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A general model for the permeability of fibrous porous media based on fluid

flow simulations using the lattice Boltzmann method

Aydin Nabovati a,*, Edward W. Llewellin b, Antonio C.M. Sousa a,c

a Department of Mechanical Engineering, University of New Brunswick, Fredericton, NB, Canadab Department of Earth Sciences, Durham University, Durham, UKc Department of Mechanical Engineering, University of Aveiro, Aveiro, Portugal

a r t i c l e i n f o

Article history:

Received 22 September 2008

Received in revised form 24 February 2009

Accepted 12 April 2009

Keywords:

A. Fibres

B. Mechanical properties

C. Computational modelling

E. Resin transfer moulding (RTM)

a b s t r a c t

Fluid flow analyses for porous media are of great importance in a wide range of industrial applications

including, but not limited to, resin transfer moulding, filter analysis, transport of underground water

and pollutants, and hydrocarbon recovery. Permeability is perhaps the most important property that

characterizes porous media; however, its determination for different types of porous media is challenging

due its complex dependence on the pore-level structure of the media. In the present work, fluid flow in

three-dimensional random fibrous media is simulated using the lattice Boltzmann method. We deter-

mine the permeability of the medium using the Darcy law across a wide range of void fractions

(0.086 /6 0.99) and find that the values for the permeability that we obtain are consistent with avail-

able experimental data. We use our numerical data to develop a semi-empirical constitutive model for

the permeability of fibrous media as a function of their porosity and of the fibre diameter. The model,

which is underpinned by the theoretical analysis of flow through cylinder arrays presented by [Gebart

BR. Permeability of unidirectional reinforcements for RTM. J Compos Mater 1992; 26(8): 110033], gives

an excellent fit to these data across the range of/. We perform further simulations to determine the

impact of the curvature and aspect ratio of the fibres on the permeability. We find that curvature has

a negligible effect, and that aspect ratio is only important for fibres with aspect ratio smaller than 6:1,

in which case the permeability increases with increasing aspect ratio. Finally, we calculate the permeabil-

ity tensor for the fibrous media studied and confirm numerically that, for an isotropic medium, the per-

meability tensor reduces to a scalar value.

2009 Elsevier Ltd. All rights reserved.

1. Introduction

Permeability prediction, and more generally, the investigation

of the effect of pore structure on the bulk properties of porous

media, has posed a major challenge to researchers and engineers

in a wide range of industrial and academic disciplines. These in-

clude, but are not limited to, resin transfer moulding [1,2], biomed-

ical engineering [35], subsurface flow of oil and groundwater[6,7], filter simulation [8,9] and fuel cell simulations [1012].

Macroscopic approaches for fluid flow simulation in porous

media, either using the Darcy law [13] or more complicated mod-

els [14], require as an input the permeability; however, the analy-

sis of the effect of pore-scale parameters on the macroscopic bulk

properties is a cumbersome task. The pore structure in porous

media is often complex, and complicated flow patterns exist within

the pores and between the grains. Consequently, permeability is

found to be highly medium-specific, hence there is no general

model for permeability as a function of the bulk properties of a

medium. The determination of permeability for a specific material

typically requires time-consuming experimental work.

Most experimental methods of permeability prediction apply a

constant pressure gradient to the porous medium and determine

the average flow velocity from the measured fluid flow-rate. The

mediums permeability is subsequently determined using the

Darcy law [13], as follows:

hui K

l:rp 1

where hui, K, rp and l are the volume-averaged flow velocity, per-meability tensor, pressure gradient vector and the dynamic viscos-

ity of the fluid, respectively. This relationship is valid in the

creeping-flow regime (Reynolds number ( 1).Earlier studies of porous media flow were conducted experi-

mentally and some of the best-known models for the permeability

of porous media are based on experimental data [1315]. In these

studies, primarily due to the macroscopic nature of the experimen-

tal approach, the details of the pore-scale flow-pattern in the

1359-835X/$ - see front matter 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.compositesa.2009.04.009

* Corresponding author. Tel.: +1 506 452 6128; fax: +1 506 453 5025.

E-mail address: [email protected] (A. Nabovati).

Composites: Part A 40 (2009) 860869

Contents lists available at ScienceDirect

Composites: Part A

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p o s i t e s a
mailto:[email protected]://www.sciencedirect.com/science/journal/1359835Xhttp://www.elsevier.com/locate/compositesahttp://www.elsevier.com/locate/compositesahttp://www.sciencedirect.com/science/journal/1359835Xmailto:[email protected]

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porous medium cannot be captured. In general, studying the effect

of pore-scale parameters on bulk properties requires a large exper-

imental dataset, which is time-consuming and expensive to

generate.

As alternative approaches, analytical and numerical methods

aim to predict the permeability by solving the fluid flow inside

the pores of the porous medium. The numerical procedure echoes

the experimental approach: a pressure gradient is applied; the

fluid flow inside the pores of the medium is solved; the permeabil-

ity of the mediumis then determined using the Darcy law (Eq. (1)).

This approach has two main advantages over the experimental ap-

proach. First, the geometry of the digitally constructed medium

can be varied rapidly and arbitrarily. Second, simulated experi-

ments are typically much quicker to run than their laboratory

counterparts. Together, these benefits allow a more-rapid and

more-thorough exploration of parameter space than can be

achieved in the laboratory.

The very first challenge in the numerical simulation of permeat-

ing flow is that the pore-level structure of the medium is required

as input. Imaging techniques, such as computed tomography

[16,17], have been developed to capture the complex structure of

real porous media; however, these methods may be costly and

time consuming and impose limits on the resolution that can be

achieved. The alternative approach, which has been widely used

in the literature, is to reconstruct the pore-level structure virtually.

The level of the reconstructed structures complexity depends on

the computational resources available and the nature of the prob-

lem under study. The reconstructed medium can be in the form of

ordered or random arrangements of two and three-dimensional

obstacles.

Analytical studies of the pore-level flow, in general, employ the

Stokes equation (a simplified form of the NavierStokes equation,

which is valid for creeping flow) for a specified domain with peri-

odic boundary conditions. Due to the limitations of the methods

applied, the computational domain, which is the building block

of the pore structure, is in the form of a simplified, well-defined

structure in which the grains of the porous medium are repre-sented in the form of two dimensional obstacles [18,19], ordered

sphere packing [20], or ordered packing of cylinders [2124].

