Transport in Porous Media 6: 207-221, 1991. 9 1991 Kluwer Academic Publishers. Printedin the Netherlands.

207

Flow Through Isotropic Granular Porous Media J. PRIEUR DU PLESSIS

Department of Applied Mathematics, University of Stellenbosch, Stellenbosch 7600, South Africa and JACOB H. MASLIYAH

Department of Chemical Engineering, University Of Alberta, Edmonton, Alberta, T6G 2G6, Canada (Received: 22 December 1988; revised: 30 October 1990)

Abstract. A generalization of the Navier-Stokes equation is developed to include laminar flow through a rigid isotropic granular porous medium of spatially varying permeability. The model is based on a theory of interspersed continua and the mean geometrical properties of an idealized granular porous microstructure. The derived momentum transport equations are applicable to granular porous media over the entire porosity range from zero through unity. No restriction with respect to flow velocity is imposed, except for the assumption of laminar flow within the pores. The results provide useful and versatile equations and substantiate many of the empirical equations currently in use. One of the major advantages of the generalized momentum equation is its adaptability to numerical simulation. Key words. Porous medium, REV, inertial effects, isotropic, granular.

1. N o t a t i o n

Ap

pore cross-sectional flow area,

Dh d,

hydraulic radius, 4Ap/Pw, microscopic characteristic length,

de as f

total t o r t u o u s flow length within R U C , Vn/Ap, cube side width, friction factor,

fapp

a p p a r e n t ..friction factor, 2(d - ds)6p/(pV2p6X),

g I K

g r a v i t a t i o n a l b o d y force vector per u n i t mass, vector integral expression, h y d r o d y n a m i c permeability when Re <~ Rec, frictional flow length, porosity (void fraction), Vn/Vo, wetted perimeter, pressure,

n Pw P

208 q

Re Rec Reqs R e qa S T t

V V Vn

vp X X +

# V

P ~m

(r

J. P R I E U R D U PLESSIS A N D JACOB H. M A S L I Y A H

specific discharge vector, (v), pore Reynolds number, 2pvp(d- ds)ll~, central Reynolds number, particle Reynolds number, pqds/#, RUC Reynolds number, pqd/#, surfacc area, tortuosity, d/de, time, volume, fluid velocity vector within Vn, mean pore velocity within Vn, mean pore velocity within pore section, axial distance along surface in pore, dimensionless axial distance, x/(DhRe), fluid dynamic viscosity, normal vector pointing into surface, fluid mass density, average wall shear stress over flow length, generic variable, volumetric phase average of 4,

Voo

C d V,

(4). volumetric intrinsic phase average of ~b, E

6 V

gbdV,

deviation, q5 - (qS),, finite increment, vector operator 'nabla', tensor inner product.

Subscripts central value at intersection of asymptotes, C total tortuous dimension, e fluid phase, f fluid-solid interface, L void volume, H total volume of REV or RUC, 0 pore section, P solid phase. S

FLOW THROUGH ISOTROPIC GRANULARPOROUS MEDIA

209

2. Introduction The flow of a liquid through a porous medium forms an important basis for many present-day engineering processes. This paper explicitly addresses the flow of an incompressible viscous fluid through granular porous media of spatially varying permeability. This type of flow frequently occurs naturally, such as in saturated ground water flow through granular soils. Also, this specific phenomenon is often observed in engineering practice, especially in the fields of filtration and percolation. For many decades, empirical modelling attempts have been made for granular porous media. Most of the studies concentrated on one particular aspect of the general field, such as the introduction of global viscous forces (Brinkman, 1947) for flow past a dense swarm of particles. The present approach uses volumetric averaging (Slattery, 1969; Whitaker, 1967) to obtain full transport equations. The philosophy is to deduce results that are more accurate than, for example, the Blake-Kozeny equation (Bird et al. 1960) where the analytical results have to b e multiplied by a factor of 25/6 to comply with experimental results. A major contribution of the present study is the incorporation of nonlinear effects (Forchheimer, 1901) for higher Reynolds number flows in a justifiable way. In this paper, a momentum transport equation, similar to that given by Du Plessis and Masliyah (1988), is derived. The present analysis, however, explicitly concerns porous media consisting of a stationary swarm of separate granules, whereas the previous analysis was aimed at sponge-like media where the solid material forms an interconnected porous matrix. The introduction of an explicit representative geometrical model of the porous medium microstructure allows the establishment of a direct link between the porosity and the tortuosity of the porous matrix. In this manner, the description of the microstructure is limited to two physical parameters only, namely the porosity n and the microscopic characteristic length d of the porous medium. No supplementary data in the form of empirically obtained hydrodynamic permeabilities are needed. Inertial effects within the microstructure are introduced through consideration of flow development along the solid surface sections, in accordance with experimental results by Dybbs and Edwards (1984). The present analysis is developed along the same lines as that of Du Plessis and Masliyah (1988). Parts of their derivation are repeated here to clarify subtle points and differences between the two cases.

