pore-scale modelling for fluid transport in 2d porous media

Pore-scale modelling for fluidtransport in 2D porous media

by

Maret Cloete

Thesis presented in partial fulfilment of the requirements for the degree of

Masters of Engineering Science at the University of Stellenbosch

Supervisor: Prof. J.P. du Plessis

December 2006

Declaration

I, the undersigned, hereby declare that the work contained in this thesis is my ownoriginal work and that I have not previously in its entirety or in part submitted itat any university for a degree.

Signature: Date:

Abstract

In the present study, a model to predict the hydrodynamic permeability of viscousflow through an array of solid phase rectangles of any aspect ratio is derived. Thisalso involves different channel widths in the streamwise and the transverse flowdirections which may be chosen irrespectively to the rectangular shape itself. Itis shown how, with the necessary care taken during description of the interstitialgeometry, a volume averaged approach can be used to obtain results identical to adirect method. Insight into the physical situation is gained during the modelling ofthe two-dimensional interstitial flow processes and resulting pressure distributionsand this may prove valuable when the volume averaging method is applied to morecomplex three-dimensional cases. The analytical results show close correspondenceto numerical calculations, except in the higher porosity range for which a morerealistic model is needed.

Tortuosity is studied together with its inverse. Correspondences and differencesregarding the definitions for the average straightness of pathlines, expressed inliterature, are examined. A new definition, allowing different channel widths inthe streamwise and the transverse flow directions, for the tortuosity is derived fromfirst principles.

A general relation between newly derived permeability and tortuosity expressionswas obtained. This equation incorporates many possible geometrical features for atwo-dimensional unit cell for granules. Three possible staggering configurations ofthe solid phase along the streamwise direction are also included in this relation.

Opsomming

In hierdie studie is ’n model geskep vir die voorspelling van die hidrodinamiese per-meabiliteit vir viskeuse vloei deur ’n reeks reghoekige vastestof-materiale, waarvandie sye enige arbitrere verhouding mag inneem. Hierdie model neem die moont-likheid in ag dat die kanaalbreedtes in die stroomsgewyse rigting en dwarsrigting kanverskil. Hierdie verskil kan egter onafhanklik van die reghoek se syverhouding gespe-sifiseer word. Daar word aangetoon hoe, met die nodige sorg tydens die definieringvan die interstisiele geometrie, ’n volumegemiddelde aanslag gebruik kan word omdieselfde resultate te verkry as wat deur middel van ’n direkte metode verkry word.Gedurende die modellering van tweedimensionele interstisiele vloeiprosesse en re-sulterende drukverspreidings, word insig aangaande die fisiese situasie verkry, wathandig te pas kan kom indien die volumegemiddelde-metode toegepas sou word opmeer komplekse driedimensionele gevalle. Die analitiese resultate toon ’n sterk kor-relasie met die numeriese berekeninge, behalwe in hoe porositeitsgebiede waarvoor’n meer realistiese model benodig word.

Die kronkeling van baanlyne (tortuositeit) asook die inverse daarvan is bestudeer.Ooreenkomste en verskille aangaande die definisies vir die gemiddelde reglynigheidvan baanlyne, soos voorgestel in die literatuur, is ondersoek. ’n Nuwe definisie, watook in ag neem dat die kanaalbreedtes in die stroomsgewyse rigting en dwarsrigtingkan verskil, vir die tortuositeit is vanuit eerste beginsels afgelei.

’n Algemene verhouding tussen die nuutgedefinieerde uitdrukkings vir permeabiliteiten tortuositeit is verkry. Hierdie vergelyking neem baie moontlike geometriese ver-houdings van ’n tweedimensionele eenheidsel vir korrels in ag. Drie moontlike ver-springende rangskikings van die vaste stof langs die stroomsgewyse rigting word ookin hierdie verhouding ingesluit.

Acknowledgements

I would like to acknowledge:

• my supervisor, Prof. J.P. du Plessis, for the hours of discussions at any timeduring the day. I would like to thank him for his interest in my study, his help,his patience and his continuous motivation and encouragement. Without himthis thesis would definitely not have been possible.

• Dr G.J.F. Smit for words of encouragement when they were most needed.

• my parents, for their moral and financial support during my studies and foralways believing in me.

• my teacher, Mrs E. van der Westhuizen, who nurtured my love for mathematicsand who inspired me towards making this choice of study.

• the NRF for their funding during my final year of study.

• God, who gave me the strength to complete this thesis and without Whomnothing is possible.

Contents

1 Introduction 1

1.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Background to this study . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Overview of this study . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Analytical modelling 5

2.1 The Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Volume averaging of transport equations . . . . . . . . . . . . . . . . 9

2.2.1 The REV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 The REA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.3 Volume averaging by means of the REV . . . . . . . . . . . . 14

2.3 The Rectangular Representative Unit Cell . . . . . . . . . . . . . . . 18

3 Derivation of the permeability in the Darcy regime 23

3.1 Direct analytical modelling . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.1 Streamwise regular array . . . . . . . . . . . . . . . . . . . . . 29

3.1.2 Streamwise staggered array . . . . . . . . . . . . . . . . . . . 30

3.2 Volume averaging and closure of momentum equations . . . . . . . . 34

3.3 Asymptotic conditions for a regular array . . . . . . . . . . . . . . . . 44

3.3.1 Plane Poiseuille flow approximation . . . . . . . . . . . . . . . 44

3.3.2 Solid walls restricting the flow . . . . . . . . . . . . . . . . . . 45

3.3.3 Porosity tending to unity . . . . . . . . . . . . . . . . . . . . . 45

i

3.4 Asymptotic conditions for a staggered array . . . . . . . . . . . . . . 47

3.4.1 Solid walls restricting the flow . . . . . . . . . . . . . . . . . . 47

3.5 The dimensionless permeability where the aspect ratio of both thesolid phase and the SUC is α . . . . . . . . . . . . . . . . . . . . . . 49

3.6 Dimensionless permeability when the transverse and parallel channelwidths are equal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.6.1 Regular array . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.6.2 Staggered array . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4 Different interpretations of lineality 56

4.1 Lineality in terms of velocity and geometrical properties . . . . . . . 61

4.1.1 Geometrical lineality . . . . . . . . . . . . . . . . . . . . . . . 61

4.1.2 Kinematic lineality . . . . . . . . . . . . . . . . . . . . . . . . 62

4.1.3 Dynamic lineality . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 Lineality and the average channel speed . . . . . . . . . . . . . . . . 67

5 Linealities from literature 72

5.1 Lineality as derived by Bear and Bachmat . . . . . . . . . . . . . . . 72

5.1.1 Derivation by Bear and Bachmat . . . . . . . . . . . . . . . . 73

5.1.2 Case study of spherical REV by Bear and Bachmat . . . . . . 76

5.1.3 Case study of cubical REV by Bear and Bachmat . . . . . . . 78

5.1.4 Two-dimensional example where the flow lines cross the bor-ders obliquely . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.1.5 The credibility of the assumptions made by Bear and Bachmat 82

5.2 Lineality as derived by Diedericks andDu Plessis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2.1 The credibility of the assumptions made by Diedericks and DuPlessis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2.2 Derivation of LD by means of the Green’s theorem generalisedfor dyadics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

ii

5.2.3 The lineality defined by Bear and Bachmat versus the linealitydefined by Diedericks and Du Plessis . . . . . . . . . . . . . . 90

5.2.4 Two-dimensional example where the flow lines cross the bor-ders obliquely revisited . . . . . . . . . . . . . . . . . . . . . . 92

5.3 Comparison between the linealities . . . . . . . . . . . . . . . . . . . 94

5.3.1 The lineality as derived by Bear and Bachmat . . . . . . . . . 94

5.3.2 The geometric lineality . . . . . . . . . . . . . . . . . . . . . 95

5.3.3 The kinematic lineality . . . . . . . . . . . . . . . . . . . . . . 95

5.3.4 The dynamic lineality . . . . . . . . . . . . . . . . . . . . . . 95

6 The permeability-tortuosity relation 98

6.1 The general relation between permeability and tortuosity . . . . . . . 101

6.1.1 Regular array . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.1.2 Staggered configuration . . . . . . . . . . . . . . . . . . . . . . 103

6.1.3 Evaluating the permeability-tortuosity relation . . . . . . . . . 104

7 Conclusions 107

7.1 Achievements of this study and recommendation for further study . . 111

A Assumptions made before closure 113

A.1 Assumption 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

A.2 Assumption 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

B Evaluating the Model 118

B.1 Steady mass flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

C Green’s vector theorem generalised for dyadics 123

D Two-dimensional case studies 124

D.1 Example 1 – Regular Array . . . . . . . . . . . . . . . . . . . . . . . 125

D.1.1 Lineality as defined by Bear and Bachmat . . . . . . . . . . . 125

D.1.2 Lineality as defined by Diedericks and Du Plessis . . . . . . . 126

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D.1.3 Geometric Lineality . . . . . . . . . . . . . . . . . . . . . . . . 126

D.1.4 Kinematic Lineality . . . . . . . . . . . . . . . . . . . . . . . . 127

D.1.5 Dynamic Lineality . . . . . . . . . . . . . . . . . . . . . . . . 128

D.2 Example 2 – Over-staggered Array . . . . . . . . . . . . . . . . . . . 129






D.3 Example 3 – Fully staggered Array . . . . . . . . . . . . . . . . . . . 133






D.4 Example 4 – Flow lines cross border obliquely . . . . . . . . . . . . . 138






D.5 Example 5 – Limit where lineality tends to zero . . . . . . . . . . . . 141






D.6 Example 6 – Recirculators . . . . . . . . . . . . . . . . . . . . . . . . 145

iv






E Example clarifying a relation obtained by Lloyd et al. and illus-trating the error in Bear and Bachmat’s assumption 149

E.1 The gradient of an average to the average of a gradient relation . . . 151

E.2 The second part of the second assumption made by Bear and Bachmat152

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List of Figures

2.1 Duplicating and stacking unit cells. . . . . . . . . . . . . . . . . . . . 7

2.2 Piece-wise straight streamlines are assumed in a unit cell. . . . . . . . 8

2.3 A spherical REV inside a porous medium. . . . . . . . . . . . . . . . 10

2.4 A schematic representation of an REV showing the different unitvectors and velocity variables. . . . . . . . . . . . . . . . . . . . . . . 11

2.5 A schematic representation of a portion of the REV illustrating driftvelocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6 A circular REA inside a porous medium. . . . . . . . . . . . . . . . . 14

2.7 RRUCs inside their respective spherical REVs, where the dark greyindicates the volume where the two RRUCs overlap. . . . . . . . . . . 19

2.8 The different dimensions defined for the RRUC. . . . . . . . . . . . . 19

2.9 Weighted shifting of the RRUC. . . . . . . . . . . . . . . . . . . . . . 20

2.10 Different simplified RRUC structures encountered. . . . . . . . . . . 22

3.1 Notation for the SUC and SRRUC with respect to the streamwise (ormean flow) direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 An illustration of the different arrays studied, as well as the SUCschosen for the different scenarios. . . . . . . . . . . . . . . . . . . . . 26

3.3 The present model considered for a regular configuration where thepressure values in the different zones are pUtA

= p + δp‖, pUtB= p,

pUtC=p− δp‖ and pUtD

=p− 2δp‖ respectively. . . . . . . . . . . . . 26

3.4 The present model considered for a fully staggered configuration wherethe pressure values in the different zones are pUtA

=p+ 12δp⊥, pUtB

=p,pUtC

=p− δp‖ and pUtD=p− δp‖ − 1

2δp⊥ respectively. . . . . . . . . . 27

vi

3.5 The present model considered for an over-staggered configurationwhere the pressure values in the different zones are pUtA

= p + δp⊥,pUtB

=p, pUtC=p− δp‖ and pUtD

=p− δp‖ − δp⊥ respectively. . . . . 27

3.6 The fluid-solid interfaces on which wall shear stress acts. . . . . . . . 28

3.7 The shifting method of the SRRUC for an over-staggered configuration. 34

3.8 Configuration when dc‖ << dc⊥. . . . . . . . . . . . . . . . . . . . . . 44

3.9 Configuration where dc⊥ << d. . . . . . . . . . . . . . . . . . . . . . 45

3.10 Configuration when dc⊥ → d and dc‖ → d. . . . . . . . . . . . . . . . 46

3.11 Configuration when dc⊥ << d for a streamwise staggered array. . . . 47

3.12 SCASA and RCARA configurations . . . . . . . . . . . . . . . . . . . 50

3.13 RCSSA and SCSRA configurations . . . . . . . . . . . . . . . . . . . 50

3.14 RCSRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.15 Numerical data points versus equation (3.74) for different unit cells. 52

3.16 Porosity ranges for different γ-values. (Note that γ ≤ ǫ ≤ 1.) . . . . . 54

3.17 The dimensionless permeability against porosity for a regular arrayas defined by equation (3.77) with ξ = 0. . . . . . . . . . . . . . . . . 54

3.18 The dimensionless permeability for a fully staggered array in termsof porosity, as defined by equation (3.77) with ξ = 1

2, for various

acceptable values of γ. . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.19 The dimensionless permeability for an over-staggered array in termsof porosity, as defined by equation (3.77) with ξ = 1, for variousacceptable values of γ. . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1 The actual distance travelled by a fluid particle, Le, and the dimen-sion of the porous medium in the direction of the total fluid displace-ment, L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2 The geometrical tortuosity of a pathline. . . . . . . . . . . . . . . . . 59

4.3 An illustration of the actual distances travelled (solid arrows) in thetime interval [t, t+ δt], considered in the calculation of the kinematictortuosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4 An illustration of some pathlines considered in the calculation of thedynamic tortuosity as well as their actual distances in the specifictime interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

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4.5 The geometrical lineality of a pathline. . . . . . . . . . . . . . . . . . 61

4.6 Drift velocity and average channel speed. . . . . . . . . . . . . . . . . 70

5.1 The position vector of P relative to the centroid, C, of the REV. . . 73

5.2 The spherical REV considered by Bear and Bachmat. . . . . . . . . 77

5.3 The REV considered in the example by Bear & Bachmat (1991). . . 78

5.4 A porous medium where the flow lines cross the border obliquely. . . 81

5.5 An arbitrary REV consisting of fluid channels. . . . . . . . . . . . . 83

5.6 Summary on different linealities discussed in Chapters 4 and 5. . . . . 97

6.1 The two SUCs or SRRUCs considered in evaluating equation (6.29). . 104

6.2 Case study A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.3 Case study B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

A.1 The pressures at the transfer volumes are: p1 = p+2δp‖+2δp⊥; p2 =p+ δp‖ + 2δp⊥; p3 = p+ δp‖ + δp⊥; p4 = p+ δp⊥; p5 = p; p6 = p− δp‖;p7 = p− δp‖ − δp⊥; p8 = p− 2δp‖ − δp⊥ and p9 = p− 2δp‖ − 2δp⊥. . 114

B.1 Fluid entering and exiting a tube over a time period ∆t = t2 − t1. . . 119

B.2 Examining the mass flow through the transfer volumes. . . . . . . . . 120

D.1 A schematic representation of piece-wise straight streamlines. . . . . . 124

D.2 The RRUC for granules stacked in a regular array. . . . . . . . . . . 125

D.3 The RRUC of an over-staggered array of granules. . . . . . . . . . . 129

D.4 The RRUC of a fully staggered array of granules . . . . . . . . . . . 133

D.5 A porous medium where the flow lines cross the border obliquely. . . 138

D.6 In this figure d2 >> L. . . . . . . . . . . . . . . . . . . . . . . . . . . 141

D.7 Recirculator: a part of U‖ consists of fluid flowing anti-parallel tothe streamwise direction. . . . . . . . . . . . . . . . . . . . . . . . . 145

E.1 The considered porous medium together with the value of pressureand the pressure gradient in the streamwise direction. . . . . . . . . . 150

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Nomenclature

Abbreviations

FD model the direct analytical model of Firdaouss & Du Plessis (2004)

LDH model the analytical model of Lloyd et al. (2004) by means of volume averaging

RCARA FD model: Rectangular Cells Aligned in a Rectangular Array

RCSRA FD model: Rectangular Cells Staggered in a Rectangular Array

RCSSA FD model: Rectangular Cells Staggered in a Square Array

REA Representative Elementary Area

REV Representative Elementary Volume

RRUC Rectangular Representative Unit Cell

SCASA FD model: Square Cells Aligned in a Square Array

SCSRA FD model: Square Cells Staggered in a Rectangular Array

SRRUC Simplified Rectangular Representative Unit Cell

SUC Simplified Unit Cell

Subscripts and other symbols

⋄‖ parallel to the streamwise direction, thus length, or

the streamwise channel itself

⋄⊥ perpendicular to the streamwise direction, thus width, or

the transverse channel itself

⋄B Bear & Bachmat (1991)

⋄C Carman (1937)

⋄D Diedericks & Du Plessis (1995)

⋄f fluid phase

⋄ff fluid-fluid interface

ix

⋄fs fluid-solid interface or interphase

⋄g stagnant region

⋄K Kozeny (1927)

⋄L perpendicular to both the streamwise and transverse direction, thus depth

⋄o entire region, thus fluid as well as solid phase

⋄s solid phase

⋄t transfer region

⋄ vector parallel to the interstitial flow direction at each point

⋄ vector parallel to the streamwise direction

⋄ vector parallel to the transverse direction

⋄ average value over a specified region

⋄ deviation from average value in each and every channel section

Miscellaneous

⋄ vector

⋄ dyadic

〈⋄ 〉f intrinsic phase average:1

Uf

∫∫∫

Uf

⋄ dU

〈⋄ 〉o phase average:1

Uo

∫∫∫

Uf

⋄ dU

⋄f deviation relative to the intrinsic phase average of the REV: ⋄f = ⋄ − 〈⋄ 〉f⋄o deviation relative to the centroid of the REV: ⋄o = ⋄ − ⋄o‖⋄‖ norm (or magnitude) of the vector: ⋄∇ [m−1] del operator: ∂

∂xi+ ∂

∂yj + ∂

∂zk

∇‖ [m−1] scalar operator representing a gradient in the streamwise channel: n · ∇∇⊥ [m−1] scalar operator representing a gradient in the transverse channel: n · ∇

Roman symbols

Axi [m2] the cross-sectional area of the i’th streamtube intersecting each

pair of opposite faces normal to the x-axis of the cubical REV

considered by Bear & Bachmat (1991)

x

Af [m2] area of void space in the REA

Ao [m2] area of the REA

As [m2] area of the solid phase in the REA

a b an arbitrary dyadic inserted into Green’s theorem

b [m] the outer dimensions of the cubical REV considered by

Bear & Bachmat (1991)

bi [m] the length of the i’th streamtube in the cubical REV considered by

Bear & Bachmat (1991)

C the centroid of the REV

c an arbitrary scalar field inserted into Green’s theorem

d [m] outer dimensions of a square SRRUC or SUC

d‖ [m] side length of the SRRUC or SUC in the streamwise direction

d⊥ [m] side length of the SRRUC or SUC perpendicular to the streamwise direction

dL [m] depth of a 3D SRRUC or SUC, set equal to unity in the present study

dc [m] width of the transverse and the streamwise channels if they are equal

dc‖ [m] width of the transverse channel

dc⊥ [m] width of the streamwise channel

dcL [m] depth of 3D channels, set equal to unity in the present study

ds [m] outer dimensions of a square solid particle

ds‖ [m] side length of a solid particle in the streamwise direction

ds⊥ [m] side length of a solid particle perpendicular to the streamwise direction

dsL [m] depth of 3D particles, set equal to unity in the present study

F a frictional coefficient: 1/K

G a scalar function, except if explicitly referred to otherwise

g [m.s−2] gravitation

i a counter, usually running from 1 to the number of streamlines or

streamtubes

i unit vector parallel to the x-axis in the rectangular Cartesian

coordinate system

j unit vector parallel to the y-axis in the rectangular Cartesian

coordinate system

xi

K dimensionless Darcy permeability: k/(d‖d⊥)

K ′ dimensionless Darcy permeability: k/(dLd⊥)

k [m2] hydrodynamic Darcy permeability

k unit vector parallel to the z-axis in the rectangular Cartesian

coordinate system

kK a factor defined by Kozeny (1927) which depends on the cross-sectional shape

of a channel

L [m] length of a porous medium in the streamwise direction

Le [m] actual path length of fluid travelling over a streamwise displacement L

L [m] length scale of the porous medium

L lineality: a measurement of the straightness of pathlines: L/Le

LB lineality defined by Bear & Bachmat (1991)

LD lineality defined by Diedericks & Du Plessis (1995) (equivalent to LGeo)LDyn dynamic lineality

LGeo geometric lineality (equivalent to LD)

LKin kinematic lineality

Lff [m] total fluid-fluid length on the circumference of the REA

Lfs [m] total fluid-solid length in the REA

Lof [m] boundary of the fluid phase in the REA: Lff + LfsLss [m] total solid-solid length on the circumference of the REA

l [m] length of the pore scale

m [m] the mean hydraulic radius, as defined by Kozeny (1927):Uf

Sfs

N number of particles on Af at time t or passing through Af during the

time interval [t, t+ δt], or

the number of streamlines considered by Zhang & Knackstedt (1995)

Nx total number of streamtubes intersecting the opposite faces of the cubical

REV, considered by Bear & Bachmat (1991), normal to the x-axis

n unit vector perpendicular to Sof or Sof directed into the solid phase

or to the outside of an REV, SRRUC or RRUC

n unit vector defined at each point within the fluid phase directed in the

direction of the interstitial flow at that point

xii

n unit vector in the streamwise direction

n unit vector perpendicular to the streamwise direction

p [Pa] absolute pressure inclusive of the body forces

p′ [Pa] microscopic pressure

pw [Pa] average channel surface section pressure

pw [Pa] wall pressure deviation for each and every channel surface section

δp [Pa] pressure drop over an SRRUC or SUC

δp‖ [Pa] pressure drop in the parallel channel of an SRRUC or SUC

δp⊥ [Pa] pressure drop in the transverse channel of an SRRUC or SUC

Q [m3.s−1] volumetric flow rate: q (d⊥dL)

q [m.s−1] superficial or Darcy velocity

R [m] radius of the spherical REV considered by Bear & Bachmat (1991)

r [m] position vector: x i+ y j + z k

ro [m] position vector of the centroid of the REV

ro [m] position vector relative to the centroid of the REV, r − ro

S [m2] outer surface of the SRRUC or SUC

S‖ [m2] fluid-solid interface of the parallel channels in the SRRUC or SUC

on which wall shear stress exerts

S⊥ [m2] fluid-solid interface of the transverse channels in the SRRUC or SUC

on which wall shear stress exerts

SCK [m] surface presented to fluid in a pipe per unit volume as defined

by Kozeny (1927) and Carman (1937):Sfs

Uo

Sff [m2] total fluid-fluid interface on S

Sfs [m2] total fluid-solid interface in the SRRUC or SUC

Sfs‖ [m2] fluid-solid interface in the parallel channels of the SRRUC or SUC

Sfs⊥ [m2] fluid-solid interface in the transverse channels of the SRRUC or SUC

Sof [m2] boundary of the fluid phase in the SRRUC or SUC: Sff + Sfs

Sss [m2] total solid-solid interface on S

S [m2] outer surface of the REV

Sff [m2] total fluid-fluid interface on S

xiii

SffRS [m2] relevant fluid-fluid interface

Sfs [m2] total fluid-solid interface in the REV

Sof [m2] boundary of the fluid phase in the REV: Sff + SfsSss [m2] total solid-solid interface on Sδs [m] average distance travelled by a particle over a time interval [t, t+ δt]

δs‖ [m] average streamwise displacement of a particle over a time interval [t, t+ δt]

T proportionality constant between the average of the gradient and the

gradient of an average

t [s] time

tR [s] residence time

δt [s] time period

U‖ [m3] volume of the SRRUC or SUC occupied by fluid flowing parallel to n,

where wall shear stress acts on the boundaries

U ′‖ [m3] volume of the SRRUC or SUC occupied by fluid flowing parallel

to n: U‖ + Ut ‖

U⊥ [m3] volume of the SRRUC or SUC occupied by fluid flowing perpendicular to n,


U ′⊥ [m3] volume of the SRRUC or SUC occupied by fluid flowing perpendicular

to n: U⊥ + Ut⊥

Uf [m3] volume of the void space in the SRRUC or SUC

Ug [m3] total stagnant region in the SRRUC or SUC

Uo [m3] volume of the SRRUC or SUC

Us [m3] volume of the solid phase in the SRRUC or SUC

Ut [m3] total transfer volume in the SRRUC or SUC: Ut‖ + Ut⊥

Ut ‖ [m3] that part of the transfer volume in the SRRUC or SUC occupied by

fluid flowing parallel to n

Ut⊥ [m3] that part of the transfer volume in the SRRUC or SUC occupied by

fluid flowing perpendicular to n

U‖ [m3] volume of the REV occupied by fluid flowing parallel to n,


xiv

U⊥ [m3] volume of the REV occupied by fluid flowing perpendicular to n,


Uf [m3] volume of the void space in the REV

Ug [m3] total stagnant region in the REV

Uo [m3] volume of the REV

Us [m3] volume of the solid phase in the REV

Ut [m3] total transfer volume in the REV: Ut‖ + Ut⊥Ut‖ [m3] that part of the transfer volume in the REV occupied by fluid flowing

parallel to n

Ut⊥ [m3] that part of the transfer volume in the REV occupied by fluid flowing

perpendicular to n

Uǫ [m3] the total void space volume of all the intersections of the streamtubes in

the cubical REV considered more than once by Bear & Bachmat (1991)

u [m.s−1] intrinsic phase average of the interstitial velocity

utR [m.s−1] drift velocity

v [m.s−1] speed defined at a point

v [m.s−1] velocity defined at a point

v [m.s−1] interstitial velocity defined at a point

vSof[m.s−1] velocity of the porous structure

w [m.s−1] average speed

w‖ [m.s−1] average interstitial speed in the parallel channels

w⊥ [m.s−1] average interstitial speed in the transverse channels

w [m.s−1] the average velocity field parallel to n at each point

w [m.s−1] the average streamwise channel velocity (equivalent to wGeo)

wDyn [m.s−1] u/LDynwGeo [m.s−1] average velocity in the parallel channels: u/LGeo (equivalent to w)

wKin [m.s−1] average channel speed taken in the streamwise direction: u/LKin=〈v 〉f n

xv

Greek symbols

α in the FD model: α = d⊥/d‖ = ds⊥/ds‖ = dc⊥/dc‖

χ tortuosity: a measurement of the waviness of pathlines: Le/L

χB tortuosity defined by Bear & Bachmat (1991)

χC tortuosity defined by Carman (1937), referring to a set of equivalent tubes

χD tortuosity defined by Diedericks & Du Plessis (1995) (equivalent to χGeo)

χDyn dynamic tortuosity

χGeo geometric tortuosity (equivalent to χD)

χKin kinematic tortuosity

δ denotes a small quantity

ǫ porosity: the fraction of a porous medium occupied by void space

ǫS the fraction of the fluid-fluid interface relative to the outer surface

of an REV: Sff/Sφ constant scalar quantity

γ ratio of parallel channel width to cross-stream side length of the

SUC or SRRUC: dc⊥/d⊥

ϕ aspect ratio of the SRRUC or SUC: d⊥/d‖

λA [m−2] number of particles per unit area in the fluid phase

λV [m−3] number of particles per unit volume in the fluid phase

µ [Pa.s] fluid viscosity

θ [rad] an arbitrary angle

ρ [kg.m−3] fluid density

τ [N.m−2] local shear stress

τ‖ [N.m−2] wall shear stress on S‖

τ⊥ [N.m−2] wall shear stress on S⊥

τw [N.m−2] wall shear stress

Ω the potential function of the uniform vector field, g: ∇Ω = ρ g

ξ staggering parameter

Ψ any arbitrary tensorial quantity

ψ relation between channel widths: dc⊥/dc‖

xvi

Chapter 1

Introduction

1.1 Preface

What is a porous medium? According to Dullien (1979) at least one of the followingtwo conditions must be abided by for a material or structure to be regarded asporous.

1. “It must contain spaces, so called pores or voids, free of solids, imbedded inthe solid or semisolid matrix. The pores usually contain some fluid, such asair, water, oil, etc., or a mixture of different fluids.

2. “It must be permeable to a variety of fluid, i.e., fluids should be able to penetratethrough one face of a septum made of the material and emerge on the otherside. In this case one refers to a ‘permeable porous material’.” (Dullien (1979),p.1, (sic))

In everyday life there are many examples of porous media. If it were not for theporous nature of soil enabling it to hold water in its pores, plant growth would havebeen impossible. Another substance holding a large amount of water is snow andpaper towels are produced for that specific trait. The human body basically consistsof porous media, e.g. kidneys have to hold a large amount of fluid, lungs exchangeslarge amount of oxygen and carbon dioxide between blood vessels and an almostinfinite amount of alveoli, all the muscles, bones, organs, etc. have to be porous to acertain extend allowing blood transfer and the porous nature of the digestive systemhelps to enlarge the absorbance surface. Bricks, sandstone, etc., are porous makingthem better insulators. It also allows for moderate expansions or contractions due toheat exchange. Porous media are also used in filtering processes, and flow throughpacked beds is equally important in operations involving chemical reactions and sep-aration of chemical substances. In the latter, the ability to determine the residencetime accurately is very important, since this is also the reaction time. This however

1

could be a very tedious and complex task, because, as the reaction takes place, thesize of the granules will change, influencing the permeability and therefore also theresidence time.

Porous media thus play an important role in many practical fields such as civil engi-neering, chemical engineering, reservoir engineering, sanitary engineering, petroleumengineering, drainage and irrigation engineering, agricultural engineering, groundwater hydrology and soil sciences.

The interstitial structure of a porous medium is highly complex, and only three setsof solutions for flow problems through porous media are possible, namely: analyticalsolutions, numerical solutions by means of computer codes and experimental solu-tions obtained in laboratories. At this point, in this field of study, different simplifiedversions of a real porous media systems and the transport phenomena occurring inthem, namely analytical models, are still scrutinized by different researchers. Forsome practical problems, some analytical solutions are more efficient to use or easierto implement than others. For numerical analysis, different packages are availableon the market solving different types of fluid mechanical problems. For examiningflow through porous media, the user needs to initiate a grid inside a particular fluiddomain, enclosed by boundaries on which certain boundary conditions are specified.After an number of iterations the initial conditions should converge to the final so-lution at a particular time. Temperature distributions, concentration variations andflow patterns are some of the data obtainable from such packages. The third solu-tion set should be viewed as the most trustworthy, because, though experimentalerrors are bound to exist, this is the most direct connection to real life situations.Both numerical and experimental methods can be expensive in regards to finance,computer memory, time, et cetera. The ultimate solution would thus be a set of ele-mentary rules and simple analytical equations, describing different flow phenomenathrough different types of porous media.

