Forch Hydrol Processes

HYDROLOGICAL PROCESSESHydrol. Process. 21, 534–554 (2007)Published online 23 August 2006 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/hyp.6264

Determination of Forchheimer equationcoefficients a and b

Melina G. Sidiropoulou, Konstadinos N. Moutsopoulos and Vassilios A. Tsihrintzis*Laboratory of Ecological Engineering and Technology, Department of Environmental Engineering, School of Engineering, Democritus University of

Thrace, 67100 Xanthi, Greece

Abstract:

This study focuses on the determination of the Forchheimer equation coefficients a and b for non-Darcian flow in porousmedia. Original theoretical equations are evaluated and empirical relations are proposed based on an investigation of availabledata in the literature. The validity of these equations is checked using existing experimental data, and their accuracy versusexisting approaches is studied. On the basis of this analysis, some insight into the physical background of the phenomenonis also provided. The dependence of the coefficients a and b on the Reynolds number is also detected, and potential futureresearch areas, e.g. investigation of inertial effects for consolidated porous media, are pointed out. Copyright 2006 JohnWiley & Sons, Ltd.

KEY WORDS Forchheimer equation; experimental data analysis; phenomenological coefficients; inertial effects; non-Darcianflow

Received 16 May 2005; Accepted 22 November 2005

INTRODUCTION

The classical assumption for the description of a largenumber of problems related to flow in porous media isthat, at the microscopic scale, a creeping flow takes place,which, at the macroscopic scale, is equivalent to a linearrelationship between the flow rate Q and the piezometrichead h, expressed as

Q D �KAdh

dx�1a�

or equivalently as

q D �Kdh

dx�1b�

Equation (1a) is the well-known Darcy law, whereA [L2] is the cross section of the porous medium,K [LT�1] is the hydraulic conductivity (which dependson porous medium and fluid properties), q [LT�1] is theDarcy velocity (defined as the mean velocity in a rep-resentative elementary volume), h [L] is the piezometrichead, and x is the flow direction (for unidirectional flow).

For non-unidirectional flow, and for isotropic andhomogeneous porous media, the following generalexpression can be used:

q D �Krh �1c�

For situations where the inertial effects in the porescale are not negligible (i.e. in practice, Reynolds number

* Correspondence to: Vassilios A. Tsihrintzis, Laboratory of EcologicalEngineering and Technology, Department of Environmental Engineering,School of Engineering, Democritus University of Thrace, 67100 Xanthi,Greece. E-mail: [email protected]

Re D qD/� > 10, with D [L] the porous medium particlediameter and � [L2T�1] the kinematic viscosity of thefluid), the macroscopic hydraulic behaviour is describedby the Forchheimer law:

rh D ��aq C bjq jq� �2�

The coefficient a [TL�1] of the linear term in the Forch-heimer equation (Equation (2)) depends on the propertiesof both the porous medium and the fluid. It representsenergy losses due to viscous forces (viscous friction)at the fluid–solid interface and is equal to 1/K, whereK is the hydraulic conductivity. Coefficient b [T2L�2]depends on the properties of the porous medium only. Itis related to inertial forces, which are irrelevant to vis-cous forces. Although, theoretically, Equation (2) is moreappropriate to simulate the flow processes in any porousmedium, for simplicity, in practice, its use is limited tocoarse granular porous media (for illustrative examples,see Moutsopoulos and Tsihrintzis (2005)), fractured orkarstified aquifers.

Numerous analytical solutions, numerical methods andsoftware packages are available for the simulation ofDarcy flows. Similar tools are also available for the sim-ulation of non-linear inertial flows, although restricted innumber (e.g. Volker, 1975; Zissis and Terzidis, 1991; Wu,2002a,b; Terzidis, 2003; Moutsopoulos and Tsihrintzis,2005). Their use, however, requires knowledge of thephenomenological coefficients a and b of Equation (2).

Various studies have suggested expressions for a andb. For example, Ward (1964) analysed experimental dataof 20 different porous media and suggested the following

Copyright 2006 John Wiley & Sons, Ltd.

FORCHHEIMER EQUATION COEFFICIENTS 535

equations for the estimation of a and b:

a D �

gk�3a�

a D 360�

gD2 �3b�

b D 10Ð44

gD�3c�

where D [L] is the particle diameter, g [LT�2] is theacceleration due to gravity, and k [L2] is the permeabilityof the porous medium given by the following equationk D D2/360.

Blick (1966) considered a mixed model of a bundle ofparallel capillary tubes with orifice plates spaced alongeach tube, and proposed the following relations:

a D 32�

gnD2 �3d�

b D CD

2gn2D�3e�

where CD is an appropriate phenomenological coefficient.Ergun (1952), referred to in Bear (1979), extended

the Kozeny–Carman model, originally developed forcreeping flows, and suggested the following expressions:

a D 150��1 � n�2

gn3D2 �4a�

b D 1Ð75�1 � n�

gn3D�4b�

where n is the porosity of the medium. Similar expres-sions to Ergun’s (1952) were derived by Kovacs (1981),who analysed a set of 300 data in the range of 10 < Re <100, and derived the following formulae for the case ofhomodisperse spherical particles:

a D 144�

gD2

�1 � n�2

n3 �4c�

b D 2Ð4gD

�1 � n�

n3 �4d�

The basic assumption of the original Kozeny–Carman approach and Ergun’s (1952) extensions is thatflow in porous media can be simulated by a bunch of con-duits. The computation of the non-linear term is basedon the hypothesis that turbulent flow takes place (Birdet al., 1960). A similar approach has also been suggestedby Ahmed and Sunada (1969).

Kadlec and Knight (1996) suggested the followingequations for the estimation of coefficients a and b:

a D 1

KD 255��1 � n�

gn3Ð7D2 �5a�

b D 2�1 � n�

gn3D�5b�

The above are typical examples of the equations avail-able in the literature for evaluating the Forchheimer coef-ficients a and b. They are based on assumptions andsimplifications of the geometry of the pore space. Conse-quently, these equations have varying degrees of accuracyin their application, depending also on the number andquality of data used to derive them.

The purpose of this article is to compare variousexisting equations predicting Forchheimer coefficients aand b with experimental data available in the literature,and to propose alternative ways of estimating a and b. Aninvestigation of the physics of the phenomena examinedis also performed in an attempt to derive a theoreticalequation. Finally, descriptions of several features of thephenomenon, not previously referred to, are presented.

METHODS AND MATERIALS

Theoretical background and proposed relations

The relations presented above (Ergun’s approach,Equations (4a) and (4b), or a similar relation suggestedby Ahmed and Sunada (1969)) assumed that the energylosses depend solely on the size of the pore gaps (orequivalently on the grain diameter). The shape of thepore space is not taken into account. The assumptionbehind the development of Ergun’s equations (Equations(4a) and (4b)), i.e. that the pore space can be simulated bycircular pipes, is not compatible with the energy balanceof the flow, for which the characteristics for the case ofcircular conduits and porous media are as follows:

ž For high Reynolds number flows in conduits, turbu-lence is produced near the walls and is transferred tothe interior of the pipe, where it is transformed to heat(Rodi, 1984).

ž Inertial flows in porous media are characterized byrecirculation zones, which are delimited from the mainarea of flow by closed streamlines. In these areas,no macroscopic transfer of the fluid particles takesplace. As demonstrated by Panfilov et al. (2003), theenergy for the eddies in these zones is provided by jetbunches, issuing from the main flow area. By arguingthat the energy of these bunches is proportional tothe kinetic energy of the mean flow, Panfilov et al.(2003) associated the energy losses induced by theabove-mentioned procedure to the quadratic terms ofthe Forchheimer equation.

Owing to the complexity of the flow, it is obviousthat a description of the hydrodynamic characteristics bymeans of numerical simulation can give some insightinto these phenomena. Flow computations in porousmedia, performed by means of conventional numericalschemes (methods of finite differences and finite ele-ments), were associated solely with a simple geometry ofthe pore space (Latinopoulos, 1980; Coulaud et al., 1988;Ganoulis et al., 1989; Panfilov and Fourar, 2006). It isobvious that the flow behaviour in the above-mentioned

Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp

536 M. G. SIDIROPOULOU, K. N. MOUTSOPOULOS AND V. A. TSIHRINTZIS

‘theoretical’ porous media is not identical with that inreal-world media. However, the simulations provided‘theoretical verification’ of the Forchheimer law and gaveuseful information concerning inertial flows:

ž As is depicted in Coulaud et al. (1988: Figure 6), theinfluence of the porosity on non-linear head losses issignificant (a result also compatible with the findingsby Koch and Ladd (1997)).

ž The same simulations demonstrated that the head lossesdo not depend solely on the porosity of the medium andthe Reynolds number, but also on the shape of the porespace.

More realistic flow simulations in three dimensionswere performed by Hill and Koch (2002), by meansof the lattice-Boltzmann method (which makes use ofthe relation between fluid flow and kinetic gas theory),and also by Fourar et al. (2004). Fourar et al. (2004), intheir numerical study of high-velocity effects in periodicporous media, state that viscous dissipation in the recir-culation area is not preponderant. They state that inertialeffects in porous media are mainly caused by deviation ofthe streamlines induced by aforementioned recirculationeddies. Since it is generally accepted that the behaviourof real-world, random, porous media can be quite dif-ferent from that of artificial ones, their conclusions maynot be definitive. Other mechanisms related to non-linearenergy dissipation cannot be excluded. In their theoreti-cal analysis, Skjetne and Auriault (1999) state that inertialenergy losses are strongly localized around the boundarylayer, which induces the flow separation. Anyway, allthree inertia-related mechanisms cited above by Skjetneand Auriault (1999), Panfilov et al. (2003) and Fouraret al. (2004) are related to the formation of boundarylayer separation and recirculation eddies; thus, the basicstatements of the present study persist:

1. The hydraulic behaviour in granular porous media isessentially different from that of closed pipes.

2. Separation mechanisms of the boundary layer areimportant and, subsequently, the shape of the particlemay be crucial for the inertial losses.

