Pore-scale modelling for fluid transport in 2D porous media

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(1)Pore-scale modelling for fluid transport in 2D porous media. by. Marèt 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.

(2) Declaration. I, the undersigned, hereby declare that the work contained in this thesis is my own original work and that I have not previously in its entirety or in part submitted it at any university for a degree.. Signature:. Date:.

(3) Abstract In the present study, a model to predict the hydrodynamic permeability of viscous flow through an array of solid phase rectangles of any aspect ratio is derived. This also involves different channel widths in the streamwise and the transverse flow directions which may be chosen irrespectively to the rectangular shape itself. It is shown how, with the necessary care taken during description of the interstitial geometry, a volume averaged approach can be used to obtain results identical to a direct method. Insight into the physical situation is gained during the modelling of the two-dimensional interstitial flow processes and resulting pressure distributions and this may prove valuable when the volume averaging method is applied to more complex three-dimensional cases. The analytical results show close correspondence to numerical calculations, except in the higher porosity range for which a more realistic model is needed. Tortuosity is studied together with its inverse. Correspondences and differences regarding the definitions for the average straightness of pathlines, expressed in literature, are examined. A new definition, allowing different channel widths in the streamwise and the transverse flow directions, for the tortuosity is derived from first principles. A general relation between newly derived permeability and tortuosity expressions was obtained. This equation incorporates many possible geometrical features for a two-dimensional unit cell for granules. Three possible staggering configurations of the solid phase along the streamwise direction are also included in this relation..

(4) Opsomming In hierdie studie is ’n model geskep vir die voorspelling van die hidrodinamiese permeabiliteit vir viskeuse vloei deur ’n reeks reghoekige vastestof-materiale, waarvan die sye enige arbitrêre verhouding mag inneem. Hierdie model neem die moontlikheid in ag dat die kanaalbreedtes in die stroomsgewyse rigting en dwarsrigting kan verskil. Hierdie verskil kan egter onafhanklik van die reghoek se syverhouding gespesifiseer word. Daar word aangetoon hoe, met die nodige sorg tydens die definiëring van die interstisiële geometrie, ’n volumegemiddelde aanslag gebruik kan word om dieselfde resultate te verkry as wat deur middel van ’n direkte metode verkry word. Gedurende die modellering van tweedimensionele interstisiële vloeiprosesse en resulterende drukverspreidings, word insig aangaande die fisiese situasie verkry, wat handig te pas kan kom indien die volumegemiddelde-metode toegepas sou word op meer komplekse driedimensionele gevalle. Die analitiese resultate toon ’n sterk korrelasie met die numeriese berekeninge, behalwe in hoë 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 reglynigheid van baanlyne, soos voorgestel in die literatuur, is ondersoek. ’n Nuwe definisie, wat ook in ag neem dat die kanaalbreedtes in die stroomsgewyse rigting en dwarsrigting kan verskil, vir die tortuositeit is vanuit eerste beginsels afgelei. ’n Algemene verhouding tussen die nuutgedefinieerde uitdrukkings vir permeabiliteit en tortuositeit is verkry. Hierdie vergelyking neem baie moontlike geometriese verhoudings van ’n tweedimensionele eenheidsel vir korrels in ag. Drie moontlike verspringende rangskikings van die vaste stof langs die stroomsgewyse rigting word ook in hierdie verhouding ingesluit..

(5) Acknowledgements I would like to acknowledge: • my supervisor, Prof. J.P. du Plessis, for the hours of discussions at any time during 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 him this 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 for always believing in me. • my teacher, Mrs E. van der Westhuizen, who nurtured my love for mathematics and 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 Whom nothing is possible..

(6) 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 3.1. 23. 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.

(7) 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 the solid phase and the SUC is α . . . . . . . . . . . . . . . . . . . . . . 49. 3.6. Dimensionless permeability when the transverse and parallel channel widths are equal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.6.1. Regular array . . . . . . . . . . . . . . . . . . . . . . . . . . . 54. 3.6.2. Staggered array . . . . . . . . . . . . . . . . . . . . . . . . . . 54. 4 Different interpretations of lineality 4.1. 4.2. 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. Lineality and the average channel speed. 5 Linealities from literature 5.1. 5.2. 56. . . . . . . . . . . . . . . . . 67 72. 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 borders obliquely . . . . . . . . . . . . . . . . . . . . . . . . . . . 80. 5.1.5. The credibility of the assumptions made by Bear and Bachmat 82. Lineality as derived by Diedericks and Du Plessis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.2.1. The credibility of the assumptions made by Diedericks and Du Plessis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87. 5.2.2. Derivation of LD by means of the Green’s theorem generalised for dyadics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88. ii.

(8) 5.3. 5.2.3. The lineality defined by Bear and Bachmat versus the lineality defined by Diedericks and Du Plessis . . . . . . . . . . . . . . 90. 5.2.4. Two-dimensional example where the flow lines cross the borders obliquely revisited . . . . . . . . . . . . . . . . . . . . . . 92. Comparison between the linealities . . . . . . . . . . . . . . . . . . . 94 5.3.1. The lineality as derived by Bear and Bachmat . . . . . . . . . 94. 5.3.2. The geometric lineality. 5.3.3. The kinematic lineality . . . . . . . . . . . . . . . . . . . . . . 95. 5.3.4. The dynamic lineality . . . . . . . . . . . . . . . . . . . . . . 95. . . . . . . . . . . . . . . . . . . . . . 95. 6 The permeability-tortuosity relation 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 7.1. 98. 107. 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. iii.

(9) 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.2.1 Lineality as defined by Bear and Bachmat . . . . . . . . . . . 130 D.2.2 Lineality as defined by Diedericks and Du Plessis . . . . . . . 130 D.2.3 Geometric Lineality . . . . . . . . . . . . . . . . . . . . . . . . 131 D.2.4 Kinematic Lineality . . . . . . . . . . . . . . . . . . . . . . . . 131 D.2.5 Dynamic Lineality . . . . . . . . . . . . . . . . . . . . . . . . 132 D.3 Example 3 – Fully staggered Array . . . . . . . . . . . . . . . . . . . 133 D.3.1 Lineality as defined by Bear and Bachmat . . . . . . . . . . . 134 D.3.2 Lineality as defined by Diedericks and Du Plessis . . . . . . . 135 D.3.3 Geometric Lineality . . . . . . . . . . . . . . . . . . . . . . . . 135 D.3.4 Kinematic Lineality . . . . . . . . . . . . . . . . . . . . . . . . 136 D.3.5 Dynamic Lineality . . . . . . . . . . . . . . . . . . . . . . . . 136 D.4 Example 4 – Flow lines cross border obliquely . . . . . . . . . . . . . 138 D.4.1 Lineality as defined by Bear and Bachmat . . . . . . . . . . . 138 D.4.2 Lineality as defined by Diedericks and Du Plessis . . . . . . . 139 D.4.3 Geometric Lineality . . . . . . . . . . . . . . . . . . . . . . . . 139 D.4.4 Kinematic Lineality . . . . . . . . . . . . . . . . . . . . . . . . 139 D.4.5 Dynamic Lineality . . . . . . . . . . . . . . . . . . . . . . . . 140 D.5 Example 5 – Limit where lineality tends to zero . . . . . . . . . . . . 141 D.5.1 Lineality as defined by Bear and Bachmat . . . . . . . . . . . 141 D.5.2 Lineality as defined by Diedericks and Du Plessis . . . . . . . 142 D.5.3 Geometric Lineality . . . . . . . . . . . . . . . . . . . . . . . . 142 D.5.4 Kinematic Lineality . . . . . . . . . . . . . . . . . . . . . . . . 143 D.5.5 Dynamic Lineality . . . . . . . . . . . . . . . . . . . . . . . . 143 D.6 Example 6 – Recirculators . . . . . . . . . . . . . . . . . . . . . . . . 145. iv.