Rapid increase in available computing power and the develop-

ment of advanced numerical algorithms mean that detailed

numerical simulations of flow in porous media are now feasible.

Removing the constraints of the analytical approaches, more com-

plex pore geometries, which resemble the real porous-media struc-

tures more closely, can be used in fluid flow simulations. Ordered

or random packing of different geometric configurations, such as

square blocks, spheres, cylinders, and parallelepipeds [25] have

been used in the literature to reconstruct the pore structure. The

choice of the constructing elements depends on the nature and

application of the porous medium to be modeled. Random arrange-

ments of spheres with mono-dispersed, bi-dispersed, or distrib-uted diameter are often used in simulations of flow in geological

materials, including studies of groundwater flow and hydrocarbon

recovery, and flow in packed beds, and rocks [6,7,2628]. Simula-

tions of flow in the preform matrix in resin transfer moulding

(RTM) [24,29,30], and through paper fibres [31,32] and woven

materials [9,3336] often use random arrangements of fibres.

1.1. Porositypermeability relationships for fibrous materials

The key parameter controlling the permeability of fibrous mate-

rials and, indeed, all porous materials is the porosity / = Vpore/

Vsample, where Vsample is the total volume of the sample and Vpore is

the volume not occupied by solid fibres. Several workers have pub-

lished relationships for the permeability of fibrous materials as afunction of their porosity.

Gebart [24] presents a combined theoretical, numerical and

experimental investigation of the permeability of ordered arrays

of fibres. The analytical treatment of creeping flow perpendicular

to the long axis of the fibres is predicated upon the assumptions

that permeability is controlled by the narrow slots formed between

the fibres at their closest approach and that the widthof these slots

varies only slowly. These assumptions are most valid in the limit of

close-packed fibres. Gebart derives the following functional form

for K(/):

K

a2 C

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 /c1 /

s 1

!5=22

where a is the fibre radius, /c is the critical value of porosity below

which there is no permeating flow (the percolation threshold) and C

is a geometric factor (Gebart calculates C 16=9ffiffiffi

2p

p and/c = 1 p/4 for a square array, C 16=9

ffiffiffi

6p

p and /c 1 p=2ffiffiffi

3p

for a hexagonal array). Gebart presents numerical results, obtained

using a finite difference solution of the NavierStokes equations,

that show excellent agreement with the relationship up to at least

/ = 0.65.

Koponen et al. [31] used the LBM on a D3Q19 lattice to studycreeping flow through three-dimensional random-fibre sheets,

analogous to paper and non-woven fabrics. They report that the

permeability of such materials is exponentially dependent on the

porosity and independent of whether the fibres were placed ran-

domly or not. They present an empirical relationship for the per-

meability as a function of the porosity, based on a fit to their

data, for porosities in the range 0.4 < / < 0.95:

K

a2

5:55

e10:11/ 13

Clague et al. [37] also studied the permeability of three-dimensional

ordered and disordered fibrous media. They used the lattice Boltz-

mann method (LBM) on a D3Q15 and D3Q19 lattice to simulate

creeping flow through fully three-dimensional random fibre net-

works, in which free overlapping of the fibres was allowed. Both

wall-bounded and unbounded media were considered and the ef-

fect of the wall on the overall permeability of the fibrous media

was investigated. They use a scaling analysis to develop a phenom-

enological relationship between permeability and porosity for both

the bounded and unbounded fibrous media. For the case of an un-

bounded medium, they find:

K

a2 b1


s 1

!2eb2 1/ 4

where b1 and b2 are curve fitting parameters. For a disordered (ran-

dom) arrangement of fibres, Clague et al. calculate a typical value of

/c = 1p/4 % 0.21 but with a 1r variation that gives minimum andmaximum values of /c = 0 and /c 2 ffiffiffiffipp p % 0:4, respectively.They show an excellent fit to their data, which span the range

0.33 < / < 0.95 (i.e. they do not cover the region near the percola-

tion threshold).

In the present work, fluid flow is simulated in 3D random fi-

brous media at the pore level. We employ the lattice Boltzmann

method for the fluid flow simulation and calculate the permeabil-

ity of the medium using the Darcy law. We cover a wide range of

porosity, from near the percolation threshold to very dilute sys-

tems (0:08 / 0:99). Based on curve fitting of our numericalexperimental results, we propose a semi-empirical constitutive

relationship for the permeability as a function of porosity. We also

investigate the effect of various other pore-level parameters,

including the curvature, diameter and aspect ratio of the fibres,

on the predicted permeability.

A. Nabovati et al./ Composites: Part A 40 (2009) 860869 861

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2. Methodology

The complex structure of the pore-level geometry, especially in

media of low porosity, yields small pores and narrow flow pas-

sages. As a rule of thumb, the local flow velocity in these narrow

pores is proportional to the volume-averaged flow velocity divided

by the porosity. The narrow flow passages and locally high veloci-

ties limit the applicability of the conventional computational fluiddynamics approaches. Mesoscopic methods such as the smoothed

particle hydrodynamics (SPH) [38,39], lattice gas automata (LGA)

[40,41] and lattice Boltzmann method (LBM) [4244] have been

successfully used for macroscopic fluid flow simulations, which re-

quire the mesoscopic details of the flow to be considered. The LGA

and SPH methods, in their current state-of-the-art, tend to be com-

putationally costly to perform three-dimensional simulations of

flow in porous media that are of a size adequate to yield physically

meaningful results. Although the early versions of LBM [45] suf-

fered from similar difficulties, later developments of the LBM have

seen dramatic improvements in the computational efficiency, mak-

ing it a suitable tool for mesoscopic, three-dimensional simulations

[42,46,47]. Other attractive features of the LBM are that numerical

operations are spatially local, easing implementation, while solid

boundaries of arbitrary complexity can be included without perfor-

mance penalty. Furthermore, the LBM is well suited to flow simu-

lation at the mesoscopic scale, and is amenable to parallelization.

These characteristics have made the LBM the most popular method

for numerical pore-level analysis; indeed this was one of the first

applications of LBM [48]. The LBM has been shown to be a more

efficient tool for flow simulation in such complex geometries than

conventional fluid dynamic approaches [49,50].

