T h e L a p l a c e t r a n s f o r m s o l u t i o n o f a o n e d i m e n s i o n a l g r o u n d w a t e r r e c h a r g e bv s p r e a d i n g A . P . VERMA (*) R i c e v u t o il 21 G e n n a i o 19(>9 R I A S S U N T O . — U n ' e s p r e s s i o n e a n a l i t i c a p e r la d i s t r i b u z i o n e d e l c o n - t e n u t o di u m i d i t à , in u n p r o b l e m a di carico u n i d i m e n s i o n a l e v e r t i c a l e di a c q u a s o t t e r r a n e a , è s t a t a o t t e n u t a m e d i a n t e il m e t o d o di t r a s f o r m a z i o n e di L a p l a c e . Il c o e f f i c i e n t e m e d i o di d i f f u s i b i l i t à , c a l c o l a t o sull'intera fiam- m a di v a l o r i di u m i d i t à c o n t e n u t a si c o n s i d e r a c o m e c o s t a n t e , m e n t r e si a s s u m e u n a v a r i a z i o n e lineare di p e r m e a b i l i t à c o n c o n t e n u t o di u m i d i t à . S U M M A R Y . — A n a n a l y t i c a l e x p r e s s i o n for t h e m o i s t u r e c o n t e n t distri- b u t i o n , in a p r o b l e m of o n e d i m e n s i o n a l v e r t i c a l g r o u n d w a t e r recharge, h a s been o b t a i n e d b y u s i n g t h e L a p l a c e t r a n s f o r m m e t h o d . T h e a v e r a g e dif- f u s i v i t v c o e f f i c i e n t o v e r t h e w h o l e r a n g e of m o i s t u r e c o n t e n t is regarded as c o n s t a n t , a n d a linear v a r i a t i o n of p e r m e a b i l i t y w i t h m o i s t u r e c o n t e n t is a s s u m e d . 1 . - I N T R O D U C T I O N . R e c e n t l y K l u t e i 1 - 2 ) a n d S a r m a (:1) h a v e d i s c u s s e d t h e n u m e r i c a l m e t h o d s of s o l u t i o n f o r t h e f l o w of w a t e r i n p a r t i a l l y s a t u r a t e d p o r o u s m e d i a . I n t h e p r e s e n t p a p e r w e h a v e o b t a i n e d a n a n a l y t i c a l s o l u t i o n of a o n e d i m e n s i o n a l p r o b l e m of g r o u n d w a t e r r e c h a r g e b y s p r e a d i n g . W e c o n s i d e r h e r e t h a t t h e r e c h a r g e t a k e s p l a c e o v e r a l a r g e b a s i n of s u c h g e o l o g i c a l l o c a t i o n t h a t t h e s i d e s a r e l i m i t e d b y r i g i d b o u n d - a r i e s , a n d t h e b o t t o m b y a t h i c k l a y e r of w a t e r t a b l e . U n d e r t h e s e c i r c u m s t a n c e s , w a t e r , f r o m t h e s p r e a d i n g g r o u n d s , w i l l f l o w v e r t i c a l l y (*) D e p a r t m e n t of M a t h e m a t i c s , F a c u l t y of T e c h n o l o g y a n d E n g i n e e r - ing, M. S. U n i v e r s i t y of B a r o d a , B a r o d a , I n d i a . :hj A . P . V E R M A d o w n w a r d s t h r o u g h t h e u n s a t u r a t e d p o r o u s m e d i a . I t is a s s u m e d t h a t t h e d i f f u s i v i t y coefficient is e q u i v a l e n t t o its a v e r a g e v a l u e o v e r t h e whole r a n g e of m o i s t u r e c o n t e n t , a n d t h e p e r m e a b i l i t y of t h e m e d i a is a c o n t i n u o u s l i n e a r f u n c t i o n of t h e m o i s t u r e c o n t e n t . T h e n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n f o r m o i s t u r e c o n t e n t h a s b e e n solved b y e m p l o y i n g t h e L a p l a c e t r a n s f o r m t e c h n i q u e , a n d a n a n a l y t i c a l e x p r e s s i o n f o r m o i s t u r e d i s t r i b u t i o n is g i v e n . 