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Forchheimer metod
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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
Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp
FORCHHEIMER EQUATION COEFFICIENTS 537
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.
Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp
538 M. G. SIDIROPOULOU, K. N. MOUTSOPOULOS AND V. A. TSIHRINTZIS
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)
Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp
FORCHHEIMER EQUATION COEFFICIENTS 539
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)
Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp
540 M. G. SIDIROPOULOU, K. N. MOUTSOPOULOS AND V. A. TSIHRINTZIS
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)
Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp
FORCHHEIMER EQUATION COEFFICIENTS 541
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).
Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp
542 M. G. SIDIROPOULOU, K. N. MOUTSOPOULOS AND V. A. TSIHRINTZIS
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
Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp
FORCHHEIMER EQUATION COEFFICIENTS 543
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
Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp
544 M. G. SIDIROPOULOU, K. N. MOUTSOPOULOS AND V. A. TSIHRINTZIS
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
Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp
FORCHHEIMER EQUATION COEFFICIENTS 545
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
Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp
546 M. G. SIDIROPOULOU, K. N. MOUTSOPOULOS AND V. A. TSIHRINTZIS
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�
Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp
FORCHHEIMER EQUATION COEFFICIENTS 547
y = 0.3174x
R2 = 0.84191
10
100
1000
10000
1 100 10000
bexp.(s2/m2) bexp.(s
2/m2) bexp.(s2/m2)
bexp.(s2/m2) bexp.(s
2/m2) bexp.(s2/m2)
bexp.(s2/m2) bexp.(s
2/m2) bexp.(s2/m2)
bexp.(s2/m2) bexp.(s
2/m2) bexp.(s2/m2)
bexp.(s2/m2) bexp.(s
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�
Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp
548 M. G. SIDIROPOULOU, K. N. MOUTSOPOULOS AND V. A. TSIHRINTZIS
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
Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp
FORCHHEIMER EQUATION COEFFICIENTS 549
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
Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp
550 M. G. SIDIROPOULOU, K. N. MOUTSOPOULOS AND V. A. TSIHRINTZIS
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
Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp
FORCHHEIMER EQUATION COEFFICIENTS 551
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
bexp.(s2/m2) bexp.(s
2/m2)
bexp.(s2/m2) bexp.(s
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
Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp
552 M. G. SIDIROPOULOU, K. N. MOUTSOPOULOS AND V. A. TSIHRINTZIS
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
Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp
FORCHHEIMER EQUATION COEFFICIENTS 553
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.
Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp
554 M. G. SIDIROPOULOU, K. N. MOUTSOPOULOS AND V. A. TSIHRINTZIS
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Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 534–554 (2007)DOI: 10.1002/hyp