Non-linear laminar flow of fluid through a vertically 
stratified cylindrical porous aquifer 

S . K . J A I N ( * ) 

Received on May 24th, 1982 

A B S T R A C T 

T h i s n o t e p r e s e n t s the study of non-linear l a m i n a r flow of fluid through a 
v e r t i c a l l y s t r a f i f i e d cylindrical porous a q u i f e r in the steady state condition. The 
i n f l u e n c e of n o n - l i n e a r l a m i n a r flow on the discharge of the fluid and its 
d e p e n d e n c e on t h e related physical quantities, is discussed. In particular, results 
f o r n o n - l i n e a r l a m i n a r flow of fluid into a concentric well fully penetrating the 
h o m o g e n e o u s a q u i f e r h a v e been deduced and c o m p a r e d . 

R I A S S U N T O 

La p r e s e n t e nota r i g u a r d a lo studio del flusso l a m i n a r e non-lineare di un fluido 
a t t r a v e r s o u n " a q u i f e r " v e r t i c a l m e n t e stratificato poroso in condizioni stazionarie. 

L ' i n f l u e n z a del flusso l a m i n a r e non-lineare sul processo di discarica viene 
d i s c u s s a con la s u a d i p e n d e n z a dalle q u a n t i t à fisiche correlate. In particolare 
v e n g o n o d e d o t t i e c o m p a r a t i i risultati per il flusso non l a m i n a r e del fluido in un 
p o z z o c o n c e n t r i c o p e n e t r a n t e l ' a q u i t e r omogeneo. 

(*) 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). 



1 1 0 S.K. JAIN 

1. - I N T R O D U C T I O N 

T h e n a t u r e of flow mainly depends upon the fluid velocity a n d 
t h e s t r u c t u r a l c o n s t i t u t i o n of the porous m a t r i x through which it 
f l o w s . On t h e basis of fluid velocity, however, the flow (Bear, 1972) 
c a n b e c h a r a c t e r i z e d into three different regimes — laminar, 
n o n - l i n e a r 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 
c r i t e r i a 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 
p o r o u s m e d i a does not always justify the n a t u r a l flow of fluid to 
b e p u r e l y l a m i n a r a n d it a p p e a r s m o r e desirable to be either 
n o n - l i n e a r 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 , 1 9 5 3 , A n a n d k r i s h a n a n d V a r a d a r a j u l u ( 1 9 6 3 ) , Whright, 
1 9 6 8 , A h m a d 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 and 
U p a d h y a y ( 1 9 7 6 ) , U p a d h y a y ( 1 9 7 5 , 1 9 7 7 ) a n d others obtained 
s p e c i f i c s o l u t i o n s of certain non-linear flow problems. 

In t h e p r e s e n t p a p e r , we consider non-linear l a m i n a r steady 
s t a t e flow of fluid into an uncased well, concentrically established 
w i t h r e s p e c t to the c o n t o u r of intake a n d p e n e t r a t i n g fully the 
v e r t i c a l l y s t r a t i f i e d p o r o u s aquifer of finite thickness. It is found 
t h a t t h e flow p a t t e r n is c h a r a c t e r i z e d by two different zones, in 
w h i c h t h e d i s c h a r g e exhibits opposite c h a r a c t e r as regards its 
d e p e n d e n c e o n g r a i n size of the m e d i u m , viscosity of the fluid and 
r a d i u s of t h e well, is concerned. F u r t h e r , it is observed t h a t as the 
p e r m e a b i l i t y of the region in the vicinity of the well decreases or 
a s t h e region becomes n a r r o w e r , in b o t h the situations, the 
i n f l u e n c e of n o n - l i n e a r l a m i n a r flow is to increase the discharge. 

As a p a r t i c u l a r case, the results for non-linear l a m i n a r flow of 
fluid i n t o a concentric well fully p e n e t r a t i n g the homogeneous 
a q u i f e r h a v e been deduced a n d c o m p a r e d with those obtained by 
U p a d h y a y , 1977. 

2 . - E Q U A T I O N S O F F L U I D FLOW IN P O R O U S M E D I U M 

T h e l a w s governing l a m i n a r a n d non-linear l a m i n a r flow of 
f l u i d in p o r o u s m e d i a are (Poluvarinova-Kochina, 1 9 6 2 ) . 



