Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation...
Transcript of Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation...
![Page 1: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/1.jpg)
J.-F. Bloch
1
Homogenisation techniques
dedicated to paper industry:
theory and applications.
Bloch, J.-F.
Paris, 30 septembre 2010
![Page 2: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/2.jpg)
J.-F. Bloch
2
Part I –
Homogeneization theory applied
to paper
Part II –
REV and X-Ray microtomography
Outline
![Page 3: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/3.jpg)
J.-F. Bloch
3
Part I - CONTENT
• I - Introduction
• II - Method presentation
• III - Main Results
• IV - Example
• V - Conclusion
![Page 4: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/4.jpg)
J.-F. Bloch
4
z
y
x
Papier
Feutre
I - INTRODUCTION
Paper
AIM :
Temperature field in paper during hot
pressing
Felt
![Page 5: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/5.jpg)
J.-F. Bloch
5
I - INTRODUCTION (2)
Paper and Felt: deformable porous media
Non saturated: (non /) wetting phases
Complex structure at the pore scale
Macroscopic modelling:
Homogeneization
![Page 6: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/6.jpg)
J.-F. Bloch
6
I - INTRODUCTION (3)
Homogeneization:
microscale to macroscale
![Page 7: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/7.jpg)
J.-F. Bloch
7
II – METHOD
PRESENTATION
The medium is assumed as periodic.
Random media yield similar
macroscopic description modelling.
AURIAULT J.-L. (1991) ”Heterogeneous medium. Is an equivalent macroscopic
description possible?”, Int. J. Engng. Sci., 29, 7, pages 785-795.
![Page 8: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/8.jpg)
J.-F. Bloch
8
II – METHOD
PRESENTATION:
Multiscale expansion method
- without macroscopic prerequisites,
- the macroscopic law,
- the effective parameters,
- the validity domain of the model.
![Page 9: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/9.jpg)
J.-F. Bloch
9
II – METHOD
PRESENTATION:
Multiscale expansion method
pore dimension / sample size
model quality.
![Page 10: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/10.jpg)
J.-F. Bloch
10
II – METHOD
PRESENTATION:
Main steps
1 – Physical phenomena at the microscopic scale.
2 – Two different scales are defined ( ).
3 – Physical Equations at microscopic scale.
4 – The dimensionless parameters are expressed in
function of .
![Page 11: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/11.jpg)
J.-F. Bloch
11
II – METHOD
PRESENTATION:
Main steps (2)5 – Asymptotic expansions in power of are
introduced.
6 – Problems at different orders are solved to
determine the successive approximations of the
variables.
7 – Physical quantities are evaluated from the
dimensionless ones.
![Page 12: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/12.jpg)
J.-F. Bloch
12
II – METHOD
PRESENTATION
• The scale ratio = l / L
• For a paper web, the characteristic lengths
l =10 m and L = 1 mm
= 10-2
• Dimensionless parameters are evaluated in
term of .
![Page 13: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/13.jpg)
J.-F. Bloch
13
II – METHOD
PRESENTATION
lk TT
kkkpkk
kpkTλ.TCρ.v
t
TCρ
k
k
Energy balance for a domain (k = w, a, s):
Flux conservation of heat on the interface:
NX
TN
X
Ti
j
lli
j
kk ijij
Temperatures are continuous on each interface
(no resistance) :
![Page 14: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/14.jpg)
J.-F. Bloch
14
II – METHOD
PRESENTATION
- in k :
Dimensionless variables: X = X*. XR
Dimensionless parameters:
i
kkkke
j
kk
i
kkk
y
TCVP
y
T
yt
TCP
iij
Ny
TN
y
TNi
j
lli
j
kkl k ij
ij
- in kl:
Rw
Ra
λλ
λN
aw
Rw
Rs
swN
R
jij
i
R
p
X
T
X
t
TC
P (R)
(*)
R
jij
i
R
iip
X
T
X
X
TVC
Pe
![Page 15: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/15.jpg)
J.-F. Bloch
15
II – METHOD
PRESENTATION- Reference Values
(J . m-1 . s-1 . K-1) (J . kg-1 . K-1) (kg.m-3)
s Cps s
0.33 1.33.10 3 1.5.10
w Cpw w
0.602 4.18.10 3 10 3
a Cpa a
0.026 10 3 1.23
![Page 16: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/16.jpg)
J.-F. Bloch
16
II – METHOD
PRESENTATION
1O0.548NεO0.043Nswaw λλ
a1-
w
2w
w PεOτλ
lCP
ws
2s
s POC
Pl
![Page 17: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/17.jpg)
J.-F. Bloch
17
II – METHOD
PRESENTATIONEach physical quantity is then looked as:
....φε.φεε.φφφ 332210
Homogenisation process
approximated macroscopic models.
