Abstract

The flow in porous media has received a great deal of attention due to its importance and many unresolved problems in science and engineering such as geophysics, soil science, underground water resources, petroleum engineering, fibrous composite manufacturing, biophysics (tissues and organs), etc. It has been shown that natural and some synthetic porous media are fractals, and these media may be called fractal porous media. The flow and transport properties such as flow resistance and permeability for fractal porous media have steadily attracted much attention in the past decades. This review article intends to summarize the theories, methods, mathematical models, achievements, and open questions in the area of flow in fractal porous media by applying the fractal geometry theory and technique. The emphases are placed on the theoretical analysis based on the fractal geometry applied to fractal porous media. This review article shows that fractal geometry and technique have the potentials in analysis of flow and transport properties in fractal porous media. A few remarks are made with respect to the theoretical studies that should further be made in this area in the future. This article contains 220 references.

References

1.
Bear
,
J.
, 1972,
Dynamics of Fluids in Porous Media
,
Elsevier
,
New York
.
2.
Katz
,
A. J.
, and
Thompson
,
A. H.
, 1986, “
Quantitative Prediction of Permeability in Porous Rocks
,”
Phys. Rev. B
0163-1829,
34
, pp.
8179
8181
.
3.
Guo
,
S. P.
,
Huang
,
Y. Z.
,
Zou
,
J.
,
Hu
,
Y.
,
Zou
,
Y. R.
, and
Yu
,
D. S.
, 1990,
Microscopic Mechanisms of Physical-Chemical Seepages
,
Sciences, Beijing
,
China
.
4.
Sahimi
,
M.
, 1993, “
Flow Phenomena in Rocks: From Continuum Models to Fractals, Percolation, Cellular Automata, and Simulated Annealing
,”
Rev. Mod. Phys.
0034-6861,
65
, pp.
1393
1534
.
5.
Sahimi
,
M.
, 1995,
Flow and Transport in Porous Media and Fractured Rocks
,
VCH Verlagsgesellshaft mbH
,
Weinheim, Germany
.
6.
Acuna
,
J. A.
,
Ershaghi
,
I.
, and
Yortsos
,
Y. C.
, 1995, “
Pratical Application of Fractal Pressure-Transient Analysis in Naturally Fractured Reservoirs
,”
SPEFE
, September, pp.
173
179
.
7.
Kaviany
,
M.
, 1995,
Principles of Heat Transfer in Porous Media
, 2nd ed.,
Springer-Verlag
,
New York
.
8.
Adler
,
P. M.
, and
Thovert
,
J.-F.
, 1998, “
Real Porous Media: Local Geometry and Macroscopic Properties
,”
Appl. Mech. Rev.
0003-6900,
51
, pp.
537
585
.
9.
Bekri
,
S.
,
Xu
,
K.
,
Yousefian
,
F.
,
Adler
,
P. M.
,
Thovert
,
J.-F.
,
Muller
,
J.
,
Iden
,
K.
,
Psyllos
,
A.
,
Stubos
,
A. K.
, and
Ioannidis
,
M. A.
, 2000, “
Pore Geometry and Transport Properties in North Sea Chalk
,”
J. Pet. Sci. Eng.
0920-4105,
25
, pp.
107
134
.
10.
Adler
,
P. M.
,
Thovert
,
J.-F.
,
Bekri
,
S.
, and
Yousefian
,
F.
, 2002, “
Real Porous Media: Local Geometry and Transports
,”
J. Eng. Mech.
0733-9399,
128
(
8
), pp.
829
839
.
11.
Karacan
,
C. Q.
, and
Halleck
,
P. M.
, 2003, “
A Fractal Model for Predicting Permeability Around Perforation Tunnels Using Size Distribution of Fragmented Grains
,”
J. Pet. Sci. Eng.
0920-4105,
40
, pp.
159
176
.
12.
Liu
,
Z. C.
,
Yue
,
X. G.
,
Hou
,
J. R.
, and
Zhang
,
L. J.
, 2004, “
A Study of Microscale Flows in Pores in Low-Permeability Reservoirs
,”
The Second International Symposium on Multiphase, Non-Newtonian and Reacting Flows’04
,
Hongzhou
,
China
, pp.
454
458
.
13.
Hanano
,
M.
, 2004, “
Contribution of Fractures to Formation and Production of Geothermal Resources
,”
Renewable Sustainable Energy Rev.
1364-0321,
8
, pp.
223
236
.
14.
van Dijke
,
M. I. J.
,
Sorbie
,
K. S.
,
Sohrabi
,
M.
, and
Danesh
,
A.
, 2006, “
Simulation of WAG Floods in an Oil-Wet Micromodel Using a 2-D Pore-Scale Network Model
,”
J. Pet. Sci. Eng.
0920-4105,
52
, pp.
71
86
.
15.
Brailovsky
,
I.
,
Babchin
,
A.
,
Frankel
,
M.
, and
Sivashinsky
,
G.
, 2006, “
Fingering Instability in Water-Oil Displacement
,”
Transp. Porous Media
0169-3913,
63
, pp.
363
380
.
16.
Gebart
,
B. R.
, 1992, “
Permeability of Unidirectional Reinforcements for RTM
,”
J. Compos. Mater.
0021-9983,
26
(
8
), pp.
1100
1133
.
17.
Parnas
,
R. S.
, and
Salem
,
A. J.
, 1993, “
A Comparison of the Unidirectional and Radial In-Plane Flow of Fluids Through Woven Composite Reinforcements
,”
Polym. Compos.
0272-8397,
14
(
5
), pp.
383
394
.
18.
Wu
,
C.-H.
,
Wang
,
T. J.
, and
Lee
,
L. J.
, 1994, “
Trans-Plane Fluid Permeability Measurement and Its Application in Liquid Composite Molding
,”
Polym. Compos.
0272-8397,
15
, pp.
289
298
.
19.
Cai
,
Z.
, 1995, “
Analysis of the Non-Viscous Flow Effect in Liquid Molding Process
,”
J. Compos. Mater.
0021-9983,
29
(
2
), pp.
257
278
.
20.
Gebart
,
B. R.
, and
Lidstrom
,
P.
, 1996, “
Measurement of In-Plane Permeability of Anisotropic Fiber Reinforcements
,”
Polym. Compos.
0272-8397,
17
(
1
), pp.
43
51
.
21.
Shih
,
C. H.
, and
Lee
,
L. J.
, 1998, “
Effect of Fiber Architecture on Permeability in Liquid Composite Molding
,”
Polym. Compos.
0272-8397,
19
, pp.
626
639
.
22.
Yu.
,
B. M.
, and
Lee
,
L. J.
, 2000, “
A Simplified In-Plane Permeability Model for Textile Fabrics
,”
Polym. Compos.
0272-8397,
21
, pp.
660
685
.
23.
Yu
,
B. M.
, and
Lee
,
L. J.
, 2000, “
Analysis of Heat Transfer and Flow in Liquid Composite Molding
,”
Int. Polym. Process.
0930-777X,
15
(
3
), pp.
273
283
.
24.
Yu
,
B. M.
,
Lee
,
L. J.
, and
Cao
,
H. Q.
, 2002, “
A Fractal In-Plane Permeability Model for Fabrics
,”
Polym. Compos.