(10) D.6.1 Lineality as defined by Bear and Bachmat . . . . . . . . . . . 145 D.6.2 Lineality as defined by Diedericks and Du Plessis . . . . . . . 146 D.6.3 Geometric Lineality . . . . . . . . . . . . . . . . . . . . . . . . 147 D.6.4 Kinematic Lineality . . . . . . . . . . . . . . . . . . . . . . . . 147 D.6.5 Dynamic Lineality . . . . . . . . . . . . . . . . . . . . . . . . 148 E Example clarifying a relation obtained by Lloyd et al. and illustrating 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. v.

(11) 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 unit vectors and velocity variables. . . . . . . . . . . . . . . . . . . . . . . 11. 2.5. A schematic representation of a portion of the REV illustrating drift velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. 2.6. A circular REA inside a porous medium. . . . . . . . . . . . . . . . . 14. 2.7. RRUCs inside their respective spherical REVs, where the dark grey indicates 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 (or mean flow) direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24. 3.2. An illustration of the different arrays studied, as well as the SUCs chosen for the different scenarios. . . . . . . . . . . . . . . . . . . . . 26. 3.3. The present model considered for a regular configuration where the pressure values in the different zones are pUtA = p + δpk , pUtB = p, pUtC = p − δpk and pUtD = p − 2δpk respectively. . . . . . . . . . . . . 26. 3.4. The present model considered for a fully staggered configuration where the pressure values in the different zones are pUtA = p+ 21 δp⊥ , pUtB = p, pUtC = p − δpk and pUtD = p − δpk − 12 δp⊥ respectively. . . . . . . . . . 27. vi.

(12) 3.5. The present model considered for an over-staggered configuration where the pressure values in the different zones are pUtA = p + δp⊥ , pUtB = p, pUtC = p − δpk and pUtD = p − δpk − 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 dck << dc⊥ . . . . . . . . . . . . . . . . . . . . . . 44. 3.9. Configuration where dc⊥ << d. . . . . . . . . . . . . . . . . . . . . . 45. 3.10 Configuration when dc⊥ → d and dck → 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 array as defined by equation (3.77) with ξ = 0. . . . . . . . . . . . . . . . . 54 3.18 The dimensionless permeability for a fully staggered array in terms of porosity, as defined by equation (3.77) with ξ = 12 , for various acceptable values of γ. . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.19 The dimensionless permeability for an over-staggered array in terms of porosity, as defined by equation (3.77) with ξ = 1, for various acceptable values of γ. . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.1. The actual distance travelled by a fluid particle, Le , and the dimension of the porous medium in the direction of the total fluid displacement, L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56. 4.2. The geometrical tortuosity of a pathline. . . . . . . . . . . . . . . . . 59. 4.3. An illustration of the actual distances travelled (solid arrows) in the time interval [t, t + δt], considered in the calculation of the kinematic tortuosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59. 4.4. An illustration of some pathlines considered in the calculation of the dynamic tortuosity as well as their actual distances in the specific time interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60. vii.

(13) 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+2δpk +2δp⊥ ; p2 = p + δpk + 2δp⊥ ; p3 = p + δpk + δp⊥ ; p4 = p + δp⊥ ; p5 = p; p6 = p − δpk ; p7 = p − δpk − δp⊥ ; p8 = p − 2δpk − δp⊥ and p9 = p − 2δpk − 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 Uk consists of fluid flowing anti-parallel to the streamwise direction. . . . . . . . . . . . . . . . . . . . . . . . . 145. E.1 The considered porous medium together with the value of pressure and the pressure gradient in the streamwise direction. . . . . . . . . . 150. viii.

(14) 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 ⋄k. parallel to the streamwise direction, thus length, or. ⋄⊥. perpendicular to the streamwise direction, thus width, or. ⋄B. Bear & Bachmat (1991). ⋄D. Diedericks & Du Plessis (1995). ⋄f f. fluid-fluid interface. the streamwise channel itself the transverse channel itself. ⋄C. Carman (1937). ⋄f. fluid phase. ix.

(15) ⋄f s. fluid-solid interface or interphase. ⋄g. stagnant region. ⋄L. perpendicular to both the streamwise and transverse direction, thus depth. ⋄K. Kozeny (1927). ⋄o. entire region, thus fluid as well as solid phase. ⋄s. solid phase. ⋄e. vector parallel to the interstitial flow direction at each point. ⋄ˇ. vector parallel to the transverse direction. ⋄t. transfer region. ⋄b. vector parallel to the streamwise direction. ⋄. average value over a specified region. ⋄e. deviation from average value in each and every channel section. Miscellaneous vector. ⋄. ⋄. dyadic. 1 ZZZ ⋄ dU intrinsic phase average: Uf Uf ZZZ 1 ⋄ dU phase average: Uo Uf. h⋄ if h⋄ io {⋄}f. deviation relative to the intrinsic phase average of the REV: {⋄}f = ⋄ − h⋄ if. {⋄}o k⋄k. deviation relative to the centroid of the REV: {⋄}o = ⋄ − ⋄o. norm (or magnitude) of the vector: ⋄ ∂ ∂x. ∂ ∂y. ∂ ∂z. ∇. [m−1 ]. del operator:. [m−1 ]. ∇⊥. [m−1 ]. scalar operator representing a gradient in the streamwise channel: bn · ∇. ∇k. i+. j+. k. scalar operator representing a gradient in the transverse channel: n ˇ·∇. 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.