Model accuracies = O( )
The larger the scale separation is,
the better is the result (approximation).
![Page 18: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/18.jpg)
J.-F. Bloch
18
III - MAIN RESULTS
A - Small Péclet number:
diffusion - convection
εOx
Tλ
xt
TC
j
effij
isw,
dy
I1
swj
kIkjij
effij
sw Ω sΩ wsw, dΩCdΩCΩ
1C
A-1: Pe O ( 2) P = O ( 2)
With :
![Page 19: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/19.jpg)
J.-F. Bloch
19
III – MAIN RESULTS (2)
Ox
T
x0
j
effij
i
A-2 : Pe O ( 2) P O ( 3)
A-3 : Pe = O ( ) P = O ( 2)
Ox
TVC
x
T
xt
TC
ii
j
effij
iw,s
J.-F. BLOCH, J.-L. AURIAULT, (1998) ‘Heat Transfer in Nonsaturated Porous Media:
Modelling by Homogenisation’, TiPM, 30: 301–321.
![Page 20: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/20.jpg)
J.-F. Bloch
20
III - MAIN RESULTS (3)
Ox
TVC
x
T
x0
ii
j
effij
i
A-4 : Pe = O ( ) P O ( 3)
A-5 : Pea,w = O ( ) Pes = O ( 2) P = O ( 2)
Ox
TVC
x
T
xt
TC
i+i
j
effij
iw,s wa
![Page 21: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/21.jpg)
J.-F. Bloch
21
III - MAIN RESULTS (4)
B - 2 : Pe = O (1) P = O ( 2)
B - 3 : Pe = O (1) P O ( 3)
2
iΩi
j
*eff**
isw, εO
x
TCV
x
Tλ
xε
t
TCε
ij
dVCy
I1
sw,
jIII(0)i
0
l
jIII
ljijeff***
ij
2
iΩi
j
*eff**
i
εOx
TCV
x
Tλ
xε0
ij
B - Higher Péclet number : dispersion
![Page 22: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/22.jpg)
J.-F. Bloch
22
III - MAIN RESULTS (5)
B - 1 : Pe = O (1) P = O ( )
B - Higher Péclet number : dispersion
2
iΩi
j
*eff*
i
i
2
ΩΩiΩΩΩ
εOx
TCV
x
Tλ
xε
xt
TχCε
t
TCεC
ij
swIIasw
dVCy
I1
w,s
j
(0)i
0
l
jljij
*eff*ij II
II
![Page 23: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/23.jpg)
J.-F. Bloch
23
III - MAIN RESULTS (6)
Pe O ( -1)
This case is NOT homogeneizable
no equivalent medium exists !
![Page 24: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/24.jpg)
J.-F. Bloch
24
z
y
x
Papier
Feutre
IV - EXAMPLE:
Hot pressing of a paper web
O10O5.0
10.10.10.10#
.V.C.Pe 2
6233p l
ii
j
effij
i x
TVC
x
T
x0
![Page 25: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/25.jpg)
J.-F. Bloch
25
IV – EXAMPLE:
Assumptions
ii
j
effij
i x
TVC
x
T
x0
ii
jijw
i x
TVC
x
TI
x0
ws
ii
jijw
i x
TVC
x
TI
x0
ii
jijsww
i x
TVC
x
TInn
x0
sw O
wpwwapaa VCVC
ns + nw # 1.
![Page 26: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/26.jpg)
J.-F. Bloch
26
IV - EXAMPLE:
Temperature field in paper and felt
Troll = 50 °C
Mach. Speed:10 m.s-1,
Paper thick.: 320 m,
Felt thick.: 2.5 mm,
Nip Length: 2.5 cm.
![Page 27: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/27.jpg)
J.-F. Bloch
27
IV - EXAMPLE
Detail of temperature field in paper during hot
pressing.
PAPER
FELT
Troll = 50 °C
![Page 28: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/28.jpg)
J.-F. Bloch
28
V – CONCLUSIONS
/ Part I• Same microscopic physics, different law
structures, with different effective coefficients.
• Evaluating dimensionless numbers ( ) :
macroscopic model catalogue, their effective
parameters and their validity domains are
obtained.