0272-8397,
23
, pp.
201
221
.
25.
Abrate
,
S.
, 2002, “
Resin Flow in Fiber Performs
,”
Appl. Mech. Rev.
0003-6900,
55
, pp.
579
599
.
26.
Belov
,
E. B.
,
Lomov
,
S. V.
,
Verpoest
,
I.
,
Peters
,
T.
,
Roose
,
D.
,
Parnas
,
R. S.
,
Hoes
,
K.
, and
Sol
,
H.
, 2004, “
Modelling of Permeability of Textile Reinforcements: Lattice Boltzmann Method
,”
Compos. Sci. Technol.
0266-3538,
64
, pp.
1069
1080
.
27.
Gokce
,
A.
,
Chohra
,
M.
,
Advani
,
S. G.
, and
Walsh
,
S. M.
, 2005, “
Permeability Estimation Algorithm to Simultaneously Characterize Distribution Media and Fabric Permeability Values in Vacuum Assisted Resin Transfer Molding Process
,”
Compos. Sci. Technol.
0266-3538,
65
(
14
), pp.
2129
2139
.
28.
Bowles
,
J.
, 1984,
Physical and Geotechnical Properties of Soil
, 2nd ed.,
McGraw-Hill
,
New York
.
29.
Tyler
,
S. W.
, and
Wheatcraft
,
W.
, 1989, “
Application of Fractal Mathematics to Soil Water Retention Estimation
,”
Soil Sci. Soc. Am. J.
0361-5995,
53
, pp.
987
996
.
30.
Perfect
,
E.
, and
Kay
,
B. D.
, 1991, “
Fractal Theory Applied to Soil Aggregation
,”
Soil Sci. Soc. Am. J.
0361-5995,
55
, pp.
1552
1558
.
31.
Rieu
,
M.
, and
Sposito
,
G.
, 1991, “
Fractal Fragmentation, Soil Porosity, and Soil Water Properties: I. Theory
,”
Soil Sci. Soc. Am. J.
0361-5995,
55
, pp.
1231
1238
.
32.
Giménez
,
D.
,
Perfect
,
E.
,
Rawls
,
W. J.
, and
Pachepsky
,
Ya.
, 1997, “
Fractal Models for Predicting Soil Hydraulic Properties: A Review
,”
Eng. Geol. (Amsterdam)
0013-7952,
48
(
3–4
), pp.
161
183
.
33.
Perrier
,
E.
,
Bird
,
N.
, and
Rieu
,
M.
, 1999, “
Generalizing the Fractal Model of Soil Structure: The Pore-Solid Fractal Approach
,”
Geoderma
0016-7061,
88
, pp.
137
164
.
34.
Moldrup
,
P.
,
Olesen
,
T.
,
Komatsu
,
T.
,
Schjønning
,
P.
, and
Rolston
,
D. E.
, 2001, “
Tortuosity, Diffusivity, and Permeability in the Soil Liquid and Gaseous Phases
,”
Soil Sci. Soc. Am. J.
0361-5995,
65
, pp.
613
623
.
35.
Berkowitz
,
B.
, 2002, “
Characterizing Flow and Transport in Fractured Geological Media: A Review
,”
Adv. Water Resour.
0309-1708,
25
, pp.
861
884
.
36.
Perfect
,
E.
, 2005, “
Modeling the Primary Drainage Curve of Prefractal Porous Media
,”
Vadose Zone J.
,
4
, pp.
959
966
.
37.
Wang
,
K.
,
Zhang
,
R. D.
, and
Wang
,
F. G.
, 2005, “
Testing the Pore-Solid Fractal Model for the Soil Water Retention Function
,”
Soil Sci. Soc. Am. J.
0361-5995,
69
, pp.
776
782
.
38.
Schaap
,
M. G.
, 2006, “
Percolation Theory for Flow in Porous Media
,”
Soil Sci. Soc. Am. J.
0361-5995,
70
, pp.
1036
-a-1037-
a
.
39.
Bird
,
N.
,
Cruz
,
D. M.
,
Saa
,
A.
, and
Tarquis
,
A. M.
, 2006, “
Fractal and Multifractal Analysis of Pore-Scale Images of Soil
,”
J. Hydrol.
0022-1694,
322
, pp.
211
219
.
40.
Majumdar
,
A.
, 1992, “
Role of Fractal Geometry in the Study of Thermal Phenomena
,”
Annual Review of Heat Transfer
, Vol.
IV
,
C. L.
Tien
, ed.,
Hemisphere Publishing
,
Paris
, pp.
51
110
.
41.
Baban
,
D. F.
, and
Seymour
,
L. W.
, 1998, “
Control of Tumour Vascular Permeability
,”
Adv. Drug Delivery Rev.
0169-409X,
34
, pp.
109
119
.
42.
Kitaok
,
H.
,
Takaki
,
R.
, and
Suki
,
B.
, 1999, “
A Three-Dimensional Model of the Human Airway Tree
,”
J. Appl. Physiol.
8750-7587,
87
, pp.
2207
2217
.
43.
Maeda
,
H.
,
Wu
,
J.
,
Sawa
,
T.
,
Matsumura
,
Y.
, and
Hori
,
K.
, 2000, “
Tumor Vascular Permeability and the EPR Effect in Macromolecular Therapeutics: A Review
,”
J. Controlled Release
0168-3659,
65
, pp.
271
284
.
44.
Egan
,
W. J.
, and
Lauri
,
G.
, 2002, “
Prediction of Intestinal Permeability
,”
Adv. Drug Delivery Rev.
0169-409X,
54
, pp.
273
289
.
45.
Dash
,
A. K.
, and
Elmquist
,
W. F.
, 2003, “
Review: Separation Methods That Are Capable of Revealing Blood-Brain Barrier Permeability
,”
J. Chromatogr., B: Biomed. Appl.
0378-4347,
797
, pp.
241
254
.
46.
Khaled
,
A.-R. A.
, and
Vafai
,
K.
, 2003, “
The Role of Porous Media in Modeling Flow and Heat Transfer in Biological Tissues
,”
Int. J. Heat Mass Transfer
0017-9310,
46
, pp.
4989
5003
.
47.
Fangang
,
M.
,
Hanmin
,
Z.
,
Yansong
,
L.
,
Xingwen
,
Z.
, and
Fenglin
,
Y.
, 2005, “
Application of Fractal Permeation Model to Investigate Membrane Fouling in Membrane Bioreactor
,”
J. Membr. Sci.
0376-7388,
262
, pp.
107
116
.
48.
Koponen
,
A.
,
Kataja
,
M.
, and
Timonen
,
J.
, 1996, “
Tortuous Flow in Porous Media
,”
Phys. Rev. E
1063-651X,
54
, pp.
406
410
.
49.
Koponen
,
A.
,
Kataja
,
M.
, and
Timonen
,
J.
, 1997, “
Permeability and Effective Porosity of Porous Media
,”
Phys. Rev. E
1063-651X,
56
, pp.
3319
3325
.
50.
Benzi
,
R.
,
Succi
,
S.
, and
Vergassola
,
M.
, 1992, “
The Lattice Boltzmann Equation: Theory and Applications
,”
Phys. Rep.
0370-1573,
222
, pp.
145
197
.
51.
