 9) a

34 sca fo ba soa occus fs 5 as of sulao ca a ao a sa. Du sco 5 a po sa sas cass u o aval of floo pa a s cas a flo vsal oas aquf (Fu 9). -D Rv oa ls -D oua oa ls M po Rv o Goua o

35 Fu 5. Pscal sup of pocal oa sl cal 8 Rv a Goua Ha Rv Dsca 500 Goua Ha a Rv a () Rv Dsca a Upsa oua (cs) 0 9//97 0:00 9/7/97 0:00 9//97 0:00 9/9/97 0:00 9/5/97 0:00 0//97 0:00 T 0 Fu 6. Goua a a v sa a po a v sca a upsa boua (P-)

36 5 0as 5as 0as 5as 0as 5as 0as Goua Ha () Dsac fo quf Cl () Fu 7. Goua a pofl a vaous s alo po (P-) 5

37 8 Rv a Goua Ha Rv Dsca 500 Goua Ha a Rv a () Rv Dsca a Upsa oua (cs) 0 9//97 0:00 9/7/97 0:00 9//97 0:00 9/9/97 0:00 9/5/97 0:00 0//97 0:00 T Fu 8. Goua a a v sa a po a v sca a upsa boua (P-) 0 6

38 as 5as 0as 5as 0as 5as 0as Goua Ha () Dsac fo quf Cl () Fu 9. Goua a pofl a vaous s alo po (P-) 7

39 .. Hpocal Cas: Cal o I sco applcao coupl cal/oua flo ol s appl o a pocal sa o-aquf ss o osa pfoac of popos sulaous soluo alo o ulpl cal applcaos. T pscal sup of pocal oa s so Fu 0. I s applcao o sa cals (.. cals a ) a fo a la cal (.. cal ) a uco po so Fu 0. T o upsa buas a 0 a 707 lo ufo caula cals a cosa slop of /. T osa sa s 5 a 5000 lo ufo caula cal a cosa slop of 0.000/. T o upsa cals cofluc a -po of oa a cas uco. T Ma s ouss coffcs of all cals a ufo ou ou oa a av a valu of sa flo coos o upsa cals ca 00 /s as osa cal cas 00 /s. T css of ss a boo of cal s 0. a aulc coucv of poss s.0e-6/s. To vsualz suls asl 0 lo a ucof aquf s assu o av a ufo a soopc aulc coucv of.0e-/s a aquf bas s s a a sa lvl. T sa flo ol s scz b vaabl l ls v a oal of 55 os. T oua flo ol oa s scz b qualaal ls vaabl s ls v a oal of 6 os a 0 ls. cosa sp of s us sulaos. I cal flo ol upsa boua coos a v b a apzoal sca oap a bas sca of 00 /s a pa sca of 50 /s a a o pa of 0 as (Fu ). T osa boua coo s v b a sl-valu a cuv a aps sca o s oal p. I oua flo ol bouas paalll o sa a spcf as cosa a coo a bouas ppcula o sa a spcf as o-flu coo. Moov al boua sa us ou s spcf as a ap l souc. T al coos sa flo ol a v as ufo flo coos a all os. I oua flo ol o ff ss of al aulc a sufacs a us. I fs sulao al oua a aquf s cos o b a a all os. Ts sulao llusas a coo laal flo occus fo sa o aquf. I sco sulao oppos scao s sula a al oua a aquf s cos o b 5 llusa a coo laal flo occus fo aquf o sa. Ts o sulaos of Pobl- a f o as cao- a cao- spcvl a a abbva as P- a P- follo scusso. I bo scaos po copasos of oua a a sa sa a ps fus a 5 a pos aalss oa. Ts pos a so Fu 0. Of s pos pos a a o lf upsa cal a a sua 75 a 57 fo os upsa po of cal. O o 8