In the present work, three-dimensional fluid flow was simu-

lated in fibrous porous media using LBflow,1 an implementation

of the single-relaxation-time (SRT) LBM on a D3Q15 lattice [51,52].

As in all such implementations, the flow is represented by the prop-

agation of fluid mass through a lattice. The lattice is a discrete

representation of physical space; in the present case a three-dimen-

sional cubic lattice. At any time t, fluid mass can propagate from anode with position r to any of its six orthogonal neighbours or eight

long-diagonal neighbours, or it can remain at the present node. Since

time is also discrete, the propagating fluid arrives at its new location

at time t+ 1, hence, there are i = 15 possible fluid velocities at each

node, represented by the vector ei. The spatial discretization Dx

and the time step Dt define the units of the simulation. The total

density at each node is given by:

qr; t X

i

fir; t 5

where q is also in simulation units (typically initialized to q = 1throughout the lattice). Similarly, the average fluid velocity at each

node is given by:

ur; t P

ifieir; t

qr; t6

In addition to the propagation step, at each node r, at each time step

t, the incoming fluid masses fi undergo collision, in which they re-

lax towards the equilibrium distribution:

feqi qxi 1 ei u

c2su u

2c2s

ei u2

2c4s

" #7

where the weightsx0 = 2/9,x1. . . 6 = 2/9 and x7. . .14 = 1/72, and thelattice pseudo-sound speed cs Dx=

ffiffiffi

3p

Dt. This equilibrium distri-

bution is a discrete analogue of the Maxwell-Boltzmann distribu-

tion for a population of fluid particles having the same density q

and average velocity u as the incoming fluid masses in the simula-

tion. The propagation and collision steps are encapsulated in the

lattice Boltzmann equation:

fir eiDt; t Dt

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}propagation

fir; t fir; t f

eqi r; t

s

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}collision

8

where s is a relaxation parameter, related to the fluid viscosity. Inthe implementation of the LBM adopted in the present work, flowis driven by imposing a constant, uniform body force G on the fluid

at every point, which is physically analogous to a gravitational force

acting on the fluid. This is achieved by adding an extra term gi to

each of the mass components fi prior to propagation. The term giis formulated in such a way that the total mass is conserved but

the momentum is adjusted to account for the force acting on the

mass at the node for the duration of the time step:

gir; t xiDtc2s

G ei 9

This term is added to the right hand side of Eq. (8). The halfway

bounceback method [44] was used to implement the solid wall

boundary condition.

LBflow uses a scripting language to set up the flow simulation;

via an interpreted text file, the user can specify the geometry and

parameters of the simulation. In this study, the geometries of the

porous media of interest are specified as a three-dimensional bin-

ary mask of the simulation lattice. Dimensional quantities for the

simulation are specified in SI units, rather than simulation units.

In this study, the working fluid has the properties of water at

20 C: kinematic viscosity m = 1.004 106 m2 s1, densityq = 998.29kg m3. The driving pressure gradient rp is specifiedin units of Pa m1. Once the flow has settled to steady state, theaverage velocity of the fluid nodes u is output in SI units. The vol-

ume-averaged fluid velocity hui /u is determined allowing thepermeability of the medium to be calculated using the Darcy law,

Eq. (1). Note that the choice of the working fluid properties is

somewhat arbitrary, since the permeability is independent of vis-

cosity, density and driving pressure gradient for creeping flow

(i.e. Reynolds number Re ( 1). This was confirmed numericallyby repeating a typical simulation multiple times for a range of

rp spanning several orders of magnitude, all at low Reynolds num-ber; the calculated permeability was indeed independent of pres-

sure gradient.

The fluid viscosity is related to the relaxation parameters of theSRT LBM:

m s1

2

c2sDt 10

The relaxation parameter represents the degree to which the fluid

populations are relaxed towards the equilibrium value during thecollision step (Eq. (8)). Consequently, the larger the value of the

relaxation parameter, the more rapid is the flow settlement. How-

ever, we should note the use of the halfway bounceback method

for the solid wall boundary condition with the SRT LBM, yields pre-

dictions of the flow field in porous media, and therefore of their per-

meability, that are dependent on the choice of the relaxation

parameter. Consequently, the accuracy of the simulation is depen-

dent on the relaxation parameter. Pan et al. [53] evaluated different

LB methods and solid wall boundary condition treatment methods

for flow simulations in porous media. They showed that results ob-

tained using the SRT LBM with the halfway bounceback method for

the solid boundaries are in good agreement with the results of the

MRT LBM with interpolated bounceback boundaries, and that the

effect of the relaxation parameter dependency is negligible for

s = 1. We set the relaxation parameter equal to 1 in all the1 Available from http://www.lbflow.co.uk.

862 A. Nabovati et al./ Composites: Part A 40 (2009) 860869
http://www.lbflow.co.uk/http://www.lbflow.co.uk/

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simulations performed. This value represents the best compromise

between simulation accuracy and the rate of flow settlement.

3. Results and discussion

3.1. Validation

To validate our methodology, we simulate creeping fluid flow in

a hexagonal array of infinite, parallel cylinders. The radius of the

cylinders was kept constant and equal to 40 lattice units in a do-

main of l ffiffiffi

3p

l 1, where l was varied between 82 and 230 latticeunits; this yields a medium of porosity between 0.15 and 0.89.

Fig. 1 depicts the schematic of the computational domain unit cell

with periodic boundary conditions on both sides; the gray colour

identifies the simulation domain. Due to the invariance of the do-

main geometry under translation in thez-direction, a fully 3D sim-

ulation can be achieved with a domain 1 lattice unit thick in that

direction by using periodic boundary conditions.

The results of these simulations are presented in Fig. 2, which

compares the permeability, calculated according to the methodol-

ogy presented in Section 2, with the value determined analytically

using the relationship for this geometry presented in [24] (Eq. (2)).

This model, described in Section 1.1, has been widely used for flow

in regular arrays of fibres [54,55]. The agreement over the range of

porosities for which that relationship is valid, i.e. 0.10 < / < 0.65, is

excellent. It is noteworthy that the fit between the data and the

analytical solution is good even beyond the upper limit of validity

claimed by Gebart (dotted line in Fig. 2).

