Keywords Open channel flow; Adjoint sensitivity analysis; Numerical models; Flash floods.

Transcription

1 Elevenh Inernaonal Waer Technoloy onference, IWT 7 Sharm El-Shekh, Ey MITIGATION OF FLAS FLOODS IN ARID REGIONS USING ADJOINT SENSITIVITY ANALYSIS ossam Elhanafy * and Graham J.M. oeland ** * PhD suden, vl Enneern Dearmen, Srahclyde Unversy, Scoland, U.K. ** Reader, vl Enneern Dearmen, Srahclyde Unversy, Scoland, U.K. ABSTRAT Ths aer resens an analyss of he sensves of flood wave roaaon o varaons n ceran conrol varables and boundary condons by means of he adon mehod. Ths uses a varaonal echnue o fnd he relaonshs beween chanes n redced flood waer levels and chanes n conrol varables such as he nflow hydrorah, bed rouhness, and bed elevaon. The sensves can be used for omal conrol of hydraulc srucures, for daa assmlaon, for decson makers' rocedures, for he analyss of he effecs of unceranes n conrol varables on he redcons of floods waer levels, and for nvesan boh he sensves of model flood forecass o model arameers, boundary and nal condons. Eamle of he las alcaon of he sensvy analyss s resened and dscussed. These mehods are develoed and mlemened hrouh a numercal hydraulc model of channel flow based on he Shallow Waer Euaons (SWEs) and he corresondn adon model. The euaons are neraed usn fne dfference mehods and a new modfed mehod of characerscs s used o defne he oen boundares. Resuls of valdaon ess on boh he forward hydraulc model and on he adon model are resened. Keywords Oen channel flow; Adon sensvy analyss; Numercal models; Flash floods. INTRODUTION Floods are consdered one of he mos danerous envronmenal haards ha hreaen lves, roeres, and culvaed lands. Flood wave roaaon models are ofen used when lannn flood manaemen sraees, Moussa e al. [] and s moran o consder wha conrol acons could mae flood mac. Such conrols could be hydraulc srucures such as aes, locks, and wers Sanders & Kaaodes [] or he dverson of waer no canals and floodlan sorae facles, Sanders & Kaaodes [3]. owever, he erm conrol also s used for a user-defned value ha deermnes he resul of a model forecas, raher han offern an acual enneern conrol. Eamles of hese conrols are he values of he nflow hydrorah, downsream waer

2 Elevenh Inernaonal Waer Technoloy onference, IWT 7 Sharm El-Shekh, Ey level, bed rouhness, and bed sloe. Ths aer resens an analyss of how a model redcon of flood waer level a a ceran locaon s sensve o varaons n some of hese conrol values. These sensves can be used o selec he mos arorae locaon and rae of absracon for flood conrol, Dn & Wan [4], o ome waer absracon for rraon, Sanders & Kaaodes [3] or o denfy Mannn's rouhness coeffcen, Dn e al. [5]. The sensves can also be used for daa assmlaon, acuc [6] and Navon [7] and o uanfy he conseuenal effecs of sensves n some conrol values on redced flood levels or flood volume as shown n hs aer. The adon sensvy analyss has been mlemened hrouh he develomen of wo numercal models; a forward model, based on he nonlnear Shallow Waer Euaons (SWEs) used o smulae he roaaon of he flood wave, and an adon model used o evaluae he me-deenden sensves wh resec o a varey of conrol varables under dfferen flow condons. The adon mehod reures ha he forward roblem and s assocaed adon roblem are solved seuenally. Sensvy eressons whch are funcons of he forward and adon varables can be aled o assess he chane n oucome resuln from chanes n conrol values. The adon sensvy analyss s used here o esablsh relaonshs beween ceran conrols and he sysem resonses. A arcular conrol roblem s defned by selecon of an arorae obecve funcon. Ths may reure o be mnmed n he case of daa assmlaon or, for eamle, may measure flood waer levels n ecess of some hreshold as dscussed n hs aer. Once hs obecve funcon s defned, he adon sensvy analyss s used o evaluae he raden of hs funcon wh resec o he conrol varables or n oher word he sensves. GOVERNING EQUATIONS FOR TE OPEN ANNEL FLOW The one dmensonal Shallow Waer Euaons (SWEs) form a sysem of aral dfferenal euaons whch reresens mass and momenum conservaon alon he channel and nclude source erms for he bed sloe and bed frcon. These euaons may be wren as: = () ( u) ( S S f ) = () Where: : s he dschare er un wdh (m s - ). : s he ravaonal acceleraon. : s he oal waer deh S : s he bed sloe = -. : vercal dsance beween he horonal daum and he channel bed as funcon (,). : s he me (s).