40 a po s o osa cal a s sua a po b uco a cal s os osa po ( fo uco). Ts pos a also poso o ascs pc Fu 0 ( a 7500 fo upp boua of aquf). Ts ascs a us o ps spaal sbuos of oua as alo aquf. alss of oua a ss fus ou a 6 ou 8 val a passa of floo av cas a cas oua as b ca a ou a v as lo as sa sas a a oua as fo a suffcl lo po of. Ts ou s sul of laal flo o aquf. I s also s a ou subss a ba soa s a bac o sa sa sa falls blo oua as. T spaal sbuo of oua as ascs a llusa a sc spos bavo sc pscal caacscs of upsa cals a boua coos a acl cal as a fuco of. ffc b s caacscs oul clal ca a asc aulc a sbuo upp alf of aquf. I P- al oua a all ascs a blo al v sas. Ts suao cas a laal ouflo fo sa cals oas oua oa ca a cas oua as as s fus a. T floo av avs a s cas s v o poouc. Oc floo av sas c oua as sa fall. c a sufac lvao cas cal s uc o ac a oua a cas a flo vsal s obsv ca a laal flo o sa cals fo oua oa. Ts bavo s ps all ascs af 09/5/997 fus a. I P- ov al oua a all ascs a abov al v sas. Hc a a laal flo o sa cals sas o vlop. I absc of a floo av s suao cas a cas oua as a vc of cals. Tfo s possbl o obsv ao assoca s bavo all ascs ll 09/05/997 fus 6 7 a 8. f 09/05/97 aval of floo av focs a cas oua as u o laal ouflo fo cal. I s poa o o fac a a lavl aulc coucv valu a a lavl sool-cas upsa sca oap a us o poo a ap spos bavo so a suls coul b aalz a spl a alz faso. I al sulaos ov aquf coucv valus a all uc sall a oaps a cool uc sp o s lb. uc a suao s ps applcao of ol. 9

41 -D v oa ls Tasc- Po : o-7 (RIV) o- (GW) Tasc- Po : o- (RIV) o-0 (GW) -D oua oa ls Juco Tasc- Po : o-0 (RIV) o-0 (GW) Rv o Goua o Fu 0. Pscal sup of pocal oa cal o 0

42 0 500 Goua Ha a Rv a () Rv Dsca a Upsa oua (cs) 0 9//97 0:00 9/7/97 0:00 9//97 0:00 9/9/97 0:00 9/5/97 0:00 0//97 0:00 T Rv a Po (o 7) Rv a Po (o ) Rv a Po (o 0) Goua Ha Po (o ) Goua Ha Po (o 0) Goua Ha Po (o 0) Rv Dsca a Upsa oua 0 Fu. Goua a a v sa a vaous pos oa a v sca a upsa boua (P-)

43 5 0as 5as 0as 5as 0as 5as 0as Goua Ha () Dsac fo quf Cl () Fu. Goua a pofl a vaous s alo asc- (P-)

44 6 5 0as 5as 0as 5as 0as 5as 0as Goua Ha () Dsac fo quf Cl () Fu. Goua a pofl a vaous s alo asc- (P-)

45 5 0as 5as 0as 5as 0as 5as 0as Goua Ha () Dsac fo quf Cl () Fu. Goua a pofl a vaous s alo asc- (P-)

46 0 500 Goua Ha a Rv a () Rv Dsca a Upsa oua (cs) 0 9//97 0:00 9/7/97 0:00 9//97 0:00 9/9/97 0:00 9/5/97 0:00 0//97 0:00 T 0 Rv a Po (o 7) Rv a Po (o ) Rv a Po (o 0) Goua Ha Po (o ) Goua Ha Po (o 0) Goua Ha Po (o 0) Rv Dsca a Upsa oua Fu 5. Goua a a v sa a vaous pos oa a v sca a upsa boua (P-) 5

47 as 5as 0as 5as 0as 5as 0as Goua Ha () Dsac fo quf Cl () Fu 6. Goua a pofl a vaous s alo asc- (P-) 6

48 as 5as 0as 5as 0as 5as 0as Goua Ha () Dsac fo quf Cl () Fu 7. Goua a pofl a vaous s alo asc- (P-) 7

49 as 5as 0as 5as 0as 5as 0as Goua Ha () Dsac fo quf Cl () Fu 8. Goua a pofl a vaous s alo asc- (P-) 8