2 . - F O R M U L A T I O N OF T H E B O U N D A R Y V A L U E PROBLEM. F o l l o w i n g K l u t e (*), we m a y w r i t e t h e f u n d a m e n t a l e q u a t i o n s as below. T h e e q u a t i o n of c o n t i n u i t y f o r a n u n s a t u r a t e d m e d i u m is given b y I M ) = - V - M , [2.11 w h e r e q, is t h e b u l k d e n s i t y of t h e m e d i u m , 0 is i t s m o i s t u r e c o n t e n t on a d r y w e i g h t basis, a n d M is t h e m a s s f l u x of m o i s t u r e . D a r c y ' s l a w g o v e r n i n g t h e m o t i o n of w a t e r hi a p o r o u s m e d i u m m a y b e w r i t t e n a s : V = —KV, [ 2 . 2 ] w h e r e r e p r e s e n t s t h e g r a d i e n t of t h e t o t a l m o i s t u r e p o t e n t i a l , V t h e v o l u m e flux of m o i s t u r e , a n d K t h e coefficient of a q u e o u s c o n d u c t i v i t y . F r o m e q u a t i o n s [2.1] a n d [2.2], we g e t : * (Q.0) = V • (qKV(/>) , [2.3] w h e r e o is t h e fluid d e n s i t y . Since, in t h e p r e s e n t p r o b l e m , flow t a k e s p l a c e only i n t h e v e r t i c a l d i r e c t i o n , t h e r e f o r e e q u a t i o n [2.3] r e d u c e s t o : W 3 / \ S m = * r * ) ~ * { e K g ) - [ 2 - 4 ] w h e r e y> is t h e p r e s s u r e (capillary) p o t e n t i a l , g is t h e g r a v i t a t i o n c o n t e n t , a n d (j> — — gZ. T h e positive d i r e c t i o n of Z axis is t h e s a m e as t h a t of t h e g r a v i t y . T H E L A P L A C E T R A N S F O R M S O L U T I O N O F O N E D I M E N S I O N A L , E T C . 2 7 C o n s i d e r i n g 0 a n d ip t o b e c o n n e c t e d b y a single v a l u e d f u n c t i o n , we m a y w r i t e e q u a t i o n [4] a s : W 3 / W \ 0 J A ' M = ^ R ~dz) + ¿ ; G * z ' [ 2 - 0 ] w h e r e D — - K \ , a n d is called t h e d i f f u s i v i t y coefficient. a» ¡>0 ' J R e p l a c i n g D b y i t s a v e r a g e v a l u e D„, a n d a s s u m i n g K = Ko0, Ko = ' 2 3 2 (as i n [3J a n d [1]), we h a v e : = l > „ — • K o — = - . 2 . ( > i t M 2 qs C o n s i d e r i n g t h e w a t e r t a b l e t o b e s i t u a t e d a t a d e p t h L, a n d p u t t i n g : = | t D " . = T L L2 ' we m a y w r i t e t h e b o u n d a r y v a l u e p r o b l e m a s : j > 0 q K o <>0 i>T p. Da ôf ' [2.7] 0 ( 0 , 2 ' ) = 0„, 0(1,T) = 1 , [ 2 . 8 ] 0(1,0) = 0, [2.9] w h e r e t h e m o i s t u r e c o n t e n t t h r o u g h o u t t h e region is zero i n i t i a l l y , a t t h e l a y e r Z = 0 i t is 0o, a n d a t t h e w a t e r t a b l e (Z = L) i t is a s s u m e d t o r e m a i n 1 0 0 % t h r o u g h o u t t h e p r o c e s s of i n v e s t i g a t i o n . I t m a y b e r e m a r k e d t h a t t h e effect of c a p i l l a r y a c t i o n a t t h e s t a t i o n a r y g r o u n d w a t e r level, b e i n g small, is n e g l e c t e d . 