Martys
,
N. S.
, and
Chen
,
H.
, 1996, “
Simulation of Multicomponent Fluids in Complex Three-Dimensional Geometries by the Lattice Boltzmann Method
,”
Phys. Rev. E
1063-651X,
53
, pp.
743
750
.
52.
Chen
,
S.
, and
Doolen
,
G. D.
, 1998, “
Lattice Boltzmann Method for Fluid Flows
,”
Annu. Rev. Fluid Mech.
0066-4189,
30
, pp.
329
364
,
Chen
,
S.
, and
Doolen
,
G. D.
, 1998, “
Lattice Boltzmann Method for Fluid Flows
,”
Annu. Rev. Fluid Mech.
0066-4189,
30
, pp.
329
364
.
53.
Xu
,
Y.-S.
, and
Wu
,
F.-M.
, 2002, “
A New Method for the Analysis of Relative Permeability in Porous Media
,”
Chin. Phys. Lett.
0256-307X,
19
(
12
), pp.
1835
1837
.
54.
Wang
,
J.
,
Zhang
,
X. X.
,
Bengough
,
A. G.
, and
Crawford
,
J. W.
, 2005, “
Domain-Decomposition Method for Parallel Lattice Boltzmann Simulation of Incompressible Flow in Porous Media
,”
Phys. Rev. E
1063-651X,
72
, p.
016706
.
55.
Ansumali
,
S.
, and
Karlin
,
I. V.
, 2005, “
Consistent Lattice Boltzmann Method
,”
Phys. Rev. Lett.
0031-9007,
95
, pp.
260605
.
56.
Tang
,
G. H.
,
Tao
,
W. Q.
, and
He
,
Y. L.
, 2005, “
Gas Slippage Effect on Microscale Porous Flow Using the Lattice Boltzmann Method
,”
Phys. Rev. E
1063-651X,
72
, p.
056301
.
57.
Tang
,
G. H.
,
Tao
,
W. Q.
, and
He
,
Y. L.
, 2005, “
Three-Dimensional Lattice Boltzmann Model for Gaseous Flow in Rectangular Microducts and Microscale Porous Media
,”
J. Appl. Phys.
0021-8979,
97
, p.
104918
.
58.
Hyväluoma
,
J.
,
Raiskinmäki
,
P.
,
Jäsberg
,
A.
,
Koponen
,
A.
,
Kataja
,
M.
, and
Timonen
,
J.
, 2006, “
Simulation of Liquid Penetration in Paper
,”
Phys. Rev. E
1063-651X,
73
, p.
036705
.
59.
Jeong
,
N. G.
,
Choi
,
D. H.
, and
Lin
,
C.-L.
, 2006, “
Prediction of Darcy–Forchheimer Drag for Micro-Porous Structures of Complex Geometry Using the Lattice Boltzmann Method
,”
J. Micromech. Microeng.
0960-1317,
16
(
10
), pp.
2240
2250
.
60.
Guo
,
Z. L.
, and
Zhao
,
T. S.
, 2003, “
Explicit Finite-Difference Lattice Boltzmann Method for Curvilinear Coordinates
,”
Phys. Rev. E
1063-651X,
67
, p.
066709
.
61.
Adler
,
P. M.
, and
Berkowitz
,
B.
, 2000, “
Effective Medium Analysis of Random Lattices
,”
Transp. Porous Media
0169-3913,
40
, pp.
145
151
.
62.
Fokker
,
P. A.
, 2001, “
General Anisotropic Effective Medium Theory for the Effective Permeability of Heterogeneous Reservoirs
,”
Transp. Porous Media
0169-3913,
44
, pp.
205
218
.
63.
Zijl
,
W.
, and
Trykozko
,
A.
, 2001, “
Numerical Homogenization of the Absolute Permeability Using the Conformal-Nodal and Mixed-Hybrid Finite Element Method
,”
Transp. Porous Media
0169-3913,
44
, pp.
33
62
.
64.
Chen
,
Z. X.
, and
Huan
,
G. R.
, 2003, “
Numerical Experiments With Various Formulations for Two Phase Flow in Petroleum Reservoirs
,”
Transp. Porous Media
0169-3913,
51
, pp.
89
102
.
65.
Myroshnychenko
,
V.
, and
Brosseau
,
C.
, 2005, “
Finite-Element Method for Calculation of the Effective Permittivity of Random Inhomogeneous Media
,”
Phys. Rev. E
1063-651X,
71
, p.
016701
.
66.
Xu
,
Y. S.
,
Zhong
,
Yi. J.
, and
Huang
,
G. X.
, 2004, “
Lattice Boltzmann Method for Diffusion-Reaction-Transport Processes in Heterogeneous Porous Media
,”
Chin. Phys. Lett.
0256-307X,
21
, pp.
1298
1301
.
67.
Adler
,
P. M.
, 1992,
Porous Media: Geometry and Transports
,
Butterworth/Heinemann
,
Stoneham, MA
.
68.
Xu
,
Y. S.
,
Liu
,
Y.
, and
Huang
,
G. X.
, 2004, “
Using Digital Imaging to Characterize Threshold Dynamic Parameters in Porous Media Based on Lattice Boltzmann Method
,”
Chin. Phys. Lett.
0256-307X,
21
, pp.
2454
2457
.
69.
Dullien
,
F. A. L.
, 1992,
Porous Media: Fluid Transport and Pore Structure
,
Academic
,
San Diego, CA
.
70.
Petruk
,
W.
, 1989, Short Course on Image Analysis Applied to Mineral and Earth Science, Mineralogical Association of Canada, Ottawa.
71.
Okabe
,
H.
, and
Blunt
,
M. J.
, 2004, “
Prediction of Permeability for Porous Media Reconstructed Using Multiple-Point Statistics
,”
Phys. Rev. E
1063-651X,
70
, p.
066135
.
72.
Yu
,
B. M.
,
Zou
,
M. Q.
, and
Feng
,
Y. J.
, 2005, “
Permeability of Fractal Porous Media by Monte Carlo Simulations
,”
Int. J. Heat Mass Transfer
0017-9310,
48
, pp.
2787
2794
.
73.
Darcy
,
H.
, 1856, Les Fontaines Publiques de la Ville de Dijon, Victor Dalmont, Paris.
74.
Kandhai
,
D.
,
Koponen
,
A.
,
Hoekstra
,
A.
,
Kataja
,
M.
,
Timonen
,
J.
, and
Sloot
,
P. M. A.
, 1999, “
Implementation Aspects of 3D Lattice-BGK: Boundaries, Accuracy, and a New Fast Relaxation Method
,”
J. Comput. Phys.
0021-9991,
150
, pp.
482
501
.
75.
Mandelbrot
,
B. B.
, 1982,
The Fractal Geometry of Nature
,
W. H. Freeman
,
San Francisco, CA
.
76.
Katz
,
A. J.
, and
Thompson
,
A. H.
, 1985, “
Fractal Sandstone Pores: Implications for Conductivity and Pore Formation
,”
Phys. Rev. Lett.
0031-9007,
54
, pp.
1325
1328
.
77.
Krohn
,
C. E.
, and
Thompson
,
A. H.
, 1986, “
Fractal Sandstone Pores: Automated Measurements Using Scanning-Electron-Microscope Images
,”
Phys. Rev. B
0163-1829,
33
, pp.