1.2 Background to this study

In literature, different analytical modelling techniques are studied for flow throughporous media. Researchers from different fields of study have over the years im-proved the different models, created new ones and eliminated some. Thus far, dif-ferent analytical models still yield different answers to the same physical problems.Idealistically all the research will lead to a single unified analytical theory, describingthe transport process of fluid traversing different kinds of porous media.

In this study two modelling techniques are used to derive the Darcy permeabilityin terms of parameters describing the specific porous structure, namely: the directanalytical model, describing the flow process through a unit cell by means of mi-croscopic transport equations, and secondly, volume averaging of the microscopictransport equations by means of an REV and then closure by means of an RRUC.

2

These two modelling techniques were found to yield the same permeability, andtherefore, for slow flow traversing a low porosity structure, these two techniquesmay be regarded as equivalent.

Besides the different models considered in literature, researchers also have differentopinions on some entities in porous media. One such entity causing debate through-out the field is tortuosity. According to Clennel (1997), tortuosity can be classifiedinto four groups. Firstly there is the geometrical tortuosity that is an objective cha-racteristic of the pore structure and measurements of a transport property cannotbe used. Secondly tortuosity can be seen as a retardation factor extracted from thetransport properties of the porous medium, like electrical tortuosity and diffusiontortuosity. Zhang & Knackstedt (1995) observed, by tracing numerically obtainedstreamlines between two parallel plates, that the electrical tortuosity is smaller thanthe hydraulic tortuosity of a specific porous structure. Thirdly, there is tortuosityparameters that enters into some simplified constructions of real pore space, suchas the network model and the RRUC model. Finally, there are tortuosity measuresthat are nothing more than correction factors (or fudge factors) in empirical models.Since the definitions of tortuosity differ widely in published literature, it is under-standable why the relationship of permeability to tortuosity also differs in literature.In this study, yet another equation was derived for the determination of tortuosity.We hope that the arguments given are strong enough to eliminate some of the dis-crepancies found in literature on hydraulic and geometrical tortuosity. Using thepermeability and the tortuosity expressions obtained in this study, their relationwere determined and found to be rather complex in general.

Hopefully, at the end, this study would have contributed to the general quest forthe ultimate analytical model.

1.3 Overview of this study

In this study only homogeneous, anisotropic, granular porous media are consideredof which the surfaces presented to the fluid phase are stationary. The fluid phasetraversing the void space of such a stationary porous structure is incompressibleand Newtonian. The velocity field is assumed to be time independent. This studycould be beneficial to, for example, ground water hydrology, reservoir engineering,soil sciences, etc., where water (or any other Newtonian fluid) traverses thoughrather dense granular particles at a very slow pace. This study is kept as general aspossible to include the possibility of implementing the same permeability expressionto various sizes and shapes of granular porous media. This equation provides aprediction of the superficial velocity of a fluid, with a known, constant viscosity, givena specific external pressure gradient over anyone of these different granular porousmedia. This could aid decision making with regards to the a system’s developmentor its operation.

3

In Chapter 2 an overview of two analytical methods is given. Firstly, the unit cellis considered. This method may only be used in an anisotropic porous mediumwhere (in two dimensions) the solid phase is non-staggered in at least one of the twoprinciple directions. In this study, this method is referred to as the direct method,because the continuity and Navier-Stokes equations are directly implemented with-out involving volume averaging. The second method discussed was volume averagingof microscopic transport equations by means of an REV (e.g. Bear (1972)) and thenclosure of the volume averaged transport equation is done via an RRUC, as was in-troduced by Du Plessis & Masliyah (1988). Theoretically, this technique may be usedfor either isotropic or anisotropic porous media, since, with this method, only a sta-tistical geometrical average is studied. This average is not dependent on whether thegranules are randomly distributed (thus representing an isotropic porous medium)or packed in a specific order (thus representing an anisotropic porous medium).

A number of simplified unit cells and RRUCs were studied in Chapter 3. Thiswas accomplished by introducing three independent geometrical parameters whichoperates in close relation to one another and porosity. A general hydrodynamicDarcy permeability is obtained in terms of those geometrical parameters and afourth parameter, describing the staggeredness of the solid phase granules along thestreamwise direction.

Chapters 4 and 5 were devoted to tortuosity. The tortuosities defined by Bear &Bachmat (1991) and Diedericks & Du Plessis (1995) as well as their differencesand correspondences were examined. A new definition for tortuosity is derived,eliminating the discrepancies found in the above definitions. Finally, in Chapter6, the dependence of the general permeability obtained in Chapter 3 to the newlydefined tortuosity of Chapter 4 is obtained.

4

Chapter 2

Analytical modelling

The equations governing the various transport phenomena in fluid mechanics may bewritten at a microscopic level, where the equations describe the physical phenomenaat a (mathematical) point within a particular phase inside a certain domain.

Two important microscopic equations governing the flow process are the Navier-Stokes equation, describing the momentum transport of an incompressible fluid,and the continuity equation, derived from the interstitial mass conservation of thefluid phase. The Navier-Stokes equation is given by

ρ∂ v

∂t+ ρ v · ∇ v − ρ g + ∇p′ − ∇· τ = 0 . (2.1)

For the remainder of this study, since g is a uniform vector field, −ρ g may bewritten as −∇Ω. The gravitational term is then included in the pressure gradientterm, i.e. ∇p refers to the vector sum of the gravitational term and the externalpressure gradient: ∇p = ∇(p′ − Ω). In the case of an incompressible, Newtonianfluid where the viscosity of the fluid is constant, equation (2.1) reduces to

ρ∂ v

∂t+ ρ v · ∇ v + ∇p− µ∇2 v = 0 . (2.2)

Here ρ∂ v∂t

is the density times the local acceleration or ∂ρ v∂t

can also be regarded asthe rate of the change in momentum per unit volume, ρ v ·∇ v is the convection termor v · ∇ (ρ v) can be viewed as the change in momentum due to the velocity field,v, per unit volume and µ∇2 v is the diffusion term where the viscosity of the fluid isthe diffusion coefficient. The source term, ∇p, is the external force responsible forthe fluid transport.

The general continuity equation is given by

∂ρ

∂t+ ∇· (ρ v) = 0 . (2.3)

5

If the fluid is incompressible, the time and the spatial derivative of the density iszero and from equation (2.3) it then follows that

∇· v = 0 . (2.4)

From equation (2.4) it follows that, for an incompressible Newtonian fluid, theNavier-Stokes equation can be rewritten as

ρ∂ v

∂t+ ρ∇· ( v v) + ∇p− µ∇2 v = 0 . (2.5)

6

2.1 The Unit Cell

To describe the process of fluid flowing through a porous medium, different modellingtechniques can be used. One technique is representing the porous medium by anensemble of identical unit cells. Duplicating and stacking the unit cells, the porousmedium can be reconstructed and a unit cell could thus be seen as an elementarybuilding block of the porous medium, as is shown in Figure 2.1. Note, however, thatit is physically impossible to reconstruct an isotropic porous medium of elementarybuilding blocks, since even though the medium might be constructed similarly in theaverage flow direction and the direction perpendicular to it, in the diagonal directionit will differ and thus the medium would be anisotropic. Therefore only porous mediathat is non-staggered in at least one direction can be modelled analytically usingthis technique. In the case of an isotropic porous medium where the solid materialis randomly distributed, other modelling methods have to be considered.

O

r1r2

r3r4

Figure 2.1: Duplicating and stacking unit cells.

In the case of a porous medium consisting of granules, for the sake of simplicity,the solid phase in the unit cell may be represented by a rectangular, rather thana circular or ellipsoidal, particle, without making too much of an error. A unitcell should be a representation of the porous medium on microscopic scale. Sincethis is the case, the unit cell should contain both an ‘north-flowing’ and ‘south-flowing’ transverse channel of the same width. This will ensure that the averageflow direction of the unit cell is that of the entire porous medium. This is shown inFigure 2.2. The unit cell must also have the same porosity as the porous medium.

The interstitial flow is assumed to be stationary (time independent), the fluid incom-pressible, Newtonian and free of body forces (or the body forces are incorporatedin the pressure gradient term) and creep flow is assumed. Therefore the flow isgoverned by the interstitial continuity equation:

∇· v = 0 , (2.6)

7

ds

dc

average flow direction

Figure 2.2: Piece-wise straight streamlines are assumed in a unit cell.

and the interstitial equation for creep flow (following from equations (2.5) and (2.6)):

∇p = ∇·τ = µ∇2 v .

Here v is the interstitial velocity defined at each point within the fluid phase of theunit cell.

Piece-wise straight streamlines are assumed as shown in Figure 2.2. In two dimen-sions this analytical modelling technique is based on a fully developed, piece-wiseplane Poiseuille flow approximation for interstitial flow between neighbouring parti-cles. In such an approximation, the wall shear stress and corresponding channel-wisepressure gradient are respectively given by

τw =6µw

dcand ‖−∇p‖ =

12µw

dc2. (2.7)

Here w is the average velocity of the assumed fully developed velocity profile in thechannel and dc is the normal distance between the facing surfaces. If the length ofthe two facing parallel plates is ds in the average flow direction, combining equations(2.7) yields

δp dc− τw(2ds) = 0 , (2.8)

where δp = ds ‖−∇p‖ represents the pressure drop along the length of the pair ofparallel plates. Note that equation (2.8) is the equilibrium equation of the forces inthe interstitial streamwise direction in a channel.

8

2.2 Volume averaging of transport equations

Another modelling technique for flow through porous media is volume averaging ofthe microscopic equations as was discussed by e.g. Bear (1972).

In porous media, the geometry of the bounding surface of the fluid phase, on whichthe boundary conditions are defined, is in general too complex to describe or is notobservable. Furthermore there could be values that varies from point to point withinthe phase under consideration. Therefore, microscopic equations governing the flowcannot be solved. Describing the transport process through a porous medium ata microscopic level is thus impossible, unless the medium can be represented bya regular array where, in that case, a unit cell may be used, as was discussed insection 2.1. If the solid material of a porous medium is randomly distributed, anotherapproach is necessary where continuous quantities may be determined and problemswith particular boundary conditions can be solved. Thus, describing the transportphenomena in a porous medium has to be, in most cases, on a macroscopic level.Each term in the microscopic equation has to be averaged over a certain volume.This method is called volume averaging and the volume over which the averaging isdone is known as the REV.

2.2.1 The REV

The first step in passing from the microscopic level, where we consider phenomenaat each point within a phase, to a macroscopic level, where we consider volumeaveraged quantities describing phenomena in the vicinity of a point, is accomplishedby introducing the REV. The REV (Representative Elementary Volume) is definedas a volume that is large enough to be statistically representative of the physicalproperties in the immediate vicinity of a point, i.e. the volume average over theREV of any physical quantity does not change abruptly with a small change in thepositioning of the REV within the porous medium. On the other hand it must bemuch smaller than the exterior domain of the porous medium. The total volume ofthe REV, Uo, consists of both fluid and solid parts, Uf and Us respectively. Thus

l3 << Uo << L3 (2.9)

where l is the average diameter of the particles and L3 represents the volume ofthe whole porous medium. The order of magnitude of these dimensions is shownschematically in Figure 2.3.

The position vector ro points to the centroid of the REV and r is the position vectorof any point within Uo. The porosity of the REV with centroid at ro is

ǫ| ro=

Uf | ro

Uo| ro

. (2.10)

9

Sff

Sfs

Sss

O( L)

O(l)

O(U

1

3o

)

O

ro

r

Figure 2.3: A spherical REV inside a porous medium.

We may, on a notational basis, drop the explicit referral to the particular positionvector, ro, and write the porosity of an REV as

ǫ=UfUo

. (2.11)

Let S denote the total outer surface of the REV. This surface consists of a fluid-fluid interface, Sff , and a solid-solid interface with the total area given by Sss. Theboundary of the fluid phase of the REV is denoted by Sof and it consists of Sffand Sfs, the latter being the area where there is a fluid-solid phase contact insidethe REV (thus the outer surface of the solid phase particles inside the REV). Theseareas are shown in Figure 2.3.

The phase average of Ψ, a tensorial quantity of any order defined within Uf , is givenby

〈Ψ 〉o =1

Uo

∫∫∫

Uf

Ψ dU . (2.12)

10

From its definition, it is evident that porosity is the phase average of unity. Theintrinsic phase average of Ψ is given by

〈Ψ 〉f =1

Uf

∫∫∫

Uf

Ψ dU . (2.13)

From equations (2.11), (2.12) and (2.13) it follows that

〈Ψ 〉o = ǫ 〈Ψ 〉f . (2.14)

The deviation of Ψ at any point within Uf , with respect to the particular REV, isdefined as

Ψf = Ψ − 〈Ψ 〉f , (2.15)

where Ψ is the actual value at that point.

v

v

v

Uf

Us

Us

UsUs

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

q

u

Figure 2.4: A schematic representation of an REV showing the different unit vectorsand velocity variables.

Unit vectors

With respect to any REV two important unit vector fields can be defined. Firstlythere is n, the unit vector in the average flow direction. From here on, all the hatted

11

vectors will denote vectors in the average flow (or streamwise) direction. Secondlyn is a unit vector defined at all the points within the fluid phase, directed in theinterstitial flow direction at that point within Uf . All the vectors directed parallelto the interstitial flow direction at that point will be denoted by a tilde. Anotherimportant unit vector is n, a vector normal to Sof , directed into the solid phase orto the outside of the REV. These different unit vectors are shown in Figure 2.4.

Velocities defined within an REV

Let v denote the interstitial velocity at each point in a fluid phase at a certain time.The phase average of v is the superficial (or Darcy) velocity and is given by

q = 〈 v 〉o =1

Uo

∫∫∫

Uf

v dU . (2.16)

The streamwise direction, n, is defined at any point as the direction of the superficialvelocity of the REV pertaining to that point, namely

n=q

‖ q‖ . (2.17)

From equations (2.16) and (2.17) it is evident that∫∫∫

Uf

v dU

is a vector in the streamwise direction.

The intrinsic phase average of v (the average of the intrinsic velocity in the channelstaken over Uf ) is

u = 〈 v 〉f =1

Uf

∫∫∫

Uf

v dU . (2.18)

Considering equations (2.16) and (2.18) it follows from equation (2.14) that

q= ǫ u (2.19)

and this is known as the Dupuit-Forchheimer relationship, Dupuit (1863), as referredto by Carman (1937).

Let the term drift velocity denote the average streamwise displacement per unit timeat which a particle meanders through the porous medium. In Figure 2.5 the drift

12

Ug

UgIn flow Out flow

n

A B

Figure 2.5: A schematic representation of a portion of the REV illustrating driftvelocity.

velocity is the distance AB divided by the time (also referred to as the residencetime) the particle took to travel the distance AB while meandering through thechannels. Thus, if there are stagnant regions of volume Ug forming part of Uf in theREV, the drift velocity, utR , is defined as

utR =1

Uf − Ug

∫∫∫

Uf

v dU , (2.20)

where Ug is the volume in which the interstitial velocity of the fluid is zero. If thereare no stagnant regions, it follows that

utR = u = 〈 v 〉f . (2.21)

2.2.2 The REA

Instead of a volumetric REV, a Representative Elementary Area (an REA), withthe same centroid, may also be used to define areal average quantities relating to theporous medium. For homogeneous, isotropic media, areal and volumetric averagesof geometric entities will be the same.

The total area of the plane, Ao, consists of both fluid parts, Af , and solid parts, As.Again the shape of the boundary is unimportant, as long as

l2 << Ao << L2 , (2.22)

where l is again defined as the length of the pore scale and L is the length scale ofthe physical boundaries of the porous medium, as is shown in Figure 2.6.

The areal porosity is defined as

ǫA =Af

Ao

(2.23)

13

Lff

Lfs

Lss

O( L)

O(l)

O(Ao

1

2

)

O

ro

r

Figure 2.6: A circular REA inside a porous medium.

and it has been shown by Bear (1972) that

ǫA ≡ ǫ .

The two-dimensional position vectors are defined similarly as in section 2.2.1. Thedifferent surfaces discussed for the REV are replaced by lines in the REA and thesame phase operators are applicable. The integrals taken over Uo and Uf are also tobe replaced by integrals taken respectively over Ao and Af .

2.2.3 Volume averaging by means of the REV

Describing the transport phenomena in a porous medium has to be on a macroscopicscale, as was already mentioned. Each term in a microscopic equation has to beaveraged over an REV. This is known as volume averaging.

Volume averaging of the Continuity Equation

The microscopic equation for the conservation of mass for a fluid is given by

∂ρ

∂t+ ∇·(ρ v) = 0 . (2.24)

Volume averaging of this equation yields⟨∂ρ

∂t

⟩

o

+ 〈∇·(ρ v) 〉o = 〈0 〉o , (2.25)

14

which, since Sof has zero velocity and the no-slip boundary condition holds, reducesto

∂〈ρ 〉o∂t

+ ∇· 〈ρ v 〉o = 0 . (2.26)

It has already been assumed that the fluid is incompressible (thus ρ is spatiallyand temporally constant) and again using the no-slip boundary condition, equation(2.26) can be written as

∇·(〈ρ 〉o 〈 v 〉o

ǫ

)+ ∇·

⟨ρf vf

⟩o

= 0 . (2.27)

There is no deviation in the density of the fluid phase and thus

〈ρ 〉o[∇·〈 v 〉o

ǫ

]= 0 . (2.28)

If the porosity is spatially constant, it follows that

∇· 〈 v 〉o = 0

and the macroscopic continuity equation for a porous medium under these circum-stances is

∇· q = 0 . (2.29)

Volume averaging of the Navier-Stokes Equation

The momentum transport equation for an incompressible fluid is governed by theinterstitial Navier-Stokes equation that can be written as:

ρ∂ v

∂t+ ∇· (ρ v v) + ∇p− ∇·τ = 0 . (2.30)

Here ρ is the density of the fluid and v is the velocity of the fluid at a particularpoint and equation (2.30) is thus a microscopic equation. After volume averagingof equation (2.30), the following equation is obtained:

⟨ρ∂ v

∂t

⟩

o

+ 〈∇· (ρ v v) 〉o + 〈∇p 〉o −⟨∇·τ

⟩o

= 0 . (2.31)

It is assumed that the density is spatially as well as temporally constant and thatthe solid phase of the porous medium, Sof , is stationary, thus vSof

= 0. Equation

15

(2.31) therefore reduces to

0 = ρ∂〈 v 〉o∂t

− ρ

Uo

∫∫

Sfs

n · vSofv dS − ρ

Uo

∫∫

Sff

n · vSofv dS

+ρ∇·(〈 v 〉o 〈 v 〉o

ǫ+⟨ vf vf

⟩o

)+

ρ

Uo

∫∫

Sfs

n · v v dS

+ǫ∇〈p 〉f +1

Uo

∫∫

Sfs

n pf dS − ∇·⟨τ⟩o− 1

Uo

∫∫

Sfs

n · τ dS

= ρ∂ q

∂t+ ρ∇·

(q q

ǫ

)+ ρ∇·

⟨ vf vf

⟩o

+ ǫ∇〈p 〉f

−∇·⟨τ⟩o

+1

Uo

∫∫

Sfs

(n pf − n · τ

)dS (2.32)

since n · v = 0 everywhere on the fluid-solid interface and 〈 v 〉o = q.

In the case of a Newtonian fluid where µ is constant, ∇·τ can be substituted by µ∇2 vin the Navier-Stokes equation. If the porous medium is stationary, i.e. vSof

= 0,volume averaging of equation (2.5) yields

0 = ρ∂〈 v 〉o∂t

+ ρ∇·(〈 v 〉o 〈 v 〉o

ǫ+⟨ vf vf

⟩o

)+ ǫ∇〈p 〉f +

1

Uo

∫∫

Sfs

n pf dS

−µ∇2 〈 v 〉o −µ

Uo

∫∫

Sfs

n · ∇ v dS − µ

Uo∇·

∫∫

Sfs

n v dS

. (2.33)

It is assumed that the no-slip boundary condition holds, thus v = 0 everywhere onSfs. Equation (2.33) therefore reduces to

0 = ρ∂ q

∂t+ ρ∇·

(q q

ǫ+⟨ vf vf

⟩o

)+ ǫ∇〈p 〉f − µ∇2 q

+1

Uo

∫∫

Sfs

(n pf − n · µ∇ v

)dS . (2.34)

Substituting equation (2.29) into equation (2.34), and using the identity∇2Ψ = ∇(∇·Ψ) −∇× (∇×Ψ), yield

0 = ρ∂ q

∂t+ ρ q · ∇ q + ρ∇·

(⟨ vf vf

⟩o

)+ ǫ∇〈p 〉f + µ

[∇×

(∇× q

)]

16

+1

Uo

∫∫

Sfs

(n pf − n · µ∇ v

)dS .

We assume a uniform field for the superficial velocity, therefore the gradient and thecurl of q are zero and, since the interstitial velocity was assumed to be stationaryin the unit cell model and vSof

= 0, it follows that q is also time independent andthus

0 = ρ∇·(⟨

vf vf⟩o

)+ ǫ∇〈p 〉f +

1

Uo

∫∫

Sfs

(n pf − n · µ∇ v

)dS . (2.35)

Equation (2.35) is still open in the sense the actual values of the interstitial velocityare required to solve the integral. To be able to determine the interstitial velocityfield the exact geometry of the porous medium should be known. Since this is notthe case, further modelling of the porous microstructure is needed to close thisequation. Closure will be done by means of a rectangular representative unit cell.

17

2.3 The Rectangular Representative Unit Cell

Since the integral over the fluid-solid interface in the macroscopic momentum trans-port equation, equation (2.35), still has to be evaluated on a microscopic scale,closure is needed to determine this integral. In this thesis the pore-scale mod-elling technique used to solve this integral is based on the RRUC (RectangularRepresentative Unit Cell), as was introduced by Du Plessis & Masliyah (1988).(Note that here, the ‘unit cell’ in the abbreviation ‘RRUC’ does not refer to the unitcell as it was defined in section 2.1 in any way.) For this modelling technique onlythe morphology of the fluid volume is required from which, with some assumptions,the interstitial flow pattern can be modelled analytically.

Fluid traversing a porous medium can be viewed as fluid flowing through manystreamtubes, since, due to the incompressibility of the fluid phase, if fluid ‘escapes’a particular streamtube, fluid from the other streamtube should exchange positionwith that fluid. This is unlikely to happen and the RRUC modelling technique isbased on this streamtube assumption. For mathematical simplicity, the RRUC has arectangular shape. Four surface areas are chosen parallel to the enclosed streamtubeand the other two are orientated perpendicular to it. It is thus assumed that no fluidexits or enters the RRUC except through the cross-stream surfaces. An RRUC isthe smallest possible cell in which the statistical average geometrical properties of anREV are imbedded. Its porosity, tortuosity (see Chapter 4), streamwise direction,pressure gradient, superficial velocity and thus also its permeability should be thesame as the corresponding REV, which is on its turn a representation of the entireporous structure.

Since the properties of an REV (for example porosity – see equation (2.10)) aredefined in the vicinity of a point indicated by the position vector ri, the RRUCrepresenting that REV is also positioned with its centroid at ri. Figure 2.7 is aschematic representation of the positioning of the RRUCs inside their two respectiveREVs. Note that, contrary to unit cells, RRUCs do overlap.

In this thesis the RRUC differs from the RRUC defined for granular porous mediaby Du Plessis & Masliyah (1991) in the sense that the streamwise direction of theflow through the porous medium is the same as the streamwise direction throughthe RRUC. In the RRUC defined by Du Plessis & Masliyah (1991), two adjacent,complimentary RRUCs (in a two-dimensional case) have to be considered to obtainthe same streamwise direction and to eliminate unwanted induced swirl effects. Inthe present work it is preferred to refer to both adjacent RRUCs as the RRUC. TheRRUC defined by Du Plessis & Masliyah (1991) will from here on be referred to asthe simplistic RRUC.

Let 2Uo denote the volume of the RRUC. The volume occupied by the fluid phase isindicated by 2Uf and 2Us represents the volume of the solid phase. The fluid-solidinterface is represented by Sfs. The outer dimensions of both the solid particles

18

O

ror1

Figure 2.7: RRUCs inside their respective spherical REVs, where the dark greyindicates the volume where the two RRUCs overlap.

Us

Us

ds⊥

dc⊥

d⊥

ds‖ dc‖

2d‖

Figure 2.8: The different dimensions defined for the RRUC.

are ds‖ × ds⊥ × dsL where the subscripts indicate whether the surface is parallel orperpendicular to the streamwise direction. In this thesis only two dimensions areconsidered and dsL is set equal to unity. Let the outer dimensions of the RRUCbe 2d‖ × d⊥ × 1. The reason for choosing the streamwise dimension as 2d‖ will bediscussed in more detail in Chapter 3. For now it is sufficient to note that the ratioof the sum of the streamwise dimensions of the solid phases to that of the RRUC

isds‖d‖

. The streamwise channel widths are denoted by dc⊥ and the widths of the

transverse channels by dc‖.

The basic structure of the RRUC is thus exactly the same as that of the unit cell,as is depicted in Figure 2.2. However, where the unit cell can be seen as a repetitivebuilding block, because by duplicating and stacking unit cells the whole porousmedium can theoretically be reconstructed, the RRUC should not be viewed in that

19

Initial position Final position

A AB BC CD D

streamwise direction

E F G

Figure 2.9: Weighted shifting of the RRUC.

manner. The RRUC should rather be seen as a small control volume that representsthe average pore-scale properties of the porous medium in the close vicinity of thecentroid of the RRUC. Since the pressure gradient over the RRUC should be thesame as over the REV, equation (2.35) can be rewritten as follows for an RRUC:

−ǫ∇〈p 〉f = ρ∇·(⟨

vf vf⟩o

)+

1

2Uo

∫∫

Sfs

(n pf − n · µ∇ v

)dS . (2.36)

It is assumed that

ρ∇·(⟨

vf vf⟩o

)and

∫∫

Sfs

n 〈p 〉f dS

are both equal to zero. These two assumptions are discussed in the Appendix A.Equation (2.36) therefore reduces to

−∇〈p 〉f =1

2Uf

∫∫

Sfs

(n p− n · µ∇ v) dS . (2.37)

Lloyd (2003) introduced the idea of shifting the RRUC in the streamwise directionby weighing each possible RRUC structure by its relative frequency of occurrence.To obtain each possible RRUC structure, it should be shifted until the upstreamcross-sectional face lies in the position where the downstream cross-sectional facewas before the shifting took place. This is illustrated in Figure 2.9.

Equation (2.37), evaluated over the RRUC when its cross-stream faces lie somewhereon A or C, should be multiplied by the ratio of the streamwise dimension of the

20

solid phase to the streamwise dimension of the RRUC. The values obtained for theintegral when the cross-stream faces lie somewhere on B or D, should be multipliedby the ratio of the width of the transverse channel to the streamwise dimensionof the RRUC. Addition of these four terms and dividing it by the volume of thefluid phase of the RRUC will yield the gradient of the intrinsic phase average of thepressure:

−∇〈p 〉f =1

2Uf

ds‖2d‖

∫∫

SfsAA

(n p− n · µ∇ v) dS +dc‖2d‖

∫∫

SfsBB

(n p− n · µ∇ v) dS

+ds‖2d‖

∫∫

SfsCC

(n p− n · µ∇ v) dS +dc‖2d‖

∫∫

SfsDD

(n p− n · µ∇ v) dS

. (2.38)

The integrals over BB and DD, as well as the integrals over AA and CC, yield thesame answer, but both are two times the integral evaluated over a simplistic RRUC,since the streamwise length is twice as long for the RRUC and the pressure gradientremains constant. Equation (2.38) may thus be rewritten for a simplistic RRUC as

−∇〈p 〉f =1

Uf

ds‖2

· 1

d‖

∫∫

SfsAC

(n p− n · µ∇ v) dS +ds‖2

· 1

d‖

∫∫

SfsCA

(n p− n · µ∇ v) dS

+dc‖d‖

∫∫

SfsBD/DB

(n p− n · µ∇ v) dS

. (2.39)

The integrals over SfsAC and SfsCA are kept separate, because their respective wallshear stress terms will differ, depending on the direction of the interstitial flow inthe transverse channels. The weighted shifting method given by equation (2.38) istherefore equivalent to shifting only half the RRUC (referred to as the simplisticRRUC) over the length of an RRUC. Considering equation (2.39) it is evident thatthis weighted shifting of the control volume should start with its cross-stream sur-faces situated halfway through the two solid phases (indicated with lines E and Fon Figure 2.9) and is then shifted in the streamwise direction until the upstreamcross-sectional face again lies in the position where the downstream cross-sectionalface was situated before the shifting took place (in other words, lines F and G onFigure 2.9). This will again be discussed in Chapter 3, section 3.2.

The word ‘shifting’ should not be seen as physically translating the RRUC relativeto the porous structure as was schematically illustrated in Figure 2.9. The centroidof the RRUC is always situated at the centroid of the REV it is representing. Figure

21

2.9 should thus merely be seen as an explanation to the specific weighing coefficientschosen for the respective RRUC structures encountered in travelling along the steam-wise direction. Figure 2.10 shows the different simplified RRUC structures that mayoccur. The weighted average of these structures thus represents the best possibleaverage structure of the simplified RRUC that should be considered in the closureof the volume averaged Navier-Stokes equation.

roO

roO

roO

roO

ds‖2d‖

×

dc‖d‖

×

ds‖2d‖

×

ordc‖d‖

×

Figure 2.10: Different simplified RRUC structures encountered.

22

Chapter 3

Derivation of the permeability inthe Darcy regime

In this chapter an equation for the dimensionless permeability for the streamwisedirection of an anisotropic porous medium is derived. Firstly the permeability is ob-tained by means of a direct method, similarly to the analytical method discussed byFirdaouss & Du Plessis (2004), where the pressure drop over an SUC (Simplified UnitCell) is determined directly. This model is referred to as the direct method. Second-ly the same equation is derived by volume averaging the interstitial Navier-Stokesequation and performing closure by means of an SRRUC (Simplified RectangularRepresentative Unit Cell).

