N o n - l i n e a r l a m i n a r f l o w i n t o e c c e n t r i c a l l y p l a c e d w e l l K . K . U P A D H Y A Y ( * ) R e c e i v e d o n S e p t e m b e r 2 n d , 1 9 7 5 SUMMARY. — I n s t e a d y s t a t e c o n d i t i o n , n o n - l i n e a r l a m i n a r flow of f l u i d i n t o a n e c c e n t r i c a l l y p l a c e d w e l l is c o n s i d e r e d . I t s i n f l u e n c e o n t l i c d i s c h a r g e a n d t h e d e p e n d e n c e o n r e l a t e d p h y s i c a l q u a n t i t i e s is i n v e s t i g a t e d . I t is o b s e r v e d t h a t a s t h e w e l l a p p r o a c h e s t o w a r d s t h e c o n t o u r of i n t a k e , t h e d i s c h a r g e i n c r e a s e s , w h i c h is a n o b v i o u s r e s u l t c o n s i s t e n t w i t h t h a t o b t a i n e d b y P o l u b a r i n o v a - K o c h i n a i n c a s e of l a m i n a r flow. A s a p a r t i c u l a r c a s e , r e s u l t f o r c o n c e n t r i c w e l l h a s a l s o b e e n d e d u c e d . RIASSUNTO. — V i e n e p r e s o i n c o n s i d e r a z i o n e u n flusso d i fluido l a m i - n a r e n o n - l i n e a r e — i n c o n d i z i o n i d i s t a t o s t a z i o n a r i o — d e n t r o u n p o z z o d i s p o s t o e c c e n t r i c a m e n t e . Si è i n o l t r e s t u d i a t a s i a l a s u a i n f l u e n z a s u l g e t t o c h e l a d i p e n d e n z a d a l l e r e l a t i v e q u a n t i t à fisiche. È s t a t o o s s e r v a t o c h e , c o m e ci si a v v i c i n a a l c o n t o r n o d e l l o s b o c c o , il flusso a u m e n t a , il clie è u n r i s u l t a t o o v v i o , i n a c c o r d o c o n q u a n t o o t t e n u t o d a i r i c e r c a t o r i P o l u b a r i n o v a e K o c h i n a , n e l c a s o d i u n flusso l a m i n a r e . I l r i s u l t a t o r e l a t i v o a d u n p o z z o c o n c e n t r i c o n o n ò q u i n d i c h e u n c a s o p a r t i c o l a r e d e l p r o b l e m a a f f r o n t a t o i n q u e s t a n o t a . 1 . - I N T R O D U C T I O N The intricacy in the n a t u r e of porous media does not always j u s t i f y t h e n a t u r a l flow of fluid through it to be purely laminar. However, it appears more justifiable to consider the liow through porous media to be either non-linear laminar or t u r b u l e n t (J). Consequently J a i n and U p a d h y a y (2), Elenbaas and K a t z (3), Engelund (4) obtained specific solutions of some non-linear laminar and t u r b u l e n t flow problems. I n the present paper, we consider the non-linear laminar steady state flow of fluid into an eccentrically places well fully penetrating ( * ) 7 9 , Mill R o a d , D e w a s , ( M . P . ) I N D I A 3 1 2 K. K. U P A D H Y A Y t h e porous aquifer. I t is found t h a t t h e flow p a t t e r n is characterised b y t w o different zones, in which discharge exhibits opposite c h a r a c t e r as regards its dependence on grain size of t h e m e d i u m , viscosity of t h e fluid and r a d i u s of t h e well. F u r t h e r , it is observed t h a t as t h e well approaches t h e contour of i n t a k e , t h e discharge increases a b r u p t l y as compared t o t h a t into a concentrically placed well, which is obvious f r o m physical considerations. The results for a concentric well h a v e been deduced and compared with those obtained b y U p a d h y a y (5). 