N o n - linear laminar flow of fluid into an open bottom well S . K . J A I N * Received on February 19th, 1982 A B S T R A C T In s t e a d y s t a t e condition, non - linear l a m i n a r flow of fluid into an open b o t t o m well j u s t p e n e t r a t i n g the semi-infinite porous aquifer is considered. The i n f l u e n c e of non-linear l a m i n a r flow on discharge and its dependance on related p h y s i c a l q u a n t i t i e s is examined. It is found that an open bottom well actually b e h a v e s like a h e m i s p h e r i c a l well, which is an obvious practical phenomenon. R I A S S U N T O Si c o n s i d e r a il flusso l a m i n a r e non-lineare di un fluido in un pozzo a fondo a p e r t o c h e p e n e t r i un acquifero poroso semi infinito. Si e s a m i n a poi l'influenza del flusso laminare non-lineare sulla discarica e la s u a d i p e n d e n z a da q u a n t i t à fisiche correlate. Si trova che un pozzo a fondo aperto si c o m p o r t a c o m e un pozzo emisferico che è un fenomeno pratico ovvio. 1 . I N T R O D U C T I O N In t h e s t u d y of flow through porous media, there exists a large n u m b e r of investigations seeking steady state solutions of Darcy's * D e p a r t m e n t of Mathematics, Rivers S t a t e College of Education, Rumuolu- m e n i , P.M.B. 5047, PORT HARCOURT (NIGERIA). 8 6 S.K. JAIN l a w w i t h respect to certain applications. One of the most impor- t a n t a p p l i c a t i o n is the study of flow of fluid from s t r a t a into wells. M a n y i n v e s t i g a t o r s obtained a series of such solutions (cf. [1], [2], [3]). T h e well p e n e t r a t i n g fully the fluid bearing s t r a t a of finite t h i c k n e s s is t e r m e d as fully p e n e t r a t i n g well otherwise partially p e n e t r a t i n g well. But in a semi-infinite s t r a t u m , a well is always c o n s i d e r e d to be p a r t i a l l y penetrating. In such cases, the flow in t h e re g ion of sa n d not p e n e t r a t e d by the well will have an u p w a r d c o m p o n e n t of velocity tending to bring the fluid into the well, w h i l e t h e flow in the u p p e r portion of the sand will mainly be r a d i a l as it will have a small negligible vertical component. The m o s t i n t e r e s t i n g limiting case of a partially penetrating well is an ' o p e n b o t t o m well', which corresponds to a well just tapping a u n i f o r m s a n d of great thickness [1]. Beside t h e n a t u r e of a well the other i m p o r t a n t aspect is the t y p e of flow which mainly depends upon the fluid velocity and the s t r u c t u r a l constitution of the porous m a t r i x through which it f l o w s . On the basis of fluid velocity, however, the flow [3] can be c h a r a c t e r i s e d into three different regimes — l a m i n a r , nonlinear l a m i n a r a n d t u r b u l e n t , Reynolds n u m b e r being the criteria for s u c h d e m a r c a t i o n . But, the intricacy of the n a t u r e of porous media d o e s n o t a l w a y s justify the n a t u r a l flow of fluid to be purely l a m i n a r a n d it a p p e a r s more desirable to be either non-linear l a m i n a r or t u r b u l e n t . Consequently, UCHIDA, 1952, E N G E L U N D , 1953, A N A N D K R I S H A N a n d VARADARAJULU, 1 9 6 3 , W R I G H T , 1 9 6 8 , AHMAD a n d S U N A D A , 1 9 6 9 , K H A N a n d RAZA, 1 9 7 2 , JAIN a n d UPADHYAY, 1 9 7 6 , UPADHYAY, 