(16) Af. [m2 ]. area of void space in the REA. [m2 ]. area of the REA. As. [m2 ]. area of the solid phase in the REA. b. [m]. Ao ab. an arbitrary dyadic inserted into Green’s theorem 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. dk. [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. dck. [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. dsk. [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 i. [m.s−2 ] gravitation 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.

(17) K. dimensionless Darcy permeability: k/(dk d⊥ ). K′. dimensionless Darcy permeability: k/(dL d⊥ ). 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. LD. lineality defined by Diedericks & Du Plessis (1995) (equivalent to LGeo ). LGeo. geometric lineality (equivalent to LD ). LB. lineality defined by Bear & Bachmat (1991). LDyn. dynamic lineality. LKin. kinematic lineality. Lf f. [m]. total fluid-fluid length on the circumference of the REA. Lf s. [m]. total fluid-solid length in the REA. Lof. [m]. Lss. [m]. boundary of the fluid phase in the REA: Lf f + Lf s. l. [m]. length of the pore scale. m. [m]. the mean hydraulic radius, as defined by Kozeny (1927):. N. total solid-solid length on the circumference of the REA Uf Sf s. 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. e n. unit vector defined at each point within the fluid phase directed in the direction of the interstitial flow at that point. xii.

(18) b n. unit vector in the streamwise direction. n ˇ. unit vector perpendicular to the streamwise direction. p. [P a]. absolute pressure inclusive of the body forces. p′. [P a]. microscopic pressure. pw. [P a]. average channel surface section pressure. pfw. [P a]. wall pressure deviation for each and every channel surface section. δp. [P a]. pressure drop over an SRRUC or SUC. δpk. [P a]. pressure drop in the parallel channel of an SRRUC or SUC. δp⊥. [P a]. pressure drop in the transverse channel of an SRRUC or SUC. Q. [m3 .s−1 ]. qb. [m.s−1 ]. 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. { r}o. [m]. position vector relative to the centroid of the REV, r − ro. S. Sk. [m2 ] 2. [m ]. volumetric flow rate: qb (d⊥ dL ) superficial or Darcy velocity. outer surface of the SRRUC or SUC. 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):. Sf s Uo. Sf f. [m2 ]. total fluid-fluid interface on S. Sf s. [m2 ]. total fluid-solid interface in the SRRUC or SUC. Sf sk. [m2 ]. fluid-solid interface in the parallel channels of the SRRUC or SUC. Sf s⊥. [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: Sf f + Sf s. Sss. [m2 ]. total solid-solid interface on S. S. [m2 ]. outer surface of the REV. [m2 ]. total fluid-fluid interface on S. Sf f. xiii.

(19) Sf f RS. [m2 ]. relevant fluid-fluid interface. Sf s. [m2 ]. total fluid-solid interface in the REV. Sof. [m2 ]. Sss. [m2 ]. boundary of the fluid phase in the REV: Sf f + Sf s. δs. [m]. average distance travelled by a particle over a time interval [t, t + δt]. δsk. [m]. average streamwise displacement of a particle over a time interval [t, t + δt]. T. total solid-solid interface on S. 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. Uk. [m3 ]. volume of the SRRUC or SUC occupied by fluid flowing parallel to bn,. where wall shear stress acts on the boundaries Uk′ U⊥. [m3 ]. volume of the SRRUC or SUC occupied by fluid flowing parallel. [m3 ]. to bn: Uk + Ut k. volume of the SRRUC or SUC occupied by fluid flowing perpendicular to bn, where wall shear stress acts on the boundaries. U⊥′. [m3 ]. volume of the SRRUC or SUC occupied by fluid flowing perpendicular. Uf. [m3 ]. to bn: U⊥ + Ut⊥. 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: Utk + Ut⊥. Ut k. [m3 ]. that part of the transfer volume in the SRRUC or SUC occupied by. [m3 ]. fluid flowing parallel to bn. that part of the transfer volume in the SRRUC or SUC occupied by. [m3 ]. fluid flowing perpendicular to bn. Ut⊥ Uk. volume of the REV occupied by fluid flowing parallel to bn, where wall shear stress acts on the boundaries. xiv.

(20) U⊥. [m3 ]. 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 ]. Utk. [m3 ]. total transfer volume in the REV: Utk + Ut⊥. Ut⊥. [m3 ]. that part of the transfer volume in the REV occupied by fluid flowing. Uǫ. [m3 ]. perpendicular to bn. ub. [m.s−1 ]. intrinsic phase average of the interstitial velocity. [m.s−1 ]. drift velocity. volume of the REV occupied by fluid flowing perpendicular to bn, where wall shear stress acts on the boundaries. that part of the transfer volume in the REV occupied by fluid flowing parallel to bn. the total void space volume of all the intersections of the streamtubes in the cubical REV considered more than once by Bear & Bachmat (1991). ubtR. v. [m.s ]. speed defined at a point. v. [m.s−1 ]. velocity defined at a point. ve. [m.s−1 ]. interstitial velocity defined at a point. [m.s−1 ]. velocity of the porous structure. w. [m.s−1 ]. average speed. wk. [m.s−1 ]. average interstitial speed in the parallel channels. w⊥. [m.s−1 ]. average interstitial speed in the transverse channels. f w. [m.s−1 ]. the average velocity field parallel to en at each point. c wDyn. [m.s−1 ]. c wKin. [m.s−1 ]. v Sof. −1. c w. [m.s−1 ]. c wGeo. [m.s−1 ]. the average streamwise channel velocity (equivalent to c wGeo ) ub/LDyn. average velocity in the parallel channels: ub/LGeo (equivalent to c w). average channel speed taken in the streamwise direction: ub/LKin = hv if bn. xv.

(21) Greek symbols α. in the FD model: α = d⊥ /dk = ds⊥ /dsk = dc⊥ /dck. 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: Sf f /S. φ. 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⊥ /dk. 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. [P a.s]. fluid viscosity. θ. [rad]. an arbitrary angle. ρ. [kg.m−3 ] fluid density. τ. [N.m−2 ]. local shear stress. τk. [N.m−2 ]. wall shear stress on Sk. τ⊥. [N.m−2 ]. wall shear stress on S⊥. τw. [N.m−2 ]. wall shear stress. Ω ξ. the potential function of the uniform vector field, g: ∇Ω = ρ g staggering parameter. Ψ. any arbitrary tensorial quantity. ψ. relation between channel widths: dc⊥ /dck. xvi.

