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Jou rnal of Hydrology, 105 (1989) 35%367Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands[31O N T H E R E L A T I O N S H I P B E T W E E N T H E T I M E C O N D E N S A T I O NA P P R O X I M A T I O N A N D T H E F L U X - C O N C E N T R A T I O N R E L A T I O N

MURUGESU SI VA PA LA N and P.C.D. MI LL YW a t e r R e s o u r c e s P r o g r a m , D e p a r t m e n t o f C i v i l E n g i n e e r i n g a n d O p e r a t i o n s R e s e a r c h , P r i n c e t o nUniversity, Princeton, N J 08544 (U.S.A.)(Received September, 1987; accepted after revision April 12, 1988)

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Sivapalan, M. and Milly, P.C.D., 1989. On ~he relationship betwee ,~ the time condensation approxi.mation and the flux-concentration relation. J. Hydrol., 105: 357-367.The time condensation approximation (TCA) has been used widely in hydrological modelingsince it was introduced in the 1940's. This paper proposes a theoretical foundation for the validityof the TCA applied to constant-rainfall infiltration by recourse to the flux-concentration relation(FCR) used in soil physics. When the FCR is independent of boundary conditions, the TCA is exact.The success of the TCA is the result of two important properties of the FCR. The first is the factthat for a variety of flow processes the FCR is nearly the same regardless of the actual boundaryconditions. Secondly, the FCR-based solutmns of the Richards equation are relatively insensitive

to the form of the FCR. For a delta-function soil, the TCA is exact. The largest errors associatedwith the TCA occur in the case of linear soil, where it leads to a 19% error in the calculation oftime to ponding.

I N T R O D U C T I O N

T h e p h e n o m e n o n o f s o il m o i s t u re m o v e m e n t h a s b e e n s t u d i e d t o t h e e x t e n tth a t bo th Ch i lds (1968) and Ph i l ip (1973) hav e sa id (W hi te e t a l. , 1979) th a t" f u r t h e r r e f i n e m e n t s o f i t s c o n c e p t u a l b a se s a r e u n l i k e l y t o b e p r o fi t a b le . "Ev e n so , t h e b e s t e x i s t i n g so l u t i o n s o f i n f i lt r a t io n p r o b l e m s , f o r e x a m p l e , a r et h o s e t h a t h a v e b e e n o b t a i n e d f o r o n l y t h e s i m p l e st o f b o u n d a r y c o n d i t io n s ,n a m e l y t h a t o f s u r f a c e s a t u r a t i o n . M o r e r e c e n t ly , g r e a t p r o g r e s s h a s a l s o b e e nm a d e o n t h e p r o b l e m o f c o n s t a n t -f l u x i n f i l tr a t i o n ( B r o a d b r i d g e a n d W h i t e , 1 98 8;W h i t e a n d B r o a d b r i d g e , 1 98 8). H o w e v e r , d u e t o th e f e e d b a c k e f fe c ts i n h e r e n ti n so i l m o i s t u r e m o v e m e n t , t h e b o u n d a r y c o n d i t i o n s a r e q u i t e c o m p l e x a n dt i m e v a r i a b l e . Th u s , t h e g o v e r n i n g b o u n d a r y c o n d i t i o n s c a n n o t b e a s s i g n e d apr ior i b u t h a v e t o b e u p d a t e d c o n t i n u a l l y . Ta k i n g t h e e x a m p l e o f i n f i lt r a ti o n ,s ince in i t i a l ly th e r a t e o f r a in fa l l i s l e ss tha n the in f i l t r a t ion c apac i ty , '~hea c t u a l i n f i l t r a t i o n r a t e i s e q u a l to t h e ra i n f a l l r a t e a n d t h e m o i s t u r e c o n t e n t a tt h e su r f a c e w i l l b e d e t e r m i n e d a s p a r t o f t h e so l u t i o n o f t h e p r o b le m . A f t e rp o n d i n g h a s t a k e n p l a c e a t t h e s u r f a c e , t h e g o v e r n i n g b o u n d a r y c o n d i t i o n

0022-1694/89/$03.50 1989 Elsevier Science Publishers B.V.