1977a b, a n d others obtained solutions of certain specific n o n - l i n e a r l a m i n a r flow problems. In the p r e s e n t paper, we consider non-linear l a m i n a r radially s y m m e t r i c a l flow of an incompressible fluid into an open bottom well in s t e a d y s t a t e condition. The influence of non-linear laminar f l o w on d i s c h a r g e a n d its d e p e n d a n c e on related physical q u a n t i t i e s is e x a m i n e d . It is found that an open bottom well a c t u a l l y b e h a v e s like a hemispherical well, as in the former case w h e r e t h e well just p e n e t r a t e s the u p p e r surface of semi-infinite a q u i f e r , a n h e m i s p h e r i c a l cavity is automatically formed to pro- v i d e a s u r f a c e for discharge. This is an obvious practical phenome- n o n . NON-LINEAR LAMINAR FLOW, ECC. 8 7 2 . B A S I C EQUATIONS OF FLUID FLOW IN POROUS MEDIUM T h e D a r c y ' s law governing the l a m i n a r flow of fluid in porous m e d i a is [2] v = K J " , [1] d h v, k a n d —— d e n o t e the seepage velocity, seepage coefficient and d s h y d r a u l i c g r a d i e n t respectively; flow being in the opposite direc- t i o n of i n c r e a s i n g h. I n c a s e of l a m i n a r flow, the total head Hc at an infinitely large d i s t a n c e f r o m t h e axis of the circular perforation of radius rw in t h e p l a n e i m p e r v i o u s boundary of semi-infinite homogeneous a q u i f e r is H c = « „ + P I w h e r e Q d e n o t e s the flow r a t e and Ht0 the head at the well surface. Besides, relations [1] a n d [2], the law for non-linear laminar flow is [2] dh = av + bv\ [3] d s a a n d b b e i n g c o n s t a n t s , which according to Engelund are 2 0 0 0 J L b , Pgd*' gd' where P and |jl being density and viscosity of the fluid respec- tively and d the grain size of the medium. 8 8 S.K. JAIN Fig. 1 3 . S T A T E M E N T OF T H E PROBLEM In s t e a d y s t a t e condition, we consider radially symmetrical flow of a n incompressible fluid into an open b o t t o m well, which is a c i r c u l a r p e r f o r a t i o n w i t h radius r w of an impervious boundary of s e m i - i n f i n i t e p o r o u s aquifer (Fig. 1). The aquifer is assumed to be h o m o g e n e o u s a n d isotropic. The pressure at the contour of well a n d a t t h e c o n t o u r of intake are prescribed as pw and pc respec- t i v e l y . Let r be the r a d i a l distance measured f r o m the axis of well. It is a s s u m e d t h a t the flow is (i) non-linear l a m i n a r within the c i r c u l a r zone T̂J ^ Y VT, and (ii) l a m i n a r in the region r, < Y ^ oo. Let p, be t h e p r e s s u r e at the transition boundary r = r,. T h e p r o b l e m is to examine the influence of non-linear l a m i n a r f l o w on discharge, its dependance on the grain size of the m e d i u m a n d t h e viscosity of the fluid. NON-LINEAR LAMINAR FLOW, ECC. 8 9 4 . S O L U T I O N S i n c e p = Pgh, we write [2] as QPg _ O P g Pc - Pos = 4 K r„ I d p = - = C - 4 K Q Pg 4 K Q P g 4 K r ' ( 1 - 1 ). [ 5 ] [6] [ 7 ] w h e r e C is the c o n s t a n t of integration determinable by boundary c o n d i t i o n s . Thus, equation [7] describes the pressure distribution p f o r a n y a r b i t r a r y r a d i u s r (rw r ») in the system for which e q u a t i o n [2] holds good. H e n c e , f r o m [7] the expression for pressure distribution at a r a d i a l d i s t a n c e r in the l a m i n a r zone r, < r is obtainable in t h e f o r m P, = Pc — Q P g 4 K r , For r a d i a l l y s y m m e t r i c a l flow, equation [1] becomes K dp [8] V = Jg dr [ 9 ] S u b s t i t u t i n g t h e value of d p d r from [7], equation [9] reduces to V = 4 r 2 [ 1 0 ] w h e r e v is t h e seepage velocity at a radial distance r. Thus, v is i n d e p e n d e n t of the well radius rw and z — the vertical coordinate. 