2 . - E Q U A T I O N S O F F L U I D F L O W I N P O R O U S M E D I U M The D a r c y ' s law governing t h e l a m i n a r flow of fluid i n porous m e d i a is Ah where v, k and —— denote t h e seepage velocity, seepage coefficient d o and h y d r a u l i c g r a d i e n t respectively; flow being in t h e opposite direction of increasing h. I n case of an eccentrically placed circular well fullly p e n e t r a t i n g t h e cylindrical s t r a t u m (radius R) of u n i t thickness, t h e pressure p a t a n y point with complex coordinate z is obtained in t h e f o r m (6) Q/j. , R {z—zi) V = o 7 ~ l o 8 ' F T + 0 2 ] 2nk0 (it2—zz i) where Q, k0 and ¡i represent flow r a t e , permeability of t h e m e d i u m a n d viscosity of t h e fluid respectively; zi denotes t h e centre of well and zi is t h e corresponding inverse point. The c o n s t a n t G is t o be d e t e r m i n e d by b o u n d a r y conditions. Besides relations [1] and [2], t h e law for non-linear l a m i n a r flow is(°) ~ = av + bv2, [ 3 ] where a and b are constants. According t o Engelund (4) N O N - L I N E A R LAMINAR FLOW INTO ECCENTRICALLY PLACED W E L L 3 1 3 2 0 0 0 a a = ; , ggd2 » = ^ gd M i M . q and d being density of t h e fluid and grain size of t h e m e d i u m . 3 . - F O R M U L A T I O N O F T H E P R O B L E M I n s t e a d y s t a t e condition, we consider t h e flow of fluid into an uncased circular cylindrical well of radius r w eccentrically established a t a distance Ri f r o m t h e centre of t h e contour of intake. I t is assumed t h a t t h e well is completely p e n e t r a t i n g t h e porous aquifer of thickness T. The aquifer is considered to be homogeneous and isotropic bounded b y horizontal impervious layers. The pressure a t t h e contour of well and a t t h e contour of i n t a k e are prescribed as 'p w and pc respectively. L e t r be t h e radial distance measured f r o m t h e axis of intake. As t h e effect of non-linear laminar or t u r b u l e n t flow is observed t o be appreciable event if such flow is restricted to a comparatively n a r r o w zone (4), we consider t h e flow t o be non-linear laminar within a narrow cylindrical zone of radius rt surrounding t h e well and laminar beyond this zone. L e t t h e pressure a t t h e transition b o u n d a r y is pt [Fig. 1]. K g . l I n t h e present situation, we h a v e («,) = (»,) = Ri 3 1 4 K. K. U P A D H Y A Y Therefore, expression for pressure distribution in t h e l a m i n a r zone (of. [2]) t a k e s t h e f o r m Qf* , R (r—Ri) . _ rK1 P = L O G - Í M - T + C [ 5 ] The problem is to examine t h e influence of non-linear l a m i n a r flow 011 discharge of fluid and its dependence on t h e related physical q u a n t i t i e s . 