The SUC and the SRRUC are only half the length of the unit cell and RRUC inthe streamwise direction. Also, the porous medium cannot be reconstructed by du-plicating and stacking SUCs. In the model that will be considered in this chapter,all the solid ‘particles’ will have the same dimension, from which it follows that allthe parallel channels have the same dimensions and all the transverse channels havethe same dimensions. Thus, half the RRUC or unit cell contains all the microscopicproperties needed for the closure of the volume averaged Navier-Stokes equation.In the discussion that follows, the solid phase and the SUC (as well as the SRRUCchosen for the closure of equation (2.35)) are represented by rectangles with any in-dependent randomly chosen aspect ratios. Following Firdaouss & Du Plessis (2004),the special case where the solid phase and the SUC have the same aspect ratiowill be examined and compared to their numerical results. (The direct analyticalapproach by Firdaouss & Du Plessis (2004) will be referred to as the FD model.)Another special case which will be studied is when the transverse channel width isequal to the streamwise channel width. Some asymptotic conditions will also beconsidered.

The fluid-solid interface parallel to the streamwise direction is represented by ds‖and the perpendicular interface by ds⊥. The dimensions of the SUC (as well asthe SRRUC) are represented by d‖ and d⊥ in the streamwise and the transverse

23

directions respectively, as are shown in Figure 3.1. (Thus the dimensions of theunit cell and the RRUC are 2d‖ × d⊥.) The width of the channel in which the flowis in the streamwise direction is represented by dc⊥ and the width of the channeloccupied by transversely flowing fluid is represented by dc‖. Therefore dc⊥=d⊥−ds⊥and dc‖ =d‖−ds‖.

nmean flow

ds‖

ds⊥

d‖

d⊥

dc‖2

dc⊥2

Figure 3.1: Notation for the SUC and SRRUC with respect to the streamwise (ormean flow) direction.

Some aspect ratios have to be defined to keep the model as general as possible.Firstly the aspect ratio of the SUC and the relation of the streamwise channel widthto the transverse channel width are given by

d⊥d‖

≡ ϕ , 0 < ϕ <∞ (3.1)

and

dc⊥dc‖

≡ ψ , 0 < ψ <∞ (3.2)

respectively. The relation of the parallel channel width to the dimension of theperpendicular side of the SUC is given by

dc⊥d⊥

≡ γ , 0 < γ < 1 (3.3)

and it then follows that

dc‖d‖

≡ ϕγ

ψ, 0 <

ϕγ

ψ< 1 . (3.4)

(Later in this chapter it is shown that, for a staggered array, the range of γ is actually0 < γ ≤ 1

2for the model, that is still to be discussed, to be valid.) If the widths of

the respective channels are the same,

ψ = 1. (3.5)

From the above relations, the dimensions of the solid phase particles are given by

ds⊥ = d⊥ − dc⊥ = dc⊥

(1 − γ

γ

)(3.6)

24

and

ds‖ = d‖ − dc‖ = dc‖

(ψ − ϕγ

ϕγ

). (3.7)

The aspect ratio of the rectangular solid phase particles is thus given by

ds⊥ds‖

≡ ϕψ(1 − γ)

ψ − ϕγ. (3.8)

In the FD model, where the aspect ratios of the SUC and the solid phase are equaland defined as α, the above relations simplify to

ϕ = ψ = α and γ = 1 −√

1 − ǫ . (3.9)

Also note that, in this particular case, equation (3.8) reduces, as expected, to

ds⊥ds‖

= α .

The porosity of the porous medium, which should be equal to the porosity of theSUC, is given by

ǫ ≡ dc‖ds⊥ + dc⊥d‖d‖d⊥

= γ

[ϕ

ψ(1 − γ) + 1

](3.10)

and useful relations for later use, are

ϕ(ǫ, γ, ψ) =ψ(ǫ− γ)

γ − γ2and ψ(ǫ, γ, ϕ) =

ϕ(γ − γ2)

ǫ− γ(3.11)

as these equation will allow elimination of either ϕ or ψ from equations in favour ofthe porosity, which is a measurable parameter.

Three different levels of staggering of the solid phase in the streamwise directionwill be studied: the regular, the over-staggered and the fully staggered array. Theseconfigurations are shown in Figure 3.2. In nature, a porous medium of which thestreamwise staggering is represented by an over-staggered array, is rarely encoun-tered. In practise such a configuration can be obtained mechanically by restrictingfluid flow as is shown in Figure 3.2(c). The over-staggered array is studied herepurely for academic purposes.

A parameter ξ which relates to the cross-stream staggeredness of the solid material,is defined as follows:

ξ =

0 Regular array

12

Fully staggered array

1 Over−staggered array .

(3.12)

25

(a)

Regular arraywhere ǫ = 0.75.

(b)

Fully staggered arraywhere ǫ ≈ 0.38.

(c)

Over-staggered arraywhere ǫ ≈ 0.66.

n

Figure 3.2: An illustration of the different arrays studied, as well as the SUCs chosenfor the different scenarios.

For the derivations of the permeability in this chapter, the configurations depictedin Figures 3.3, 3.4 and 3.5 are considered. The volume (the length in the thirddimension is unity) of an SUC (or an SRRUC) occupied by fluid flowing parallel tothe net streamwise direction is given by U‖, and U⊥ represents the volume of thetransverse channel.

UtA UtBU‖

Ug

U‖

Ug

UtC UtDU‖

Ug

U‖

Ug

Figure 3.3: The present model considered for a regular configuration where thepressure values in the different zones are pUtA

= p + δp‖, pUtB= p, pUtC

= p − δp‖and pUtD

=p− 2δp‖ respectively.

For a staggered array (Figures 3.4 and 3.5), in volume Ut, one side is bounded bya fluid-solid interface on which the wall shear stresses are neglected. (This is themain difference between the present model and the FD model.) The rest of Ut isbounded by fluid only. The shear stress induced by the surrounding fluid over Utis assumed negligible to first order accuracy. Under such conditions, it is shown in

26

UtB

4

UtB

4UtC

4

UtC

4

UtA

2UtD

2U‖

U⊥

2

U⊥

2

U⊥

2

U⊥

2

U‖

2

U‖

2Ut

4

Ut

4Ut

4

Ut

4

U‖

U⊥

2

U⊥

2

U‖

2

U‖

2

U⊥

2

U⊥

2

Ut

2Ut

2

Figure 3.4: The present model considered for a fully staggered configuration wherethe pressure values in the different zones are pUtA

=p+ 12δp⊥, pUtB

=p, pUtC=p−δp‖

and pUtD=p− δp‖ − 1

2δp⊥ respectively.

UtA2

UtB2

UtC2

UtD2

U‖

U⊥

U‖

U⊥

Ut2

Ut2

Ut2

Ut2

U‖

U⊥

U‖

U⊥

Figure 3.5: The present model considered for an over-staggered configuration wherethe pressure values in the different zones are pUtA

=p+ δp⊥, pUtB=p, pUtC

=p− δp‖and pUtD

=p− δp‖ − δp⊥ respectively.

Appendix B that, for a staggered array, in the transverse channels, the net effect ofthe pressure gradient over Ut is zero, and Ut merely acts as a transfer volume. InFigure 3.6, the difference between the assumptions made regarding the wall shearstresses and the transfer volumes in the present model to that of the FD model, isshown. Note that in the present model, for a staggered configuration, part of Utconsists of fluid flowing perpendicular to the streamwise direction, whereas in theFD model, Ut consists entirely of streamwise flowing fluid.

27

ds⊥−dc⊥

U‖

U⊥

Ut ‖ + Ut⊥2

Ut ‖ + Ut⊥2

(a) The present model

ds⊥

U‖

U⊥

Ut ‖2

Ut ‖2

(b) The FD model

Figure 3.6: The fluid-solid interfaces on which wall shear stress acts.

3.1 Direct analytical modelling

A direct analytical method for the determination of the hydrodynamic permeabilityin the Darcy regime in terms of porosity was used in the FD model. Firdaouss &Du Plessis (2004) examined SUCs with different dimensions representing differentlevels of streamwise staggering for which a general expression for the dimensionlesspermeability was obtained. This analytical model was based on a piece-wise planePoiseuille flow approximation for the interstitial flow between neighbouring particles.In the FD model, however, the pressure gradient in the transverse channel was takenover the entire ds⊥ (since the wall shear stress over the entire perpendicular fluid-solid interface was considered – see Figure 3.6), and not over ds⊥ − dc⊥ as is donehere. In this section an identical analytical method to the FD model is used toobtain a general expression (thus the aspect ratio of the SUC is not dependent onthe aspect ratio of the solid phase) for the permeability of the model presented inFigures 3.3, 3.4 and 3.5.

The Darcy permeability in the streamwise direction is given by

k ≡ µq

‖∇p‖ =µq

δp/d‖(3.13)

and the dimensionless permeability is defined as

K ≡ k

d‖d⊥=

µq

d⊥δp. (3.14)

It might make more sense physically to divide the permeability by the area of theSUC perpendicular to the streamwise direction since the Darcy (or superficial) velo-city, q, is the volumetric flow rate per cross-sectional area. In these two-dimensionalcase studies, this area is d⊥ since the third dimension is taken as unity. Division bythis area then leads to the expression

K ′ ≡ µQ

d2⊥ ‖∇p‖ , (3.15)

28

where K ′ is a dimensionless permeability, but non-dimensionalised differently thanK in equation (3.14), and Q is the volumetric flow rate. The general expressionfor K ′ would be more complex, since it would not be possible to write it in termsof geometrical ratios. Therefore, for the remainder of this thesis, the dimensionlesspermeability refers to K as defined in equation (3.14).

3.1.1 Streamwise regular array

The streamwise flux is conserved due to the assumed fluid incompressibility andtherefore the magnitude of the average interstitial velocity in the parallel channels,presented in Figure 3.3, can be expressed in terms of the flux as

w‖ =qd⊥dc⊥

=q

γ. (3.16)

The streamwise pressure gradient for plane Poiseuille flow is

−∇‖p =12µw‖dc2⊥

=12µq

d2⊥γ

3. (3.17)

Here ∇‖ is defined as a scalar gradient operator in the streamwise direction. Thepressure gradient given in equation (3.17) is only applicable to U‖, where wall shearstress is acting in on the traversing fluid due to the fluid-solid interfaces. The totalpressure drop over the SUC is thus given by

δp = δp‖ = −∇‖pds‖ =12µq

d⊥γ2ψ

[ψ − ϕγ

ϕγ

]. (3.18)

Substituting equation (3.18) into equation (3.14), and eliminating one of the para-meters by means of equation (3.11), the dimensionless permeability for a regulararray can be written as follows:

K =γ2ψ(ϕγ)

12(ψ − ϕγ)=γ2ψ(ǫ− γ)

12(1 − ǫ)=γ3ϕ(1 − γ)

12(1 − ǫ). (3.19)

Substituting equations (3.9) into equation (3.19), it follows that, for a model wherethe aspect ratios of the SUC and solid phase are equal, the dimensionless per-meability is

K =(1 −

√1 − ǫ)3α

√1 − ǫ

12(1 − ǫ)=

(1 −√

1 − ǫ)3α

12√

1 − ǫ. (3.20)

This expression for the dimensionless permeability for a regular array was also ob-tained in the FD model – Firdaouss & Du Plessis (2004), their equation (15).

29

If the width of the transverse channel is equal to the width of the parallel channel,equation (3.19) reduces to

K =γ2(ǫ− γ)

12(1 − ǫ). (3.21)

3.1.2 Streamwise staggered array

From flux-conservation and fluid incompressibility it follows for a streamwise stag-gered array, Figures 3.4 and 3.5, that

w⊥ = ξw‖dc⊥dc‖

= ξψw‖ . (3.22)

The total pressure drop over the SUC consists as follows of pressure drops in boththe parallel and the transverse channels:

δp = δp‖ + ξδp⊥ . (3.23)

The pressure drop in the parallel channel is given by equation (3.18). The pressuredrop in the transverse channel consists of both the pressure drop over ds⊥−dc⊥and the pressure drops at the fluid-solid interfaces bounding Ut, where the wallshear stresses were neglected – see Figure 3.6. The two pressure drops over dc⊥ areexamined in Appendix B where it is shown that these two pressure drops are equalin magnitude, but orientated in opposite directions and thus cancel vectorially. Thepressure drop in the transverse channel (where ∇⊥ is defined as a scalar gradientoperator in the transverse direction of flow) is thus given by

δp⊥ =−∇⊥p (ds⊥ − dc⊥)

=12µw⊥dc2‖

(2ds⊥ − d⊥)

= ξψ3 12µw‖dc2⊥

(2ds⊥ − d⊥)

=−ξψ3∇‖p (2ds⊥ − d⊥)

=−ξψ3∇‖p

(2ds‖ϕψ(1 − γ)

ψ − ϕγ− d⊥ds‖

ds‖

)

=−ξψ3∇‖pds‖

(2ϕψ(1 − γ)

ψ − ϕγ− d⊥ϕγψ

dc⊥(ψ − ϕγ)

)

=−ξψ3∇‖pds‖

(2ϕψ(1 − γ)

ψ − ϕγ− ϕψ

(ψ − ϕγ)

)

30

=−ξϕψ4∇‖pds‖

(1 − 2γ

ψ − ϕγ

). (3.24)

From equations (3.18), (3.23) and (3.24) it follows that the total pressure drop overthe SUC is given by

δp = −∇‖pds‖ + ξδp⊥ = −∇‖pds‖

[1 + ξ2ϕψ4

(1 − 2γ

ψ − ϕγ

)]. (3.25)

Substitution of equations (3.25) and (3.17) into equation (3.14) yields the followingdimensionless permeability in the Darcy regime:

K =µq

d⊥δp

=µq

−d⊥∇‖pds‖[1 + ξ2ϕψ4

(1−2γψ−ϕγ

)]

=γ3ψϕ

12(ψ − ϕγ)[1 + ξ2ϕψ4

(1−2γψ−ϕγ

)]

=γ3ψϕ

12 [ψ − ϕγ + ξ2ϕψ4 (1 − 2γ)]. (3.26)

If the aspect ratios are α, the dimensionless permeability is given by

K =α2(1 −

√1 − ǫ)3

12[α− α(1 −

√1 − ǫ) + ξ2α5

(1 − 2(1 −

√1 − ǫ)

)]

=α(1 −

√1 − ǫ)3

12[√

1 − ǫ+ ξ2α4(2√

1 − ǫ− 1)] , (3.27)

as was obtained by Cloete & Du Plessis (2006), their equation (17).

If the parallel and the transverse channels have the same width, the dimensionlesspermeability can be written as follows in terms of the porosity and γ:

K =γ3ϕ

12 [1 − ϕγ + ξ2ϕ (1 − 2γ)]=

γ3(ǫ− γ)

12γ(1 − ǫ) + 12ξ2(ǫ− γ)(1 − 2γ). (3.28)

To better understand the physical origin of some of the coefficients in equation(3.26), the gradient of the pressure is examined. It can be written as follows interms of the wall shear stresses in the parallel and the transverse channels:

‖−∇p ‖ =δp

d‖

31

=δp‖ + ξδp⊥

d‖(3.29)

=δp‖ds‖

· ds‖d‖

+δp⊥

2ds⊥ − d⊥· ξ(2ds⊥ − d⊥)

d‖

= −∇‖p ·ds‖d‖

−∇⊥p ·ξ(2ds⊥ − d⊥)

d‖

=12µw‖dc2⊥

· ds‖d‖

+12µw⊥dc2‖

· ξ(2ds⊥ − d⊥)

d‖

=12µw‖ds‖dc2⊥d‖

1 +

w⊥w‖

.ξ(2ds⊥ − d⊥)

ds‖

(dc⊥dc‖

)2 (3.30)

=1

d‖dc⊥· 2ds‖

6µw‖dc⊥

1 +

w⊥w‖

.ξ(2ds⊥ − d⊥)

ds‖

(dc⊥dc‖

)2

=1

d‖dc⊥

[τ‖S‖ + ξτ⊥S⊥

(dc⊥dc‖

)]

=1

d‖dc⊥

[τ‖S‖ + ψξτ⊥S⊥

]. (3.31)

Here S‖ and S⊥ represent the fluid-solid interfaces on which a wall shear stress exertsand are equal to (2ds‖) and (4ds⊥ − 2d⊥) respectively.

From equation (3.25) it is evident that the ratio between the pressure drop generatedin the transverse part and that generated in the streamwise part of the SUC, is

ξ2ϕψ4

(1 − 2γ

ψ − ϕγ

). (3.32)

From equation (3.30), the origin of the respective variables in expression (3.32) canbe determined. The extra staggering parameter, ξ, as well as one of the ψ factorsoriginates from the velocity magnitude ratio. A ψ2 factor is the squared ratio of thedimensions of the transverse and the streamwise channels in equation (3.30). Theonly part of the second term of equation (3.30) which has not yet been consideredis

2ds⊥ − d⊥ds‖

. (3.33)

It is thus expected that expression (3.33) is equivalent to ϕψ

(1 − 2γ

ψ − ϕγ

)and indeed,

substitution of the geometrical parameters into expression (3.33) yields:

2ds⊥ − d⊥ds‖

= 2

[ϕψ(1 − γ)

ψ − ϕγ

]− d⊥ϕγ

dc‖(ψ − ϕγ)

32

= 2

[ϕψ(1 − γ)

ψ − ϕγ

]− ϕψ

(ψ − ϕγ)

= ϕψ

(1 − 2γ

ψ − ϕγ

). (3.34)

The dimensionless permeability is inversely proportional to the pressure gradient andhence the numerator of expression (3.32), ξ2ϕψ4(1−2γ), appears in the denominatorof the dimensionless permeability, equation (3.26).

33

3.2 Volume averaging and closure of momentum

equations

Closure of the volume averaged Navier-Stokes equation by means of an SRRUC(equation (2.39)) and shifting it as depicted in Figure 3.7 yield

−∇〈p 〉o =1

Uo

∫∫

Sfs

n p dS − 1

Uo

∫∫

Sfs

µn · ∇ v dS (3.35)

=1

Uf

∫∫

Sfs‖

n pw dS +ds‖

2d‖Uf

∫∫

Sfs⊥AA

n pw dS +dc‖d‖Uf

∫∫

Sfs⊥BB

n pw dS

+ds‖

2d‖Uf

∫∫

Sfs⊥CC

n pw dS +1

Uf

∫∫

Sfs

n pw dS

− 1

Uf

∫∫

Sfs‖

µn · ∇ v dS − dc‖d‖Uf

∫∫

Sfs⊥BB

µn · ∇ v dS

− ds‖2d‖Uf

∫∫

Sfs⊥AA

µn · ∇ v dS − ds‖2d‖Uf

∫∫

Sfs⊥CC

µn · ∇ v dS . (3.36)

Note that in this equation the tilde on the wall pressure term indicates a deviation,whereas the tilde on the velocity term still refers to the fact that this velocity isthe interstitial velocity and is directed in the interstitial fluid flow direction at aparticular point in the fluid phase.

ds‖2

ds‖2

A A

dc‖ dc‖

B B

nstreamwise

ds‖2

ds‖2

CC

Figure 3.7: The shifting method of the SRRUC for an over-staggered configuration.

34

The pressure term was split into an average channel surface section pressure anda wall pressure deviation for each and every channel surface section. In the modelpresented by Lloyd et al. (2004) it has been assumed that: “ . . . the pressure devia-tions are caused by shear stress at the transverse surfaces and the pressure deviationintegral thus provides the streamwise effect of the transverse integral deleted fromequation (9). . . ” (Lloyd et al. (2004), (sic)). (The transverse integral in their equa-tion (9) corresponds to the last three terms of equation (3.36) in this present work.)If it is assumed (as is assumed in this study) that this pressure deviation refers tothe deviation with respect to the average pressure on each surface section, this isquite an unconvincing assumption by Lloyd et al. (2004), since the deviation of anyentity from its mean value defined on a surface should be zero if integrated overthat surface. The shear stress term over the transverse surfaces can therefore notoriginate from the pressure deviation term. (Note that in the conference proceedingsby Lloyd et al. (2004), if their definition of pw however refers to the average pressureover both the channel surface sections in the transverse channel, their integral overthe pressure deviation correlates with the integral over the average pressure in thisstudy.)

In accordance with the definition of deviation, the integral of the wall pressuredeviation is set equal to zero in the present model. The integral of the average wallchannel pressure is then split into an integral over the fluid-solid interface in the fluidchannels parallel to the streamwise direction and one over the fluid-solid interfacein the transverse fluid channels. The integral over the parallel channel will be zero,since the average wall channel pressures on the upper and the lower surfaces will beequal and thus cancel vectorially. The integral over the transverse channels is thensplit into three integrals which are weighed according to their relative frequency ofoccurrence if the SRRUC is shifted in the streamwise direction. The Sfs⊥BB termcorresponds to the instances when the boundaries of the SRRUC are situated in thetransverse fluid channels. The Sfs⊥AA term and the Sfs⊥CC term correspond to theinstances when the transverse boundaries of the SRRUC intercept the second halfof the solid phase and the first half of the solid phase, respectively. These differentSRRUC orientations are shown in Figure 3.7. This shifting method is equivalentto shifting an RRUC over 2d‖ so that its initial position and final position aregeometrically equivalent, as was discussed in Chapter 2.

The shear stress integral is split in a similar manner as the pressure term. Themagnitudes of the wall shear stress on all fluid-solid interfaces in the transversechannels are equal. For a fully staggered array, the Sfs⊥AA, Sfs⊥BB and Sfs⊥CCterms should thus be zero, since the wall shear stresses on their two surfaces willcancel vectorially. If the solid phase is not fully staggered in the streamwise directionand the length of the transverse channel section, where the interstitial flow directionis n, differs from the length of the transverse channel section wherein the flow is inthe −n direction (with n× n = 0), the integrals over Sfs⊥AA and Sfs⊥CC will not bezero respectively (as is the case in Figure 3.7). If an REV is considered, there shouldonly exist a pressure drop in the mean streamwise direction n, and the net pressure

35

drop in the direction perpendicular to n should be zero. The net force acting on thefluid in a particular channel is zero if piece-wise straight streamlines (as illustratedin Figure 3.7) are assumed. The magnitude of the wall shear stress times the surfacearea it is acting on should be equal to the magnitude of the pressure drop in thatchannel times the cross-sectional area of the channel. Since the net pressure dropin the transverse direction is zero, the net wall shear stress on the surfaces of thosechannels should also be zero. The SRRUC should be a representative unit cell andtherefore it is expected that, in equation (3.36), the part of the integral of the shearstress term which is taken over Sfs⊥ should be zero. Since the integral over Sfs⊥BB iszero for all levels of streamwise staggering, the net effect of the integrals taken overSfs⊥AA and Sfs⊥CC should also be zero. The wall shear stresses on the surfaces ofSfs⊥AA will cancel vectorially with the wall shear stresses on the surfaces of Sfs⊥CC ,if the two terms are weighed equally.

In the article by Lloyd et al. (2004), from here onwards referred to as the LDHmodel, the integral containing microscopic terms in the volume averaged Navier-Stokes equation was also closed with an SRRUC (though referred to as an RRUCby Lloyd et al. (2004)) for granules as was introduced by Du Plessis & Masliyah(1991). The major difference between the LDH model and the model by Du Plessis& Masliyah (1991) lies in the fact that, in the LDH model, the boundaries of theSRRUC were shifted in the streamwise direction which resulted in two positionsof the outer boundary of the SRRUC. One position is where the two transverseboundaries (referred to as AA in the LDH model) cross the solid phase and theother one is where the transversely orientated edges (BB) lie within the fluid phase.These two positions are weighed according to there relative frequency of occurrence.Yet, as was already mentioned, in the LDH model it is assumed (as is also done inthis present model) that

1

Uo

∫∫

Sfs⊥

µn · ∇ v dS = 0

which will not be the case if the SRRUC is shifted as mentioned above. Over AAthe wall shear stresses on both the S⊥ surfaces will be in the −n direction. Over BBone side’s shear stress is in the −n direction, and the other in the n direction andthese two will cancel vectorially. There is thus a resultant wall shear stress in the−n direction, since the transverse channel where the interstitial flow is in the −ndirection has not been considered in the LDH model. To overcome this shortcoming,in the present model, the SRRUC is shifted and weighed over three positions wherethe first and last position (the instance where the transversely orientated edges of theSRRUC cuts through the solid phase) are weighed equally as was already mentionedand shown in Figure 3.7.

36

Equation (3.36) thus reduces to

−∇〈p 〉o =ds‖

2d‖Uo

∫∫

Sfs⊥AA

n pw dS +dc‖d‖Uo

∫∫

Sfs⊥BB

n pw dS

+ds‖

2d‖Uo

∫∫

Sfs⊥CC

n pw dS − 1

Uo

∫∫

Sfs‖

µn · ∇ v dS . (3.37)

Considering the model shown in Figures 3.3, 3.4 and 3.5, where the transfer volumeof an SRRUC is 2dc⊥dc‖, and shifting the SRRUC in the streamwise direction aswas depicted in Figure 3.7, equation (3.37) reduces to

−∇〈p 〉o =ds‖

2d‖Uo[dc⊥ (p+ ξδp⊥ − [p])] n

+dc‖d‖Uo

[dc⊥

(p+ ξδp⊥ − [p− δp‖ − ξδp⊥]

)]n

+dc‖d‖Uo

[(ds⊥ − dc⊥)

(p+

ξδp⊥2

−[p− δp‖ −

ξδp⊥2

])]n

+ds‖

2d‖Uo

[dc⊥

(p− δp‖ − [p− δp‖ − ξδp⊥]

)]n+

τ‖S‖Uo

n

=ds‖

2d‖Uo[dc⊥ (ξδp⊥)] n

+dc‖d‖Uo

[dc⊥

(2ξδp⊥ + δp‖

)+ (ds⊥ − dc⊥)

(δp‖ + ξδp⊥

)]n

+ds‖

2d‖Uo[dc⊥ (ξδp⊥)] n+

τ‖S‖Uo

n (3.38)

=ds‖d‖Uo

[dc⊥ (ξδp⊥)] n+δp‖dc⊥Uo

n

+dc‖d‖Uo

[dc⊥

(2ξδp⊥ + δp‖

)+ (2ds⊥ − d⊥)

(δp‖ + ξδp⊥

)]n

= ξδp⊥

[dc‖d⊥d‖Uo

+ds‖dc⊥d‖Uo

]n+ δp‖

[dc‖ds⊥d‖Uo

+d‖dc⊥d‖Uo

]n

=UfUo

[ξδp⊥ + δp‖

d‖

]n

= ǫ

[ξδp⊥ + δp‖

d‖

]n . (3.39)

37

Note that since 〈Ψ 〉o = ǫ 〈Ψ 〉f , it follows from equation (3.39) that

−∇〈p 〉f =ξδp⊥ + δp‖

d‖n . (3.40)

This result is the same as equation (3.29) where the pressure gradient was determineddirectly.

The gradient of the intrinsic phase average of the pressure can also be determinedby means of volume averaging as follows:

∇〈p 〉f =〈pCC 〉f − 〈pAA 〉f

d‖n . (3.41)

The intrinsic phase average of the pressure in the SRRUC orientated with its boun-daries at AA is given by

〈pAA 〉f =1

Uf

([p+ ξδp⊥ +

δp‖4

] [dc⊥ds‖

2

]+ [p+ ξδp⊥]

[dc‖dc⊥

])

+1

Uf

([p+

ξδp⊥2

] [dc‖(2ds⊥ − d⊥)

]+ [p]

[dc‖dc⊥

])

+1

Uf

([p− δp‖

4

] [dc⊥ds‖

2

])

=1

Uf

(p [Uf ] + ξδp⊥

[Uf2

])

= p+ξδp⊥

2. (3.42)

The intrinsic phase average of the pressure in the SRRUC orientated with its boun-daries at CC is given by

〈pCC 〉f =1

Uf

([p− 3δp‖

4

] [dc⊥ds‖

2

]+[p− δp‖

] [dc‖dc⊥

])

+1

Uf

([p− δp‖ −

ξδp⊥2

] [dc‖(2ds⊥ − d⊥)

]+[p− δp‖ − ξδp⊥

] [dc‖dc⊥

])

+1

Uf

([p− 5δp‖

4− ξδp⊥

] [dc⊥ds‖

2

])

=1

Uf

(p [Uf ] + ξδp⊥

[−Uf2

]+ δp‖ [−Uf ]

)

= p− ξδp⊥2

− δp‖ . (3.43)

Substituting equations (3.42) and (3.43) into equation (3.41) yields

∇〈p 〉f =〈pCC 〉f − 〈pAA 〉f

d‖n =

−(ξδp⊥ + δp‖

)

d‖n , (3.44)

38

which is identical to equations (3.29) and (3.40).

From equation (3.38) it follows that

1

Uf

∫∫

Sfs⊥

n pw dS =ds‖

2d‖Uf[dc⊥ (ξδp⊥)] n+

ds‖2d‖Uf

[dc⊥ (ξδp⊥)] n

+dc‖d‖Uf

[dc⊥

(2ξδp⊥ + δp‖

)+ (ds⊥ − dc⊥)

(δp‖ + ξδp⊥

)]n

=dc‖d‖Uf

[dc⊥

(2ξδp⊥ + δp‖

)+ (2ds⊥ − d⊥)

(δp‖ + ξδp⊥

)]n

+ds‖d‖Uf

[dc⊥ (ξδp⊥)] n

=ξδp⊥d‖

n+ δp‖

(ds⊥dc‖d‖Uf

)n . (3.45)

From an identity based on Slattery’s Averaging Theorem and the Divergence Theo-rem it also follows as a point of interest that

〈∇p 〉f = ∇〈p 〉f +1

Uf

∫∫

Sfs

n p dS (3.46)

=−(ξδp⊥ + δp‖

)

d‖n+

ξδp⊥d‖

n+ δp‖

(ds⊥dc‖d‖Uf

)n

= δp‖

[−Ufd‖Uf

+ds⊥dc‖d‖Uf

]n

=δp‖d‖Uf

[−dc⊥(d‖)

]n

=1

Uf

(−δp‖dc⊥

)n . (3.47)

This equation (3.47) could also have been obtained directly by means of volumeaveraging, as follows:

〈∇p 〉f =1

Uf

(−δp‖ds‖

[dc⊥ds‖

])n

=1

Uf

(−δp‖dc⊥

)n . (3.48)

Dividing equation (3.37) by the porosity and after substitution of equation (3.45),the gradient of the intrinsic phase average of the pressure can be written in terms

39

of the shear stresses in the parallel and the transverse channels. It thus follows that

−∇〈p 〉f = ξδp⊥1

d‖n+ δp‖

ds⊥dc‖d‖Uf

n+τ‖S‖Uf

n

=ξδp⊥d‖

[ds⊥dc‖ + ψdc‖d‖

Uf

]n+ δp‖

ds⊥dc‖d‖Uf

n+τ‖S‖Uf

n

= ξδp⊥ds⊥dc‖d‖Uf

n+ δp‖ds⊥dc‖d‖Uf

n+τ‖S‖ + ψξτ⊥S⊥

Ufn

=ds⊥dc‖d‖Uf

(ξδp⊥ + δp‖

)n+

τ‖S‖ + ψξτ⊥S⊥Uf

n

= −ds⊥dc‖Uf

∇〈p 〉f n+τ‖S‖ + ψξτ⊥S⊥

Ufn

=Uf

d‖dc⊥


Uf

]n (3.49)

=1

d‖dc⊥


]n , (3.50)

where τ⊥S⊥ = δp⊥dc‖ .

