INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, VOL. 13, 443450 (1989)

SHORT COMMUNICATION

FREE CONVECTION ALONG A VERTICAL WALL IN A POROUS MEDIUM WITH PERIODIC PERMEABILITY VARIATION

P. SINGH A N D J. K. MISRA

Applied Mathematics Division

A N D

K. A. NARAYAN*

Department of Chemical Engineering, Indian Institute of Technology, Kunpur, 208016, India

SUMMARY

This communication investigates the effect of permeability variation on free convection flow and heat transfer in a porous medium bounded by a vertical porous wall. A transverse periodic variation in permeability is considered to study the effect on heat transfer rate and skin friction. The problem becomes three-dimensional due to the variation in permeability. Analytical expressions for velocity, temperature, skin friction and rate of heat transfer are obtained using the perturbation technique. Effects of permeability and suction parameters on skin friction and heat transfer is studied and it is found that the increase in permeability creates a higher heat transfer rate and greater skin friction. However, the increase in suction parameter causes the skin friction values to decrease. The results obtained will be useful in the design of steam displacement processes in oil recovery and various geothermal systems.

INTRODUCTION

The flow of fluids through porous media has attracted a considerable interest in recent years because of energy crisis. This has resulted in an unabated exploration of new ideas and avenues in harnessing various conventional energy sources like tidal waves, wind power and geothermal energy. In order to harness maximum geothermal energy, one should have complete and precise knowledge of the quanta of perturbations needed to initiate convection currents in geothermal fluids. Moreover, the knowledge of convection currents in mineral fluids embedded in earth’s crust enables one to use minimal energy to extract the minerals. For example, the use of thermal processes in the recovery of hydrocarbons from under ground petroleum reservoirs is becoming important to ease the recovery of dense crude. In all such processes fluid flow takes place through a porous medium and convection currents are of utmost importance. Combarnous and Bories’ and Cheng’ have provided extensive reviews on free convection in fluid filled porous media.

The inhomogeneities which can be present in a porous medium are numerous. For example, in the recovery of oil from underground reservoirs drilling or steam injection may cause well damage, and permeability around the wellbore becomes a function of reservoir distance and thickness.

* Author to whom correspondence should be addressed.

0363-906 1/89/040443-08$05.00 0 1989 by John Wiley & Sons, Ltd.

Received 8 March 1988 Revised I 1 October 1988

444 SHORT COMMUNICATION

Hever et aL3 reported that the permeability due to damage was greatest near the well and decreased as one moved away. Chandrasekhara et aL4 and Vedha-Nayagam et al.' have incorporated the permeability variation to study the flow behaviour along a heated horizontal surface in a porous medium and have shown that variable permeability has a greater influence on velocity distribution and heat transfer. Among other works on convection in porous media with continuous permeability variation, we mention Gheorghitza,6 Ribando and T ~ r r a n c e , ~ and Gjerde and Tyvand.' These authors have considered linear6* as well as periodic' permeability variations to study the stability in a horizontal porous layer heated from below.

The objective of this short note is to study the effect of periodic permeability variation on free convection flow and heat transfer in a porous medium bounded by an infinite vertical porous wall. The permeability of the porous bed is assumed to be of the form:

KO (1 + c: cos nz/L)

K ( z ) =

where K O is the mean permeability of the medium, L is the wave length of permeability distribution and E is a constant ( 6 1) representing the amplitude of the oscillations.

The flow in the medium becomes three-dimensional due to the variation in permeability. A series expansion method is used to analyse the flow and heat transfer behaviour in the medium.

GOVERNING EQUATIONS

We consider free convection flow of a viscous incompressible fluid through a porous medium bounded by an infinite vertical porous wall. The wall is considered on the x - z plane and the y- axis perpendicular to it is directed into the medium. The physical model of the porous medium is shown in Figure 1. Since the wall is infinite in the x-direction, all the physical quantities are assumed to be independent of x. However, the flow remains three-dimensional due to variation in permeability. All the fluid properties are assumed to be constant except that the influence of the

X 6

Figure 1. Co-ordinate system of the porous medium

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density variation with temperature is considered in the body force term. Under these conditions the problem is governed by the following system of equations.'-"

av aw ay aZ -.+-=o

where u, v and w are the velocity components in the x, y and z directions respectively, T is the fluid temperature, T,, is temperature of ambient fluid away from the wall, p is the fluid pressure and g is acceleration due to gravity. The quantities a, p, v and p are thermal diffusivity, coefficient of thermal expansion, kinematic viscosity and fluid density respectively. It should be noted that the basic flow in the medium is entirely due to buoyancy force caused by the temperature difference between the wall and the porous medium.

