1

Permeability Estimation through Micro-Macro Coupling in

Particulate and Fibrous Porous Media

Bamdad Barari, Saman Beyhaghi and Krishna M. Pillai

Laboratory for Flow and Transport Studies in Porous Media

Department of Mechanical Engineering, University of Wisconsin-Milwaukee

1. Abstract

The closure formulation, developed as a part of the derivation of Darcys law proposed by

Whitaker [1], is used to develop a novel numerical method for estimating the permeability of a

porous medium with a given pore-level microstructure. The permeability of two distinct porous

media created from cellulose nano-fibers (CNF) and sintered polymer beads were then estimated

numerically. In order to use real micrograph in such simulations, 2D SEM pictures of the CNF

and polymer-wick porous media were considered. The falling head permeameter was used for

measuring the experimental permeability in order to test the accuracy of the permeability results

obtained by numerical simulation. The Permeability values were also compared with the

theoretical models of Kozeny-Carman and Rumpf-Gupte, and a good agreement was observed. A

good agreement between the numerical permeability results and the results obtained from the

experimental/theoretical means confirm the accuracy and usefulness of the proposed method in

estimating in this crucial property through an effective micro-macro coupling.

Keyword: Porous media, permeability, closure formulation, cellulose nano-fiber, polymer wick

2

2. Introduction

The problem of flow and transport of fluids in porous media has been studied extensively in the

last hundred years. In the last few decades, the volume averaging method has emerged as a

credible method to develop the macroscopic governing equations for modeling fluid flow in

porous media [1]. Using these equations, one can predict the volume-averaged velocity, pressure,

liquid density, and species concentrations inside any porous medium [2-5].

The basic idea in volume averaging is that the equations for flow and transport in porous

media can be extracted using spatial averaging with the help of representative elementary

volume or REV [6-8]. Efforts towards developing the volume averaging method were made by

several researchers during last few decades. Whitaker provided perhaps the most comprehensive

derivation of volume averaging theorems based on geometric ideas [9, 10]. Later, Whitaker [1]

outlined in detail the volume averaging method and development of the closure formulations for

Darcys law and mass transport in single- and multi-scale porous media. Gray et al. used a

generalized function to formulate and prove the volume averaging theorems [11]. The volume

averaging method as a powerful technique for upscaling was subsequently used by several

authors. Hassanizadeh and Gray [12] applied the volume averaging theorem for developing

continuum models for multiphase flows in porous media. Bennethum and Cushman [13] used the

multiscale hybrid mixture theory along with the volume averaging method to develop continuum

equations for swelling porous media. In general, the size of the REV must be much greater than

the nominal size of the solid particles and the characteristic inter-particle distances. Further

studies on upscaling was done by Gray and Miller [14] where averaging of quantities from

micro- to macro-scale using a thermodynamically constrained approach was conducted.

3

A great advantage of the volume average method is the development of an effective micro-macro

coupling, i.e., the coefficients of the macroscopic equations are derived from solving closure

formulations in unit cells of real porous media, thus incorporating the effects of pore-scale

geometry in the overall upscaled equations. As a result, physical properties such as tortuosity,

dispersion, and permeability are estimated after including the features of the microscopic flow,

transport and geometry. This approach was used to propose single- and two-equation models for

conduction and convection in porous media [15, 16]. A theoretical derivation of the governing

equations for the single-phase flow in swelling porous media was investigated by Pillai [17]

where novel forms of the mass balance (continuity) and the momentum balance (Darcy's law)

were developed using the volume averaging method. Bear and Bachmat [18] comprehensively

presented the volume averaging equations for single-phase, multi-phase and multicomponent

systems using some unique and rather complicated proofs of the averaging theorems. Dormieux

et al. [19] estimated the tortuosity after using the homogenization method by assuming the

porous medium to be consisting of parallel constant-thickness horizontal and vertical channels.