4 . - S O L U T I O N As in t h e vicinity of well t h e lines of equal pressure are closed to circles, therefore, we assume t h e contour of well as one of t h e isobars close t o t h e circle of r a d i u s rw (c). Along t h e b o u n d a r y of transition, which is close t o t h e contour of well, pressure pt m a y be obtained f r o m [5] b y using t h e b o u n d a r y conditions p = pc a t r = R, p = pt a t r = Ri + rh (rt « Ri < R). [6] H e n c e ^Z4 i R r> m ~2nlcaT l 0 g R^R^ [ 7 ] Since p = qgli, pressure distribution in t h e non-linear laminar zone is obtainable f r o m [3] as — ~ = av + bv* [8] gg dr I n general, discharge Q f r o m a n y cylindrical surface of radius A and height T is Q = 2 n I Tv. [9] Consequently, in this situation r = R1 + A, rw < A < r,, [ 1 0 ] N O N - L I N E A R LAMINAR FLOW INTO ECCENTRICALLY PLACED W E L L 3 1 5 i t follows f r o m [8], [9] and [10] t h a t f t Ri-\-rt Q / - « J (£ bQ1 , , Pu Ri+r, [11] i.e. pt = p„ Q9 aQ l o 2 J I T \ - R I + R Ri+rt\ , bQ2 •in2T2 \i?i+r,„ Ri+rt [12] A t t h e b o u n d a r y of t r a n s i t i o n f r o m l a m i n a r t o non-linear l a m i n a r flow, t h e r e l a t i o n b e t w e e n critical R e y n o l d ' s n u m b e r f c = 0.07 a n d critical velocity vc is given b y (") Q t « [ 1 3 ] dli w h e r e — as given b y [1] a n d [3] yield t h e s a m e v a l u e . Accordingly Vc m i b \ — = a vc ( 1 + — Vc), l{> a or, —- = 1.07 a. k Since k = — , i t follows t h a t i" JL ko = 1.07 agg [14] Using [14] in [7] and t h e n c o m p a r i n g w i t h [12], we g e t Pc—p„ ~qìt aQ_ 2tzT log + Bi+n Ri+rw bQ2 4n2T2 \Ri+r — 1.07 log Rr, R2—Ri2 1 Ri+rt [15] C o m b i n i n g e q u a t i o n s [4]i [4], a n d [13] w i t h [15], we o b t a i n gd3 {pc—pw) H 2 r w 8000 rt log + 0.07 r, Ri+r, Ri+r„ — 1.07 lop Rrt R2-R,2 Í + ( » ' i — r , c ) (Ri+r,c) (Ri+rt) [ 1 0 ] 3 1 6 K. K . U P A D H Y A Y If we a s s u m e p u r e l y laminar flow in t h e entire flow regio n t h e n t h e flow r a t e Qu,m m a y be obtained f r o m [5] b y using t h e corresponding b o u n d a r y conditions a t t h e well a n d t h e counter of i n t a k e . H e n c e Q lam — 2 nkT Q9 iPo — Pw) lOR R2—R i2 R rw [17] Therefore, f r o m [13] and [17], we obtain t h e ratio Q = 8 5 6 0 ^ Q' rw [ q(P {pc — p,„) I n t r o d u c i n g dimensionless q u a n t i t y X and r a t i o Y such t h a t gd3 (pc—pw) X = j u 2 r w Q Qlan [19] i [19], and combining [18] with [16], we obtain an implicit relation 1.07 X XY Z i I B i , A " r l o g t + liEooz •log ( f + 1 ) + — 1.07 log X Y 8560 Z + 1.07 Z + 0.07 XY XY 8560 Z [20] 8560 Z (Ri \ / Ri I I r,o + j1 rw 8560 Z where Z = log R2 — Ri R r,0 I t m a y be inferred f r o m [19]i t h a t t h e value of X which is possible f r o m physical considerations is X > 0 , hence equation [20] becomes g N O N - L I N E A R LAMINAR FLOW INTO ECCENTRICALLY PLACED W E L L 3 1 7 1.07 Z = Y 0 1 ra 8560 Z — 1.07 log X Y 8560 Z l o g g + l ] + + 1.07 Z + 0.07 X Y 8560 Z X Y 8560 Z — 1 Ih r w + XY 8560 Z [21] 5 . - P A R T I C U L A R C A S E If E i = 0 t h a t is, when the well is established concentrically with respect t o t h e contour of intake, equation [21] reduces t o 1.07 log R = Y 0.07 X Y 8560 log R + X Y — 0.07 log I Q _ n ( R 8560 log - \ r w 1.07 log 0.07 [22] which corresponds t o t h e non-linear l a m i n a r flow of fluid into a fully p e n e t r a t i n g concentric well discussed b y U p a d h y a y (5). 6 . - D I S C U S S I O N F r o m [19]i, it is evident t h a t X depends on t h e density of t h e fluid, grain size of t h e medium, ju'essure difference of t h e system, viscosity of t h e fluid and well radius. Since d and ¡i occur in higher powers in expression for X, t h e y highly affect t h e discharge. Moreover, f r o m physical considerations it is obvious t h a t X and Y are b o t h positive. 