N O N - L I N E A R LAMINAR FLOW, ECC. Ill 

a n d 

^ = av + bv2 [2] 
d s 

w h e r e v, k, ^ d e n o t e 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; c o n s t a n t s a a n d b, according to 
E n g e l u n d (1953) a r e 

a 2000 ii , 3 5 . . 
Y = — ¿ r - H = Jd [ 3 A ' B ] 

p a n d ĴL b e i n g d e n s i t y a n d viscosity of t h e fluid respectively and d 
t h e g r a i n size of t h e m e d i u m . 

B e s i d e r e l a t i o n s [1] a n d [2], we consider t h e expression for 
p r e s s u r e d i s t r i b u t i o n in t h e case of a concentrically placed circular 
w e l l f u l l y p e n e t r a t i n g a two-layered vertically stratified cylindrical 
p o r o u s a q u i f e r of thickness T in t h e f o r m (Poluvarinova-Kochina, 
1962) 

Q \i 
P = - Jfc. T 

log ( — ) + log ( - ) 
rw k 2 r o 

+ C , [ 4 1 

w h e r e C is t h e c o n s t a n t , to be d e t e r m i n e d w i t h the help of 
b o u n d a r y c o n d i t i o n s consistent w i t h t h e system. 

3 . - S T A T E M E N T O F T H E P R O B L E M 

I n t h e s t e a d y s t a t e condition, we consider t h e flow of fluid into 
a n u n c a s e d well of r a d i u s rw , concentrically established with 
r e s p e c t t o t h e c o n t o u r of i n t a k e of r a d i u s rc. It is a s s u m e d t h a t the 
w e l l is fully p e n e t r a t i n g t h e p o r o u s a q u i f e r of thickness T. The 
a q u i f e r is c o n s i d e r e d to be vertically s t r a t i f i e d into a d j a c e n t 
c y l i n d r i c a l r e g i o n s of p e r m e a b i l i t i e s k\ a n d ki s e p a r a t e d by the 
b o u n d a r y r = rQ. The p r e s s u r e at the c o n t o u r of well a n d t h a t of 
i n t a k e a r e p r e s c r i b e d as a n d pc respectively. 



112 S.K. JAIN 

In t h e p r e s e n t system, we a s s u m e the flow to be non-linear 
l a m i n a r r, w i t h i n a n a r r o w cylindrical zone r w > r « r, and 
l a m i n a r in t h e region r, < r =£ rc. Hence it becames obvious from 
p h y s i c a l c o n s i d e r a t i o n s t h a t k1 k.2. Let p, be the pressure at the 
t r a n s i t i o n b o u n d a r y r = r, [cf. fig. 1 nel testo]. 

Fig. 1 

T h e p r o b l e m is to e x a m i n e the influence of non-linear l a m i n a r 
flow on t h e d i s c h a r g e of fluid a n d its dependence on the related 
p h y s i c a l q u a n t i t i e s . 

4 . - S O L U T I O N 

F o r l a m i n a r zone, the b o u n d a r y conditions are 

p = pc at r = rc, 
[5] 

p = p, at r = r, 



N O N - L I N E A R LAMINAR F L O W , ECC. 1 1 3 

w h i c h on s u b s t i t u t i o n in [4] gives 

P, = Pc + 
Q |x 

2 -IT it, T E ( t t ) 
[6] 

As p = p g/z a n d Q = 2 -ir r 7v, equation [2] becomes 

d r 
= Pt aQ 

bQ2 

2-arT 
[7] 

I n t e g r a t i n g [7] a n d then evaluating the constant of integration 
w i t h t h e h e l p of b o u n d a r y conditions 

P = Pw at r = r w , 

p = p, at r = r„ 

w e o b t a i n 

[8] 

A = + P g 
Q Q 

2 TT T [9] 

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 Reynolds n u m b e r 
{; e = 0.07 a n d critical velocity vc is given by 

Q a . 
2 it r, T b 

[10] 

S i n c e at t h e b o u n d a r y of transition as given by [1] and [2] 
d s 

y i e l d t h e s a m e value, it follows from [10] that 

1.07 a ki P g = n. [ 1 1 J 

Using [11] in [6] a n d then e q u a t i n g the value of p , so obtained 
w i t h t h a t given by [9], we get 