In most studies on scaling up done till now using the closure formulation, a simple square

shaped unit-cell with circular solid phase is most often considered [20]. Kim et al. used different

aspect-ratio rectangular unit-cells with rectangular inclusions for this purpose as well [21]. Davit

et al. [22] worked on local non-equilibrium models for mass transport in dual-porosity, dual-

phase porous media where they observed that the volume-averaging-method formulation with

the closure and the method of matching spatial moments were equivalent in the one-equation

non-equilibrium case. Golfier et al. [23] investigated mass transport in a porous medium

containing bio-film to develop the Darcy-scale equations and the effective dispersion tensor

using a closure formulation was solved for both cases of simple and complex unit cells.

4

The preforms made from cellulose nano-fibers (CNFs) were one of the two porous materials

used in this study. CNFs are made purely from cellulose molecules and have very good

mechanical property compared to other natural fibers, and even carbon or glass fibers. Recent

studies establish that the films or "nano-paper" made from CNFs are the strongest man-made,

cellulosic materials [24]. Recently, CNFs have begun receiving serious consideration as potential

reinforcement materials. The present research springs from the efforts by the authors on using

CNF preforms for making CNF composites by infiltrating them with thermosetting polymers

[25]. CNF preforms used in this study were prepared according to the technique developed by

Saito et al. [26].

The other porous medium used in this paper is the polymer wick. These kinds of wicks,

made from sintering polymer beads, are often used for delivering active liquids such as

fragrances and insecticides from a reservoir to ambient air. Such wicks have been used

previously in several research efforts: for example, in the estimation of the tortuosity tensor

using closure formulation in Beyhaghi and Pillai [27], in the development of a new model for

wicking based on the single-phase Darcys law after using the capillary suction pressure at the

moving liquid-air interface in Masoodi et al. [28].

In the present study, the closure formulation, developed as a part of the proof for Darcys

law by Whitaker [1], is used to measure the permeability through the two above-mentioned

porous media with realistic, irregular pore structures. Scanning Electron Microscopy (SEM) is

used to take micrographs of the cross sectional areas of the CNF preform and polymer wick. The

SEM micrographs are then transformed to binary pictures (containing only two colors: one

representing the solid phase, and the other the void region) and then they are imported into

COMSOL multi-physics software. The governing equations for the closure formulation along

5

with the assigned boundary conditions are solved within a unit cell placed at the center of the

micrograph. The permeability tensor can then be obtained from integrating the field of mapping

variables inside the unit-cell. In order to establish the independence of permeability tensor from

the unit-cell size and consequently find a proper size of the unit cell for our permeability

calculations, four concentric square-shaped unit cells with different areas are chosen. For the

CNF sample, an additional step is done. After estimating the size of the optimum REV, the

permeability is calculated as an average of the permeability obtained for four different REV

locations in the CNF micrograph. Finally, for both the CNF and wick specimens, the

permeability obtained by the numerical simulation is compared with the experimental

permeability obtained using the falling head permeameter and the theoretical/empirical formulae

for permeability from literature such as the Kozeny-Carman relation.

NOMENCLATURE

B Closure or mapping variable for velocity fV Pore volume or fluid volume

b Closure or mapping variable for pressure V REV volume

D Velocity closure variable in terms of permeability x Height of liquid during experiment

d Pressure closure variable in terms of permeability Greek Symbols

f Arbitrary continuous function Arbitrary flow variable

g Acceleration due to gravity Gradient vector

I Unity tensor 2 Laplace operator

K Permeability tensor Fluid density

L Porous sample's thickness in experiment Fluid viscosity

il 1,2,3i Lattice vectors Porosity

n Unit vector normal to fluid-solid interface Subscripts p Pressure f Fluid phase

r Position vector fs fluid-solid

R Radius fe Fluid entrance-exit

Sfs fluid-solid interface area on REV B Burette

Sfe fluid entrance and exit area on REV P Porous

t Time 0 Initial

v Velocity

6

3. Theory

3.1. Permeability Estimation using Closure Formulation of Volume Averaging Method

The formulation developed by Whitaker [1], based on the volume averaging method, is used in

this theoretical model employed for estimating the porous-medium permeability. The flow

variables in the pore region of a representative elementary volume (REV) are averaged and the

values are used in the macroscopic flow-field. Two types of averaging can be defined for an

arbitrary flow variable as follow.