3. Transport Equations The present analysis is based upon the following conditions concerning the porous medium: The granular and unconsolidated porous medium is rigid, stationary and locally isotropic with respect to average geometrical properties. The porosity and the characteristic microscopic length are continuous variable functions of position. Maximum pore to pore interconnectivity is assumed.

J. PRIEUR DU PLESSIS AND JACOB H. MASLIYAH

210

The traversing fluid is subject to the following constraints: The fluid consists of a single fluid phase with constant physical properties. The flow conditions are specified as follows: The flow is laminar with no local flow separation within the pores. A no-slip boundary condition applies to all fluid-solid interfaces. There is no restriction on averaged flow separation or recirculation due to the presence of external boundaries or regions of different permeabilities. The continuity equation for conservation of mass of the fluid everywhere locally within the void volume Vn is given by

V.v=0.

(1)

The Navier-Stokes equation, governing the transport of momentum locally within Vn, may be written as g p ~ v + pV" (vv) + Vp - pg -/~V2v = 0.

(2)

The transport equation is to be averaged volumetrically over a Representative Elementary Volume (REV). An REV is defined as a control volume Vo, which is small enough to be regarded as an elementary volume in the calculus, but large enough to be statistically representative of the porous medium. An REV is shown schematically in Figure 1. The specific discharge q (inter alia, also called filter velocity) is a variable commonly used to denote the discharge though a porous medium and it will therefore be used here as a variable in the averaged equations. Its relationship with the phase averaged velocities is given by q = (v) = n(V)n.

(3)

The volumetric phase averaging (Bachmat and Bear, 1986; Bear and Bachmat, 1986) of the continuity equation yields the following equation: V ' q = 0,

(4) f

/

G'

Sfs \\

' | /

I

j-

Fig. 1. Schematicrepresentation of an REV.

FLOW THROUGH ISOTROPIC GRANULAR POROUS MEDIA

211

A similar procedure gives the following averaged form of the Navier-Stokes equation: p ~ q + pV" (qq/n) + nVp,, - npg -/~V2q

+pV " ( n ( ~ ) , ) - ~o l ~ ~s+s( - k v + l t v ' V v )

dS=O.

(5)

The last two terms on the the left-hand side of this equation still need to be transformed into more direct physical quantities to be of practical use. As was done in the case of consolidated porous media (Du Plessis and Masliyah, 1988), the first of these terms will be neglected, being of importance only in cases of large gradients in mean pore velocities. The quantity ~ is the vectorial difference between the real velocity v at a point within Vn and the velocity average over Vn within the REV surrounding the point. The evaluation of the surface integral in the last term is subject to a description of the real velocity gradients at the pore surfaces. This in turn warrants a fairly accurate description of the porous microstructure, which is established in the next section. The quantity .3 is the difference between the pressure at a point within the void volume and the average pressure over V, of the REV surrounding the point. The term involving /~ is considered together with the frictional term as part of a total surface integral. This allows the evaluation of the frictional effects in transverse pore sections at a later stage of the analysis.