3.2. Flow in fibrous media

To create the fibrous medium structure, randomly oriented,

straight, cylindrical fibres of constant diameter are randomly

placed in a cubic domain with free overlapping. By allowing the fi-

bres to overlap freely, we are able to investigate flow in fibrous

media with porosities across the full range, right down to the per-

colation threshold. Media in which the fibres are not able to over-

lap have a minimum porosity, which is higher than the percolation

threshold. Koponen et al. [31] studied the fluid flow in three

dimensional fibre webs, where flexible fibres were placed ran-

domly in the computational domain without overlapping, the min-

imum reported porosity was higher than 0.4. Nabovati and Sousa

[56] investigated the permeability of sphere packs with and with-

out free overlapping. They found that overlapping has a negligible

impact on permeability for media with porosities higher than 0.85

and leads to a decrease of less than 35% in permeability for low

porosity media. The minimum reported porosity for random pack-

ing of spheres without overlapping was 0.55.

The computational domain comprises a cube of 128 lattice sites

on each side and the periodic boundary condition was applied to

all faces of the cube. The fibres extend to the boundaries of the do-

main. The fibres are placed in the computational domain by using

the following algorithm: (1) a random position vector is chosen

within the computational domain, or one fibre radius of it, as the

origin of the fibres core, (2) a random vector representing the spa-

tial orientation of the fibre core is generated, (3) the fibre core line

is extended from the origin point along the randomly determined

orientation in both directions until it intersects the domain bound-

aries, and (4) lattice sites that are closer to the core line than the

radius of the fibre are designated as solid. The radius of the fibres

is constant along their length and equal to 2, 3, 4, 5 or 6 lattice

units, Dx, depending on the experimental run. Fig. 3 shows a high

porosity sample of the reconstructed fibrous medium; the porosity

of the sample is 0.80.

Fig. 1. Schematic of two dimensional section through the computational unit cell

used for the permeability prediction in hexagonal arrangement of cylinders; flow

domain with periodic boundary conditions on both sides is shown in gray colour.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.910

-6

10-4

10-2

100

102

Porosity ()

Present Work

Gebart [24]

K/a2

Fig. 2. Normalized permeability, calculated using the methodology presented in Section 2, is plotted with the analytically-determined relationship (Eq. (2)) presented byGebart in [24] for a hexagonal arrangement of solid cylinders. Gebarts relationship is shown dotted outside the range of values of / for which Gebart claims validity.

Fig. 3. Reconstructed medium with straight fibres and porosity/ = 0.80; the radius

of the fibres is four lattice units and their length is such that they span the

computational domain.


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A pressure gradient is applied in thex-, y- orz-direction and the

velocity field is calculated in the pores of the reconstructed med-

ium. Fig. 4 shows the velocity vectors in a slice of a three-dimen-

sional fibrous medium which has a porosity equal to 0.2; the

pressure gradient, and the mean flow direction, are in the positive

x-direction. We should note that, whilst some pores appear to be

dead-end or isolated in the two-dimensional slice, in the third

dimension, these pores are effectively connected to each other

and they contribute to the fluid flow pattern. As a result, the per-

colation threshold reported to be equal to 0.33 for two-dimen-

sional media made up of random arrangements of squareobstacles [57], in fact, takes a lower value for three-dimensional

media. In the present study, permeating pathways for fluid flow

exist for porosity as low as 0.08.

3.2.1. Effect of fibre radius

Permeability has dimensions of length squared, and it is usual

to normalize permeability by the square of a length scale that is

characteristic of the system; for fibrous media, this is typically

the fibre radius. To validate this approach, we determine the per-

meability of random networks of fibres with radii of 2, 3, 4, 5

and 6 lattice units. Results are shown in Fig. 5. Fig 5 demonstrates:

(1) permeability increases as porosity increases for constant fibre

radius; (2) permeability increases as fibre radius increases for con-

stant porosity. Fig. 5b demonstrates that normalizing the perme-ability by the square of the fibre radius causes the curves for

different fibre radius to collapse onto a single curve, hence, that

this non-dimensionalization is appropriate. Note that the normal-

ized permeability is not completely independent of fibre radius;

the two show an inverse relationship with the normalized perme-

ability for the thinnest fibre typically about 20% higher than for the

fattest on average. We postulate that this dependence is due to the

finite length of the fibres in our simulations giving rise to a variable

aspect ratio as the fibre radius is increased. This is supported by the

results presented later in Section 3.2.4.

3.2.2. Constitutive permeabilityporosity relationship for random fibre

networks

We determine the permeability of around 50 random fibre net-works in the x-, y- and z-direction (giving $150 data points in to-

tal). For each network, the fibres all have the same radius: either 2

or 4 lattice nodes. The permeability of the networks is in the range

0.08 < / < 0.99. Results are plotted in Fig. 6 (note the logarithmic

scale on the permeability axis). The normalized permeability is

approximately exponentially dependent on porosity in the range

0.2 < / < 0.9. At lower and higher porosities, the dependence is

stronger. As expected, the permeability tends towards infinity in

the limit /? 1 and drops towards zero at low (but finite) porosity.

We note that the variation in the data is greatest at very low poros-

ities, where random differences in the placement of the fibres be-

tween different domains lead to large relative changes in thepredicted permeability.

We find that a modified version of the Gebart [24] relationship

provides an excellent fit to data across the full range of porosity

(Fig. 6). We adapt Gebarts original relationship (Eq. (2)) by allow-

ing the three constants it contains to vary:

K

a2 C1


s 1

!C211

where /c is the critical value of porosity above which permeating

flow can occur (the percolation threshold). C1 and C2 are related

to the geometry of the network (compare with the values deter-

mined by Gebart for a regular array of fibres, presented in Section

1.1). We use the freely-available statistical analysis package SimFit2

to fit Eq. (11) to the data presented in Fig. 6. The fit is carried out in

logarithmic space to avoid biasing the fit towards large values of per-

meability at high porosity. Table 1 shows the best fit values obtained

(R2 = 0.999).

The values are consistent with those determined analytically by

Gebart for a hexagonal array of aligned fibres (Section 1.1); differ-

ences are due to the different geometries of the networks. The va-

lue of/c falls within the range calculated, on statistical grounds, by

Clague et al. [37], for a random network of fibres (Section 1.1).