50 .. Lo laaa Was T popos ol s also appl o laaa Rv bas sou Goa. T ol oa clus a poo of laaa a Ooop vs. I covs aa aa bou b UG sa a saos a al Rsvll a Docoo. Ts aa s scz b 70 oal pos a 688 qualaal f ls oua flo zo a 9 oal pos cal flo zo. T ava l s l alo v scos vas fo 50 o 00 a abou 000 ls. T scz ol oa s so Fu 9. ucof sufacal aquf ovl Upp Floa aquf s cos o b ps aa a ava css of abou 0. T aquf cosss pal of ucosola ll so sa a sl sols. T sol ps aquf us a ols Goapc Daabas (TTGO) of Goa vlop b U Dpa of culu (TTGO 998). T cospo saua aulc coucvs of s sols a assu o follo sascall ava valus pov b Casl a Pas (988). T coucv valus us ol as.5e-6/s fo sl loa sols.05e-5/s fo loa sa sols a.e-5 /s fo sa loa sols. I ao a 0. css of v ss s cos o b ufol ps alo cal ss a aulc coucv of 6.9E-7/s ps sl aal pos cal boos. T laaa v ss s ol as a a-p l souc a cas laal /ou flo o/fo oua flo oa acco o lav valus of v a oua as. T aual a afcal las a pos bas a ol as cosa-a boua coos. Moov al as boua s sula as a o-flu p boua coo cp fo a vc of laaa Rv a Docoo a a s osl caacz as asla a ol as a cosa a boua coo. T coss-scoal aas of copuaoal os cal flo oa a oba b us () asus a a a saos b UG; () pofls of a bs alo v cals; a () opoapc aps of aa. T Ma s ouss coffcs us sulaos va b 0.00 o 0.00 a cal a 0.00 o alo floopla. T sulao po covs a o-o po sa 09/0/997 ou /0/997 a coo sp of 6 s fo bo oua a cal flo oas. T sca oaps a al a Rsvll a us as upsa boua coos of cal ol a sa-sca a cuv a Docoo a us as osa boua coo. T sca oaps a al a Rsvll ca b s Fu 0. T sl-valu sa-sca a cuv a Docoo s v Fu. 9

51 Ooop Rv a Rsvll G UG 0500 laaa Rv a al G UG laaa Rv a Docoo G UG Fu 9. Dscz oa of Lo laaa Rv as 0

52 00 al Rsvll 00 Dsca (cs) /8/97 9/7/97 9/7/97 0/7/97 /6/97 T Fu 0. Dsca oaps obsv al a Rsvll a saos

53 5 Dp () Dsca (cs) Fu. Docoo a sao a cuv

54 T sa a sca oap co a Docoo s us as calbao paa. copaso of sulao suls vs. Docoo a asus a so Fu a. s s fo fu ol sulaos a fou o b v clos o a asus. T au o b asu a sula sa valus s calcula o b.% a a o s calcula o b blo %. lal au o b asu a sula sca valus s calcula o b 0% a a o s calcula o b blo %. Ts valus a ll blo accp vaos po lau (Fa 985). Pa of s os s assoca cobuos fo buas of laaa Rv b al a Docoo as. Ts a o quaf a o ol sulaos. Hc sca v s ol o b lss a asu. o possbl souc of o ca b abu o qual of pu aa us sulaos clu cal coss-scos a v boo slops. Ts aass a ca osl fo asus a a a saos b UG a pofls of a bs alo v cals; a pal fo opoapc aps of aa. possbl os cao of s aass a flc ol suls. T oua a coous oba a of pa floo (.. 09//997) a so Fu. T a coous sufacal aquf osl follo sufac lvao. T oua a coous a coss v posos oa. Mos of clos coous a abu o aual a afcal pos a las a a clu sulaos as cosa a boua coos. lo all o-flu bouas of oa oua coous a pfc als ps absc of a flo ou of s bouas. Du o la oal spac us alo cal vc (abou 00) sall aulc coucv valus of aquf a o-sc oal posos bo bas of v bavo of ba-soa ffc alo cal s o clal ps as so pvous pobls.