Hill and Koch (2002) investigated numerically theflow processes in a closely packed, face-centred arrayof spheres. In the present study, their theory was used todevelop relations for both coefficients a and b, as follows:

1. Using their equations (3), (4), (5), (10) and (11),the following relations for coefficients a and b wereobtained.For 10 < Re � 80:

a D 6570��1 � n�

gD2 �6a�

b D 98Ð1�1 � n�

gD�6b�

For Re > 80:

a D 8316��1 � n�

gD2 �6c�

b D 88Ð65�1 � n�

gD�6d�

2. Using Equations (6a)–(6d), and considering that theporosity in the porous medium examined is n D 0Ð26,the following equations result.For 10 < Re � 80:

a D 4861Ð8�

gD2 �7a�

b D 72Ð594

gD�7b�

For Re > 80:

a D 6153Ð84�

gD2 �7c�

b D 65Ð60

gD�7d�

Since a unique configuration of spheres was takeninto consideration, the accuracy of Equations (6a)–(6d) for different porosity values has to be examined.3. An alternative way to estimate the values of the coef-

ficients a and b is also proposed, by considering, inaddition to the equations developed by Hill and Koch(2002), a relation linking the force F (acting on a rigidobject) and the hydraulic head losses h induced byit (Naudascher, 1987):

F D �qA?�h�

where � is the fluid density and A? is the cross-sectionof flow in which no obstacles are present (Naudascher,1987: equation (4Ð1)). In the present work, A? cannotbe defined exactly; therefore, two extreme cases areconsidered (Figure 1):

(a) By assuming that A? is equal to 2D2 (Figure 1a) andthat �∂h/∂x D h/l, where l is the distance betweentwo spheres, one obtains the following relations.

(a) (b)

z

yy

z

Figure 1. A projection of the close-packed face-centred cubic unit cell,with the flow directed along the x-axis (perpendicular to the page) and thecross-section of flow (shaded area) A? in which no obstacles are present,

assumed to be: (a) equal to 2D2; (b) equal to �D2/4



For 10 < Re � 80:

a D 1215Ð62�

gD2 �8a�

b D 18Ð15

gD�8b�

For Re > 80:

a D 1538Ð60�

gD2 �8c�

b D 16Ð39

gD�8d�

(b) By assuming that A? is equal to �D2/4 (Figure 1b),one obtains the following relations.

For 10 < Re � 80:

a D 3097Ð12�

gD2 �9a�

b D 46Ð24

gD�9b�

For Re > 80:

a D 3920Ð20�

gD2 �9c�

b D 41Ð79

gD�9d�

Owing to the fact that the term A? cannot be esti-mated exactly, one may assume that Equations (7a)–(7d)are more accurate, although this issue remains open todiscussion. Equations (6a)–(9d), however, suggest thatForchheimer coefficients a and b are not constants, butdepend on the bulk velocity, i.e. the Reynolds num-ber. The influence of the flow field may not be crucialfor a large number of practical problems (deviations areapproximately 10% for coefficient b), but may be of someimportance for simulation approaches for which a highaccuracy level is required. A plausible explanation forthis dependence of the coefficients a and b is that the posi-tion for which the boundary layer separation occurs, andsubsequently the characteristics of the recirculation zone,depends on the Reynolds number (Batchelor, 1990). Theinfluence of the recirculation zone on the inertial losseshas already been presented. Eventually, the above-citeddependence might be less important for porous media forwhich the solid phase has sharp edges. It is known thatfor such media the separation point of the boundary layeris not dependent on the Reynolds number, i.e. for isolatedrigid bodies (Batchelor, 1990; Kotsovinos, 2003), and forporous media as well (Latinopoulos, 1980).

For the case of granular unconsolidated porous media,a large amount of information concerning experimentaldata and simulation results is available. Such informationis rather sparse for consolidated porous media. Althoughto a certain extent the analysis of the mechanisms thatinduce energy losses in granular media might also berelevant for consolidated porous media, it is believed

that consolidated porous media might exhibit specificfeatures, which deserve a special research effort.

A potential guideline for the adequate estimation ofthe non-linear energy losses in fractured or karst aquifersmight be the simulation approach by Lao et al. (2004).They used the pore diameter distribution proposed byYanuka et al. (1986) to create artificial porous media,composed of straight pipes of cylindrical cross-sectionand random orientation in space. The hydraulic behaviourof this network was simulated assuming that in eachpipe the flow is described by the Poiseuille law, andadditional head losses were taken into account due tosudden contraction or expansion of the conduit diameterand pipe splitting or bending. For these minor losses, thecoefficients proposed by Bird et al. (1960) were used.The computation of the flow field of the above-mentionednetwork demonstrated that, on a macroscopic scale, theflow is described by the Forchheimer equation, where thecoefficient a is related to energy losses in the straight pipesections and the Poiseuille law, and where the inertialcoefficient b depends on minor losses that are inducedat pipe junctions. A serious drawback of the approachby Lao et al. (2004) is that the use of the relations byBird et al. (1960), which are valid for fully developedturbulence and large Reynolds numbers, is not compatiblewith the use of the Poiseuille law, which is valid forcreeping flow; Lao et al. (2004) could not reproduce theexperimental data by Jones (1987).

Experimental data from previous studies

In this study, the literature was searched and experi-mental data were collected from previous studies on theForchheimer coefficients a and b. A total of 115 datapoints were collected, which are presented in Table I.This table contains information on the medium type,particle size D, porosity n, permeability k, Forchheimercoefficients a and b, researcher providing the data, andthe reference.

Ward (1964) evaluated experimentally the permeabilityk for various spherical and granular porous media. Hedid not present porosity n or coefficient a or b values,but provided k values. Thus, parameter a in Table I wascomputed using Equation (3a).

Arbhabhirama and Dinoy (1973) presented two sets ofdata: one set with porosity n and permeability k, andanother set with porosity n and coefficients a and b. Inthe first set, parameter a (Table I) was computed usingEquation (3a).

Ranganadha Rao and Suresh (1970) plotted their exper-imental data in the form of a graphical plot of �∂h/∂x�/qversus q. If the Forchheimer law holds, then the data fallon a straight line, and the coefficients a and b can beevaluated by the relation: �∂h/∂x�/q D a C bq. They pre-sented data of porosity n, permeability k and coefficientsa and b.

Tyagi and Todd (1970), using data by Dudgeon (1966),presented values of k, a and b. The porosity n of theporous medium was not available.



Tabl

eI.

Exp

erim

enta

lda

tafo

rFo

rchh

eim

erco

effic

ient

sa

and

ban

dpe

rmea

bili

tyk

No.

Med

ium

type

Part

icle

size

D(m

)Po

rosi

tyn

a(s

m�1

)b

�s2

m�2

�k

�m2�

Dat

aby

Ref

eren

ce

1G

lass

bead

sa0.

0002

73—

b1

766Ð6

7c—

b5Ð7

7ð

10�1

1W

ard

War

d(1

964)

2G

lass

bead

sa0.

0003

22—

b1

600Ð2

6c—

b6Ð3

777

ð10

�11

War

dW

ard

(196

4)3

Gla

ssbe

adsa

0.00

0322

—b

121

9Ð34c

—b

8Ð367

7ð

10�1

1W

ard

War

d(1

964)

4G

lass

bead

sa0.

0003

22—

b1

131Ð3

7c—

b9Ð0

177

ð10

�11

War

dW

ard

(196

4)5

Gla

ssbe

adsa

0.00

038

—b

886Ð4

1c—

b1Ð1

5ð

10�1

0W

ard

War

d(1

964)

6G

lass

bead

sa0.

0004

58—

b54

5Ð12c

—b

1Ð87

ð10

�10

War

dW

ard

(196

4)7

Gla

ssbe

adsa

0.00

0545

—b

389Ð0

7c—

b2Ð6

2ð

10�1

0W

ard

War

d(1

964)

8G

ranu

lar

acti

vate

dca

rbon

0.00

061

—b

312Ð6

9c—

b3Ð2

6ð

10�1

0W

ard

War

d(1

964)

9Sa

nd0.

0006

25—

b34

2Ð07c

—b

2Ð98

ð10

�10

War

dW

ard

(196

4)10

Gla

ssbe

adsa

0.00

065

—b

293Ð7

7c—

b3Ð4

7ð

10�1

0W

ard

War

d(1

964)

11Io

nex

chan

gere

sin

0.00

073

—b

426Ð5

1c—

b2Ð3

9ð

10�1

0W

ard

War

d(1

964)

12G

ranu

lar

acti

vate

dca

rbon

0.00

114

—b

50Ð46

c—

b2Ð0

2ð

10�9

War

dW

ard

(196

4)

13Sa

nd0.

0012

6—

b74

Ð95c

—b

1Ð36

ð10

�9W

ard

War

d(1

964)

14G

rave

l0.

0018

8—

b34

Ð21c

—b

2Ð98

ð10

�9W

ard

War

d(1

964)

15A

nthr

acite

coal

0.00

236

—b

30Ð25

c—

b3Ð3

7ð

10�9

War

dW

ard

(196

4)16

Ant

hrac

iteco

al0.