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becomes the specification of satu ration moisture content at the surface and therate of infiltration is obtained as part of the solution. An analogous situationexists in the case of evaporat ion from the soil surface. The problem is even morecomplicated in practice since rainfall and evaporation events are quitenonuniform and random and since hysteresis effects also need to be taken intoConsideration.At present, numerical methods have advanced to the stage that permits theaccurate solution of the nonl inear and hysteretic governing equations for thequite general initial and boundary conditions. But these are far too expensiveand time consuming for practical applications in field-scale modelling of therainfa ll- runoff process; what is needed is a simple and reasonably accuratemethod for predicting infiltration and evaporation amounts on a continuousbasis during periods of in:ermi ttent storm and interstorm periods. It was in thisregard that the time con~lensation approximation (TCA) was introduced intohydrological practice i~ the early 1940's to be used in conjunction withempirical field-scale infiltration functions. The physical basis for the validityof TCA was never analyzed though numerical tests have shown it to bereasonably accurate and useful when applied to the problem of one-dimension-al infiltration into a homogeneous soil.Significant progress on assigned.flux infiltration has recently followed thedevelopment by Parlange (1971) and Philip (1973) of the fl~lx-concentrationrelation (FCR). The purpose of the present exercise is to explore the theoreticalunderpinnings of the TCA by use of the FCR.Time condensation approximation

Apparently, the concept of time condensation or time compression was firstintroduced by Sherman (1943) in the context of part ition ing ra infall intoinfiltration and overland flow during erratic rainfall events. The underlyingassumption of the TCA as originally applied was that the infi ltration capacityof the catchment at any given time within a storm depends only on the volumeof infiltration from the storm until that time, regardless of its distribution intime. The same assumption is implicit in the application of the TCA to ahomogeneous one-dimensional soil profile. In the la tt er case, it is equivalent toassuming that the soil moisture profile, once the soil surface is saturated, isdependent only on cumulative infiltration and is independent of the actualrainfall history. If this is true it means that to predict infiltration capacityresulting from any time-varying boundary condition, one needs only to solvethe infiltration problem for a single boundary condition. The simplest suchsolution is for a continuously ponded surface.

A good illustration of the application of the TCA m variable-rainfall infiltra-tion is given by Milly (1986), who also applied the YCA to the analogousproblem in evaporation. He used a form of Philip 's (1957) infil tration equationto derive an equation for the infiltration capac ity as a function of cumulativeinfiltrat;on for a ponded surface and used the la tter as the base equation for thedetermination of rainfall infiltration. For field applications of the TCA the base

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curve could be obtained by field experiments under saturated surfaceconditions.Reeves and Miller (1975) and Ibrahim and Brutsaert (1968) have performednumerical tests of the TCA for erratic rainfall by comparing the cumulativeinfiltration and the moisture profile obtained using the TCA against thoseobtained by numerical solution of the hysteretic governing equations. Theyfound generally good agreement. Reeves and Miller found t hat the infiltrationcapacities are slightly underestimated when the TCA is used; the errorsobserved in their comparisons were due to the redistribution taking placeduring breaks in the rainfall and due to the hysteresis accompanying it. Theirresults imply that the error at the end of a break is related to the amount ofredistribu tion and t ha t if a correction is made in the continuous accounting ofthe cumulative infiltration, the errors might be great ly eliminated. More workon this aspect of the problem is needed to improve TCA-based models.T h e f l u x - c o n c e n t r a t i o n r e la t io n

The work of Parlange (1971) led to Philip' s (1973) introduction of the flux-concentration relation (FCR). The FCR expresses the dependence of fluxdensity on moisture content during various unsteady flow processes in un-satura ted soils. The governing equation considered here is the one-dimensionalform of the Richards equation for homogeneous soil:dO d [ D dO K ] (la)e-7 = &where 0 is the volumetric moisture content. D ( O ) is the moisture diffusivity,K ( O ) is the hydraulic conductivity, and z is the vertical coordinate, takenpositive downward. Both D and K are usually strongly varying functions of 0.For one-dimensional systems which are horizontal or in which the effect ofmoisture gradients dominates that of gravity, eqn. (la) can be simplified to:0 [ 0 0 ]0-7 = d-~ d-~ (lb)These governing equations will be subject, in general, to the initial condition[eqn. (2)] and one of the boundary conditions [eqn. (3) or (4)]:t = 0 z >I 0 0 = 0 , ( 2 )t > o z = o o = 00 (0 ( 3 )

00t > 0 z = 0 D (O ) o---z - K (O ) = - Vo(t) (4a)or :

00t ) 0 z = 0 D ( O ) - ~ - = - V o ( t )Equation (4a) applies to eqn. (la), and eqn. (4b) to eqn. (lb).