3 Elevenh Inernaonal Waer Technoloy onference, IWT 7 Sharm El-Shekh, Ey : s he horonal dsance alon he channel (m). S f : s he bed frcon sloe = k 3 k : s a frcon facor = / accordn o hey or = n / (/3) accordn o Mannn. ( u) : s he momenum flu erm, or convecve acceleraon In smulan an unseady channel flow durn a flood wave even he one dmensonal SWEs, Euaon () and Euaon () are subeced o nal and boundary condons. Inal condons are (,) and (,) and he boundary condons are (,) and (L,) where = L s he downsream lm of he model doman. Values (,) comrse he nflow hydrorah and (L,) are nerolaed from whn he doman usn he mehod of characerscs (MO), Abbo [3] and French [4] afer modfyn o su he case of slon rouh bed descrbed below o rovde a ransaren downsream boundary hrouh whch he flood wave can ass whou reflecon. ADJOINT SENSTIVITY ANALYSIS FOR TE (SWEs). Defnn he obecve funcon If he alcaon of he analyss s o he conrol or assessmen of rsk of flood waer levels hen s convenen o comare redced levels wh known hreshold values above whch floodn could occur. A measure of he dfference beween hese levels can be used o uanfy he effec of a conrol. In arcular, we are neresed n waer levels ha eceed hreshold values a ceran locaons ( o ) and a ceran mes ( o ). As a surroae for level we can use local deh (rovded he local bed level s known) and so defne a uadrac obecve funcon ha uanfes he waer dehs reaer hen some secfed hreshold flood dehs d. r =.5{( - d) - d } ( o ) ( o ) (3) Where: (, ) s he waer deh calculaed by he forward hydraulc model, and d ( o, o ) are hreshold values a = o, and = o.. Governn euaon for he sensvy analyss Adon sensvy analyss evaluaes he sensves of he obecve funcon o ceran conrol varables. Ths s acheved by crean he Laranan and akn he frs varaon. The mehod follows closely ha descrbed by Sanders & Kaoodes [3], Geade & oeland [8], and oeland& El-hanafy [9] as follow:

4 Elevenh Inernaonal Waer Technoloy onference, IWT 7 Sharm El-Shekh, Ey The cos funcon J s defned by neran he wehed sum of he obecve funcon and he resduals of he SWEs over he enre comuaonal doman as follows: = T L f d d u S S r J ) ( ) ( φ (4) where he wehs and are Larane mullers laer o be revealed as he adon varables. The sensves and he adon euaons are evaluaed by akn he frs varaon of J n Euaon (4) wh resec o all flow varables and conrol varables, akn no consderaon ha = -So, u =, S f = 3 k, k = and usn neraon by ars, he fnal eresson for he varaon n J, J, s ven by he sum of he follown 6 nerals:- [ ] { } ( ) d d d d d d r r d d J T L T L T L L T T L φ φ φ φ = 3 3 We are seekn varaons n J ha are caused by ossble varaons n and as he nal condons and a he usream and downsream boundares. We also are lookn for he effecs of varaons n conrols, and Z n he doman. These are eressed by nerals,, 4, and 5 resecveley n Euaon (5). We are no lookn for he effecs of varaons n and whn he doman as eressed by neral 3 n Euaon (5). ence hs neral s reured o eual ero for any varaons n and. Ths condons leads o he denfcaon of he wo adon euaons from he kernal of hs neral as follows: (5)