6366
6374
.
78.
Smidt
,
J. M.
, and
Monro
,
D. M.
, 1998, “
Fractal Modeling Applied to Reservoir Characterization and Flow Simulation
,”
Fractals
0218-348X,
6
, pp.
401
408
.
79.
Yu
,
B. M.
, and
Cheng
,
P.
, 2002, “
A Fractal Model for Permeability of Bi-Dispersed Porous Media
,”
Int. J. Heat Mass Transfer
0017-9310,
45
, pp.
2983
2993
.
80.
Arya
,
L. M.
, and
Paris
,
J. F.
, 1981, “
A Physicoempirical Model to Predict the Soil Moisture Characteristic From Particle-Size Distribution and Bulk Density Data
,”
Soil Sci. Soc. Am. J.
0361-5995,
45
, pp.
1023
1030
.
81.
Young
,
I. M.
, and
Crawford
,
J. W.
, 1991, “
The Fractal Structure of Soil Aggregates: Its Measurement and Interpretation
,”
J. Soil Sci.
0022-4588,
42
, pp.
187
192
.
82.
West
,
G. B.
,
Brown
,
J. H.
, and
Enquist
,
B. J.
, 1997, “
A General Model for the Origin of Allometric Scaling Laws in Biology
,”
Science
0036-8075,
276
, pp.
122
126
.
83.
Brown
,
J. H.
,
West
,
G. B.
, and
Enquist
,
B. J.
, 2000,
Scaling in Biology
,
Oxford University Press
,
Oxford, UK
.
84.
Mauroyl
,
B.
,
Filoche
,
M.
,
Weibel
,
E. R.
, and
Sapoval
,
B.
, 2004, “
An Optimal Bronchial Tree May Be Dangerous
,”
Nature (London)
0028-0836,
427
, pp.
633
636
.
85.
Adler
,
P. M.
, 1985, “
Transport Processes in Fractals—I. Conductivity and Permeability of a Leibniz Packing in the Lubrication Limit
,”
Int. J. Multiphase Flow
0301-9322,
11
, pp.
91
108
.
86.
Adler
,
P. M.
, 1985, “
Transport Processes in Fractals—II. Stocks Flow in Fractal Capillary Networks
,”
Int. J. Multiphase Flow
0301-9322,
11
, pp.
213
239
.
87.
Adler
,
P. M.
, 1985, “
Transport Processes in Fractals—III. Taylor Dispersion in Two Examples of Fractal Capillary Networks
,”
Int. J. Multiphase Flow
0301-9322,
11
, pp.
241
253
.
88.
Adler
,
P. M.
, 1985, “
Transport Processes in Fractals—IV. Nonlinear Flow Problems in Fractal Capillary Networks
,”
Int. J. Multiphase Flow
0301-9322,
11
, pp.
853
871
.
89.
Bejan
,
A.
, 1997, “
Constructal-Theory Network of Conducting Paths for Cooling a Heat Generating Volume
,”
Int. J. Heat Mass Transfer
0017-9310,
40
, pp.
799
816
.
90.
Lorente
,
S.
, and
Bejan
,
A.
, 2006, “
Heterogeneous Porous Media as Multiscale Structures for Maximum Flow Access
,”
J. Appl. Phys.
0021-8979,
100
, p.
114909
.
91.
Gosselin
,
L.
, and
Bejan
,
A.
, 2005, “
Tree Networks for Minimal Pumping Power
,”
Int. J. Therm. Sci.
1290-0729,
44
, pp.
53
63
.
92.
Chen
,
Y. P.
, and
Cheng
,
P.
, 2002, “
Heat Transfer and Pressure Drop in Fractal Tree-Like Microchannel Nets
,”
Int. J. Heat Mass Transfer
0017-9310,
45
, pp.
2643
2648
.
93.
Wechsatol
,
W.
,
Lorente
,
S.
, and
Bejan
,
A.
, 2002, “
Optimal Tree-Shaped Networks for Fluid Flow in a Disc-Shaped Body
,”
Int. J. Heat Mass Transfer
0017-9310,
45
, pp.
4911
4924
.
94.
Yu
,
B. M.
, and
Li
,
B. W.
, 2006, “
Fractal-Like Tree Networks Reducing the Thermal Conductivity
,”
Phys. Rev. E
1063-651X,
73
, p.
066302
.
95.
Xu
,
P.
,
Yu
,
B. M.
,
Feng
,
Y. J.
, and
Liu
,
Y. J.
, 2006, “
Analysis of Permeability for the Fractal-Like Tree Network by Parallel and Series Models
,”
Physica A
0378-4371,
369
, pp.
884
894
.
96.
Xu
,
P.
,
Yu
,
B. M.
,
Feng
,
Y. J.
, and
Zou
,
M. Q.
, 2006, “
The Permeability of the Fractal Disc-Shaped Branched Network With Tortuosity Effect
,”
Phys. Fluids
1070-6631,
18
, p.
078103
.
97.
Xu
,
P.
, and
Yu
,
B. M.
, 2006, “
The Conductivity Scaling Laws of the Fractal-Like Tree Networks
,”
J. Appl. Phys.
0021-8979,
100
, p.
104906
.
98.
Bejan
,
A.
, and
Lorente
,
S.
, 2006, “
Constructal Theory of Generation of Configuration in Nature and Engineering
,”
J. Appl. Phys.
0021-8979,
100
, p.
041301
.
99.
Stauffer
,
D.
, 1979, “
Scaling Theory of Percolation Clusters (Review)
,”
Phys. Rep.
0370-1573,
54
, pp.
1
74
.
100.
Feder
,
J.
, 1988,
Fractals
,
Plenum
,
New York
.
101.
Majumdar
,
A. A.
, and
Bhushan
,
B.
, 1990, “
Role of Fractal Geometry in Roughness Characterization and Contact Mechanics of Surfaces
,”
ASME J. Tribol.
0742-4787,
112
, pp.
205
216
.
102.
Yu
,
B. M.
, and
Li
,
J. H.
, 2001, “
Some Fractal Characters of Porous Media
,”
Fractals
0218-348X,
9
, pp.
365
372
.
103.
Xu
,
J.
,
Yu
,
B. M.
,
Zou
,
M. Q.
, and
Xu
,
P.
, 2006, “
A New Model for Heat Conduction of Nanofluids Based on Fractal Distributions of Nanoparticles
,”
J. Phys. D: Appl. Phys.
0022-3727,
39
, pp.
4486
4490
.
104.
Wyllie
,
M. R. J.
, and
Gregory
,
A. R.
, 1955, “
Fluid Flow Through Unconsolidated Porous Aggregates
,”
Ind. Eng. Chem.
0019-7866,
47
, pp.
1379
1388
.
105.
Dullien
,
F. A. L.
, 1979,
Porous Media, Fluid Transport and Pore Structure
,
Academic
,
San Diego
.
106.
Comiti
,
J.
, and
Renaud
,
M.
, 1989, “
A New Model for Determining Mean Structure Parameters of Fixed Beds From Pressure Drop Measurements: Application to Beds Packed With Parallelepipedal Particles
,”
Chem. Eng. Sci.
0009-2509,
44
, pp.
1539
1545
.
107.