( 4 b )

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I t i s a l s o r e q u i r e d t h a t O o ( t ) a n d v o ( t ) a r e s u c h t h a t h y s t e r e s i s e f f e c t s a r ea v o i d e d . U s i n g t h e t e r m i n o l o g y o f P h i l i p ( 1 9 69 , 1 97 3), t h e s e t o f c o n d i t i o n s o fe q n (3 ) m a y b e d e s i g n a t e d a s v a r i a b l e - c o n c e n t r a t i o n c o n d i t i o n s a n d t h o s e o fe q n ( 4 ) a s v a r i a b l e - f l u x c o n d i t i o n s . W h e n O o ( t ) a n d v o ( t ) a r e t i m e i n d e p e n d e n t ,t h e n t h e y w i ll b e c a l le d c o n s t a n t - c o n c e n t r a t i o n a n d c o n s ta n t -f lu x c o n d i t io n sr e s p e c t iv e l y . I n g e n e r a l O o ( t ) a n d v o ( t ) w i l l d e n o t e s u r f a c e m o i s t u r e c o n t e n t a n df lu x d e n si t y , re g a r d l e s s o f t h e c h o i c e o f b o u n d a r y c o n d i t i o n s .I t is a s s u m e d t h a t f o r a ll t > 0 t h e m o i s t u r e p r o f i le v a r i e s m o n o t o n i c a l l yf r o m t h e v a l u e 0 0 ( 0 a t z = 0 to the va lue 0n as z - - , oo . S imi la r ly , the f luxd e n s i t y o r D a r c y v e l o c i t y v ( O , t ) , c h a n g e s m o n o t o n i c a l l y fr o m V o ( t ) t o z e r o i n t h ec a s e o f e q n . ( l b ) a n d f r om V o ( t ) t o K n i n t h e c a s e o f eq n . ( l a ) . P h i l i p (1 97 3)i n t r o d u c e d :

0 -- 0 n = O o ( t ) - O , (5 )a n d t h e f l u x - c o n c e n t r a t i o n r e l a t io n ,

v (O, t ) - g .F(O, t ) = V o ( t ) - K , (6a)o r:

v ( e , t )F(, t ) = V o ( t ~ (6b)T h e f ir s t e x p r e ~ s i o n f o r F ( , t) a p p l i e s t o p r o c e s s e s ( i n f i l t r a t i o n o r e v a p o r a t i o n )g o v e r n e d b y e q n . ( l a ) a n d t h e s e c o n d e x p r e s s i o n a p p l i e s t o p r o c e s s e s( a b s o r p t i o n o r d e s o r p t i o n ) d e s c r i b e d b y e qn . (l b ). W i t h t h e a s s u m p t i o n s a l r e a d ym a d e , b o t h a n d F ( O , t) a r e n o n i n c r e a s i n g f u n c t i o n s o f z , a n d t h u s F { ,t ) i sn e c e s s a r i l y a n o n d e c r e a s i n g f u n c t i o n o f EJ f o r a n y t i m e t a n d s a t i sf i e s:F(1 .t) = 1; F(0,t) = 0 (7)P h i l i p (19 73~ h a s s t a t e d t w o p r o p o s i t i o n s o n t h e u t i l i t y o f t h e F C R . F i r s t l y , i fF ( ,t ) i s k n o w n e x a c t l y f o r a n y g i v e n p r o c e s s a m o n g t h o s e d i s c u s s ed a b o v e , t h er e l e v a n t e x a c t s o l u t i o n o f e q n. ( l a ) o r ( l b ) c a n b e f o u n d i m m e d i a t e l y . S e c o n d l y ,w h e n F ( @ ,t) i s n o t k n o w n e x a c t l y , a f i rs t e s t i m a t e o f a n y s o l u t i o n m a y b eo b t a i n e d b y u s i n g a g u e s s e d ( F ( , t ) f u n c t i o n . I t t u r n s o u t , a s P h i l i p (1 97 3) h a ss h o w n , t h a t F ( , t ) d o e s n o t v a r y g r e a t l y o v e r a w i d e r a n g e o f f lo w p r o c e s s e sa n d s o i l c h a r a c t e r i s t i c s , a n d f u r t h e r , t h a t f ir st s o l u t i o n e s t i m a t e s a r e r e l a t i v e l yi n s e n s i ti v e t o e r r o r s i n t h e g u e s s e d F ( , t ) .