• Physical characteristic values dedicated to the
studied process.
![Page 29: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/29.jpg)
J.-F. Bloch
29
V - CONCLUSION / Part I (2)
• The equivalent description corresponds to a
medium that reacts globally to the considered
physical excitation like the microscopic medium
studied.
• If the medium cannot be homogenised,
macroscopic properties are not intrinsic to the
media.
• Applicable to any porous medium that satisfies
the microscopic hypotheses in use here.
![Page 30: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/30.jpg)
J.-F. Bloch
30
Part II
3D structure
of Papers
/ VER
(X-Ray Microtomography)
![Page 31: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/31.jpg)
J.-F. Bloch
31
Estimation of paper physical properties
based on synchrotron
X-ray microtomography.
Jean-Francis Bloch
Sabine Rolland du Roscoat
Maxime Decain, Christian Geindreau
![Page 32: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/32.jpg)
J.-F. Bloch
32
Content
(Partie Non présentée)
I. Context
II. Materials and Methods
III. Estimation of paper structure / VER
Deterministic approach
Statistical approach
I. Estimation of paper properties
II. Conclusions and perspectives
![Page 33: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/33.jpg)
J.-F. Bloch
33
I. Context
• Synchrotron X-ray microtomography gives
access to the inner structure of samples at a
micrometric scale
– Quantification of the structure
– Estimation of physical properties
• Comparison to experimental data
![Page 34: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/34.jpg)
J.-F. Bloch
34
I. Position of the problem
• May Physical properties be simulated from
X-Ray microtomography?
• Analysed volume is much smaller than the
characteristic lengths / physical properties?
Problem of the representativity for:
• Porosity and specific surface area
• Effective thermal conductivity and
permeability
![Page 35: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/35.jpg)
J.-F. Bloch
35
Tomographic data
REV ? No …
Yes
Estimation of physical properties
![Page 36: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/36.jpg)
J.-F. Bloch
36
How Long is Long enough
/ structural properties ?
Property
(porosity)
REV
Volume
size
Small samples (mm) !?
II.4. Representative Elementary
VolumeREV
• Definition
![Page 37: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/37.jpg)
J.-F. Bloch
3737
II.4. Applied methods
Two approachs
Deterministic (classical)1
Statistical2
1 W.J. Drugan, J.R. Willis, "A micromecanics-based nonlocal constitutive equation and estimates of
representative volume element size for elastic composites.", J. Mech, Phys Solids, Vol. 44, No 4, 1996,
pp 497-524.
2 T. Kanit, S. Forest, I. Galliet, V. Mounoury, D. Jeulin, "Determination of the size of the representative
volume element for random composites: statistical and numerical approach", International Journal of
Solids and Structures 40, 2003, pp 3647-3679.
![Page 38: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/38.jpg)
J.-F. Bloch
38
III.5. Conclusions
/ Statistical REV
• Estimation of properties on volumes smaller
than the deterministic REV
saving computing time in the case of the
estimation of physical properties
Rolland du Roscoat, S., Decain, M., Thibault, X., Geindreau, C., Bloch J.-F. Estimation of
microstructural properties from synchrotron X-Ray microtomography and determination of
the REV size in paper materials. Acta Materiala, 55(8), 2007, 2841-2850
![Page 39: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/39.jpg)
J.-F. Bloch
39
Content
I. Context
II. Materials and Methods
III. Estimation of paper structure / VER
Deterministic approach
Statistical approach
IV. Estimation of paper properties
V. Conclusions & perspectives
![Page 40: Homogenisation techniques dedicated to paper industry ... · J.-F. Bloch 1 Homogenisation techniques dedicated to paper industry: theory and applications. Bloch, J.-F. Paris, 30 septembre](https://reader033.fdocuments.us/reader033/viewer/2022060405/5f0f227b7e708231d442a911/html5/thumbnails/40.jpg)
J.-F. Bloch
40
IV.1. Estimation of permeability
• Estimation of tensor of permeability
– No penetration of fluid into fibres
– Pressure gradient is imposed at the interface
– Estimation of the local fluid velocity
– Deducing the permeability
from Darcy Law
Koivu, V., Geindreau, C., Decain, M., Mattila, K., Bloch, J.-F., Kataja, M., Transport
properties of heterogeneous materials combining computerized x-ray micro-tomography and
direct numerical simulations, International Journal of Computational Fluid Dynamics,
23(10), 2010, 713-721