0044

2—

b8Ð0

3c—

b1Ð2

7ð

10�8

War

dW

ard

(196

4)17

Gra

vel

0.00

504

—b

6Ð03c

—b

1Ð69

ð10

�8W

ard

War

d(1

964)

18A

nthr

acite

coal

0.00

882

—b

2Ð88c

—b

3Ð54

ð10

�8W

ard

War

d(1

964)

19G

rave

l0.

0092

1—

b1Ð9

4c—

b5Ð2

6ð

10�8

War

dW

ard

(196

4)20

Gra

vel

0.01

61—

b0Ð5

7c—

b1Ð8

ð10

�7W

ard

War

d(1

964)

21Sa

nd0.

0016

0.39

985

Ð23c

—b

1Ð196

ð10

�9A

rbha

bhir

ama

Arb

habh

iram

aan

dD

inoy

(197

3)22

Sand

0.00

160.

391

95Ð18

c—

b1Ð0

71ð

10�9

Arb

habh

iram

aA

rbha

bhir

ama

and

Din

oy(1

973)

23A

ngul

argr

avel

0.00

640.

467

8Ð80c

—b

1Ð158

1ð

10�8

Arb

habh

iram

aA

rbha

bhir

ama

and

Din

oy(1

973)

24A

ngul

argr

avel

0.00

640.

4711

Ð98c

—b

8Ð51

ð10

�9A

rbha

bhir

ama

Arb

habh

iram

aan

dD

inoy

(197

3)25

Ang

ular

grav

el0.

0283

0.46

51Ð1

6c—

b8Ð8

255

ð10

�8A

rbha

bhir

ama

Arb

habh

iram

aan

dD

inoy

(197

3)26

Ang

ular

grav

el0.

013

0.46

12Ð9

6c—

b3Ð4

425

ð10

�8A

rbha

bhir

ama

Arb

habh

iram

aan

dD

inoy

(197

3)



Tabl

eI.

(Con

tinu

ed)

No.

Med

ium

type

Part

icle

size

D(m

)Po

rosi

tyn

a(s

m�1

)b

�s2

m�2

�k

�m2�

Dat

aby

Ref

eren

ce

27Sa

nd0.

0010

10Ð4

99Ð00

263

07Ð3

ð10

�10

Subb

aA

rbha

bhir

ama

and

Din

oy(1

973)

28Sa

nd0.

0010

10Ð3

8111

5Ð00

345

06Ð5

3ð

10�1

0Su

bba

Arb

habh

iram

aan

dD

inoy

(197

3)29

Sand

0.00

170Ð4

3632

Ð501

100

2Ð254

ð10

�9Su

bba

Arb

habh

iram

aan

dD

inoy

(197

3)30

Sand

0.00

170Ð4

1747

Ð501

990

1Ð59

ð10

�9Su

bba

Arb

habh

iram

aan

dD

inoy

(197

3)31

Sand

0.00

170Ð4

0340

Ð001

640

1Ð878

ð10

�9Su

bba

Arb

habh

iram

aan

dD

inoy

(197

3)32

Sand

0.00

286

0Ð43

13Ð50

720

5Ð665

ð10

�9Su

bba

Arb

habh

iram

aan

dD

inoy

(197

3)33

Sand

0.00

286

0Ð423

22Ð50

880

3Ð46

ð10

�9Su

bba

Arb

habh

iram

aan

dD

inoy

(197

3)34

Sand

0.00

404

0Ð384

7Ð50

530

9Ð79

ð10

�10

Subb

aA

rbha

bhir

ama

and

Din

oy(1

973)

35Sa

nd0.

0040

40Ð3

6710

Ð5078

06Ð8

47ð

10�9

Subb

aA

rbha

bhir

ama

and

Din

oy(1

973)

36G

rave

l0.

0055

0Ð372

4Ð30

430

1Ð658

ð10

�8Su

bba

Arb

habh

iram

aan

dD

inoy

(197

3)37

Gra

vel

0.00

550Ð3

567Ð5

055

01Ð0

28ð

10�8

Subb

aA

rbha

bhir

ama

and

Din

oy(1

973)

38G

rave

l0.

0055

0Ð346

10Ð50

780

7Ð338

ð10

�9Su

bba

Arb

habh

iram

aan

dD

inoy

(197

3)39

Rou

ndri

ver

grav

el0.

0010

10Ð4

99Ð00

263

07Ð3

ð10

�10

Ran

gana

dha

Ran

gana

dha

etal

.(1

970)

40R

ound

rive

rgr

avel

0.00

101

0Ð381

115Ð0

03

450

6Ð53

ð10

�10

Ran

gana

dha

Ran

gana

dha

etal

.(1

970)

41R

ound

rive

rgr

avel

0.00

170Ð4

3632

Ð501

100

2Ð254

ð10

�9R

anga

nadh

aR

anga

nadh

aet

al.

(197

0)42

Rou

ndri

ver

grav

el0.

0017

0Ð417

47Ð50

199

01Ð5

9ð

10�9

Ran

gana

dha

Ran

gana

dha

etal

.(1

970)

43R

ound

rive

rgr

avel

0.00

170Ð4

0340

Ð001

640

1Ð878

ð10

�9R

anga

nadh

aR

anga

nadh

aet

al.

(197

0)44

Rou

ndri

ver

grav

el0.

0017

0Ð392

51Ð50

333

01Ð4

88ð

10�9

Ran

gana

dha

Ran

gana

dha

etal

.(1

970)

45R

ound

rive

rgr

avel

0.00

286

0Ð43

13Ð50

720

5Ð665

ð10

�9R

anga

nadh

aR

anga

nadh

aet

al.

(197

0)46

Rou

ndri

ver

grav

el0.

0028

60Ð4

2322

Ð5088

03Ð4

6ð

10�9

Ran

gana

dha

Ran

gana

dha

etal

.(1

970)

47R

ound

rive

rgr

avel

0.00

286

0Ð403

34Ð00

400

2Ð21

ð10

�9R

anga

nadh

aR

anga

nadh

aet

al.

(197

0)48

Rou

ndri

ver

grav

el0.

0040

40Ð3

847Ð5

053

09Ð7

9ð

10�1

0R

anga

nadh

aR

anga

nadh

aet

al.

(197

0)49

Rou

ndri

ver

grav

el0.

0040

40Ð3

6710

Ð5078

06Ð8

47ð

10�9

Ran

gana

dha

Ran

gana

dha

etal

.(1

970)

50R

ound

rive

rgr

avel

0.00

550Ð3

724Ð3

043

01Ð6

58ð

10�8

Ran

gana

dha

Ran

gana

dha

etal

.(1

970)

51R

ound

rive

rgr

avel

0.00

550Ð3

567Ð5

055

01Ð0

28ð

10�8

Ran

gana

dha

Ran

gana

dha

etal

.(1

970)

52R

ound

rive

rgr

avel

0.00

550Ð3

4610

Ð5078

07Ð3

38ð

10�9

Ran

gana

dha

Ran

gana

dha

etal

.(1

970)

53B

lue

met

al0.

0019

—b

16Ð61

959

8Ð05

ð10

�9D

udge

onTy

agi

and

Todd

(197

0)54

Riv

ergr

avel

0.00

2—

b19

Ð042

174

7Ð07

ð10

�9D

udge

onTy

agi

and

Todd

(197

0)55

Nep

ean

sand

0.00

027

—b

811Ð6

196

11Ð6

6ð

10�1

0D

udge

onTy

agi

and

Todd

(197

0)56

Blu

em

etal

0.00

47—

b7Ð7

957

31Ð7

2ð

10�8

Dud

geon

Tyag

ian

dTo

dd(1

970)

57R

iver

grav

el0.

0009

5—

b78

Ð912

232

1Ð69

ð10

�9D

udge

onTy

agi

and

Todd

(197

0)



Tabl

eI.

(Con

tinu

ed)

No.

Med

ium

type

Part

icle

size

D(m

)Po

rosi

tyn

a(s

m�1

)b

�s2

m�2

�k

�m2�

Dat

aby

Ref

eren

ce

58B

lue

met

al0.

0105

—b

1Ð43

220

9Ð2ð

10�8

Dud

geon

Tyag

ian

dTo

dd(1

970)

59B

lue

met

al0.

0105

—b

0Ð51

972Ð3

ð10

�7D

udge

onTy

agi

and

Todd

(197

0)60

Blu

em

etal

0.01

1—

b1Ð1

516

21Ð0

3ð

10�7

Dud

geon

Tyag

ian

dTo

dd(1

970)

61R

iver

grav

el0.

012

—b

1Ð89

262

7ð

10�8

Dud

geon

Tyag

ian

dTo

dd(1

970)

62M

arbl

es0.

0156

—b

1Ð10

103

1Ð2ð

10�7

Dud

geon

Tyag

ian

dTo

dd(1

970)

63M

arbl

es0.

0156

—b

0Ð50

632Ð4

7ð

10�7

Dud

geon

Tyag

ian

dTo

dd(1

970)

64M

arbl

es0.

0156

—b

0Ð76

951Ð5

6ð

10�7

Dud

geon

Tyag

ian

dTo

dd(1

970)

65M

arbl

em

ixtu

re0.

0158

—b

0Ð73

771Ð7

5ð

10�7

Dud

geon

Tyag

ian

dTo

dd(1

970)

66R

iver

grav

el0.