Equation (3.50) is the same as equation (3.31) where the direct method was used.Note that equation (3.49) corresponds to the following equation obtained in theLDH model (their equation (15))

−∇〈p 〉o =Uf

U‖ + Ut ‖· τ‖S‖ + ξτ⊥S⊥

Uon , (3.51)

where squares, rather than rectangles, were considered for both the solid phase andthe SRRUC and ψ was thus set equal to unity. For a porous medium where theaverage speed in each channel is the same (referred to as an isotropic porous mediumby Diedericks & Du Plessis (1995)), it has been shown by Diedericks & Du Plessis(1995) that the tortuosity is

χD =Uf

U‖ + Ut ‖. (3.52)

After substitution of equation (3.52) and dividing by porosity, equation (3.51) maybe rewritten as

∇〈p 〉f = −χD · τ‖S‖ + ξτ⊥S⊥Uf

n

= χD ·∇‖p[ds‖dc⊥] + ξ∇⊥p

[(ds⊥ − dc⊥)dc‖

]

Ufn

= χD 〈∇p 〉f . (3.53)

In Appendix E equation (3.53) is discussed. There will again be referred to equation(3.53) later in this study. Tortuosity is scrutinized in the following chapters, and it

40

is shown that equation (3.52) fails to predict the tortuosity correctly if the averagemagnitudes of the streamwise and transverse velocities differ. Since the averagespeed of flow in each channel section had to be the same in the porous structureconsidered by Lloyd et al. (2004) (also referred to as isotropic by Lloyd et al. (2004))to be able to use χD, from equation (3.53) it follows that the gradient of the phaseaverage of the pressure is directly proportional to the tortuosity. (Note that inLloyd et al. (2004), the magnitude of the transverse and streamwise velocities werenot required to be equal and a β factor was defined as the relation between thetransverse and streamwise velocity.)

In Chapter 6, equation (3.51) is examined more thoroughly, and it is shown thatthe shear stress part also provides a tortuosity term and from equation (3.51) itthen follows that the gradient of the phase average of the pressure is proportionalto the tortuosity squared. Considering the definition of the Darcy permeability(equation (3.13)), it was concluded that the permeability is inversely proportionalto the tortuosity squared. This, however, was a very special case, and is in generalnot true. The general relation between tortuosity and permeability of the presentmodel will be discussed in Chapter 6, after an extensive study on tortuosity iscompleted in the following two chapters.


−∇〈p 〉f =1

d‖dc⊥


]n

=

[τ‖S‖dc⊥

· 1

d‖+ψξτ⊥S⊥dc⊥

· 1

d‖

]n

(3.54)

and, considering equation (3.39), yields

−∇〈p 〉f d‖ =(δp‖ + ξδp⊥

)n =

[τ‖S‖dc⊥

+ ξτ⊥S⊥dc‖

]n (3.55)

which is simply the interstitial force equilibrium equation of the streamwise andtransverse channels. Thus, in general, equation (3.50) has no connection to thetortuosity whatsoever, and it was purely coincidence that a tortuosity-like coefficientwas obtained in the case studied by Lloyd et al. (2004) where the average speed ofeach channel was the same.

Though the third dimension of a two-dimensional model is taken as unity, it isassumed that the interstitial flow in the channels between neighbouring solid phaseparticles is the same as the fully developed plane Poiseuille flow profile, where thelatter describes flow between infinitely wide parallel plates. Therefore, the followingexpressions hold for the shear forces in the transverse and the streamwise channelsrespectively:

τ⊥S⊥ =6µw⊥dc‖

(4ds⊥ − 2d⊥) (3.56)

41

and

τ‖S‖ =6µw‖dc⊥

(2ds‖) . (3.57)

The relation of the magnitude of the average interstitial velocities is given byw⊥w‖

= ξψ . (3.58)


−∇〈p 〉f =1

d‖dc⊥

[τ‖S‖ + ψξτ⊥S⊥)

]n

=1

d‖dc⊥

[6µw‖dc⊥

(2ds‖) + ψξ6µw⊥dc‖

(4ds⊥ − 2d⊥)

]n

=1

d‖dc⊥

[12µw‖dc⊥

(ds‖) + ψ3ξ2 12µw‖dc⊥

(2ds⊥ − d⊥)]n

=12µw‖d‖dc2⊥

[ds‖ + ψ3ξ2

(2ds‖

[ϕψ(1 − γ)

ψ − ϕγ

]− ϕd‖

)]n

=12µqd⊥dc3⊥d‖

[ds‖ + ψ3ϕξ2

(2ds‖

[ψ(1 − γ)

ψ − ϕγ

]− d‖

)]n . (3.59)

From the definition of the Darcy permeability in the streamwise direction and equa-tion (3.59) it follows that

k ≡ µq∥∥∥∇〈p 〉f∥∥∥

=dc3⊥d‖

12d⊥[ds‖ + ψ3ϕξ2

(2ds‖

[ψ(1−γ)ψ−ϕγ

]− d‖

)] . (3.60)

The dimensionless hydrodynamic permeability, K, is given by

K ≡ k

Uo

=dc3⊥

12d2⊥[ds‖ + ψ3ϕξ2

(2ds‖


]− d‖

)]

=d⊥γ

3

12[ds‖ + ψ3ϕξ2

(2ds‖


]− d‖

)]

=d⊥γ

3

12[dc‖

(ψ−ϕγϕγ

)+ ψ3ϕξ2

(2dc‖

[ψ−ϕγϕγ

] [ψ(1−γ)ψ−ϕγ

]− d⊥

ϕ

)]

=d⊥γ

3

12[dc⊥ψ

(ψ−ϕγϕγ

)+ ψ4ϕξ2

ψ

(2dc⊥

[ψ−ϕγϕγ

] [1−γψ−ϕγ

]− d⊥

ϕ

)]

=ϕψγ3

12[dc⊥d⊥

(ψ−ϕγγ

)+ ψ4ϕξ2

(2dc⊥d⊥

[1−γγ

]− d⊥

d⊥

)]

=ϕψγ3

12 [ψ − ϕγ + ψ4ϕξ2 (1 − 2γ)], (3.61)

42

which is the same as equation (3.26). This is considered an important contributionof this study namely to show that identical results are possible by using either thedirect or the volumetric approach to determine the permeability of porous media.

43

3.3 Asymptotic conditions for a regular array

From equation (3.19) it follows that the general equation obtained for the Darcypermeability of a regular array is

k =γ3ψϕ(d⊥d‖)

12 [ψ − ϕγ]=

γ3ψd2⊥

12 [ψ − ϕγ]. (3.62)

In this paragraph all the SRRUCs (or SUCs) considered are prismatic squares whered = d⊥ = d‖. Equation (3.62) thus reduces to

k =γ3ψd2

12 [ψ − γ]. (3.63)

3.3.1 Plane Poiseuille flow approximation

In the first case study the solid ‘particles’ are elongated in the streamwise directionto form a parallel array of streamwisely orientated laminae.

dc⊥ d

d

dc‖

Figure 3.8: Configuration when dc‖ << dc⊥.

For this limiting situation, presented in Figure 3.8, the following condition holds:

dc‖ << dc⊥ ⇒ ψ → ∞ ⇒ γ << ψ . (3.64)

From this assumption the streamwise flux can be written as

qd = w‖dc⊥ ⇒ q = γw‖ . (3.65)

From equation (3.65) it thus follows that, for plane Poiseuille flow, the permeabilityis given by:

k =µq

‖∇p‖ =µqdc2⊥12µw‖

=γdc2⊥

12(3.66)

where dc⊥ represents the distance between the parallel plates.

44

For a regular array, it follows from equation (3.63) that

k =γ3ψd2

12(ψ − γ)=γ3ψd2

12ψ=γ3d2

12=γdc2⊥

12(3.67)

which is, as expected, the same as equation (3.66), the expression obtained for thepermeability of plane Poiseuille flow.

3.3.2 Solid walls restricting the flow

The second limiting condition considered, as is shown in Figure 3.9, is the transverseelongation of solid material irrespective of the streamwise thickness, yielding

dc⊥ << d ⇒ γ → 0 . (3.68)

dc⊥

d

d

dc‖

Figure 3.9: Configuration where dc⊥ << d.

For a regular array, it thus follows that

limγ→0

k =γ3ψd2

12 [ψ − γ]= 0 . (3.69)

which corresponds to the physical situation where the flow cannot traverse a fixedwall. Since there were no restrictions regarding the width dc‖, equation (3.69) alsoholds for dc‖ = 0, i.e. when the porosity tends to zero.

3.3.3 Porosity tending to unity

The next limiting situation studied is the combined transverse and streamwiseshortening of the solid material so that ǫ→ 1.

For this limiting situation, shown in Figure 3.10, the following conditions prevail:

(a) dc‖ → d ⇒ γ

ψ→ 1 and thus ψ → γ

(b) dc⊥ → d ⇒ γ → 1 .

45

dc⊥ d

d

dc‖

Figure 3.10: Configuration when dc⊥ → d and dc‖ → d.

For a regular array, it follows that

limγ, ψ→1

k =d2

12 [1 − 1]→ ∞ (3.70)

and, as expected, the permeability tends to infinity since there is no solid materialrestricting the flow.

46

3.4 Asymptotic conditions for a staggered array

From the expression obtained for the dimensionless Darcy permeability, equation(3.26), it follows that a frictional coefficient (the inverse of the permeability) exertedby the porous medium on the fluid is

F = 12

(ψ − ϕγ

γ3ϕψ

)+ 12ξ2ψ3

(1 − 2γ

γ3

). (3.71)

In equation (3.71), the first term is friction due to the streamwise channels, and thesecond term is friction exerted by the transverse channels. Both these terms shouldbe larger than or equal to zero. If the SRRUC or the SUC is square, equation (3.71)reduces to

F = 12

(ψ − γ

γ3ψ

)+ 12ξ2ψ3

(1 − 2γ

γ3

)(3.72)

from which it follows that friction due to the parallel channels will add no constric-tion to the geometry, since dc‖ ≤ d. From the second term in equation (3.72) itfollows that 0 < γ ≤ 1

2and therefore, limiting situations where dc⊥ → d cannot be

studied for staggered arrays for the present geometrical model.

3.4.1 Solid walls restricting the flow

A similar situation as the one studied in section 3.3.2 is examined. Let the solidmaterial be elongated in the transverse direction to form a porous medium thatconsists of transversely orientated walls.

dc⊥ d

d

dc‖

dc⊥

d

d

dc‖

fully staggered over-staggered

Figure 3.11: Configuration when dc⊥ << d for a streamwise staggered array.

For this limiting situation it follows that

dc⊥ << d ⇒ γ → 0 . (3.73)

For a staggered array, under these circumstances, the permeability is

limγ→0

k =γ3ψd2

12 [ψ − γ + ξ2ψ4(1 − 2γ)]= 0

47

as expected. Since there were no restrictions regarding the width of the transversechannel, this case study also includes situations where the porosity tends to zero.

48

3.5 The dimensionless permeability where the as-

pect ratio of both the solid phase and the SUC

is α

Equation (3.27) is the general expression derived by means of both a direct approachand volume averaging for the dimensionless permeability in terms of porosity, whenthe aspect ratios of the solid phase and the SUC (or SRRUC) is α, i.e. ψ = ϕ = α.This equation was also obtained by Cloete & Du Plessis (2006) and is rewrittenbelow:

K =α(1 −

√1 − ǫ)3

12[√

1 − ǫ+ α4ξ2(2√

1 − ǫ− 1)] . (3.74)

From equation (3.71) it follows for this α-model that a frictional coefficient exertedby the porous structure on the fluid is

F = 12

(1 − γ

αγ3

)+ 12ξ2α3

(1 − 2γ

γ3

), (3.75)

where the first term represents friction due to the parallel channels and the secondterm, friction due to the transverse channels. From equation (3.75) it follows that,for a regular array, since γ ≤ 1, the only restriction is γ 6= 0 and porosities up tounity may be considered. For a staggered array, it follows from the second termthat 0 < γ ≤ 1

2and substituting it into equation (3.9), it follows that ǫ ≤ 0.75. This

restriction corresponds to the geometrical situation where the length of the solidphase in the transverse channel must be larger than or equal to the width of thestreamwise fluid channel, thus

ds⊥ ≥ dc⊥

whence

√1 − ǫ ≥ 1 −

√1 − ǫ

yielding

ǫ ≤ 3

4. (3.76)

Numerical results were obtained by Firdaouss & Du Plessis (2004) for the dimen-sionless permeability of flow through different unit cells. These data point are shownwith their corresponding graphs obtained from equation (3.74).

The five different unit cells studied were:

49

• SCASA: Square Cells Aligned in a Square Array, (Figure 3.12)

– Both directions: ξ = 0 and α = 1

• RCARA: Rectangular Cells Aligned in a Rectangular Array, (Figure 3.12)

– x-direction: ξ = 0 and α = 0.5

– y-direction: ξ = 0 and α = 2

x

y

Figure 3.12: SCASA and RCARA configurations

• RCSSA: Rectangular Cells Staggered in a Square Array, (Figure 3.13)

– x-direction: ξ = 0.5 and α = 2

– y-direction: ξ = 0 and α = 0.5

• SCSRA: Square Cells Staggered in a Rectangular Array, (Figure 3.13)

– x-direction: ξ = 0.5 and α = 1

– y-direction: ξ = 0 and α = 1

x

y

Figure 3.13: RCSSA and SCSRA configurations

• RCSRA: Rectangular Cells Staggered in a Rectangular Array, (Figure 3.14)

– x-direction: ξ = 0 and α = 0.25

– y-direction: ξ = 0.5 and α = 4

50

x

y

Figure 3.14: RCSRA

The numerical data points obtained by Firdaouss & Du Plessis (2004) are shownin Figure 3.15 along with the corresponding graphs of the present model, equation(3.74). From these graphs it is evident that the expression for the dimensionlesspermeability is not valid for high porosities and other modelling techniques shouldbe considered in that region. Equation (3.74) gives a good approximation for thedimensionless permeability of low porosities (ǫ < 0.5). One of the reasons for thatis that, though the equation is valid for porosities up to 0.75, fully developed planePoiseuille flow is not a good description of the flow pattern between neighbouringcells for high porosities. Also note that the narrower the width of the streamwisechannel (thus for a smaller α), the smaller the area of the fluid-solid interface onwhich the shear stress was neglected (as was illustrated in Figure 3.6) and the lessthe simplifying assumption plays a role. Furthermore, the smaller the α-value, thelonger the streamwise channels relative to the transverse channels and the betterplane Poiseuille flow would approximate the flow in the predominating streamwisechannels. Considering the three staggered examples (Figures 3.15 (e), (h) and (l)where α is 2, 1 and 4 respectively), it is evident that equation (3.74) approximatesthe permeability more accurate as α decreases.

51

0 0.2 0.4 0.6 0.8 110

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Porosity, ∈

Dim

ensi

on

less

Per

mea

bili

ty, K

Present ModelFirdaouss and Du Plessis − numerical

(a) SCASA

0 0.2 0.4 0.6 0.8 110

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Porosity, ∈

Dim

ensi

on

less

Per

mea

bili

ty, K


(b) RCARA where n is in the x-

direction

0 0.2 0.4 0.6 0.8 110

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Porosity, ∈

Dim

ensi

on

less

Per

mea

bili

ty, K


(c) RCARA where n is in the y-

direction

0 0.2 0.4 0.6 0.8 110

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Porosity, ∈

Dim

ensi

on

less

Per

mea

bili

ty, K


(d) RCSSA where n is in the x-

direction

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Porosity, ∈

Dim

ensi

on

less

Per

mea

bili

ty, K


(e) RCSSA where n is in the x-

direction with ǫ < 0.75

0 0.2 0.4 0.6 0.8 110

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Porosity, ∈

Dim

ensi

on

less

Per

mea

bili

ty, K


(f) RCSSA where n is in the y-

direction

0 0.2 0.4 0.6 0.8 110

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Porosity, ∈

Dim

ensi

on

less

Per

mea

bili

ty, K


(g) SCSRA where n is in the x-

direction

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Porosity, ∈

Dim

ensi

on

less

Per

mea

bili

ty, K


(h) SCSRA where n is in the x-


0 0.2 0.4 0.6 0.8 110

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Porosity, ∈

Dim

ensi

on

less

Per

mea

bili

ty, K


(i) SCSRA where n is in the y-

direction

0 0.2 0.4 0.6 0.8 110

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Porosity, ∈

Dim

ensi

on

less

Per

mea

bili

ty, K


(j) RCSRA where n is in the x-

direction

0 0.2 0.4 0.6 0.8 110

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Porosity, ∈

Dim

ensi

on

less

Per

mea

bili

ty, K


(k) RCSRA where n is in the y-

direction

0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Porosity, ∈

Dim

ensi

on

less

Per

mea

bili

ty, K


(l) RCSRA where n is in the y-


Figure 3.15: Numerical data points versus equation (3.74) for different unit cells.

52

3.6 Dimensionless permeability when the trans-

verse and parallel channel widths are equal

For the situation where all the fluid channels have the same width, thus ψ = 1,the following expression for the dimensionless permeability (equation (3.28)) in theDarcy flow regime has been obtained:

K =γ3(ǫ− γ)

12γ(1 − ǫ) + 12ξ2(ǫ− γ)(1 − 2γ). (3.77)


F = 12

(1 − ϕγ

γ3ϕ

)+ 12ξ2

(1 − 2γ

γ3

). (3.78)

From the part of the frictional coefficient due to the parallel channels, it followsthat dc⊥ ≤ d‖, where dc⊥ = d‖ represents the situation where ǫ = 1. From equation(3.10), the porosity for this particular case, where the widths of the streamwisechannels and the transverse channels are equal, is

ǫ = γ[ϕ(1 − γ) + 1] . (3.79)

If γ = 1 then it also follows that ǫ = 1. From the first term of equation (3.78) it

follows that, if γ = 1, the frictional coefficient in the parallel channels is 12(

1ϕ− 1

).

Therefore, if d‖ < d⊥, this frictional coefficient will be negative, and thus γ 6= ǫ.If, however, d‖ ≥ d⊥, then the friction is zero or positive, and the dimensionlesspermeability when γ = ǫ = 1 is defined. From the second term of equation (3.78) itfollows that, for a staggered array, 0 < γ ≤ 1

2.

Considering equation (3.79) and plotting the porosity for various values of γ , where0 < γ ≤ 1

2, against ϕ, where 0 ≤ ϕ ≤ 16, it is evident from Figure 3.16 that porosity

up to unity is valid for all the γ-values if ϕ has a large enough range.

If ǫ < γ then

γ > γ[ϕ(1 − γ) + 1]

yielding

γ > 1 (3.80)

which is impossible, because dc⊥ ≤ d⊥ and therefore ǫ ≥ γ. Note however that, fora regular array, γ 6= ǫ = 1 if d‖ < d⊥.

53

0 2 4 6 8 10 12 14 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Aspect ratio of SUC or SRRUC, α

Po

rosi

ty, ∈

γ = 1/16γ = 1/8γ = 3/16γ = 1/4γ = 5/16γ = 3/8γ = 7/16γ = 1/2

Figure 3.16: Porosity ranges for different γ-values. (Note that γ ≤ ǫ ≤ 1.)

3.6.1 Regular array

The permeability expression for a regular array (equation (3.77), with ξ = 0) wherethe parallel and the transverse channels have the same width, is undefined whend‖ < dc⊥, which is clearly an impossible geometrical situation, and thus holds norestriction for the porous medium. If d‖ ≥ d⊥, then 0 < γ ≤ 1 and γ ≤ ǫ. If

d‖ < d⊥, then 0 < γ ≤ d‖d⊥

and again γ ≤ ǫ. Porosity may have a value up to unityfor any γ if ϕ has a large enough range, as was illustrated in Figure 3.16. Figure3.17 represents the log-scale of the dimensionless permeability against the porosity.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

−6

10−5

10−4

10−3

10−2

10−1

100

101

102

Porosity, ∈

Dim

ensi

on

less

Per

mea

bili

ty, K

γ = 1/4γ = 1/2γ = 3/4

Figure 3.17: The dimensionless permeability against porosity for a regular array asdefined by equation (3.77) with ξ = 0.

3.6.2 Staggered array

Similarly to the regular array, the permeability expression (equation (3.77), withξ = 1

2or ξ = 1) for a staggered array, where the parallel and the transverse channels

54

have the same width, is undefined when d‖ < dc⊥, which is geometrically impossible,and thus holds no restriction. For a staggered array, 0 < γ ≤ 1

2and γ ≤ ǫ ≤ 1 if ϕ

has a large enough range for a small γ-value.

0 0.2 0.4 0.6 0.8 110

−8

10−6

10−4

10−2

100

102

Porosity, ∈

Dim

ensi

on

less

Per

mea

bili

ty, K

γ = 1/16γ = 1/8γ = 3/16γ = 1/4γ = 5/16γ = 3/8γ = 7/16γ = 1/2

Figure 3.18: The dimensionless permeability for a fully staggered array in terms ofporosity, as defined by equation (3.77) with ξ = 1

2, for various acceptable values of

γ.

0 0.2 0.4 0.6 0.8 110

−8

10−6

10−4

10−2

100

102

Porosity, ∈

Dim

ensi

on

less

Per

mea

bili

ty, K

γ = 1/16γ = 1/8γ = 3/16γ = 1/2γ = 5/16γ = 3/8γ = 7/16γ = 1/2

Figure 3.19: The dimensionless permeability for an over-staggered array in terms ofporosity, as defined by equation (3.77) with ξ = 1, for various acceptable values ofγ.

Considering the graphs in Figures 3.18 and 3.19, it is evident that the permeabilityfor a fully staggered array is larger than the permeability of an over-staggered array.This makes sense since, for an over-staggered array, the path length over which thewall shear stress is acting on the fluid in the transverse channel is twice as long asthe length of a fully staggered array. Thus, the pressure drop over an SUC is largerin the case of an over-staggered array and therefore (since the pressure gradientis inversely proportional to the permeability) the permeability of a fully staggeredarray is larger than the permeability of an over-staggered array of larger tortuosity.It is thus clear that permeability is tortuosity dependent. This relation will bestudied later on in this work.

55

Chapter 4

Different interpretations oflineality

It is necessary to develop a good understanding of the geometrical properties ofporous media, because it plays an important role in the analysis of the transport pro-cess through any particular medium. Among these properties is tortuosity (loosely,a measure of the waviness of a bundle of pathlines (or streamlines for stationaryflow) through a porous medium). Many times in literature (i.e. Bear & Bachmat(1991)) referral is made to tortuosity, when the straightness of a bundle of pathlinesis implied.

n

L

Le

Figure 4.1: The actual distance travelled by a fluid particle, Le, and the dimensionof the porous medium in the direction of the total fluid displacement, L.

The tortuosity is defined as:

χ=LeL, χ ≥ 1. (4.1)

56

Tortuosity is a very vague concept and there exist many confusing interpretationsfor it, e.g. as reported by Clennel (1997). Some interpretations will be consideredand we will try to find their physical meanings and compare it with equation (4.1),the definition of tortuosity. Sometimes, for convenience, the square of the tortuosityis used rather than tortuosity itself, i.e. Zhang & Knackstedt (1995).

Frequently, and especially in groundwater research, the inverse of the tortuositynamely L

Leis referred to as the tortuosity. Since this is semantically problematic,

because LLe

decreases with increase in the waviness of the bundle, the concept of‘lineality’ was introduced by Diedericks & Du Plessis (1995), namely

L=L

Le, 0 ≤ L ≥ 1. (4.2)

According to Bear (1972) there are two interpretations of lineality as defined inequation (4.2): “When Le is defined as the average path of fluid particles throughthe porous medium, two interpretations of this average are possible. We may averagethe actual lengths of pathlines – disregarding the fact that fluid particles move alongthese pathlines at different velocities, following the velocity distribution across theelementary passages. In this case tortuosity is a purely geometrical concept. Suchaverage pathlines may be obtained (conceptually) by averaging the pathlines of allthe particles passing at a certain instant through a given porous cross-section. Onthe other hand, the average may be obtained by averaging lengths of pathlines of allthe particles passing through the area of a given medium cross-section during a giventime period. In this case tortuosity becomes a kinematical property as the velocitydistribution, affected by the shape of the elementary channels and by the slip or no-slip boundary condition at the fluid-solid interface, will effect the resulting averagepathline.” (Bear (1972), p.117, (sic))

The lineality as defined by Bear and Bachmat as a geometrical concept does not makeany sense as it stands. The actual distances (or pathlines) have to be taken over aspecific time period (which will render to the inverse of the kinematic interpretationof tortuosity discussed below) or per unit streamwise displacement (the inverse ofthe geometric tortuosity discussed below). The inverse of the kinematic definitionfor lineality by Bear and Bachmat will be discussed below as the dynamic tortuosity.The different corresponding linealities are listed in Table 4.1.

Zhang & Knackstedt (1995) considered stationary flow through a porous mediumand, from numerical simulations, the velocity at each point were determined andstreamlines between two parallel surface areas, chosen a distance L apart, wereobtained. For each streamline its streamline tortuosity was calculated by taking theratio of the actual length of the streamline between the parallel surface areas and L.To define the tortuosity they then weighed the streamline tortuosities by the overallvolumetric flow associated with each streamline, which is proportional to t−1

i . The

57

Bear (1972) Present study

Geometrical - per unit streamwise displacement: Geometric

- over a specific time period: Kinematic

Kinematical Dynamic

Table 4.1: Some equivalent linealities or tortuosities.

macroscopic tortuosity factor

χ2 =(LeL

)2

for a random medium has been defined as

χ2 =

[∑Ni=1 t

−1iχi∑N

i=1 t−1i

]2

. (4.3)

In equation (4.3), i denotes a particular streamline and N is the number of stream-lines considered in the averaging. The reasoning behind this equation is that aparticle with a larger average velocity through the porous medium, will spend lesstime in the medium. In case of stationary flow, where streamlines and pathlinesare equivalent, more particles will travel along this route and this streamline shouldthus have a predominant effect on the average tortuosity.

We will consider three different points of view on tortuosity:

• The geometrical tortuosity is the average path length per unit streamwise dis-placement of all the particles in an REV at time t. In two dimensions it isthe average path length per unit streamwise displacement of all the particlescrossing the REA at time t.

In Figure 4.2 the tortuosities, χi−1, χi and χi+1 of pathlines, i− 1, i and i+ 1respectively, are shown. To obtain the geometric tortuosity of the REA, theaverage of the tortuosities of all the pathlines has to be calculated.

In case of stationary flow this is also the average streamline tortuosity. Allpath or streamlines are weighed equally and particle velocities are not takeninto account.

• We define the kinematic tortuosity as the average distance travelled per stream-wise displacement in a time interval [t, t+ δt] by all particles which are in theREV at time t. In two dimensions it is the average distance travelled perstreamwise displacement in a time interval [t, t + δt] by all particles crossingthe REA at time t.

In Figure 4.3 the actual distances travelled in the time period [t, t+δt], in caseof parabolic channel flow, are illustrated by the solid arrows. Note that in the

58

n

n

n

REA

1

1

1

χi

χ i+1

χi−

1

n

one unit streamwise displacement

Figure 4.2: The geometrical tortuosity of a pathline.

case of an REV, some of the particles considered in calculating the kinematictortuosity might be outside the REV at time (t+ δt). This happens, becausean REV is a control volume and not a material volume.

REA

n

n

Figure 4.3: An illustration of the actual distances travelled (solid arrows) in thetime interval [t, t+ δt], considered in the calculation of the kinematic tortuosity.

• The dynamic tortuosity is defined as the average distance travelled per stream-wise displacement, over a certain time interval [t, t + δt], by all particles tra-versing any part of the REV at any time within the interval [t, t+ δt]. Again,

59

in two dimensions, it is the average distance travelled per streamwise displace-ment in a certain time interval [t, t+ δt] by all particles traversing the REA atany time within the interval [t, t+ δt]. This interpretation is similar to that ofthe kinematic point of view expressed by Bear (1972).

In Figure 4.4 the solid arrows indicate the actual distances travelled by thoseparticles crossing the REA in the time interval [t, t+ δt] in case of a parabolicvelocity profile in an oblique channel section. Note that in the case of an REV,some of the particles considered in calculating the dynamic tortuosity mightbe outside the REV for some portion of the time interval δt. Contrary to thekinematic tortuosity where the particles considered have to be inside the REVat time t, this time portion includes the beginning of the time interval.

n

n

REA

Figure 4.4: An illustration of some pathlines considered in the calculation of thedynamic tortuosity as well as their actual distances in the specific time interval.

60

4.1 Lineality in terms of velocity and geometrical

properties

In this section expressions for lineality will be derived for the three definitions oftortuosity given above. For notational simplicity, in these derivations, we will onlywork in two dimensions.

The definitions of tortuosity as well as the definitions by Bear (1972) for linealityare in terms of particles. It is impossible to work with actual particles in practiceand therefore the reference to particles has to be replaced by geometrical propertiesof the porous medium. In an attempt to move away from particle definitions, inthis section, areal expressions will be derived directly from the particle definitionsabove.

We consider an REA through which particles with different velocities travel. Thearea of the REA occupied by the fluid phase is denoted by Af .

4.1.1 Geometrical lineality

n

n

n

REA1

1

1

Li−1

Li+1

Lin

Figure 4.5: The geometrical lineality of a pathline.

This lineality is not a function of velocity and is the average streamwise displacementper unit distance travelled by all the particles traversing the REA at time t. Thenumber of particles at Af at time t is

N = λAAf , (4.4)

where λA is defined as the number of particles per unit area on Af . The streamwisedisplacement per unit distance in the direction of n is thus merely the streamwisecomponent of n.

61

If we consider only one unit displacement per particle parallel to n, it follows thatthe average distance travelled by all the particles is

Le =

∫ N

01dN

N=

1

λAAf

∫∫

Af

λA dA = 1 (4.5)

and the average streamwise component of these displacements is

L =1

λAAf

∫∫

Af

(λA n · n) dA =1

Af

∫∫

Af

( n · n) dA . (4.6)

It therefore follows from equation (4.2) that

LGeo =L

Le=

1

Af

∫∫

Af

( n · n) dA . (4.7)

From here on equation (4.7) will be referred to as the definition of the magnitudeof the geometric lineality in two dimensions.