F i g u r e 1 i l l u s t r a t e s t h e c h a r a c t e r i s t i c s o f F ( , t) f o r w e t t i n g p r o c e s s e s a n d i sb a s e d o n t h e w o r k o f P h i l i p (19 73 ). F o r a b s o r p t i o n F ( , t) i s i n d e p e n d e n t o f t im e .T h e F ( O ,t ) r e l a t i o n s f o r a b s o r p t m n w i t h t h e c o n s t a n t - c o n c e n t r a t i o n b o u n d a r yc o n d i t i o n f o r v a r i o u s s of t t y p e s a r e g i v e n b y t h e c u r v e s m a r k e d A , B, a n d C .T h e r e l a t i o n s h i p g i v e n b y c u r v e C i s fo r l i n e a r s o i l ( c o n s t a n t d i f fu s i v it y ) a n dw a s d e r i v e d a n a l y t i c a l l y b y P h i l i p (1 9 73 ). C u r v e A i s f o r t h e s o - c a l le d d e l t a-

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3 6 11.0

0 .8

0.~F ( O , t )

0 .4

0 .2

, ~ ' ~ , ~J

I ! ,/ / / / "C t , . / / /

/

I I I ,0.2 0.4 0.6 I- 0 . 8 1 . 0F i g . 1. F l u x c o n c e n t r a t i o n r e l at i o ns , F ( , t ) , f o r a b s o r p t i o n w i t h c o n s t a n t - c o n c e n t r a t i o n ( s u b s c r i p tc ) a n d c o n s t a n t - f l u x ( s u b s c r i p t f ) c o n d i t i o n s . A i s f o r a d e l t a - f u n c t i o n s o il f o r b o t h c o n d i t i o n s . C ~a n d C f a r e f o r l i n e ar s o i ls a n d B c a n d B f a r ~ f o r a " r e a l " s oi l.

f u n c t i o n s oi l, o r w h i c h D ( ) i s a D i r a c d e l t a f u n c t i o n ; i n t h i s c a s e F ( , t ) = @ .C u r v e s C c a n d A r e p r e s e n t u p p e r a n d l o w e r b o u n d s o f F ( @ , t ) f o r t h e c o n s t a n t -c o n c e n t r a t i o n b o u n d a r y c o n d i t i o n . T h e F ( , t ) c u r v e s f o r " r e a l " s o i l s w o u l d b ee x p e c t e d t o b e w i t h i a t h e s e b o u n d s ( e . g . , c u r v e B e ) a n d t h i s i s b o r n e o u t b ye x p e r i m e n t a l a n d n m a e r i c a l o b s e r v a t i o n s . T h e F ( , t ) r e l a t i o n s f o r a b s o r p t i o nw i t h t h e c o n s t a n t - f l u x b o u n d a r y c o n d i t i o n f o r t h e t h r e e s o i l c u r v e s a r e g i v e nb y c u r v e s m a r k e d A , B f, a n d C f. F ( , t ) f o r t h e d e l t a - f u n c t i o n s o i l i s a g a i n g i v e nb y c u r v e A . C u r v e C f is f o r l i n e a r s o i l s a n d c u r v e B f i s f o r t h e " r e a l " s oi l. N o t et h a t B f a n d C f a r e c l o s e r t o A t h a n a r e B e a n d C r e s p e c t i v e l y . I n t h e c a s e o fi nf i l tr a t io n , a t t = 0 , t h e F ( , t ) r e l a t i o n s a r e e q u a l t o t h e i r c o u n t e r p a r t s f o ra b s o r p t i o n e x p r e s s e d b y c u r v e s C a n d C f f o r l i n e a r s o i l a n d B c a n d B f f o r " r e a l "s oi ls . A ~ t - ~ o o, a ll t h e c u r v e s i n t h e i n t e r v a l A t o C c m o v e t o w a r d s A . F o r t h ed e l t a - f u n c t i o n s o i l, u r v e A h o l d s f o r a ll t. I t i s c l e a r t h a t t h e d i f f e r e n c e b e t w e e nt h e c o n s t a n t - c o n c e n t r a t i o n a n d c o n s t a n t - f l u x i nf i l t r at i o n u r v e s i s a m a x i m u mf o r l i n e a r s o i l s a t e a r l y t i m e s ; it i s s m a l l e r f o r " r e a l " s o i l s a n d i s z e r o f o r t h ed e l t a - f u n c t i o n s o i l; w i t h t i m e t h e d i f f e r e n c e s g e t e v e n s m a l l e r .