9 0 S.K. JAIN Using p = P gh in equation [3) and then combining it with r e l a t i o n [10], the expression for pressure distribution in the n o n - l i n e a r l a m i n a r zone is obtainable in the form dp n / aQ b Q2 \ —r— = pg —— + —— , « r a t r = rto. [ 1 2 ] p, at r ~ rt, Í aQ ( \ 1 \ b Q1 / 1 1 \] [131 At t h e b o u n d a r y of transition f r o m l a m i n a r to non-linear l a m i n a r flow, the relation between critical Reynold's n u m b e r = 0.07 a n d critical velocity vc is given by " " t V - T * - 1141 d h S i n c e a t t h i s b o u n d a r y — — a s given [1] a n d [3] yield the same d s v a l u e , it follows f r o m [14] that 1.07 a AT = 1 [15] U s i n g [15] in [8] a n d then equating it to [13], we get \±Q ( 1 + 0 . 0 7 1 ) + b-91 (1 - 1)1 | 1 6 , Pg L 4 \ r w r, ) 48 I tf. r > ) j 1 1 0 1 NON-LINEAR LAMINAR FLOW, ECC. 9 1 C o m b i n i n g e q u a t i o n s [4 a, b] a n d [14] w i t h [16], we obtain ?d>(pc-pj _ l^2 = 8 0 0 0 ( - i L ) [ - Ü - + 0 . 0 2 3 3 ( - Ü - ) > + 0 . 0 4 6 7 1 [ 1 7 ] rw L rw r(ii J w h e r e r, = y / U J 1 6 (JL If w e a s s u m e p u r e l y l a m i n a r flow in the region r t h e n t h e f l o w r a t e Q]am is o b t a i n a b l e f r o m [7] in the form n 4 K R r Ï Q | a m = T V ^ " [18] T [ 1 9 ] H e n c e f r o m [14] a n d [19], we o b t a i n the r a t i o 0 Qlan = 8 5 6 0 M-2 ^ P d } (p, - p w l (-Ï—)2 [ 2 0 ] I n t r o d u c i n g d i m e n s i o n l e s s q u a n t i t y X a n d r a t i o 7 such t h a t [21 a, b] x = Pd>(pc-pJ F = 1¿2 'lam a n d c o m b i n i n g [20] w i t h [17], we o b t a i n an implicit relation X = 8000 I + 0 . 0 2 3 3 ( - ^ - ) 2 + 0 . 0 4 6 7 ( — ) V 4 1 ( 2 2 ] L 8 5 6 0 8 5 6 0 8 5 6 0 J T h i s e x p r e s s i o n is the s a m e as t h a t o b t a i n e d by the a u t h o r [ J A I N , 1 9 8 1 ] , w h i c h describes the n o n - l i n e a r l a m i n a r flow into a h e m i s p h e r i c a l w e l l . 9 2 S.K. JAIN 5 . C O N C L U S I O N It m a y t h u s be concluded that an open b o t t o m well actually b e h a v e s like a h e m i s p h e r i c a l well, as in the f o r m e r case where the well j u s t p e n e t r a t e s the u p p e r surface of a semi-infinite porous a q u i f e r , a h e m i s p h e r i c a l cavity is automatically formed to provide a s u r f a c e for discharge. This is an obvious practical p h e n o m e n o n , a s o u r conclusion based on theoretical discussion justifies the s t a t e m e n t : "... t h a t is a practical case when well just taps the s a n d , t h e well s u r f a c e is really a hemisphere'.'. N O N - L I N E A R L A M I N A R F L O W , E C C . 9 3 R E F E R E N C E S A H M A D , N. a n d S U N A D A , D.K., 1969 - Non - Linear laminar flow in porous media, "J. Hy. D n . " , Proc. A S C E , 95, p p . 1847-1857. A N A N D K R I S H A N , M. VARADARA-JULU , G.H. 1963 - Laminar and turbulent flow of water through sand. " J . Soil Mech.", Proc. 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