The published relationships of Koponen et al. [31] and Clague

et al. [37] Eqs. (3) and (4), respectively in Section 1.1 were sim-

ilarly adapted and fitted to the data in Fig. 6. The Koponen relation-

ship provided a very poor best fit to data, and was abandoned. The

Fig. 4. Velocity vectors in a slice of the three-dimensional fibrous medium; the porosity of the sample is 0.2 and the pressure gradient drives flow in the positive x direction.

2 Available from: http://www.simfit.man.ac.uk.

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best fit of the Clague relationship was as good as the Gebart fit. We

favour the Gebart relationship because it has a sound theoretical

basis, whereas the Clague relationship is purely phenomenological.

To assess the variation of the predicted permeability values due

to the random nature of the reconstructed fibrous porous media,

we performed multiple repeated simulations for media at three

different porosities: $0.1, $0.5 and $0.95, respectively. Due tothe discrete nature of the fibre placement procedure, it is not pos-

sible to create different random media with exactly the same

porosity; hence, there is a variation of around 0.5% for each pre-

scribed porosity. For each porosity, we created 21 media with dif-

ferent random fibre placement and simulated flow in x, y, and z

directions, resulting in 63 permeability determinations for each

porosity. Each dataset is separately plotted against porosity inFig. 7; the permeability predicted using Eq. (11), with the parame-

ter values presented in Table 1, is shown on each plot as a solid

line. Table 2 shows the mean permeability for each dataset and

the permeability normalized by the value predicted using Eq.

(11). The standard deviation is also presented, expressed as a per-

centage of the mean permeability.

From Fig. 7 and Table 2, it can be seen that both the extent of

the variation in permeability and the quality of fit of Eq. (11) de-

pend on the porosity. The standard deviation is small (around

10% of the mean) for mid and high-range porosity; at very lowporosity, the standard deviation is larger (around 50% for /

% 0.1), reflecting the large changes in permeability that arise fromsmall structural differences near the percolation threshold. Eq. (11)

has a tendency to slightly under-predict permeability at lowporos-

ity and to over-predict at high porosity. Given the$6 order of mag-nitude difference in permeability between / % 0.1 and / % 0.95,however, we consider Eq. (11) to provide an accurate and flexible

tool for permeability prediction across the porosity range and note

that the model value is within one standard deviation of the

numerical data across the porosity range.

The literature contains permeabilityporosity data from a num-

ber of laboratory investigations into fibrous media; Jackson and

James [23] provide a summary. In Fig. 8, we compare our perme-

abilityporosity relationship (Eq. (11)) with experimental datafor high porosity fibrous media with randomly oriented straight

0.2 0.4 0.6 0.8 110

-10

10-9

10

-8

10-7

10-6

10-5

Porosity ()

K

(m2)

a = 2 lu

3

4

5

6

0.2 0.4 0.6 0.8 110

-3

10-2

10-1

100

101

Porosity ()

K/a2

(b)(a)

Fig. 5. (a) Numerically-determined permeability as a function of theporosity for fibres with a range of radii. Within each simulation suite, all fibres have thesame radius; (b)

numerically-determined permeability normalized by fibre radius as a function of the porosity.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

-6

10-4

10-2

100

102

K/a2

Numerically Predicted Values

Proposed Relation, Eq. 11

Porosity ()

Fig. 6. Dimensionless permeabilityas a function of porosity for randomnetworks of straight fibres. Fibre radius is 2 or 4 lattice units in allcases. Thesolid line shows thefit of

our semi-empirical relationship (Eq. (11)) which is adapted from Gebart [24]. Best fit parameters are given in Table 1. The fit (R2 = 0.999) indicates a percolation threshold of

/c% 7.4%.

Table 1

Best fit parameters of Eq. (11) to data presented in Fig. 6; regression coefficient

R2 = 0.999. See main text for details of fitting procedure.

Parameter Best fit value

/c 0.0743

C1 0.491

C2 2.31


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cylindrical fibres of constant diameter presented in that work. The

data were obtained using a range of experimental methods, a

broad spectrum of fibrous materials including filter pads, nylon fi-

bres, Kapron fibres, collagen, metal fibres and polymer fibres, and

various working fluids including water, glycerol and air. Despite

the diversity of these investigations, there is a broad agreement

(within an order of magnitude) among the resulting datasets. Over-

all the agreement between Eq. (11) and the experimental data is

good, especially for porosities higher than approximately 0.75.

3.2.3. Effect of fibre curvature

We investigate the effect of fibre curvature on the permeability

of fibrous media, by replacing straight fibres with randomly curved

fibres as the constituting elements of the medium. We generate the

fibres by constructing a cylinder of constant diameter around a

randomly curved fibre core. The origin of the fibres is chosen ran-

domly. Three different third-order polynomials represent the three

coordinates defining the orientation of the fibre core and they are a

function of a single variable, t:

xi ai;3t3 ai;2t

2 ai;1t ai;0 i 1; 2; 3 12

where the coefficients of these polynomials are chosen randomly in

the range of [1,1], this ensures that fibres are smoothly curved and

avoids tight spirals. The variable tis incremented and decrementedin appropriate steps to extend the fibre in both directions until the

fibre reaches the domain boundaries, producing a smoothly curved

fibre, which crosses the domain.

Fig. 9, shows the permeability we determine from simulations

of flow in media composed of curved fibres. The results are almost

indistinguishable from the results for straight fibres; and the effect

of the fibres curvature on the overall permeability of the medium

can be considered negligible.

3.2.4. Effect of fibre aspect ratio

We investigate the effect of fibre aspect ratio a (length to diam-eter ratio of straight cylindrical fibres of finite length) on perme-

ability for a range of aspect ratios between 1 and 20, for twodifferent values of porosity, namely 0.51 and 0.73. The fibres diam-

4.0

5.0

6.0

7.0

8.0

9.0

0.946 0.947 0.948 0.949 0.950

Porosity ( )

K/a2

0.030

0.035

0.040

0.045

0.050

0.055

0.060

0.497 0.498 0.499 0.500

Porosity ( )

K/a2

(a) (b)

0.0E+00

2.0E-05

4.0E-05

6.0E-05

8.0E-05

1.0E-04

1.2E-04

1.4E-04

0.0995 0.0996 0.0997 0.0998 0.0999 0.1

Porosity ( )

K/a2

(c)

Fig. 7. Dimensionless permeability for three different porosities: (a) $0.95, (b) $0.5, and (c) $0.1. At each porosity, the permeability was determined for 21 different randomnetworks of straight fibres to determine the random variation in permeability. The solid line represents Eq. (11) with the parameters given in Table 1. Note the dramatically

expanded scales compared with Fig. 6. See Section 3.2.2 for discussion.