W h i t e e t a l . , ( 1 9 7 9 ) , w o r k i n g w i t h B u n g e n d o r e s a n d , c o n f i r m e d s o m e o fPh i l i p ' s ( 1973 ) conc l us i ons ; t h e c a l cu l a t ed m o i s t u r e p r o f il e s w er e r ea s ona b l yi n s ens i t i ve t o t he a s s umed f o r m o f , ~ ( , t ) i n t he ca s e o f abs o r p t i on w i t h t hecons t an t - f l ux bo un da r y cond i t i on . W l i t e (1979) d i d s i mi l a r w or k f o r abs o r p t i oni n B u n g e n d o r e s a n d f o r c o n s t a n t - f l u x a n d c o n s t a n t c o n c e n t r a t i o n b o u n d a r ycond i t i ons . H i s expe r i men t a l mo i s t u r e p r o f i l e s con f i r med Ph i l i p ' s ( 1973 )p r ed i c t i on t ha t f o r cons t an t - f l ux abs o r p t i on F ( O , t ) i s t i me- dependen t . W hi t ea l s o f ound , a s Ph i hp ha d p r ed i c ted , t ha t t h e FC R a t co ns t a n t f l ux , F f( O ,t ), l aybe l ow F~ ( O , t ) , t he FC R a t cons t an t cow cen t r a t i on , avd t ha t t he d i f f e r enceb e t w e e n F c a n d Ff i s ve r y s ma l l . H e t he r e f o r e p r opos ed t ha t f o r p r ac t i ca lpu r p os es F~ can be u s ed a s an ap p r ox i ma t i on f o r F f. W h i t e a l s o con fi rmedP h i l i p ' s p r o p o s i t io n t h a t t h e d i v e r g e n c e b e tw e e n p r e d i c t i o n s u s i n g F a n d t h e

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362correct F~ can be expected to be maximum for linear soil and will decrease formaterials whose diffusivity is more strongly dependent on .In summary, the important proper ty of F(O,$) is that during inf iltration evenin the worst case of the linear soil, the difference between F~ and Fc is small andthat for more "real" soil and with the passage of time, this difference becomeseven smaller.Out l ine o f the analys i s

A limited special case of the application of the TCA is the problem ofinfil tration due to a cons tant applied flux v0f at the surface where it is assumedtha t v0f is greater than/(8, the saturated hydraulic conductivity of the soil. Insuch a case during a certain in itial period all of the applied flux infil trates andthe surface moisture content gradually increase until it reaches the saturationvalue. From that time onward ponding takes place. After the time of ponding,tp, the boundary condition becomes a constant-concentrat ion condition and theinfiltration ra te falls below the applied flux, decreasing with time. In th is paper,the validity of the application of the TCA to this special case is explored byrecourse to the FCR-based solutions of the problem. An important par t of anysolution is the determination of tp and the moisture profile at this time; thisp'cofile will then be used as the initial condition with the constant-concentra-tion boundary condition to obtain the solution after ponding. In the nextsection, the moisture profile at the time of ponding due to the constant.fluxboundary condition is shown to be the same (or nearly the same) as tha t whichwould have been obtained for the same cumulative infiltration had infiltrationbeen taking place due to the saturation boundary condition. The solution fort greater than t p c a n then be o utained using either of the two cases since theboundary condition as well as the initial condition would be the same (ornearly the same) in both cases. This will establish the validity of the ~pplica-tion of the TCA to the infiltration problem with the constant-flux boundarycondition, In the solutions outlined below the phenomenon of tension.satura-tion has been ignored.A N A L Y S I S O F I N F I L T R A T I O N U S I N G T H E F C RInf i l t rat ion wi th the cons tant-concentrat ion boundary condi t ion