55 0 Masu ula a () /7/97 9/7/97 9/7/97 0/7/97 /6/97 T Fu. ula vs. asu sa valus Docoo a sao

56 00 Masu ula 00 Dca (cs) /7/97 9/7/97 9/7/97 0/7/97 /6/97 T Fu. ula vs. asu sca valus Docoo a sao 5

57 Y-Cooa () X-Cooa () Fu. Goua a sbuo poc aa a 09//997 6

58 5. Coclusos soluo sa s popos o solv sufac a subsufac flo sss a coupl a. I s a a o-soal cal flo ol s coupl a o-soal vcall-ava oua flo ol va laal flo acos ba cal boo. Isa of pl cool appl av soluo cqu popos o s bas o a of solv o sss a sulaous a a sl lobal a foa. T ol s applcabl o sss o-loop cal os ovl a o-ufo asoopc ucof aquf. T l b o oas s pov b laal flo. T applcaos of ol a pfo o pocal a s sss o s s spos caacscs o aa sss as ll as capabls o sula al sss. T ol suls ca oo pcv capabl of cob sufac-subsufac sss. Tfo s popos as a alav ool fo aalz oloc pas la ass. 7

59 Rfcs a.o. a Y.C. (98a). Maacal ol of sallo a flo ov poous a. Joual of Haulcs Dvso CE Vol. 07 o. HY pp a.o. a Y..C. (98b). Dffuso-av floo ou cal os. Joual of Haulcs Dvso CE Vol. 07 o. HY 6 pp M. a Fa C.. (970). Iplc floo ou aual cals. Joual of Haulcs Dvso CE Vol. 96 o. HY pp al M.M. (990). Goua ol ulla aqufs: Usa flo. Ls Publss Ic. p. Casl R. F. a Pas R.. (988). Dvlop o pobabl sbuos of sol a o caacscs. Wa Rsoucs Rsac Vol. o. 5 pp Cau Y. M. a Coaco D.. (97). pplcao of plc o o sus cals. Wa Rsoucs Rsac Vol. 9 o. 6 pp Cua.. a cla P.J. (979). pplcao a aalss of a coupl sufac a oua ol. Joual of Holo Vol. pp Fa D. L. (97). ucal pops of plc fou po f ffc quaos of usa flo. O Tccal Mo W HYDRO-8 U.. Dpa of Coc aoal Wa vc lv p Mala. Fa D. L. (985). Cap : Cal Rou. Holocal Focas. M. G. so a T. P. u Jo Wl a os L pp Fa D.L. (99). Flo Rou. Haboo of Holo. D.R. Ma McGa-Hll Ic. pp Fz R.. (97). Rol of subsufac flo a sufac uoff:. as flo cobuos o cal flo. Wa Rsoucs Rsac Vol. 8 o. pp Guuz O. a al M.M. (00). ulaous soluo of coupl sufac a / oua flo sss. Rv as Maa 00 pp. 5- WIT Pss U. Has J.W. a oc H Haboo of Maacs a Copuaoal cc. p-vla Ic. Yo 08p. 8