Phase Average:

1

fVdV

V (1)

Intrinsic Phase-Average (also known as Pore Average):

1

f

f

Vf

dVV

(2)

Here V represents the REV volume and fV represents pore volume (or fluid volume in case of

single-phase flow) within the REV of a porous medium. For low-Reynolds number flows, the

boundary value problem for solving the velocity and pressure fields associated with the Stokes

and continuity equations can be expressed as

20 f f fp g v (3)

. 0f v (4)

B.C.1 0f v on fsS (5)

B.C.2 ( , )f f tv r on feS (6)

7

where fsS represents the fluid-solid interface area while feS stands for the fluid entrance and

exit areas in the REV. Using the volume averaging method [1], one can apply the volume

averaging theorems to all terms of Eqns. (3) and (4) to obtain the macroscopic momentum

balance equation in the form of Darcys law. During this process of upscaling, the pressure and

velocity terms are decomposed in terms of their intrinsic phase averages, and

, and

corresponding perturbations, fp and fv , as

f

f f fp p p (7)

f

f f f v v v (8)

The use of Eqns. (7) and (8) during the volume averaging process results in the following

equation set to solve for pressure and velocity perturbations:

21

0 .

fs

f f f fs f f f

f S

p p dSV

v n I v (9a)

. 0f v (9b)

B.C.1 f

f f v v on fsS (9c)

B.C.2 ( , )f f tv r on feS (9d)

Note that Eqns. (9a) and (9b) result from the momentum and continuity equations. In order to

close the system of equations, fp and fv are defined in terms of the corresponding averaged

quantities as

8

.

.

f

f f f f

f

f f f

p

b v

v B v

(10)

On substituting Eqn. (10) in Eqn. (9a) and (9b), the governing equations for the closure

variables, and can be developed as

21

0 .

fs

f f f fs f f

f S

dSV

b n Ib (11a)

0f (11b)

The closure formulation allows the permeability tensor to be estimated in terms of the closure

variables as

11

.

fs

fs f f f

f S

dSV

n Ib K (12)

Use of this definition in Eqn. (11a) results in

2 10 f f f

b B K (13)

At this point, fb and fB can be replaced by fd and fD [1] through the transformations

1

1 0

0

.

.

f f f

f f f

f f

d b K

D B K

B B I

(14)

On using the expressions given by Eqn. (14) in Eqn. (13) transforms the pore level momentum

equation as

20 f f d D I (15)

9

and the pore level continuity equation can be stated as

0f D (16)

Note that the closure formulation described by Eqns. (15) and (16) is similar to the Stokes

boundary value problem that can be solved in an REV [1].

In Stokes flow, the pressure and velocity fields within the REV are controlled by

boundary conditions Eqns. (5) and (6) at the fluid-solid interface and at the entrance and exit

regions of REV. The first boundary condition on fluid-solid interface, Eqn. (9c), in terms of the

closure variables can be expressed as

B.C. 1 0f D on fsS (17)

The second boundary condition given by Eqn. (9d) is problematic since it entails the use of an

unknown function, f, to assign deviations at the entrance of the REV [1]. The problem was made

tractable by considering a periodic unit cell instead of a full-fledged REV. The corresponding

periodicity conditions for the velocity and pressure deviations can be described in terms of the

closure variables as

( ) ( )f i fl D r D r (18a)

( ) ( )f i fl d r d r (18b)

These periodicity conditions state that the opposite sides of a square-shaped unit cell have the

same values of the D and d components in the considered 2-D domain. Here r represents the

position vector of any point on the unit-cell boundary, while li are the lattice vectors with

{1, 2} that express the spatially periodic nature of the unit cell in the 2-D space. Enforcing

the velocity deviations to add to zero within the REV results in the average-value condition of

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the form

0f

f B (19)

This is consistent with definition of the spatial deviation as defined in the volume-averaging

formulation [1]. On using this result with the definition of permeability tensor stated in Eqn.