019

—b

0Ð82

145

1Ð62

ð10

�7D

udge

onTy

agi

and

Todd

(197

0)67

Blu

em

etal

0.01

9—

b0Ð6

111

72Ð2

ð10

�7D

udge

onTy

agi

and

Todd

(197

0)68

Mar

bles

0.02

46—

b0Ð5

866

2Ð24

ð10

�7D

udge

onTy

agi

and

Todd

(197

0)69

Blu

em

etal

0.02

5—

b0Ð3

312

14

ð10

�7D

udge

onTy

agi

and

Todd

(197

0)70

Mar

bles

0.02

85—

b0Ð3

649

3Ð18

ð10

�7D

udge

onTy

agi

and

Todd

(197

0)71

Riv

ergr

avel

0.04

—b

0Ð24

515Ð4

ð10

�7D

udge

onTy

agi

and

Todd

(197

0)72

Riv

ergr

avel

0.08

4—

b0Ð0

615

2Ð04

ð10

�6D

udge

onTy

agi

and

Todd

(197

0)73

Sand

0.00

107

—b

230Ð0

03

080

6Ð91

ð10

�10

Ahm

edA

hmed

and

Suna

da(1

969)

74Sa

nd0.

0007

64—

b38

0Ð00

454

03Ð9

6ð

10�1

0A

hmed

Ahm

edan

dSu

nada

(196

9)75

Sand

0.00

14—

b14

9Ð00

240

01

ð10

�9A

hmed

Ahm

edan

dSu

nada

(196

9)76

Sand

0.00

054

—b

739Ð0

07

450

2Ð1ð

10�1

0A

hmed

Ahm

edan

dSu

nada

(196

9)77

Sand

0.00

199

—b

93Ð80

179

01Ð6

9ð

10�9

Ahm

edA

hmed

and

Suna

da(1

969)

78Sa

nd0.

0025

8—

b69

Ð401

650

2Ð21

ð10

�9A

hmed

Ahm

edan

dSu

nada

(196

9)79

Sand

0.00

105

—b

116Ð4

02

920

8ð

10�1

0L

indq

uist

Ahm

edan

dSu

nada

(196

9)80

Sand

0.00

492

—b

6Ð74

368

1Ð38

ð10

�8L

indq

uist

Ahm

edan

dSu

nada

(196

9)81

Otta

wa

sand

0.00

07—

b1

660Ð0

079

600

8Ð277

ð10

�11

Fanc

her

Ahm

edan

dSu

nada

(196

9)82

Sand

0.00

3—

b1Ð2

39Ð2

7Ð6ð

10�8

Forc

hhei

mer

Ahm

edan

dSu

nada

(196

9)83

Sand

0.00

5—

b0Ð4

15

2Ð3ð

10�7

Forc

hhei

mer

Ahm

edan

dSu

nada

(196

9)84

Gla

sssp

here

sa0.

003

—b

14Ð50

648

6Ð45

ð10

�9Su

nada

Ahm

edan

dSu

nada

(196

9)85

Gla

ssbe

ads

0.00

32—

b14

Ð9062

36Ð7

ð10

�9B

lake

Ahm

edan

dSu

nada

(196

9)86

Nic

kel

sadd

les

0.00

334

—b

8Ð90

210

1Ð12

ð10

�8B

row

nell

Ahm

edan

dSu

nada

(196

9)87

Gla

ssbe

ads

0.00

53—

b6Ð4

718

31Ð5

ð10

�8B

row

nell

Ahm

edan

dSu

nada

(196

9)88

Gra

nula

rab

sorb

ent

0.00

0855

—b

147Ð0

01

420

8Ð6ð

10�1

0A

llen

Ahm

edan

dSu

nada

(196

9)



Tabl

eI.

(Con

tinu

ed)

No.

Med

ium

type

Part

icle

size

D(m

)Po

rosi

tyn

a(s

m�1

)b

�s2

m�2

�k

�m2�

Dat

aby

Ref

eren

ce

89Sa

nd0.

005

—b

18Ð90

137

04Ð9

4ð

10�9

Mob

ashe

riA

hmed

and

Suna

da(1

969)

90M

arbl

e0.

016

—b

0Ð911

71Ð1

9ð

10�7

Kir

kham

Ahm

edan

dSu

nada

(196

9)91

Gra

vel

0.01

2—

b1Ð2

635

Ð18Ð0

9022

ð10

�8c

Bor

dier

–Z

imm

erB

ordi

eran

dZ

imm

er(2

000)

92G

rave

l0.

03—

b0Ð6

330

Ð81Ð6

0784

ð10

�7c

Bor

dier

–Z

imm

erB

ordi

eran

dZ

imm

er(2

000)

93C

r.R

ock

0.00

288

0Ð42

35Ð15

c94

02Ð9

ð10

�9N

iran

jan

Van

kata

ram

anan

dR

ao(1

998)

94C

r.R

ock

0.00

925

0Ð43

1Ð90c

258

5Ð36

ð10

�8N

iran

jan

Van

kata

ram

anan

dR

ao(1

998)

95C

r.R

ock

0.01

440Ð4

152Ð2

5c11

54Ð5

3ð

10�8

Nir

anja

nV

anka

tara

man

and

Rao

(199

8)96

Gl.

Sps.

a0.

020Ð3

832Ð0

3c44

Ð85Ð0

2ð

10�8

Nir

anja

nV

anka

tara

man

and

Rao

(199

8)97

Gl.

Sps.

a0.

025

0Ð392

1Ð53c

27Ð5

6Ð67

ð10

�8N

iran

jan

Van

kata

ram

anan

dR

ao(1

998)

98C

r.R

ock

0.02

990Ð4

660Ð3

8c29

2Ð69

ð10

�7N

iran

jan

Van

kata

ram

anan

dR

ao(1

998)

99C

r.R

ock

0.00

690Ð4

722Ð4

0c30

44Ð2

4ð

10�8

Nas

ser

Van

kata

ram

anan

dR

ao(1

998)

100

Cr.

Roc

k0.

0168

0Ð445

0Ð64c

128

01Ð5

9ð

10�7

Nas

ser

Van

kata

ram

anan

dR

ao(1

998)

101

Cr.

Roc

k0.

0106

50Ð4

411

1Ð2c

210

8Ð5ð

10�8

Jaya

chan

dra

Van

kata

ram

anan

dR

ao(1

998)

102

Cr.

Roc

k0.

014

0Ð444

1Ð13c

122

9ð

10�8

Jaya

chan

dra

Van

kata

ram

anan

dR

ao(1

998)

103

Cr.

Roc

k0.

019

0Ð440

80Ð7

2c10

81Ð4

2ð

10�7

Jaya

chan

dra

Van

kata

ram

anan

dR

ao(1

998)

104

Rou

ndgr

avel

0.01

20Ð3

736Ð0

3c20

71Ð6

9ð

10�8

Arb

habh

iram

aV

anka

tara

man

and

Rao

(199

8)10

5R

ound

grav

el0.

012

0Ð357

5Ð63c

187

1Ð81

ð10

�8A

rbha

bhir

ama

Van

kata

ram

anan

dR

ao(1

998)

106

Ang

ular

grav

el0.

013

0Ð479

2Ð71c

130

3Ð76

ð10

�8A

rbha

bhir

ama

Van

kata

ram

anan

dR

ao(1

998)

107

Cr.

Roc

k0.

0131

0Ð47

4Ð45c

672Ð2

9ð

10�8

Prad

ipK

umar

Van

kata

ram

anan

dR

ao(1

998)

108

Gl.

Sps.

a0.

0156

0Ð395

1Ð20c

66Ð7

8Ð49

ð10

�8Pr

adip

Kum

arV

anka

tara

man

and

Rao

(199

8)10

9G

l.Sp

s.a

0.01

840Ð3

827

1Ð25c

568Ð1

4ð

10�8

Prad

ipK

umar

Van

kata

ram

anan

dR

ao(1

998)

110

Cr.

Roc

k0.

0201

0Ð458

81Ð5

5c56

Ð76Ð5

9ð

10�8

Prad

ipK

umar

Van

kata

ram

anan

dR

ao(1

998)

111

Gl.

Sps.

a0.

0289

0Ð413

10Ð6

0c26

Ð91Ð6

9ð

10�7

Prad

ipK

umar

Van

kata

ram

anan

dR

ao(1

998)

112

Cr.

Roc

k0.

0289

0Ð487

31Ð1

c31

9Ð28

ð10

�8Pr

adip

Kum

arV

anka

tara

man

and

Rao

(199

8)11

3G

l.Sp

s.a

0.01

560Ð3

553

1Ð42c

97Ð9

7Ð2ð

10�8

Shar

ma

Van

kata

ram

anan

dR

ao(1

998)

114

Gl.

Sps.

a0.

0156

0Ð355

80Ð7

6c14

51Ð3

5ð

10�7

Shar

ma

Van

kata

ram

anan

dR

ao(1

998)

115

Gl.

Sps.

a0.

0289

0Ð398

20Ð3

1c38

Ð63Ð3

1ð

10�7

Shar

ma

Van

kata

ram

anan

dR

ao(1

998)

aSp

heri

cal

grai

ns.

bD

ata

not

avai

labl

e.c

Dat

aco

mpu

ted

from

Equ

atio

n(3

a).



Ahmed and Sunada (1969) presented data of parame-ters k, a and b without reference to the porosity n. Thedata were based on their own studies and on studies byForchheimer (1901), Blake (1922), Fancher and Lewis(1933), Lindquist (1933), Allen (1944), Brownell andKatz (1947), Mobasheri and Todd (1963) and Kirkham(1966). They evaluated experimentally the values of aand b based on the graphical plot of �∂h/∂x�/q versus q.