In equation (6) the heat due to viscous dissipation is neglected for small velocities." Also in the same equation the Darcy dissipation term is neglected because it is of the same order of magnitude as the viscous dissipation term.

The boundary conditions of the problem are

y=O: u=O, V = - v , , w = O , T = T , Y + C O : u=O, w = O , p = p m , T = T , (7)

where p , is the fluid pressure far away from the wall, T, is the temperature at the wall and u , is the suction velocity at the wall and is a positive quantity. The suction velocity corresponds to the flow rate per unit area of the porous wall.

METHOD OF SOLUTION

When the amplitude of oscillation of permeability is small, we assume the main flow velocity u to take the following form

The similar equations hold for other variables v, w, p and 0, where 6' = ( T - T,)/( T, - T,). When E = 0, the problem reduces to two-dimensional free convection flow in an infinite porous medium with constant permeability K O . In this case equations (2) to (6) reduce to

u = u o + & U 1 + & 2 U 2 + . . . (8)

= 0, dv, a Y

(9)

d6' d28, dy dy2 '

v O = g -

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The solutions of these equations are

where

a

When EZO, the series expansion (8) along with (1) is substituted into equations (2H6) and like powers of E are equated to get the perturbation equations in various orders of E. For small values of E it is sufficient to consider the perturbation equations only of WE), which are

ae, ae, aze, a2e,

a Y a Y ( ay2 +=) o,---+u,---=a __

with boundary conditions

y=o: u1 =o, u1 =o, w1 =o, el =o y+oo; ul=o, wl=o, P1=o, e,=o

This is a set of linear partial differential equations which describe the three-dimensional cross flow which is due to the variation in permeability.

First of all we consider equations (14), (16) and (17). The solutions for u, (y , z), wl(y,z) and pl(y, z) are independent of the main flow component u1 and temperature field 8,.

We now assume u l , w1 and p1 to be

where J = y/L and Z=z/L. Here prime denotes differentiation with respect to j 7 . Equations (20) and (21) have been chosen so that the continuity equation (14) is satisfied.

Substituting these expressions into (16) and (17) and applying the corresponding transformed

SHORT COMMUNICATION 447

boundary conditions, we get expressions for v l , w1 and p , :

(ne-'! - Ae-ny) cos nZ+ vwN cos nZ vwN1 V 1 ( J , a= -__

n-A

where

K l N , = l + - -(n2 + K 1) ' R n

K , N -

112 A =;+ (:+ n2 + K l )

LVW L2 KO

R = - K -- 9 1-

V

We now consider equations (1 5) and (1 8). To reduce the partial differential equations (1 5) and (18) into ordinary differential equations, the following forms for u1 and O1 are assumed:

U,(J, 2) = u1 1 @) cos nz

e,(j, z) = el ,(j) cos n~

(26)

(27)

The expressions for O1 and u1 which satisfy the boundary conditions (19) are:

448

where

SHORT COMMUNICATION

P , P 2 P , = A ( l + P ) + K , / R , P , = -

RA(P- 1)- K ,

1 P 4 = R2 P(P- 1)-n2 - K , , N - - ( R 2 +4K1)l i2

, - r n

n(l + P i P / P , R ) P R + 2 A - R

N 4 = 1 + K , / 2 m A , A , =

A3 = P,(x/PZ - A/nP) AP ( R + P , /n) A,= 2 2 P R - P R 2 + 2 n R P - ~ R - K , '

P R A4=PR(1- P P , / n 2 ) , X=--+(z2 2 +p2R2/4 ) ' /2

m = m , L

RESULTS

We now study the important physical characteristics of the problem. The expression for the shear stress component in the x-direction is given by

G = - ( P R - m) - EGF 1 (P , R) cos ~2

p , where

nm A F i ( P , R ) = (n + m - A) + A , P R + A , (A - P R - n) + A , (A - x)

P 3 R 71 - 1 --?[ ;(i-m) +-(A A4 - P R ) +--I-@- 2) 1 p4

+- When the suction parameter R+O, the skin friction factor F , ( P , R ) reduces to

However, it is found that for R - + GO the function F , -+O. The skin friction factor F , is calculated for various values of permeability parameter K , . An examination of the data reveals that the skin friction factor F , increases significantly with increase in K , .

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The heat flux at the wall in terms of Nusselt number Nu is given by

= 1 +~cosn .?-~F, (P , R) C O S ~ ~ . ? (33)

where qw is heat flux at the wall and c p is the specific heat at the constant pressure. The heat transfer factor F , is given by

F , in the limiting cases, R-+O and R-m, approaches K , / ( n 2 + K , ) which is a function of the permeability parameter only.

