Non - linear laminar flow of fluid into an open bottom well

S . K . JAIN *

Received on February 19th, 1982

A B S T R A C T

In steady state condition, non - linear laminar flow of fluid into an open bo t tom well just penetrating the semi-infinite porous aquifer is considered. The influence of non-linear laminar flow on discharge and its dependance on related physical quanti t ies is examined. It is found that an open bottom well actually behaves like a hemispherical well, which is an obvious practical phenomenon.

R I A S S U N T O

Si considera il flusso laminare non-lineare di un fluido in un pozzo a fondo ape r to che penetri un acquifero poroso semi infinito.

Si esamina poi l'influenza del flusso laminare non-lineare sulla discarica e la sua dipendenza da quanti tà fisiche correlate. Si trova che un pozzo a fondo aperto si compor ta come un pozzo emisferico che è un fenomeno pratico ovvio.

1 . INTRODUCTION

In the study of flow through porous media, there exists a large n u m b e r of investigations seeking steady state solutions of Darcy's

* Department of Mathematics, Rivers State College of Education, Rumuolu-meni, P.M.B. 5047, PORT HARCOURT (NIGERIA).

8 6 S.K. JAIN

law with respect to certain applications. One of the most impor-tant application is the study of flow of fluid from strata into wells. Many investigators obtained a series of such solutions (cf. [1], [2], [3]). The well penetrating fully the fluid bearing strata of finite thickness is termed as fully penetrating well otherwise partially penetra t ing well. But in a semi-infinite stratum, a well is always considered to be partially penetrating. In such cases, the flow in the region of sand not penetrated by the well will have an upward component of velocity tending to bring the fluid into the well, while the flow in the upper portion of the sand will mainly be radial as it will have a small negligible vertical component. The most interesting limiting case of a partially penetrating well is an 'open bottom well', which corresponds to a well just tapping a uni form sand of great thickness [1].

Beside the nature of a well the other important aspect is the type of flow which mainly depends upon the fluid velocity and the s t ructura l constitution of the porous matrix through which it flows. On the basis of fluid velocity, however, the flow [3] can be characterised into three different regimes — laminar, nonlinear l aminar and turbulent, Reynolds number being the criteria for such demarcation. But, the intricacy of the nature of porous media does not always justify the natural flow of fluid to be purely l amina r and it appears more desirable to be either non-linear l amina r or turbulent. Consequently, UCHIDA, 1952, ENGELUND, 1953, ANANDKRISHAN a n d VARADARAJULU, 1 9 6 3 , WRIGHT, 1 9 6 8 , AHMAD a n d

SUNADA, 1 9 6 9 , KHAN a n d RAZA, 1 9 7 2 , JAIN a n d UPADHYAY, 1 9 7 6 ,

UPADHYAY, 1977a b, and others obtained solutions of certain specific non-linear laminar flow problems.

In the present paper, we consider non-linear laminar radially symmetr ical flow of an incompressible fluid into an open bottom well in steady state condition. The influence of non-linear laminar flow on discharge and its dependance on related physical quant i t ies is examined. It is found that an open bottom well actually behaves like a hemispherical well, as in the former case where the well just penetrates the upper surface of semi-infinite aquifer , an hemispherical cavity is automatically formed to pro-vide a surface for discharge. This is an obvious practical phenome-non.

NON-LINEAR LAMINAR FLOW, ECC. 8 7

2 . B A S I C EQUATIONS OF FLUID FLOW IN POROUS MEDIUM

The Darcy's law governing the laminar flow of fluid in porous media is [2]

v = K J " , [1]

d h v, k and —— denote the seepage velocity, seepage coefficient and ds

hydraul ic gradient respectively; flow being in the opposite direc-tion of increasing h.

In case of laminar flow, the total head Hc at an infinitely large dis tance from the axis of the circular perforation of radius rw in the plane impervious boundary of semi-infinite homogeneous aqui fer is

H c = « „ + PI

where Q denotes the flow rate and Ht0 the head at the well surface. Besides, relations [1] and [2], the law for non-linear laminar

flow is [2]

dh = av + bv\ [3] ds

a and b being constants, 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 . STATEMENT OF THE PROBLEM

In steady state condition, we consider radially symmetrical flow of an incompressible fluid into an open bottom well, which is a circular perforation with radius r w of an impervious boundary of semi-infinite porous aquifer (Fig. 1). The aquifer is assumed to be homogeneous and isotropic. The pressure at the contour of well and at the contour of intake are prescribed as pw and pc respec-tively.

Let r be the radial distance measured from the axis of well. It is assumed that the flow is (i) non-linear laminar within the c i rcular zone T̂J ^ Y VT, and (ii) laminar in the region r, < Y ^ oo. Let p, be the pressure at the transition boundary r = r,.

The problem is to examine the influence of non-linear laminar flow on discharge, its dependance on the grain size of the medium and the viscosity of the fluid.

NON-LINEAR LAMINAR FLOW, ECC. 8 9

4 . SOLUTION

Since p = Pgh, we write [2] as

QPg _ O P g Pc - Pos = 4 K r„

I dp = -

= C -

4 K

Q Pg

4 K

Q P g 4 K r '

( 1 - 1 ). [ 5 ]

[6]

[ 7 ]

where C is the constant of integration determinable by boundary condit ions. Thus, equation [7] describes the pressure distribution p for any arbi t rary radius r (rw r ») in the system for which equat ion [2] holds good.