4.1.2 Kinematic lineality

The distance travelled by each particle in a time period [t, t+ δt] is given by

δs = v δt, (4.8)

where v is the actual interstitial speed of the particle. The number of particles, dN ,passing through a cross-sectional area, dA, at time t is directly proportional to thesize of dA, thus

dN = λAdA .

The total number of particles passing through the REA is therefore

N =∫∫

Af

λA dA ,

where Af is the area occupied by the fluid phase of an REA.

It is assumed that λA is constant over Af and therefore the total number of particles,N , passing through the cross-sectional area, Af , at time t is

N = λAAf . (4.9)

62

The average distance of the particles passing through the REA at time, t, is

Le =

∫ N

0δs dN

N

=1

λAAf

∫∫

Af

(λA)(v δt) dA

=1

Af

∫∫

Af

v δt dA . (4.10)

Similarly L can be calculated by taking the average streamwise component of thedistance travelled by all the particles traversing the REA at time t. The averagestreamwise displacement is

δs‖ = ( v · n) δt, (4.11)

where ( v · n) is the streamwise component of the interstitial velocity at each point.Thus

L =

∫ N

0δs‖ dN

N

=1

λAAf

∫∫

Af

(λA)( v · n δt) dA

=1

Af

∫∫

Af

( v · n) δt dA . (4.12)

From equation (4.2) the following equation for the kinematic lineality is obtained:

LKin =L

Le=

∫∫

Af

v ( n · n) dA

∫∫

Af

v dA. (4.13)

From here on equation (4.13) will be referred to as the magnitude of the kinematiclineality in two dimensions.

63

4.1.3 Dynamic lineality

In this section a formula will be derived for the dynamic lineality by referring directlyto the definition of Le as a kinematic property by Bear (1972).

The distance travelled by each particle in a time period [t, t+ δt] is given by

δs = v δt, (4.14)

where v is the point-wise magnitude of the actual interstitial velocity field. Thenumber of particles, dN , passing through a cross-sectional area, dA, in a time period[t, t+ δt], is

dN = λV v δt dA, (4.15)

where λV is defined as the number of particles per unit volume occupied by the fluidphase. Thus

N =∫∫

Af

λV v δt dA , (4.16)

where Af is the area occupied by the fluid phase on an REA. If λV is constant overthe volume occupied by the fluid phase, it follows from equation (4.16) that thetotal number of particles, N , passing through the REA is

N = λV δt∫∫

Af

v dA . (4.17)

According to the definition of Bear (1972), Le can be obtained by taking the averageof the distances travelled by all the particle passing through an area, A, in a timeperiod [t, t + δt]. Thus, the average distance of the particles passing through theREA during the time interval [t, t+ δt] is

Le =

∫ N

0δs dN

N

=1

λV δt∫∫

Af

v dA

∫∫

Af

(λV vδt)(v δt) dA

=δt∫∫

Af

v dA

∫∫

Af

v2 dA . (4.18)

64

Similarly L can be calculated by taking the average streamwise component of thedistance travelled by all the particles traversing the REA in a time period [t, t+ δt].The average streamwise displacement is

δs‖ = ( v · n) δt, (4.19)

where ( v · n) is the streamwise component of the interstitial velocity at each point.Thus

L =

∫ N

0δs‖ dN

N

=1

λV δt∫∫

Af

v dA

∫∫

Af

(λV v δt)( v · n δt) dA

=δt∫∫

Af

v dA

∫∫

Af

( v · n)( v · n) dA

=δt∫∫

Af

v dA

∫∫

Af

v2( n · n) dA . (4.20)

From equation (4.2) the following equation for the dynamic lineality is obtained:

LDyn =L

Le=

∫∫

Af

v2( n · n) dA

∫∫

Af

v2 dA. (4.21)

From here on equation (4.21) will be referred to as the magnitude of the dynamiclineality in two dimensions.

4.1.4 Conclusion

We have thus introduced three different interpretations for lineality. If an REV isconsidered, the areal integrals are replaced by volume integrals. Since the lineality(as well as the tortuosity) refers to a displacement length relative to the streamwisedirection, this direction has to be assigned to it.

65

• Geometric Lineality

LGeo =n

Uf

∫∫∫

Uf

( n · n) dU = n 〈 n · n 〉f =

(n ·

〈 n 〉f〈1 〉f

)n (4.22)

• Kinematic Lineality

LKin =

n∫∫∫

Uf

v( n · n) dU

∫∫∫

Uf

v dU= n

〈 v · n 〉f〈v 〉f

=

(n ·

〈 v 〉f〈v 〉f

)n (4.23)

• Dynamic Lineality

LDyn =

n∫∫∫

Uf

v 2( n · n) dU

∫∫∫

Uf

v 2 dU= n

〈v v · n 〉f〈v 2 〉f

=

(n ·

〈v v 〉f〈v 2 〉f

)n (4.24)

Note that, should the magnitude of the interstitial velocity, i.e. v, be a uniformscalar field in Uf , these three definitions are equivalent.

66

4.2 Lineality and the average channel speed

Let us define an average velocity field, w, which has a constant magnitude through-out the fluid phase of the REV and which is parallel to v at each point so that

∥∥∥∥∥∥∥∥∥∥

∫∫∫

Uf

w dU

∥∥∥∥∥∥∥∥∥∥

=

∥∥∥∥∥∥∥∥∥∥

∫∫∫

Uf

v dU

∥∥∥∥∥∥∥∥∥∥

. (4.25)

Note that∫∫∫

Uf

w dU (4.26)

is not necessarily a vector in the streamwise direction, because, to obtain a resultantvector in the direction of n, the vectors n are weighed with different interstitialvelocities at each point. Simply replacing v by w will therefore not necessarily yielda streamwise discharge.

However, according to Diedericks & Du Plessis (1995),

n∫∫∫

Uf

n · n dU = n n ·∫∫∫

Uf

n dU (4.27)

will always be a vector in the streamwise direction. They defined a new velocityparameter, w = w n, so that ‖ w‖ ≥ ‖ w‖ and

w n n ·∫∫∫

Uf

n dU ≡∫∫∫

Uf

v dU . (4.28)

We define U‖ + Ut‖ as that the part of Uf in which the interstitial flow direction isparallel to the streamwise direction. Thus

U‖ + Ut‖ =∫∫∫

Uf

n · n dU (4.29)

and from equations (4.28) and (4.29) it then follows that

w =1

U‖ + Ut‖

∫∫∫

Uf

v dU =Uf

U‖ + Ut‖〈 v 〉f . (4.30)

67

Considering equations (2.18) and (4.30) it follows that

u = wU‖ + Ut‖

Uf. (4.31)

Substitution of u from equation (4.31) into the Dupuit-Forchheimer relationship,(2.19), yields

q = ǫ wU‖ + Ut‖

Uf= ǫ w ·LGeo n = ǫ w ·L

Geo= ǫwLGeo n . (4.32)

From here on we shall refer to this w as wGeo and LGeo

is the geometric lineality,as is defined in equation (4.22), times the streamwise direction. In this work wewill only be interested in the tortuosity (or lineality) for flow in the direction ofthe geometrical principle axes of porous structures and not in oblique directions.Lineality and tortuosity will thus be dyadic quantities for the remainder of thisstudy, and the two unit vectors assigned to them, are vectors in the streamwisedirection.

If we want to write q in terms of LKin

rather than LGeo

we have to find wKin sothat

q = ǫ wKin ·LKin . (4.33)

The streamwise kinematic lineality is given by

LKin

=

n n∫∫∫

Uf

v( n · n) dU

∫∫∫

Uf

v dU. (4.34)

The superficial velocity can be written as

q =1

Uo

∫∫∫

Uf

v dU =n

Uo

∫∫∫

Uf

v ( n · n) dU , (4.35)

because∫∫∫

U⊥ + Ut⊥

v dU = 0 , (4.36)

where U⊥+Ut⊥ is the volume of the fluid phase where the direction of the interstitialvelocity is perpendicular to n, thus where

U⊥ + Ut⊥ = Uf − U‖ − Ut‖ − Ug . (4.37)

68

In equation (4.37), Ug represents the volume of any possible stagnant regions in theREV, therefore where v = 0, as was shown in Figure 2.5.


n

Uo

∫∫∫

Uf

v ( n · n) dU =UfUo

wKin ·

n n∫∫∫

Uf

v ( n · n) dU

∫∫∫

Uf

v dU

which then reduces to

1 =UfwKin∫∫∫

Uf

v dU, (4.38)

since wKin is in the streamwise direction. From equation (4.38) it follows that

wKin =n

Uf

∫∫∫

Uf

v dU = 〈v 〉f n . (4.39)

The inner product of the kinematic lineality and the intrinsic phase average of thechannel speed taken in the streamwise direction times the porosity therefore yieldsthe superficial velocity.

In a similar manner one can derive that

wDyn =

n

Uf

∫∫∫

Uf

v( n · n) dU∫∫∫

Uf

v2 dU

∫∫∫

Uf

v2( n · n) dU=

n · 〈 v 〉f 〈v 2 〉f nn · 〈v v 〉f

(4.40)

where

q = ǫ wDyn ·LDyn . (4.41)

Substitution of equations (4.30), (4.39) and (4.40) into equations (4.22), (4.23) and

(4.24) yield, as expected,(LGeo =

u

wGeo

),(LKin =

u

wKin

)and

(LDyn =

u

wDyn

)

respectively.

69

Consider

LKin =u

wKin. (4.42)

From equation (4.39) it follows that wKin = 〈v 〉f and substituting this equation intoequation (4.42) yields

LKin =u

〈v 〉f=

utR(Uf − Ug)∫∫∫

Uf

v dU. (4.43)

In the discussion that follows on Figure 2.5 it is stated that utR (the drift velocity)is the displacement, L, of fluid through a porous medium divided by the residence

time, tR. In equation (4.43),1

Uf − Ug

∫∫∫

Uf

v dU is the average speed of the fluid

travelling a distance Le through the porous medium, also in a time period tR.

To illustrate this, consider Figure 4.6. From equation (4.39) it follows that

wKin = 〈v 〉f =2w‖(U‖ + Ut ‖) + 2w⊥(U⊥ + Ut⊥)

2Uf(4.44)

for the RRUC shown in Figure 4.6. Note that the streamlines are assumed to bestraight and w‖ and w⊥ represents the average speeds in the parallel and transversechannels of an assumed fully developed plane Poiseuille flow profile:

w‖ = −∇‖pdc2⊥12µ

and w⊥ = −∇⊥pdc2‖12µ

. (4.45)

w‖

w‖

w⊥ w⊥

n

streamwise

Figure 4.6: Drift velocity and average channel speed.

70

From equations (3.22) and (4.44) it follows that

〈v 〉f = w‖

[dc⊥d‖ + ξψdc‖ds⊥

Uf

]=w‖dc⊥Uf

[d‖ + ξds⊥

]. (4.46)

The residence time of particle with an average interstitial speed given in equation(4.46), is

tR =Le〈v 〉f

=Uf

w‖dc⊥. (4.47)

Again, considering Figure 4.6, the intrinsic phase average of the interstitial velocity(which is equivalent to the drift velocity if there are no stagnant regions) is given by

〈 v 〉f =2w‖(U‖ + Ut ‖)

2Ufn =

w‖dc⊥d‖Uf

n . (4.48)

The residence time times 〈 v 〉f is thus the total displacement in the streamwisedirection. Therefore

tR 〈 v 〉f = L n

tR

(w‖dc⊥d‖Uf

n · n)

= L n · n

⇒ tR =Uf

w‖dc⊥. (4.49)

This corresponds to equation (4.47) as expected.

Therefore, from equation (4.42) it follows that

LKin =〈 v 〉f〈v 〉f

=L n/tRLe/tR

=L

Len (4.50)

which is indeed equivalent to equation (4.2), the definition of lineality.

71

Chapter 5

Linealities from literature

5.1 Lineality as derived by Bear and Bachmat

Equations that describe the transport process of fluids at a microscopic level containderivatives in time as well as in space. By volume averaging these equations overan REV, the transport process can be described at a macroscopic level. Bachmat& Bear (1986) derived equations for the volume average of both the time derivativeand the spatial derivative. In this section a summary is given of the derivation givenby Bear & Bachmat (1991) for an equation for the average of the spatial derivativeof a tensorial quantity.

Applying the divergence and the volume averaging theorem to the gradient of atensorial quantity, G, Bachmat & Bear (1986) derived (their equation (49))

ǫ 〈∇G 〉f = ∇(ǫ 〈G 〉f

)+

1

Uo

∫∫

Sfs

nGdS . (5.1)

If the porosity is spatially constant, this equation can also be written as (Bachmat& Bear (1986), their equation (50))

〈∇G 〉f = ∇〈G 〉f +1

Uf

∫∫

Sfs

nGdS . (5.2)

According to Bear and Bachmat information on the derivative of a quantity, normalto the fluid-solid interface inside the REV, is often known, but no information isavailable about the quantity itself on that surface. This was the reason for derivinga general equation for the spatial derivative of a tensorial quantity in terms thegradient of the quantity normal to Sfs.

72

5.1.1 Derivation by Bear and Bachmat

Let G=G(r, t) be a scalar function defined within the fluid region Uf . In therectangular Cartesian coordinate system where the position vector has been definedas r ≡ x i+ y j + z k , a deviation, ro, was defined (Bear & Bachmat (1991), justbelow their equation (2.3.36)) as

ro = r − 1

Uo

∫∫∫

Uo

r dU = r − ro . (5.3)

In other words, ro is a position vector of point P , relative to the centroid, C, ofthe REV as shown in Figure 5.1.

Sff

Sfs

Sss

ro

ro r

O

PC

Figure 5.1: The position vector of P relative to the centroid, C, of the REV.

This is contrary to the more frequent definition of deviation rf which is in termsof the intrinsic phase average and defined as

rf = r − 1

Uf

∫∫∫

Uf

r dU = r − 〈r 〉f . (5.4)

After substituting the vector ro and the scalar G into Green’s vector theorem,Appendix C, equation (C.4), the following is obtained:

∫∫

Sof

n · [G∇ro− (∇G) ro] dS =∫∫∫

Uf

[G∇2 ro−

(∇2G

)ro

]dU . (5.5)

73

Since

∇ro =

1 0 0

0 1 0

0 0 1

= i i+ j j + k k

in the rectangular Cartesian coordinate system, it follows that

∇2 ro = 0 .

Equation (5.5) thus reduces to∫∫

Sof

GndS =∫∫

Sof

n ·(∇G) ro dS −∫∫∫

Uf

(∇2G

)ro dU . (5.6)

At this point Bear & Bachmat (1991) introduced the following assumptions:

• Assumption 1:

In the fluid phase, Uf ,

∇2G = 0 . (5.7)

In other words, there exist no local maximum or minimum value for G withinUf and the gradient is constant.

• Assumption 2:∫∫

Sff

n ·(∇G) ro dS ≈ 1

Sff

∫∫

Sff

∇GdS ·∫∫

Sff

n ro dS

≈ ∇〈G 〉f ·∫∫

Sff

n ro dS .(5.8)

Here Bear & Bachmat (1991) assume that the average of the gradient of Gon the fluid-fluid interface on the outer surface of the REV, is equal to thegradient of the average of G over Uf .

This in itself is not a good approximation. In Appendix E it is shown by meansof an example that, if G represents the pressure and ∇2p = 0, this assumptiononly holds if the lineality (or tortuosity) of the momentum transport is unity.The same arguments hold for molecular diffusion.

By substituting equations (5.7) and (5.8) into equation (5.6), it follows that∫∫

Sof

Gn dS =∫∫

Sfs

n ·(∇G) ro dS + ∇〈G 〉f ·∫∫

Sff

n ro dS . (5.9)

74

The LHS of equation (5.9) may be simplified as follows by means of the divergencetheorem:

∫∫

Sof

Gn dS =∫∫∫

Uf

∇GdU = Uf 〈∇G 〉f . (5.10)

Substitution of equation (5.10) in equation (5.9) and division by Uf , leads to thefollowing:

〈∇G 〉f = ∇〈G 〉f ·1

Uf

∫∫

Sff

n ro dS +1

Uf

∫∫

Sfs

n · (∇G) ro dS . (5.11)

Defining a dyadic factor LB

(in Bear & Bachmat (1991), T ∗αij) as

LB

=1

Uf

∫∫

Sff

n ro dS , (5.12)

it then follows from equation (5.11) that

〈∇G 〉f = ∇〈G 〉f · LB +1

Uf

∫∫

Sfs

n · (∇G) ro dS . (5.13)

Examining equation (5.13) it is evident that Bear and Bachmat managed to writethe intrinsic phase average of a partial derivative of a quantity as the gradient of

the intrinsic phase average multiplied by the factor LB

=1

Uf

∫∫

Sff

n ro dS plus a

surface integral over Sfs on which the gradient of a quantity, and not the quantityitself, is required.

Neglecting adsorption of the fluid phase on Sfs and diffusion into the solid phase,

n · ∇G = 0 (5.14)

corresponds to the situation of mass transport by molecular diffusion of a fluid in aporous medium, where the entire void space of the porous medium is occupied bythat single fluid phase. Substitution of equation (5.14) into equation (5.13) yields

〈∇G 〉f = ∇〈G 〉f · LB . (5.15)

In deriving equation (5.13), it was stated in Appendix E that, in the second assump-tion, Bear and Bachmat in fact assumed that the proportionality constant of equa-tion (5.15) is unity. Since both the gradient of the average and the average of thegradient is directed in the streamwise direction, it therefore follows that

LB

= n n . (5.16)

75

It is thus evident from equation (5.16) that LB

fails to predict the straightness ofthe average molecular path route by molecular diffusion.

To obtain an equation for the lineality of the average pathlines of molecules travellingby means of momentum transport due to an external pressure gradient, Bear &Bachmat (1991) (their equation (2.6.6)) assumed that the fluid pressure inside Ufvaries monotonically and that

∇2p = 0 . (5.17)

Substituting pressure into G of equation (5.13), Bear & Bachmat (1991) (theirequation 2.6.7) then obtained

〈∇p 〉f = ∇〈p 〉f · LB +1

Uf

∫∫

Sfs

n · (∇p) ro dS . (5.18)

Note that in deriving equation (5.18) the second assumption, stated above, wasagain made, from which it follows, considering Appendix E, that L

Bwas in fact

already assumed to be unity. Furthermore, if it is assumed that:

• the porous medium is stationary (thus vSof= 0),

• the channel width remains constant throughout the porous structure,

• the interstitial flow can be represented by fully developed plane Poiseuille flow,and

• that the channels or streamtubes through which the interstitial flow takes placeare straight and thus have no curvature,

the gradient of the pressure will always be directed parallel to Sfs and equation(5.18) can be rewritten as

〈∇p 〉f = ∇〈p 〉f · LB . (5.19)

From Appendix E and equation (5.19) it thus follows, similarly to equation (5.16)for molecular diffusion, that

LB

= n n (5.20)

for momentum transport due to an external pressure gradient.

5.1.2 Case study of spherical REV by Bear and Bachmat

In Bear & Bachmat (1991), an alternative expression for LB

was derived as followsby applying a spherical REV to equation (5.12). For such an REV all the ro

76

n

n ro=R

ro=R

O

ro

Figure 5.2: The spherical REV considered by Bear and Bachmat.

position vectors have a magnitude R, the radius of the sphere, and lie parallel ton on the fluid-fluid interface. Therefore, no off-diagonal terms are relevant in thiscase study. This REV is shown in Figure 5.2.

For this REV equation (5.12) reduces to

LB

=RSffUf

1

Sff

∫∫

Sff

nn dS

=3ǫS

ǫ

1

Sff

∫∫

Sff

nn dS , (5.21)

where

ǫS =SffS . (5.22)

77

Verifying that equation (5.21) holds for other geometrical REVs

A cube with side lengths b is now considered. From equation (5.12), if the off-diagonal terms are ignored, it follows that

LB

=ǫS SǫUo

b

2

1

Sff

∫∫

Sff

nn dS

=3ǫS

ǫ

1

Sff

∫∫

Sff

nn dS , (5.23)

which is identical to equation (5.21). We thus assume that in general, the diagonalterms of equation (5.12) can be written as equation (5.21).

5.1.3 Case study of cubical REV by Bear and Bachmat

To clarify the meaning of LB

, Bear and Bachmat considered the example, as shownin Figure 5.3, of a cube with side lengths b, where the opposite sides are connectedwith a number of streamtubes, N . In this example, “ . . .Axi, Ayi, Azi, are thecross-sectional areas of tubes intersecting each pair of opposite faces normal to thex, y and z axes, respectively . . . ” (Bear & Bachmat (1991), p.131, (sic)).

b

b

b

i

j

k

Figure 5.3: The REV considered in the example by Bear & Bachmat (1991).

Streamtube i has length bi where bi > b. Thus

ǫS =

∑Nxi=1Axi +

∑Ny

i=1Ayi +∑Nzi=1Azi

3b2, (5.24)

where Nκ is the number of tubes entering the REV on the outer surface which isperpendicular to the κ-axis.

78

If the porous medium is quasi-isotropic (identical only along the three principleaxes i, j and k as is shown in Figure 5.3, referred to by Bear & Bachmat (1991) asisotropic), it follows from equation (5.24) that

ǫS =

∑Ni=1Aib2

. (5.25)

The porosity of the REV is given by

ǫ =3∑Ni=1Aibi − Uǫ

b3, (5.26)

where Uǫ is the volume of the intersections of the fluid channels which was accountedfor more than once in the first term. Although this term may be relatively large ifthe porosity is high, Bear and Bachmat assumed that it is negligible. Substitutionof equations (5.25) and (5.26) into equation (5.23) then yields

LB

=

(b∑Ni=1Ai∑N

i=1Aibi

)1

Sff

∫∫

Sff

nn dS . (5.27)

For a quasi-isotropic medium

1

Sff

∫∫

Sff

nn dS =1

3( i i+ j j + k k) (5.28)

in the perpendicular Cartesian coordinate system. After substituting equation (5.28)into equation (5.27), Bear and Bachmat obtained

LB

=

(b∑Ni=1Ai

3∑Ni=1Aibi

)( i i+ j j + k k) . (5.29)

Therefore according to equation (5.29), the lineality in the direction of any base vec-

tor is less or equal to1

3, (since bi ≥ b ∀ i) as was stated in Bear & Bachmat (1991)

(their equation (2.3.72)). Furthermore,3ǫS

ǫrepresents the average straightness of

the streamtubes (in three dimensions) and1

Sff

∫∫

Sff

nn dS represents the anisotropy

of the medium.

The latter however does not make sense, because, if bi = b ∀ i, the lineality shouldbe unity in a specific direction and not a third. Bear & Bachmat (1991) onlyconsidered the geometrical properties of an REV in their derivation and not theinterstitial flow direction. Because lineality is defined as the length L of the porous

79

medium divided by the distance Le of a streamtube traversing through the porousmedium, the interstitial flow direction plays an important role. Let us for exampletake the x-axis parallel to the flow direction n. Thus, if the REV is shifted toanother region inside the porous medium, the flow direction in the j and k directionsshould change, and over the whole porous medium it should cancel to zero. AnREV as defined by Bear & Bachmat (1991), as described above, is thus not arepresentation of the entire porous medium. In a representative REV, where Ufconsists of non-interconnected channels, the streamtubes in the j and k directionsshould be neglected, and those streamtubes can be viewed as part of the solid phase.In such an REV, the integral in equation (5.28) is in effect taken only over the fluid-fluid interface on the outer surface perpendicular to the x-axis. This integral should

therefore only be divided by SffRS =Sff3

, where SffRS is the relevant surface.

Thus, if only the streamwise direction is considered (where n is chosen parallel to abase vector),

1

SffRS

∫∫

SffRS

nn dS = n n (5.30)

from which it follows that

LB

=

(b∑Ni=1Ai∑N

i=1Aibi

)n n . (5.31)

From equation (5.31) it is evident that, in this case, the lineality in the streamwisedirection is less or equal to unity.

5.1.4 Two-dimensional example where the flow lines crossthe borders obliquely

Consider the following example, represented in Figure 5.4, where flow lines cross theouter boundary, Sff , obliquely.

Examining Figure 5.4, it is evident that, according to equation (4.2), the streamwiselineality is given by

L =L

Len n = L

[2(

L

2 cos θ

)]−1

n n = n n cos θ . (5.32)

From equation (5.13), the definition of lineality according to Bear and Bachmat,

LB

=1

Uf

∫∫

Sff

n ro dS

80

L

L2

d

θ θ

n

Figure 5.4: A porous medium where the flow lines cross the border obliquely.

=1

Ld

[d((

−L2n)

(− n) +(L

2n)

( n))]

= n n , (5.33)

as, from equation (5.20), is expected. Equation (5.33) is not the same as equation(5.32) and L

Bthus fails to predict the lineality correctly.

This example is two-dimensional. For such a square REV of side lengths L anequation similar to equation (5.23) can be derived for the lineality in two dimensions.From equation (5.12), if the off-diagonal terms are ignored, it follows that

LB

=ǫS Sǫ Uo

L

2

1

Sff

∫∫

Sff

nn dS

=

(2ǫS

ǫ

)1

Sff

∫∫

Sff

nn dS (5.34)

=2(

2d4L

)

(d cos θ L

cos θ

L2

) 1

Sff

∫∫

Sff

nn dS

=1

Sff

∫∫

Sff

nn dS . (5.35)

But, as was already mentioned in equation (5.33), this should be equal to n n. Thus,

n n =1

Sff

∫∫

Sff

nn dS (5.36)

81

which confirms equation (5.30). Note that in this example Sff = SffRS. Substi-tuting equation (5.36) into equation (5.34) yields

LB =

(2ǫS

ǫ

)n n (5.37)

which is the general expression for equation (5.12) in two dimensions where onlytwo opposing surfaces are interconnected with streamtubes.

5.1.5 The credibility of the assumptions made by Bear andBachmat

In the example of the cubical REV presented in Figure 5.3, a few assumptions weremade by Bear and Bachmat to simplify the example. These simplifications cannotbe made to an arbitrary REV representing a porous medium.

The following phrase from Bear & Bachmat (1991) “. . .Axi, Ayi, Azi, are the cross-sectional areas of tubes intersecting each pair of opposite faces normal to the x, yand z axes, respectively . . . ” (Bear & Bachmat (1991), p.131, (sic)) is ambiguous,since it either means that the faces, which the tubes intersects, are normal to theprinciple axes, or it implies that the tubes intersects the faces normal to the principleaxes. Had the first interpretation of this phrase been implied, thus the tubes neednot intersect the faces orthogonally, considering the example in Figure 5.4 as well asall the other examples in Appendix D, it is evident that equation (5.12) fails to pre-dict the correct lineality under such circumstances. The second interpretation thatthe streamtubes intercepts the fluid-fluid interface orthogonally was thus intended,though, in general, for an arbitrary REV, this assumption is rarely met.

In the quoted phrase, two important assumption were made, namely:

1. the interstitial flow direction on the outer surface of the REV is perpendicularto the plane in which that surface lies, and

2. each streamtube is connected to opposite sides of the REV.

A third assumption by Bear and Bachmat was that Uǫ << Uf , thus:

3. the total volume of all the intersections of the streamtubes, Uǫ, is negligible incomparison to the total volume of the fluid phase, Uf .

The first two assumptions will be discussed first. In Figure 5.5 an REV of a porousmedium consisting of some individual fluid channels is shown.

Note that only channels 3 and 4 satisfy the first two assumptions made by Bearand Bachmat. In channels 1 and 5, n is not parallel to n everywhere on Sff andchannel 2 does not satisfy the second assumption, as the fluid does not enter andexit opposite sides of the REV. Clearly, in general, it seldom happens that the firsttwo assumptions are met.

82

L

L

L

i

j

k

n

nn

n nn

n

n

n

nn

n

n

n

n

n

n n

n

n

n

n

nn

nθ

1

2

3

4

5

Figure 5.5: An arbitrary REV consisting of fluid channels.

Assumption 1:

Channel 5 in Figure 5.5 has a large surface area where n is more or less perpendicularto n and therefore this channel is going to be examined to test the effect that thefirst assumption has on this arbitrary REV. The following assumptions were made:

• only channel 5 is present in the REV with volume L3;

• Le(5) =L

L(5)

;

• n ⊥ n over the whole fluid-fluid interface with an area Sff(5Right) on the rightboundary of the REV;

• n is anti-parallel to n on the fluid-fluid interface with an area Sff(5Lower) onthe lower boundary of the REV;

• n makes an angle of θ radians with n on the fluid-fluid interface on the upperboundary, Sff(5Upper);

• the volume of channel 5 is denoted by Uf(5);

• n ‖ j and considering the above assumptions, it follows that

Le(5) ≈ Uf(5)

[Sff(5Lower) + Sff(5Upper) cos θ

2

]−1

83

and therefore

L(5)

= L(5) n n ≈ L(Sff(5Lower) + Sff(5Upper) cos θ)

2Uf(5)

.

If the off-diagonal terms of the dyadic are ignored, it follows from equation (5.12)that

LB

=1

Uf

∫∫

Sff

n ro dS

=1

Uf(5)

[(L

2i i) (

Sff(5Right)

)+(L

2n n

)(Sff(5Upper)

)

+(−L

2n)

(− n)(Sff(5Lower)

)]

=LSff(5Right)

2Uf(5)

i i+L(Sff(5Upper) + Sff(5Lower))

2Uf(5)

n n . (5.38)

Thus, in the streamwise direction, sinceLe(Sff(5Upper) + Sff(5Lower))

2will be larger

than Uf(5) for θ 6= 0, the lineality predicted, according to the equation of Bear andBachmat, could be larger than unity, which does not make any sense. Further-more, L

iishould approximately be 0, but, due to the large outer surface, Sff(5Right),

this is not the case. If Sff(5Right) is much larger than the sum of Sff(5Upper) andSff(5Lower), as is the case in channel 5, the first term of equation (5.38) might evenbe predominant.

Assumption 2:

The effect of the second assumption can be evaluated by considering flow throughchannels 2 and 3 of Figure 5.5. We make the following assumptions regarding thesechannels:

• to eliminate any errors that might occur if the first assumption does not hold,if n exists, let it be either parallel or anti-parallel to n on all the fluid-fluidinterfaces on the boundary of the REV;

• Sff(3Right) = Sff(3Left) = Sff(3) and Sff(2Left) = Sff(2);

• the total volume occupied by the fluid phase, Uf , consists of both Uf(3) andUf(2), the volumes of channels 3 and 2 respectively;

• n ‖ i and since no fluid traverses channel 2, Le(3) =L

L ≈ Uf(3)

Sff(3)

, and therefore

L =LSff(3)

Uf(3)

n n .