(12) and the definitions of the closure variables given in Eqn. (14), we can estimate the

permeability tensor in much simpler form as

f

f fK D (20)

The mapping variable fD is first estimated within the unit cell using the closure formulation

given by Eqns. (15) to (19), then fD is integrated in the pore region of the unit-cell to obtain K

through Eqn. (20).

3.2. Experimental method for estimating permeability

The falling-head permeameter is employed to measure the permeability of porous medium using

a liquid level at the inlet that is gradually decreasing with time due to the leakage of liquid

through the sample. As shown schematically in Fig. 1, the continuously decreasing head at the

inlet implies that the inlet pressure driving the flow is decreasing. As a result, the Darcy velocity

inside the porous sample is decreasing with time as well. If the initial height of water column in

the burette is 0x at time t = 0, and the height at each subsequent time t is x during the

experiment, then the permeability of the porous sample, K, can be estimated through the formula

2

0

2ln( )

p

p B

K gRxt

x L R

(21)

11

where pL and pR are the porous sample's length and radius, respectively; BR is the radius of the

burette; and are the liquid density and viscosity, respectively. If the ordered pairs of (x,t)

are recorded during the experiment and the 0ln( )x

xvs. t plot developed, the permeability can be

calculated from the slope of the plot.

Fig. 1. A Schematic of Falling-Head Permeameter

3.3. Theoretical/empirical model for permeability

As mentioned in many well-known texts on porous media, several permeability models (both

empirical and theoretical) have been developed for different types of porous media such as

particulate (soil-like or sand-like) media, fibrous media, etc. Dullein [29] categorized

permeability models into different types based on flow patterns and porous-media structure that

included particulate or fibrous beds, tube-flow models, network models, etc. Most of the

permeability formulas or models listed in literature are of the form

2 ( )f fK D (22)

x

x0

12

where ( )f is a function of the porosity and fD is the mean particle or fiber diameter. In

general, predictions from these correlations cannot match the measured permeability perfectly;

however, by choosing a model that fits the case, the obtained permeability values will be within

an order-of-magnitude of the measured values [30, 31]. In the present paper, the Cozeny-Carman

[29] and Rumpf- Gupte [29] empirical correlations are used to estimate permeability for the two

porous media considered, i.e., the polymer wick and the CNF porous sample. These relations for

the permeability are:

3

2

2

2 5.5

180(1 )

1, 1 0.35 0.7

5.6

f

f

f

f f f

K D Cozeny Carman

K D K for Rumpf GupteK

(23)

4. Results and discussion

4.1 Initial validation of Closure Formulation Method

In order to evaluate the accuracy of our proposed simulation technique for estimating the

permeability using COMSOL, results of a recent study by Lasseux et al. on a similar

determination of the permeability of an idealized porous medium were used [32]. A simple

square-shaped unit cell (with side of length l) with two different square-shaped solid particles in

the center (representing porosities of 0.5 and 0.75) was considered. The set of closure equations

defined in Eqns. (15) and (16), subject to the boundary conditions Eqns. (17) and (18), were then

solved within the void regions of the two unit cells. As shown in Table (1), there is a very good

agreement between the dimensionless permeability values for the unit cells predicted in [32] and

in the present study. This clearly establishes the accuracy of our method.