Bordier and Zimmer (2000) obtained data for a and bon the basis of a graphical plot of macroscopic velocityq versus the experimentally measured hydraulic gradient∂h/∂x. Porosity n values were not available.

Venkataraman and Rao (1998) presented experimentaldata provided by Nasser (1970), Arbhabhirama andDinoy (1973), Niranjan (1973), Pradip Kumar (1994),Jayachandra (1995) and Sharma (1995). They studied

spherical and non-spherical porous media. Where valuesof a were not available, they were computed usingEquation (3a).

RESULTS OF EXPERIMENTAL DATA ANALYSIS

Empirical equations from experimental values of a and b

The experimental data in Table I were used to deriveempirical equations, of various forms, relating coeffi-cients a and b to D and/or n. In an effort to deriveequations similar to Equations (3b) and (3c), graphs ofthe experimental values of a or b as a function of par-ticle size D are presented in Figure 2. In Figure 2a andb, all the data are used and the following two regression

a = 0.000859D-1.726567

R2 = 0.924634

0.01

0.1

1

10

100

1000

10000

0.0001 0.001 0.01 0.1D (m)

a exp

.(s/m

)

1

10

100

1000

10000

0.0001 0.001 0.01 0.1

D (m)

b exp

.(s2 /

m2 )

bsph. = 0.237922D-1.385925

R2 = 0.905440

bn-sph. = 0.679715D-1.218168

R2 = 0.909655

1

10

100

1000

10000

0.0001 0.001 0.01 0.1

D (m)

b exp

.(s2 /

m2 )

(a)

(b)

(c)

b = 0.546692D-1.253135

R2 = 0.915710

b exp (Non Sph.)

b exp.(Sph.)

Figure 2. (a) Correlation of aexp and D, (b) correlation of bexp and D, and (c) correlation of bexp and D for spherical and non-spherical porous media



equations are derived:

a D 0Ð000 859D�1Ð726 567 �R2 D 0Ð9246; N D 115�

�10a�

b D 0Ð546 692D�1Ð253 135 �R2 D 0Ð9157; N D 89�

�10b�

where N is the total number of experimental data pointsused in the derivation of the empirical equations and R2

is the coefficient of determination.In Figure 2c, the following two regression lines, con-

cerning b values, are shown separately for non-sphericaland spherical porous media:

b D 0Ð679 715D�1Ð218 168 �R2 D 0Ð9097; N D 80;

non-spherical� �10c�

b D 0Ð237 922ÐD�1Ð385 925 �R2 D 0Ð9054; N D 9;

spherical� �10d�

The difference between Equation (10c) and Equation(10d) is not significant, and the number of data pointsused to derive Equation (10d) is limited; therefore, mostinterest is given to the general equation, Equation (10b).

In an attempt to derive more accurate empiricalequations, n was also introduced in the regressionanalysis. By multiple regression analysis of experimentaldata of coefficients a and b versus particle size D andporosity n, the following equations were obtained:

a D 0Ð003 333D�1Ð500 403n0Ð060 350 �R2 D 0Ð9108;

N D 55� �11a�

b D 0Ð194 325D�1Ð265 175n�1Ð141 417 �R2 D 0Ð8715;

N D 49� �11b�

In an attempt to derive equations similar to Equations(6a)–(6d), multiple regression analysis was performedbetween the experimental data of a and b versus par-ticle size D and parameter (1 � n), and the followingequations were obtained:

a D 0Ð002 789D�1Ð502 361�1 � n��0Ð216 014 �R2 D 0Ð9142;

N D 55� �12a�

b D 1Ð228 873D�1Ð263 314�1 � n�1Ð532 475 �R2 D 0Ð8762;

N D 49� �12b�

Finally, in an attempt to derive equations similar toEquations (4a)–(4d) or (5a) and (5b), multiple regressionanalysis was performed between the experimental data ofa and b versus particle size D, porosity n and parameter(1 � n), and the following equations were obtained:

a D 6Ð527 953 ð 10�15D�1Ð547 45n�16Ð068 711

�1 � n��23Ð157 232 �R2 D 0Ð9188; N D 55�

�13a�

b D 1Ð107 768 ð 10�10D�1Ð301 82n�13Ð836 369

�1 � n��18Ð365 290 �R2 D 0Ð8806; N D 49�

�13b�

In the above empirical Equations (10a)–(13b) the unitsare metres for D, seconds per meter for a and secondssquared per meter squared for b.

The different number of data N used in deriving theprevious equations is a result of the lack of n and/or bvalues in some data sets.

Validity testing of various equations used to predict aor b

The experimental data of Table I were used in testingthe validity and in comparing the various theoretical andempirical relations presented. The following two methodswere used to test the validity of various equations:

1. The root-mean-square error (RMSE) and the normal-ized objective function (NOF) of the theoretical andexperimental values of a and b were computed. RMSEis defined as

RMSE D

√√√√√√N∑

iD1

�xi � yi�2

N�14�

where xi are the experimental values of a or b (Table I),yi are the values computed by Equations (3a)–(13b), andN is the total number of points. The parameter RMSEhas to be as close to 0Ð0 as possible for good prediction.NOF is the ratio of the RMSE to the overall mean X ofthe experimental data, defined as:

NOF D RMSE

X�15�

where X D �1/N�∑N

iD1 xi is the average value of theexperimental data (for a or b). NOF has to be as close to0Ð0 as possible. However, when parameter NOF is lessthan 1Ð0, then the theoretical method is reliable and canbe used with sufficient accuracy (Hession et al., 1994;Kornecki et al., 1999).2. The validity is also tested through scattergrams of

computed (Equations (3a)–(13b)) versus experimentalvalues of a and b. The best match occurs when allpoints fall on a 1 : 1 slope line. Deviation from thatline is measured by fitting through the points a straightregression line of equation

y D �x �16�

where y implies computed values and x experimentalvalues. The slope � of this straight line should be equalto 1Ð0 for a perfect match. If this slope � is less than 1Ð0,then the theoretical equation underestimates the experi-mental data. If the slope � is greater than 1Ð0, then thetheoretical equation overestimates the experimental val-ues. Another parameter that evaluates the accuracy of the



agreement is the coefficient of determination R2, whichshows how well a straight regression line fits the data.The closer R2 is to 1Ð0, the less the points are scatteredaround the straight line.

A comparison between the various theoretical valuesof a and b and the experimental values is presentedin Figures 3 and 4, where experimental and computeddata points are plotted as a function of particle diameter.Figure 3a uses 115 data points, Figure 3b uses 55 datapoints, Figure 4a uses 89 data points and Figure 4b uses49 data points. As mentioned, the different number ofdata points used in Figures 3 and 4 is a result of thelack of values of porosity n and/or b in some datasets. In Figures 3a and 4a, the best agreement with the

experimental data is seen for Equations (10a)–(10b) andin Figures 3b and 4b for Equations (11a) and (11b), (12a)and (12b) and (13a) and (13b).

Scattergrams of predicted versus experimental valuesof a and b are presented in Figures 5 and 6 respectively.A linear regression equation (Equation (16)) and the bestfit line (1 : 1 slope) are also shown. Best agreement for avalue, based on Figure 5, is observed for Equation (11a),with slope � D 0Ð8547 and R2 D 0Ð9160. Good agree-ment is also observed for Equation (12a) and Equation(13a). Based on Figure 6, for the b value, good agree-ment show the empirical Equations (11b), (12b) and (13b)with slopes � close to 0Ð98 and R2 close to 0Ð87. It isalso noted that the range of values of diameter D hasbeen also divided into smaller ranges to see whether the

0.001

0.01

0.1

1

10

100

1000

10000

0.0001 0.001 0.01 0.1D (m)

a (s

/m)

Exper.Eq.3bEq.7aEq.7cEq.8aEq.8cEq.9aEq.9cEq.10a

(a)

0.001

0.01

1.0

1

10

100

1000

10000

0.0001 0.001 0.01 0.1D (m)

a (s

/m)

Exper.Eq.4aEq.5aEq.6aEq.6cEq.11aEq.12aEq.13a

(b)

Figure 3. Comparison of experimental and computed a values as a function of particle size D



1

10

100

1000

10000

100000

0.0001 0.001 0.01 0.1

D (m)

b (s

2 /m

2 )

Exper.Eq.3cEq.7bEq.7dEq.8bEq.8dEq.9bEq.9dEq.10b

(a)

1

10

100

1000

10000

0.0001 0.001 0.01 0.1

D (m)

b (s

2 /m

2 )

Exper.Eq.4bEq.5bEq.6bEq.6dEq.11bEq.12bEq.13b

(b)

Figure 4. Comparison of experimental and computed b values as a function of particle size D

slope � and the coefficient of determination R2 are betterfor some of those ranges. No improvement was detectedbased on these tests, so only the general results are pre-sented (Figures 5 and 6).

For spherical porous media, a comparison betweenthe theoretical values of a and b and the experimentalvalues is presented in Figures 7 and 8 for various particlediameters. Figure 7a uses 17 data points, Figure 7b useseight data points, Figure 8a uses nine data points andFigure 8b uses eight data points.