(22) Chapter 1 Introduction 1.1. Preface. What is a porous medium? According to Dullien (1979) at least one of the following two conditions must be abided by for a material or structure to be regarded as porous. 1. “It must contain spaces, so called pores or voids, free of solids, imbedded in the solid or semisolid matrix. The pores usually contain some fluid, such as air, 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 penetrate through one face of a septum made of the material and emerge on the other side. 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 the porous nature of soil enabling it to hold water in its pores, plant growth would have been impossible. Another substance holding a large amount of water is snow and paper towels are produced for that specific trait. The human body basically consists of porous media, e.g. kidneys have to hold a large amount of fluid, lungs exchanges large amount of oxygen and carbon dioxide between blood vessels and an almost infinite amount of alveoli, all the muscles, bones, organs, etc. have to be porous to a certain extend allowing blood transfer and the porous nature of the digestive system helps to enlarge the absorbance surface. Bricks, sandstone, etc., are porous making them better insulators. It also allows for moderate expansions or contractions due to heat exchange. Porous media are also used in filtering processes, and flow through packed beds is equally important in operations involving chemical reactions and separation of chemical substances. In the latter, the ability to determine the residence time accurately is very important, since this is also the reaction time. This however. 1.

(23) could be a very tedious and complex task, because, as the reaction takes place, the size of the granules will change, influencing the permeability and therefore also the residence time. Porous media thus play an important role in many practical fields such as civil engineering, chemical engineering, reservoir engineering, sanitary engineering, petroleum engineering, drainage and irrigation engineering, agricultural engineering, ground water hydrology and soil sciences. The interstitial structure of a porous medium is highly complex, and only three sets of solutions for flow problems through porous media are possible, namely: analytical solutions, numerical solutions by means of computer codes and experimental solutions obtained in laboratories. At this point, in this field of study, different simplified versions of a real porous media systems and the transport phenomena occurring in them, namely analytical models, are still scrutinized by different researchers. For some practical problems, some analytical solutions are more efficient to use or easier to implement than others. For numerical analysis, different packages are available on the market solving different types of fluid mechanical problems. For examining flow through porous media, the user needs to initiate a grid inside a particular fluid domain, enclosed by boundaries on which certain boundary conditions are specified. After an number of iterations the initial conditions should converge to the final solution at a particular time. Temperature distributions, concentration variations and flow patterns are some of the data obtainable from such packages. The third solution set should be viewed as the most trustworthy, because, though experimental errors 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 elementary rules and simple analytical equations, describing different flow phenomena through different types of porous media.. 1.2. Background to this study. In literature, different analytical modelling techniques are studied for flow through porous media. Researchers from different fields of study have over the years improved the different models, created new ones and eliminated some. Thus far, different analytical models still yield different answers to the same physical problems. Idealistically all the research will lead to a single unified analytical theory, describing the transport process of fluid traversing different kinds of porous media. In this study two modelling techniques are used to derive the Darcy permeability in terms of parameters describing the specific porous structure, namely: the direct analytical model, describing the flow process through a unit cell by means of microscopic transport equations, and secondly, volume averaging of the microscopic transport equations by means of an REV and then closure by means of an RRUC.. 2.

(24) These two modelling techniques were found to yield the same permeability, and therefore, for slow flow traversing a low porosity structure, these two techniques may be regarded as equivalent. Besides the different models considered in literature, researchers also have different opinions on some entities in porous media. One such entity causing debate throughout the field is tortuosity. According to Clennel (1997), tortuosity can be classified into four groups. Firstly there is the geometrical tortuosity that is an objective characteristic of the pore structure and measurements of a transport property cannot be used. Secondly tortuosity can be seen as a retardation factor extracted from the transport properties of the porous medium, like electrical tortuosity and diffusion tortuosity. Zhang & Knackstedt (1995) observed, by tracing numerically obtained streamlines between two parallel plates, that the electrical tortuosity is smaller than the hydraulic tortuosity of a specific porous structure. Thirdly, there is tortuosity parameters that enters into some simplified constructions of real pore space, such as the network model and the RRUC model. Finally, there are tortuosity measures that are nothing more than correction factors (or fudge factors) in empirical models. Since the definitions of tortuosity differ widely in published literature, it is understandable 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 discrepancies found in literature on hydraulic and geometrical tortuosity. Using the permeability and the tortuosity expressions obtained in this study, their relation were determined and found to be rather complex in general. Hopefully, at the end, this study would have contributed to the general quest for the ultimate analytical model.. 1.3. Overview of this study. In this study only homogeneous, anisotropic, granular porous media are considered of which the surfaces presented to the fluid phase are stationary. The fluid phase traversing the void space of such a stationary porous structure is incompressible and Newtonian. The velocity field is assumed to be time independent. This study could be beneficial to, for example, ground water hydrology, reservoir engineering, soil sciences, etc., where water (or any other Newtonian fluid) traverses though rather dense granular particles at a very slow pace. This study is kept as general as possible to include the possibility of implementing the same permeability expression to various sizes and shapes of granular porous media. This equation provides a prediction of the superficial velocity of a fluid, with a known, constant viscosity, given a specific external pressure gradient over anyone of these different granular porous media. This could aid decision making with regards to the a system’s development or its operation.. 3.

(25) In Chapter 2 an overview of two analytical methods is given. Firstly, the unit cell is considered. This method may only be used in an anisotropic porous medium where (in two dimensions) the solid phase is non-staggered in at least one of the two principle directions. In this study, this method is referred to as the direct method, because the continuity and Navier-Stokes equations are directly implemented without involving volume averaging. The second method discussed was volume averaging of microscopic transport equations by means of an REV (e.g. Bear (1972)) and then closure of the volume averaged transport equation is done via an RRUC, as was introduced by Du Plessis & Masliyah (1988). Theoretically, this technique may be used for either isotropic or anisotropic porous media, since, with this method, only a statistical geometrical average is studied. This average is not dependent on whether the granules 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. This was accomplished by introducing three independent geometrical parameters which operates in close relation to one another and porosity. A general hydrodynamic Darcy permeability is obtained in terms of those geometrical parameters and a fourth parameter, describing the staggeredness of the solid phase granules along the streamwise 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 differences and correspondences were examined. A new definition for tortuosity is derived, eliminating the discrepancies found in the above definitions. Finally, in Chapter 6, the dependence of the general permeability obtained in Chapter 3 to the newly defined tortuosity of Chapter 4 is obtained.. 4.

(26) Chapter 2 Analytical modelling The equations governing the various transport phenomena in fluid mechanics may be written at a microscopic level, where the equations describe the physical phenomena at a (mathematical) point within a particular phase inside a certain domain. Two important microscopic equations governing the flow process are the NavierStokes equation, describing the momentum transport of an incompressible fluid, and the continuity equation, derived from the interstitial mass conservation of the fluid phase. The Navier-Stokes equation is given by ρ. ∂v + ρ v · ∇ v − ρ g + ∇p′ − ∇· τ = 0 . ∂t. (2.1). For the remainder of this study, since g is a uniform vector field, −ρ g may be written as −∇Ω. The gravitational term is then included in the pressure gradient term, i.e. ∇p refers to the vector sum of the gravitational term and the external pressure gradient: ∇p = ∇(p′ − Ω). In the case of an incompressible, Newtonian fluid where the viscosity of the fluid is constant, equation (2.1) reduces to ρ. ∂v + ρ v · ∇ v + ∇p − µ∇2 v = 0 . ∂t. (2.2). Here ρ ∂∂tv is the density times the local acceleration or ∂ρ∂tv can also be regarded as the rate of the change in momentum per unit volume, ρ v ·∇ v is the convection term or 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 is the diffusion coefficient. The source term, ∇p, is the external force responsible for the fluid transport. The general continuity equation is given by ∂ρ + ∇· (ρ v) = 0 . ∂t. 5. (2.3).