3 . - A N A L Y T I C A L S O L U T I O N . S e t t i n g e K ° _ » Qs D„ P in e q u a t i o n [2.7], we g e t : W **0 ft 00 R11 :hj A . P . V E R M A O n m u l t i p l y i n g e a c h t e r m of e q u a t i o n [3. J J b y e~STdT, i n t e g r a t - i n g t h e r e s u l t f r o m z e r o t o i n f i n i t y , a n d u s i n g c o n d i t i o n [2.9], w e o b t a i n d*6 dB dp P d£ 86 = 0. [3.2] w h e r e 0(£,S) = I e-sr d(£,T) dT, r e p r e s e n t s t h e L a p l a c e t r a n s f o r m of 0(i,T). Tlie L a p l a c e t r a n s f o r m a t i o n of t h e b o u n d a r y c o n d i t i o n s [2.8] y i e l d s 0(0,8) Oo 8 6(1,8) = 8 [3.3] S i n c e e q u a t i o n [3.2] is a l i n e a r e q u a t i o n w i t h c o n s t a n t c o e f f i c i e n t , we m a y w r i t e i t s g e n e r a l s o l u t i o n a s : 0(Ç,S) = E c o s h (f J//?»/4 + 8) + F s i n h (f J / p * ] i +8) P e2 , [3.4] w h e r e E a n d F a r e c o n s t a n t s of i n t e g r a t i o n . F o r e v a l u a t i n g E a n d F, w e a p p l y c o n d i t i o n s [3.3] t o e q u a t i o n [3.4], so t h a t , a f t e r s o m e s i m p l i f i c a t i o n , w e h a v e : P S IO_ 8 c o s h (l /?2M + 8 ) sinh (j//? z /4 + # ) S u b s t i t u t i n g t h e s e v a l u e s i n e q u a t i o n [3.4], w e h a v e : 0 ( f , 0 ) = e •{ ( do 8 sinh (1 — i ) + 8 s i n h j//5 2 /4 + 8 ~ + + e - W s i n h f j ' ^ / l + 8 8 s i n h | > / 4 + 8 [3.5] T H E L A P L A C E T R A N S F O R M S O L U T I O N O F O N E D I M E N S I O N A L , E T C . 2 9 T h e i n v e r s e t r a n s f o r m (L~l) of t h e r i g h t h a n d side t e r m s i n e q u a t i o n [3.5] m a y b e d e t e r m i n e d b y r e c a l l i n g a s t a n d a r d r e s u l t [4] viz.. L 1 m r,(8) = X C( / 4 + 8) s i n h ( i f f > / 4 + £ ) V ('S>) 8 s i n h (yp'li +S) 8 sinh (t J//j»/4 + , [3.7] a n d n o t i n g t h a t t h e r o o t s of e q u a t i o n sinh ( | ' p / i + S ) = 0 , a r e g i v e n b y = — /S2/4 — n* Tt2, we m a y w r i t e : *(Sn) = sin (n,3t(), C(0) = i sinh ' £ = ^i(O) = i s i n h (/?/2) . [3.8] F r o m e q u a t i o n s [3.6], [3.7] a n d [3.8], we g e t : sinh & L-1 + 271 2 n = l sinh f | /32/4 + i B _ , /? 2/4 + 11*71' [3.11] T h i s is t h e desired a n a l y t i c a l e x p r e s s i o n f o r t h e m o i s t u r e c o n t e n t d i s t r i b u t i o n . I t follows i m m e d i a t e l y f r o m e q u a t i o n [3.11] t h a t t h e g r a p h of t h e m o i s t u r e c o n t e n t v e r s u s d i s t a n c e (for g i v e n v a l u e s of t i m e , s a y t = 1, 4, 16 etc.) m a y b e easily d r a w n (as i n [1]). A n u m e r i c a l i l l u s t r a t i o n is e q u a l l y o b v i o u s . H o w e v e r , t h e s e a r e n o t i n c l u d e d h e r e d u e t o o u r p a r t i c u l a r i n t e r e s t i n o n l y a n a n a l y t i c a l s o l u t i o n . 