For spherical grains, scattergrams of theoretical ver-sus experimental values of a and b are presented inFigures 9 and 10 respectively. Again, a linear regressionline (Equation (16)) and a best fit (1 : 1 slope) line arealso presented. Figure 9 indicates that the best methodfor evaluating the parameter a is Equation (8a) with

slope � D 0Ð9203 and R2 D 0Ð9765. Figure 10 indicatesthat the best method for the evaluation of coefficient b isEquation (8b) with slope � D 0Ð9768 and R2 D 0Ð9474. InFigure 9, owing to the small number of available experi-mental data, Equations (4a), (5a), (6a), (6c), (11a), (12a)and (13a) have small values of R2. For the same reason,Equations (4b), (5b), (6b), (6d), (11b), (12b) and (13b)in Figure 10 present small values of R2. Owing to thesmall number of data sets, further checks are necessaryto ensure the reliability of the results.

Table II summarizes values of slope � and coef-ficient of determination R2 from the scattergrams ofFigures 5, 6, 9 and 10, and of parameters RMSE(Equation (14)) and NOF (Equation (15)) for allmethods used for evaluating a and b (Equations(3a)–(13b)). As is shown in Table II, the best RMSE



y=0.2859xR2=0.8928

0.01

1

100

10000

0.01 1 100 10000

aexp.(s/m)

a (E

q.3b

)(s/

m)

1:1

(a)

y =0.7075xR2=0.8832

0.01

1

100

10000

0.01 1 100 10000

aexp.(s/m)

a (E

q.4a

)(s/

m)

1:1

(b)

y = 3.8125xR2=0.8829

0.01

1

100

10000

0.01 1 100 10000

aexp.(s/m)

a (E

q.5a

)(s/

m)

1:1

(c)

y = 3.1151xR2=0.8982

0

1

100

10000

0.01 1 100 10000

aexp.(s/m)

a (E

q.6a

)(s/

m)

1:1

(d)

y=3.9430xR2=0.8982

0

1

100

10000

0.01 1 100 10000

aexp.(s/m)

a (E

q.6c

)(s/

m)

1:1

(e)

y = 3.8350xR2=0.8899

0.01

1

100

10000

0.01 1 100 10000

aexp.(s/m)

a (E

q.7a

)(s/

m)

1:1

(f)

y=4.8541xR2=0.8899

0.01

1

100

10000

0.01 1 100 10000

aexp.(s/m)

a (E

q.7c

)(s/

m)

1:1

(g)

y=0.6619xR2=0.7685

0.01

1

100

10000

0.01 1 100 10000

aexp.(s/m)

a (E

q.10

a)(s

/m)

1:1

(l)

y=0.8547xR2=0.9160

0.01

1

100

10000

0.01 1 100 10000

aexp.(s/m)

a (E

q.11

a)(s

/m)

1:1

(m)

y=0.8546xR2=0.9149

0.01

1

100

10000

0.01 1 100 10000

aexp.(s/m)

a (E

q.12

a)(s

/m)

1:1

(n)

y=0.8473xR2=0.9159

0.01

1

100

10000

0.01 1 100 10000

aexp.(s/m)

a (E

q.13

a)(s

/m)

1:1

(o)

y=0.8055xR2=0.7526

0.01

1

100

10000

0.01 1 100 10000

aexp.(s/m)

a (E

q.8a

)(s/

m)

1:1

(h)

y=1.0195xR2=0.7526

0.01

1

100

10000

0.01 1 100 10000

aexp.(s/m)a (

Eq.

8c)(

s/m

)

1:1

(i)

y = 2.0523xR2=0.7526

0.01

1

100

10000

0.01 1 100 10000

aexp.(s/m)

a (E

q.9a

)(s/

m)

1:1

(j)

y=2.5977xR2=0.7526

0.01

1

100

10000

0.01 1 100 10000

aexp.(s/m)

a (E

q.9c

)(s/

m)

1:1

(k)

Figure 5. Scattergrams of computed and experimental values of coefficient a: (a) Equation (3b); (b) Equation (4a); (c) Equation (5a); (d) Equation(6a); (e) Equation (6c); (f) Equation (7a); (g) Equation (7c); (h) Equation (8a); (i) Equation (8c); (j) Equation (9a); (k) Equation (9c); (l) Equation

(10a); (m) Equation (11a); (n) Equation (12a); (o) Equation (13a)

and NOF values for coefficient a are for Equation (11a)and for coefficient b for Equation (13b). A general com-parison of all the relations is presented on Table III, withthe best results for a and b estimation being Equations(11a) and (11b), Equations (12a) and (12b) and Equations(13a) and (13b).

It is also interesting to compare the semi-empiricalrelations of Ergun (1952) (Equations (4a) and (4b)) andKovacs (1981) (Equations (4c) and (4d)) with thoseproposed in this study, i.e. Equations (6a) and (6b). Thiscan be done by building the following ratios:

(a) ratio of Equation (4a) to Equation (6a)

f1 D 150�1 � n�

6570n3 �17a�

(b) ratio of Equation (4c) to Equation (6a)

f2 D 144�1 � n�

6570n3 �17b�

(c) ratio of Equation (4b) to Equation (6b)

f3 D 1Ð75

98Ð1n3 �17c�



y = 0.3174x

R2 = 0.84191

10

100

1000

10000

1 100 10000

bexp.(s2/m2) bexp.(s

2/m2) bexp.(s2/m2)


2/m2) bexp.(s2/m2)


2/m2) bexp.(s2/m2)


2/m2) bexp.(s2/m2)


2/m2) bexp.(s2/m2)

b (E

q.3c

) (s

2 /m

2 )b (

Eq.

6b)(

s2 /m

2 )

b (E

q.6d

) (s

2 /m

2 )

b (E

q.7b

) (s

2 /m

2 )

b (E

q.7d

) (s

2 /m

2 )

b (E

q.8b

) (s

2 /m

2 )

b (E

q.8d

) (s

2 /m

2 )

b (E

q.9b

) (s

2 /m

2 )

b (E

q.9d

) (s

2 /m

2 )

b (E

q.10

b) (s

2 /m

2 )

b (E

q.11

b) (s

2 /m

2 )

b (E

q.12

b) (s

2 /m

2 )

b (E

q.13

b) (s

2 /m

2 )

b (E

q.4b

) (s

2 /m

2 )

b (E

q.5b

) (s

2 /m

2 )

1:1

(a)

y = 0.5335x

R2 = 0.8589

1

10

100

1000

10000

1 100 10000

1:1

(b)

y = 0.6097x

R2 = 0.8589

1

10

100

1000

10000

1 100 10000

1:1

(c)

y = 1.8466x

R2 = 0.8406

1

10

100

1000

10000

1 100 10000

1:1

(d)

y = 1.6688x

R2 = 0.8406

1

10

100

1000

10000

1 100 10000

1:1

(e)

y = 2.2076x

R2 = 0.84191

10

100

1000

10000

1 100 10000

1:1

(f)

y = 1.9949x

R2 = 0.84191

10

100

1000

10000

1 100 10000

1:1

(g)

y = 0.5519x

R2 = 0.84191

10

100

1000

10000

1 100 10000

1:1

(h)

y = 1.4062x

R2 = 0.8419

1

10

100

1000

10000

1 100 10000

1:1

(j)

y = 0.9353x

R2 = 0.87841

10

100

1000

10000

1 100 10000

1:1

(l)

y = 0.9774x

R2 = 0.8653

1

10

100

1000

10000

1 100 10000

1:1

(m)

y = 0.9776x

R2 = 0.86431

10

100

1000

10000

1 100 10000

1:1

(n)

y = 0.9743x

R2 = 0.866

1

10

100

1000

10000

1 100 10000

1:1

(o)

y = 0.4984x

R2 = 0.84191

10

100

1000

10000

1 100 10000

1:1

(i)

y = 1.2709x

R2 = 0.84191

10

100

1000

10000

1 100 10000

1:1

(k)

Figure 6. Scattergrams of computed and experimental values of coefficient b: (a) Equation (3c); (b) Equation (4b); (c) Equation (5b); (d) Equation(6b); (e) Equation (6d); (f) Equation (7b); (g) Equation (7d); (h) Equation (8b); (i) Equation (8d); (j) Equation (9b); (k) Equation (9d); (l) Equation

(10b); (m) Equation (11b); (n) Equation (12b); (o) Equation (13b)

(d) ratio of Equation (4d) to Equation (6b)

f4 D 2Ð498Ð1n3 �17d�

(e) ratio of Equation (4a) to Equation (6c)

f5 D 150�1 � n�

8316n3 �17e�

(f) ratio of Equation (4c) to Equation (6c)

f6 D 144�1 � n�

8316n3 �17f�

(g) ratio of Equation (4b) to Equation (6d)

f7 D 1Ð75

88Ð65n3 �17g�



0.01

0.1

1

10

100

1000

10000

0.0001 0.001 0.01 0.1

D (m)

a (s

/m)

Exper.Eq.3aEq.7aEq.7cEq.8aEq.8cEq.9aEq.9cEq.10a

(a)

0.01

0.1

1

10

100

1000

10000

0.0001 0.001 0.01 0.1

D (m)

a (s

/m)

Exper.Eq.4aEq.5aEq.6aEq.6cEq.11aEq.12aEq.13a

(b)

Figure 7. Comparison of experimental and computed values of coefficient a, as a function of particle size D, for spherical porous media

(h) ratio of Equation (4d) to Equation (6d)

f8 D 2Ð488Ð65n3 �17h�

As presented in Table IV, the values of the coeffi-cients f1, f2, f3 and f7 are close to 1Ð0 for n D 0Ð26(minimum porosity); for higher values of the porositythe coefficients fi assume values smaller than 1Ð0. Aswill be discussed in the following section, the coeffi-cients a and b also decay with increasing porosity. Theexplanation for this is that, in the denominator of the coef-ficients fi is a theoretical coefficient that corresponds tothe minimum porosity. This effect would be even morepronounced if Equations (7a)–(7d) were used to computethe fi coefficients. Although the physical background ofErgun’s (1952) and Kovacs’s (1981) approximations is

not clear, they give a reasonable to excellent comparisonwith the numerical results obtained using the minimumvalue of porosity. Therefore, it would be interesting toperform further numerical simulations for different valuesof the porosity and compare them with the semi-empiricalEquations (4a)–(4d) and (5a) and (5b).