5 Elevenh Inernaonal Waer Technoloy onference, IWT 7 Sharm El-Shekh, Ey φ τ φ τ r 3 = r = (6) Where τ = (T ), measured n reverse me drecon. Soluon of adon euaons, Euaon (6), for ven flow condons and ensures ha neral 3 n Euaon (5) euals ero and rovdes values for and everywhere n he doman. These values can be used n condons derved from nerals,, 4, and 5 n Euaon (5) o uanfy he sensvesas shown below. 3. Dervaon of he sensves 3.. Sensvy o he usream channel flow Sensves o nal condons are revealed by neral n Euaon (5). If we mose he condons φ(,t) = (,T) =, ha s no sensvy nformaon roaaes no he doman a = T, hen he sensvy o nal condons s us J J = φ and = a each dscree locaon alon he channel. If we do (,) (,) no wan o conrol he soluon o any nal condon hen we may se (,) = (,) = hen neral vanshes. Sensves o boundary condons are revealed by neral n Euaon (5). If we mose he condon (L,) =, ha s (L,) s no used as a conrol, hen he sensvy o nflow (,) s shown o be J (, ) { φ(, ) u(, ) (, )} = (7) a each me se or erurbaon me from = o T. Smlarly by mosn condon (,) =, ha s (,) s no used as a conrol, hen he sensvy o downsream deh s shown o be J ( L, ) { ( L, ) ( L, ) ( u )( L, )} ( L, ) = (8) Fure () shows all of he reured boundary condons. We noe ha he boundary condons for (,) and φ(l,) are boh reured and mus be defned such ha boh boundares are ransaren o he ouon eurbaons creaed whn he adon doman. Ths s acheved by nerolaon from avalable values whn he doman usn he mehod of characerscs (MO), he formulaon wll be ven below.

6 Elevenh Inernaonal Waer Technoloy onference, IWT 7 Sharm El-Shekh, Ey φ(,t) = (,T) = T (,) = (,) s obaned usn MO (L,) = φ(l,) s obaned usn MO L Fure () Boundares and nal condons for he soluon doman 3.. Sensvy o he Bed elevaon The bed elevaon can be consdered as a conrol varable whch can affec he flood level a = o. Sensves o he bed elevaon a boh he usream and boundary condons are revealed by neral n Euaon (5). If we mose he condon Z(L,) = Z(,) =, ha s Z(L,) and Z(,) are no used as a conrol, hen he sensvy wh resec o he bed elevaon whn he whole doman, from neral 6 n Euaon (5), s wren as: J T L = ( ) d d (9) From Euaon (9) he saal and emoral varaons of he sensvy wh resec o he bed elevaon can be evaluaed afer he wo adon euaons, Euaon (6) are solved Sensvy o he Bed frcon The bed frcon, n erms of hey coeffcen can be also consdered as a conrol varable whch should affec he flood level, hence he sensvy s wren as: J T L = 3 d d () From Euaon () he saal and emoral varaons of he sensvy wh resec o hey coeffcen can be evaluaed afer he wo adon euaons, Euaon (6) are solved

7 Elevenh Inernaonal Waer Technoloy onference, IWT 7 Sharm El-Shekh, Ey NUMERIAL APPROA. Dscresn he Forward Model The fne dfference mesh for dscresn he doman s deced n Fure () ha follows a smle sace and me saered. A reular mesh of dmenson (), sacn of rd ons n -drecon by (), sacn n he me drecon, so now f [,L] s L dscresed by (n) eually saced hen = n, and by he same conce T = n, so he dscresed values of a funcon (h) a (, ) wll be denoed h = h(,) = h(, ) he suerscron () refer o me dscreaon and s called me se or me level, whle he subscr () refers o sace dscreaon and s h called sace se or sace level. The aromaon of he dervaves s now h h,a frs order uwnd scheme s used o sable he soluon of he ( u ) shallow waer euaon, he convecve erm s dscresed wh wo on uwnd dfference eresson or a wehed averae of cenered and uwnd dfference eressons: ( u) = u( ) u( ).5 ( u( ) 3u( ) 3u( ) u( ) ) 3 See Flecher [], Leonard [] and Falconer & Lu [] for more deals. The dschare s marched forward n me usn he momenum euaon as follow: [ ]. ( ). k. ( ) =. ( u). () The waer deh s marched forward n me usn he connuy euaon: [ ] = () The nal condons are and whle he boundary condons are a he usream boundary and n a he downsream boundary, he usream condon s he nflow hydrorah; he downsream condon mus be nerolaed usn he mehod of characerscs (MO) as descrbed n Abbo [3] and French [4].