84

Again, if the off-diagonal terms of the dyadic are ignored, it follows from equation(5.12) that

LB

=1

Uf

∫∫

Sff

n ro dS

=1

Uf

[(−L

2n)

(− n)(Sff(2)

)+ 2

(L

2n n

)(Sff(3)

)]

=L(Sff(2) + 2Sff(3))

2Ufn n . (5.39)

Whether channel 2 is a dead end pore or exits on any other boundary than theright, the same value for L is obtained in both cases. If the length of channel 2inside the REV is approximately the half of the length had it continued throughwith the same curvature to the other side of the REV, a sensible approximation forthe streamwise straightness of that tube will be obtained from equation (5.12). Ifequation (5.12) is applied to both channels 2 and 3, the contribution of the dead endpore to the overall lineality could have a predominant effect if its volume and/or itsouter surface is larger than the volume and the sum of the outer surfaces of channel3 respectively, though this term should play no role in the lineality at all, since nofluid traverses through it.

Assumption 3:

The third assumption made by Bear and Bachmat was that the total volume ofthe intersections of the streamtubes, Uǫ, is negligible in comparison to the totalvolume occupied by the fluid phase, Uf . The validity of this assumption is porositydependent and therefore the porosity subscript in Uǫ. If the porosity is low, the fluidcarrying channels are small and/or few, and therefore this assumption is relevant.Thus, for a porous medium similar to the one represented in Figure 5.3, for lowporosities, equation (5.21) predicts the correct lineality, according to equation (5.12),in three-dimensions and equation (5.34) for two-dimensions. If however the porosityis high, this assumption presents a problem. However, in the derivation of equation(5.12) no assumptions in this regard was made, and the validity of this equation willnot depend on Uǫ.

Note that in the RRUC model, even for very low porosities,

LB

=1

Uf

∫∫

Sff

n ro dS 6= 3ǫS

ǫn n ,

because, for an RRUC, Uf = Uǫ.

85

5.2 Lineality as derived by Diedericks and

Du Plessis

Diedericks & Du Plessis (1995) formulated an alternative expression for the linea-lity of an isotropic medium, assuming that the direction of flow on the fluid-fluidinterfaces of a particular REV is known. They considered the following identity:

∇· ( n n ro) = (∇· n n) ro + ( n n) ·∇ ro= [(∇· n) n+ n ·∇ n] ro + n n

= (∇· n) n ro + n n . (5.40)

The vector n is constant and therefore ∇ n = 0. As indicated earlier on, ∇ro = 1.After volume averaging of equation (5.40), division by the porosity and applicationof the divergence theorem to 〈∇· ( n n ro) 〉f , the following equation is obtained:

〈 n n 〉f =1

Uf

∫∫

Sof

n · n n ro dS − 〈(∇· n) n ro 〉f . (5.41)

In a porous medium, n ⊥ n at all the fluid-solid interfaces, therefore the integralin equation (5.41) vanishes over Sfs part of Sof and the above equation reduces to

〈 n n 〉f =1

Uf

∫∫

Sff

n · n n ro dS − 〈(∇· n) n ro 〉f . (5.42)

Diedericks & Du Plessis (1995) further states, somewhat unconvincingly, that thelatter term takes into account the porosity changes that might occur. Accordingto Diedericks (1999), ∇· n takes into account the possibility that the number ofstreamlines entering and exiting the considered volume may differ. Therefore, thelast term of equation (5.42) is zero except at stagnation points where streamlinesdivide or converge. If one accepts these statements, this last term may only beneglected for a porous medium with constant porosity where the stagnation pointsare ignored.

The LHS of equation (5.42) is the “effective streamwise volume” defined by Die-dericks & Du Plessis (1995), their equation (9), divided by Uf . This intrinsic phaseaverage is thus a measure of the straightness of streamlines as a fluid traverses aporous medium. (Note that the flow is considered to be stationary from which itfollows that the streamlines and pathlines are equivalent. Therefore the LHS is alsoa measure of the straightness of pathlines, and thus a measure of the lineality.)

According to Diedericks & Du Plessis (1995), in an isotropic medium, the streamwise

86

lineality is thus given as

LD

=1

Uf

∫∫

Sff

n · n n ro dS . (5.43)

Here, an isotropic medium refers to a medium where the average speed in everychannel section is equal. Therefore, a porous medium where, for a regular and over-staggered array all the channels have the same width and for a fully staggered arraythe width of the streamwise channels is half the width of the transverse channels,though the latter was incorrectly not assumed.

The difference between LD

and LB

is that for LD

the interstitial flow direction onSff plays a role. Note that since

n · n =

−1 if − n = n

1 if n = n

LD

will reduce to LB

if the interstitial flow direction at each point on Sff is parallelto the streamwise direction.

5.2.1 The credibility of the assumptions made by Diedericksand Du Plessis

For confirmation of the fact that this equation will predict the lineality correctlyfor any arbitrary REV in an isotropic porous medium, we will start off by alsoexamining the example on channel 5 of Figure 5.5. The same assumptions stated insection 5.1.5 regarding this channel holds. From equation (5.43) it follows that

LD

=1

Uf

∫∫

Sff

n ro n · n dS

=1

Uf(5)

[nL

2i (0)

(Sff(5Right)

)+ n

L

2n (cos θ)

(Sff(5Upper)

)

+ n(−L

2n)

(−1)(Sff(5Lower)

)]

=L(Sff(5Lower) + Sff(5Upper) cos θ)

2Uf(5)

n n

≈ L(5) n n . (5.44)

Therefore, flow crossing the fluid-fluid interface obliquely does not present a problemfor L

D.

87

If equation (5.43) is applied to the second example, where channels 2 and 3 wereexamined, then

LD

=1

Uf

∫∫

Sff

n ro n · n dS

=1

Uf

[2 n

L

2n(Sff(3)

)]

=LSff(3)

Ufn n , (5.45)

since n is not defined in channel 2. Though the outer surface of channel 2 does notplay a role in the determination of the lineality, as was the case in the definitionby Bear and Bachmat, the denominator of equation (5.45) is the sum of Uf(2) andUf(3), and not Uf(3) only. Considering Example 1 of Appendix D, it is evident thatstagnant regions indeed present a problem for the lineality as defined by Diedericks& Du Plessis (1995). It could thus be concluded that if Uf is substituted withUf −Ug in equation (5.43), a better approximation for the lineality would have beenobtained.

5.2.2 Derivation of LD by means of the Green’s theoremgeneralised for dyadics

To ensure streamwise lineality, let (a b) ≡ n ro in equation (C.3) (Green’s theoremgeneralised for dyadics), and substitute c with G, where G is an arbitrary scalar fielddefined within the fluid phase. Thus

∫∫

Sof

n · [G∇( n ro) − (∇G) ( n ro)] dS

=∫∫∫

Uf

[G∇2( n ro) −∇2G( n ro)

]dU .

(5.46)

As mentioned earlier, ∇ro = 1 and ∇ n = 0, from which it follows that

∇2 ro = ∇· 1 = 0

and

∇2 n = 0 .

It is also important to note that ∇1 = 0 and that ∇· n = 0.

88

The first term on the RHS of equation (5.46) thus reduces to

∫∫∫

Uf

[G∇2( n ro)

]dU =

∫∫∫

Uf

[G∇· (∇ ( n ro))] dU

=∫∫∫

Uf

[G∇· ( n (∇ro))] dU

=∫∫∫

Uf

[G∇·

(n 1)]dU

=∫∫∫

Uf

[G(∇· n 1 + n · ∇1

)]dU

= 0 . (5.47)

The first term of equation (5.46) reduces to

∫∫

Sof

n · [G∇( n ro)] dS =∫∫∫

Uf

∇· [G∇( n ro)] dU

=∫∫∫

Uf

(∇G) · ∇( n ro) +G∇·∇( n ro) dU

=∫∫∫

Uf

(∇G) · (∇ n) ro+ n(∇G) · ∇ ro+G∇· ( n 1) dU

=∫∫∫

Uf

n (∇G) · 1 dU

=∫∫∫

Uf

n (∇G) dU . (5.48)


∫∫∫

Uf

n (∇G) dU =∫∫

Sof

n · (∇G) n ro dS −∫∫∫

Uf

∇· (∇G) n ro dU . (5.49)

Comparing equation (5.49) and equation (5.42) it is evident that if we substitute(∇G) ≡ φ n, where φ is any constant scalar quantity, into equation (5.49) and after

89

division by φUf , the following is obtained:

〈 n n 〉f =1

Uf

∫∫

Sff

n · n n ro dS − 〈(∇· n) n ro 〉f . (5.50)

Clearly this is exactly the same as equation (5.42). This means that, if G is ascalar field so that ∇G is parallel to n (thus ∇G ≡ ∇G), we obtain the equationformulated for lineality by Diedericks & Du Plessis (1995). To obtain equation(5.18), describing momentum transport due to an external pressure gradient, Bearand Bachmat substituted G form equation (5.13) with p. Note that ∇p = φ n whereφ < 0, everywhere within Uf , except at Ut where the pressure gradient is zero.

Therefore, the equation derived by Diedericks & Du Plessis (1995) for lineality,equation (5.42), was again derived, this time via Green’s theorem generalised fordyadics in Appendix C. We substituted a scalar, G = p, of which the gradient isparallel to the direction of the interstitial velocity at any point in the fluid phase,and the dyadic term, n ro, into Green’s dyadic theorem. Note that after equation(5.42) has been obtained, Diedericks and Du Plessis assumed that ∇· n is zero, whichis equivalent to ∇2p = 0, since φ is a constant. This assumption thus correspondsto assumption (5.17) made by Bear & Bachmat (1991). In Appendix E it is shownthat if ∇G = ∇p = φ n,

〈∇p 〉f = LD∇〈p 〉f , (5.51)

for the RRUC model considered by Lloyd (2003) where the transfer volume is rectan-gular, as is presented in Figure 3.6(b).

5.2.3 The lineality defined by Bear and Bachmat versus thelineality defined by Diedericks and Du Plessis

In section 5.1.1 the derivation by Bear & Bachmat (1991) for lineality was discussed.They also substituted a scalar, G = p, of which the second spatial derivative isassumed to be zero (equation (5.17)), this time into Green’s vector theorem. Theother vector substituted was ro. Bear and Bachmat obtained equation (5.13)which is rewritten below:

〈∇p 〉f = ∇〈p 〉f · LB +1

Uf

∫∫

Sfs

n · (∇p) ro dS . (5.52)

In that section, two assumptions made by Bear & Bachmat (1991) were mentioned.The second part of the second assumption was that

1

Sff

∫∫

Sff

∇p dS ≈ ∇〈p 〉f , (5.53)

90

which, from Appendix E, only holds if

∇〈p 〉f = 〈∇p 〉f . (5.54)

From equation (5.51) that also follows from the discussion in Appendix E, it isevident that equation (5.54) is in general false and therefore assumption (5.53) onlyholds if LD = 1. Cancelling this assumption by substituting approximation (5.53)back into equation (5.52) yields

〈∇p 〉f =1

Sff

∫∫

Sff

∇p dS · LB

+1

Uf

∫∫

Sfs

n · (∇p) ro dS . (5.55)

After substituting ∇p = φ n into equation (5.55), we obtain

〈φ n 〉f =1

Sff

∫∫

Sff

φ n dS · LB

+1

Uf

∫∫

Sfs

n · (φ n) ro dS

⇒ 〈 n 〉f =1

Sff

∫∫

Sff

n dS · LB

+1

Uf

∫∫

Sfs

n · n ro dS . (5.56)

From a volume averaging identity, it follows that

〈 n n 〉f = 〈 n 〉f 〈 n 〉f +⟨ nf nf

⟩f

= 〈 n 〉f 〈 n 〉f , (5.57)

and therefore, multiplying equation (5.56) from the left with 〈 n 〉f = n yields

〈 n n 〉f =n

Sff

∫∫

Sff

n dS · LB

+n

Uf

∫∫

Sfs

n · n ro dS

=1

Sff

∫∫

Sff

n n dS · LB

+n

Uf

∫∫

Sfs

n · n ro dS . (5.58)

Since n is perpendicular to n everywhere on Sfs, the last term in equation (5.58)is zero and

〈 n n 〉f =1

Sff

∫∫

Sff

n n dS · LB. (5.59)

Diedericks & Du Plessis (1995) assumed that ∇· n = 0 and from equation (5.42) wethus obtain:

LD

=1

Sff

∫∫

Sff

n n dS · LB. (5.60)

91

In section 5.1.1 the first part of the second assumption made by Bear & Bachmat(1991) was

∫∫

Sff

n · φ n ro dS ≈ 1

Sff

∫∫

Sff

φ n dS ·∫∫

Sff

n ro dS . (5.61)

Multiplying this assumption from the left with n and dividing it by the Uf , equation(5.60) follows directly.

5.2.4 Two-dimensional example where the flow lines crossthe borders obliquely revisited

To evaluate relation (5.60), consider the example presented in Figure 5.4. Accordingto equation (5.43) it follows that

LD

=1

Uf

∫∫

Sff

n ro n · n dS

=1

Ld

[d([

(− n) ·(cos θ n+ sin θ j)] [−L

2n]

[ n])]

+1

Ld

[d([

( n) ·(cos θ n− sin θ j)] [L

2n]

[ n])]

=1

L

[L

2cos θ n n+

L

2cos θ n n

]

= n n cos θ , (5.62)

which is the same as equation (5.32) and thus predicts the correct lineality for thisexample.

According to equation (5.60) and substitution of equation (5.33), we obtain

LD

=1

Sff

∫∫

Sff

n n dS · LB

=1

Sff

∫∫

Sff

n n dS · n n

=1

Sff

∫∫

Sff

n n · n n dS

92

=1

Sff

∫∫

Sff

cos θ n n dS

= n n cos θ , (5.63)

which is also the same as equation (5.32). The integral over the fluid-fluid interface ofequation (5.60) thus compensates for the interstitial flow direction on the boundarywhich is not incorporated in L

B.

93

5.3 Comparison between the linealities

In Appendix D different SRRUCs and porous structures have been examined. Weonly considered two-dimensional examples and the SRRUCs and porous structureshad unit length in the third dimension.

Examples from Appendix D LB

LD

LGeo

LKin

LDyn

Example 1: Regular Array constant ‖ v‖√ × × √ √

Example 2: Over-staggered Arrayconstant ‖ v‖ × √ √ √ √

variable ‖ v‖ × × × √ ×

Example 3: Fully staggered Arrayconstant ‖ v‖ × √ √ √ √

variable ‖ v‖ × × × √ ×

Example 4: Flow lines cross border obliquely constant ‖ v‖ × √ √ √ √

Example 5: Limit where lineality tends to zero constant ‖ v‖ × √ √ √ √

Example 6: Recirculators constant ‖ v‖ × √ √ √ √

Table 5.1: Results from Appendix D where ‘√

’ and ‘×’ indicate whether themethod predicted the lineality (as defined by equation (4.2)) correctly or incorrectlyrespectively.

5.3.1 The lineality as derived by Bear and Bachmat

Clearly equation (5.12) is not a good approximation for the lineality of an SRRUC.The only case where this equation will predict the lineality correctly is when theintrinsic flow direction on Sff is parallel to the streamwise direction and remains soeven if the SRRUC is shifted relative to the solid phase. From this it immediatelyfollows that there may not exist transversely orientated channels which then resultsin a lineality equal to unity. Also, the average interstitial speed should be the samein each channel section throughout the medium. An example of this is a regulararray.

After examining all the examples presented in this work, one finds that the differencebetween this lineality and the lineality as derived by Diedericks and Du Plessis (orthe geometric lineality) is when the fluid-fluid interface is not perpendicular to theinterstitial flow direction at the surface of the RRUC. Every time when the controlvolume (SRRUC) is shifted so that its edges lie in the transverse channels, part ofSff is parallel to n. Example 4 was chosen to illustrate this deficiency. As mentioned

94

earlier on, if this interstitial flow direction on Sff is in the streamwise direction, LD

reduces to LB

.

Similarly to these SRRUC examples, if equation (5.12) is applied to an arbitraryREV chosen anywhere inside the porous medium, it will also fail to predict thelineality correctly. Parts of Sff will not lie in a plane perpendicular to the flowdirection and therefore an error is bound to occur, as was discussed in section 5.1.5.

5.3.2 The geometric lineality

Referring to Table 5.1, it is evident that the scalar components of equations (5.43)and (4.22) are equivalent and a good approximation for the lineality of porous mediawhere the magnitude of the average streamwise interstitial velocity and the averageinterstitial velocity in the direction perpendicular to n are equal and remain con-stant. (Under such circumstances the kinematic and dynamic lineality will alsoreduce to this lineality.) Considering equations (4.22) and (4.29) it follows that thislineality is equivalent to

U‖ + Ut‖Uf

. (5.64)

Stagnant regions, however, still present a problem for this definition, since the de-nominator of expression (5.64) also includes volumes where n is not defined.

5.3.3 The kinematic lineality

It is evident that this definition predicts the correct lineality under all the circum-stances studied. If the magnitude of the average interstitial velocity in each channelis the same, this equation reduces to the geometric lineality. If there are dead endpores, the velocities of the particle inside these channels are small or zero and thecontribution of the part of Uf consisting of dead end pore volumes is neglected.Therefore this will not present a problem.

5.3.4 The dynamic lineality

The only case for which the dynamic lineality predicts the correct lineality is whenit reduces to the geometric lineality, except for a regular array, where there existstagnant regions. If the magnitude of the average interstitial velocities are not equaland constant, the lineality predicted is incorrect. We derived the equation for thedynamic lineality directly from the definition of the lineality as a kinematic propertyexpressed by Bear (1972). Clearly this definition maybe interpreted erroneously.A better definition for the average path of the fluid particles, Le, as a kinematic

95

property, would therefore be to consider particles passing through a cross-sectionalarea of a given medium at time t and not over a time period as was stated by Bear(1972). If only these particles are considered, this definition reduces to that of thekinematic lineality defined in this present study.

Figure 5.6 summarises the different linealities, as well as their corresponding restric-tions, as was discussed in this Chapters 4 and 5.

96

Lineality

• w differs in streamwise and transverse channels: (w⊥ = ξψw‖)

• n is directed in any arbitrary direction on Sff

present work: equation (4.23)

LKin =

[〈 n · v v 〉f〈v2 〉f

]n

• w is a uniform scalar field in Uf : (w⊥ = w‖, thus ξψ = 1)

Diedericks & Du Plessis (1995): p.275, their equation (39)

present work: equations (5.43), (5.60) and (4.22)

LD

=1

Uf

∫∫

Sff

n ro n · n dS

=1

Sff

∫∫

Sff

n n dS · LB

= 〈 n · n 〉f n n= LGeo n

• n is directed parallel to streamwise direction on Sff : ( n ‖ n on Sff )

Bear & Bachmat (1991): p.126, their equation (2.3.49)

present work: equation (5.12)

LB

=1

Uf

∫∫

Sff

n ro dS

Figure 5.6: Summary on different linealities discussed in Chapters 4 and 5.

97

Chapter 6

The permeability-tortuosityrelation

In this chapter the objective is to write the dimensionless Darcy permeability, de-termined in Chapter 3 by both the direct and volume averaging method (equations(3.26) and (3.61) respectively), in terms of the kinematic tortuosity, χKin, which isthe inverse of the kinematic lineality, LKin, discussed in Chapters 4 and 5.

In literature, numerous attempts are reported to find such a relation between per-meability and the average waviness or straightness of pathlines through a porousstructure. Bear & Bachmat (1991) (p.174, their equation (2.6.55)) stated that

k ∝ LB . (6.1)

Contrary to Bear and Bachmat, Lloyd et al. (2004) (their equation (20)) stated that

K ∝(

1χD

)2

, (6.2)

where K is the dimensionless permeability defined by substituting equation (3.13)into equation (3.14). From equations (3.49) and (3.52) it follows for plane Poiseuilleflow that

−∇〈p 〉o = χDτ‖S‖ + ξψτ⊥S⊥

Uon

=χDUo

(6µw‖dc⊥

S‖ + ξψ6µw⊥dc‖

S⊥

)n . (6.3)

In Lloyd et al. (2004) only situations were considered where the width of the stream-wise and transverse channels are equal, thus dc‖ = dc⊥ = dc. Equation (6.3) there-fore reduces to

−∇〈p 〉o =χDUo

(6µw‖dc

)(S‖ + ξ2S⊥

)n . (6.4)

98


w‖ =wKinUf

U‖ + Ut ‖ + ξψ(U⊥ + Ut⊥). (6.5)

For an RRUC, as the one shown in Figure 4.6, the inverse of the kinematic lineality(equation (4.13)), the kinematic tortuosity, can be written as

χKin =U‖ + Ut ‖ + ξψ(U⊥ + Ut⊥)

U‖ + Ut ‖. (6.6)

Combining equations (4.50), (5.64), (6.5) and (6.6) yields

w‖χD

=wKinχKin

= 〈 v 〉f · n . (6.7)

From the Dupuit-Forchheimer relation, equation (2.19) and equation (6.7) it followsthat

q = ǫw‖χD

. (6.8)

Substituting equation (6.8) into equation (6.4) yields

−∇〈p 〉o =χDUo

(6µ q χD

ǫdc

)(S‖ + ξ2S⊥

),

hence

−∇〈p 〉f =S‖ + ξ2S⊥

Uo(χD)2

(6

ǫ2dc

) (µ q). (6.9)

Substituting equation (6.9) into the definition for the Darcy permeability, equation(3.13), where the pressure gradient is replaced by the gradient of the intrinsic phaseaverage of the pressure, it follows that

k =Uo

S‖ + ξ2S⊥

(1χD

)2(ǫ2dc

6

). (6.10)

Equation (6.10) was also obtained by Lloyd et al. (2004) in their equation (19).From this, relation (6.2) follows.

Similarly to Lloyd et al. (2004), Carman (1937) also arrived at

k ∝(

1χC

)2

. (6.11)

The following law of Darcy,

q = k1

µ

∆p

Ln , (6.12)

99

where k is the permeability, was examined to obtain the permeability-tortuosityrelation. Kozeny (1927), as referred to by Carman (1937), assumed that a granularbed can be represented by a group of similar, parallel channels where the ratio of thetotal internal volume to total internal surface is equal to that of the pore volumesto the total surfaces of the particles. Thus the mean hydraulic radius is denoted by

m =volume of fluid in pipe

surface presented to fluid=

ǫ

SCK. (6.13)

Kozeny pointed out that due to the tortuous character of flow through a granularbed, the length of the equivalent channels should be Le as is depicted in Figure 4.1.The general law of streamline flow through a channel is given by

u =m2

µ kK· ∆p

Le(6.14)

where kK is a constant that depends on the shape of the cross-section of the channel.

According to Carman, substitution of u and Le in equation (6.14) is not sufficient,

and has to be modified even further. Carman stated thatq

ǫis the “average velocity

parallel to the direction of flow” (Carman (1937), p.152 (sic)) and since the pathof an element of the fluid is sinuous, u represents only the velocity parallel to thestreamwise direction. According to Carman, the time taken to pass over a distanceL at a velocity, u, corresponds to fluid travelling a length Le at a velocity χC u,

with χC =LeL

. It was thus stated that the “true” value for u isχC q

ǫ, and therefore,

after substituting equation (6.13), equation (6.14) becomes

q =ǫ3

kK µS2CK

∆p

L

(1χC

)2

, (6.15)

corresponding to their equation (8), and thus, considering the Darcy equation (6.12),relation (6.11) followed.

Note that, not u, but w‖ n= wGeo defined in equation (4.32), is the “average velocityparallel to the direction of flow”. From equations (6.7) and (6.8) it follows that

q = ǫwGeoχGeo

= ǫwKinχKin

= ǫ u . (6.16)

For equivalent parallel channels, as the ones examined by Kozeny, wGeo= wKin andχGeo = χKin. Thus, a more sensible derivation would have been if the subject ofKozeny’s equation (6.14) was wGeo, since then Carman’s substitution would nothave been questionable.

From equations (6.3), (6.2) and (6.11) it is evident that a clear definition regardingthe permeability-tortuosity relation is still needed. In the following section such adefinition is attempted.

100

6.1 The general relation between permeability and

tortuosity

Rewriting equations (3.26) and (3.61), the dimensionless Darcy permeability is givenby

K =γ3ψϕ

12 [ψ − ϕγ + ξ2ϕψ4(1 − 2γ)], (6.17)

where ϕ, ψ and γ are geometrical properties of the specific SUC or SRRUC consid-ered. These geometrical parameters are defined in equations (3.1), (3.2) and (3.3)respectively. In equation (3.12) the various values of ξ, the constant that describesthe streamwise staggeredness of the solid material, are given.

From equations (5.64) and (6.6) it follows that

χDUf

=χKin

U‖ + Ut ‖ + ξψ(U⊥ + Ut⊥). (6.18)


−∇〈p 〉f =Uf

d‖dc⊥


Uf

]n

=χDUf


]n

=χkin

U‖ + Ut ‖ + ψξ(U⊥ + Ut⊥)


]n

=χkin

U‖ + Ut ‖ + ψξ(U⊥ + Ut⊥)

[12µw‖ds‖dc⊥

+12µψξw⊥(ds⊥ − dc⊥)

dc‖

]n

=χkin

U‖ + Ut ‖ + ψξ(U⊥ + Ut⊥)

[12µqd⊥ds‖

dc2⊥+

12µψ2ξ2qd⊥(ds⊥ − dc⊥)

dc‖dc⊥

]n

=12µqχkin

U‖ + Ut ‖ + ψξ(U⊥ + Ut⊥)

[d⊥ds⊥(ψ − ϕγ)

dc2⊥ϕψ(1 − γ)+ψ3ξ2d⊥(ds⊥ − dc⊥)

dc2⊥

]n

=12µqχkin

U‖ + Ut ‖ + ψξ(U⊥ + Ut⊥)

[(1 − γ)(ψ − ϕγ)

γ2ϕψ(1 − γ)+ ψ3ξ2 1

γ

(1 − γ

γ− 1

)]n

=12µqχkin

U‖ + Ut ‖ + ψξ(U⊥ + Ut⊥)

[ψ − ϕγ

γ2ϕψ+ψ3ξ2(1 − 2γ)

γ2

]n

=12µqχkin

U‖ + Ut ‖ + ψξ(U⊥ + Ut⊥)

[ψ − ϕγ + ξ2ϕψ4(1 − 2γ)

γ2ϕψ

]n . (6.19)

Substitution of equation (6.19) into the definition for the Darcy permeability, equa-tion (3.13), where the pressure is replaced by the intrinsic phase average of the

101

pressure, non-dimensionalised as described in equation (3.14), yields

K =µq

d‖d⊥‖∇ 〈p 〉f ‖

=U‖ + Ut ‖ + ψξ(U⊥ + Ut⊥)

12d‖d⊥χkin

[γ2ϕψ

ψ − ϕγ + ξ2ϕψ4(1 − 2γ)

]

=U‖ + Ut ‖ + ψξ(U⊥ + Ut⊥)

d‖d⊥χkinγ

[γ3ϕψ

12[ψ − ϕγ + ξ2ϕψ4(1 − 2γ)]

]

=U‖ + Ut ‖ + ψξ(U⊥ + Ut⊥)

d‖d⊥χkinγK .

Hence

χKin =U‖ + Ut ‖ + ψξ(U⊥ + Ut⊥)

d‖d⊥γ

=dc⊥d‖ + ψξdc‖ds⊥

d‖d⊥γ

= 1 + ξϕ(1 − γ) . (6.20)

In the α-model (equations (3.9)), equation (6.20) reduces to

χKin = 1 + ξα√

1 − ǫ . (6.21)

If α = 1, thus dc‖ = dc⊥, following from equation (3.22), the average speed relationbetween the streamwise and transverse channels is

w⊥ = ξw‖ . (6.22)

The tortuosity in equation (6.21) therefore differs from that of Lloyd et al. (2004), intheir Table 1, where α = 1, because the interstitial speed is not a uniform scalar fieldthroughout Uf in a fully staggered configuration. From their respective equationsfor χD(ǫ), it follows that

χD = 1 + ξ2√

1 − ǫ , (6.23)

which does not yield χD =LeL

in a fully staggered configuration. Note that, when

α = 1, the difference between equations (6.20) and (6.23) only materializes whenthe solid material in a porous medium is fully staggered in the streamwise directionand the average interstitial speeds in the parallel and transverse channels differ.

6.1.1 Regular array

From equation (6.17) it follows that the dimensionless Darcy permeability of a re-gular array is

K =γ3ψϕ

12 [ψ − ϕγ]. (6.24)

102

For an SUC or SRRUC with specific geometrical properties, its corresponding per-meability can be determined for a regular array.

The tortuosity of a non-staggered configuration is always unity and therefore equa-tion (6.24) is indisputably independent of tortuosity.

6.1.2 Staggered configuration


γ =ξϕ− χkin + 1

ξϕ. (6.25)

From a frictional coefficient (the inverse of the dimensionless permeability, equation(3.71)) exerted by the porous medium on the fluid:

F = 12

(ψ − ϕγ

γ3ϕψ

)+ 12ξ2ψ3

(1 − 2γ

γ3

), (6.26)

it follows that

0 < γ ≤ min

1

2;ψ

ϕ

. (6.27)

Substitution of equation (6.25) into equation (6.27) yields the following bounds ontortuosity values:

max

1 +

ξϕ

2; 1 + ξ(ϕ− ψ)

≤ χKin < 1 + ξϕ . (6.28)

Eliminating γ from equation (6.17) by substitution of equation (6.25), it follows that

K =γ3ϕψ

12[ψ − ϕγ + ξ2ϕψ4(1 − 2γ)]

=

(ξϕ−χKin+1

ξϕ

)3ϕψ

12[ψ − ϕ(ξϕ−χKin+1

ξϕ

)+ ξ2ϕψ4

(1 − 2

(ξϕ−χKin+1

ξϕ

))]

=(ξϕ− χKin + 1)3 ψξϕ

12ξ3ϕ2[ψξϕ− ϕ (ξϕ− χKin + 1) + ξ2ϕψ4 (ξϕ− 2 (ξϕ− χKin + 1))]

=(ξϕ− χKin + 1)3 ψ

12ξ2ϕ2[ψξ − ξϕ+ χKin − 1 + ξ2ψ4 (2χKin − 2 − ξϕ)]. (6.29)

For the α-model, equation (6.29) reduces to

K =(αξ − χKin + 1)3

12αξ2 [χKin − 1 + ξ2α4(2χKin − 2 − αξ)], (6.30)

103

and substituting

αξ =χKin − 1√

1 − ǫ, (6.31)

following from equation (6.21), into equation (6.30), yields

K(χKin, ǫ, ξ) =(χKin − 1)

[1 −

√1 − ǫ

]3

12[ξ(1 − ǫ) + (χKin−1)4

ξ(√

1−ǫ)3 (2√

1 − ǫ− 1)] . (6.32)

Note that since ǫ ≤ 0.75 for this model, it follows from equation (6.31) that thebounds on tortuosity are 0.5αξ + 1 ≤ χkin ≤ αξ + 1.