10-3

10-2

10-1

100

10-4

10-3

10-2

10-1

100

101

102

103

Solid Fraction (1-)

K/a2

Chen (1995)

Ingmanson et al. (1959)

Kirsch & Fuchs(1967)

Stenzel et al. (1971)

Kostornov & Shevchuck (1977)

Jackson and James (1982)

Proposed relation, Eq. 11

Fig. 8. Comparison of dimensionless permeability of fibrous media, calculated

using Eq. (11) with the parameters given in Table 1, with the experimental data

reported by Jackson and James [23]. Readers are directed to that work for full

references to the original experimental studies.

Table 2

Mean permeability and standard deviation for three different porosities: $0.1, $0.5,

and $0.95. See Fig. 7.

Porosity Mean permeability (K/a2) Mean permeability

(normalized to Eq. (11))

Standard

deviation (%)

$0.1 2.58

105 2.09 52

$0.5 4.62 102 1.01 10$0.95 7.50 0.877 11


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eter is constant and equal to 6 lattice units and the aspect ratio is

varied by changing the fibres length. The numerically-determined

values of the permeability are shown in Fig. 10 as a function of the

fibre aspect ratio; the permeability values are normalized by the

permeability calculated using Eq. (11). It can be seen that perme-

ability increases with aspect ratio for a < 6; for a > 6, permeabilityis largely independent of aspect ratio. This finding explains the

imperfect collapse of the data for fibres of varying radius, pre-

sented in Fig. 2, when normalized by the fibre radius. Since the

length of the fibres is finite limited by intersection with the do-

main boundaries the aspect ratio of the fibres in that suite of sim-

ulations varies somewhat with fibre radius.

For a cylinder of diameter d, the ratio of the surface area to the

volume of the fibre (specific surface area) is equal to 1d4 1a. An

increase in aspect ratio therefore yields a decrease of the specific

surface area of the fibres, leading to a reduction in the frictional

drag force and higher values for the permeability.

3.2.5. Permeability tensor

Any experimental or numerical attempt to determine the per-meability tensor and the principle directions of a porous medium

should be performed in three dimensions. This is an expensive

and time consuming task. Neuman [58] proved analytically that

the permeability tensor is a symmetric second order tensor, which

has six distinct elements in general. For isotropic materials, the

three diagonal elements are equal and non-zero and all off-diago-

nal elements are zero. In this case, the permeability can be repre-

sented as a single, scalar value.

Ahn et al. [59], Weitzenbck et al. [60,61] and Parnas et al. [62]

discuss experimental methods and approaches for permeability

tensor measurements. Kolodziej et al. [63] analytically investigated

the permeability tensor of high porosity fibrous porous media. In

their study, fibres had a unidirectional arrangement with non-uni-

form spacing. Nedanov and Advani [64] studied fluid flow in two-

scale fibrous porous media numerically; governing equations of

the fluid flow around and inside the solid and permeable fibres

were developed using the homogenization method and were

solved numerically. They report the diagonal elements of the per-

meability tensor for single and three-ply fabrics. Song et al. [35]calculated the permeability tensor of a three-dimensional, woven

fibrous medium using the finite volume method.

In this study, to evaluate the viability of the LBM for permeabil-

ity tensor determination, the permeability tensor elements of three

samples of different porosities of 0.90, 0.70 and 0.50, respectively,

are determinedbased on the simulated flowfield. For eachporosity,

a pressure gradient is applied in one of the three principle direc-

tions. The mean flow is in the direction of the applied pressure gra-

dient, which builds the diagonal element of the permeability tensor

in the specified direction. We calculated the off-diagonal elements

of the permeability tensor using the net flow in each of the other

two directions, and repeated this procedure for each of the three

directions to obtain the nine elements of the permeability tensor.

The predicted permeability tensors for these media are as follows:

K0:90

1:0866 0:0073 0:0203

0:0073 1:0526 0:0138

0:0206 0:0138 1:0820

264

375 107 m2

K0:70

1:1643 0:0262 0:0021

0:0262 1:1726 0:0006

0:0021 0:0006 1:1632

264

375 108 m2

K0:50

1:8851 0:0478 0:0395

0:0478 1:9017 0:0029

0:0395 0:0029 1:8484

264

375 109 m2

As the media are created with randomly-oriented fibres, there is no

preferential flow direction and the media are expected to be isotro-

pic. In each case, we find that the diagonal elements of the perme-

ability tensor are the same to within $3%; this small difference caneasily be accounted for by the random nature of the network. Fur-

thermore, the off-diagonal elements are approximately symmetri-

cal, and are smaller than the diagonal elements by around two

orders of magnitude. These results support the analysis of Neuman

[58] and demonstrate that it is valid to report a scalar value for per-

meability of these random porous media, and a determination of

the full second order tensor is unnecessary.

4. Conclusion

Three-dimensional fluid flow simulations in fibrous media are

conducted using the SRT LBM; the fibrous media are recon-structed by random placement of cylindrical fibres, with random

0 5 10 15 200.5

0.6

0.7

0.8

0.9

1

Fiber Aspect Ratio

K/Kmodel(Eq.

11)

Phi = 0.505

Phi = 0.730

Fig. 10. Effect of the fibre aspect ratio on the numerically-determined permeability

for two different porosities, 0.51 and 0.73, respectively; permeability values are

normalized by the permeability calculated using Eq. (11) with theparameters givenin Table 1.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

-4

10-2

100

102

Straight Fibres - Equation 11

Curved Fibres

K/a2

Porosity ()

Fig. 9. Numerically-determined permeabilityof fibrous media as a function of the porosity for media formed of straight (solidline, Eq. (11) with parameters given in Table 1)

and randomly curved fibres (circles, simulation results).


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orientations, within the computational domain. The radius, curva-

ture and length of the fibres are varied systematically. We find that

dividing the permeability by the square of the fibre radius yields an

appropriate non-dimensionalization. We also find that fibre curva-

ture has a negligible impact on the permeability of the medium.

For fibres of finite length with aspect ratios smaller than $6,permeability increases with increasing aspect ratio; this effect is

negligible for values of the aspect ratio greater than 6.