8 Elevenh Inernaonal Waer Technoloy onference, IWT 7 Sharm El-Shekh, Ey. Dscresn he Adon Model The adon model whch s reresened by Euaon (6) s dscreed usn a smle sace and me saered elc fne dfference scheme as llusraed n Fure (). The adon varable φ s marched backwards n me usn he dscree form as follown [ ] ( ) 3... ). ( u c φ φ = (3) Whle he adon varable s marched backwards n me usn he dscree form as follown: [ ] ( ) ( ).. u φ φ = (4) Fure () The dscreaon scheme for he forward and adon model 3. Mehod of haracerscs (MO) Follown a sandard e such as Abbo [3] and French [4], he characerscs of he lneared (SWEs) are denfed as: ) ( = ± K c u (5) Whle characerscs of he adon model are denfed as: ) )( ( ) ( 3 = ± ± c u c u φ (6) Where ndcaes a oal chane n varable alon he characersc ah.

9 Elevenh Inernaonal Waer Technoloy onference, IWT 7 Sharm El-Shekh, Ey MODELS VERIFIATIONS. Forward model verfcaons. Inroducon Develon a comlee es o check and valdae an eac soluon for he nonlnear Shallow Waer Euaons (SWEs) s no ossble. I s ossble however o develo smle ess o comare he model resuls wh analycal soluons of ceran dealed cases. Several ess have been carred ou o verfy he model from unform seady flow o non-unform unseady flow; we wll menon here us he wo mos moran ess.. Valdaon es non-unform unseady flow The man obecves of hs es are o assure he follown: - The value of boh he dschare () and he waer deh () a he usream roaae downsream whou any chane. - The relaonsh beween and follow he analycal soluon of he shallow waer wave. The analycal soluon of he shallow waer wave The analycal soluon of he shallow waer euaon n dee waer nally, m. wh a drvn usream hydrorah follown snusodal wave conce of amlude. m. as llusraed a Fure (3), where; a: s he amlude of he wave, T: s wave erod, : me, c: wave seed =.., and : s he oal waer deh s = a.{ sn()} =. m whch lead o ma=. m and = a. (. ). (sn ()) whch lead o ma = 8. m 3 /s/m. and he raveln seed s eual o (. ) = 4.69 m/s..5.5 ( m ) (_cycles ) number of wave cycle Fure (3) The drvn hydrorah shae

10 Elevenh Inernaonal Waer Technoloy onference, IWT 7 Sharm El-Shekh, Ey The resuls of he model are a drvn usream boundary hydrorah of eak dschare = 8.4 m 3 /s/m and he calculaed usream boundary hydrorah of eak value, ma =.96 m. whle he wave seed s 4.74 m/s. so, he frs concluson s ha he relaonsh calculaed by he model ycally follow he shallow wave euaon and alhouh here s dscreancy beween he calculaed values from he model and he analycal soluon bu hs dscreancy could be nerreed due o he values of dsance se = 3.5 Km and me se of 8 s, he second concluson s ha he hydrorah raveled from he usream boundary o he down sream boundary wh a small chane n he eak dschare from 8. m 3 /s/m o 8.4 m 3 /s/m and from.96 m o.94 m for he eak waer deh as llusraed a Fure (4) and hs acceable dffuson s duo o he numercal dssaon of he used elc scheme. The las concluson s ha he wave raveled a dsance of 5.6 Km. whn 6 sec. so s seed s 4.74 m/s. whle he seed of he wave should eual o (. ) = 4.69 m/s whch s nearly he same. So fnally, s clear here s a ood areemen beween he analycal soluon and he develoed model and also here s no numercal dssaon. Fure (4) Waer deh () whn he doman.3 Valdaon es - Unseady flow whn a slon channel and rouh bed: There are wo man obecves of hs es; he frs obecve s smly o look for he whole channel as a conrol volume o assure here s no snfcan losses or accumulaon n volume whn he smulaed doman and he resuls of hs ess wll no be comared wh he analycal soluon only, bu wll be comared wh oher model resuls as well, he second obecve s o assure he volumerc conservaon rncal a dfferen me ses s always vald and no numercal oscllaon a he wave fron. If we consdered he nal waer deh s and a he end of he