4. Modelling of the Granular Microstructure A consistent mathematical model of the porous microstructure has to differentiate between at least two basic types of porous media, namely a consolidated spongelike medium and a granular medium consisting of an unconsolidated swarm of particles. The concept of a Representative Unit cell (RUC), introduced by Du Plessis and Masliyah (1988), will again be used here to facilitate evaluation of the surface integral in Equation (5). An RUC for an isotropic porous medium is defined as the rectangular volume of minimum dimensions into which the geometric properties of the REV may be embedded. It provides the facility to consider flow conditions within the most elementary control volume of the particular porous medium and still have all the geometrical properties of the medium at hand for modelling of physical phenomena. In each principal direction, the side length d corresponds with the average pore centre to pore centre (or solid centre to solid centre). The assumption of mean geometrical isotropy allows the introduction of a cubic RUC of linear dimension d and volume V0, so that its void part can be written in terms of the porosity as

V,, = n Vo =nd 3.

(6)

212

J. PRIEUR DU PLESSIS AND JACOB H. MASLIYAH

/

f

/

~---

f Vo- Vn

v

/ ./

f

1 /

Fig. 2. Geometric representation of an RUC.

It is assumed that the average geometrical properties of the solid structure within the RUC can be resembled by a cube of solid material, located centrally within and aligned with the cubic RUC, as is shown in Figure 2. Furthermore, the arrangement of neighbouring cubes is required to provide maximum possible staggering of resulting duct sections within the porous medium. This requirement ensures that isotropy is maintained and that fluid is therefore forced to traverse all void sections of the medium. If the sides of the cube are of length ds, the volume Vs of solid material within the R U C is given by Vs = 110- V, = d 3 = ( 1 - n ) d

3.

(7)

The assumption of a cubic solid within the RUC also leads to a pore area within the R U C of Ap =

d 2 --

dZs,

(8)

with Ap defined as the minimum cross-sectional flow area available in any of the three principle directions of the RUC. From Equation (7), it then follows that

Ap = [1 - (1 - n)2/3]d 2.

(9)

Defining the total tortuous path length, available within the R U C for flow under the constant cross-sectional Ap, as de = V,,/Ap and the tortuosity as T = d/de, lead to

Ap = n Td 2.

( 1O)

The following relationship between the porosity and tortuosity of the granular porous medium can also be deduced from Equations (9) and (10): T = [1 - (1 - n)Z/3]/n.

(11)

213

FLOW T H R O U G H ISOTROPIC G R A N U L A R POROUS M E D I A 1_0

I

,

i

l

I

I

I

I

I

0.9 0.8 I,,--

>.:

GRANULAR

0.7

I--

0

0.6

s 0.5 ;ONSOLIDATED

J

J

0.4 0.3

0

I

I

I

I

I

]

I

I

I

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

POROSITY, n Fig. 3.

Tortuosity as a function of porosity.

In Figure 3, the tortuosity is shown as a function of porosity. Interesting features are the infinite slope at the maximum value of porosity and the fairly constant value for rather impermeable granular media. The tortuosity of the granular medium is shown in Figure 3 together with that for the consolidated medium, previously derived. It is interesting to note that at porosity, n, close to unity the variation of the tortuosity, T, with the porosity is fairly sharp. Indeed, the slope of the T vs n function is infinite at n = 1. The gradient of this tortuosity function at the limit of zero porosity is 1/9, a value readily obtained by expanding the square-bracketed factor in Equation (11) into a truncated binomial series. It is evident from Fig. 3 that for highly permeable media a very accurate evaluation of porosity is needed to provide the correct tortuosity of the medium. Also presented in Figure 3 for the sake of comparison is the tortuosity function for consolidated porous media and the basic differences are evident. The cubic representation of the granular microstructure also facilitates the calculation of surface areas and orientations for the solid material. This facility is exploited extensively in the following sections of the derivation. The total wetted surface within the RUC is given by

Sfs

=

6d~ = 6(1 -

n)2/3d2.

(12)

The total flow length, under a wetted perimeter of 4d~, along which fluid-solid friction occurs within the RUC, is given by

lf=~ds=

d(1-n) '/3

(13)

214

J. PRIEUR DU PLESSIS AND JACOB H. MASLIYAH

Two-thirds of this flow length are orientated streamwise and one-third transversally. Furthermore, the cubic RUC establishes the following relationship between the velocity definitions (Du Plessis and Masliyah, 1988): q = nTvp,

(14)

with the mean pore velocity vp being defined as v, --~-~p

vdA.