Hsu
,
C. T.
, and
Cheng
,
P.
, 1990, “
Thermal Dispersion in Porous Media
,”
Int. J. Heat Mass Transfer
0017-9310,
33
, pp.
1587
1597
.
108.
Westhuizen
,
J. V.
, and
Pless
,
J. P. D.
, 1994, “
Quanlification of Unidirectional Fiber Bed Permeability
,”
J. Compos. Mater.
0021-9983,
28
, pp.
619
637
.
109.
Moldrup
,
P.
,
Oleson
,
T.
,
Komatsu
,
T.
,
SchJønning
,
P.
, and
Rolston
,
D. E.
, 2001, “
The Tortuosity, Diffusivity, and Permeability in the Soil Liquid and Gaseous Phases
,”
Soil Sci. Soc. Am. J.
0361-5995,
65
, pp.
613
623
.
110.
Gerasimos
,
S.
,
Armatas
,
P.
, and
Pomonis
,
J.
, 2004, “
A Monte Carlo Pore Network for the Simulation of Porous Characteristics of Functionalized Silica: Pore Size Distribution, Connectivity Distribution and Mean Tortuosities
,”
Chem. Eng. Sci.
0009-2509,
59
, pp.
5735
5749
.
111.
Yu
,
B. M.
, and
Li
,
J. H.
, 2004, “
A Geometry Model for Tortuosity of Flow Path in Porous Media
,”
Chin. Phys. Lett.
0256-307X,
21
, pp.
1569
1571
.
112.
Yu
,
B. M.
, and
Li
,
J. H.
, 2004, “
A Fractal Model for the Transverse Thermal Dispersion Conductivity in Porous Media
,”
Chin. Phys. Lett.
0256-307X,
21
(
1
), pp.
117
120
.
113.
Yun
,
M. J.
,
Yu
,
B. M.
,
Zhang
,
B.
, and
Huang
,
M. T.
, 2005, “
A Geometry Model for Tortuosity of Streamtubes in Porous Media with Spherical Particles
,”
Chin. Phys. Lett.
0256-307X,
22
, pp.
1464
1467
.
114.
Yun
,
M. J.
,
Yu
,
B. M.
,
Xu
,
P.
, and
Wu
,
J. S.
, 2006, “
Geometrical Models for Tortuosity of Streamlines in Three Dimensional Porous Media
,”
Can. J. Chem. Eng.
0008-4034,
84
, pp.
301
309
.
115.
Yu
,
B. M.
,
Feng
,
Y. J.
,
Zou
,
M. Q.
, and
Huang
,
M. T.
, 2006, “
An Investigation on Transport Properties Near the Wall in Porous Media by Fractal Models
,”
Heat Transfer Eng.
0145-7632,
27
(
1
), pp.
54
62
.
116.
Pech
,
D.
, 1984, “
Étude de la perméabilité des lits compressibles constitués de copeaux de bois partiellement destructurés
,” Ph.D. thesis, INP Grenoble, France.
117.
Boudreau
,
B. P.
, 1996, “
The Diffusive Tortuosity of Fine-Grained Unlithified Sediments
,”
Geochim. Cosmochim. Acta
0016-7037,
60
(
16
), pp.
3139
3142
.
118.
Lerman
,
A.
, 1979,
Geochemical Processes: Water and Sediment Environments
,
Wiley
,
New York
.
119.
Sen
,
P. N.
,
Scala
,
C.
, and
Cohen
,
M. H.
, 1981, “
A Self-Similar Model for Sedimentary-Rocks with Application to the Dielectric-Constant of Fused Glass-Beads
,”
Geophysics
0016-8033,
46
, pp.
781
795
.
120.
Millington
,
R. J.
, 1959, “
Gas Diffusion in Porous Media
,”
Science
0036-8075,
130
, pp.
100
102
.
121.
Iversen
,
N.
, and
Jorgensen
,
B. B.
, 1993, “
Difussion Coefficiens of Sulfate and Methane in Marine Sediments: Influence of Porosity
,”
Geochim. Cosmochim. Acta
0016-7037,
57
, pp.
571
578
.
122.
Attia
,
A. M.
, 2005, “
Effects of Petrophysical Rock Properties on Tortuosity Factor
,”
J. Pet. Sci. Eng.
0920-4105,
48
, pp.
185
198
.
123.
Wheatcraft
,
S. W.
, and
Tyler
,
S. W.
, 1988, “
An Explanation of Scale-Dependent Dispersivity in Heterogeneous Aquifers Using Concepts of Fractal Geometry
,”
Water Resour. Res.
0043-1397,
24
, pp.
566
578
.
124.
Yu
,
B. M.
, 2005, “
Fractal Character for Tortuous Streamtubes in Porous Media
,”
Chin. Phys. Lett.
0256-307X,
22
(
1
), pp.
158
160
.
125.
Feng
,
Y. J.
, and
Yu
,
B. M.
, 2007, “
Fractal Dimension for Tortuous Streamtubes in Porous Media
,”
Fractals
0218-348X,
15
, pp.
386
390
.
126.
Yu
,
B. M.
, and
Cheng
,
P.
, 2002, “
A Fractal Model for Nucleate Pool Boiling Heat Transfer
,”
ASME J. Heat Transfer
0022-1481,
124
, pp.
1117
1124
.
127.
Xiao
,
B. Q.
, and
Yu
,
B. M.
, 2007, “
A Fractal Model for Critical Heat Flux in Pool Boiling
,”
Int. J. Therm. Sci.
1290-0729,
46
, pp.
426
433
.
128.
Yu
,
B. M.
, 2006, “
Fractal Dimensions of Multiphase Fractal Media
,”
Fractals
0218-348X,
14
(
2
), pp.
111
118
.
129.
Chatzis
,
L.
,
Kantzas
,
A.
, and
Dullien
,
F. A. L.
, 1988, “
on the Investigation of Gravity Assisted Insert Gas Injection, Using Micromodels, Long Berea Cores and Computer Assisted Tomography
,” SPE 18284, SPE63rD Meeting, Houston, TX.
130.
Yu
,
B. M.
, and
Li
,
J. H.
, 2004, “
Fractal Dimensions for Unsaturated Porous Media
,”
Fractals
0218-348X,
12
(
1
), pp.
17
22
.
131.
Malcai
,
O.
,
Lidar
,
D. A.
, and
Biham
,
O.
, 1997, “
Scaling Range and Cutoffs in Empirical Fractals
,”
Phys. Rev. E
1063-651X,
56
, pp.
2817
2828
.
132.
Pfeifer
,
P.
, and
Avnir
,
D.
, 1983, “
Chemistry in Noninteger Dimensions Between Two and Three. I. Fractal Theory of Heterogeneous Surfaces
,”
J. Chem. Phys.
0021-9606,
79
, pp.
3558
3565
.
133.
Sze
,
S. J.
, and
Lee
,
T. Y.
, 1995,
Phys. Rev. B
0163-1829,
51
, pp.
8709
8714
.
134.
Hansen
,
J. P.
, and
Skjeltorp
,
A. T.
, 1988, “
Fractal Pore Space and Rock Permeability Implications
,”
Phys. Rev. B
0163-1829,
38
, pp.
2635
2638
.
135.