60 Huao P.. p E.P. Guvaas V. a Waso T.D. (986). soal f l ol fo sula a flo vaabl-saua poous a. Wa Rsoucs Rsac Vol. o. pp Ja R. Ha. a Musa. (000). Rv o soluo fo a sbu olocal ol a applcaos. Holocal Pocsss Vol. pp McDoal M.G. a Habau.W. (998). oula -soal f ffc oua flo ol. U.. Golocal uv Tcqus of Wa Rsoucs Ivsaos oo 6 Cap 586p. Moa M. a Y.C. (00). Mol of coucv o-soal sufac- soal subsufac flos. Joual of Haulc E CE Vol. 8 o. pp P G.F. a au.p. (97). ucal sulao of floo av ofcao u o ba soa ffcs. Wa Rsoucs Rsac Vol. 7 o. pp Pssa. (96). Popaao of aslao avs cals a vs. Pocs of Fs Coss of Fc ssocao fo Copuao Gobl Fac pp. -. R.E. a Wools D.. (97). Ovla flo o a fla sufac. Wa Rsoucs Rsac Vol. 7 o. pp TTGO (998). a sol oapc (TTGO) aabas of Goa. Goa GI Daa Claous: p://s.sa.a.us u T. W. (00). Op Cal Haulcs. McGa Hll Ic. 9p. a E.D. a Wl E.J. (99). coupl sufac a a oua ol. Pocs of 99 aoal Cofc o Iao a Daa Hoolulu Haa pp Vaaa J.E. a Loau. (00). Holoc-spos sulaos fo R-5 cac a copsv pscs-bas ol. Wa Rsoucs Rsac Vol. 7 o. pp Zcz O.C. a Talo R.L. (989). T F El Mo: Volu asc Foulao a La Pobls. McGa-Hll oo Copa 68p. 9

61 pp I. Dvao of F Dffc Equaos of Cal Flo T f ffc quaos of cou ou a boua coo quaos of cal flo a v s app. Fo ac cal of o cou a ou quaos v Equaos () a () a scz - pla us fou-po ffc plc sc. To aoal quaos a us o ps coos upsa a osa bouas of cal. W s pocu s o fo all cals a ss of quaos s fo c s solv o valua uo sca a sa a scz os. T scz fos of cou ou a boua coo quaos a v follo scos. Cal o I o o asss vao sapl o so Fu 5 s us follo scusso. Ts o coas 5 cals a ucos. T cal ub sc sas fo os upsa cal a follos co of flo. W a uco s ac o ub cous fo cal s os upsa o a follos squall up o uco ul all flo cals a ub squall. W all flo cals a ub fo a pacula uco o ub cous os upsa o of ouflo cal. Ts pocu s cou ul ss s ub. Dscz fos of cou a ou quaos T f ffc sczao of cou a ou quaos s o fo all cals as so Fu 6. I ca b s fo Fu 6 a soluo pla fo cal s ps b a oal of os local o ubs sa fo a u ou. I fou-po sc appoaos of vavs a cosa s a v Equaos (55) (56) a (57) fo a u paa E: E E E E E ψ ( ψ ) (55) Δ Δ E E E E θ ( θ ) (56) Δ Δ E E E E E E θ ( θ ) (57) 50

62 a a subscps fo a as spcvl ψ a θ a facos b 0 a a Δ a Δ a ac l b os a a sp b ls a spcvl. I s possbl o ca ff ofcaos of sc us ff valus fo facos. θ valu of 0.0 a 0.5 cospo o plc sc a bo sc spcvl. lal a sc a θ valu of.0 s o as full-plc sc spac. Ma sacs pf us a ψ valu of 0.5 a appoa vav a c of b () a () ls ( a Fa 970; Cau a Coaco 97) as os us va valus p o pacula applcao (Fa 985). Cal Cal 5 6 Juco : Juco Cal Cal 5 Cal Juco Juco : Global o ub Juco Dals Fu 5. apl cal o a ub sc 5

63 T o Rac Δ () () Δ () () G - - Upsa oua Dsac Dosa oua Fu 6. T sac- us o foula plc f ffc sc fo cal (f Fa 97) Fa (97) as so a fou-po plc sc s ucooall sabl fo a sp f valu of θ s slc b 0.5 a.0. I ao o sabl ca also aalz fluc of faco o accuac of copuaos a fou ou a accuac cass as θ pas fo 0.5 a appoacs o.0. H po a s ffc bca o poouc as au of copuaoal sp cas. Fuo s aalss val a a θ valu of b 0.55 a 0.6 pov ucooal sabl a oo accuac c as s sc supo copa o plc sc a qus sps of lss a a ccal valu b Coua coo. W pla v Equaos (55) (56) a (57) spac vavs Equaos () a () a appoa as: θ ( θ ) (58) Δ Δ 5