(15)

P

5. Developing Flow Between Parallel Plates Now we need to focus our attention on Equation (5) where velocity field gradients and pressure effects in the last integral term on the left-hand side need to be evaluated. To this end, the geometry of the RUC shown in Figure 2 will be used to arrive at an expression for the two terms within the surface integral. Facing surfaces of two neighbouring staggered cubes present the equivalent of shifted parallel plates between which the flow is developing. This is schematically shown in Figure 4. In the limiting case of low porosity, the relative shift between the plates is very small and the flow conditions may be approximated by flow between parallel plates. The friction factor for a fully developed laminar flow between parallel plates is given by Shah and London (1978) as f Re = 24.

(16)

This is the theoretical limit OffappRe for a set of long parallel plates when x + ~ ~ . In the case of high porosity, the facing surfaces of two neighbouring cubes are so far removed from one another, relative to their streamwise dimensions, that the flow conditions resemble flow develo 3ment along a short flat plate. According to an d$

1

l

d,

j

Fig. 4. Parallel plate configurationcreated by facing cube sides.

FLOW T H R O U G H ISOTROPIC G R A N U L A R POROUS MEDIA

215

analytical flat plate analysis reported by Shah and London (1978), the limit, of x + ~ 0 , is favv Re = 3.44(x +) -1/2,

(17)

where the dimensionless axial distance x + is defined by x

x + - 2(d - d,) Re"

(18)

The asymptotes corresponding to the two limiting conditions mentioned above intersect at a central point. x + -- 0.0205.

(19)

Interpolation of the numerical values for/app Re, provided by Shah and London (1978) (in their Equation (289)), gives a corresponding critical value of

(fapp Re)c =

32

(20)

The two asymptotes are now matched according to the method outlined by Churchill and Usagi (1972) and lead to fapp Re = 24[ 1 + 0.0205/x +]0.5.

(21)

Equation (21) requires a power exponent of about 1/2.4 for the proper matching of the asymptotes in the transitional region around xc. This value is, however, rounded to 1/2 here to keep the final equations as simple as possible. The discrepancy with (20) introduced by this simplification is less than 6.1% at the central point and it diminishes monotonically to zero at either axymptotic limit. Equation (21) is now applicable to flow developing between parallel plates from a uniform inlet profile and over a distance x +. For the particular case of parallel plates a distance ( d - ds) apart, it then follows from Equation (21) that over a length of x:

fapp Re =

2411 + 0.041 l(d - ds) Re/x] 1/2.

(22)

This expression will be used in the next section to provide information on the frictional effects per unit width of solid on the traversing fluid.

6. Modelling of the Intra-Pore Fluid-Solid Interaction The surface integral in Equation (5) over the fluid-solid interface is a representation of the momentum transfer between the two phases. The success of any porous flow modelling is directly related to the proper evaluation of this integral expression, which may be written as I-w;/

dN,

(23)

J. PRIEUR DU PLESSIS AND JACOB H. MASLIYAH

216

where the last term of the integrand is the directional derivative along the normal vector v. This integral has to be evaluated for the specific porous medium. In the case of granular porous media, the RUC described in the previous section may be utilized to approximate the surface integral in Equation (23). In a further simplifying assumption it is now stipulated that the cross-sectional mean pore velocity Vpin each duct section is directed axially along that specific duct section. The duct sections are orientated perpendicularly and transversally with respect to the locally average velocity. Those shear stresses which occur along transverse surfaces, will therefore not contribute to the streamwise component of the second term on the right-hand side of the expression for I given in Equation (23). These stresses will be manifested through the pressure deviation term also present in the full expression. If, again for the sake of simplicity, it is assumed that the transverse flow velocities are equal in magnitude to the velocities in the streamwise section, the pressure deviation term in Equation (23) will contribute maximally half as much as the viscosity term to the value of I. This is due to the total flow development length given in Equation (13). It then follows from Equation (23) that

T3 ffsssP-~v ~ vdS =-34dsfdsl~J-vVdl" 2 o

I-=-2. o

(24)

Through the definition of average wall shear stress, Zm, over the streamwise flow length d,, the expression for I can be written as 32 I=2v0

" 3 Sfs'Cm =

(25)

o "Cm'

~Sfs

where 2 refers to the entire streamwise aligned part of the surface within the RUC. If it is further assumed that ~m gives rise to the total apparent Fanning friction factor between two facing surfaces and over the flow developing length d,, it follows that I = ~ 89

vpf.pp.