Bale
,
H. D.
, and
Schmidt
,
P. W.
, 1984, “
Small-Angle X-Ray-Scattering Investigation of Submicroscopic Porosity with Fractal Properties
,”
Phys. Rev. Lett.
0031-9007,
53
, pp.
596
599
.
136.
Radliński
,
A. P.
,
Radlińska
,
E. Z.
,
Agamalian
,
M.
,
Wignall
,
G. D.
,
Lindner
,
P.
, and
Randl
,
O. G.
, 1999, “
Fractal Geometry of Rocks
,”
Phys. Rev. Lett.
0031-9007,
82
, pp.
3078
3081
.
137.
Ma
,
J.
,
Qi
,
H.
, and
Wong
,
P. Z.
, 1999, “
Experimental Study of Multilayer Adsorption on Fractal Surfaces in Porous Media
,”
Phys. Rev. E
1063-651X,
59
, pp.
2049
2059
.
138.
Al-Raoush
,
R.
, and
Alshibli
,
K. A.
, 2006, “
Distribution of Local Void Ratio in Porous Media Systems From 3D X-Ray Microtomography Images
,”
Physica A
0378-4371,
359
, pp.
713
728
.
139.
Müller
,
H.-P.
,
Kimmich
,
R.
, and
Weis
,
J.
, 1996, “
NMR Flow Velocity Mapping in Random Percolation Model Objects: Evidence for a Power-Law Dependence of the Volume-Averaged Velocity on the Probe-Volume Radius
,”
Phys. Rev. E
1063-651X,
54
, pp.
5278
5285
.
140.
Klemm
,
A.
,
Müller
,
H.-P.
, and
Kimmich
,
R.
, 1997, “
NMR Microscopy of Pore-Space Backbones in Rock, Sponge, and Sand in Comparison with Random Percolation Model Objects
,”
Phys. Rev. E
1063-651X,
55
, pp.
4413
4422
.
141.
Stallmach
,
F.
,
Vogt
,
C.
,
Kärger
,
J.
,
Helbig
,
K.
, and
Jacobs
,
F.
, 2002, “
Fractal Geometry of Surface Areas of Sand Grains Probed by Pulsed Field Gradient NMR
,”
Phys. Rev. E
1063-651X,
88
, p.
105505
.
142.
Feng
,
Y. J.
,
Yu
,
B. M.
,
Zou
,
M. Q.
, and
Zhang
,
D. M.
, 2004, “
A Generalized Fractal Geometry Model for the Effective Thermal Conductivity of Porous Media
,”
J. Phys. D: Appl. Phys.
0022-3727,
37
, pp.
3030
3040
.
143.
Wu
,
J. S.
, and
Yu
,
B. M.
, 2007, “
A Fractal Resistance Model for Flow Through Porous Media
,”
Int. J. Heat Mass Transfer
0017-9310,
50
, pp.
3925
3932
.
144.
Ergun
,
S.
, 1952, “
Fluid Flow Through Packed Columns
,”
Chem. Eng. Prog.
0360-7275,
48
, pp.
89
94
.
145.
Bird
,
R. B.
,
Stewart
,
W. E.
, and
Lightfoot
,
E. N.
, 1962,
Transport Phenomena
,
Wiley
,
New York
, Chap. 6.
146.
Hicks
,
R. E.
, 1970, “
Pressure Drop in Packed Beds of Spheres
,”
Ind. Eng. Chem. Fundam.
0196-4313,
9
, pp.
500
502
.
147.
Leva
,
M.
, 1959,
Fluidization
,
McGraw-Hill
,
New York
.
148.
MacDonald
,
T. F.
,
El-Sayer
,
M. S.
,
Mow
,
K.
, and
Dullien
,
F. A. L.
, 1979, “
Flow Through Porous Media—the Ergun Equation Revisited
,”
Ind. Eng. Chem. Fundam.
0196-4313,
18
, pp.
199
208
.
149.
Nemec
,
D.
, and
levec
,
J.
, 2005, “
Flow Through Packed Bed Reactors:1. Single-Phase Flow
,”
Chem. Eng. Sci.
0009-2509,
60
, pp.
6947
6957
.
150.
Du Plessis
,
J. P.
, 1994, “
Analytical Quantification of Coefficients in the Ergun Equation for Fluid Friction in a Packed Bed
,”
Transp. Porous Media
0169-3913,
16
, pp.
189
207
.
151.
Wu
,
J. S.
, and
Yu
,
B. M.
, 2008, “
A Resistance Model for Flow Through Porous Media
,”
Transp. Porous Media
0169-3913,
71
, pp.
331
343
.
152.
Denn
,
M. M.
, 1980,
Process Fluid Mechanics
,
Prentice-Hall
,
Englewood Cliffs, NJ
, p.
35
.
153.
Zhang
,
Y. Y.
, 1999,
Fluid Mechanics
,
Higher Education
,
Beijing, China
, Chaps. 3 and 6.
154.
Borodich
,
F. M.
, and
Mosolov
,
A. B.
, 1992, “
Fractal Roughness in Contact Problems
,”
J. Appl. Math.
1110-757X,
56
, pp.
681
690
.
155.
Warren
,
T. L.
, and
Krajcinovic
,
D.
, 1996, “
Random Cantor Set Models for the Elastic-Perfectly Plastic Contact of Roughness Surfaces
,”
Wear
0043-1648,
196
, pp.
1
15
.
156.
Turcotte
,
D. L.
, 1988, “
Fractals in Fluid Mechanics
,”
Annu. Rev. Fluid Mech.
0066-4189,
20
, pp.
5
16
.
157.
Sreenivasan
,
K. R.
, 1991, “
Fractals and Multifractals in Fluid Turbulence
,”
Annu. Rev. Fluid Mech.
0066-4189,
23
, pp.
539
600
.
158.
Mandelbrot
,
B. B.
,
Passoja
,
D. E.
, and
Paullay
,
A. J.
, 1984, “
Fractal Character of Fracture Surfaces of Metals
,”
Nature (London)
0028-0836,
308
, pp.
721
722
.
159.
Adler
,
P. M.
, and
Thovert
,
J.-F.
, 1999,
Fractures and Fracture Networks
,
Kluwer
,
Dordrecht
.
160.
Berkowitz
,
B.
, 2002, “
Characterizing Flow and Transport in Fractured Geological Media: A Review
,”
Adv. Water Resour.
0309-1708,
25
, pp.
861
884
.
161.
Drazer
,
G.
,
Auradou
,
H.
,
Koplik
,
J.
, and
Hulin
,
J. P.
, 2004, “
Self-Affine Fronts in Self-Affine Fractures: Large and Small-Scale Structure
,”
Phys. Rev. Lett.
0031-9007,
92
, p.
014501
.
162.
Xie
,
H.
,
Bhaskar
,
R.
, and
Li
,
J.
, 1993, “
Generation of Fractal Models for Characterization of Pulverized Materials
,”
Minerals and Metallurgical Processing
,
10
, pp.
36
42
.
163.
Fan
,
L. T.
,
Kang
,
Y.
,
Neogi
,
D.
, and
Yashima
,
M.
, 1993, “
Fractal Analysis of Fluidized Particle Behavior in Liquid-Solid Fluidized Beds
,”
AIChE J.
0001-1541,
39
, pp.