64 θ ( θ ) (59) Δ Δ ( β / ) ( β / ) ( β / ) β / θ Δ ( ) ( ) β / θ Δ (60) lal vavs a appoa as: s c ( ) s ( ) s ( ) o c / s o c / Δ c / ( ) s ( ) o Δ o c / o (6) s s s s s / / / / Δ Δ (6) Fall cosa s suc as q f L a a appoa as: q ql ql / θ L / θ (6) f f / ( ) θ θ (6) f / / θ / θ (65) L L / θ L / θ (66) θ (67) ( θ ) θ / ( θ ) / vaabls subscps (½) a f fo ac b os () a () as a ava of o oal valus. T follo plc foulaos a us o f abov vaabls: / / ( q ) L / / (68) / / 5

65 / (69) / c / R / / / / / / f / c / / (70) Δ / / L / / q L (7) / (7) / / R (7) / / / / (7) / (75) / (76) / (77) / (78) / (79) / (80) T f ffc fo of cou quao s oba Equaos (58) ou (80) a subsu o () a aa fo ac cal o: 5

66 55 0 / / / / / / / / / / Δ Δ Δ Δ o c o c o c o c s s s s θ θ (8) lal f ffc fo of ou quao s oba Equaos (58) ou (80) a subsu o () a aa fo ac cal o (.. fo laal ouflo cas): [ ] [ ] 0 / / / / / / / / / / / / / / / / / / / / / / / / Δ Δ Δ Δ Δ Δ Δ Δ f f s s s s β β θ β β θ (8) c s o cobuo o ou fo cas of laal flo las s s sco a bacs op ou.

67 Dscz fos of al boua coo quaos a upsa al boua of a cal a sca o a sa oap ca b us as boua coo quao. T scz fos of s quaos a v as: ( ) 0 u (8) ( ) 0 u (8) subscp pss lobal upsa o ub of cal. Fo a sl cal ss as valu. I sapl o cals a av al upsa coos a as valus 5 a spcvl. c popos ol os o allo loop os ol o al osa boua coo ca b clu ol. T boua coo a osa al boua ca also b f as a sca o a sa oap. T scz fos of s quaos a v as: ( ) 0 (85) ( ) 0 (86) pss las o o. I cas of sapl o so Fu 5 as valu of 0. I ao o sa a sca coos al osa boua coo ca also b spcf as a slvalu a cuv a loop-a cuv a a ccal p sco. If a sl-valu a cuv s us as osa boua coo scz fo bcos: [ ] 0 (87) subscp pss valus fo a cuv aa. If a loop-valu a cuv s us as osa boua coo scz fo bcos: / R c f / ( / ) 0 (88) fco slop s appoa us o valus of sca a sa a osa ac: 56

68 ( / ) ( / ) f / / (89) Δ Δ Δ Fall f a ccal p sco s us as osa boua coo scz fo bcos: ( / ) 0 (90) Dscz fos of al boua coo quaos W al boua coos a pl s obsv a ca cals o o av a upsa a osa boua coo. Ts ss coos occu a uco pos of s cals. Tfo al boua coos a o sasf ass a ou balac a s ucos. Fo a uco flo cals s qu o spcf a oal of al boua coo. Ts coos a spcf as osa boua coos fo ac flo cal a o upsa boua coo fo ouflo cal. I s a o ou quao s fo ac flo cal sasf cou sas. o 0 (9) subscp o ps las o of pacula flo cal o uco a subscp o ps fs o of ouflo cal fo uco. W (9) s s fo all flo cals a oal of quaos a fo uco a ss al osa boua coo of all flo cals a copl. Fall o las coo s spcf o ss al upsa boua coo of ouflo cal. Ts coo s sasf b cou quao fo uco: 0 (9) pss las o of cal. Fo fs uco of pacula o so Fu 5 al osa boua coos of flo cals a s spcf b: o 9 0 (9) 57

69 8 9 0 (9) a al upsa boua coo of ouflo cal s spcf b: (95) lal fo sco uco of pacula o so Fu 5 al osa boua coos of flo cals a s spcf b: 8 0 (96) (97) a al upsa boua coo of ouflo cal 5 s spcf b: (98) 58