Here the relevant governing equation is eqn. (1) with the initial conditiongiven by eqn. (2) and the boundary condition given by eqn. (3) where 00(0 = 08,a constant. 08 is the saturation moisture content of the soil. Let v0(t) denote thesurface flux in this case. Then:v ( 0 , 0 -

F o ( e , 0 = v 0 o ( t ) - K (8 )

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3630 - - 0 n = 0 , - 0 . ( 9 )

The flux density v(O,t) is given by the Darcy-Buckingham law as:O0v (O,t ) = - D(O) ~ z + K(O) (i0)

Letg c( O ,O a ) = ( v o c ( t ) - K . )zc ( O , O a ) ( 1 1 )where zc(O,O~) represents the variation of 0 with depth z for the constant-concen tra tion boundary condition. Substi tution of eqns. (10) and (11) in eqn. (8)and integration yield:

0.D(O)dOZ o ( 0 , 0 8 ) = F . ( , O - [ K ( 0 ) - K . ] l [ v o c ( t ) - K . ] ( 1 2 )

Let Qc be the net storage increase at any time due to infiltration with theconstant-concentra tion boundary condition. Then:0.

Qc = [ z,:(O,O~)dO (13)Ab

or:08

Qc [ v o c (t ) - g , , ] = | Z c ( O , O . ) s o (14)i/On

Substituting eqn. (12) into eqn (14), integrating by parts and rearranging:O~1 f ( 0 - ( ),) D O ) dOQ = [v0c(t)- Kn] Y c ( , t ) - [K(0) - K , ] / [ V o c ( t ) - K , ] (15)

Given D(O ) , K(O) , and Fc (,t), eqn. (15) is a unique relationship between Q andvo~( t) - K , . Back-substituting in eqn. (11):

0.f D(O)dOQc F ~ ( o , t ) - [K(0) - K.] / [V0c ( t ) - K . ]0z ( O ,O , ) = 0 , ( 1 6 )( 0 - O.)D(O)dOf F c ( , t) - [ K ( 0 ) - K . ] l [ v o c ( t ) - K . ]On

Equation (16) can be further rearranged by expressing v oc ( t) - K , in the twointegrands above in terms of Q~ from eqn. (15), which shows that z~(O,O,) is aunique function of Q. The corresponding relationship for the case of

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eqn . (23 ) . In add i t ion , z f (0 ,O~) i s u n i q u e l y r e l a t e d t o Qfp b y e q n . ( 22 ?. E q u a t i o n( 23 ) i s s a t i s fi e d a t t h e t i m e o f p o n d i n g tp a n d c a ~ b e s o l v e d t o o b t a i n tp , a n d a tt h i s t i m e t h e m o i s t u r e p r o f i l e i s g i v e n b y e q n . ( 2 2 ) .O n c e a g a i n t h e c o r r e s p o n d i n g e x p r e s si o n fo r z f (O,0~) f o r a b s o r p t i o n i s g iv e nby:

0 sQfp ~ D(O)dOFf (O, t )oz f ( O , O ~ ) = o~ ( 2 4 )f (O - On)D (O)dOF , . ( , t )O.

E v a l u a t io n o f t h e T C AE a r l i e r i n t h i s s e c t i o : ~ t w a s s h o w n t h a t d u r i n g p o n d i n g i n f i l t r a t i o n Q c i s au n i q u e ~ a n c t i o n o f v uc(0 - K ' . S i m i l a r ly , it w a s s h o w n t h a t a t t h e t i m e o fp o n d i n g d u r i n g c o n s t a n t - f l u x i n f i l t r a t i o n Q f , i s a u n i q u e f u n c t i o n o f v0r - K n .I f Qc i s equ a l to Qf , , th en f rom eqn . (15) ~nd (23 ) Voc( t ) i s e qu al to v0f i f Ff ( , t )a n d F c (O , t) a r e i d e n t i c a l .C o m p a r e t h e m o i s t u r e p r o fi le a t t h e t im e o f p o n d i n g o f t h e c o n s t a n t - fl u x