(27) If the fluid is incompressible, the time and the spatial derivative of the density is zero 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, the Navier-Stokes equation can be rewritten as ρ. ∂v + ρ ∇· ( v v) + ∇p − µ∇2 v = 0 . ∂t. 6. (2.5).

(28) 2.1. The Unit Cell. To describe the process of fluid flowing through a porous medium, different modelling techniques can be used. One technique is representing the porous medium by an ensemble of identical unit cells. Duplicating and stacking the unit cells, the porous medium can be reconstructed and a unit cell could thus be seen as an elementary building block of the porous medium, as is shown in Figure 2.1. Note, however, that it is physically impossible to reconstruct an isotropic porous medium of elementary building blocks, since even though the medium might be constructed similarly in the average flow direction and the direction perpendicular to it, in the diagonal direction it will differ and thus the medium would be anisotropic. Therefore only porous media that is non-staggered in at least one direction can be modelled analytically using this technique. In the case of an isotropic porous medium where the solid material is randomly distributed, other modelling methods have to be considered.. r2. r1. r4. r3 O. 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 than a circular or ellipsoidal, particle, without making too much of an error. A unit cell should be a representation of the porous medium on microscopic scale. Since this is the case, the unit cell should contain both an ‘north-flowing’ and ‘southflowing’ transverse channel of the same width. This will ensure that the average flow direction of the unit cell is that of the entire porous medium. This is shown in Figure 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 incompressible, Newtonian and free of body forces (or the body forces are incorporated in the pressure gradient term) and creep flow is assumed. Therefore the flow is governed by the interstitial continuity equation: ∇· v = 0 , 7. (2.6).

(29) 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 the unit cell. Piece-wise straight streamlines are assumed as shown in Figure 2.2. In two dimensions this analytical modelling technique is based on a fully developed, piece-wise plane Poiseuille flow approximation for interstitial flow between neighbouring particles. In such an approximation, the wall shear stress and corresponding channel-wise pressure gradient are respectively given by τw =. 6µw dc. and. k−∇pk =. 12µw . dc2. (2.7). Here w is the average velocity of the assumed fully developed velocity profile in the channel and dc is the normal distance between the facing surfaces. If the length of the 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 k−∇pk represents the pressure drop along the length of the pair of parallel plates. Note that equation (2.8) is the equilibrium equation of the forces in the interstitial streamwise direction in a channel.. 8.

(30) 2.2. Volume averaging of transport equations. Another modelling technique for flow through porous media is volume averaging of the microscopic equations as was discussed by e.g. Bear (1972). In porous media, the geometry of the bounding surface of the fluid phase, on which the boundary conditions are defined, is in general too complex to describe or is not observable. Furthermore there could be values that varies from point to point within the phase under consideration. Therefore, microscopic equations governing the flow cannot be solved. Describing the transport process through a porous medium at a microscopic level is thus impossible, unless the medium can be represented by a regular array where, in that case, a unit cell may be used, as was discussed in section 2.1. If the solid material of a porous medium is randomly distributed, another approach is necessary where continuous quantities may be determined and problems with particular boundary conditions can be solved. Thus, describing the transport phenomena 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 is done is known as the REV.. 2.2.1. The REV. The first step in passing from the microscopic level, where we consider phenomena at each point within a phase, to a macroscopic level, where we consider volume averaged quantities describing phenomena in the vicinity of a point, is accomplished by introducing the REV. The REV (Representative Elementary Volume) is defined as a volume that is large enough to be statistically representative of the physical properties in the immediate vicinity of a point, i.e. the volume average over the REV of any physical quantity does not change abruptly with a small change in the positioning of the REV within the porous medium. On the other hand it must be much smaller than the exterior domain of the porous medium. The total volume of the 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 of the whole porous medium. The order of magnitude of these dimensions is shown schematically in Figure 2.3. The position vector ro points to the centroid of the REV and r is the position vector of any point within Uo . The porosity of the REV with centroid at ro is ǫ| ro =. Uf | r. o. Uo | ro. 9. .. (2.10).

(31) Sf f. . 1. O Uo3. Sf s. . O(L) O(l). ro r. Sss. O. Figure 2.3: A spherical REV inside a porous medium. We may, on a notational basis, drop the explicit referral to the particular position vector, ro , and write the porosity of an REV as ǫ=. Uf . Uo. (2.11). Let S denote the total outer surface of the REV. This surface consists of a fluidfluid interface, Sf f , and a solid-solid interface with the total area given by Sss . The boundary of the fluid phase of the REV is denoted by Sof and it consists of Sf f and Sf s , the latter being the area where there is a fluid-solid phase contact inside the REV (thus the outer surface of the solid phase particles inside the REV). These areas are shown in Figure 2.3. The phase average of Ψ, a tensorial quantity of any order defined within Uf , is given by 1 ZZZ hΨ io = Ψ dU . Uo Uf 10. (2.12).

(32) From its definition, it is evident that porosity is the phase average of unity. The intrinsic phase average of Ψ is given by 1 ZZZ hΨ if = Ψ dU . Uf Uf. (2.13). From equations (2.11), (2.12) and (2.13) it follows that hΨ io = ǫ hΨ if .. (2.14). The deviation of Ψ at any point within Uf , with respect to the particular REV, is defined as {Ψ}f = Ψ − hΨ if ,. (2.15). where Ψ is the actual value at that point. ub. qb. b n. n n e n. Us n. n. ve. Uf n. e n. n. ve. e n. n. ve. Us. Us. n Us n. e n. n. Figure 2.4: A schematic representation of an REV showing the different unit vectors and velocity variables.. Unit vectors With respect to any REV two important unit vector fields can be defined. Firstly there is bn, the unit vector in the average flow direction. From here on, all the hatted 11.