6.1.3 Evaluating the permeability-tortuosity relation

For the evaluation of equation (6.29) two case studies are considered. The two SUCsor SRRUCs under consideration are depicted in Figure 6.1 where the respectiveratios are presented on scale.

ϕ =1

3; ψ =

1

12; γ =

1

6

B

ϕ =1

3; ψ =

5

4; γ =

5

12

A

n

Figure 6.1: The two SUCs or SRRUCs considered in evaluating equation (6.29).

From Figures 6.2 and 6.3 it is evident that the permeability-tortuosity relation isnot as simple as claimed by many authors (e.g. Bear & Bachmat (1991), Lloyd et al.(2004) and Carman (1937)), and it differs for different geometrical situations.

In the α-model, equation (6.32) represents the dimensionless Darcy permeability interms of χKin, ǫ, and ξ. It is however impossible to write the general equation forthe latter permeability, equation (6.17), as a function of χKin, ǫ, and ξ. Eliminating

104

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.40

0.5

1

1.5

2

2.5

3

3.5

4x 10

−3

Kinematic tortuosity

Dim

ensi

onle

ss D

arcy

per

mea

bilit

y, K

ϕ = 1/3; ψ = 5/4; γ = 5/12

over−staggered, 1.17 ≤ χKin

< 1.33

fully staggered, 1.08 ≤ χKin

< 1.17

regular array

The total applicable region for the tortuosity.

Figure 6.2: Case study A

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.40

0.5

1

1.5

2

2.5

3

3.5

4


Dim

ensi

onle

ss D

arcy

per

mea

bilit

y, K

ϕ = 1/3; ψ = 1/12; γ = 1/6


< 1.33


< 1.17

regular array

(a) The total applicable region for the tor-tuosity.

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.40

0.5

1

1.5

2

2.5

3

3.5

4x 10

−3


Dim

ensi

onle

ss D

arcy

per

mea

bilit

y, K

ϕ = 1/3; ψ = 1/12; γ = 1/6


< 1.33


< 1.17

regular array

(b) Zoomed in on a lower permeability re-gion.

Figure 6.3: Case study B

ψ from equation (6.29) through substitution of

ψ(ǫ, γ, ϕ) =ϕ(γ − γ2)

ǫ− γ=

(ξϕ− χkin + 1) (χkin − 1)

ϕξ2ǫ− ξ (ξϕ− χkin + 1)(6.33)

yields

(6.34)

K(χKin, ǫ, ξ, ϕ) =(ξϕ− χkin + 1)3 (χkin − 1)

12ξϕ2

(−ϕξ3 (ǫ− 1) +

(ξϕ−χkin+1

ϕξ(ǫ−1)+χkin−1

)3(χkin − 1)4 (2χkin − 2 − ξϕ)

) .

Here the permeability is still dependent on the aspect ratio of the SUC or SRRUCconsidered. Note that substituting equation (6.33) into the restriction on tortuosity,

105

equation (6.28), one is confronted with the dilemma where the lower bound of thetortuosity is in itself tortuosity dependent.

From equations (6.29) and (6.34) it is evident that no simple permeability-tortuosityrelation exists. This relation is dependent on the geometrical properties (in equations(6.29), ψ and ϕ, and in equation (6.34), ϕ) still found in the permeability equation.

106

Chapter 7

Conclusions

In this study a broad overview was given on mathematical aspects of pore-scale mod-elling in two-dimensional porous media. In Chapter 2, two methods were consideredin applying the Navier-Stokes and continuity equations to a single incompressible,Newtonian fluid traversing the void space of a porous structure. Firstly a directmethod was followed, whereby the microscopic transport equation was applied di-rectly to a unit cell and the interstitial flow was assumed to be stationary.

Secondly, volume averaging of the microscopic transport equation was done by meansof an REV. A uniform field for the superficial velocity, q, was assumed. Correspond-ing to the unit cell, where the interstitial velocity was assumed to be stationary, qwas assumed to be time independent and the porous structure was assumed to befixed yielding the volume averaged transport equation. This volume averaged equa-tion still had interstitial parameters and closure was done by means of an RRUC.

Using a redefined RRUC (having twice the streamwise length of the original RRUC),it followed from the volume averaged transport equation that

−∇〈p 〉f =1

2Uf

∫∫

Sfs

(n p− n · µ∇ v) dS .

Shifting this redefined RRUC, by weighing each possible structure with its frequencyof occurrence (equation (2.38)), yielded an identical equation to the one obtained(equation (2.39)) if an SRRUC (equivalent to the original RRUC) was considered.The reason for the change in the RRUC definition was to incorporate the streamwisedirection of flow through an REV in the RRUC as an extra predictive requirement.Therefore, the redefined RRUC was not only defined as the minimum size into whichthe statistical average geometry of its corresponding REV could be imbedded, butalso where the statistical average velocities of its corresponding REV were imbedded.

In obtaining the latter equation, the weighted shifting method used, was similar tothe idea introduced by Lloyd (2003). This weighted shifting of the control volume,

107

however, differed from Lloyd (2003), since three different possible SRRUC positionswere considered in this study, contrary to their two positions. These three positionsfollowed directly from the weighted shifting method applied to the redefined RRUC.In the new shifting equation obtained, two transverse channels were considered ofwhich the two interstitial flow directions were anti-parallel. Since the wall shearstresses on the fluid-solid interfaces of these two channels were directed in oppositedirections, and since they were weighed equally, they cancelled vectorially to zero.Though it was also assumed by Lloyd et al. (2004) that the total contribution of thewall shear stresses over Sfs⊥ are zero, mathematically it was not obtainable, sincethey only considered one transverse channel.

Furthermore, in the present work, the pressure deviation at each channel sectionwas set equal to zero. It was not assumed to be caused by shear stress at thetransverse surfaces and therefore it was not assumed that the pressure deviationintegral provided the streamwise effect of the wall shear stress in the transversechannels, as was assumed by Lloyd et al. (2004). In the model presented in Chapter3, the transverse wall shear stress term resulted from the integral of the averagepressure over Sfs. This term, however, did not follow from the model discussedby Lloyd et al. (2004), since there, the gradient of the pressure over the entireds⊥ was uniform. Therefore, in the LDH model, the average pressures on the twotransverse fluid-solid interfaces was assumed to be equal and to cancel vectorially ifthe SRRUC’s outer cross-stream faces were situated in the solid phase and if thosefaces were situated in the transverse channels, the pressure drop over the transverse

channels were weighed only bydc‖ds⊥d‖Uo

, and notdc‖d⊥d‖Uo

, as was the case in the model

presented in Chapter 3.

From both the direct method and the volume averaging method, the same gene-ral equation for the dimensionless Darcy permeability was obtained in terms ofgeometrical ratios (equations (3.26) and (3.61)):

K =γ3ψϕ

12 [ψ − ϕγ + ξ2ϕψ4 (1 − 2γ)].

This general permeability equation could thus be implemented for a wide range ofpossible SRRUC structures as long as 2dc⊥ ≤ d⊥ for staggered arrays. Asymptoticconditions were also examined and the permeabilities obtained were satisfactory.

In Chapter 4, the two definitions for lineality (referred to as tortuosity by Bear),as defined by Bear (1972) in terms of particles, were quoted. From these particlerelated definitions, three different lineality equations were derived in terms of voidspace volumes and the average interstitial velocities of an assumed incompressiblefluid in those different volumes. The geometric and kinematic linealities related totwo different interpretations for the particle definition, where lineality was viewedas a geometrical concept by Bear and Bachmat. The dynamic lineality related tothe particle definition, whereas lineality was viewed as a kinematical concept. Three

108

different w velocity fields were defined, namely

q = ǫ wGeo/Kin/Dyn · LGeo/Kin/Dyn .

It was found that wGeo was the average velocity in the parallel channels, w‖ n, whichwas the same as w, the “average streamwise channel velocity within Uf”, definedby Diedericks & Du Plessis (1995), their equation (11). It was stated by Diedericksand Du Plessis that

q = ǫ w · LD,

yieldingLGeo

≡ LD,

which was without doubt the case, considering Table 5.1.

The average interstitial speed in Uf taken in the streamwise direction was denotedby wKin. The same residence time was obtained for a particle travelling a distanceLe at a speed wKin, than for a particle travelling over a streamwise displacementL n at a drift velocity, utR . From this it was assumed that

LKin

=

(n ·

〈 v 〉f〈v 〉f

)n n

would predict the correct lineality, as was indeed found, for all the examples studiedin Appendix D.

In Chapter 5, the lineality defined by Bear & Bachmat (1991) (their equation(2.3.49)) was examined. It was found that this definition only predicted the correctlineality if the average interstitial speed in each channel section was equal and if nwas parallel to n everywhere on Sff . If the lineality of an RRUC was determined,using this definition, n n was always obtained, i.e. a tortuosity of unity. This wasdue to the second part of the second assumption made by Bear & Bachmat (1991),where they assumed that

1

Sff

∫∫

Sff

∇GdS ≈ ∇〈G 〉f ,

with ∇2G=0, which (as was shown by means of an example in Appendix E) is onlytrue if L

B= n n.

The lineality defined by Diedericks & Du Plessis (1995) (their equation (39)) wasalso examined. It was found that this lineality only predicted the correct linealityif the average interstitial speed in each channel section was equal, but n need notbe parallel to n everywhere on Sff . This definition for lineality was again derived,following a similar route to that of Bear and Bachmat in their derivation of the equa-tion (Bear & Bachmat (1991), their equation (2.3.48)) from which their definition

109

for lineality followed. This derivation was done by means of Green’s vector theorem,expanded in Appendix C for dyadics. It was shown that, had the second part of thesecond assumption by Bear and Bachmat (mentioned above) not been made, thelineality defined by Diedericks and Du Plessis followed directly from equation (5.13)(equation (2.3.48) of Bear & Bachmat (1991)) and the relation between these twolinealities was found to be

LD

=1

Sff

∫∫

Sff

n n dS · LB.

An attempt was made, in Chapter 6, to find a relation between χKin and the dimen-sionless permeability, derived in Chapter 3. The following permeability tortuosityrelation was obtained:

K =(ξϕ− χKin + 1)3 ψ

12ξ2ϕ2[ψξ − ξϕ+ χKin − 1 + ξ2ψ4 (2χKin − 2 − ξϕ)].

This is much more complex than the relationships portrayed in literature (e.g. Bear& Bachmat (1991), Lloyd et al. (2004) and Carman (1937)).

An equation for the kinematic tortuosity in terms of porosity was also derived. Asexpected, if the average interstitial speed in each channel section was not equal, thisequation differed from the equation obtained for χD(ǫ) by Lloyd et al. (2004). Undersuch circumstances, χD(ǫ) fails to predict the correct tortuosity. Since this is thecase, one could thus argue that defining tortuosity as a geometrical concept doesnot make any sense. Though tortuosity might seem like just another geometricalproperty, one cannot define a suitable definition for it, without involving the ratio ofthe average interstitial speed and the interstitial channel velocities, since this ratioyields the relation between the actual path length and the total displacement of thefluid traversing a porous structure.

110

7.1 Achievements of this study and recommenda-

tion for further study

To summarise, the following new contributions are presented in this study:

1. The RRUC’s geometry is redefined, yielding a new weighted shifting method ofthe SRRUC from which it follows that the wall shear stresses in the transversechannels cancels to zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . equation (2.39), p.21

2. Contrary to Lloyd et al. (2004), the pressure deviation was assumed to be zeroon each channel surface section . . . . . . . . . . . . . . . . . . . . . . . . equation (3.36), p.34

3. The same result for the Darcy permeability was obtained through means ofa direct method as well as through volume averaging and then closure via anRRUC . . . . . . . . . . . . . . . . . . . . . . . equation (3.26), p.31 and equation (3.61), p.42

4. This permeability was evaluated by implementing it to different asymptoticconditions for the unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sections 3.3 and 3.4

5. The lineality as defined by Bear & Bachmat (1991) was found to fail if thestreamlines cross the outer surface of an REV obliquely or if the interstitialspeeds in the different channels differed. . . . . . . . . . . . . . . . . . . . . . . . . . .section 5.1

6. The lineality as defined by Diedericks & Du Plessis (1995) was found to fail ifthe interstitial speeds in the different channels differed. . . . . . . . . . . .section 5.2

7. Green’s vector theorem were expanded for dyadics . . . . . . . . . . . . . . .Appendix C

8. The lineality as defined by Diedericks & Du Plessis (1995) was rederived bymeans of Green’s dyadic theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . section 5.2.2

9. The relation between the linealities by Bear & Bachmat (1991) and Diedericks& Du Plessis (1995) was obtained . . . . . . . . . . . . . . . . . . . . . equation (5.60), p.90

10. A new definition for lineality is derived from first principles. This linealitypredicts the correct lineality according to equation (4.2) even if the streamlinescross the outer surface of an REV obliquely or if the interstitial speeds in thedifferent channels differed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .section 4.1.2

11. It is shown that, contrary to many proposals in literature, the newly derivedpermeability is not inversely proportional to the newly derived tortuosity norto the square of the tortuosity . . . . . . . . . . . . . . . . . . . . . . . .equation (6.29), p.102

Further study is still needed to incorporate wall shear stresses on the surfaces inthe transverse channels facing the parallel channels, as is shown in Figure 3.6 (b),(in this study it was assumed to be negligible as was depicted in Figure 3.6 (a)) to

111

derive an identical equation for the dimensionless permeability by means of volumeaveraging as the one obtained by Firdaouss & Du Plessis (2004), using the directmethod (referred to as the FD model in this study).

Furthermore, a proper mathematical proof is still needed to show that the assump-tion made by Bear and Bachmat,

1

Sff

∫∫

Sff

∇GdS ≈ ∇〈G 〉f ,

only holds if the tortuosity is unity. In this study this fact was demonstrated onlyby means of example, presented in Appendix E.

112

Appendix A

Assumptions made before closure

A.1 Assumption 1

For a uniform vector field q, it follows from Slattery’s averaging theorem that:

ρ∇·(⟨

vf vf⟩o

)=

ρ

2Uo∇·∫∫∫

2Uf

( vf vf

)dU

=ρ

2Uo

∫∫

Sff

n · vf vf dS . (A.1)

If the cross-stream surfaces of the RRUC lie within the transverse channel, vfis perpendicular to n except over dc⊥ as will also be the case if the cross-streamsurfaces of the RRUC intercepts the solid phase particles. Therefore the RRUCneed not be shifted. For stationary flow, the deviation in all the parallel channelswill be equal and therefore equation (A.1) will be zero since the integral evaluatedon the two sides of the RRUC will cancel vectorially.

It therefore follows that

ρ∇·(⟨

vf vf⟩o

)= 0 (A.2)

holds - even for flow at higher Reynolds numbers.

113

A.2 Assumption 2

It was assumed that∫∫

Sfs

n 〈p 〉f dS = 0 . (A.3)

To examine the validity of this assumption, consider the over-staggered configurationin Figure A.1. (Here only an over-staggered array is considered, but the followingalso holds for a regular and fully staggered array.)

RRUC 1 RRUC 2

Out FlowIn Flow

A AB BC CD D

Streamwise direction

1 2

3 4

5 6

7 8

9

Figure A.1: The pressures at the transfer volumes are: p1 = p + 2δp‖ + 2δp⊥;p2 = p + δp‖ + 2δp⊥; p3 = p + δp‖ + δp⊥; p4 = p + δp⊥; p5 = p; p6 = p − δp‖;p7 = p− δp‖ − δp⊥; p8 = p− 2δp‖ − δp⊥ and p9 = p− 2δp‖ − 2δp⊥.

The intrinsic phase average over RRUC 1 is given by:

〈p 〉f =1

ǫ(2Uo)

[(p+

3

2δp‖ + 2δp⊥

)U‖ +

(p+ δp‖ + 2δp⊥

)Ut

+(p+ δp‖ +

3

2δp⊥

)U⊥ +

(p+ δp‖ + δp⊥

)Ut

+(p+

1

2δp‖ + δp⊥

)U‖ + (p+ δp⊥)Ut

+(p+

1

2δp⊥

)U⊥ + (p)Ut

]

= p+1

ǫ(2Uo)

[δp‖

(2dc⊥ds‖ + d⊥dc‖

)+ δp⊥

(3dc⊥ds‖ + 2d⊥dc‖

)].

114

The intrinsic phase average over RRUC 2 is given by:

〈p 〉f =1

ǫ(2Uo)

[(p− 1

2δp‖

)U‖ +

(p− δp‖

)Ut

+(p− δp‖ −

1

2δp⊥

)U⊥ +

(p− δp‖ − δp⊥

)Ut

+(p− 3

2δp‖ − δp⊥

)U‖ +

(p− 2δp‖ − δp⊥

)Ut

+(p− 2δp‖ −

3

2δp⊥

)U⊥ +

(p− 2δp‖ − 2δp⊥

)Ut

]

= p+1

ǫ(2Uo)

[δp‖

(−2dc⊥ds‖ − 3d⊥dc‖

)+ δp⊥

(−dc⊥ds‖ − 2d⊥dc‖

)].

A linear graph for the decrease in the intrinsic phase average is obtained by fitting aline through the centre points of the RRUCs, thus through the following coordinates:

(d‖; p+

1

ǫ(2Uo)

[δp‖

(2dc⊥ds‖ + d⊥dc‖

)+ δp⊥

(3dc⊥ds‖ + 2d⊥dc‖

)])

and(

3d‖; p+1

ǫ(2Uo)

[δp‖

(−2dc⊥ds‖ − 3d⊥dc‖

)+ δp⊥


)]).

It follows that the equation describing the linear decrease of the intrinsic phaseaverage of the pressure is:

P −(p+

1

ǫ(2Uo)

[δp‖

(−2dc⊥ds‖ − 3d⊥dc‖

)+ δp⊥


)])

=−1

4ǫUod‖

[(δp‖ + δp⊥

) (4dc⊥ds‖ + 4d⊥dc‖

)] (X − 3d‖

)

P = p+

(δp‖ + δp⊥

d‖

)(3d‖ −X

)

+1

ǫ(2Uo)

[δp‖

(−2dc⊥ds‖ − 3d⊥dc‖

)+ δp⊥


)]

= −(δp‖ + δp⊥

d‖

)X + p+

δp‖ǫ(2Uo)

(6ǫUo − 2dc⊥ds‖ − 3d⊥dc‖

)

+δp⊥ǫ(2Uo)

(6ǫUo − dc⊥ds‖ − 2d⊥dc‖

)

= −(δp‖ + δp⊥

d‖

)X + p+ δp‖

(3

2+dc⊥ds‖2ǫd⊥d‖

)+ δp⊥

(2 +

dc⊥ds‖2ǫd⊥d‖

). (A.4)

115

Equation (A.4) thus represents the decrease of the intrinsic phase average of thepressure in the streamwise direction. In equation (A.3), that part of the integraltaken over the fluid-solid interphase in the channels parallel to the streamwise di-rection, will cancel vectorially. Therefore only the fluid-solid interfaces in the trans-verse channels are examined. The RRUC is shifted in the streamwise direction, asis shown in Figure A.1. Since the P -intercept is constant for any X-value, it cancelsvectorially on the fluid-solid interfaces in the transverse channels. Thus only thefirst term on the RHS of equation (A.4) will be considered to solve the integral inequation (A.3).

If the cross-stream surfaces of the RRUC intercept A on Figure A.1, it follows that:

n 〈p 〉f = −P (ds‖) n+ P (d‖) n− P (d‖ + ds‖) n+ P (2d‖) n

= −(δp‖ + δp⊥

d‖

)(2d‖ − 2ds‖

)n

=

(δp‖ + δp⊥

d‖

)(2ds‖ − 2d‖

)n ,

and if they intercept B, it follows that

n 〈p 〉f = P (d‖) n− P (d‖ + ds‖) n+ P (2d‖) n− P (2d‖ + ds‖) n

= −(δp‖ + δp⊥

d‖

)(−2ds‖

)n

=

(δp‖ + δp⊥

d‖

)(2ds‖

)n .

Similarly it follows that, if the cross-stream surfaces of the RRUC intercept C,

n 〈p 〉f = −P (d‖ + ds‖) n+ P (2d‖) n− P (2d‖ + ds‖) n+ P (3d‖) n

= −(δp‖ + δp⊥

d‖

)(2d‖ − 2ds‖

)n

=

(δp‖ + δp⊥

d‖

)(2ds‖ − 2d‖

)n

and if they are intercepting D,

n 〈p 〉f = P (2d‖) n− P (2d‖ + ds‖) n+ P (3d‖) n− P (3d‖ + ds‖) n

= −(δp‖ + δp⊥

d‖

)(−2ds‖

)n

=

(δp‖ + δp⊥

d‖

)(2ds‖

)n .

To determine the LHS of equation (A.3), the integral over AA, BB, CC and DD is

116

weighed according to their relative frequency of occurrence. Thus

∫∫

Sfs

n 〈p 〉f dS = ds⊥

[ds‖2d‖

(δp‖ + δp⊥

d‖

)(2ds‖ − 2d‖

)+dc‖2d‖

(δp‖ + δp⊥

d‖

)(2ds‖

)

+ds‖2d‖

(δp‖ + δp⊥

d‖

)(2ds‖ − 2d‖

)+dc‖2d‖

(δp‖ + δp⊥

d‖

)(2ds‖

)]n

= ds⊥

[(δp‖ + δp⊥

d‖

)(ds‖d‖

(2ds‖ − 2d‖

)+dc‖d‖

(2ds‖

))]n

= ds⊥

[(δp‖ + δp⊥

d‖

)(−2dc‖ds‖

d‖+

2dc‖ds‖d‖

)]n

= 0

and equation (A.3) holds.

117

Appendix B

Evaluating the Model

It has already been mentioned that the main difference between the FD model andthe model considered in Chapter 3 is that, besides for the lack in generality of theFD model, in the FD model, the wall shear over the entire fluid-solid interface inthe transverse channels of the SUC is considered. The same equation to the oneobtained for the dimensionless permeability through means of analytical modellingwith an SUC by Firdaouss & Du Plessis (2004), was obtained by Lloyd et al. (2004)through means of volume averaging and closure with an SRRUC, though some ofthe assumptions made by Lloyd et al. (2004) are questionable.

B.1 Steady mass flow

In Figure B.2 two SRRUCs of an over-staggered configuration of granules are shown.It is assumed that over the transfer volumes (as indicated with Ut in Figure B.2)the pressure is constant. Furthermore, as is stated above, in the present model, itis also assumed that the wall shear stresses over the fluid-solid interfaces of volumes1, 2, 3 and 4, also depicted in Figure B.2, are negligible.

From the conservation of mass, it follows that

ρA1ds1

dt= ρA2

ds2

dt=

dm

dt(B.1)

at two cross-sectional areas, where m represents the mass of the fluid, and s is thedistance the fluid travels in an infinitesimally small time interval. It follows fromNewton’s second law that

∑F = G = m∇ v (B.2)

where G represents the time derivative of momentum.

118

t1t2

t1

t2

∑F

∑F

∆m v2

∆m v1

mwmw

Figure B.1: Fluid entering and exiting a tube over a time period ∆t = t2 − t1.

Consider Figure B.1 where the total mass displaced in ∆t is ∆m, m is the totalmass in the tube (which is constant) and w is the average interstitial velocity insidethe tube. It follows from the impulse-momentum-equation that

∆m v1 +mw +∑

F∆t = ∆m v2 +mw

⇒∑

F =∆m

∆t( v2 − v1) (B.3)

and if ∆t→ 0 it follows that

∑F =

dm

dt( v2 − v1) = m∇ v (B.4)

yielding equation (B.2).

The four bullets in Figure B.2 represent stagnation points. A stagnation pointis defined as a point at which the fluid velocity is zero, in particular, a point onthe boundary at which the velocity is zero in an ideal fluid or at which there isno tangential stress in a viscous fluid. To determine the definite velocity profileanalytically at the diagonal intersection between the transfer volume and volumes1, 2, 3 or 4 is tedious (or may even be impossible) since the direction and themagnitude of the velocity vectors will be a function of a variable geometry. Thus,to evaluate equation (B.2), volumes 1, 2, 3 and 4 will be examined together withtheir adjacent transfer volumes.

The pressure at the bottom left hand side of Figure (B.2) is p+δp‖+δp⊥+δp1 +δp2.Thus, the following forces are relevant:

• F (a) = dc⊥(p+ δp⊥ + δp1 + δp2) i

• F (b) = dc‖(p+ δp⊥ + δp1 + δp2) j

• F (c) = −dc‖(p+ δp⊥ + δp2) j

119

U‖

U⊥

U‖

U⊥

Ut2

Ut2

Ut2

Ut2

1

2 3

4(a)

(b)

(c)

(f)

(d)

(e)

(g)

(i)

(h)

(k)

(j)

(l)

x

y

Figure B.2: Examining the mass flow through the transfer volumes.

• F (d) = −dc⊥(p) i

• F (e) = −dc‖(p) j

• F (f) = dc‖(p+ δp2) j

• F (g) = dc⊥(p− δp‖) i

• F (h) = −dc‖(p− δp‖) j

• F (i) = dc‖(p− δp‖ − δp3) j

• F (j) = −dc⊥(p− δp‖ − p⊥ − δp3 − δp4) i

• F (k) = dc‖(p− δp‖ − p⊥ − δp3 − δp4) j

• F (l) = −dc‖(p− δp‖ − p⊥ − δp3) j .

Over volume 1 and its adjacent transfer volume it follows from equations (B.1) and(B.2) that

∑F = m( v2 − v1)

= ρw‖dc⊥(w⊥ j − w‖ i) . (B.5)

Considering the y-components of equation (B.5) yields:

dc‖(p+ δp⊥ + δp1 + δp2) − dc‖(p+ δp⊥ + δp2) = ρw‖dc⊥w⊥

⇒ δp1 = ψρw‖w⊥ .

120

Similarly, for volume 3 and its adjacent transfer volume, equations (B.1) and (B.2)yields:

∑F = m( v2 − v1)

= ρw‖dc⊥(−w⊥ j − w‖ i) (B.6)

and it follows from the y-components of equation (B.6) that

−dc‖(p− δp‖) + dc‖(p− δp‖ − δp3) = −ρw‖dc⊥w⊥

⇒ δp3 = δp1 = ψρw‖w⊥ .

From equations (B.1) and (B.2) it follows for volume 2 and its adjacent transfervolume that

∑F = m( v2 − v1)

= ρw‖dc⊥(w‖ i− w⊥ j) . (B.7)

Consideration of the y-components of equation (B.7) yields:

−dc‖(p) + dc‖(p+ δp2) = −ρw‖dc⊥w⊥

⇒ δp2 = −ψρw‖w⊥

from which it follows that δp1 = −δp2,

Similarly, it follows from equations (B.1) and (B.2) for volume 4 and its adjacenttransfer volume that

∑F = m( v2 − v1)

= ρw‖dc⊥(w‖ i+ w⊥ j) (B.8)

and again the y-components of equation (B.8) yields:

dc‖(p− δp‖ − δp⊥ − δp3 − δp4) − dc‖(p− δp‖ − δp⊥ − δp3) = ρw‖dc⊥w⊥

⇒ δp4 = δp2 = −ψρw‖w⊥ .

Solving equation (3.37) for Figure (B.2), where δp3 = δp1 and δp4 = δp2, it followsthat

−∇〈p 〉o =ds‖

2d‖Uo

∫∫

Sfs⊥AA

npw dS +dc‖d‖Uo

∫∫

Sfs⊥BB

npw dS

+ds‖

2d‖Uo

∫∫

Sfs⊥CC

npw dS − 1

Uo

∫∫

Sfs‖

µn · ∇ v dS

121

=ds‖

2d‖Uodc⊥

[p+ δp⊥ +

δp1

2+ δp2 −

(p+

δp2

2

)]n

+dc‖d‖Uo

dc⊥

[p+ δp⊥ +

δp1

2+ δp2 −

(p− δp⊥ − δp‖ − δp1 −

δp2

2

)]n

+dc‖d‖Uo

(ds⊥ − dc⊥)

[p+

δp⊥2

+ δp2 −(p− δp⊥

2− δp‖ − δp1

)]n

+ds‖

2d‖Uodc⊥

[p− δp‖ −

δp1

2−(p− δp⊥ − δp‖ − δp1 −

δp2

2

)]n

+τ‖S‖Uo

n

=ds‖dc⊥d‖Uo

[δp⊥ +

δp1

2+δp2

2

]n+

dc‖dc⊥d‖Uo

[2δp⊥ + δp‖ +

3δp1

2+

3δp2

2

]n

+dc‖(ds⊥ − dc⊥)

d‖Uo

[δp⊥ + δp‖ + δp1 + δp2)

]n+

δp‖dc⊥Uo

n

= δp⊥

[ds‖dc⊥d‖Uo

+2dc‖dc⊥d‖Uo


d‖Uo

]n

+δp‖

[dc‖dc⊥d‖Uo


d‖Uo+d‖dc⊥d‖Uo

]n

+δp1

[ds‖dc⊥2d‖Uo

+3dc‖dc⊥2d‖Uo


d‖Uo

]n

+δp2

[ds‖dc⊥2d‖Uo

+3dc‖dc⊥2d‖Uo


d‖Uo

]n

= ǫ

(δp⊥ + δp‖ + δp1 + δp2

d‖

)n− δp1

(ds‖dc⊥ + dc‖dc⊥

2d‖Uo

)n

−δp2

(ds‖dc⊥ + dc‖dc⊥

2d‖Uo

)n .

Since δp1 = −δp2, it follows that

−∇〈p 〉o = ǫ

(δp⊥ + δp‖

d‖

)n

which is exactly the same as equation (3.39) (for an over-staggered array) where thepressure drops over volumes 1, 2, 3 and 4 were neglected.