The permeability values that we obtain are compatible with the

available experimental data and, based on the determined values,

we develop a relationship (Eq. (11)) for the permeability as a

function of the materials porosity and the fibre diameter. The

relationship that we present results from a semi-empirical

parameterization of a published analytical relationship for ordered

arrays of fibres [24]. The fit of the relationship to the data is excel-

lent across the range of porosities investigated (0.08 < / < 0.99).

Our analysis shows that the percolation threshold for a three-

dimensional network of randomly oriented fibres is /c = 0.0743 .

The media in this study were created randomly, hence there

was no preferential direction for the flow through the medium;

therefore, the media can be assumed isotropic. Prediction of the

permeability tensor for media of three different porosities sup-

ported this assumption, as the diagonal elements of the numeri-

cally-determined permeability tensors differ by less than $3%,and the off-diagonal elements were two orders of magnitude smal-

ler than the average value of the diagonal elements. This finding

indicates that it is valid to report a scalar value, instead of a second

order tensor, for the permeability of the fibrous media studied.

The results obtained in this study and the general relationship

proposed for the permeability, can be fed to the macroscopic flow

modelling approaches for the industrial applications, e.g. resin

transfer moulding process, where the pore level approach is not

applicable due to the large computational resources requirement.

Acknowledgment

The authors acknowledge the support received from NSERC(Natural Sciences and Engineering Research Council of Canada)

Discovery Grant 12875 (ACMS) and from the Foundation for Sci-

ence and Technology (FCT, Portugal) through the research grant

POCTI/EME/59728/2004 (ACMS). EWL is supported by NERC (UK)

Research Fellowship NE/D009758/2. We thank two anonymous

reviewers for their helpful comments.

References

[1] Phelan Jr FR. Simulation of the injection process in resin transfer molding.

Polym Compos 1997;18(4):46076.

[2] Abrate S. Resin flow in fiber preforms. Appl Mech Rev 2002;55(6):57999.

[3] Khanafer K, Vafai K. The role of porous media in biomedical engineering as

related to magnetic resonance imaging and drug delivery. Heat Mass Transfer

2006;42(10):93953.

[4] Vafai K, Ai L. A coupling model for macromolecule transport in a stenosed

arterial wall. Int J Heat Mass Transfer 2006;49(910):156891.

[5] Yang N, Vafai K. Modeling of low-density lipoprotein (LDL) transport in the

artery-effects of hypertension. Int J Heat Mass Transfer 2006;49(56):85067.

[6] Pan C, Hilpert M, Miller CT. Pore-scale modeling of saturated permeabilities in

random sphere packings. Phys Rev E 2001;64(6):066702-1.

[7] Pan C. Use of pore-scale modeling to understand transport phenomena in

porous media. PhD dissertation in environmental science and engineering.

University of North Carolina at Chapel Hill; 2003.

[8] Lu WM, Tung KL, Hwang KJ. Fluid flow through basic weaves of monofilament

filter cloth. Textile Res J 1996;66(5):31123.

[9] Tung KL, Shiau J, Chuang C. CFD analysis on fluid flow through multifilament

woven filter cloths. Separat Sci Technol 2002;37(4):799821.

[10] Koido T, Furusawa T, Moriyama K. An approach to modeling two-phase

transport in the gas diffusion layer of a proton exchange membrane fuel cell. J

Power Sources 2008;175(1):12736.

[11] Djilali N. Computational modelling of polymer electrolyte membrane (PEM)

fuel cells: challenges and opportunities. Energy 2007;32(4):26980.

[12] Wang L, Afsharpoya B. Modeling fluid flow in fuel cells using the lattice-Boltzmann approach. Math Comput Simul 2006;72(2-6):2428.

[13] Vafai K. Handbook of porous media. Taylor & Francis; 2005.

[14] Vafai K, Tien CL. Boundary and inertia effects on flow and heat transfer in

porous media. Int J Heat Mass Transfer 1981;24(2):195203.

[15] Bear J, Bachmat Y. Introductionto modeling of transport phenomena in porous

media. Dordrecht: Kluwer Academic Publishers; 1990.

[16] Hazlett RD. Simulation of capillary-dominated displacements in

microtomographic images of reservoir rocks. Transport Porous Med

1995;20(12):2135.

[17] Kerckhofs G, Schrooten J, Van Cleynenbreugel T. Validation of X-ray

microfocus computed tomography as an imaging tool for porous structures.

Rev Sci Instrum 2008;79(1):013711.[18] Larson RE, Higdon JJL. Microscopic flow near the surface of two-dimensional

porous media. Part 1: axial Flow. J Fluid Mech 1986;166:44972.

[19] Larson RE, Higdon JJL. Microscopic flow near the surface of two-dimensional

porous media. Part 2: transverse flow. J Fluid Mech 1987;178:11936.

[20] Sangani AS, Acrivos A. Slow flow through a periodic array of spheres. Int J

Multiphase Flow 1982;8(4):34360.

[21] Sangani AS, Acrivos A. Slow flow past periodic arrays of cylinders with

application to heat transfer. Int J Multiphase Flow 1982;8(3):193206.

[22] Drummond JE, Tahir MI. Laminar viscous flow through regular arrays of

parallel solid cylinders. Int J Multiphase Flow 1984;10(5):51540.

[23] Jackson GW, James DF. Permeability of fibrous porous media. Can J Chem Eng

1986;64(3):36474.

[24] Gebart BR. Permeability of unidirectional reinforcements for RTM. J Compos

Mater 1992;26(8):110033.

[25] Nabovati A, Sousa ACM. Fluid flow simulation at open-porous medium

interface using the lattice Boltzmann method. Int J Numer Method Fluids

2008;56(8):144956.

[26] Van Der Hoef MA, Beetstra R, Kuipers JAM. Lattice-Boltzmann simulations

of low-Reynolds-number flow past mono- and bidisperse arrays of

spheres: results for the permeability and drag force. J Fluid Mech

2005;528:23354.

[27] Yang A, Miller CT, Turcoliver LD. Simulation of correlated and uncorrelated

packing of random size spheres. Phys Rev E 1996;53(2):1516.

[28] Martys NS, Torquato S, Bentz DP. Universal scaling of fluid permeability for

sphere packings. Phys Rev E 1994;50(1):4038.