(33) vectors will denote vectors in the average flow (or streamwise) direction. Secondly e n is a unit vector defined at all the points within the fluid phase, directed in the interstitial flow direction at that point within Uf . All the vectors directed parallel to the interstitial flow direction at that point will be denoted by a tilde. Another important unit vector is n, a vector normal to Sof , directed into the solid phase or to the outside of the REV. These different unit vectors are shown in Figure 2.4. Velocities defined within an REV Let ve denote the interstitial velocity at each point in a fluid phase at a certain time. The phase average of ve is the superficial (or Darcy) velocity and is given by qb = h ve io =. 1 ZZZ ve dU . Uo Uf. (2.16). The streamwise direction, bn, is defined at any point as the direction of the superficial velocity of the REV pertaining to that point, namely b n=. qb . b k qk. (2.17). From equations (2.16) and (2.17) it is evident that ZZZ. Uf. ve dU. is a vector in the streamwise direction. The intrinsic phase average of ve (the average of the intrinsic velocity in the channels taken over Uf ) is ub = h ve if =. 1 ZZZ ve dU . Uf Uf. (2.18). Considering equations (2.16) and (2.18) it follows from equation (2.14) that qb = ǫ ub. (2.19). and this is known as the Dupuit-Forchheimer relationship, Dupuit (1863), as referred to by Carman (1937). Let the term drift velocity denote the average streamwise displacement per unit time at which a particle meanders through the porous medium. In Figure 2.5 the drift. 12.

(34) Ug. b n. Ug In flow. Out flow. A Figure 2.5: velocity.. B A schematic representation of a portion of the REV illustrating drift. velocity is the distance AB divided by the time (also referred to as the residence time) the particle took to travel the distance AB while meandering through the channels. Thus, if there are stagnant regions of volume Ug forming part of Uf in the REV, the drift velocity, ubtR , is defined as ZZZ 1 ubtR = ve dU , Uf − Ug Uf. (2.20). where Ug is the volume in which the interstitial velocity of the fluid is zero. If there are no stagnant regions, it follows that ubtR = ub = h ve if .. 2.2.2. (2.21). The REA. Instead of a volumetric REV, a Representative Elementary Area (an REA), with the same centroid, may also be used to define areal average quantities relating to the porous medium. For homogeneous, isotropic media, areal and volumetric averages of 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 of the physical boundaries of the porous medium, as is shown in Figure 2.6. The areal porosity is defined as ǫA =. Af Ao. 13. (2.23).

(35) Lf f. Lf s. . 1. O Ao 2. . O(L) O(l). ro. Lss. r O. 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. The different surfaces discussed for the REV are replaced by lines in the REA and the same phase operators are applicable. The integrals taken over Uo and Uf are also to be 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 macroscopic scale, as was already mentioned. Each term in a microscopic equation has to be averaged 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 + ∇·(ρ ve) = 0 . ∂t Volume averaging of this equation yields *. ∂ρ ∂t. +. o. + h ∇·(ρ ve) io = h0 io , 14. (2.24). (2.25).

(36) which, since Sof has zero velocity and the no-slip boundary condition holds, reduces to ∂hρ io + ∇· hρ ve io = 0 . ∂t. (2.26). It has already been assumed that the fluid is incompressible (thus ρ is spatially and temporally constant) and again using the no-slip boundary condition, equation (2.26) can be written as hρ io h ve io ∇· ǫ. !. D. + ∇· {ρ}f { ve}f. E. o. = 0.. (2.27). There is no deviation in the density of the fluid phase and thus hρ io. ". #. h ve io ∇· = 0. ǫ. (2.28). If the porosity is spatially constant, it follows that ∇· h ve io = 0. and the macroscopic continuity equation for a porous medium under these circumstances is ∇· qb = 0 .. (2.29). Volume averaging of the Navier-Stokes Equation The momentum transport equation for an incompressible fluid is governed by the interstitial Navier-Stokes equation that can be written as: ρ. ∂ ve + ∇· (ρ ve ve) + ∇p − ∇·τ = 0 . ∂t. (2.30). Here ρ is the density of the fluid and ve is the velocity of the fluid at a particular point and equation (2.30) is thus a microscopic equation. After volume averaging of equation (2.30), the following equation is obtained: *. ∂ ve ρ ∂t. +. o. D. + h ∇· (ρ ve ve) io + h∇p io − ∇·τ. E. o. = 0.. (2.31). It is assumed that the density is spatially as well as temporally constant and that the solid phase of the porous medium, Sof , is stationary, thus v Sof = 0. Equation. 15.

(37) (2.31) therefore reduces to 0 = ρ. ∂h ve io ρ ZZ ρ ZZ − n · v Sof ve dS − n · v Sof ve dS ∂t Uo Uo Sf s Sf f. ! E h ve io h ve io D ρ ZZ +ρ ∇· + + { ve}f { ve}f n · ve ve dS o ǫ Uo Sf s ZZ D E 1 1 ZZ n {p}f dS − ∇· τ − n · τ dS +ǫ∇ hp if + o Uo Uo Sf s Sf s !. E D qb qb + ρ ∇· { ve}f { ve}f + ǫ∇ hp if o ǫ ZZ D E 1 − ∇· τ + n {p}f − n · τ dS o Uo Sf s. ∂ qb = ρ + ρ ∇· ∂t. (2.32). b since n · ve = 0 everywhere on the fluid-solid interface and h ve io = q.. In the case of a Newtonian fluid where µ is constant, ∇·τ can be substituted by µ∇2 ve in the Navier-Stokes equation. If the porous medium is stationary, i.e. v Sof = 0, volume averaging of equation (2.5) yields ! E ∂h ve io h ve io h ve io D 1 ZZ 0 = ρ + ρ ∇· + { ve}f { ve}f + ǫ∇ hp if + n {p}f dS o ∂t ǫ Uo Sf s . .  ZZ    µ ZZ µ  n · ∇ ve dS − ∇·  n ve dS  −µ∇ h ve io − . Uo Uo   Sf s Sf s 2. (2.33). It is assumed that the no-slip boundary condition holds, thus ve = 0 everywhere on Sf s . Equation (2.33) therefore reduces to !. E ∂ qb qb qb D 0 = ρ + ρ ∇· + { ve}f { ve}f + ǫ∇ hp if − µ∇2 qb o ∂t ǫ 1 ZZ + n {p}f − n · µ∇ ve dS . Uo Sf s. Substituting equation (2.29) into equation (2.34), and using the identity ∇2 Ψ = ∇( ∇·Ψ) − ∇× (∇× Ψ), yield 0 = ρ. (2.34). D E h i ∂ qb + ρ qb · ∇ qb + ρ ∇· { ve}f { ve}f + ǫ∇ hp if + µ ∇× ∇× qb o ∂t. 16.