(26)

Introduction of the Reynolds number, Re, then yields ]ASfsfa pp R e

I - 4Vo(d - ds) vp.

(27)

Application of Equations (12), (14) and (22) leads to 36 d~ I = ~-~pq a3(d--

as) [1 + 0.041 l(a -

as) Re/as] ,/2.

(28)

This expression for I is now written in terms of the porosity, n, and the microscopic

FLOW THROUGH ISOTROPIC GRANULAR POROUS MEDIA

217

characteristic length, d: 36 ( 1 - n) 2/3 I =/zq ~-5 [ 1 - ( 1 - n)1/3][ 1 - ( 1 - n) 2/3] [1 +0.0411 Re[(1 - n ) -1/3- 1] 1/2. (29) The above expression, relating the shear influence of the solid on the fluid is given in terms of variables defining the microstructure. It could be substituted into Equation (5) to yield a general momentum transport equation. In the next section such general transport equations are discussed, where the intra-pore Reynolds number Re is substituted by the semi-macroscopic Reynolds number Reqa according to Re =

2( 1 - ds/d)

nT

Reqa

(30)

7. Final Equations The final set of equations, which is applicable for any porosity and characteristic length, becomes

Mass conservation:

V.q=0.

(31)

Momentum transport: p ~ q + pV" (qq/n) + nVpn - npg -/~V2q +/~Fq = 0.

(32)

The frictional effects introduced by the presence of the porous medium are governed by the term I~Fq. The factor F is given by: 3 6 ( 1 - n ) 2/3 { O.0822Reqa[(1-n) 1/3-1]} 1/2 Fd2 = [1 - (1 -- n-)~3~-i~ (-1 - n) 2/3] 1 + [1 + (1 - n) '/3] (33) In case of small Reynolds number flow, the factor enclosed by the curly brackets on the right-hand side of Equation (33) approaches unity, rendering the term #Fq linear in velocity. The hydrodynamic permeability,~ inclusive of the nonlinear microscopic inertial effects, is given by n/F. This leads to a velocity-independent Darcy-permeability, K, for very low Reynolds number flow, where

nd 2

K - 36(1 -- n) 2/3 [1 -- (1 -- n) 1/3] -[1 - (1 -- n)2/3].

(34)

In Equation (33), the factor in curly brackets establishes the velocity dependent part of the microscopic shear influence of the porous medium on the transversing fluid. The location of the transition region between the Darcy region of velocity

J. PRIEUR DU PLESSIS AND JACOB H. MASLIYAH

218

independent F a n d the Forchheimer region, where nonlinearity is introduced by the velocity-dependent inertial effects, is given by the central Reynolds n u m b e r Rec. It corresponds with the central point o f Equations (18) and (19) and can be expressed in terms o f porosity as 1 + (1

-

n ) 1/3

(35)

Re~ - 0.0822[( 1 - n ) - 1/3 _ 1] M a k i n g use o f (34) and (35) the last term o f E q u a t i o n (32) becomes # F q = - ~ q[1 + Reqd/Rec] 1/2.

(36)

Numerical values for Rec are provided in Table I for different porosities. Well into the F o r c h h e i m e r region (i.e. where Reqd >>Rec) E q u a t i o n (36) exhibits a power law behaviour o f 1.5 for the velocity partly due to the inertial part o f F. W h e n the central Reynolds n u m b e r is used to predict the transition region, it should be kept in mind that deviation f r o m the constant D a r c y permeability already commences gradually at a m u c h lower Reynolds number. A n illustration o f such behaviour was given by D u Plessis and Masliyah (1987) (in their Figure 4), where [ 1 + Reqd/Rec] 1/2 is graphically presented as a function o f Reqd/Rec. This figure is equally valid in the present case o f flow t h r o u g h granular p o r o u s media. D e p e n d i n g on the application, it is sometimes convenient to express the equations in terms o f the linear dimension ds o f the separate granules. To this end, the following expressions can be derived as respective substitutes for Equations (33),