[29] Yang J, Jia Y, Sun S. Mesoscopic simulation of the impregnating process of

unidirectional fibrous preform in resin transfer molding. Mater Sci Eng A

2006:51520.

[30] Shojaei A, Ghaffarian SR, Karimian SMH. Modeling and simulation approaches

in the resin transfer molding process: a review. Polym Compos 2003;24(4):

52544.

[31] Koponen A, Kandhai D, Hellen E. Permeability of three-dimensional random

fiber webs. Phys Rev Lett 1998;80(4):716.

[32] Koponen A. Simulations of fluid flow in porous media by lattice gas and lattice

Boltzmann methods. PhD dissertation, Jyvaskyla, Department of Physics:

University of Jyvaskyla; 1998.

[33] Wang Q, Maze B, Tafreshi HV. A note on permeability simulation ofmultifilament woven fabrics. Chem Eng Sci 2006;61(24):80858.

[34] Wang Q, Maze B, Tafreshi HV. On the pressure drop modeling of

monofilament-woven fabrics. Chem Eng Sci 2007;62(17):481721.

[35] Song YS, Chung K, Kang TJ. Prediction of permeability tensor for three

dimensional circular braided preform by applying a finite volume method to a

unit cell. Compos Sci Technol 2004;64(1011):6291636.

[36] Song YS, Youn JR. Asymptotic expansion homogenization of permeability

tensor for plain woven fabrics. Compos A: Appl Sci Manuf

2006;37(11):20807.

[37] Clague DS, Kandhai BD, Zhang R. Hydraulic permeability of (un)bounded

fibrous media using the lattice Boltzmann method. Phys Rev E 2000;61(1):

61625.

[38] Jiang F, Sousa ACM. Smoothed particle hydrodynamics modeling of transverse

flow in randomly aligned fibrous porous media. Transport Porous Med

2008;75(1):1733.

[39] Jiang FM, Oliveira MSA, Sousa ACM. Mesoscale SPH modeling of fluid flow in

isotropic porous media. Comput Phys Commun 2007;176(7):47180.

[40] Frisch U, Hasslacher B, Pomeau Y. Lattice-gas automata for the NavierStokes

equation. Phys Rev Lett 1986;56(14):15058.[41] Krafczyk M, Schulz M, Rank E. Lattice-gas simulations of two-phase flow in

porous media. Commun Numer Methods Eng 1998;14(8):70917.

[42] Qian YH, DHumieres D, Lallemand P. Lattice BGK models for NavierStokes

equation. Europhys Lett 1992;17(6):47984.

[43] Chen S, Doolen GD. Lattice Boltzmann method for fluid flows. Ann Rev Fluid

Mech 1998;30:32964.

[44] Succi S. The lattice Boltzmann equation for fluid dynamics and

beyond. Oxford: Oxford University Press; 2001.

[45] McNamara GR, Zanetti G. Use of the Boltzmann equation to simulate lattice-

gas automata. Phys Rev Lett 1988;61(20):23325.

[46] Higuera FJ, Succi S, Benzi R. Lattice gas dynamics with enhanced collisions.

Europhys Lett 1989;9(4):3459.

[47] Higuera FJ, Jimenez J. Boltzmann approach to lattice gas simulations. Europhys

Lett 1989;9(7):6638.

[48] Succi S, Foti E, Higuera F. Three-dimensional flows in complex geometries with

the lattice Boltzmann method. Europhys Lett 1989;10(5):4338.

[49] Bernsdorf J, Durst F, Schfer M. Comparison of cellular automata and finite

volume techniques for simulation of incompressible flows in complexgeometries. Int J Numerical Method Fluids 1999;29(3):25164.


8/6/2019 Nab Ova Ti 09

10/10

[50] Breuer M, Bernsdorf J, Zeiser T. Accurate computations of thelaminar flowpast

a square cylinder based on two different methods: lattice-Boltzmann and

finite-volume. Int J Heat Fluid Flow 2000;21(2):18696.

[51] Llewellin EW. LB flow: an extensible lattice Boltzmann framework for the

simulation of geophysical flows. Part I: theory and implementation. Comput

Geosci, accepted for publication.

[52] Llewellin EW. LBflow: an extensible lattice-Boltzmann framework for the

simulation of geophysical flows. Part II: usage and validation. Comput Geosci,

accepted for publication.

[53] Pan C, Luo L-S, Miller CT. An evaluation of lattice Boltzmann schemes

for porous medium flow simulation. Comput Fluids 2006;35(89):898909.

[54] Wang JF, Hwang WR. Permeability prediction of fibrous porous media in a bi-

periodic domain. J Compos Mater 2008;42(9):90929.

[55] Grujicic M, Chittajallu KM, Walsh S. Lattice Boltzmann method based

computation of the permeability of the orthogonal plain-weave fabric

preforms. J Mater Sci 2006;41(23):79898000.

[56] Nabovati A, Sousa ACM. LBM mesoscale modeling of porous media. Int Rev

Mech Eng 2008;2(4).

[57] Koponen A, Kataja M, Timonen J. Permeability and effective porosity of porous

media. Phys Rev E 1997;56(3):331925.

[58] Neuman SP. Theoretical derivation of Darcys law. Acta Mech 1977;25(34):

15370.

[59] Ahn SH, Lee WI, Springer GS. Measurement of the three-dimensional

permeability of fiber preforms using embedded fiber optic sensors. J Compos

Mater 1995;29(6):71433.

[60] Weitzenbock JR, Shenoi RA, Wilson PA. Measurement of three-dimensional

permeability. Compos Part A: Appl Sci Manuf 1998;29(12):15969.

[61] Weitzenbock JR, Shenoi RA, Wilson PA. A unified approach to determine

principal permeability of fibrous porous media. Polym Compos2002;23(6):113250.

[62] Parnas RS, Liu Q, Giffard HS. New Set-Up for in-plane permeability

measurement. Compos Part A (Appl Sci Manuf) 2007;38(3):95462.

[63] Kolodziej JA, Dziecielak R, Konczak Z. Permeability tensor for heterogeneous

porous medium of fibre type. Transport Porous Med 1998;32(1):119.

[64] Nedanov PB, Advani SG. Numerical computation of the fiber preform

permeability tensor by the homogenization method. Polym Compos

2002;23(5):75870.


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