(38) 1 ZZ + n {p}f − n · µ∇ ve dS . Uo Sf s. We assume a uniform field for the superficial velocity, therefore the gradient and the curl of qb are zero and, since the interstitial velocity was assumed to be stationary in the unit cell model and v Sof = 0, it follows that qb is also time independent and thus 0 = ρ ∇·. D. { ve}f { ve}f. E o. 1 ZZ n {p}f − n · µ∇ ve dS . + ǫ∇ hp if + Uo Sf s. (2.35). Equation (2.35) is still open in the sense the actual values of the interstitial velocity are required to solve the integral. To be able to determine the interstitial velocity field the exact geometry of the porous medium should be known. Since this is not the case, further modelling of the porous microstructure is needed to close this equation. Closure will be done by means of a rectangular representative unit cell.. 17.

(39) 2.3. The Rectangular Representative Unit Cell. Since the integral over the fluid-solid interface in the macroscopic momentum transport 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 modelling technique used to solve this integral is based on the RRUC (Rectangular Representative 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 unit cell as it was defined in section 2.1 in any way.) For this modelling technique only the 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 many streamtubes, since, due to the incompressibility of the fluid phase, if fluid ‘escapes’ a particular streamtube, fluid from the other streamtube should exchange position with that fluid. This is unlikely to happen and the RRUC modelling technique is based on this streamtube assumption. For mathematical simplicity, the RRUC has a rectangular shape. Four surface areas are chosen parallel to the enclosed streamtube and the other two are orientated perpendicular to it. It is thus assumed that no fluid exits or enters the RRUC except through the cross-stream surfaces. An RRUC is the smallest possible cell in which the statistical average geometrical properties of an REV are imbedded. Its porosity, tortuosity (see Chapter 4), streamwise direction, pressure gradient, superficial velocity and thus also its permeability should be the same as the corresponding REV, which is on its turn a representation of the entire porous structure. Since the properties of an REV (for example porosity – see equation (2.10)) are defined in the vicinity of a point indicated by the position vector ri , the RRUC representing that REV is also positioned with its centroid at ri . Figure 2.7 is a schematic representation of the positioning of the RRUCs inside their two respective REVs. Note that, contrary to unit cells, RRUCs do overlap. In this thesis the RRUC differs from the RRUC defined for granular porous media by Du Plessis & Masliyah (1991) in the sense that the streamwise direction of the flow through the porous medium is the same as the streamwise direction through the RRUC. In the RRUC defined by Du Plessis & Masliyah (1991), two adjacent, complimentary RRUCs (in a two-dimensional case) have to be considered to obtain the same streamwise direction and to eliminate unwanted induced swirl effects. In the present work it is preferred to refer to both adjacent RRUCs as the RRUC. The RRUC defined by Du Plessis & Masliyah (1991) will from here on be referred to as the simplistic RRUC. Let 2Uo denote the volume of the RRUC. The volume occupied by the fluid phase is indicated by 2Uf and 2Us represents the volume of the solid phase. The fluid-solid interface is represented by Sf s . The outer dimensions of both the solid particles. 18.

(40) r1. ro. O Figure 2.7: RRUCs inside their respective spherical REVs, where the dark grey indicates the volume where the two RRUCs overlap. dsk ds⊥. dck. Us d⊥ Us. dc⊥ 2dk. Figure 2.8: The different dimensions defined for the RRUC. are dsk × ds⊥ × dsL where the subscripts indicate whether the surface is parallel or perpendicular to the streamwise direction. In this thesis only two dimensions are considered and dsL is set equal to unity. Let the outer dimensions of the RRUC be 2dk × d⊥ × 1. The reason for choosing the streamwise dimension as 2dk will be discussed in more detail in Chapter 3. For now it is sufficient to note that the ratio of the sum of the streamwise dimensions of the solid phases to that of the RRUC dsk is . The streamwise channel widths are denoted by dc⊥ and the widths of the dk transverse channels by dck . 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 repetitive building block, because by duplicating and stacking unit cells the whole porous medium can theoretically be reconstructed, the RRUC should not be viewed in that. 19.

(41) A. B. C. E. D. A. F. B. C. D. G. Initial position. Final position streamwise direction. Figure 2.9: Weighted shifting of the RRUC. manner. The RRUC should rather be seen as a small control volume that represents the average pore-scale properties of the porous medium in the close vicinity of the centroid of the RRUC. Since the pressure gradient over the RRUC should be the same as over the REV, equation (2.35) can be rewritten as follows for an RRUC: −ǫ∇ hp if = ρ ∇·. D. { ve}f { ve}f. E o. 1 ZZ n {p}f − n · µ∇ ve dS . + 2Uo Sf s. (2.36). It is assumed that ρ ∇·. D. { ve}f { ve}f. E o. and. ZZ. n hp if dS. Sf s. are both equal to zero. These two assumptions are discussed in the Appendix A. Equation (2.36) therefore reduces to 1 ZZ −∇ hp if = ( n p − n · µ∇ ve) dS . 2Uf Sf s. (2.37). Lloyd (2003) introduced the idea of shifting the RRUC in the streamwise direction by weighing each possible RRUC structure by its relative frequency of occurrence. To obtain each possible RRUC structure, it should be shifted until the upstream cross-sectional face lies in the position where the downstream cross-sectional face was 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 somewhere on A or C, should be multiplied by the ratio of the streamwise dimension of the. 20.

(42) solid phase to the streamwise dimension of the RRUC. The values obtained for the integral when the cross-stream faces lie somewhere on B or D, should be multiplied by the ratio of the width of the transverse channel to the streamwise dimension of the RRUC. Addition of these four terms and dividing it by the volume of the fluid phase of the RRUC will yield the gradient of the intrinsic phase average of the pressure: 1 −∇ hp if = 2Uf. .  ZZ  dsk dck ZZ  ( n p − n · µ∇ ve) dS + ( n p − n · µ∇ ve) dS  2dk  2dk. Sf sAA. Sf sBB.  .  dck ZZ dsk ZZ ( n p − n · µ∇ ve) dS + ( n p − n · µ∇ ve) dS  + . (2.38) 2dk 2dk  Sf sCC Sf sDD. The integrals over BB and DD, as well as the integrals over AA and CC, yield the same 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 gradient remains constant. Equation (2.38) may thus be rewritten for a simplistic RRUC as. −∇ hp if =. 1 Uf. .  ZZ  dsk 1 dsk 1 ZZ  e · ( n · ( n p − n · µ∇ ve) dS p − n · µ∇ v ) dS +  dk 2 dk  2. Sf sAC.  .  dck ZZ ( n p − n · µ∇ ve) dS  + . dk  Sf sBD/DB. Sf sCA. (2.39). The integrals over Sf sAC and Sf sCA are kept separate, because their respective wall shear stress terms will differ, depending on the direction of the interstitial flow in the transverse channels. The weighted shifting method given by equation (2.38) is therefore equivalent to shifting only half the RRUC (referred to as the simplistic RRUC) over the length of an RRUC. Considering equation (2.39) it is evident that this weighted shifting of the control volume should start with its cross-stream surfaces situated halfway through the two solid phases (indicated with lines E and F on Figure 2.9) and is then shifted in the streamwise direction until the upstream cross-sectional face again lies in the position where the downstream cross-sectional face was situated before the shifting took place (in other words, lines F and G on Figure 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 relative to the porous structure as was schematically illustrated in Figure 2.9. The centroid of the RRUC is always situated at the centroid of the REV it is representing. Figure. 21.