DISCUSSION AND CONCLUSIONS

In this study, the application field of the Forchheimerequation was presented, and the range of values and thephysical significance of its parameters were analysed. Theprocedure included the analysis of existing experimentaldata, and the exploitation of existing research, based onnumerical simulation approaches (Hill and Koch, 2002).Original relations for the parameters a and b are presented



1

10

100

1000

10000

0.001 0.01 0.1

D (m) (a)

b (s

2 /m

2 )

Exper.Eq.3cEq.7bEq.7dEq.8bEq.8dEq.9bEq.9dEq.10b

1

10

100

1000

10000

0.001 0.01 0.1

D (m)

b (s

2 /m

2 )

Exper.Eq.4bEq.5bEq.6bEq.6dEq.11bEq.12bEq.13b

(b)

Figure 8. Comparison of experimental and computed values of coefficient b, as a function of particle size D, for spherical porous media

in this study based on analysis of experimental andnumerical data available in the literature.

The main conclusions of our experimental data analysisare as follows. Concerning the estimation of a, by theuse of the relations presented in the Introduction (i.e. theinverse of hydraulic conductivity), the Kozeny–Carmanapproximation included in Ergun’s (1952) approach(Equation (4a)) gives excellent results, a conclusioncompatible with previous findings. The reason forthe discrepancies between Ergun’s approach and thereported experimental data is that, as already discussed,its theoretical background is not consistent with thephysical processes taking place in porous media. Ward’s(1964) and Kadlec and Knight’s (1996) approaches,which provide a better approximation, are based on an

experimental data evaluation. Thereafter, the agreementwith the data is of a statistical nature. Since in the presentstudy a larger amount of data have been analysed, it isbelieved that the empirical relations derived are morereliable. Concerning the estimation of b, by the use ofrelations presented in the Introduction, the Kadlec andKnight (1996) approximation gives the best results.

Equations (7a)–(7d) are based on Hill and Koch’s(2002) numerical simulations, who investigated flow phe-nomena in a closely packed bed of spheres. Since it isknown that this type of porous formation exhibits thelowest possible porosity, and that the inertial resistancecoefficient b is inversely proportional to the porosity(Blick, 1966), the relation above is consequently ade-quate rather for the estimation of the upper bound of the



y= 0.2725x

R2=0 .97650.01

0.11

10100

100010000

0.01 1 100 10000

aexp. (s/m)

a (E

q.3b

) (s

/m)

1:1

(a)

y = 0.2348x

R2 = -0.2151

0.010.1

110

1001000

10000

0.01 1 100 10000

aexp. (s/m)

a (E

q.4a

) (s

/m)

1:1

(b)

y = 1.2707x

R2 = -0.21350.010.1

110

1001000

10000

0.01 1 100 10000

aexp.(s/m)

a (E

q.5a

) (s

/m)

a (E

q.7a

) (s

/m)

a (E

q.6c

) (s

/m)

a (E

q.6a

) (s

/m)

a (E

q.8c

) (s

/m)

a (E

q.8a

) (s

/m)

a (E

q.7c

) (s

/m)

1:1

(c)

y = 0.8722x

R2 = -0.34930.010.1

110

1001000

10000

0.01 1 100 10000

aexp. (s/m)

1:1

(d)

y = 1.1040x

R2 = -0.34930.010.1

110

1001000

10000

0.01 1 100 10000

aexp. (s/m)

1:1

(e)

y = 3.6806x

R2 = 0.9765

0.010.1

110

1001000

10000

0.01 1 100 10000

aexp. (s/m)

1:1

(f)

y = 4.6587x

R2 = 0.97650.010.1

110

1001000

10000

0.01 1 100 10000

aexp. (s/m)

aexp. (s/m)

aexp. (s/m) aexp. (s/m) aexp. (s/m)

aexp. (s/m) aexp. (s/m)

1:1

(g)

y = 0.7403x

R2 = 0.96830.010.1

110

1001000

10000

0.01 1 100 10000

a (E

q.10

a)(s

/m)

1:1

(l)

y = 0.9203x

R2 = 0.97650.010.1

110

1001000

10000

0.01 1 100 10000

aexp. (s/m)

1:1

(h)

y = 1.1648x

R2 = 0.97650.010.1

110

1001000

10000

0.01 1 100 10000

aexp. (s/m)

1:1

(i)

y = 2.3447x

R2 = 0.97650.010.1

110

1001000

10000

0.01 1 100 10000

a (E

q.9a

)(s

/m)

1:1

(j)

y = 2.9678x

R2 = 0.97650.010.1

110

1001000

10000

0.01 1 100 10000

a (E

q.9c

)(s

/m)

1:1

(k)

y = 0.8791x

R2 = -0.76610.010.1

110

1001000

10000

0.01 1 100 10000

a (E

q.12

a)(s

/m)

1:1

(n)

y = 0.8381x

R2 = -0.50490.010.1

110

1001000

10000

0.01 1 100 10000

a (E

q.13

a)(s

/m)

1:1

(o)

y = 0.8869x

R2 = -0.75610.010.1

110

1001000

10000

0.01 1 100 10000

a (E

q.11

a)(s

/m)

1:1

(m)

Figure 9. Scattergrams of computed and experimental values of coefficient a for spherical porous media: (a) Equation (3b); (b) Equation (4a);(c) Equation (5a); (d) Equation (6a); (e) Equation (6c); (f) Equation (7a); (g) Equation (7c); (h) Equation (8a); (i) Equation (8c); (j) Equation (9a);

(k) Equation (9c); (l) Equation (10a); (m) Equation (11a); (n) Equation (12a); (o) Equation (13a)

coefficient b than for estimation purposes. Porosity val-ues of 0Ð26 are seldom obtained in laboratory columns ofhomogeneous spheres. A better approach is obtained ifthe influence of the porosity is taken into account (i.e. byusing Equations (6a)–(6d)). However, further numericalchecks are necessary to ensure the reliability of the aboverelation.

The determination of the range of values of the Forch-heimer law coefficients, presented in the previous section,

may be used by models dealing with uncertainty inaquifers, e.g. using Monte Carlo and fuzzy analysisapproaches (de Marsily, 1986). For the latter approach,Equations (11a) and (11b) can be used to estimate the‘most likely values’ of the parameters a and b.

To our knowledge, a quantitative dependence of theresistance coefficient on the Reynolds number has notbeen previously reported. However, further investigationis necessary to detect the influence of porosity, particle



y = 0.5616x

R2 = 0.94741

10

100

1000

10000

1 100 10000

bexp.(s2/m2)

b (E

q.3c

) (s

2 /m

2 )b (

Eq.

6b) (s

2 /m

2 )

b (E

q.4b

) (s

2 /m

2 )b (

Eq.

6d) (s

2 /m

2 )

b (E

q.5b

) (s

2 /m

2 )b (

Eq.

7b) (s

2 /m

2 )

1:1

(a)

y = 1.4662x

R2 = 0.55441

10

100

1000

10000

1 100 10000


2/m2)


2/m2)

bexp.(s2/m2)

bexp.(s2/m2)

bexp.(s2/m2)

bexp.(s2/m2)

bexp.(s2/m2)

bexp.(s2/m2)

bexp.(s2/m2)

bexp.(s2/m2)

bexp.(s2/m2)

1:1

(b)

y = 1.6756x

R2 = 0.55441

10

100

1000

10000

1 100 10000

1:1

(c)

y = 4.1463x

R2 = -0.86731

10

100

1000

10000

1 100 10000

1:1

(d)

y = 3.7469x

R2 = -0.86731

10

100

1000

10000

1 100 10000

1:1

(e)

y = 3.9071x

R2 = 0.94741

10

100

1000

10000

1 100 10000

1:1

(f)

y = 3.5307x

R2 = 0.94741

10

100

1000

10000

1 100 10000

b (E

q.7d

) (s

2 /m

2 )b (

Eq.

9b) (s

2 /m

2 )b (

Eq.

11b)

(s2 /

m2 )

b (E

q.8b

) (s

2 /m

2 )b (

Eq.

9d) (s

2 /m

2 )b (

Eq.

12b)

(s2 /

m2 )

b (E

q.8d

) (s

2 /m

2 )b (

Eq.

10b)

(s2 /

m2 )

b (E

q.13

b) (s

2 /m

2 )1:1

(g)

y = 0.9768x

R2 = 0.94741

10

100

1000

10000

1 100 10000

1:1

(h)

y = 2.4887x

R2 = 0.94741

10

100

1000

10000

1 100 10000

1:1

(j)

y = 1.2034x

R2 = 0.98341

10

100

1000

10000

1 100 10000

1:1

(l)

y =1.1702x

R2 = 0.12911

10

100

1000

10000

1 100 10000

bexp.(s2/m2)

1:1

(m)

y = 2.2492x

R2 = 0.94741

10

100

1000

10000

1 100 10000

1:1

(k)

y = 1.1608x

R2 = 0.0385

1

10

100

1000

10000

1 100 10000

1:1

(n)

y = 1.1663x

R2 = 0.60371

10

100

1000

10000

1 100 10000

1:1

(o)

y = 0.8821x

R2 = 0.94741

10

100

1000

10000

1 100 10000

1:1

(i)

Figure 10. Scattergrams of computed and experimental values of coefficient b for spherical porous media: (a) Equation (3c); (b) Equation (4b);(c) Equation (5b); (d) Equation (6b); (e) Equation (6d); (f) Equation (7b); (g) Equation (7d); (h) Equation (8b); (i) Equation (8d); (j) Equation (9b);

(k) Equation (9d); (l) Equation (10b); (m) Equation (11b); (n) Equation (12b); (o) Equation (13b)

shape and flow regime. A dependence of the coefficientsa and b on the Reynolds number will eventually beimportant for consolidated porous media.