114 S.K. JAIN 

Pc -P(0 _ aQ 
p g 

= ^ [ l o g ( - 5 - ) - 1 . 0 7 2 IT T L \ ro> ' 
k, 

, O G ( ^ ) L 
feQ2 

( ± _ ± ) r, / 

[12] 

C o m b i n i n g e q u a t i o n s [3a, b] a n d [10] with [12], we obtain 

= 8000 - A - f l o g ( - 5 - ) - 1.07 

T + 0 0 7 ( ¿ - - O : 

1131 

If w e a s s u m e the flow of fluid to be purely l a m i n a r in the 
e n t i r e regi o n r w r =£ rc, then the flow r a t e Qiam may be obtained 
f r o m [4] by using t h e corresponding b o u n d a r y conditions both at 
t h e c o n t o u r of intake a n d at the well. Thus 

Qla 
2 - i t A:, T (Pc-P<Q > 

log + log ( - A 
7 V ra 

a n d t h e r e f o r e , using [10] a n d [14], we o b t a i n 

Q = 8 5 6 0 - i l 
Qiam 'W 

H-2 ^ 
P d 2 ( P c ) 

[ * m + A . l o g / 

[14] 

[15] 

k2 ~ \ ri 

I n t r o d u c i n g dimensionless q u a n t i t y X a n d r a t i o Y such that 

Q x = P ipc - p( 0 ) Y = 
¿lam 

[16 a, b] 

a n d c o m b i n i n g [15] w i t h [13], we o b t a i n an implicit relation 



N O N - L I N E A R LAMINAR FLOW, ECC. 1 1 5 

1.07 X = 
XY 

1 - 1 . 0 7 l o g 
XY 

8560 2 

1.07 A _ log (JL_) + 0.07 — 0.07 
ki rM 8560 Z 

[17] 

w h e r e 

Z = H i + —— log 
I n view of physical considerations it may, however, be inferred 

f r o m [16 a] t h a t X > 0, hence equation [17] becomes 

1.07 Z = Y [ ( • - , . 07 £ log 
XY 

8560 Z 
[18] 

1.07 A 
k2 

log • K 
' w 

+ 0.07 
XY 

8560 Z 
0.07 

5 . - P A R T I C U L A R C A S E 

If = k2, t h a t is, the entire flow region is homogeneous with 
u n i f o r m p e r m e a b i l i t y , equation [18] reduces to 

1.07 log = 7 [ 1.07 log 0.07-

[19] 

— 0.07 log I - 0 . 0 7 ] , 
8560 log ( — ) ' 8560 log ( — ) J 

ra> r w 

w h i c h r e p r e s e n t s the non-linear l a m i n a r flow of fluid into a 
c o n c e n t r i c well fully p e n e t r a t i n g the homogeneous porous aquifer 
d i s c u s s e d by U p a d h y a y (1977). 



1 1 6 S . K . JAIN 

6 . - D I S C U S S I O N 

F r o m [16a] it is evident t h a t X d e p e n d s on the density of fluid, 
g r a i n size of the m e d i u m , pressure difference of the system, 
viscosity of the fluid a n d well radius. Since d a n d p. occur in 
h i g h e r p o w e r s in the expression for X, they highly affect the 
d i s c h a r g e . 

In ca se of n o n - l i n e a r l a m i n a r flow of fluid, the inertial forces 
a r e p r e d o m i n a n t which causes more resistance and consequently 
s m a l l e r d i s c h a r g e t h a t might be expected in l a m i n a r flow. Obvi-
o u s l y t h e effect of non-linear l a m i n a r flow can be observed only 
w h e n Y < 1. 

To get a definite idea of result [18], we take the system for 
w h i c h rc = 3 x 103 r , t h a t is, the radius of contour of intake is 
e q u a l t o 3 0 0 0 t i m e s t h e r a d i u s of t h e w e l l . A s s u m i n g 
rQ = 1.5 x 10

3 r , we analyse the effect of permeabilities of the 
s t r a t i f i e d zones on the discharge, in the p a r t i c u l a r cases. 

kt = 3 k2, 2 k2, k2. [20] 

T h e f i r s t t w o of which imply t h a t the permeability of the inner 
z o n e is respectively 3 a n d 2 t i m e s t h a t of the outer zone, where as 
t h e last v a l u e c o r r e s p o n d s to the case of u n i f o r m permeability. 