(43) 2.9 should thus merely be seen as an explanation to the specific weighing coefficients chosen for the respective RRUC structures encountered in travelling along the steamwise direction. Figure 2.10 shows the different simplified RRUC structures that may occur. The weighted average of these structures thus represents the best possible average structure of the simplified RRUC that should be considered in the closure of the volume averaged Navier-Stokes equation.. dsk × 2dk O. ro. dck × dk O. or. ro. dck × dk O. ro. dsk × 2dk O. ro. Figure 2.10: Different simplified RRUC structures encountered.. 22.

(44) Chapter 3 Derivation of the permeability in the Darcy regime In this chapter an equation for the dimensionless permeability for the streamwise direction of an anisotropic porous medium is derived. Firstly the permeability is obtained by means of a direct method, similarly to the analytical method discussed by Firdaouss & Du Plessis (2004), where the pressure drop over an SUC (Simplified Unit Cell) is determined directly. This model is referred to as the direct method. Secondly the same equation is derived by volume averaging the interstitial Navier-Stokes equation and performing closure by means of an SRRUC (Simplified Rectangular Representative Unit Cell). The SUC and the SRRUC are only half the length of the unit cell and RRUC in the streamwise direction. Also, the porous medium cannot be reconstructed by duplicating 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 all the parallel channels have the same dimensions and all the transverse channels have the same dimensions. Thus, half the RRUC or unit cell contains all the microscopic properties 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 SRRUC chosen for the closure of equation (2.35)) are represented by rectangles with any independent randomly chosen aspect ratios. Following Firdaouss & Du Plessis (2004), the special case where the solid phase and the SUC have the same aspect ratio will be examined and compared to their numerical results. (The direct analytical approach 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 is equal to the streamwise channel width. Some asymptotic conditions will also be considered. The fluid-solid interface parallel to the streamwise direction is represented by dsk and the perpendicular interface by ds⊥ . The dimensions of the SUC (as well as the SRRUC) are represented by dk and d⊥ in the streamwise and the transverse 23.

(45) directions respectively, as are shown in Figure 3.1. (Thus the dimensions of the unit cell and the RRUC are 2dk × d⊥ .) The width of the channel in which the flow is in the streamwise direction is represented by dc⊥ and the width of the channel occupied by transversely flowing fluid is represented by dck . Therefore dc⊥= d⊥−ds⊥ and dck = dk −dsk . dk. dsk b n. mean flow. dck 2. ds⊥. d⊥. dc⊥ 2. Figure 3.1: Notation for the SUC and SRRUC with respect to the streamwise (or mean 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 width to the transverse channel width are given by d⊥ ≡ ϕ, dk. 0<ϕ<∞. (3.1). dc⊥ ≡ ψ, dck. 0<ψ<∞. (3.2). and. respectively. The relation of the parallel channel width to the dimension of the perpendicular side of the SUC is given by dc⊥ ≡ γ, d⊥. 0<γ<1. (3.3). and it then follows that dck ϕγ ≡ , dk ψ. 0<. ϕγ < 1. ψ. (3.4). (Later in this chapter it is shown that, for a staggered array, the range of γ is actually 0 < γ ≤ 12 for 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⊥ 24. 1−γ γ. !. (3.6).

(46) and ψ − ϕγ ϕγ. dsk = dk − dck = dck. !. .. (3.7). The aspect ratio of the rectangular solid phase particles is thus given by ds⊥ ϕψ(1 − γ) ≡ . dsk ψ − ϕγ. (3.8). In the FD model, where the aspect ratios of the SUC and the solid phase are equal and defined as α, the above relations simplify to √ (3.9) ϕ = ψ = α and γ = 1 − 1 − ǫ . Also note that, in this particular case, equation (3.8) reduces, as expected, to ds⊥ = α. dsk The porosity of the porous medium, which should be equal to the porosity of the SUC, is given by ". #. dck ds⊥ + dc⊥ dk ϕ ǫ≡ =γ (1 − γ) + 1 dk d⊥ ψ. (3.10). and useful relations for later use, are ϕ(ǫ, γ, ψ) =. ψ(ǫ − γ) γ − γ2. and ψ(ǫ, γ, ϕ) =. ϕ(γ − γ 2 ) ǫ−γ. (3.11). as these equation will allow elimination of either ϕ or ψ from equations in favour of the porosity, which is a measurable parameter. Three different levels of staggering of the solid phase in the streamwise direction will be studied: the regular, the over-staggered and the fully staggered array. These configurations are shown in Figure 3.2. In nature, a porous medium of which the streamwise staggering is represented by an over-staggered array, is rarely encountered. In practise such a configuration can be obtained mechanically by restricting fluid flow as is shown in Figure 3.2(c). The over-staggered array is studied here purely for academic purposes. A parameter ξ which relates to the cross-stream staggeredness of the solid material, is defined as follows:. ξ =.    0    1 2.      1. Regular array Fully staggered array Over−staggered array .. 25. (3.12).

(47) b n. (a) Regular array where ǫ = 0.75.. (b) Fully staggered array where ǫ ≈ 0.38.. (c) Over-staggered array where ǫ ≈ 0.66.. Figure 3.2: An illustration of the different arrays studied, as well as the SUCs chosen for the different scenarios. For the derivations of the permeability in this chapter, the configurations depicted in Figures 3.3, 3.4 and 3.5 are considered. The volume (the length in the third dimension is unity) of an SUC (or an SRRUC) occupied by fluid flowing parallel to the net streamwise direction is given by Uk , and U⊥ represents the volume of the transverse channel.. Ug. Uk. UtA. Ug. Uk. UtB. Ug. Uk. UtC. Ug. Uk. UtD. Figure 3.3: The present model considered for a regular configuration where the pressure values in the different zones are pUtA = p + δpk , pUtB = p, pUtC = p − δpk and pUtD = p − 2δpk respectively. For a staggered array (Figures 3.4 and 3.5), in volume Ut , one side is bounded by a fluid-solid interface on which the wall shear stresses are neglected. (This is the main difference between the present model and the FD model.) The rest of Ut is bounded by fluid only. The shear stress induced by the surrounding fluid over Ut is assumed negligible to first order accuracy. Under such conditions, it is shown in. 26.

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