11 Elevenh Inernaonal Waer Technoloy onference, IWT 7 Sharm El-Shekh, Ey smulaon s f. Whle he drvn dschare usream s u and downsream s d so we could say: Toal volume eners he channel s V = u d d d, whle he oal volume leaves he channel s V = f d d, o be n eulbrum, should be V = V. The model was aled for non-unform unseady flow condons whn a slon channel and rouh bed. The nal waer deh was chosen nal =. m. The resul of he flood wave roaaon whn he doman s resened a Fure (5). V Fure (5) Waer deh () whn he doman = = d d u d = m 3 /m V = d f d = = m 3 /m So, V V m3.86 whch s acceable and s very small error comared o several revously develoed model such as Abola [5] whch was overesmaes by 8 %.. Adon verfcaons In he follown eermen, a drec smulaon s done frs wh a chosen boundary and he resuls a ceran locaon are now consdered as he observaons ( = ). Then he model s re-run aan wh a new boundary and he resuls a same locaon are recorded ( = ). The dscreancy beween he wo soluons a ( = ) s used o measure he cos funcon. Then by usn he conuae raden mnmaon he cos funcon s mnmed a each eraon o cover from as shown n Fure (6). Ths echnue where he same model s used o do he daa assmlaon and o e he observaons s called he dencal wn eermen.

12 Elevenh Inernaonal Waer Technoloy onference, IWT 7 Sharm El-Shekh, Ey Se he frs wn Se usream Run forward model Se he second wn Se = a usream Run forward model omue (= ), (= ) omue (= ), (= ) alculae cos fn. onuae raden mnmaon Run he adon model alculae raden Recover usream (k) from usream (k) Run forward model No alculae cos fn. = chosen accuracy lm (e) Yes so Fure (6) Idencal wn eermen flow char The converence was rad ha he nle hydrorah, was found afer only 4 eraons ha a reducon of he measurn funcon by a facor was acheved n abou eraons and abou 97 % of had been recovered afer only hree eraons as llusraed n Fure (7) and Fure (8). cos funcon no. of eraons Fure (7) os funcon converence

13 Elevenh Inernaonal Waer Technoloy onference, IWT 7 Sharm El-Shekh, Ey Waer level (m) Ieraon=,( ) Ieraon= Ieraon=3 ydrorah a nle,( ) Tme (ours.) Fure (8) Recovern of from as a resul of he dencal wn eermen TEST ASE In hs case, a 3 Km-lon channel wh an usream hydrorah follown a snusodal wave shae as shown n Fure (3) ha wll be used o run he forward model and hen he adon euaons (6) are solved, he values of he adon varables [, ] are hen obaned whn he whole doman as shown n Fure (9). A B Fure (9) Soluon of he adon varables [, ]

14 Elevenh Inernaonal Waer Technoloy onference, IWT 7 Sharm El-Shekh, Ey The same me se and saal ncremen wll be used for boh he forward model and he adon model, he mamum hreshold waer level s assumed o be.9 m. a 5. Km down sream from he nle. The sensvy of he drvn usream dschare J descrbed n Euaon (7), he emoral varaon of he sensvy, ) s shown n Fure () whch s conssen o he drvn hydrorah usream, shown n Fure (). ( Dschare (m 3 /s./m.) Tme (mn.) Fure () The nle drvn hydrorah Sensvy o dschare usream Tme (mn.) Fure () The sensvy wh resec o he usream hydrorah J The varaons of he sensvy o he bed elevaon descrbed n Euaon (9), n sace and me s shown n Fure (). Whle he varaons of he sensvy wh

15 Elevenh Inernaonal Waer Technoloy onference, IWT 7 Sharm El-Shekh, Ey resec o he hey coeffcen descrbed n Euaon (), shown n Fure (3). J n sace and me s Fure () The sensvy wh resec o he bed elevaon Fure (3) The sensvy wh resec o hey oeffcen

16 Elevenh Inernaonal Waer Technoloy onference, IWT 7 Sharm El-Shekh, Ey ONLUSIONS The sensves eressons relaon o he redefned obecve funcon resonse o usream drvn condons and o some moran conrol varables whch are saally and emorarly dsrbued have been derved, and s clear from Fure () ha he sensvy of he flood level a he secfed locaon ( ) o he usream dschare follow he hydrorah shae, n oher word he sensvy ncreases as he dschare ncreases and decreases as he dschare decreases. Whle he sensvy o he bed elevaon whch s llusraed n Fure () elan he effec of he bed elevaon from he usream boundary o he down sream boundary on he hreshold waer level. Fnally, Fure (3) show ha he effec of he channel rouhness from he usream boundary ll he secfed locaon ( ) s much more reaer han from he secfed locaon ( ) ll he down sream boundary duo o he backwaer effec ha aree wh he basc hydraulc conces n ha any nformaon could roaae usream only n subcrcal flow, whch s case suded n hs aer. These sensves could now be funconed for several uroses, could be used for arameers denfcaon or could be used by decson makers o hel n rorn he mos moran arameers, n he case suded n hs aer as an eamle, s clear ha he mos moran conrol varable s he drvn usream dschare comared o he bed elevaon and he channel rouhness eressed n hey coeffcen. or may be used as a ool o mae he flood haards a ceran locaons alon he channel by denfyn he hreshold waer level no only a = ( ) bu as funcon alon he suded channel and selec he mos arorae locaon for a ceran conrol acon whch may be a reservor or deenon dam or a dverson channel. The roer numercal soluon and achevn oen boundares for boh he forward model and he adon roblem lead o formulaon of an adon soluon whch s conssen wh he basc roblem. In he near fuure, he research s o be eended o evaluae boh he effec of ndvdual uncerany n each conrol varable on he flood even and he lobal uncerany from all he conrol varables on he flood mac. AKNOWLEDGMENTS The auhors would lke o eress her deees raude o Dr. Ior Geade for hs sncere concern and useful udance hrouh my PhD research. I am arcularly arecan my sonsor a my counry who ave me hs chance o comlee my sudy. Las bu no leas, I would lke o hank all my colleaues and he saff of vl Enneern Dearmen of Srahclyde Unversy.

17 Elevenh Inernaonal Waer Technoloy onference, IWT 7 Sharm El-Shekh, Ey REFERENES. Moussa, O.M., Abdelmeaal, N.., Elmony, A.E. and EL-anafy,.M., Effec Of Envronmenal hanes On Dranae Paerns And Urban Plannn Alon oasal Reons (ase Sudy), Thrd Inernaonal onference On vl & Archecure Enneern, Mlary Techncal ollee, Kobry Elkobbah, aro Ey, SU9, 999, Sanders, B.F and Kaoodes, N.D., onrol of mul-dmensonal wave moon n shallow-waer, Proceedns of he 5 h In. onf. on Esuarne and oas. Modeln, M. L. Sauldn, Ed., 998, Sanders, B.F. and Kaoodes, N.D., Adon sensvy analyss for shallowwaer wave conrol, Journal of Enneern Mechchancs, ASE, Vol. 6, No. 9, Dn, Y. and Wan, S.Y., Omal conrol of oen-channel flow usn adon sensvy analyss, Journal of ydraulc Enneern, ASE, Vol. 3, No., Dn, Y., Ja Y., and Wan, S.Y., Idenfcaon of Mannn's rouhness coeffcens n shallow waer flows, Journal of ydraulc Enneern, ASE, Vol. 3, No. 6, n ress. 6. acuc, D.G., haer 3, Sensvy and Uncerany Analyss, Volume I, haman & all R., 3 7. Navon, I.M. and Zou, X., Alcaon of he Adon Model n Meeoroloy, Proceedns of he Inernaonal onference on Auomac Dfferenaon of Alorhms: Theory, Imlemenaon and Alcaon, SIAM Publcaons, A. Grewanek and G. orlss Eds, SIAM, Phladelha, PA Geade, I.Yu and oeland, G.J.M., Adon sensvy analyss for flud flow wh free surface, In. J. Numer. Mehods n Fluds, Vol. 47, Issue 8-9, oeland, G.J.M. & El-anafy,., omuer modelln of channel flow usn an nverse mehod, Proc. 6h In. onf. on vl and Arch. En., aro, May 6.. Flecher,.J., haer 5, omuaonal Technues for Flud Dynamcs : Fundamenal and General Technues, Srner-Verla, 99. Leonard, B.P., Thrd-Order Uwndn as a Raonal Bass for omuaonal Flud Dynamcs, omuaonal Technues & Alcaons: TA-83, Eded by Noye, J. And Flecher,., Elsever.. Falconer, R.A. and Lu, S.Q., Modeln Solue Transor Usn QUIK Scheme, ASE Journal of Envronmenal Enneern, Vol. 4, M.B. Abbo, haer &3, An Inroducon o he mehod of characerscs, Thames and udson, London UK, French, R.. haer 4&6, Oen hannel ydraulcs, McGraw-ll, Abola, A.A. and Nkaloaos, D.K., Model for Flood Proaaon on Inally Dry Land. ASE Journal of ydraulc Enneern, Vol. 4, No. 7,