i n f i l t r a t i o n c a s e t o t h a t o f t h e p o n d e d i n f i l t r a t io n c a s e, c o r r e s p o n d i n g t o Q c =Qrp, i .e . , eqn (16) ve rs us eqn. (22) . I f Ff (O, t ) i s e q u a l t o F c ( , t ), i t f o l l o w s t h a tVoc(t) i s e q u a l v 0f, a n d t h e r e f o r e f r o m e q n s . ( 16 ) a n d (2 2) t h e m o i s t u r e p r o f i le s a r ei d e n t i c a l . T h u s a s u f fi c ie n t c o n d i t i o n f o r t h e v a l i d i t y o f T C A i n t h i s p r o b l e m i st h a t :F f (O , t ) = , ~ ( O , t ) (25)F o r a d e l t a - f u n c t i o n s o il , t h i s r e l a t i o n i s e x a c t , a n d t h e r e f o r e t h e T C A i s e x a c t .T h e d i f f e r e n c e b e t w e e n F c ( , t ) a n d F f ( , t) w i ll b e m a x i m u m f o r a b s o r p t i o n ( o ri n f i l t r a t i o n a t e a r l y t i m e ) i n l i n e a r s o i l s . T h e s o l u t i o n s t o t h e a b s o r p t i o np r o b l e m i n a l i n e a r s o il h a v i n g d i f f u s i v it y e q u a l t o D w e r e o b t a i n e d n u m e r i c -a l l y l t si n g a p p r o x i m a t e F ( O , t ) e x p r e s s i o n s g i v e n b y W h i t e e t a l . (1 9 7 9) . T h er e l a t i o n s h i p b e t w e e n Vow(t)and Q~ an d th a t be tw een v0f an d Qfp a re g ive n by :Vow(t) 0.636= (26a)D(O~ - 0, , )2 Q ~

Vof 0.785= (27b)D ( O ~ - 0 , , ) 2 Q f~I t c a n b e s h o w n a n a l y t i c a l l y t h a t t h e n u m e r i c a l c o n s t a n t s i n e q n s . ( 2 6 a ) a n d( 2 6 b ) a r e i n f a c t 2 / ~ a n d ~ / 4 r e s p e c t i v e l y . U s i n g t h e s e t h e T C A u n d e r e s t i m a t e stp b y a b o u t 1 9 % .

T h e t r u e m o i s t u r e p r o fi le a t t h e t im e o f p o n d i n g d u e t o c o n s t a n t - f l u za b s o r p t i o n i n l i n e a r s o i l a n d t l ~ a v p r ~ x i m a t e m o i s t u r e p r o f i l e b a s e d o n t h eT C A a r e g i v e n b y t h e c o m m o n d i m e n s i o n le s s e q u a t i o n :

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3 6 6

oo

0 .5

1.0

Z(@}vof ~ '5O 2 .0

2 .5

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~ T R U E PROFILEPROF I ,E/ (bosedon the TCA)

r

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Fig . 2 . Com par ison of d imen s ion less mo is ture p rof iles as a func t ion of dep th z du r ing c ons tan t - f luxabsorp t ion : (a ) t rue prof i le , and (b) approximate prof i le cb ta ined us ing th~ TCA.lz (O )vo f = i d OD F ( O , t ) (27)

w h e r e = ( 0 - 0 ,) /( 08 - 0 ~ ) . T h e tr u e p r o fi le w i l l b e o b t a i n e d i f F f ( , t ) i s u s e di n e q n . ( 27 ) w h e r e a s t h e u s e o f F c ( @ , t) y i e l d s t h e a p p r o x i m a t e p r o f i le . T h ec o m p u t e d d i m e n s i o n l e s s m o i s t u r e p r o f i l e s a r e p r e s v n t e d i n F i g . 2 . S i n c e t h ed i f f e r e n c e b e t w e e n F f (@ , t ) a n d F ( , t ) i s m a x i m u m f o r li n e a r s o i l s , t h e e r r o r si n tp a n d t h e m o i s t u r e p r c ,file o b t a i n e d h e r e f o r l i n e a r s o i l s r e p r e s e n t u p p e rb o u n d s t o t h e c o r r e s p o n d i n g e r r o r s i n n a t u r a l s o i l s .