I l l I I I I 1 I l l I I I I I 1 1 1 1 1 I I I 1 1 1 1 I I I I I l l I I I I 1 I l l I I I I I 1 1 1 1 1 I I I 1 1 1 1 I I I - -

4.0 Kl -1.0 2 ao.0 -

X c

u 2.0- 5.0

I l l I 1 I I 1 1 1 1 1 1 I I I l l l l I I I I

1 lo1 lo2 R

Figure 2. Skin friction variation with R for different values of permeability parameter K ,

Figure 3.

R = 0 . 1 K1 -1.0 1 . 1 -

10.0 -

25.0

-

-

0.0 0 .5 1 .o 1.5 2 .o - Z

Nusselt number variation with Z for different values of permeability parameter

450 SHORT COMMUNICATION

The values of skin friction (equation (30)) and Nusselt number (Equation (33)) have been calculated for various values of the permeability parameter K , and the suction parameter R by taking typical values of G=4.0, P=0.71 and ~=0.1. It is worth mentioning here that for the natural gas in the reservoir system the Prandtl number P is 0.71. Figure 2 shows the skin friction versus R plot for various values of K , . It can be observed that as the value of K , increases, the skin friction decreases significantly. This shows that the skin friction decreases with permeability. It is also noted from this figure that, for large values of R, the skin friction tends to zero. In Figure 3 the variation of Nusselt number with Z is shown for different values of permeability parameter K , . It is noted from this figure that Nusselt number decreases considerably with K and is periodic in z. A similar trend was noticed in the case of skin friction versus z although it is not shown here. The effect of suction parameter on the Nusselt number was not significant indicating that the heat transfer rate is not enhanced due to increase in suction velocity.

CONCLUSIONS

The analytical solutions for evaluating the heat transfer and skin friction on a vertical porous wall in a variable permeability medium are presented. The results in the present analysis demonstrate the theoretical evidence to support the field and laboratory observations showing that an increase in permeability results in higher heat transfer and skin friction. For a porous medium with periodic permeability variation it was observed that the fluctuating Nusselt number has a phase lag of 7t with respect to the permeability parameter in the medium. However, the variation of skin friction is in phase with the permeability fluctuation. With regard to the application of this analysis to a steam injection process for enhancing oil recovery or to a geothermal process, once the flow rate at the wall (i.e. the suction parameter) and the permeability variation are known, the solution developed in the present analysis could be used to obtain the rate of heat transfer and the skin friction in the porous medium.

ACKNOWLEDGEMENT

Financial assistance for this work from the Council of Scientific and Industrial Research is gratefully acknowledged. The authors are thankful to the referees for their valuable comments.

REFERENCES

1. M. A. Combarnous and S. A. Bories, ‘Hydrothermal convection in a saturated porous medium’, Adu. Hydroscience, 10,

2. P. Cheng, ‘Heat transfer in geothermal systems’, Adv. Heat Transfer 14, 1-105 (1978). 3. G. J. Hever, G. C. Clark and J. N. Dew, ‘The influence of production rate, permeability variation and well spacing on

solution gas drive performance’, Trans. AIME, 222,469482 (1961). 4. B. C. Chandrasekhara, P. M. S. Namboodiri and A. R. Hanumanthappa, ‘Similarity solutions for buoyancy induced

flows in a saturated porous medium adjacent to impermeable horizontal surfaces’, Warme-Und Stoffubertragung, 18,

5. M. Vedha Nayagam, P. Jain and G. Fairweather, ‘The effect of surface mass transfer on buoyancy-induced flow in a variable porosity medium adjacent to a horizontal heated plate’, Int. Comm. Heat Mass Transfer, 14,495-506 (1987).

6. S. I. Gheorghitza, ‘The marginal stability in porous inhomogeneous media’, Proc. Camb. Phil. Soc., 57,871-877 (1961). 7. R. Ribando and K. E. Torrance, ‘Natural convection in a porous medium: effects of confinement, variable

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8. K. M. Gjerde and P. A. Tyvand, ‘Thermal convection in a porous medium with continuous periodic stratification’, Int. J . Heat Mass Transfer, 21, 2289-2295 (1984).

9. A. A. Raptis, ‘Unsteady free convection flow through a porous medium’, Int. J . Engng. Sci., 21(4), 345-348 (1983). 10. M. Becker, Heat Transfer: A Modern Approach, Plenum Press, New York, 1986. 11. N. Rudraiah, B. C. Chandrasekhara, R. Veerabhadraiah and S. T. Nagaraj, Some Flow Problems in Porous Media,

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