Hence, from [7] the expression for pressure distribution at a radia l distance r in the laminar zone r, < r is obtainable in the form

P, = Pc — Q P g 4 Kr,

For radially symmetrical flow, equation [1] becomes

K dp

[8]

V = Jg dr [ 9 ]

Subst i tu t ing the value of dp dr

from [7], equation [9] reduces to

V = 4 r2 [10]

where v is the seepage velocity at a radial distance r. Thus, v is independent 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 relat ion [10], the expression for pressure distribution in the non-linear laminar zone is obtainable in the form

dp n / aQ b Q2 \ —r— = pg —— + —— , « r<r, 11

d r 8 U r2 16 r 4 / w 1 1

Integrating [11] and evaluating the constant of integration wi th the help of boundary conditions

P =

we obtain

Po> a t r = rto. [12]

p, at r ~ rt,

Í aQ ( \ 1 \ b Q1 / 1 1 \]

[131

At the boundary of transition from laminar to non-linear l aminar flow, the relation between critical Reynold's number = 0.07 and critical velocity vc is given by

" " t V - T * - 1141

d h Since at this boundary ——as given [1] and [3] yield the same ds

value, it follows from [14] that

1.07 a AT = 1 [15]

Using [15] in [8] and then equating it to [13], we get

\±Q ( 1 + 0 . 0 7 1 ) + b-91 (1 - 1)1 | 1 6 , Pg L 4 \ rw r, ) 48 I tf. r > ) j 1 1 0 1

NON-LINEAR LAMINAR FLOW, ECC. 91

Combin ing equat ions [4 a, b] and [14] with [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 [17]

rw L rw r(ii J

w h e r e

r, = y / U J 1 6 (JL

If we assume purely laminar flow in the region r then the flow rate Q]am is obtainable from [7] in the form

n 4 K R r Ï Q|am = T V ^ "

[18]

T

[ 1 9 ]

Hence f rom [14] and [19], we obtain the ratio

0 Qlan

= 8 5 6 0 M-2 ^

P d } (p, - pwl

(-Ï—)2 [20]

In t roducing dimensionless quanti ty X a n d ratio 7 such that

[21 a, b] x = Pd>(pc-pJ F =

1¿2 'lam

a n d combin ing [20] with [17], we obtain 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

This expression is the same as that obtained by the author [ JAIN, 1 9 8 1 ] , which describes the non-linear laminar flow into a hemispher ica l well.

9 2 S.K. JAIN

5 . CONCLUSION

It may thus be concluded that an open bottom well actually behaves like a hemispherical well, as in the former case where the well just penetrates the upper surface of a semi-infinite porous aquifer, a hemispherical cavity is automatically formed to provide a surface for discharge. This is an obvious practical phenomenon, as our conclusion based on theoretical discussion justifies the s ta tement : "... that is a practical case when well just taps the sand, the well surface is really a hemisphere'.'.

N O N - L I N E A R L A M I N A R F L O W , ECC. 9 3

R E F E R E N C E S

A H M A D , N. and SUNADA, D.K., 1969 - Non - Linear laminar flow in porous media, "J.

Hy. Dn.", Proc. A S C E , 95, pp. 1847-1857.

ANANDKRISHAN , M. VARADARA-JULU , G.H. 1963 - Laminar and turbulent flow of water through sand. "J . Soil Mech.", Proc. ASCE, 89, 1, p. 15.

BEAR, J., 1972 - Dynamics of fluids in porous media (American Elsevier Publishing Co. Inc., New York 1972).

E N G E L U N D , F., 1953 - On laminar and turbulent flow of ground water in homogeneous sand. "Trans. Dan. Acad. Tech. Sei.", 3, 1, p. 105.

J A I N , S . K . and UPADHYAY, K . K . , 1 9 7 6 - Non-linear laminar flow of fluid into a partially penetrating well. "Gerlands Beiträge Zur Geophysik", 8 5 , pp. 1 5 1 - 1 5 6 .

J A I N , S . K . , 1 9 8 1 - Non - linear laminar flow of fluid into a hemispherical well just penetrating the semi-infinite porous aquifer. "Gerlands Beiträge Zur Geophysik". KHAN, I.H. and RAZA, S.A., 1972 - A study of non-linear flow through porous media by electrical analogy method. "J. Inst, of Engg. India", C.E. Dn., 53, pp. 57-60.

M U S K A T , M . , 1946 - The flow of homogeneous fluids through porous media (J.W. Edwards Inc., Ann Arbor Michingan 1 9 4 6 ) .

PDLUBARINOVA - KOCHINA, P. Ya., 1 9 6 2 - Theory of ground water movement (Princeton University Press, Princeton, New Jersey 1 9 6 2 ) .

U C H I D A , S . , 1 9 5 2 - On the non-linear percolation at high Reynolds number. Nat. Committee for Theor. Appl. Mech., pp. 4 3 7 - 4 4 2 .

UPADHYAY, K . K . , 1 9 7 5 - Non - Linear laminar flow into eccentrically placed well. "Annali di Geofísica", XXVIII, pp. 311-319.

UPADHYAY, K . K . 1 9 7 7 - Non-linear laminar flow into well partially penetrating a porous aquifer of finite thichness. "Pure and Applied Geophysics", 1 1 5 , pp. 6 3 1 - 6 3 8 .

UPADHYAY, K . K . 1 9 7 7 - Non-Linear laminar flow of fluid into a fully penetrating cylindrical well. "Ind. Jour, Theor. Physics", 2 5 , pp. 1 4 9 - 1 5 6 .

W R I G H T , D.E., 1 9 6 8 - Non-Linear flow through granular media. "J. Hy. Dn.", Proc. ASCE, 9 4 , pp. 8 5 1 - 8 7 2 .