Table I. Tortuosity, permeability, and central Reynolds numbers for different porosities n

T

Kd -z

Kdz 2

Rec

(Reqs)c

0.00 0.01 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99

0.667 0.668 0.672 0.678 0.691 0.705 0.722 0.740 0.762 0.788 0.823 0.872 0.910 0.963

0.0 6.24 • 10 - 9 8.19 • 10 7 6.98 • 10 6 6.39 • 10-5 2.51 )< 10 4 7.06 • 10 - 4 0.00168 0.00369 0.00792 0.0178 0.0488 0.106 0.443

1.00

1.000

oo

0.0 6.29 • 10 - 9 8.48 • 10-7 7.48 • 10 - 6 7.41 • 10-5 3.18 • 10-4 9.92 • 10 4 0.00267 0.00680 0.0177 0.0519 0.226 0.782 9.55 oo

~ 7238.4 1399.0 669.0 303.8 181.9 120.8 84.0 59.2 41.1 27.2 15.4 9.71 4.06 0.0

7214.2 1375.2 645.9 282.0 161.5 101.9 66.6 43.6 27.5 15.9 7.16 3.58 0.875 0.0

FLOW THROUGH ISOTROPIC GRANULAR POROUS MEDIA

219

(34), and (35): 36( 1 - - n ) 4/3 I 0.0822Reqs[ 1 -- ( 1 -- n) 1/3]71/2 1 - - n - - ~ 3 ] ~ ~ ( 1 - - n) 2/3] L I + (~_--n~-q_--(q-_--n~iT~. j , (37)

Fd~-[1-(

nd2 [1 - (1 - n)'/3] 9 [1 - (1 - n ) 2/3] K - 36(1 - - n) 4/3

(38)

and (Reqs)c ~

(1 - n)2;3[1 + (1 - n) 1/3] 0.082211 -- (1 -- n) 1/3]

(39)

Numerical values for Kd7 2 and (Reqs)c are presented in Table I. In the Darcy regime of very low Reynolds number flow, the present results may be compared with numerical values supplied by the C a r m a n - K o z e n y equation (Bird et al., 1960). Such comparative values of d~/K are provided in Table II for the entire porosity spectrum where the C a r m a n - K o z e n y equation is applicable. It is clear from the table that the present results quantitatively yield very reasonable results in this porosity range. In the non-Darcy (Forchheimer) regime the present results may be compared to the Ergun equation (Bird et al., 1960), which is also applicable in the porosity range below 0.5. In Table III numerical values for the pressure gradient factor Fd~/n from Equation (37) are given for certain Reynolds numbers, together with corresponding notation and with coefficients as improved by Macdonald et al. (Dullien, 1979)): D2 r #q c3x

180(1 -- n) 2 1.8(1 - n) n3 k n3Reqs.

(40)

It is evident that the present theoretical result tends to underpredict the inertial effect and this may possibly be attributed to the neglect of flow recirculation on the lee side of the solid particles. A shape factor of some kind may be introduced to the present results to improve the correlation. This, however, should only be done after a substantial investigation of the behaviour at high Reynolds numbers, which is the subject of current investigations. Table II. Comparison with the Carman-Kozeny equation in the Darcy regime d~/K n

Carman-Kozeny

Presentresults

0.00 0.01 0.05 0.10 0.20 0.30 0.40 0.50

~ 1.76 x 1.30 x 1.46 x 1.44 • 3267 1012 360

oo 1.59 x 1.18 • 1.34 x 1.35 x 3144 1008 374

108 106 105 104

108 106 105 104

220

J. P R I E U R D U PLESSIS A N D JACOB H. M A S L I Y A H Table III. Comparison with the equation in the transitional regime