Using [18] a n d [20], the n u m e r i c a l values of Y are obtained in 
T a b l e I c o r r e s p o n d i n g to different values of X > 0. 

T A B L E I 

Values of Y; ra = 1.50 x 103 x r< 

ktlk2 X ... 10 10
2 103 10" 105 10" 107 10« 10' 

3.0 
2.0 
1.0 

0.2028 
0.2952 
0.9394 

0.2320 
0.3217 
0.9561 

0.2596 
0.4045 
0.9739 

0.3061 
0.4450 
0.9912 

0.3612 
0.5212 
0.9994 

0.4778 
0.6111 
0.9253 

0.4835 
0.5396 
0.5988 

0.2712 
0.3650 
0.2526 

0.1018 
0.0908 
0.0876 

T h e s e t a b u l a t e d values have been graphically represented in 
Fig. 2, a b s c i s s a a n d o r d i n a t e being X a n d Y respectively. It is seen 



N O N - L I N E A R LAMINAR F L O W , ECC. 1 1 7 

Fig. 2 

f r o m Fig. 2 t h a t in curve — I which corresponds to A:, = 3 k2, as X 
i n c r e a s e s , initially Y increases till it a t t a i n s a m a x i m u m value 
0.4970 ( c o r r e s p o n d i n g to X = 4.85 x 106); a f t e r w a r d s it decreases 
a s y m p t o t i c a l l y . Thus, in the f o r m e r region 0 < X =s 4.85 x 10" the 
d i s c h a r g e increases as X increases, that is, when the density of the 
f l u i d , g r a i n size of the m e d i u m a n d pressure difference of the 
s y s t e m increases, viscosity of the fluid a n d well radius decreases. 
In t h e l a t e r region X > 4.85 x 10" the influence of non-linear 
l a m i n a r flow is reversed. 

H e n c e , it m a y be concluded that in case of non-linear laminar 
f l o w , t h e flow p a t t e r n is characterised by two different zones in 
w h i c h d i s c h a r g e exhibits opposite c h a r a c t e r . 

7 . - C O M P A R I S I O N 

To e x a m i n e how the permeabilities of the stratified zones 
a f f e c t to discharge, we consider k¡ = 2k2 and the limiting case 



1 1 8 S . K . J A I N 

kt = k2. In Fig. 2 these cases are represented by curves II a n d III 
r e s p e c t i v e l y . It is observed t h a t the discharge increases as the 
p e r m e a b i l i t y of the region s u r r o u n d i n g the well decreases. 

N o w to investigate the effect of shrinking of the region 
of p e r m e a b i l i t y k t on discharge, we consider a p a r t i c u l a r case 
rQ = 1.0 x 10

3 x r0) . Taking k{ = 3 k2, 2 k2, values of Y are obtained 
in T a b l e II c o r r e s p o n d i n g to different values of X > 0. 

T A B L E I I 

Values of Y; r„ = 1.0 x 103 x r w 

kt I k2 x 10 10
2 103 10" 105 10" 107 10" 10' 

3.0 0.2240 0.2699 0.2990 0.3532 0.3992 0.4922 0.5257 0.2982 0.1042 
2.0 0.3415 0.3799 0.4236 0.4818 0.5574 0.6400 0.5219 0.2476 0.0930 

T h e s e values h a v e been graphically shown in Fig. 2 by dotted 
c u r v e s . It is i n f e r r e d t h a t discharge also increases when the region 
of p e r m e a b i l i t y /c, becomes n a r r o w e r . 

H e n c e , in general it may be concluded t h a t the shrinking of 
t h e z o n e s u r r o u n d i n g the well, as well as decrease, in its permea-
b i l i t y c a u s e s a larger discharge comparatively, which is an obvious 
p h y s i c a l p h e n o m e n o n . 



N O N - L I N E A R L A M I N A R F L O W , E C C . 1 1 9 

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