3 1 8 K . K . U P A D H Y A Y Now, t o get tlie definite idea of t h e r e s u l t [21], we t a k e — = 3 • 103 t h a t is t h e r a d i u s of contour of i n t a k e is 3000 times rw t h e radius of t h e well. Considering — = 102, t h e numerical values " r w of Y are obtained corresponding t o different values of X > 0 and h a v e been graphically p l o t t e d in t h e f o r m of curve - I [Fig. 2]. I t is seen f r o m c u r v a - I t h a t as X increases, initially Y increases till it a t t a i n s a m a x i m u m value 2.16 corresponding to X = 1.2 691 -107, a f t e r w a r d s it descresases asymptotically. Thus in t h e f o r m e r region O 1 . 2 6 9 1 • 107 t h e influence of non-linear l a m i n a r flow is reversed. Thus, it m a y be concluded t h a t in case of non-linear l a m i n a r flow, t h e flow p a t t e r n is characterised by t w o different zones in which, discharge exhibits opposite character. N O N - L I N E A R L A M I N A R F L O W I N T O E C C E N T R I C A L L Y P L A C E D W E L L 3 1 9 7 . - C O M P A R I S I O N To examine as t o how t h e position of t h e well affects t h e dischagre into it, we consider t h e cases — = 0 and — = 10. These cases T w Tw h a v e been graphically respresented by d o t t e d curve and c u r v e - I I respectively in Fig. 2. Hence, it is inferred t h a t as t h e well approaches t h e contour of i n t a k e t h e discharge increases a b r u p t l y as compared to t h a t into a well concentrically established with respect to t h e contour of i n t a k e . F r o m physical consideration, the result is quite obvious and consistent with t h a t obtained b y Poluberinova-Kochina (6) in case of l a m i n a r flow. A C K N O W L E D G E M E N T The a u t h o r gratefully acknowledges t h e valuable guidance of D r . S. K . J a i n , D e p a r t m e n t of Applied Mathematics, G.S. I n s t i t u t e of Technology and Science, I n d o r e in the p r e p a r a t i o n of this p a p e r . H e is also v e r y m u c h t h a n k f u l to t h e referee for his valuable suggestions for t h e i m p r o v e m e n t of t h e p a p e r . R E F E R E N C E S P ) BEAR, J . , 1 9 7 2 . - Dynamics of fluids in porous media. A m e r i c a n E l s e v i e r P u b l i s h i n g C o m p a n y I n c . , N e w Y o r k . ( 2 ) J A I N , S. K . , U P A D I I Y A Y , K . K . , 1 9 7 5 . - Non-linear laminar flow of fluid into a partially penetrating well. " G e r l a n d s B e i t r i i g e Z u r G e o p h y s i k " (in p r e s s ) . ( 3 ) E L E N B A A S , J . R . , K A T Z , D . L . , 1 9 4 8 . - A radial turbulent flow formula. " T r a n s . A I M E " , 1 7 4 , p . 25. ( 4 ) E N G E L U N D , F . , 1 9 5 3 . - On laminar and turbulent flow of ground water in homogeneous sand. " T r a n s . D a n . A c a d . T e c h . S c i " , 3 , p p . 1 - 1 0 5 . ( 5 ) U P A D I I Y A Y , K . K . , 1 9 7 6 . - Non-linear laminar flow of fluid into a fully penetrating cylindrical well. " I n d . J . T l i e o . P l i y s . " (in p r e s s ) . (6) POLUBARINOVA-KOCIIINA, P . Y A . , 1952. - Theory of Ground water Mo- vement. P r i n c e t o n U n i v e r s i t y , P r e s s , P r i n c e t o n , 3 6 4 , p . 17.