70 pp II. Dvao of Paal Dvavs of F Dffc Equaos of Cal Flo T o-rapso cqu s bas o aalcal o ucal ffao of cou a ou quaos o valua paal vav s of Jacoba a. T ffc fos of cou a ou quaos v (5) a (6) a paall ffa spc o uo s a a () l fo oal pos () a (). I follo vaos cou a ou quaos a ps b ls C a M a al upsa a al osa boua coo quaos a ps b ls U a D spcvl fo cla. lal al boua coo quaos a ps b l I. Paal Dvavs of Cou Equao T paal vavs of cou quao spc o uo s (.. a a () l) a copu as follos: C θ Δ / / Δ Δ ( ) / Δ Δ / [ s ( ) ] c / o (99) C θ Δ / / Δ Δ ( ) / Δ Δ / [ s ( ) ] c / o (00) C θ (0) C θ (0) Paal Dvavs of Mou Equao T paal vavs of ou quao spc o uo s (.. a a () l) a copu as follos: 59

71 60 Δ Δ Δ Δ Δ / / / / / / f f L M M β θ (0) : Δ Δ Δ Δ / / / / / / 5 f f (0) / / Δ c (05) Δ Δ / / / / / / / / 0 L q L (06) Δ Δ Δ Δ Δ / / / / / / f f L M M β θ (07) : Δ Δ Δ Δ / / / / / / 5 f f (08)

72 6 / / ) ( ) ( Δ c (09) Δ Δ / / / / / / / 0 L q L (0) Δ Δ Δ Δ Δ / / / / / f L s M β θ () : Δ Δ / / / / / / f f () / / ) ( Δ c () / / / 0 L q L () Δ Δ Δ Δ Δ / / / / / / / f L s M β θ (5) : Δ Δ / / / / / / f f (6)

73 / Δ c / ( ) (7) L / 0 ql ( ) / / (8) Paal Dvavs of Eal oua Coos upsa bouas a sca o a sa oap ca b pl as boua coo. W a sca oap s us as upsa boua coo paal vavs of Jacoba bco: U U 0 (9) (0) subscp ps upsa o ub of cal. Hov f a sa oap s us as upsa boua coo paal vavs bco: U 0 U () () osa boua a sca oap a sa oap a sl-valu a cuv a loop a cuv o a ccal p sco ca b pl as boua coo. If a sca oap s us as osa boua coo paal vavs bco: D () 6

74 D 0 () If a sa oap s us as osa boua coo paal vavs bco: D 0 (5) D (6) Fo sl-valu a cuv paal vavs ca bco: D (7) D (8) If a loop a cuv s us paal vavs a as: Δ ' Δ D (9) D Δ ' Δ 5 Δ Δ (0) : / R ( / ' c ) f () / Fall f a ccal sco s us a osa boua paal vavs a as: D () 6

75 D c Δ Δ () : ( / ) c () Paal Dvavs of Ial oua Coos a uco flo cals a oal of al boua coos a spcf. T paal vavs of uco ou quao so Equao (9) fo ac flo cals bco: I 0 I (5) (6) subscp pss las o of pacula flo cal o uco. lal paal vavs of uco ou quao so Equao (9) fo ouflo cal bco: I 0 o I o (7) (8) subscp o pss fs o of ouflo cal fo uco. T paal vavs of uco cou quao so Equao (9) fo ac flo cals bco: I I 0 (9) (0) 6

76 subscp aa pss las o of pacula flo cal o uco. lal paal vavs of uco cou quao so Equao (9) fo ouflo cal bco: I o () I 0 o () 65