513
517
.
164.
Chang
,
J.
, and
Yortsos
,
Y. C.
, 1990, “
Pressure-Transient Analysis of Fractal Reservoirs
,”
SPE Form. Eval.
, March, pp.
31
38
.
165.
O’Shaughnessy
,
B.
, and
Procaceia
,
I.
, 1985, “
Diffusion on Fractals
,”
Phys. Rev. A
1050-2947,
32
, pp.
3073
3083
.
166.
Acuna
,
J. A.
, and
Yortsos
,
Y. C.
, 1995, “
Application of Fractal Geometry to the Study of Networks of Fractures and Their Pressure Transient
,”
Water Resour. Res.
0043-1397,
31
, pp.
527
540
.
167.
Acuna
,
J. A.
,
Ershaghi
,
I.
, and
Yortsos
,
Y. C.
, 1995, “
Practical Application of Fractal Pressure-Transient Analysis in Naturally Fractured Reservoirs
,”
SPE Form. Eval.
, September, pp.
173
179
.
168.
Isichenko
,
M. B.
, 1992, “
Percolation, Statistical Topography, and Transport in Random Media
,”
Rev. Mod. Phys.
0034-6861,
64
, pp.
961
1043
.
169.
Adler
,
P. M.
, 1996, “
Transports in Fractal Porous Media
,”
J. Hydrol.
0022-1694,
187
, pp.
195
213
.
170.
Skjeltorp
,
A. T.
, 1990,
Fractals in Physics
,
J.
Feder
, and
A.
Aharony
, eds.,
North-Holland
,
Amsterdam
, pp.
315
321
,
see also 1989,
Skjeltorp
,
A. T.
, 1989, “
Geometrical Scaling of Microsphere-Deposited Monolayers With Holes
,”
Physica D
0167-2789,
38
, pp.
315
321
.
171.
Vicsek
,
T.
, 1989,
Fractal Growth Phenomena
,
World Scientific
,
Singapore
, p.
14
.
172.
Yu
,
B. M.
, and
Yao
,
K. L.
, 1990, “
Properties for Two-Dimensional Fractal Aggregation in External Fields
,”
Phys. Rev. A
1050-2947,
41
, pp.
5564
5567
.
173.
Shi
,
Y.
,
Xiao
,
J. S.
,
Pan
,
M.
, and
Yuan
,
Z. R.
, 2006, “
A Fractal Permeability Model for the Gas Diffusion Layer of PEM Fuel Cells
,”
J. Power Sources
0378-7753,
160
, pp.
277
283
.
174.
Shi
,
Y.
,
Xiao
,
J. S.
,
Pan
,
M.
, and
Yuan
,
Z. R.
, 2007, “
Addendum to ‘A Fractal Permeability Model for Gas Diffusion Layer of PEM Fuel Cells’
,”
J. Power Sources
0378-7753,
165
, p.
299
.
175.
Xu
,
P.
, and
Yu
,
B. M.
, 2008, “
Developing a New Form of Permeability and Kozeny–Carman Constant for Homogeneous Porous Media by Means of Fractal Geometry
,”
Adv. Water Resour.
0309-1708,
31
, pp.
74
81
.
176.
Yu
,
B. M.
,
Li
,
J. H.
,
Li
,
Z. H.
, and
Zou
,
M. Q.
, 2003, “
Permeabilities of Unsaturated Porous Media
,”
Int. J. Multiphase Flow
0301-9322,
29
, pp.
1625
1642
.
177.
Yu
,
B. M.
, and
Liu
,
W.
, 2004, “
Fractal Analysis of Permeabilities for Porous Media
,”
AIChE J.
0001-1541,
50
(
1
), pp.
46
57
.
178.
Kozeny
,
J.
, 1927, “
Ueber Kapillare Leitung Des Wassers im Boden. Stizungsber
,”
Sitzungsberchte Akademie Wisselkoers Wiener
,
136
, pp.
271
306
.
179.
Carman
,
P. C.
, 1937, “
Fluid Flow Through Granular Beds. Trans
,”
Trans. Am. Inst. Chem. Eng.
0096-7408,
15
, pp.
150
167
.
180.
Carman
,
P. C.
, 1939, “
Permeability of Saturated Sands, Soils and Clays
,”
J. Agric. Sci.
0021-8596,
29
, pp.
263
273
.
181.
Carman
,
P. C.
, 1956,
Flow of Gases Through Porous Media
,
Butterworths
,
London
.
182.
McGregor
,
R.
, 1965, “
The Effect of Rate of Flow on Rate of Dyeing. Part II—The Mechanism of Fluid Flow Through Textiles and Its Significance in Dyeing
,”
J. Soc. Dyers Colour.
0037-9859,
81
, pp.
429
438
.
183.
Bourbié
,
T.
,
Coussy
,
O.
, and
Zinszner
,
B.
, 1987,
Acoustics of Porous Media
,
Educational Technology Publications
,
Paris
.
184.
Panda
,
M. N.
, and
Lake
,
L. W.
, 1994, “
Estimation of Single-Phase Permeability from Parameters of Particle-Size Distribution
,”
AAPG Bull.
0149-1423,
78
, pp.
1028
1039
.
185.
Rodriguez
,
E. F.
, and
Giacomelli
,
V. A.
, 2004, “
Permeability-Porosity Relationship in RTM for Different Fiberglass and Natural Reinforcements
,”
J. Compos. Mater.
0021-9983,
38
, pp.
259
268
.
186.
Mavko
,
G.
, and
Nur
,
A.
, 1997, “
The Effect of a Percolation Threshold in the Kozeny–Carman Relation
,”
Geophysics
0016-8033,
62
, pp.
1480
1482
.
187.
Pape
,
H. C.
, and
Clauser
,
I. J.
, 2000, “
Variation of Permeability With Porosity in Sandstone Diagenesis Interpreted With a Fractal Pore Space Model
,”
Pure Appl. Geophys.
0033-4553,
157
, pp.
603
619
.
188.
Civan
,
F.
, 2001, “
Scale Effect on Porosity and Permeability: Kinetics, Model and Correlation
,”
AIChE J.
0001-1541,
47
, pp.
271
287
.
189.
Costa
,
A.
, 2006, “
Permeability-Porosity Relationship: A Reexamination of the Kozeny–Carman Equation Based on a Fractal Pore-Space Geometry Assumption
,”
Geophys. Res. Lett.
0094-8276,
33
, pp.
L02318
.
190.
Bayles
,
G.
,
Klinzing
,
G.
, and
Chiang
,
S.
, 1989, “
Fractal Mathematics Applied to Flow in Porous Systems
,”
Part. Part. Syst. Charact.
0934-0866,
6
, pp.
168
175
.
191.
Govier
,
G. W.
, and
Aziz
,
K.
, 1972,
The Flow of Complex Mixtures in Pipes
,
Krieger
,
New York
, p.
143
.
192.
Bird
,
R. B.
,
Stewart
,
W. E.
, and
Lightfoot
,
E. N.
, 1960,
Transport Phenomena
,
Wiley
,
New York
.
193.
Kemblowski
,
Z.
, and
Michniewicz
,
M.
, 1979, “
A New Look at the Laminar Flow of Power Law Fluids Through Granular Beds
,”
Rheol. Acta
0035-4511,
18
, pp.