For the laminar flow regime, the simulation approachby Lao et al. (2004) is adequate, but one has to considerour remarks. If turbulence occurs, one should use theapproach of Ahmed and Sunada (1969), where additional

losses due to direction changes, contractions of the con-duits, etc., should also be taken into account.

In summary, existing relations evaluating Forchheimercoefficients a and b have been presented. Equations basedon theoretical approaches and experimental analysis havebeen compared and evaluated. By comparison of allthese relations, based on RMSE (Equation (14)), NOF



Tabl

eII

.R

esul

tsof

vali

dity

test

sfo

rva

riou

seq

uatio

nsus

edto

estim

ate

aan

db

Coe

ffici

ent

aE

quat

ion

(x)

Para

met

erte

stin

gva

lidi

ty3b

4a5a

6a6c

7a7c

8a8c

9a9c

10a

11a

12a

13a

RM

SE(s

m�1

)27

3Ð97

13Ð99

116Ð4

087

Ð4212

0Ð03

992Ð5

713

32Ð55

165Ð7

519

0Ð85

535Ð6

674

6Ð88

167Ð1

49Ð6

09Ð6

49Ð7

2N

OF

2Ð112

0Ð648

5Ð390

4Ð048

5Ð558

7Ð653

10Ð27

51Ð2

781Ð4

724Ð1

305Ð7

591Ð2

890Ð4

440Ð4

460Ð4

50�

all

data

0Ð285

90Ð7

075

3Ð812

53Ð1

151

3Ð943

03Ð8

350

4Ð854

10Ð8

055

1Ð019

52Ð0

523

2Ð597

70Ð6

619

0Ð854

70Ð8

546

0Ð847

3R

2al

lda

ta0Ð8

928

0Ð883

20Ð8

829

0Ð898

20Ð8

982

0Ð889

90Ð8

899

0Ð752

60Ð7

526

0Ð752

60Ð7

526

0Ð768

50Ð9

160

0Ð914

90Ð9

159

�fo

rsp

heri

cal

grai

ns0Ð2

725

0Ð234

81Ð2

707

0Ð872

21Ð1

040

3Ð680

64Ð6

587

0Ð920

31Ð1

648

2Ð344

72Ð9

678

0Ð740

30Ð8

869

0Ð879

10Ð8

381

R2

for

sphe

rica

lgr

ains

0Ð978

5�0

Ð2151

�0Ð21

35�0

Ð3493

�0Ð34

930Ð9

765

0Ð976

50Ð9

765

0Ð976

50Ð9

765

0Ð976

50Ð9

683

�0Ð75

61�0

Ð7661

�0Ð50

49

Coe

ffici

ent

bE

quat

ion

(x)

Para

met

erte

stin

gva

lidi

ty3c

4b5b

6b6d

7b7d

8b8d

9b9d

10b

11b

12b

13b

RM

SE(s

2m

�2)

8346

Ð1360

2Ð29

523Ð8

012

37Ð57

1020

Ð7481

07Ð22

8037

Ð2182

13Ð31

8241

Ð2079

71Ð68

7983

Ð6380

92Ð68

358Ð5

736

0Ð08

356Ð8

2N

OF

4Ð755

0Ð752

0Ð654

1Ð545

1Ð275

4Ð618

4Ð579

4Ð679

4Ð695

4Ð541

4Ð548

4Ð610

0Ð448

0Ð450

0Ð446

�al

lda

ta0Ð3

174

0Ð533

50Ð6

097

1Ð846

61Ð6

688

2Ð207

61Ð9

949

0Ð551

90Ð4

984

1Ð406

21Ð2

709

0Ð935

30Ð9

774

0Ð977

60Ð9

743

R2

all

data

0Ð841

90Ð8

589

0Ð858

90Ð8

406

0Ð840

60Ð8

419

0Ð841

90Ð8

419

0Ð841

90Ð8

419

0Ð841

90Ð8

784

0Ð865

30Ð8

643

0Ð866

0�

for

sphe

rica

lgr

ains

0Ð561

61Ð4

662

1Ð675

64Ð1

463

3Ð746

93Ð9

071

3Ð530

70Ð9

768

0Ð882

12Ð4

887

2Ð249

21Ð2

034

1Ð170

21Ð1

608

1Ð166

3R

2fo

rsp

heri

cal

grai

ns0Ð9

474

0Ð554

40Ð5

544

�0Ð86

73�0

Ð8673

0Ð947

40Ð9

474

0Ð947

40Ð9

474

0Ð947

40Ð9

474

0Ð983

40Ð1

291

0Ð038

50Ð6

037



Table III. General ranking of various methods

Equation (x) Rank of parameter used for validity testing˛

� (Figures 9 and 10 R2 (Figures 9 and 10RMSE NOF � (Figures 5 and 6) R2 (Figures 5 and 6) for spherical porous media) for spherical porous media)

Coefficient a3b 11 8 8 6 10 14a 4 4 6 9 11 —b

5a 6 11 12 10 9 —b

6a 5 9 11 4, 5 5 —b

6c 7 12 14 4, 5 2 —b

7a 14 14 13 7, 8 14 2, 3, 4, 5, 6, 77c 15 15 15 7, 8 15 2, 3, 4, 5, 6, 78a 8 5 5 12, 13, 14, 15 1 2, 3, 4, 5, 6, 78c 10 7 1 12, 13, 14, 15 7 2, 3, 4, 5, 6, 79a 12 10 9 12, 13, 14, 15 12 2, 3, 4, 5, 6, 79c 13 13 10 12, 13, 14, 15 13 2, 3, 4, 5, 6, 710a 9 6 7 11 8 811a 1 1 2 1 3 —b

12a 2 2 3 3 4 —b

13a 3 3 4 2 6 —b

Coefficient b3c 15 15 12 7, 8, 9, 10, 11, 12, 13 7 2, 3, 4, 5, 6, 7, 84b 5 5 9 5–6 8 —b

5b 4 4 6 5–6 9 —b

6b 7 7 13 14–15 15 —b

6d 6 6 11 14–15 13 —b

7b 12 12 15 7, 8, 9, 10, 11, 12, 13 14 2, 3, 4, 5, 6, 7, 87d 10 10 14 7, 8, 9, 10, 11, 12, 13 12 2, 3, 4, 5, 6, 7, 88b 13 13 8 7, 8, 9, 10, 11, 12, 13 1 2, 3, 4, 5, 6, 7, 88d 14 14 10 7, 8, 9, 10, 11, 12, 13 2 2, 3, 4, 5, 6, 7, 89b 8 8 7 7, 8, 9, 10, 11, 12, 13 11 2, 3, 4, 5, 6, 7, 89d 9 9 5 7, 8, 9, 10, 11, 12, 13 10 2, 3, 4, 5, 6, 7, 810b 11 11 4 1 6 111b 2 2 2 3 5 —b

12b 3 3 1 4 3 —b

13b 1 1 3 2 4 —b

a 1: best method; 15: worst method.b Low values of R2.

Table IV. Values of ratios fi comparing Ergun’s (1952) and Kovacs’s (1981) expressions (Equations (4a)–(4d)) with the presentEquations (6a)–(6d) (based on numerical data of Hill and Koch (2002)), for: 10 < Re � 80 (coefficients f1 to f4) for coefficients a(coefficients f1 and f2) and b (coefficients f3 and f4); and Re > 80 (coefficients f5 to f8) for coefficients a (coefficients f5 and

f6) and b (coefficients f7 and f8)

n f1 f2 f3 f4 f5 f6 f7 f8

0.26 0Ð96 0Ð92 1Ð01 1Ð39 0Ð76 0Ð73 1Ð12 1Ð540.30 0Ð59 0Ð57 0Ð66 0Ð91 0Ð47 0Ð45 0Ð73 1Ð000.40 0Ð21 0Ð21 0Ð28 0Ð38 0Ð17 0Ð16 0Ð31 0Ð420.50 0Ð09 0Ð09 0Ð14 0Ð19 0Ð01 0Ð01 0Ð16 0Ð22

(Equation (15)) and linear correlation of computed ver-sus experimental values (Equation (16)), the empiricalEquations (11a) and (11b) seem to be the best for theestimation of coefficients a and b; but empirical equationsEquations (12a) and (12b) and (13a) and (13b) are alsovery good. However, one should be cautious when usingEquations (13a) and (13b) in accurately evaluating poros-ity n, because small differences in porosity n may causelarge differences in values of coefficients a and b, becauseof the large exponents of n.

ACKNOWLEDGEMENTS

The first author acknowledges a fellowship by the Mpo-dossakis Foundation, Greece, which was essential forthe completion of this study. The contributions of theanonymous referees are gratefully acknowledged. In par-ticular, we thank the second anonymous referee for hissuggestions concerning the incorporation of Blick’s andKovacs’s equations and the comparison of Ergun’s rela-tion to our equations.



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