4. - CONCLUSION. A n a n a l y t i c a l s o l u t i o n f o r t h e n o n l i n e a r d i f f e r e n t i a l e q u a t i o n g o v e r n i n g m o i s t u r e c o n t e n t d i s t r i b u t i o n lias b e e n o b t a i n e d , b y u s i n g L a p l a c e t r a n s f o r m m e t h o d , f o r t h o s e cases of g r o u n d w a t e r r e c h a r g e b y s p r e a d i n g w h e r e t h e flow is e s s e n t i a l l y one d i m e n s i o n a l , a n d in t h e v e r t i c a l d i r e c t i o n . T h o u g h no n u m e r i c a l i l l u s t r a t i o n is i n c l u d e d i n t h e p r e s e n t p a p e r (because of our p a r t i c u l a r i n t e r e s t ) y e t t h e c o n v e n i e n t f o r m of t h e m o i s t u r e c o n t e n t e x p r e s s i o n is i m m e d i a t e l y e v i d e n t . T H E L A P L A C E T R A N S F O R M S O L U T I O N O F O N E D I M E N S I O N A L , E T C . 3 1 R E F E R E N C E S ( 1 ) K L U T E A . , A numerical method for solving the flow equation for water in unsaturated materials. " S o i l S c i e n c e , , 73, 2, 105, (1952). ( 2 ) K L U T E A . , Some theoretical aspects of the flow of water in unsaturated soils. " Proc. Soil Science S o c i e t y " , 16, 2, 144. (1952). ( 3 ) S A R M A S . V . K . , Problem of partially saturated unsteady state of flow through porous media with special reference to groundwater recharge by spreading. " J . Sci. E n g g . R e s e a r c h " , I X . 1, 09, (1965). ( 4 ) M I C K L E Y H . S . , S H E R W O O D T . K . , R E E D C . E . , Applied Mathematics in chemical Engineering. .Me G r a w - I l i l l , 29(i, N e w Y o r k , 1957. N O M E N C L A T U R E D = d i f f u s i v i t y coefflcient. (cm2 s e c 1 ) I>„ = a v e r a g e v a l u e of t h e d i f f u s i v i t y coefficient over t h e whole r a n g e of m o i s t u r e c o n t e n t (cm2 sec- 1) K0 = slope of t h e p e r m e a b i l i t y vs m o i s t u r e c o n t e n t plot (cm s e c 1 ) K = p e r m e a b i l i t y coefflcient (cm s e c 1 ) L = d e p t h of p e r m e a b l e s t r a t u m (cm) M = m a s s flux of m o i s t u r e (gm) t = t i m e (sec) T = t i m e (dimensionless) V = v e l o c i t y of flow of w a t e r (cm s e c 1 ) 7J = d e p t h of p e n e t r a t i o n of w a t e r a t a n y i n s t a n t t (cm) ß = a flow p a r a m e t e r (cm2) f = p e n e t r a t i o n d e p t h (dimensionless) ft = t o t a l p r e s s u r e p o t e n t i a l (cm s e c 2 ) ip = c a p i l l a r y p o t e n t i a l (cm s e c 2 ) g = a c c e l e r a t i o n d u e t o g r a v i t y (cm s e c 2 ) o = m a s s d e n s i t y of w a t e r (gm) Qs = b u l k d e n s i t y of t h e m e d i u m on d r y w e i g h t basis (gm cm 3) Oo = m o i s t u r e c o n t e n t a t Z 0 f o r all t i m e ( g m / g m ) 0 — m o i s t u r e c o n t e n t a t a n y d e p t h Z (gm g m 1 ) V — v e c t o r o p e r a t o r = i + J + A {), j, K a r e 7>x 7)// 7>z u n i t v e c t o r s )