572
739
.
194.
Chhabra
,
R. P.
, 1993,
Bubbles, Drops, and Particles in Non-Newtonian Fluids
,
CRC
,
Boca Raton, FL
, p.
217
.
195.
Sabiri
,
N. E.
, and
Comiti
,
J.
, 1995, “
Pressure Drop in Non-Newtonian Purely Viscous Fluid Through Porous Media
,”
Chem. Eng. Sci.
0009-2509,
50
, pp.
1193
1201
.
196.
Chen
,
C.-T.
,
Malkus
,
D. S.
, and
Vanderby
,
R.
, Jr.
, 1998, “
A Fiber Matrix Model for Interstitial Fluid Flow and Permeability in Ligaments and Tendons
,”
Biorheology
0006-355X,
35
, pp.
103
118
.
197.
Smit
,
G. J. F.
, and
du Plessis
,
J. P.
, 1999, “
Modeling of Non-Newtonian Purely Viscous Flow Through Isotropic High Porosity Synthetic Foams
,”
Chem. Eng. Sci.
0009-2509,
54
, pp.
645
654
.
198.
Pearson
,
J. R. A.
, and
Tardy
,
P. M. J.
, 2002, “
Models for Flow of Non-Newtonian and Complex Fluids Through Porous Media
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
102
, pp.
447
473
.
199.
Woods
,
J. K.
,
Spelt
,
P. D. M.
,
Lee
,
P. D.
,
Selerland
,
T.
, and
Lawrence
,
C. J.
, 2003, “
Creeping Flows of Power-Law Fluids Through Periodic Arrays of Elliptical Cylinders
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
111
, pp.
211
228
.
200.
Kuzhir
,
P.
,
Bossis
,
G.
,
Bashtovoi
,
V.
, and
Volkova
,
O.
, 2005, “
Flow of Magnetorheological Fluid Through Porous Media
,”
Eur. J. Mech. B/Fluids
0997-7546,
122
, pp.
331
343
.
201.
Spelt
,
P. D. M.
,
Yeow
,
A. Y.
,
Lawrence
,
C. J.
, and
Selerland
,
T.
, 2005, “
Creeping Flows of Bingham Fluids Through Arrays of Aligned Cylinders
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
129
, pp.
66
74
.
202.
Bostan
,
M.
, and
Hild
,
P.
, 2006, “
Starting Flow Analysis for Bingham Fluids
,”
Nonlinear Anal. Theory, Methods Appl.
0362-546X,
64
, pp.
1119
1139
.
203.
Matsuhisa
,
S.
, and
Bird
,
R. B.
, 1965, “
Analytical and Numerical Solutions for Laminar Flow of the Non-Newtonian Ellis Fluid
,”
AIChE J.
0001-1541,
11
, pp.
588
594
.
204.
Steller
,
R. T.
, 2001, “
Generalized Slit Flow of an Ellis Fluid
,”
Polym. Eng. Sci.
0032-3888,
41
, pp.
1859
1870
.
205.
Myers
,
T. G.
, 2005, “
Application of Non-Newtonian Models to Thin Film Flow
,”
Phys. Rev. E
1063-651X,
72
, p.
066302
.
206.
Balhoff
,
M. T.
, and
Thompson
,
K. E.
, 2006, “
A Macroscopic Model for Shear-Thinning Flow in Packed Beds Based on Network Modeling
,”
Chem. Eng. Sci.
0009-2509,
61
, pp.
698
719
.
207.
Zhang
,
B.
,
Yu
,
B. M.
,
Wang
,
H. X.
, and
Yun
,
M. J.
, 2006, “
A Fractal Analysis of Permeability for Power-Law Fluids in Porous Media
,”
Fractals
0218-348X,
14
, pp.
171
177
.
208.
Li
,
Y. H.
,
Yu
,
B. M.
,
Chen
,
J.
, and
Wang
,
C. Y.
, 2008, “
Analysis of Permeability for Ellis Fluid Flow in Fractal Porous Media
,”
Chem. Eng. Commun.
0098-6445,
195
(
10
), pp.
1240
1256
.
209.
Yun
,
M. J.
,
Yu
,
B. M.
, and
Cai
,
J. C.
,2008, “
A Fractal Model for the Starting Pressure Gradient for Bingham Fluids in Porous Media
,”
Int. J. Heat Mass Transfer
0017-9310,
51
, pp.
1402
1408
.
210.
Kong
,
X. Y.
, 1999,
Advanced Mechanics of Fluids in Porous Media
,
Press of University of Science and Technology of China
,
Hefei, China
.
211.
Wakao
,
N.
, and
Kaguei
,
S.
, 1982,
Heat and Mass Transfer in Packed Beds
,
Gordon and Breach Science Publ.
,
New York
.
212.
Hunt
,
M. L.
, 1990, “
Comparison of Convective Heat Transfer in Packed Beds and Granular Flows
,”
Annual Review of Heat Transfer
,
C. L.
Tien
, ed.,
Hemisphere
,
Paris
, p.
163
.
213.
Hsu
,
C. T.
, and
Cheng
,
P.
, 1990, “
Thermal Dispersion in a Porous Medium
,”
Int. J. Heat Mass Transfer
0017-9310,
33
, pp.
1587
1597
.
214.
Dessemberger
,
R. B.
, and
Tucker
,
C. T.
, 1995, “
Thermal Dispersion in Resin Transfer Molding
,”
Polym. Compos.
0272-8397,
16
, pp.
495
506
.
215.
Wen
,
C. Y.
, and
Fan
,
L. T.
, 1975,
Models for Flow Systems and Chemical Reactors
,
Dekker
,
New York
, Chap 5.
216.
Kamiuto
,
K.
, and
Saitoh
,
S.
, 1994, “
Fully Developed Forced-Convection Heat Transfer in Cylindrical Packed Beds With Constant Wall Temperatures
,”
JSME Int. J., Ser. B
1340-8054,
37
, pp.
554
559
.
217.
Chiu
,
H.-T.
,
Yu
,
B. M.
,
Chen
,
S. C.
, and
Lee
,
L. J.
, 2000, “
Heat Transfer During Flow and Resin Reaction Through Fiber Reinforcement
,”
Chem. Eng. Sci.
0009-2509,
55
, pp.
3365
3376
.
218.
Yu
,
B. M.
, and
Li
,
J. H.
, 2004, “
A Fractal Model for the Transverse Thermal Dispersion Conductivity in Porous Media
,”
Chin. Phys. Lett.
0256-307X,
21
, pp.
117
120
.
219.
Cheng
,
P.
,
Chowdhury
,
A.
, and
Hsu
,
C. T.
, 1991, “
Forced Convection in Packed Tubes and Channels With Variable Porosity and Thermal Dispersion Effects
,”
Computer Applications in Production Engineering
,
S.
Kakc
,
B.
Kikic
,
F. A.
Kulacki
, and
F.
Arinc
, eds.,
Kluwer
,
Dordrecht
, pp.
625
653
.
220.
Hunt
,
M. L.
, and
Tien
,
C. L.
, 1988, “
Effects of Thermal Dispersion on Forced Convection in Fibrous Media
,”
Int. J. Heat Mass Transfer
0017-9310,
31
, pp.
301
309
.
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