Ergun

Fd~/n n

Reqs

Ergun

Present

0.2

1 10 100

14580 16200 32400

13524 13737 15712

0.3

1 10 100

3314 3734 7937

3155 3241 4002

0.4

1 10 100

1030 1182 2703

1013 1056 1419

0.5

1 10 100

367 432 1080

378 402 593

8. Conclusions A simple model hasbeen developed for laminar flow through granular porous media. The model is presented in vectorial form and is fully three-dimensional. The problem of inertial effects, which affects porous flow at all but very low Reynolds numbers, has been resolved to some degree. All limitations on the range of porosity values have been lifted. The resulting equations could be of value for numerical simulation of saturated flow through porous media. Changes needed to incorporate the present model in existing computer codes for Newtonian flow are usually minimal. Together with the model for flow through consolidated porous media by Du Plessis and Masliyah (1988), this model presents a unified approach to porous flows, allowing strong interaction between totally different engineering and scientific disciplines. Since the two models differ only in the construction of the F-term, the numerical implementation is almost the same for the two cases and flow through a stationary swarm of particles could be handled in a similar way to the flow past a consolidated porous obstruction (Du Plessis and Masliyah, 1987). Also of importance is the possibility of numerically simulating flow through composite granular media, again similar to the case of consolidated media (Du Plessis, 1988). Comparison with empirical equations based on experimental observations, indicates that the model adequately predicts flow through granular porous media at Reynolds numbers below about 20. At Reynolds numbers much higher than 20, the Forchheimer-effect is accounted for by the model, although its magnitude is underpredicted. The present model also provides information on flow through granular porous media of very high porosity, a regime where the Ergun equation does not apply.

FLOW THROUGH ISOTROPIC GRANULAR POROUS MEDIA

221

References Bachmat, Y. and Bear, J., 1986, Macroscopic modelling of transport phenomena in porous media. 1: The continuum approach, Transport in Porous Media 1, 213-240. Bear, J. and Bachmat, Y., 1986, Macroscopic modelling of transport phenomena in porous media. 2: Applications to mass, momentum and energy transport, Transport in Porous Media 1, 241-269. Bird, R. B., Stewart, W. E., and Lightfoot, E. N., 1960, Transport Phenomena, Wiley, New York. Brinkman, H. C., 1947, A calculation of the viscus force exerted by a flowing fluid on an dense swarm of particles, Appl. Sci. Res. A1, 27-34. Churchill, S. W. and Usagi, R., 1972, A general expression for the correlation of rates of transfer and other phenomena, Am. Inst. Chem. Eng. J. 18(6), 1121-1128. Dullien, F. A. L., 1979, Porous Media Fluid Transport and Pore Structure, Academic Press, New York. Du Plessis, J. P., 1988, On the computation of flow through a composite porous domain, in M. A. Celia et al. (eds.) Developments in Water Science Vol. 35, Modeling Surface and Subsurface Flows, Proceedings of the VII International Conference on Computational Methods in Water Resources, MIT, Cambridge, Mass., 13-17 June 1988, Elsevier, Amsterdam, pp. 77-82. Du Plessis, J. P. and Masliyah, J. H., 1987, Flow in a tube with a porous obstruction, Proc. 30th Heat Transfer and Fluid Mechanics Institute, Sacramento, CA, U.S.A. May 28-29, pp. 89-96. Du Plessis, J. P. and Masliyah, J. H., 1988, Mathematical modelling of flow through consolidated isotropic porous media, Transport in Porous Media 3, 145-16l. Dybbs, A. and Edwards, R. V., 1984, A new look at porous media fluid mechanics- Darcy to Turbulent, in J. Bear and M. Y. Corapcioglu (eds.), Fundamentals o f Transport Phenomena in Porous Media, Proc. NATO/ASI, Newark, Delaware, Martinus Nijhoff, Dordrecht, pp. 199-256. Forchheimer, P. H., 1901, Wasserbewegung durch boden, Zeit. Ver. Deutsch. Ing. 45, 1782-1788. Shah, R. K. and London, A. L., 1978, Laminar flow forced convection in ducts, F. I. Thomas and J. P. Hartnett (eds.), Advances in Heat Transfer, Suppl. 1, Academic Press, London. Slattery, J. C., 1969, Single phase flow through porous media, Am. Inst. Chem. Eng. J. 15, 866-872. Whitaker, S., 1967, Diffusion and dispersion in porous media, Am. Inst. Chem. Eng. J. 13(7), 420-427.