77 pp III. Ipola Fucos Gal Mo Ipola (o bass/sap) fucos fo co of f l aalss. T s a o-o-o lao b bass fucos a os scz oa. bass fuco a s f a a pacula o s zo ov a l ulss a o s assoca l of coc. Tocall bass fucos ca b lobal o local cooas. Hov s paccal o us a local cooa ss qualaal ls o splf aos a ffaos of bass fucos. I s a a local cooa ss o a as l cocp s pl s su (Fu 7). c cosquc of s appoac s css o foula a asfoao fuco b lobal a local cooas. Ufoual s asfoao s o la fo a ula qualaal l a c a ucal ao sc s oall qu o valua als f l aalss. ( ) ( ) ( ) ( ) ( ) Mas El ( ) ( ) ( ) GLOL COORDITE YTEM LOCL COORDITE YTEM Fu 7. Global a local cooa sss a as l cocp I s o o soal oa s loball scz us ula qualaal ls. T cooas of os of ac l s app o a local cooa ss va as l cocp. T as l s a X squa loca a c of local cooa as os a ac co. T co cooas of as l a (--) (-) () a (-). Tfo all aos ca b o o as l us ls - a -. T al foula fo sap fucos of a qualaal l ca b oba b a so pouc of o sap fucos fo la l l a v b psso: 66

78 67 () a f local cooa ss us as l cocp. Us s foula a local cooas of as l s possbl o fou sap fucos of qualaal l so Fu 7: () T a o poa ofcaos a soul b o bfo as l cocp ca b us f l aalss. T fs o of s ofcaos s o asfo vavs of as o local cooas. I o o pl s asfoao fo lobal o local cooas vavs of s sap fucos spc o local cooas a valua us ca ul of ffao: (5) c ca b ps a fo as: (6) T vavs of sap fucos spc o local cooas ca asl b copu as follos:

79 68 (7) T coffc a ov qus a fucoal laosp a aps lobal cooas o local cooas. Ts asfoao fo lobal o local cooas s oba b us sap fucos o pola lobal cooas. Hc o ca : (8) If vavs of s pssos a a spc o local cooas o oul oba a X a a s cool o as Jacoba a: [ ] J (9) T a of Jacoba s a poa qua a s svl us as l aos us local cooas. [ ] J J (50) I soul b o ov a a qu o asfo vavs lobal cooas o vavs local cooas s o acl Jacoba a v abov. T l b s o acs ca b sabls f a a s suc a:

80 (5) I abov quao sco a s spl aspos of Jacoba a. c a s oba b ulpl a a a s vs fs a bcos vs of Jacoba aspos. [ ] T J J (5) I s o possbl o copl asfoao of sap fuco vavs b us s a a vavs of sap fucos spc o local cooas. T sco ofcao a s qu o us as l cocp f l aalss s o cov ao vaabls o local cooas. T basc foula fo a ca of ao vaabls s v as: l as l as J f f (5) a of Jacoba s us plcl. I soul b o a s ca ao vaabls os o pos a a ffcul as lo as a of Jacoba s a cosa. Fo o-la cooa asfoaos suc as o us qualaal ls Jacoba s o a cosa a abov ao ca ol b o us a ucal ao sc.

81 70 pp IV. Dvao of Gal Fo of Goua Flo Equao T fs sp of vao of a fo s o appoa uo fuco ov oa us pola fucos of fo: (5) ĥ s appoa valu of aulc a a s oal ub of os o soal oua flo oa. I ssc poal a spaal sczao s spaa fo ac o appoa soluo. T oal valus () bcos ol a fuco of a sap fuco () s o ol a fuco of spac. I s also poa o o a sap fucos a f ol s cospo o a bo ls. T a zo ls. c soluo s ol appoa os o sasf ffal quao acl a a sual R occus. T Gal o sas a ava of s sual ov ol oa bcos zo. Fuo fucos a a o b bass fucos. If appoa soluo s subsu ffal quao o ca oal sual as: [ ] () () I q z z z z R L b b b b 0 δ δ δ δ (55) T Gal f l o s bas o s sual ov ol oa us pola fucos as fucos: 0 R (56)

82 7 us fo ou. W psso fo sual s subsu Equao (56) a squa oo psso s spl as o of a of paac vco quao al splfs o: [ ] () () 0 0 I q z z z z L b b b b δ δ δ δ (57) T ao b pas s o appl o sco o vav s abov al o uc o fs o a copoa aual boua coos: z z z z b b b b (58) z z z z b b b b (59) a ps a copos of u oal vco. ubsu s pssos a aa vs: