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Solution of the kinematic-wave equations for border irrigation
Agrwullural Wulcr Ma.agem¢'nl, 9 ( 198,4 ) 1",'7-138 Eh~evler ,.qulerlce PulJli~her~, B V , A r r l s t . e t d a m
-
S O L I . I T I O N O F ']'HE KI N E M A T I C - W A V I RRIGATI(-)N
VI,IAY P SIN(-3H I and R,A M A S
Plltlled
12'7 III 'T'h~, N e l h e r l a n d ~ ,
E EQI.I A T I O N S F O R B O R D E R
R, AM ~
[h'purlrnt'nt u f (?w~l E,n/~mecrm~, L,uu~,stunu Stub, I lmr,t,r~z~ty, Eluhm R~ugc, LA 70803 I.,'S ,4 ) Dvpa~ tm,,nl ,~1',tl,lalhemulws, ,4 It'~rn Stab' I.hm,,,rs~l y, L,,~rmun, ,~ll,.q ,'1!4t)!46 ( II ,.~ ,4 ) A c c e p t , e d ~ May 198,4)
ABSTRA("T ~llit.;h. V P a n d R a m , R S . l.qH4. S . l u t , l u n u f t h e I, in~n~ulH: w a v e equall.n=,~ l u r bu=de~ ir~lgal,l~n ,,-t~rzc Waterl~ldnug*', 9 l ~ ' 7 - l : t ~
A meth,,d i~.,devehJped t,, obt.aitl solulion~, l,, l he l.tltlemalil' w a v e ellU,Jl, l u n , ~., l'ur buI,Je~ The ac~'uracy ur th~:~ rrielhud is a~isessed by i.'(.inlpaJ'lllb.~, the tesulls wilh I he l.iltlerrlal.iL" wave tlaln (Kw"r) melhud prPvtuu,,.;ly rep,.nted h~ he accurate Experimerilal d a [ a ,~tt f r e e l y dramln 8 hurdet.~, were u l I I I z e d l'q~r t'qmlpmi,~url and evalual.i~n uf l.he:.,e melhuds Thepiedlctlunelr~r lul advance Limeie~,ultlng frum I he, lwu mel hud,~iwascum II'l'l,El, a l l l l l l
palal-~le II wa,,; b e h , w BS',";, I~Jr I hl'ee ,~ieln ~1 d a l a b u t a~,~ hll.~,h a.~ h()% fur u n ~ ,,.Jel ~pl' dal,.l T h e rJledlelll~rl erH~r l'4at i ¢rq.'e,~.~l,ilOll t.lme wa, ~, al.sl~ l ' U l r l p o u a h l e b e h l w 6 % f u r t i l e ~iame t.hleP ~.,el~ o f d a t a Llul. ] ,..I 0 % ['ul Olle .'.,el
IN'T'R(.~DLJCTION A k n o w l e d g e of advance, recession, dept, h or flow, and sod mOlSl.uJ'e dL%nbul.ton k'~ required l'or an opt, unaJ desi[.,~] u f surface trngat, ion One way l.o del.ermine LLhem Is using mat±hemal.lcaJ models which have recent±ly been surveyed l.)y Ram ( 1 9 8 2 ) . Basset,t, el. al. ( 1 9 8 1 ) have discussed l.he (-u.rrenl. st±al.e o f t±he-art± o f h y d r a u l i c s o f surface Lmgat±ion T h l s s t u d y uses a kinemat, ic wave ( K W ) model (Sherman and Singh, 1978, 1982). T w o t±yl.)es o[" solut±ic)ns t,o t,he KW model have heen sought, in t,he past,. (a) analy1,1cal; and (b) numerical. AnaJyLLicaJ solul.lons have been rel.)ort, ed for consl,ant, i n f l o w and const, ant, )nfdLLrat, lon (C, Lmge and Wooll-user, 1975; Sherman and Sblgh, 1978, 1982; Sblgh and Sherman, 1988), since bot±h inl'low and tnfdt±rat, lon are seldom constant', t±hese solut, tons are of llmlt± ed vaJue t,o border irrlgal.lOn design. Alt,hough ntunertcal ,~olut,ions |lave been obt±ab]ed l'or t,une-varyblg infilt,rat±ion and t±ime va.rylng ml'low, t,hey are corn plicat,ed and expensive ( R.am el, aJ., 1983). T h i s sLLudy develops a met±hod t,o solve t,he KW equaLLions for b o r d e r irmgat,ton w h i c h can ha.ndle t, tme varyLng ulfilt,rat,lon told t , i m e v a r y m g ird']ow. T h e met, hod is semt-a.naJyt±tcaJ and
I'~
yields ,~olut,,.ms r¢,cu.rsivel.y. ']"he accuracy of t,he m~t, hod is asse~,,.;ed by ~'om p~'m~, t,he resuJt, s wtLh t,he kinemat,~c wave tram (KW"I") meU~od wh=ch is part, ly numerlcaJ amd r),,.ul,ly a.nalyLic~, aJ}d ha,~ been round t,u be a~'~'u_rat.e ( R a n et, .,d , 1983). Experlment, aJ dat,a t'ur rour t'rt-,ely dra.ming hordt-,rs a.re ut, lhzed ror compar=son Not,e t,hat, v',lJidat, lcl,n of thP KW model ~s out,side t,he ~i~.'l.)].)e of t,hls paper. K I N EM A"I"IC WA V E MODEL For border irngat, lon t,he KW model was rurmulat,ed ~ l d discussed by Shernlan ~ l d Smeh (197L4, 1962,). We refer t,o t,heir wurk for backgruund LnforrnuLioll. Border crn~at, lon essent, lally i n v o l w s t,he flow of wat, er d o w n a I:~la.ne wlLh a sm',.dl slope and porous bud 'T'he border under conslderat, l~.m is assumed t,~.~ be feet,angular, of u n i f o r m c r u ~ ~e~'tlO, and intt, i',dly dry Let, .~ be t,he dl:~t,an(.'e ',.dung t,he harder which ext, ends t,o t.he right, of tt,s head at .~' = 1) At, t,une t = () wat, er is released air, Lhe head of t,he harder. The influw ot' wuL~r at, .~" = (.) has a knuwn t,mle.dependent, dept, h U ( t ) or rat,e q(U, t), uJld la.,st,s for a ,~pecil'led lengt, h of t,irrle TI D~pendln~ upon '/'~ a~ld t,hu b o u n d ~ ' y c'ondlt, lons at. t,he end o1" t,he border at, :~" = L, various phases Itke advance, ~t,oru~e and recession can I-Dedl~t,[JlgLiished us shown in F I ~ 1 o - Horizontal Recession b - Storage Phase i: - A d v a n c e Phase
t
t ." T 2
Phase
. . . . . . . . . . . . . . . . . . . . .
02
Time of Opportunity
O~
"
b
I
i I
t:T
. . . . . . . . . . . . . . . . . . . . . . . . . .
,,
Dt
t-'O
,,=0
C
= L
]
I?ll~ I ~(.~liJtl~arl dumairl=~ fur /.tie, I'h.~w in u I'r~ely dralrllnl:~ I,.JId=-,r
When waLer is released, t,here is a front, of water whl(:h advm~ces d o w n t,tle border 'This advance front, is t,he moving int, ert'ace bet, ween t,he waLer cuver ed and uncovered parts o1" Lhe border. Ir x = sit), or inversely I = ~(:~'),
129 denot, es t,he t,tme htst.ory o r t,hls [ronL o r Lhe advance run,:t,,on, t,hen tt, Ires t,o be det, e r m m e d aJong wJLh the dept, h h(x, t) and veloult.y u(x, t) I'ora specd'led G ( t ) . T h e qua.nt, it,y r = t - t,(x) denot, es t,he mrtlLrat, ion opport, uniLy t4rne a L a IJomt, x , t,hat, is, t,he int, erw.d o f t.lme t.hul, wat,er coveri, d t,he point, x T h e In I'dt, rat, lon rat,e/( r ) IS assumed t.o depend o n l y on t,hi-, dll't'erence 'r t)et.wPen t,he t.ot,aJ elapsed t.lme and t,he advam'e t,tme T h e dept.h ,Jr wat.er h(.~, t), t,he u n k n o w n t.lme hlst.ory ~.(x') and t.he in [llt.rat.l(.m I'(7 ) are suF)je,'l, t.o t.he rullowmg KW I'ormulat.tun'
Oh --
,) t
a +
- -
,) X
- - =
(.[~.
(~ttl")
=
-l'(t
-
(1)
~,(:~ ))
[/.lh'-'(x,~,(x))]-',
~,(()) =()
(2)
wllere n It.; ;.,it'l e×lJi.Jl'le, llt, alld va.ry, ir'll~ between 1 zul,.l 3, and (.1( , O) is {,}le rrl(, Liun pas'uniet,er; h(O, t) = ( ; ( t ) , O ' t ' T I . h(O, t) = (), t , TI N(.)t.e t,hat, t,he wlo,'lt, y o r rlow u = ( . I h ' '. T h u s n and /.I (.'an he det,ermmed f r o m MaJmlnl~'S equut, ion unlJlyln ~ n = 5/3 asu.t IJ = S l ° ' " / l l m , sr IS {,h~ I'nct, lun slope and It m I,~
MannlnB's roughness ,'()erl'lelent,. METHOD (.)F SOLLI'T'ION
Aduance. As shown in Fig. 1, t,hLs phase ts r,'present,ed by dorna.m [)j whteh is b o u n d e d by t = t-,(:~ ), ~uld t = T denut, lng t,he advmice Lime. CenLra] t,u the ~`;olut, l,.m tn t,hls d u m a m IS det,errnllmtion of t,he t, une hLst,ory or the advml,.'e front,, t = t.,(x). For ,.'um,~t,a n t , / and const,~it (J, Sherman and Singh (.19'78) deriw'd t, he analyt, i,.'al sulut, lonS USln~ t,he met, hod o f ,'tm.racterlst, l,-S: h(t, 1,,) = U - l'(t - to) :~'(t, t.,) = }: IU"
-
((~ - l(t -
(3a) t.))"l
(3b)
where tu IS u charact, erlst, lc pa.rarneLer, 0 ' to' T. Equat, lon (3) ts ,'haraet,er L'4,i,: solut, ion In t,hat, It, describes t,he surface f o r m e d by 'all the ,.'ha.ract,,-,rlst, lc curves t.hroLigh t.he Segl'lrlent. (.) ' t . ' 'T', .~' = O. By elunmaLtnl~ t,, t,he ud vance curve becomes.
"[
(4)
aJ,J ibhe d e p t h h(x, l) ui D, as.
t,(x, t) = [ G , _ I"' ] '''" Slrlee [ usually L,.SnoL ('(.)nsI,a,nL, t,he direct, use o1" Eqs 3 - 5 is not, re'J..hsLic 'T'herefore, I'or tune va.ry. LnB [ we develop a met, hod, by using eqs. 3 - 5 , t,o
130
solve eqs. 1 - 2 t,o find t,he a d v a n c e I'tmet, ion. We a s s u m e I,hat, wat, er m o v e s in a series o f c o l u m n s o f fmit,e wldt, h as s h o w n In Fig 2 T h e ~'ont, c o l u m n m o v e s wit, h k i n e m a U e v e l o c i t y . W h e n it, m o v e s t,o a new p o s i t i o n all t,he foLlowing (.'olumns m o v e forwa.rd, each o e c u p y b l g t,he place vaeat,ed by it,s I m m e d i a L e p r e d e c e s s o r T h i s m o v e m e n t , eont, inues unt, d t,he height, o1' t,he I'runt, c o l u m n ({,he dept, h o f t,he wave front,) reduces t,o a mlr|unLuTi vaJue de noLed as he. T h e n t,he pre<:edmg c o l u m n t,akes o v e r l,he r u l e a,.,~ t,he wave r r o n t . 'T'hls p r o c e s s c o n t , l n u e s Lmt,il t,he wat,er r e a c h e s t,he e n d c)f t,he b o r d e r .
h (j
~,- WATER C(.)LL)I )LUMN ~
1
13
;2
(a)
x
'','~
" X
:0
WATER L;C)LUMN ;'~ ~:0
(b)
3
51a
I
•
II
X
_-I~z_WATECOLLIMN 2
, S i k ~ II
I
X
(c) FII~ 2 MuvPmt'm. ~)1' W;Jt(-'l eulu=mlh We a,.'qsulne /' t,o be const, ant, o v e r a sm,'dl dlst,ant.'e ,Ax, ~ l d furt, lter assume t,ilat, t h e wut, er, ~ t , h an Lnltlal c u l u m n deyt, h th ;.rod u ~'unsLant, infllt, rat, l u n / i , a d v a n c e s o v e r a dlst, am.'e ,Ax on the b o r d e r in t, ime A t . E q u a L I o n 4 can be re
cast, as. At
.[
= -[, -
hi
-
(
h~ j
/.l
At, t,he e n d o f t,hls t, irue mt,erval At, t,hu w a f e r dep{,tl h . . eq 5 as:
161 can be e x p r e s s e d by
13 1
-_
(7)
F,qua{,ions (';--7 were used {,o I'l.nd |,he adva.rl(~e I'unl'Liorl I = ,LI x ) ~; I'oBows (1) We assume LhaL {.he, int'h-)w q(0, t) = q. is i~unsLaJ'lL. The ~.'orrespondlng dept, h o f flow at, t,he upst, reanl end wa~ ('aJ(:uJat,ed by Ma.nnLrlg's equal, lon C; = (~.lqu)°'~'
(81
where C; us t,he normaJ del.)t,h of fh.)w at, t.he upst, ream end In eq. ~, t,lle vaJue q.)f/.I is expressed as before wit, h SI replaced hy So ;~ = Su°"~/n n,
(t-) )
(2) q"he infllLraLiorl raLe/'('r ) was ua.IculaLed by Kost, yakov's equaLion
I'(~) = KuT "-~,
()"
o'
1
(l())
where K aJ1(l e m'e cunsLant, s Lo he det,ermu]ed e,npi.ricaJly For ('oml.)Ut,aLnun i.)l" /'(t) we assume {,hal, Lhe liffflt, ral.,ion opl,.)rt, u.nit, y Lime T alway,~ exceed,'.; ur equaJs a m i n i m u m v',.due T,, such Lha{,' I'(TI : a K ' r , Y " , = GK'? LI-I ,
(.)' 'r
7 ' '
r,,
(11)
T~,
(3) T h e wat.er c o l u m n at.x~ w a s advt:meed t,o.'t',l, i = 0, 1, 2 , . , b y eqs. 6 '7. 'T'hns pro(:ess w a s i'ont, muecl unLil t,he heighL o1" Lhe ~(.)nt e o l u r n n r e a c h e d t,he value h,. A raLnu Rt = hc/G w ~ used Lo conLrol t,he Lnp dept, h tn Lhe solu t,ion. For a h y d r o d y n a m i c model, KmcaJd (197"/0) l'ound RI Lo range bet, ween (.).10 and 0.1,5. However, in {,his a.nalysls Ri = 0.0,5 ynelded bet, Ler resulLs. (4) (.)nee h,. was reached, Lhe waLer ~'olumn immediat,ely behind Lhe fl'ont, was rJermiLt, ed I,o La.ke over and was allowed Lo adva.nue. The ulfdt, rat, ion opporLunit, y t,i.me at, a.ny point,, when t,he front, was at, t,he p(.)slLnon x . i, was c'.d~ulat,ed by ~,.~ - t,~, ~.,.~ - L.,.,..... .~,~ - ~,t_~, ~,t,~ - ~-, Thus wesee t,hat, for the t'Lrst, column Lhe irfl'ilt, rat,..)n opport, uniLy LLme is 'always ~-,~,~ - ~',i = lt, ~ - tt subject, Lo =t,s m t m m u m value ,~,.. $ i m d a r l y , I'or t,he second a.nd t,hu'd e u l u m n s t,he nnl'nlt, rat, ton o p p o r t , unJt,y t, i m e s are ,~i~,~ - ~,i-= aJ~d ~.,,,~ - l,~.,, A snmda.r pat, t,ern rollows for 'all columns. T h e uffnlt, ratJon rat,e l'or any column at, :~'~ ts an average of Lhe rat,es at, :~:t aJ~d x~_ ~. The del.)t,h o f t,he first, culumn at, :~:~,~ is hi~ and is super(:eded by t,he se~'ond column wtt, h depLh /h of h~,t he. Now t,he second column plays t,he role of l,he firsL column, Lhe Lhird ~-olumn plays Lhe role o f t,he second column and so on This l.)rot.'ess was ('ont, mued unl.,d wat,er reached Lhe dow-nsLream end oi" t,he border Havb'~g det,ermined t,he advance l'unct, ion, t,he dept, h ou~d x t i'haract, enst, ms m [)n were det,ermined by eqs 3 and ,.5 l'ollownng t,he above procedure 8/orage 1t' Lhe nffhJw q(O, I) is conLmued aft,er waLer reaches Lhe d o w n sLream end o f t,he border, Lhen Lhe waLer will conL, inue t,o a~'cumu.lat,e 'T'hi,s huddup o f wut,er c(.)rl~t,lt, uLes Lhe sL(.)rage phase As shown tn Fig I, Lhis [.)ha~e
132 Is represent, ed h y d o m a i n D,, w h i c h IS b o u n d e d b y a = (), t = T, t = T I a.nd x = L. S i n c e l,he s o l u L i o n in t,hls d o m a b ~ IS ',also given b y eqs. 4 - 5 , t,he c o r n put,at, lonal st,el..s are t,he s a m e as f o r d o m a . m [)~ R e c e s s t o n . As soorl ms t,he inl'low ts cut, e r r at, t = TI, q(U, t) = 0, [,Fie st, o r a g e I.)ha,se ends and the recession phase begins "['he dept, h o f f l o w aL :~' = (.), t = TI
g o e s t,o z e r o tnst,mlt, mleoulsly T h i s I m p l i e s t,hut, t,he KW a s s u m p L t o n d o e s n o t hold f o r vert,,.'al recession As t,mle progresses t,he zer¢~ depl.h ( d r y i n g fi'ont.) t,ravels d o w n w a r d . T h e n l o v e m e n t , o f t, ht,,; d r y i n g |rent, charact, erizes t,he h o n z o n t , a l r e c e s s i o n a.nd c,.mt, m u e s unt, iI 'all t.he w a t e r IS d r m n e d out, o f t,he b o r d e r . We let, t = i(.t') d e n o l . e t h e r e c e s s t o n c'urve, t,hat, is, t,lm m o v i n g mt, er face bel, w e e n t,he paJ't o f t,he h o r d e r wit, h hl.t', t) = () aJld t,he p a r t o f t h e chaJirlel wlt, h It(x, t) ' O T h e [,llne hlsL,ory t = l i x ) , T I ' t ' T,, tt:; a free b o u n d a r y w h o s e det, e r m m a t , lon IS a |.)art o r t,he solut, Jon for t,he req.'eSSlOll pllase. T h i s p h a s e IS r e p r e s e n t , e d I)y d o m a u l D. w h i c h IS b o u n d e d by .i = O, t = '/"~, t = t(:~) a n d x = L S h e r r n a n mid S i n g h 1 1 9 7 8 ) d e r t v e d , for <.'onstant, I'. aJmlyt, lc',d solut, l
'/'l)"
(12a)
o r X
I 'H
(121~)
aJ~d t,he dept, h o f f l o w hlx, t) In [ I j as .~ = - [ Ih
1'
+/(t
-
'T')I" - h n ]
(13)
F o r t,i.me vaJTIng I, t = ,~(.,t') ~'ml be ~'aJculut, ed for a spectl'led ~ x f r o m Eq 1 2 b as.
l,÷l =~,+
AX ] I, tl /j(/., + i,,,) , ,,
I= 1 ...... '~
(14)
2 T o sl,art, t,he comlJuL, at, lOl¢-~,e assumed t.hat, It, i = [i Since / varms f r o m i l Lo tt,J ( u r c o r r e s p o n d i n g l y f r o m x~ h., "[Itl) ror each A-~, ~i÷l W"IS t-'omlJut,ed It,eratlvely; five tt,eraL, ions were round sul'l'Jclerlt, t.o give an a c c u r a c y o f t,he o r d e r o£ 1 0 " m i n . T h e recession curve t = ~(.',~') was Lraced urlt, il .r," = L. T o obt, am h(x, t), we solved f o r the i'hm'acLerlst, ics ISSUing f r o m t.he se8 men[, t = t(x,,, T I ) = t,, ' t ' .~',,, .~' = .~', as s h o w n Irl Pig 3. At, x = -~',, = A t , t,, = t(x,,, T I ) was obLa.med f r o m [,he chazact, ertst, ic emanat, lng ['rum .~" = (), t = T I bl D , . F o r s p e c i f i e d A x = x,, con'e,,.;porlding t,o atl Incremer]ta] Lime .hi = l~,l ~, h(x~, TI + A t ) = () We a l l o w e d t,he dep|,h o f f l o w tt(x, t) t,o va~.'y l i n e a r l y IJet, ween Iz(a'~, T I ) mid h(a',, TI + - ~ t ) = ( ) i l l a speclfted n u m b e r
133
I~
' h.~ 2~1
i~I
t
/;2: ~{xE ) ,h : hix2,Tli, A l ) : 0
/
/
I
t~T
k
I*1
IZ: t l x 2 , Tl) , h : h l x 2 , T i )
ChotoctertstlC Line
t =tj ,j I'l
= ~'(x ) , Recess,on Curve
""
DED,> ~ ~
C
uDI r
1
2 x'x 2
x 0
~ v
e
~
I
I
3
i-I
~
~ I
i v,"x I
I
,~I x-'L
Fig 3 I~l'|arat'r~li.~llt" ~,~lul.l~rzs fur t.h," dunlaltV.; Dr, D~ and E)~
of stepsp
T h e t, m l e .~'z was o h t , a m e d f r o m eq. 1 2 b as'
.~,, = T, +
~l/.(,T,.),,.i
115)
T h e Lime f o r w a t e r to m o v e '~a', f o l l o w i n g t h e c h a . r a c t e r l s t l c o r i g i n a t i n g at. t = 'T',, ts t,,. T h e c h a r a c t e r i s t i c s w e r e t h e n a l l o w e d to o n g b l a t , e r r o m t h e seg rnent, t., t ' t ,,, x = x : a l o n g w h i c h t h e d e p L h el' t'hJw a n d t h e c o r r e s p o n d ing t,ime w e r e comlaut, ed as a first, a p p r o x t m a t i o r l b y '
t,,t = /,, + IPl,
J = 1,2 .... p
(16)
h,., t = h ( A x , t;,.i ) =.IP.,,
I = 1, 2 . . . . p
(17)
Pl = (~;,, - t2)/p
118)
l~ = h(x,,, T I ) / p
(19)
Llst.ng t.hL~, al.q.proxirnuLion, eq. 13 win.; used it,eratlvely to comput, e the depth o1" flow. Five Iterations were found t,o y i e l d an aCcLLracy o f the order o f ] 0 - " m The chaJ'aeterisLtcs were aJlowed Lo move A x distance d o w n s t r e a m and c',tlculat, lOnS were carried uuL in the above mmmer. This wa,.,~c o n t i n u e d untd the e n t i r e border was Included. The tune or occurrence o f zero depth h , i,i = () L,.; the recession tune at x't,i. In thts a.naJysh~ a v',.due o f p = 25 was found sutLable.
i-i I.t~. +-, @
Ii:l
I:, e£),:'~ ,:~ I " irl ~b v--4 u'=l +~1 I:'l I:P
q~) I~'1
~l
,l'l
i:1
q_-i
,,.~
l"l
I.'l I-p
IJI
+' I E.+
m:
d ~ L('.~ ~D
,~
It) U'I p+
~, ,:,,
C)
IJ'l I.'Ti
l-~
iJ~,,
ii'l
I" r-~
I"
{-)
r.-t It,i:I
.,
n
8E
~-.di 12. ,
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v ERI FICA"rlON D A T A Four sets o f dal, a, due to Roth (1971) and R ot,h el, ,,,I (19'7.4) from freely c'lrublblg borders were used. "riley referred t,o t,huse dut,a set,s as R,oUi 8, R oth 9, Rot, h 1(.} aJid Rot, h 11, which are summarled Ill Table I. 'T'hese daLa perl,aln t,o non vegetated borders (sod classified ~,~ sandy Ioa.m bL.dk denslt,y 1.,I). These herders are 5.89 m wide aJld 91.46 m long wlLh 1.24 nL of exl, ra sl~ace un each of l,lle Ul.~sl,ream wld t,h~ downstream ends of l,he border AL, (,hese ends sIUs were IJrowded t,o assure u n i f o r m ent, ry o f wat, er at, t,he up st,reum end aJld t,o elimblat,e [,he d r a w d o w n effect, or ~uLl'luw at, t,he dawn si,reanh end. Inflow was measured by a IU cr'n l.~rolJelh-,r meter aJ1d l,he uut, l'luw by a t,rla]igular crlt, lcaJ depl, h l'lume PARAMETER
BSTIM ATIC)N
"the KW model cont,abls t,wo ~ l ~ l u w n inl'dt, ratlun paramet, ers K and a of the Kost.ya.kuv equat, lon aJid t,wu u.nkJiown friction paramel, ers n a.nd /3 of t,he st,aide discharge relal, iun. K and a were esLimal, ed by a volume h',.tlaJIce analysm aJid n mid /.1 from Mannmg's equal, ion (Ram, 1982,) 'T'he values o1' t,hese paxamel,ers for each dat,a set, axe gwen Ill Tahle [. VBRI PIC'A'T'IUN Advance t,Lrr,es and recesston t,Jrnes were comput,ed using t,he recurstve solut, tons aJld t,he K W T method fur t,he rour set,s of data given m 'T'ahle I The calculat,ed and observed vaJues of advaJlce [1111e a.lld recesston l, ulle are plotted hi Figs. 4--'7. Fur aJl four dal, a sets l,here IS u t..tood agreement, be l,ween observed (obs) values and compuLed (caJ) v',dues by t,he two meLhods The absolute percenL deviat, ion, PF) [(obs - caJ)/obsl i~'mormg ,',Igebra.ic sign, bet, ween observed aJid coml.,uted advance tames raJiged bet, ween 0.(.) aJlcl 2,3% for t,lle sequentml met, hod, aJid between 0 aJid 21% for the K W T met, h ud fur aU the dut, a set,s except for R.ot,h 10 for which It, wa,~ 35',"~, and 4.8.9% respect, ively, n.,r t,he t,wo mel,hods. "rile t,wu met,hods yielded comparable values of adv~lce and recession "rile error was htgh ul L,tle inlt, lal sLa~es of adva.nce but de~'reased slgnd'l c~lLly wtt, l l l , he progress o f advance Since t,Fle advaJ'ICe l,tme IS smaJl in l,he beginning, even a smaU error m l,he comput,ed value will cause hlgh PD. The PD l'or recession t,iJne raJ'Iged heL,ween 0.0 ~ I d 6% for 'd..ll the daLa sel,s excel.d, for Rut, h IU l'or whlch It, was 14.0',.% II, is not, cleat why t,he PD m consider ably hlgher fur R,oUl 10 l,h~l for uther data seL,s. The l,wo met, hods were ident, lcaJ for [,he recession phase.
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T h e l'ollowtr'Ig ,'onclustons c~'i be d r a w n frorn l, l lts sl, udy ( I ) "l"l'~P propos ed meLhod ts suffietenl, ly aec'urate for solvlng {,he ktnemuLtc wave equaL, i,,rm ul" burder LrngaL, t(.,n rlow. (2) I{, ,.'ornl.,a.rPs very well wtl, h (,hP K W T meLh,.),J. (3) Sm,.'e l, he proposed mel,hod ts reeurstve, tL, ts stml'.,IPr and rnorP efrtcienL. t,haJ'~ Ll'le KW'T' mel, l'~od. It, may Lie of int,Prest t,u n(.,l,P l,haL l'or l,he dat.a aJra.lyzed l'IPre l, he klnemaLte wave model m surl'~emnl, ly ac'~'urat, P fur mu(:lellrlg l,he Prlt, i.re Lmgat, ton excPpl, for t,he vert, ii",.d rei'esst()n "['he mudPl predtc{,s t,he advan,L'e time SLd'l'tcti-,nt, ly ae(.'urately for l,he dal,a an',.dyzed l'iPre Usually t hP l:.)redtct, tor~ e r r o r ts heh.)w 25";,';, A laJ"ge errt.)r ri,u,rrna.]ly o(.'eurs in t,he bel.,qnrlinp, at' L,he i.rrigat, t(.)t~. T h e m u d e l predtl.'t,s the t'H)rizonl,al i'el'PSSlCm t,llTle Wlt, h iJre dl('l.,l(:ln error (.)I" less t,haJ1 ] 5".';,. A C K NOW L E D G EM EN'T'
Thi,',; study was SUlJp(.)rted m p',.u-t hy fuJ'~ds provlded by l,he Nat, ton',.d Sl'i em.'e Fourldai, ton w-il, h i,he prujPcL, Prep Buundazy Problems in Wat Pr Re s(Ju.rce EngumerLng, NSF ENG 79 23345.
REFERENCES B;s;.,,:,ell., b L , F'ullgrrleler, r) [) ...lll(-I Sll~lk4,,ll', 'T', 19NI. HydrJl, lh'~,; ul :~url'.',ce liil~01{.iq)rl In' M E ,leni.len (Edll.(Jl'), [)~.Nl~rl gild L)pel.'ll, lOll U]' P',Jlm htip,.',lu.m Sv,~..,r.ern~., A~NAE Mlllll.)~i 3, Arllerll.,Liii ~O('ll-q,V Of Ag¢l~.'ullur~l Er,l~tne~r% SI ,Io~.;eph, MI, pp ,-147-,-19N (~Ulll.~l-' , ,I. IJnd WllulhI.M-'I', D A , 1~'7!) I.rnMe.',dv I'h.)w m upl-,n i,h~Jllll~,l.~L (~,h~pl.er 1:} irl I~. M.',l'lm~.~d ;jrll"l V Yeill-,vlel"l (Edlll.m,i), Irlll~.lt.lon ~.V!.,I.erll.'.; W.'l|(.-,i R,t-'~.iUlJi P u b I , F'ipll Culllni..,, (~(-L PP ;~9~-!.~:.1'7 KIncLII("], [ ) C ' , 1~'70. H'llldlud,vIliJrnli;h Ill b u l d e l li'll~/JI,i(ll'l Th,,-,,%,; p l e h e l l l e d t.i)(.lulur,Jd~ S[,,JI.e'. I.IlllVei,i.,It.y, I?l=rl (.'ulhrl,% (."(.), trl pm'th..ll I'ull'lllmPnt i)l' Iht-, reql.llll-'ml-,ll{.I.., I'ur lilt, del,.;ree =~i Di..'lur uf Phthb.'.,uphy, ]()l-i pp Ram, l-I S , 19,':'12 M a l h e m a l t e a l rn,.~,dehril.; ul l.,Ul'l','Jee urt~;:.d..m, 'T'he~,m pre~;enled I,~ Ml.,.,ni.~.~ll::lpl Slul.e l.]mver;,;tl.9, Mt.'.~,it~,,.,.,tpptSl,'Jl,f', MS, ill p,Ji li~Jl l'ull'lllmer~l i~l l lle rel..luti~ IneliL,.; l'ut L.he deb]ree of' Dli(?|.iJl i_i[ Phdos{~phv, ,'.III'2 pp Ranl, R ~ , ~lrlEh, V P arid Pr;j~iJl"l, ~ N , |9~;'1 Malherrlatl,l.,al rrlodehlll~ c)l' bOll'lel l l l l g J l,un 'T'e~.'h Rep W R R h , WiJl.el Re,~iilUl(.'e~,, Prul~l',Jln , [.)t-'p,..u'lmelll i)l' (."lvll EIIil~ll|elJIIIl~, Li~ui,'.,lan~ ~L,,JL.*-'U n l v e l s i l V , B a l u n R.UlJi~e , L A , ;'1()~ tip R¢~th, R L , ]~'71 Roughne~.,,,, (.]lurlrl~ bl.licli.,i ilill:~il[l()ri Thc.-,hl.~; pre!.,Pnl.ed I.u I.rrllyel,,,,ll.y Ill Alll, Urld, TLIL%OlI, A Z , III p,..Irllal Illll'lllnlerll id' l.he re(lUll'l:'.lTIt-'lllh I'()i t.he del-.~,ree ,~1 M~,,l,l-'l' uf S(.'lellce, '7~ plJ Ri.)th, R I, , Funl~,en, D W , Falil~rnelel, LI I') .llll'l All'lll~,;~.)tl, K 'T', ]Lr?J [)Llla lul Ijc)ldel ir'rlgallUll rrludel~,; 'T'rurl~,, AS,,~,E, N l i b ' i - i f ' ; | ~hei'lll.'lrl, B dnd Slllgh , V P., 19'7N A klnerll,Jil.ll.' ITIorJl-,t I'lJr ,~aUll,.ll',l-' ilrit.~,,Jllllll W,..lllel R~-' ~'~uur Re.'., , 1,-1(2)' 35';'--364. ~heilTl~Jtl, B ;.,llid ~lnl..~,h, V P , ]~H2 A IIinelndl.le rr'illde] Iol ,~ull'al.'e trill~.lt.lllrl .ill I-'~.lert .~,;li.lll Waler RP~,,our I'~el~, ]N ( 3 ) I';:~.1d~6'7 ~nil.~h , V P ,Jiid Shelril;Jrl, B , I~"l:} A hln~'.lTl;JllL" ;,l.udy ur ~ulldL'e LlrlI.Jgl.IO|l. m.'qherrhJ[I ,L'at.lOl~ ~.;l=lut.lUn~.~ "l"eeh R~p W R R 4 , W;JI.e'l' R.l=lNiluli'e~,~ Prol, rum, Deparl.rnent id' C'lVli ,'Ell~llleel'll|~, I..,it.lUll.,iaria ~L,'II,P I.JrllvPrNII.V, BIJI.IIlI Rl.lu~o, LA , 'T{'I p~)
-
S O L I . I T I O N O F ']'HE KI N E M A T I C - W A V I RRIGATI(-)N
VI,IAY P SIN(-3H I and R,A M A S
Plltlled
12'7 III 'T'h~, N e l h e r l a n d ~ ,
E EQI.I A T I O N S F O R B O R D E R
R, AM ~
[h'purlrnt'nt u f (?w~l E,n/~mecrm~, L,uu~,stunu Stub, I lmr,t,r~z~ty, Eluhm R~ugc, LA 70803 I.,'S ,4 ) Dvpa~ tm,,nl ,~1',tl,lalhemulws, ,4 It'~rn Stab' I.hm,,,rs~l y, L,,~rmun, ,~ll,.q ,'1!4t)!46 ( II ,.~ ,4 ) A c c e p t , e d ~ May 198,4)
ABSTRA("T ~llit.;h. V P a n d R a m , R S . l.qH4. S . l u t , l u n u f t h e I, in~n~ulH: w a v e equall.n=,~ l u r bu=de~ ir~lgal,l~n ,,-t~rzc Waterl~ldnug*', 9 l ~ ' 7 - l : t ~
A meth,,d i~.,devehJped t,, obt.aitl solulion~, l,, l he l.tltlemalil' w a v e ellU,Jl, l u n , ~., l'ur buI,Je~ The ac~'uracy ur th~:~ rrielhud is a~isessed by i.'(.inlpaJ'lllb.~, the tesulls wilh I he l.iltlerrlal.iL" wave tlaln (Kw"r) melhud prPvtuu,,.;ly rep,.nted h~ he accurate Experimerilal d a [ a ,~tt f r e e l y dramln 8 hurdet.~, were u l I I I z e d l'q~r t'qmlpmi,~url and evalual.i~n uf l.he:.,e melhuds Thepiedlctlunelr~r lul advance Limeie~,ultlng frum I he, lwu mel hud,~iwascum II'l'l,El, a l l l l l l
palal-~le II wa,,; b e h , w BS',";, I~Jr I hl'ee ,~ieln ~1 d a l a b u t a~,~ hll.~,h a.~ h()% fur u n ~ ,,.Jel ~pl' dal,.l T h e rJledlelll~rl erH~r l'4at i ¢rq.'e,~.~l,ilOll t.lme wa, ~, al.sl~ l ' U l r l p o u a h l e b e h l w 6 % f u r t i l e ~iame t.hleP ~.,el~ o f d a t a Llul. ] ,..I 0 % ['ul Olle .'.,el
IN'T'R(.~DLJCTION A k n o w l e d g e of advance, recession, dept, h or flow, and sod mOlSl.uJ'e dL%nbul.ton k'~ required l'or an opt, unaJ desi[.,~] u f surface trngat, ion One way l.o del.ermine LLhem Is using mat±hemal.lcaJ models which have recent±ly been surveyed l.)y Ram ( 1 9 8 2 ) . Basset,t, el. al. ( 1 9 8 1 ) have discussed l.he (-u.rrenl. st±al.e o f t±he-art± o f h y d r a u l i c s o f surface Lmgat±ion T h l s s t u d y uses a kinemat, ic wave ( K W ) model (Sherman and Singh, 1978, 1982). T w o t±yl.)es o[" solut±ic)ns t,o t,he KW model have heen sought, in t,he past,. (a) analy1,1cal; and (b) numerical. AnaJyLLicaJ solul.lons have been rel.)ort, ed for consl,ant, i n f l o w and const, ant, )nfdLLrat, lon (C, Lmge and Wooll-user, 1975; Sherman and Sblgh, 1978, 1982; Sblgh and Sherman, 1988), since bot±h inl'low and tnfdt±rat, lon are seldom constant', t±hese solut, tons are of llmlt± ed vaJue t,o border irrlgal.lOn design. Alt,hough ntunertcal ,~olut,ions |lave been obt±ab]ed l'or t,une-varyblg infilt,rat±ion and t±ime va.rylng ml'low, t,hey are corn plicat,ed and expensive ( R.am el, aJ., 1983). T h i s sLLudy develops a met±hod t,o solve t,he KW equaLLions for b o r d e r irmgat,ton w h i c h can ha.ndle t, tme varyLng ulfilt,rat,lon told t , i m e v a r y m g ird']ow. T h e met, hod is semt-a.naJyt±tcaJ and
I'~
yields ,~olut,,.ms r¢,cu.rsivel.y. ']"he accuracy of t,he m~t, hod is asse~,,.;ed by ~'om p~'m~, t,he resuJt, s wtLh t,he kinemat,~c wave tram (KW"I") meU~od wh=ch is part, ly numerlcaJ amd r),,.ul,ly a.nalyLic~, aJ}d ha,~ been round t,u be a~'~'u_rat.e ( R a n et, .,d , 1983). Experlment, aJ dat,a t'ur rour t'rt-,ely dra.ming hordt-,rs a.re ut, lhzed ror compar=son Not,e t,hat, v',lJidat, lcl,n of thP KW model ~s out,side t,he ~i~.'l.)].)e of t,hls paper. K I N EM A"I"IC WA V E MODEL For border irngat, lon t,he KW model was rurmulat,ed ~ l d discussed by Shernlan ~ l d Smeh (197L4, 1962,). We refer t,o t,heir wurk for backgruund LnforrnuLioll. Border crn~at, lon essent, lally i n v o l w s t,he flow of wat, er d o w n a I:~la.ne wlLh a sm',.dl slope and porous bud 'T'he border under conslderat, l~.m is assumed t,~.~ be feet,angular, of u n i f o r m c r u ~ ~e~'tlO, and intt, i',dly dry Let, .~ be t,he dl:~t,an(.'e ',.dung t,he harder which ext, ends t,o t.he right, of tt,s head at .~' = 1) At, t,une t = () wat, er is released air, Lhe head of t,he harder. The influw ot' wuL~r at, .~" = (.) has a knuwn t,mle.dependent, dept, h U ( t ) or rat,e q(U, t), uJld la.,st,s for a ,~pecil'led lengt, h of t,irrle TI D~pendln~ upon '/'~ a~ld t,hu b o u n d ~ ' y c'ondlt, lons at. t,he end o1" t,he border at, :~" = L, various phases Itke advance, ~t,oru~e and recession can I-Dedl~t,[JlgLiished us shown in F I ~ 1 o - Horizontal Recession b - Storage Phase i: - A d v a n c e Phase
t
t ." T 2
Phase
. . . . . . . . . . . . . . . . . . . . .
02
Time of Opportunity
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,,=0
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I?ll~ I ~(.~liJtl~arl dumairl=~ fur /.tie, I'h.~w in u I'r~ely dralrllnl:~ I,.JId=-,r
When waLer is released, t,here is a front, of water whl(:h advm~ces d o w n t,tle border 'This advance front, is t,he moving int, ert'ace bet, ween t,he waLer cuver ed and uncovered parts o1" Lhe border. Ir x = sit), or inversely I = ~(:~'),
129 denot, es t,he t,tme htst.ory o r t,hls [ronL o r Lhe advance run,:t,,on, t,hen tt, Ires t,o be det, e r m m e d aJong wJLh the dept, h h(x, t) and veloult.y u(x, t) I'ora specd'led G ( t ) . T h e qua.nt, it,y r = t - t,(x) denot, es t,he mrtlLrat, ion opport, uniLy t4rne a L a IJomt, x , t,hat, is, t,he int, erw.d o f t.lme t.hul, wat,er coveri, d t,he point, x T h e In I'dt, rat, lon rat,e/( r ) IS assumed t.o depend o n l y on t,hi-, dll't'erence 'r t)et.wPen t,he t.ot,aJ elapsed t.lme and t,he advam'e t,tme T h e dept.h ,Jr wat.er h(.~, t), t,he u n k n o w n t.lme hlst.ory ~.(x') and t.he in [llt.rat.l(.m I'(7 ) are suF)je,'l, t.o t.he rullowmg KW I'ormulat.tun'
Oh --
,) t
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,) X
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~,(()) =()
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wllere n It.; ;.,it'l e×lJi.Jl'le, llt, alld va.ry, ir'll~ between 1 zul,.l 3, and (.1( , O) is {,}le rrl(, Liun pas'uniet,er; h(O, t) = ( ; ( t ) , O ' t ' T I . h(O, t) = (), t , TI N(.)t.e t,hat, t,he wlo,'lt, y o r rlow u = ( . I h ' '. T h u s n and /.I (.'an he det,ermmed f r o m MaJmlnl~'S equut, ion unlJlyln ~ n = 5/3 asu.t IJ = S l ° ' " / l l m , sr IS {,h~ I'nct, lun slope and It m I,~
MannlnB's roughness ,'()erl'lelent,. METHOD (.)F SOLLI'T'ION
Aduance. As shown in Fig. 1, t,hLs phase ts r,'present,ed by dorna.m [)j whteh is b o u n d e d by t = t-,(:~ ), ~uld t = T denut, lng t,he advmice Lime. CenLra] t,u the ~`;olut, l,.m tn t,hls d u m a m IS det,errnllmtion of t,he t, une hLst,ory or the advml,.'e front,, t = t.,(x). For ,.'um,~t,a n t , / and const,~it (J, Sherman and Singh (.19'78) deriw'd t, he analyt, i,.'al sulut, lonS USln~ t,he met, hod o f ,'tm.racterlst, l,-S: h(t, 1,,) = U - l'(t - to) :~'(t, t.,) = }: IU"
-
((~ - l(t -
(3a) t.))"l
(3b)
where tu IS u charact, erlst, lc pa.rarneLer, 0 ' to' T. Equat, lon (3) ts ,'haraet,er L'4,i,: solut, ion In t,hat, It, describes t,he surface f o r m e d by 'all the ,.'ha.ract,,-,rlst, lc curves t.hroLigh t.he Segl'lrlent. (.) ' t . ' 'T', .~' = O. By elunmaLtnl~ t,, t,he ud vance curve becomes.
"[
(4)
aJ,J ibhe d e p t h h(x, l) ui D, as.
t,(x, t) = [ G , _ I"' ] '''" Slrlee [ usually L,.SnoL ('(.)nsI,a,nL, t,he direct, use o1" Eqs 3 - 5 is not, re'J..hsLic 'T'herefore, I'or tune va.ry. LnB [ we develop a met, hod, by using eqs. 3 - 5 , t,o
130
solve eqs. 1 - 2 t,o find t,he a d v a n c e I'tmet, ion. We a s s u m e I,hat, wat, er m o v e s in a series o f c o l u m n s o f fmit,e wldt, h as s h o w n In Fig 2 T h e ~'ont, c o l u m n m o v e s wit, h k i n e m a U e v e l o c i t y . W h e n it, m o v e s t,o a new p o s i t i o n all t,he foLlowing (.'olumns m o v e forwa.rd, each o e c u p y b l g t,he place vaeat,ed by it,s I m m e d i a L e p r e d e c e s s o r T h i s m o v e m e n t , eont, inues unt, d t,he height, o1' t,he I'runt, c o l u m n ({,he dept, h o f t,he wave front,) reduces t,o a mlr|unLuTi vaJue de noLed as he. T h e n t,he pre<:edmg c o l u m n t,akes o v e r l,he r u l e a,.,~ t,he wave r r o n t . 'T'hls p r o c e s s c o n t , l n u e s Lmt,il t,he wat,er r e a c h e s t,he e n d c)f t,he b o r d e r .
h (j
~,- WATER C(.)LL)I )LUMN ~
1
13
;2
(a)
x
'','~
" X
:0
WATER L;C)LUMN ;'~ ~:0
(b)
3
51a
I
•
II
X
_-I~z_WATECOLLIMN 2
, S i k ~ II
I
X
(c) FII~ 2 MuvPmt'm. ~)1' W;Jt(-'l eulu=mlh We a,.'qsulne /' t,o be const, ant, o v e r a sm,'dl dlst,ant.'e ,Ax, ~ l d furt, lter assume t,ilat, t h e wut, er, ~ t , h an Lnltlal c u l u m n deyt, h th ;.rod u ~'unsLant, infllt, rat, l u n / i , a d v a n c e s o v e r a dlst, am.'e ,Ax on the b o r d e r in t, ime A t . E q u a L I o n 4 can be re
cast, as. At
.[
= -[, -
hi
-
(
h~ j
/.l
At, t,he e n d o f t,hls t, irue mt,erval At, t,hu w a f e r dep{,tl h . . eq 5 as:
161 can be e x p r e s s e d by
13 1
-_
(7)
F,qua{,ions (';--7 were used {,o I'l.nd |,he adva.rl(~e I'unl'Liorl I = ,LI x ) ~; I'oBows (1) We assume LhaL {.he, int'h-)w q(0, t) = q. is i~unsLaJ'lL. The ~.'orrespondlng dept, h o f flow at, t,he upst, reanl end wa~ ('aJ(:uJat,ed by Ma.nnLrlg's equal, lon C; = (~.lqu)°'~'
(81
where C; us t,he normaJ del.)t,h of fh.)w at, t.he upst, ream end In eq. ~, t,lle vaJue q.)f/.I is expressed as before wit, h SI replaced hy So ;~ = Su°"~/n n,
(t-) )
(2) q"he infllLraLiorl raLe/'('r ) was ua.IculaLed by Kost, yakov's equaLion
I'(~) = KuT "-~,
()"
o'
1
(l())
where K aJ1(l e m'e cunsLant, s Lo he det,ermu]ed e,npi.ricaJly For ('oml.)Ut,aLnun i.)l" /'(t) we assume {,hal, Lhe liffflt, ral.,ion opl,.)rt, u.nit, y Lime T alway,~ exceed,'.; ur equaJs a m i n i m u m v',.due T,, such Lha{,' I'(TI : a K ' r , Y " , = GK'? LI-I ,
(.)' 'r
7 ' '
r,,
(11)
T~,
(3) T h e wat.er c o l u m n at.x~ w a s advt:meed t,o.'t',l, i = 0, 1, 2 , . , b y eqs. 6 '7. 'T'hns pro(:ess w a s i'ont, muecl unLil t,he heighL o1" Lhe ~(.)nt e o l u r n n r e a c h e d t,he value h,. A raLnu Rt = hc/G w ~ used Lo conLrol t,he Lnp dept, h tn Lhe solu t,ion. For a h y d r o d y n a m i c model, KmcaJd (197"/0) l'ound RI Lo range bet, ween (.).10 and 0.1,5. However, in {,his a.nalysls Ri = 0.0,5 ynelded bet, Ler resulLs. (4) (.)nee h,. was reached, Lhe waLer ~'olumn immediat,ely behind Lhe fl'ont, was rJermiLt, ed I,o La.ke over and was allowed Lo adva.nue. The ulfdt, rat, ion opporLunit, y t,i.me at, a.ny point,, when t,he front, was at, t,he p(.)slLnon x . i, was c'.d~ulat,ed by ~,.~ - t,~, ~.,.~ - L.,.,..... .~,~ - ~,t_~, ~,t,~ - ~-, Thus wesee t,hat, for the t'Lrst, column Lhe irfl'ilt, rat,..)n opport, uniLy LLme is 'always ~-,~,~ - ~',i = lt, ~ - tt subject, Lo =t,s m t m m u m value ,~,.. $ i m d a r l y , I'or t,he second a.nd t,hu'd e u l u m n s t,he nnl'nlt, rat, ton o p p o r t , unJt,y t, i m e s are ,~i~,~ - ~,i-= aJ~d ~.,,,~ - l,~.,, A snmda.r pat, t,ern rollows for 'all columns. T h e uffnlt, ratJon rat,e l'or any column at, :~'~ ts an average of Lhe rat,es at, :~:t aJ~d x~_ ~. The del.)t,h o f t,he first, culumn at, :~:~,~ is hi~ and is super(:eded by t,he se~'ond column wtt, h depLh /h of h~,t he. Now t,he second column plays t,he role of l,he firsL column, Lhe Lhird ~-olumn plays Lhe role o f t,he second column and so on This l.)rot.'ess was ('ont, mued unl.,d wat,er reached Lhe dow-nsLream end oi" t,he border Havb'~g det,ermined t,he advance l'unct, ion, t,he dept, h ou~d x t i'haract, enst, ms m [)n were det,ermined by eqs 3 and ,.5 l'ollownng t,he above procedure 8/orage 1t' Lhe nffhJw q(O, I) is conLmued aft,er waLer reaches Lhe d o w n sLream end o f t,he border, Lhen Lhe waLer will conL, inue t,o a~'cumu.lat,e 'T'hi,s huddup o f wut,er c(.)rl~t,lt, uLes Lhe sL(.)rage phase As shown tn Fig I, Lhis [.)ha~e
132 Is represent, ed h y d o m a i n D,, w h i c h IS b o u n d e d b y a = (), t = T, t = T I a.nd x = L. S i n c e l,he s o l u L i o n in t,hls d o m a b ~ IS ',also given b y eqs. 4 - 5 , t,he c o r n put,at, lonal st,el..s are t,he s a m e as f o r d o m a . m [)~ R e c e s s t o n . As soorl ms t,he inl'low ts cut, e r r at, t = TI, q(U, t) = 0, [,Fie st, o r a g e I.)ha,se ends and the recession phase begins "['he dept, h o f f l o w aL :~' = (.), t = TI
g o e s t,o z e r o tnst,mlt, mleoulsly T h i s I m p l i e s t,hut, t,he KW a s s u m p L t o n d o e s n o t hold f o r vert,,.'al recession As t,mle progresses t,he zer¢~ depl.h ( d r y i n g fi'ont.) t,ravels d o w n w a r d . T h e n l o v e m e n t , o f t, ht,,; d r y i n g |rent, charact, erizes t,he h o n z o n t , a l r e c e s s i o n a.nd c,.mt, m u e s unt, iI 'all t.he w a t e r IS d r m n e d out, o f t,he b o r d e r . We let, t = i(.t') d e n o l . e t h e r e c e s s t o n c'urve, t,hat, is, t,lm m o v i n g mt, er face bel, w e e n t,he paJ't o f t,he h o r d e r wit, h hl.t', t) = () aJld t,he p a r t o f t h e chaJirlel wlt, h It(x, t) ' O T h e [,llne hlsL,ory t = l i x ) , T I ' t ' T,, tt:; a free b o u n d a r y w h o s e det, e r m m a t , lon IS a |.)art o r t,he solut, Jon for t,he req.'eSSlOll pllase. T h i s p h a s e IS r e p r e s e n t , e d I)y d o m a u l D. w h i c h IS b o u n d e d by .i = O, t = '/"~, t = t(:~) a n d x = L S h e r r n a n mid S i n g h 1 1 9 7 8 ) d e r t v e d , for <.'onstant, I'. aJmlyt, lc',d solut, l
'/'l)"
(12a)
o r X
I 'H
(121~)
aJ~d t,he dept, h o f f l o w hlx, t) In [ I j as .~ = - [ Ih
1'
+/(t
-
'T')I" - h n ]
(13)
F o r t,i.me vaJTIng I, t = ,~(.,t') ~'ml be ~'aJculut, ed for a spectl'led ~ x f r o m Eq 1 2 b as.
l,÷l =~,+
AX ] I, tl /j(/., + i,,,) , ,,
I= 1 ...... '~
(14)
2 T o sl,art, t,he comlJuL, at, lOl¢-~,e assumed t.hat, It, i = [i Since / varms f r o m i l Lo tt,J ( u r c o r r e s p o n d i n g l y f r o m x~ h., "[Itl) ror each A-~, ~i÷l W"IS t-'omlJut,ed It,eratlvely; five tt,eraL, ions were round sul'l'Jclerlt, t.o give an a c c u r a c y o f t,he o r d e r o£ 1 0 " m i n . T h e recession curve t = ~(.',~') was Lraced urlt, il .r," = L. T o obt, am h(x, t), we solved f o r the i'hm'acLerlst, ics ISSUing f r o m t.he se8 men[, t = t(x,,, T I ) = t,, ' t ' .~',,, .~' = .~', as s h o w n Irl Pig 3. At, x = -~',, = A t , t,, = t(x,,, T I ) was obLa.med f r o m [,he chazact, ertst, ic emanat, lng ['rum .~" = (), t = T I bl D , . F o r s p e c i f i e d A x = x,, con'e,,.;porlding t,o atl Incremer]ta] Lime .hi = l~,l ~, h(x~, TI + A t ) = () We a l l o w e d t,he dep|,h o f f l o w tt(x, t) t,o va~.'y l i n e a r l y IJet, ween Iz(a'~, T I ) mid h(a',, TI + - ~ t ) = ( ) i l l a speclfted n u m b e r
133
I~
' h.~ 2~1
i~I
t
/;2: ~{xE ) ,h : hix2,Tli, A l ) : 0
/
/
I
t~T
k
I*1
IZ: t l x 2 , Tl) , h : h l x 2 , T i )
ChotoctertstlC Line
t =tj ,j I'l
= ~'(x ) , Recess,on Curve
""
DED,> ~ ~
C
uDI r
1
2 x'x 2
x 0
~ v
e
~
I
I
3
i-I
~
~ I
i v,"x I
I
,~I x-'L
Fig 3 I~l'|arat'r~li.~llt" ~,~lul.l~rzs fur t.h," dunlaltV.; Dr, D~ and E)~
of stepsp
T h e t, m l e .~'z was o h t , a m e d f r o m eq. 1 2 b as'
.~,, = T, +
~l/.(,T,.),,.i
115)
T h e Lime f o r w a t e r to m o v e '~a', f o l l o w i n g t h e c h a . r a c t e r l s t l c o r i g i n a t i n g at. t = 'T',, ts t,,. T h e c h a r a c t e r i s t i c s w e r e t h e n a l l o w e d to o n g b l a t , e r r o m t h e seg rnent, t., t ' t ,,, x = x : a l o n g w h i c h t h e d e p L h el' t'hJw a n d t h e c o r r e s p o n d ing t,ime w e r e comlaut, ed as a first, a p p r o x t m a t i o r l b y '
t,,t = /,, + IPl,
J = 1,2 .... p
(16)
h,., t = h ( A x , t;,.i ) =.IP.,,
I = 1, 2 . . . . p
(17)
Pl = (~;,, - t2)/p
118)
l~ = h(x,,, T I ) / p
(19)
Llst.ng t.hL~, al.q.proxirnuLion, eq. 13 win.; used it,eratlvely to comput, e the depth o1" flow. Five Iterations were found t,o y i e l d an aCcLLracy o f the order o f ] 0 - " m The chaJ'aeterisLtcs were aJlowed Lo move A x distance d o w n s t r e a m and c',tlculat, lOnS were carried uuL in the above mmmer. This wa,.,~c o n t i n u e d untd the e n t i r e border was Included. The tune or occurrence o f zero depth h , i,i = () L,.; the recession tune at x't,i. In thts a.naJysh~ a v',.due o f p = 25 was found sutLable.
i-i I.t~. +-, @
Ii:l
I:, e£),:'~ ,:~ I " irl ~b v--4 u'=l +~1 I:'l I:P
q~) I~'1
~l
,l'l
i:1
q_-i
,,.~
l"l
I.'l I-p
IJI
+' I E.+
m:
d ~ L('.~ ~D
,~
It) U'I p+
~, ,:,,
C)
IJ'l I.'Ti
l-~
iJ~,,
ii'l
I" r-~
I"
{-)
r.-t It,i:I
.,
n
8E
~-.di 12. ,
i.)
I
ii I
4~.I
.=
,~
,~
,~
i~
,~
,_~
t~
IY,,
~'~ ~.z:
~I
{"
i
~
.,'~
,3
,S N ,~
~
,~
'~ '..-I
~ ~, = '~ = ,.~ ~
13h
v ERI FICA"rlON D A T A Four sets o f dal, a, due to Roth (1971) and R ot,h el, ,,,I (19'7.4) from freely c'lrublblg borders were used. "riley referred t,o t,huse dut,a set,s as R,oUi 8, R oth 9, Rot, h 1(.} aJid Rot, h 11, which are summarled Ill Table I. 'T'hese daLa perl,aln t,o non vegetated borders (sod classified ~,~ sandy Ioa.m bL.dk denslt,y 1.,I). These herders are 5.89 m wide aJld 91.46 m long wlLh 1.24 nL of exl, ra sl~ace un each of l,lle Ul.~sl,ream wld t,h~ downstream ends of l,he border AL, (,hese ends sIUs were IJrowded t,o assure u n i f o r m ent, ry o f wat, er at, t,he up st,reum end aJld t,o elimblat,e [,he d r a w d o w n effect, or ~uLl'luw at, t,he dawn si,reanh end. Inflow was measured by a IU cr'n l.~rolJelh-,r meter aJ1d l,he uut, l'luw by a t,rla]igular crlt, lcaJ depl, h l'lume PARAMETER
BSTIM ATIC)N
"the KW model cont,abls t,wo ~ l ~ l u w n inl'dt, ratlun paramet, ers K and a of the Kost.ya.kuv equat, lon aJid t,wu u.nkJiown friction paramel, ers n a.nd /3 of t,he st,aide discharge relal, iun. K and a were esLimal, ed by a volume h',.tlaJIce analysm aJid n mid /.1 from Mannmg's equal, ion (Ram, 1982,) 'T'he values o1' t,hese paxamel,ers for each dat,a set, axe gwen Ill Tahle [. VBRI PIC'A'T'IUN Advance t,Lrr,es and recesston t,Jrnes were comput,ed using t,he recurstve solut, tons aJld t,he K W T method fur t,he rour set,s of data given m 'T'ahle I The calculat,ed and observed vaJues of advaJlce [1111e a.lld recesston l, ulle are plotted hi Figs. 4--'7. Fur aJl four dal, a sets l,here IS u t..tood agreement, be l,ween observed (obs) values and compuLed (caJ) v',dues by t,he two meLhods The absolute percenL deviat, ion, PF) [(obs - caJ)/obsl i~'mormg ,',Igebra.ic sign, bet, ween observed aJid coml.,uted advance tames raJiged bet, ween 0.(.) aJlcl 2,3% for t,lle sequentml met, hod, aJid between 0 aJid 21% for the K W T met, h ud fur aU the dut, a set,s except for R.ot,h 10 for which It, wa,~ 35',"~, and 4.8.9% respect, ively, n.,r t,he t,wo mel,hods. "rile t,wu met,hods yielded comparable values of adv~lce and recession "rile error was htgh ul L,tle inlt, lal sLa~es of adva.nce but de~'reased slgnd'l c~lLly wtt, l l l , he progress o f advance Since t,Fle advaJ'ICe l,tme IS smaJl in l,he beginning, even a smaU error m l,he comput,ed value will cause hlgh PD. The PD l'or recession t,iJne raJ'Iged heL,ween 0.0 ~ I d 6% for 'd..ll the daLa sel,s excel.d, for Rut, h IU l'or whlch It, was 14.0',.% II, is not, cleat why t,he PD m consider ably hlgher fur R,oUl 10 l,h~l for uther data seL,s. The l,wo met, hods were ident, lcaJ for [,he recession phase.
1 :IH
i_1 i.i I;
fli i
i.Ti I
~.1
, i
I,:)
~:r.i .i.r
I.LI l-
,7 ill
i
i.lE I~-i ';
'1
r~
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~
E
I_1
i.)
'1
~_)
i i
'1 i
I-(l)
e, qJ
~5 ;;i,
i
,p q.4 'T1
CI i~J
i
,-<3 I:i "l ill
4.
I:~
I') I ~J
I:l i%J llll/J
I.'l ill
I.') ri
1_7 io J
' 3~11
-:-_:7
LI
i:11
it
';
\}
i_z 4 ,~ l'l (.1~
,J LI
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I.LI Iii ~'1
I~1 Ii~ I';l. I.LI 21
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.
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II I L ~ l
0 1.7 (z)
1
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i#l ivl igl li)
r2.~ ill
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t~J
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gu~ #1 i#1
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. . . . .
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ql,I ,,.~
I.LJ t.J
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z
,,~
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,
ij
,,~ q,9 IJ
n'~l
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,
,
q-~ q'r)
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utw
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(21 r 1'
(.7 q~,b
13N C(JN(." L U S I ( ) N S
T h e l'ollowtr'Ig ,'onclustons c~'i be d r a w n frorn l, l lts sl, udy ( I ) "l"l'~P propos ed meLhod ts suffietenl, ly aec'urate for solvlng {,he ktnemuLtc wave equaL, i,,rm ul" burder LrngaL, t(.,n rlow. (2) I{, ,.'ornl.,a.rPs very well wtl, h (,hP K W T meLh,.),J. (3) Sm,.'e l, he proposed mel,hod ts reeurstve, tL, ts stml'.,IPr and rnorP efrtcienL. t,haJ'~ Ll'le KW'T' mel, l'~od. It, may Lie of int,Prest t,u n(.,l,P l,haL l'or l,he dat.a aJra.lyzed l'IPre l, he klnemaLte wave model m surl'~emnl, ly ac'~'urat, P fur mu(:lellrlg l,he Prlt, i.re Lmgat, ton excPpl, for t,he vert, ii",.d rei'esst()n "['he mudPl predtc{,s t,he advan,L'e time SLd'l'tcti-,nt, ly ae(.'urately for l,he dal,a an',.dyzed l'iPre Usually t hP l:.)redtct, tor~ e r r o r ts heh.)w 25";,';, A laJ"ge errt.)r ri,u,rrna.]ly o(.'eurs in t,he bel.,qnrlinp, at' L,he i.rrigat, t(.)t~. T h e m u d e l predtl.'t,s the t'H)rizonl,al i'el'PSSlCm t,llTle Wlt, h iJre dl('l.,l(:ln error (.)I" less t,haJ1 ] 5".';,. A C K NOW L E D G EM EN'T'
Thi,',; study was SUlJp(.)rted m p',.u-t hy fuJ'~ds provlded by l,he Nat, ton',.d Sl'i em.'e Fourldai, ton w-il, h i,he prujPcL, Prep Buundazy Problems in Wat Pr Re s(Ju.rce EngumerLng, NSF ENG 79 23345.
REFERENCES B;s;.,,:,ell., b L , F'ullgrrleler, r) [) ...lll(-I Sll~lk4,,ll', 'T', 19NI. HydrJl, lh'~,; ul :~url'.',ce liil~01{.iq)rl In' M E ,leni.len (Edll.(Jl'), [)~.Nl~rl gild L)pel.'ll, lOll U]' P',Jlm htip,.',lu.m Sv,~..,r.ern~., A~NAE Mlllll.)~i 3, Arllerll.,Liii ~O('ll-q,V Of Ag¢l~.'ullur~l Er,l~tne~r% SI ,Io~.;eph, MI, pp ,-147-,-19N (~Ulll.~l-' , ,I. IJnd WllulhI.M-'I', D A , 1~'7!) I.rnMe.',dv I'h.)w m upl-,n i,h~Jllll~,l.~L (~,h~pl.er 1:} irl I~. M.',l'lm~.~d ;jrll"l V Yeill-,vlel"l (Edlll.m,i), Irlll~.lt.lon ~.V!.,I.erll.'.; W.'l|(.-,i R,t-'~.iUlJi P u b I , F'ipll Culllni..,, (~(-L PP ;~9~-!.~:.1'7 KIncLII("], [ ) C ' , 1~'70. H'llldlud,vIliJrnli;h Ill b u l d e l li'll~/JI,i(ll'l Th,,-,,%,; p l e h e l l l e d t.i)(.lulur,Jd~ S[,,JI.e'. I.IlllVei,i.,It.y, I?l=rl (.'ulhrl,% (."(.), trl pm'th..ll I'ull'lllmPnt i)l' Iht-, reql.llll-'ml-,ll{.I.., I'ur lilt, del,.;ree =~i Di..'lur uf Phthb.'.,uphy, ]()l-i pp Ram, l-I S , 19,':'12 M a l h e m a l t e a l rn,.~,dehril.; ul l.,Ul'l','Jee urt~;:.d..m, 'T'he~,m pre~;enled I,~ Ml.,.,ni.~.~ll::lpl Slul.e l.]mver;,;tl.9, Mt.'.~,it~,,.,.,tpptSl,'Jl,f', MS, ill p,Ji li~Jl l'ull'lllmer~l i~l l lle rel..luti~ IneliL,.; l'ut L.he deb]ree of' Dli(?|.iJl i_i[ Phdos{~phv, ,'.III'2 pp Ranl, R ~ , ~lrlEh, V P arid Pr;j~iJl"l, ~ N , |9~;'1 Malherrlatl,l.,al rrlodehlll~ c)l' bOll'lel l l l l g J l,un 'T'e~.'h Rep W R R h , WiJl.el Re,~iilUl(.'e~,, Prul~l',Jln , [.)t-'p,..u'lmelll i)l' (."lvll EIIil~ll|elJIIIl~, Li~ui,'.,lan~ ~L,,JL.*-'U n l v e l s i l V , B a l u n R.UlJi~e , L A , ;'1()~ tip R¢~th, R L , ]~'71 Roughne~.,,,, (.]lurlrl~ bl.licli.,i ilill:~il[l()ri Thc.-,hl.~; pre!.,Pnl.ed I.u I.rrllyel,,,,ll.y Ill Alll, Urld, TLIL%OlI, A Z , III p,..Irllal Illll'lllnlerll id' l.he re(lUll'l:'.lTIt-'lllh I'()i t.he del-.~,ree ,~1 M~,,l,l-'l' uf S(.'lellce, '7~ plJ Ri.)th, R I, , Funl~,en, D W , Falil~rnelel, LI I') .llll'l All'lll~,;~.)tl, K 'T', ]Lr?J [)Llla lul Ijc)ldel ir'rlgallUll rrludel~,; 'T'rurl~,, AS,,~,E, N l i b ' i - i f ' ; | ~hei'lll.'lrl, B dnd Slllgh , V P., 19'7N A klnerll,Jil.ll.' ITIorJl-,t I'lJr ,~aUll,.ll',l-' ilrit.~,,Jllllll W,..lllel R~-' ~'~uur Re.'., , 1,-1(2)' 35';'--364. ~heilTl~Jtl, B ;.,llid ~lnl..~,h, V P , ]~H2 A IIinelndl.le rr'illde] Iol ,~ull'al.'e trill~.lt.lllrl .ill I-'~.lert .~,;li.lll Waler RP~,,our I'~el~, ]N ( 3 ) I';:~.1d~6'7 ~nil.~h , V P ,Jiid Shelril;Jrl, B , I~"l:} A hln~'.lTl;JllL" ;,l.udy ur ~ulldL'e LlrlI.Jgl.IO|l. m.'qherrhJ[I ,L'at.lOl~ ~.;l=lut.lUn~.~ "l"eeh R~p W R R 4 , W;JI.e'l' R.l=lNiluli'e~,~ Prol, rum, Deparl.rnent id' C'lVli ,'Ell~llleel'll|~, I..,it.lUll.,iaria ~L,'II,P I.JrllvPrNII.V, BIJI.IIlI Rl.lu~o, LA , 'T{'I p~)