122

Appendix C

Green’s vector theoremgeneralised for dyadics

Let (a b) be an arbitrary dyadic and c a scalar field. Application of the divergencetheorem to c∇ (a b), yields the following:

∫∫

Sof

n · c∇ (a b) dS =∫∫∫

Uo

∇· (c∇ (a b)) dU

=∫∫∫

Uo

(∇c) ·∇ (a b) dU +∫∫∫

Uo

c∇2 (a b) dU . (C.1)

Likewise, application of the divergence theorem to (∇c) (a b), yields∫∫

Sof

n ·(∇c) (a b) dS =∫∫∫

Uo

(∇c) · ∇ (a b) dU +∫∫∫

Uo

(∇2c

)(a b) dU . (C.2)

Subtraction of equation (C.2) from equation (C.1) then leads to:∫∫

Sof

n · [c∇ (a b) − (∇c) (a b)] dS =∫∫∫

Uo

[c∇2 (a b) −

(∇2c

)(a b)

]dU . (C.3)

Had (a b) been substituted with an arbitrary vector field d, the following similarequation can be derived:

∫∫

Sof

n · [c∇d− (∇c) d] dS =∫∫∫

Uo

[c∇2d−

(∇2c

)d]dU , (C.4)

which is known as Green’s vector theorem. Equations (C.4) and (C.3) will be usedin sections 5.1.1 and 5.2.2 respectively.

123

Appendix D

Two-dimensional case studies

In these examples we shall assume straight streamlines as shown in Figure D.1. Ifthe Reynolds number tends to zero, in other words for creep flow, this is a reasonableassumption. Had this assumption not been made, it would have been impossible tosolve the transport process analytically and numerical methods would have to beused. We shall also assume that the magnitude of the average velocity in the parallelchannels and the magnitude of the average velocity in the transverse channels remainconstant, though these two entities need not have the same magnitude. In theseexamples plane Poiseuille flow are assumed and

w‖ ≡dc2⊥12µ

(δp‖ds‖

)and w⊥ ≡

dc2‖12µ

(δp⊥

ds⊥ − dc⊥

).

n

Figure D.1: A schematic representation of piece-wise straight streamlines.

Since only the average channel velocities and channel volumes are relevant in thecalculation of lineality, in these examples, for notational simplicity, U ′

‖ = U‖ + Ut ‖and U ′

⊥ = U⊥ + Ut⊥.

124

D.1 Example 1 – Regular Array

1

2Ug

1

2Ug

1

2U‖

1

2U‖Ut ‖

d‖

d⊥

ds⊥2

ds⊥2

ds‖2

ds‖2

n

Figure D.2: The RRUC for granules stacked in a regular array.

First consider the case where the solid particles are aligned and where stagnantregions thus occur. According to equation (4.3), the definition of tortuosity byZhang & Knackstedt (1995), the tortuosity (as well as the lineality) for this exampleshould be more or less unity if we assume that the fluid particles which enter thestagnant regions will remain in the SRRUC for a much longer time than thosenot entering stagnant regions. The path lengths of these particles which enter thestagnant regions will thus be weighed with a smaller factor than the other particles.The tortuosity of the pathlines of the particles not entering the stagnant regions isunity.

D.1.1 Lineality as defined by Bear and Bachmat


LB

=1

Uf

∫∫∫

Sff

n ro dS

=1

Uf

(2ds‖d‖

[d‖2dc⊥

]+ 2

dc‖d‖

[d‖2d⊥

])n n

=UfUf

n n

= n n ⇒ LB = 1 . (D.1)

The lineality defined by Bear & Bachmat (1991) predicts the lineality correctly.

125

D.1.2 Lineality as defined by Diedericks and Du Plessis

According to equation (5.43), the lineality is given by

LD

=1

Uf

∫∫

Sff

n · n n ro dS

=1

Uf

(2ds‖d‖

[d‖2dc⊥

]+ 2

dc‖d‖

[d‖2dc⊥

])n n

=d‖dc⊥Uf

n n

=U ′‖

Ufn n

6= n n ⇒ LD 6= 1 . (D.2)

The stagnant regions will not contribute to the path length of a particle. Since itcould be argued that there exists a relation between tortuosity and the residencetime (see Zhang & Knackstedt (1995)), considering equation (2.20), it might besensible to change equation (5.43) to

LD2

=1

Uf − Ug

∫∫

Sff

n · n ro n dS , (D.3)

where Ug is the total volume of the stagnant regions where n is not defined. Fromequation (D.3) it follows that

LD2

=U ′‖

U ′‖n n = n n = L

B

⇒ LB = LD2 = 1 , (D.4)

and thus predicts the lineality correctly.

D.1.3 Geometric Lineality

According to equation (4.22) it follows that

LGeo

=n n

Uf

∫∫∫

Uf

n · n dU

126

=

[d‖dc⊥Uf

]n n

=U ′‖

Ufn n

= LD

6= n n ⇒ LGeo 6= 1 . (D.5)

Similarly, as was done in (D.3), if Ug is neglected, the geometric lineality can bewritten as

LGeo2

=1

Uf − Ug

∫∫∫

Uf − Ug

n · n dU (D.6)

and

LGeo2

= LD2

= n n ⇒ LGeo = 1.

Thus, for equations (5.43) and (4.22) to predict the correct lineality, n has to bedefined over the entire fluid occupied volume of the porous structure.

D.1.4 Kinematic Lineality


LKin

=

n n∫∫∫

Uf

v( n · n) dU

∫∫∫

Uf

v( n · n) dU

=

[w‖ dc⊥d‖w‖ dc⊥d‖

]n n

= n n ⇒ LKin = 1. (D.7)

The stagnant regions do not contribute to the integrals, because the interstitialvelocities in those volumes are zero. Therefore the correct lineality is obtaineddirectly.

127

D.1.5 Dynamic Lineality


LDyn

=

n n∫∫∫

Uf

v 2( n · n) dU

∫∫∫

Uf

v 2( n · n) dU

=

w

2‖ dc⊥d‖

w 2‖ dc⊥d‖

n n

= n n ⇒ LDyn = 1. (D.8)

The average channel speed in each channel section remains constant throughout theporous medium and therefore it cancels out and the same equation is obtained asfor the kinematic lineality.

128

D.2 Example 2 – Over-staggered Array

Consider the SRRUC for an over-staggered array of granules as illustrated in FigureD.3.

U‖

U ′⊥

1

2Ut ‖

1

2Ut ‖

d‖

d⊥

ds⊥

ds‖

n

Figure D.3: The RRUC of an over-staggered array of granules.

From the definition of lineality as stated in equation (4.2) we know that the stream-wise lineality should be

L =d‖

d‖ + ds⊥n n (D.9)

=d‖dc⊥

d‖d⊥ − d‖ds⊥ + ds⊥d⊥ − ds2⊥n n

=U ′‖

Uf − ds⊥[dc‖ − dc⊥

] n n. (D.10)

If the areas perpendicular to the interstitial flow direction in the parallel and trans-verse channels differ, the relation between the magnitudes of the average interstitialvelocities in the two channels will be

dc⊥w‖ = dc‖w⊥

ds⊥dc⊥w‖ = ds⊥dc‖w⊥. (D.11)

By substituting equation (D.11) into equation (D.10), we obtain

L =U ′‖

Uf − ds⊥dc‖

[w‖−w⊥

w‖

] n n

129

=U ′‖

d‖dc⊥ + ds⊥dc‖

[1 − 1 + w⊥

w‖

] n n

=U ′‖

d‖dc⊥w‖+dc‖ds⊥w⊥

w‖

n n

=w‖U

′‖

d‖dc⊥w‖ + dc‖ds⊥w⊥n n

=w‖U

′‖

w‖U′‖ + w⊥U ′

⊥n n , (D.12)

where U ′⊥ = U⊥ + Ut⊥.


From equation (5.12) we obtain

LB

=1

Uf

∫∫∫

Sff

n ro dS

=1

Uf

(2ds‖d‖

[d‖2dc⊥

]+ 2

dc‖d‖

[d‖2d⊥

])n n

=ds‖dc⊥ + dc‖d⊥

Ufn n

=UfUf

n n

= n n . (D.13)

In this case the definition by Bear & Bachmat (1991) fails to predict the linealitycorrectly.


From equation (5.43) we have that

LD

=1

Uf

∫∫

Sff

n · n n ro dS

130

=1

Uf

(2ds‖d‖

[d‖2dc⊥

]+ 2

dc‖d‖

[d‖2dc⊥

])n n

=d‖dc⊥Uf

n n

=U ′‖

Ufn n . (D.14)

By comparing equations (D.10) and (D.14) it is evident that equation (5.43) is nota proper definition for lineality if the parallel and transverse channels differ in size.Thus, if the magnitude of the average channel velocity in each and every channel isthe same, equation (D.10) will reduce to equation (D.14).



LGeo

=n n

Uf

∫∫∫

Uf

n · n dU

=

[d‖dc⊥Uf

]n n

=U ′‖

Ufn n

= LD . (D.15)

For this example the geometric lineality is equivalent to the equation derived byDiedericks and Du Plessis.



LKin

=

n n∫∫∫

Uf

v( n · n) dU

∫∫∫

Uf

v( n · n) dU

131

=

[w‖ dc⊥d‖

w‖ dc⊥d‖ + w⊥ ds⊥dc‖

]n n

=

w‖ U

′‖

w‖ U ′‖ + w⊥U ′

⊥

n n (D.16)

which is the same as equation (D.12) and thus a predicts the lineality correctly.



LDyn

=

n n∫∫∫

Uf

v2( n · n) dU

∫∫∫

Uf

v2( n · n) dU

=

w2

‖dc⊥d‖

w2‖dc⊥d‖ + w2

⊥ds⊥dc‖

n n

=

w2

‖U′‖

w2‖U

′‖ + w2

⊥U⊥

n n . (D.17)

If the magnitude of the average interstitial velocities in the parallel and transversechannels differs, the dynamic lineality fails to predict the lineality correctly as thevolumes are weighed with the square of the speeds, and thus per residence timesquared and not only per residence time.

132

D.3 Example 3 – Fully staggered Array

We now consider fully staggered granules. The SRRUC of such a staggering is shownin Figure D.4.

U‖1

2Ut ‖

1

4Ut ‖

1

4Ut ‖

1

2U ′⊥

1

2U ′⊥

d‖

d⊥

ds⊥2

ds‖

n

Figure D.4: The RRUC of a fully staggered array of granules

From the definition, equation (4.2), we know that the streamwise lineality shouldbe

L =d‖

d‖ + ds⊥2

n n

=2d‖dc⊥

2d‖d⊥ − 2d‖ds⊥ + ds⊥d⊥ − ds2⊥n n

=2U ′

‖

2Uf − ds⊥[2dc‖ − dc⊥

] n n

=U ′‖

Uf − ds⊥[dc‖ − dc⊥

2

] n n. (D.18)

133

From mass conservation it follows that the relation of the magnitudes of the averageinterstitial velocities in the two channels will be

dc⊥w‖ = 2dc‖w⊥

ds⊥dc⊥w‖ = 2ds⊥dc‖w⊥. (D.19)

By substituting equation (D.19) into equation (D.18), we obtain

L =U ′‖

Uf − ds⊥dc‖

[1 − dc⊥

2dc‖

] n n

=U ′‖

Uf − ds⊥dc‖

[1 − w⊥

w‖

] n n

=U ′‖

d‖dc⊥ + ds⊥dc‖

[1 − 1 + w⊥

w‖

] n n

=U ′‖

d‖dc⊥w‖+dc‖ds⊥w⊥

w‖

n n

=U ′‖w‖

d‖dc⊥w‖ + dc‖ds⊥w⊥n n

=w‖U

′‖

w‖U′‖ + w⊥U ′

⊥n n . (D.20)

Thus, according to equation (D.20), every volume has to be weighed with the mag-nitude of the interstitial velocity within that volume. This formula, equation (D.20),for lineality is exactly the same as the one obtained in the example where the gran-ules were over-staggered, equation (D.12), as expected.


From the definition for lineality derived by Bear and Bachmat

LB

=1

Uf

∫∫∫

Sff

n ro dS

=1

Uf

(2ds‖d‖

[d‖2dc⊥

]+ 2

dc‖d‖

[d‖2d⊥

])n n

134

=ds‖dc⊥ + dc‖d⊥

Ufn n

=UfUf

n n

= n n . (D.21)

Equation (5.12) thus fails to predict the lineality for this example as well. Note thatagain a lineality of unity is obtained if the SRRUC is shifted relative to the solidphase.



LD

=1

Uf

∫∫

Sff

n ro n · n dS

=1

Uf

(2ds‖d‖

[d‖2dc⊥

]+ 2

dc‖d‖

[d‖2dc⊥

])n n

=d‖dc⊥Uf

n n

=U ′‖

Ufn n . (D.22)

Comparing equations (D.18) and (D.22), it is evident that equation (5.43) is not aproper definition for lineality unless dc‖ = dc⊥

2, where equation (D.18) will reduce

to equation (D.22). Thus for fully staggered media equation (5.43) is not a properdefinition for the lineality.



LGeo

=n n

Uf

∫∫∫

Uf

n · n dU

=

[d‖dc⊥Uf

]n n

135

=U‖Uf

n n

= LD. (D.23)

Thus, in all the above examples, the geometric lineality and the lineality defined byDiedericks and Du Plessis are equivalent.



LKin

=

n n∫∫∫

Uf

v( n · n) dU

∫∫∫

Uf

v( n · n) dU

=

[w‖ dc⊥d‖

w‖ dc⊥d‖ + w⊥ ds⊥dc‖

]n n

=

w‖ U

′‖

w‖ U ′‖ + w⊥U ′

⊥

n n (D.24)

which is the same as equation (D.20) and thus predicts the lineality correctly.



LDyn

=

n n∫∫∫

Uf

v2( n · n) dU

∫∫∫

Uf

v2( n · n) dU

=

w2

‖dc⊥d‖

w2‖dc⊥d‖ + w2

⊥ds⊥dc‖

n n

=

w2

‖U′‖

w2‖U

′‖ + w2

⊥U′⊥

n n . (D.25)

136

Again this differs from the correct lineality given in equation (D.20) as derived fromequation (4.2).

137

D.4 Example 4 – Flow lines cross border obliquelyL

L2

d

θ θ

n

Figure D.5: A porous medium where the flow lines cross the border obliquely.

In this example Figure D.5 is viewed as the whole porous medium and not just arepresentative cell. In examples 1 to 3, whether the figures represented an SRRUCor the whole porous structure, all the linealities (except the one derived by Bearand Bachmat) would yield the same answer. This is so, because the path length ofa particle remains the same whether the SRRUC is shifted or not. In the previous

examples the same result for LB

and LD

, namelyU ′‖

Uf, would have been obtained if

the figures represented the whole porous medium and were thus not shifted, since theinterstitial flow direction on Sff of these SRRUCs was either parallel or anti-parallelto n.

By examining Figure D.5 it is evident that the streamwise lineality is given by

L =L

Len n = L

[2(

L

2 cos θ

)]−1

n n = n n cos θ . (D.26)

This example was already considered in Chapter 5 for the calculation of LB

, equation(5.33) and L

D, equation (5.62). Therefore these two linealities will not be derived

here again.


From the definition of lineality (equation (5.12)) according to Bear and Bachmat,

LB

= n n . (D.27)

This is not the same as equation (D.26) and thus this definition fails to predict thecorrect lineality.

138



LD

= n n cos θ , (D.28)

which is the same as equation (D.26) and therefore predicts the lineality correctly.



LGeo

=n n

Uf

∫∫∫

Uf

n · n dU

=Ld cos θ

L dn n

= n n cos θ (D.29)




LKin

=

n n∫∫∫

Uf

v( n · n) dU

∫∫∫

Uf

v( n · n) dU

=v cos θL d

v L dn n

= n n cos θ (D.30)


139


According to equation (4.24), the dynamic lineality yields

LDyn

=

n n∫∫∫

Uf

v2( n · n) dU

∫∫∫

Uf

v2( n · n) dU

=v2 cos θL d

v2 Ldn n

= n n cos θ (D.31)

which is also the same as equation (D.26) and therefore predicts the lineality cor-rectly.

In this example all the different linealities predict the correct value, except thelineality as derived by Bear and Bachmat.

140

D.5 Example 5 – Limit where lineality tends to

zero

We assume that the average interstitial speeds in the different channels are equal.If we assume that d2 >> L, then Le >> L and the lineality tends to zero.

L

dp

d1

L

4

d2

n

Figure D.6: In this figure d2 >> L.


Viewing Figure D.6 as an SRRUC, and shifting this control volume relative to thefixed porous structure, equation (5.12) yields

LB

=1

Uf

∫∫

Sff

n ro dS

=

(L−4dpL

)[Ldp] +

(2dpL

)[L (dp+ d1)] +

(2dpL

)[L (dp+ d2)]

dp [L+ 2 (d1 + d2)]n n

=L− 4dp+ 2dp+ 2d1 + 2dp+ 2d2

L+ 2 (d1 + d2)n n

=L+ 2 (d1 + d2)

L+ 2 (d1 + d2)n n

= n n . (D.32)

This is wrong, because the lineality obtained is not a function of d2 and limd2→∞

LB6= 0.

141

If we, however, view Figure D.6 as the whole porous structure under consideration,it follows that

LB

=1

Uf

∫∫

Sff

n ro dS

=2(L2

)dp

dp [L+ 2 (d1 + d2)]n n

=L

L+ 2 (d1 + d2)n n

⇒ limd2→∞

L

L+ 2 (d1 + d2)n n = 0 . (D.33)

Thus the correct lineality is obtained. This is so, because the average interstitialspeeds in the different channels are equal and n is parallel to n on Sff .


Viewing Figure D.6 as an SRRUC, from equation (5.43) it follows that

LD

=1

Uf

∫∫

Sff

n ro n · n dS

=

(L−4dpL

)[Ldp] +

(4dpL

)[Ldp]

dp [L+ 2 (d1 + d2)]n n

=L

L+ 2 (d1 + d2)n n

⇒ limd2→∞

L

L+ 2 (d1 + d2)n n = 0 , (D.34)

which is the correct prediction for the lineality.


According to the definition of the geometric lineality,

LGeo

=n n

Uf

∫∫

Uf

n · n dU

142

=Ldp

dp [L+ 2 (d1 + d2)]n n

=L

L+ 2 (d1 + d2)n n

⇒ limd2→∞

L

L+ 2 (d1 + d2)n n = 0 . (D.35)

Again this is exactly the same as the lineality derived by Diedericks and Du Plessis.



LKin

=

n n∫∫∫

Uf

v( n · n) dU

∫∫

Uf

v( n · n) dU(D.36)

=v L dp

v dp [L+ 2 (d1 + d2)]n n

=L

L+ 2 (d1 + d2)n n

⇒ limd2→∞

L

L+ 2 (d1 + d2)n n = 0 . (D.37)

Thus it predicts the correct lineality and is equivalent to the LGeo

and LD

, becausethe magnitude of the average interstitial velocities in the different channels are equal.


Application of equation (4.24) to this example yields

LDyn

=

n n∫∫∫

Uf

v2( n · n) dU

∫∫∫

Uf

v2( n · n) dU(D.38)

143

=v2 Ldp

v2 dp [L+ 2 (d1 + d2)]n n

=L

L+ 2 (d1 + d2)n n

⇒ limd2→∞

L

L+ 2 (d1 + d2)n n = 0 . (D.39)

Thus it also predicts the correct lineality and is equivalent to the LKin

, LGeo

andLD

, since the average interstitial speeds in the different channels are equal.

144

D.6 Example 6 – Recirculators

L

l

d

dp

2dp

n

Figure D.7: Recirculator: a part of U‖ consists of fluid flowing anti-parallel to thestreamwise direction.

For this example we will assume that the interstitial speed is constant over Uf .

From the definition of lineality (equation (4.2)) is follows that

L

Le=

L

L+ 2d+ 2l + 6dp(D.40)


If Figure D.7 is seen as an SRRUC and we shift this control volume relative tothe channel in the streamwise direction, the lineality defined by Bear and Bachmatyields

LB

=1

Uf

∫∫

Sff

n ro dS

=

(dpL

)[L (dp2 + dp(2dp+ d))] +

(lL

)[L 3dp2] +

(dpL

)[L (2dp2 + dp(2dp+ d))]

dp2(L+ 2d+ 2l + 6dp)

+

(dpL

)[L 2dp2] +

(L−3dp−l

L

)[Ldp2]

dp2(L+ 2d+ 2l + 6dp)n n

=dp2 (dp+ 2dp+ d+ 3l + 2dp+ 2dp+ d+ 2dp+ L− 3dp− l)

dp2(L+ 2d+ 2l + 6dp)n n

= n n , (D.41)

145

which is not the same as equation (D.40).

If, again, the porous structure only consists of the single loop shown in Figure D.7,equation (5.12) yields

LB

=1

Uf

∫∫

Sff

n ro dS

=2(L2

)dp2

dp2(L+ 2d+ 2l + 6dp)n n

=L

L+ 2d+ 2l + 6dpn n , (D.42)

which is the same as equation (D.40) and thus predicts the lineality correctly.


From the definition of lineality by Diedericks and Du Plessis, the lineality for thisexample is given by

LD

=1

Uf

∫∫

Sff

n ro n · n dS

=

(dpL

)[Ldp2] +

(lL

)[Ldp2] +

(dpL

)[Ldp2]

dp2(L+ 2d+ 2l + 6dp)

+

(dpL

)[Ldp2] +

(L−3dp−l

L

)[Ldp2]

dp2(L+ 2d+ 2l + 6dp)n n

=dp2 (dp+ l + dp+ dp+ L− 3dp− l)

dp2(L+ 2d+ 2l + 6dp)n n

=L

L+ 2d+ 2l + 6dpn n , (D.43)

which is the same as equation (D.40). Thus the recirculatory flow does not presenta problem, because, in two dimensions considering an SRRUC where the void spaceconsists of a single channel with a width dc, L

Dwill always reduce to

LD

=Ldc

Ufn n , (D.44)

where dc is the cross-sectional area of the channel, which is, in this example, notequal to

U ′‖

Ufn n .

146


According to the geometric lineality,

LGeo

=n n

Uf

∫∫∫

Uf

n · n dU

=dp2(dp+ l + dp

2− dp

2− l − dp

2+ dp

2+ l + dp

2+ dp

2+ L− 2dp− l)

dp2(L+ 2d+ 2l + 6dp)n n

=dp2 L

dp2(L+ 2d+ 2l + 6dp)n n

= LD

(D.45)

and thus also predicts the lineality correctly.


From the kinematic interpretation of lineality,

LKin

=

n n∫∫∫

Uf

v( n · n) dU

∫∫∫

Uf

v( n · n) dU

=v dp2(dp+ l + dp

2− dp

2− l − dp

2+ dp

2+ l + dp

2+ dp

2+ L− 2dp− l)

v dp2(L+ 2d+ 2l + 6dp)n n

=v dp2 L

v dp2(L+ 2d+ 2l + 6dp)n n

=L

L+ 2d+ 2l + 6dpn n . (D.46)

This definition of lineality will thus also predict the lineality correctly. Note however:had the magnitude of the average interstitial channel velocities not been equal, theabove linealities would have failed to predict the lineality, but referring to examples1 to 5, it is safe to assume that the kinematic definition would still predict thecorrect lineality.

147


From equation (4.24), the definition of the dynamic lineality, it follows that

LDyn

=

n n∫∫∫

Uf

v2( n · n) dU

∫∫∫

Uf

v2( n · n) dU

=v2 dp2(dp+ l + dp

2− dp

2− l − dp

2+ dp

2+ l + dp

2+ dp

2+ L− 2dp− l)

v2 dp2(L+ 2d+ 2l + 6dp)n n

=v2 dp2 L

v2 dp2(L+ 2d+ 2l + 6dp)n n

=L

L+ 2d+ 2l + 6dpn n , (D.47)

which is the same as equation (D.40). Had the magnitude of the average interstitialchannel velocities not been equal, the dynamic definition would fail to predict thecorrect lineality.

148

Appendix E

Example clarifying a relationobtained by Lloyd et al. andillustrating the error in Bear andBachmat’s assumption

In this study we assume that ∇p = φ n where φ < 0, except within the defined trans-fer volumes, where the pressure gradient is zero. Figure E.1 shows the consideredporous medium together the value of p and ∇p in the streamwise direction.

The gradient of the average of p

Consider the first graph in Figure E.1. The average value of p in the fluid phase ofSRRUC 1 is

〈p 〉(SRRUC1)f =

1

Uf

[φds‖

2(ds‖dc⊥) + φds‖

(dc‖dc⊥

2

)

+φ

(ds‖ +

ξds⊥2

)(ds⊥dc‖) + φ(ds‖ + ξds⊥)

(dc‖dc⊥

2

)]

=φ

Uf

[ds2

‖dc⊥

2+ ds‖dc‖dc⊥ + ds‖ds⊥dc‖ +

ξds2⊥dc‖2

+ξds⊥dc‖dc⊥

2

],

and of SRRUC 2 is

〈p 〉(SRRUC2)f =

1

Uf

[φ

(3ds‖

2+ ξds⊥

)(ds‖dc⊥) + φ

(2ds‖ + ξds⊥

)(dc‖dc⊥2

)

+φ

(2ds‖ +

3ξds⊥2

)(ds⊥dc‖) + φ(2ds‖ + 2ξds⊥)

(dc‖dc⊥

2

)]

=φ

Uf

[3ds2

‖dc⊥

2+ 2ds‖dc‖dc⊥ + 2ds‖ds⊥dc‖

149

In flow Out flow

x

0 ds‖ d‖ d‖+ds‖ 2d‖

SRRUC 1 SRRUC 2

U‖

U‖

U⊥ U⊥

x

p

0 ds‖ d‖ d‖+ds‖ 2d‖

φds‖

φ(ds‖ + ξds⊥)

φ(2ds‖ + ds⊥)

φ(2ds‖ + 2ξds⊥)

x

‖∇p‖ i

0 ds‖ d‖ d‖+ds‖ 2d‖

φ

φ

(ξdc⊥dc‖

)

Figure E.1: The considered porous medium together with the value of pressure andthe pressure gradient in the streamwise direction.

+3ξds2

⊥dc‖2

+3ξds⊥dc‖dc⊥

2+ ξdc⊥ds⊥ds‖

].

150

Subtracting 〈p 〉(SRRUC1)f from 〈p 〉(SRRUC2)

f and dividing it by the distance betweenthe two centres of the respective SRRUCs, it follows that

∇〈p 〉f =

〈p 〉(SRRUC2)

f − 〈p 〉(SRRUC1)f

d‖

i

=φ

d‖Uf[ds‖Uf + ξds⊥Uf

]i

= φ

[ds‖ + ξds⊥

d‖

]i . (E.1)

The average of the gradient of p

Consider the second graph in Figure E.1. The average of the gradient is given by

〈∇p 〉f =1

2Uf

[2φds‖dc⊥ + 2

(φξdc⊥dc‖

)ds⊥dc‖

]i

= φ

[dc⊥(ds‖ + ξds⊥)

Uf

]i . (E.2)

The gradient of p over the fluid-fluid interfaces

The average of the pressure gradient over the fluid-fluid interfaces are given by

1

Sff

∫∫

Sff

∇p dS =ds‖d‖

[1

2dc⊥(2dc⊥[φ])

]+dc‖d‖

[1

2d⊥

(2dc⊥[0] + 2ds⊥

[φξdc⊥dc‖

])]

=ds‖φ

d‖+ ξ

(dc⊥ds⊥φ

d‖d⊥

). (E.3)

E.1 The gradient of an average to the average of

a gradient relation

Consider equations (E.1) and (E.2). The constant of proportionality, C, betweenthese two equations is

T 〈∇p 〉f = ∇〈p 〉fTdc⊥Uf

=1

d‖,

151

which implies that

T =Uf

d‖dc⊥= χD

yielding

∇〈p 〉f = χD 〈∇p 〉f . (E.4)

Note that this is equivalent to equation (3.53) obtained by Lloyd (2003).

For confirmation of equation (E.4) in the case where χ = 1, since (following fromequation (5.2))

∇〈p 〉f = 〈∇p 〉f −1

Uf

∫∫

Sfs

n p dS (E.5)

the latter term must be zero. In a staggered array, the surface integral only playsa role over Sfs⊥, since it cancels vectorially in the streamwise channels (as wasmentioned in Chapter 3). Therefore, since ds⊥= 0 if χ = 1, this surface integralis zero. In a regular array all the transverse channels are stagnant regions, i.e.there are no pressure drop over dc‖. It thus follows that the surface integral cancelsvectorially to zero in both the stagnant and streamwise channels of a regular array.Therefore, if χ = 1, it follows from both equations (E.4) and (E.5) that,

∇〈p 〉f = 〈∇p 〉f . (E.6)

E.2 The second part of the second assumption

made by Bear and Bachmat

Bear & Bachmat (1991) made the assumption that

1

Sff

∫∫

Sff

∇p dS ≈ ∇〈p 〉f (E.7)

Considering equations (E.1) and (E.3) it follows that assumption (E.7) only holdsif the tortuosity is unity, therefore if ds⊥ = 0. Under such circumstances we obtain

1

Sff

∫∫

Sff

∇p dS =ds‖φ

d‖= ∇〈p 〉f . (E.8)

Since the tortuosity of a regular array is per definition always unity, it follows fromequations (E.1) and (E.3) that, if ξ = 0, assumption (E.7) is always met.

152

Note that for a staggered array, if χD = 1, from equation (E.6) it follows thatequations (E.2) and (E.3) should also be equal. For a staggered array where χD = 1,Uf = dc⊥d‖. Considering equations (E.2) and (E.3), it follows that

1

Sff

∫∫

Sff

∇p dS =ds‖φ

d‖=dc⊥ds‖φ

Uf= 〈∇p 〉f . (E.9)

After obtaining LB

(equation (5.12)) the latter integral in both equations (5.13)and (5.18) was set equal to zero. Thus, for both molecular diffusion and momentumtransport, in equations (5.15) and (5.19) (equation (E.4) in this example) the pro-portionality constant was assumed to be lineality, as defined in equation (4.2). But,from equations (E.1) and (E.3) in this example, it follows that, in the second part ofthe second assumption made by Bear & Bachmat (1991) (assumption (E.7)), theyactually assumed that the lineality (thus also the tortuosity) is unity, even beforederiving equations (5.13) and (5.18). Thus equation (E.4) reduces to equation (E.6).This is reason why, in Appendix D, L

B= n n was always obtained.

153

Bibliography

Bachmat, Y. & Bear, J. (1986). Macroscopic modelling of transport phenomenain porous media 1: The continuum approach. Transport in Porous Media, 1,pp. 213–240.

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