F o r m o r e " r e a l " s o i l s a n d f o r i n t e r m e d i a t e t i m e s , b o t h F f ( @ , t ) a n d F c ( @ , t )a p p r o a c h a n d t h e d i ff e r e n c e b e t w e e n t h e m w i l l b e s m a l le r . A s a r e s u l t , th ed i f fe r e n c e s b e t w e e n t h e s o l u t i o n s w i l l a l s o b e e x p e c t e d t o b e l e s s fo r " r e a l" s o i l st h a n f or li n e a r s o i l s. T h i s is b o r n e o u t b y t h e n u m e r i c a l a n d e x p e r i m e n t a lr e s u l t s o f P h i l i p ( 19 7 3) , W h i t e ( 19 7 9 ) a n d W h i t e e t a l . ( 1 9 79 ).K P E R E N C E SB r o a d b r i d g e , P. a n d W h i t e , l., 9 8 8 . o n s t a n t r a t e r a i n fa l l n fi l tr a ti o n: v e r s a t i l e o n l i n e a r m o d e ] ,

I . A n a l y t i c s o l u t i o n . W a t e r R e s o u r . R e s . , 2 4 ( 1) : 1 4 5- - 15 4 .C h i l d s , E . C . , 1 9 6 8 . T h e a c h i e v e m e n t ~ o f d r a i n a g e t h e o r y i n r e l a t i o n s h i p t o p r a c t ic a l n e e d s : t a k i n g

s t o ck . 9 t h I n t. C o n g r . S o i l S c i. , r a n s . ( 1 ): n g u s a n d R o b e r t s n , S y d n e y .I b r a h i m , H . A a n d B r u t s a e r t , W . , 1 9 6 8 . I n t e r m i t t e n t i n fi l tr a ti o n n ~ o s o i ls w i t h h y s t er e s is . A m . S o c .

C i r . E n g . , J . H y d r a u l , D i v H Y I : 1 1 3 - 1 3 7 .M i l ly , P . C . D . , 1 9 8 6 . A n e v e n t . b a s e d s i m u l a t i o n m v d e | o f m o i s t u r e a n d e n e r g y f l u x e s a t a b a r e s oi l

s u r fa c e . W a t e r R e s o u r . R e ~ . , 2 2 ( 1 2 ): 1 6 8 0 - 1 6 9 2 .P ' x l a n g e , J .- Y. , 1 9 7 1 . T h e o r y o f w a t e r m o v e m e n t i n s oi ls : . O n e d i m e n s i o n a l a b s o r p t i o n . S o i l S ci .,

I 1 1 : 1 3 4 - 1 3 7 .

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36 7Ph i l ip , J .R . , 1957. Th eory o f in f i l t r a t ion , 4 . So rp t iv i ty a nd a lgeb ra ic in f i l t r a t i on equ a t ions . So i l Sc i . ,84: 257-264.Ph il ip , J .R. , 1969. Th eo ry o f inf i l t ra t io n . Adv. H ydro sci . , 5 : 215-3~J5.P h i l i p , J . R ., 1 9 73 . O n s o l v i n g t h e u n s a t u r a t e d f ln w e q u a t i o n : 1 . ? h e f l u x - c o n c e n t r a t i o n r e l a t i o n .So il S ci., 116: 328-335.R e e v e s , M . a n d M i l ~er , E . E . , 1 97 5. Es t i m a t i n g i n f i l t r a t i o n fo r e ~ i c r a i n f a l l . Wa t e r R e s o u r . R e s .,11 (1): 102-110.S h e r m a n , L . K . , t 9 43 . C o m p a r i s o n o f F - c u r v e s d e r i v e d b y t h e m e t k o d s o f S h a r p a n d H o l t a n a n d o fS h e r m a n a n d M a y e r . T r a n s . A m . G e o p hy s . U n i o n , 2 4: 4 65 -4 ~7 .W h i t e , I ., 1 9 79 . M e a s u r e d a n d a p p r o x i m a t e f l ux c o n c e n t r a t i o n r e i a t i o n s f o r a b s o r p t i o n o f w a t e r b ysoil. Soil Sci. Am. J., 43: 1074-1080.W h i t e , I . a n d E r o a d b r i d g e , P . , 1 9 88 . C o n s t v n t r a t e r a i n f a l l i n f i lt r a t i o n : A v e r s a t i l e n o n - l i n e a rmode l , 2 . App l ica t ions and so lu t ions . W ate r Resou r . Res . , 24 (1 ) : 155-162.W hi te , I . , Smi les, D .E. and Pe r roux , K.M. , 1979. Abso rp t ion o f wa te r by so i l: The cons tan t - f luxboundary condit ion . Soi l Sci . Soc. Am., J . , 43: 659-664.