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Table (1): A comparison between the results of this study and those of Lasseux et al. in terms of

dimensionless permeability K* [32]

K* (= K/l2)

Porosity Lasseux et al., 2011 Present work

0.5 0.002386 0.002301

0.75 0.013023 0.013022

4.2 Permeability of porous medium made from CNF

Most of the earlier efforts of using the closure formulation were confined to regular idealized 2D

or 3D geometries that could be recreated by repeating a unit cell. Kim et. al. [21] considered a

regular 2D porous medium made of a unit cell consisting of rectangular particles, while Quintard

[33] studied a 3D periodic medium with rectangular objects within a unit cell. It was observed

that the 3D model was closer to the experiments as compared to the 2D effort. In spite of all

these previous investigations with idealized porous media, the closure formulation based

approach was never tested on real porous media with irregular, non-ideal pore space.

Here we will use the closure formulation to estimate the permeability of CNF sample that

appears to be isotropic from micrographic studies. As seen in the SEM pictures of CNF (Fig.2),

the random distribution of CNF solid phase indicates that the closure formulation cannot be

solved in a unit cell due to the lack of periodicity in the CNF solid-phase distribution. However,

Whitaker had discovered that the use of a unit cell is permissible for such purposes because the

error arising through the use of the periodic boundary conditions is confined to the borders of the

unit cell and is not significant [1]. Therefore, in order to solve the closure equations in our pore-

level geometry, SEM micrographs of different cross-section of the CNF sample were taken.

14

These cross-sections were selected in the vertical and horizontal planes, and because the CNF

sample was determined to be isotropic from the study of such micrographs, one of the

micrographs (Fig. 2) was selected for the purpose of solving the closure-formulation equations.

Fig. 2. SEM micrograph of a representative sectioned plane from the CNF sample

Using some sort of image processing software, it is possible to create a 3D geometry of pore-

space for our CNF sample by stacking different cross-sections from consecutive parallel planes

and interpolating between them. However, in the present study, a 2D micrograph selected from

inside of the CNF sample is used as a porous-medium geometry to solve COMSOL in. In order

to use the geometry for computations, we assumed that brighter color fibers were closer to the

top surface of micrograph while darker ones belong to the lower levels. Then if a cross flow were

to occur, the brighter color surfaces will act as obstacles (particles) while the darker surfaces can

be treated as pores. Using this assumption, the SEM micrograph of the CNF sample can be

converted into a binary picture for use with COMSOL for further analysis. The 2-D geometry

extracted from the CNF micrograph is shown in Fig. 3.

15

(a) (b)

Fig. 3. (a) The CNF micrograph used in our permeability calculation. (b) The corresponding

solid (white) and pore (dark) regions of a reconstructed 2-D binary region extracted from the

micrograph, which is used to define the domain geometry for COMSOL.

Note that for 2-D samples considered in the present study, the permeability tensor can be

obtained through Eqn. (20) as

11 12

21 22

f f

f f f

D D

D D

K (24)

where each of the bracketed f

ijD terms represents the intrinsic phase-average value of that

particular Dij element, obtained after integrating it in the entire void region of the reconstructed

binary micrograph. Here f represents the porosity that is obtained by dividing the total void

area with the total area of the micrograph. Different elements of the permeability tensor, Eqn. 24,

were estimated with COMSOL after the governing equations, Eqns. 15 to 19, were solved within

the void region of the domain. After all these calculations, the final form of the permeability

tensor for our sample CNF is obtained as

11 2 61 0 23 0 38 0 0710

0 21 0 05 3 28 0 11

. . . .

. . . .

K (m2) (25)

16

The closeness of the diagonal terms in this matrix form of the permeability tensor for this 2D

medium indicates that our CNF sample can be characterized as being almost isotropic. It is also

interesting to note the presence of non-zero diagonal terms that are fairly big and are on the order

of 10% of the main diagonal terms. This clearly indicates that the x (horizontal) and y (vertical)

coordinates considered in Fig. 3(b) are not the principal or material directions of the porous

medium. (If the x and y directions match the principal directions, the permeability-tensor matrix

acquires a pure diagonal form.) One may also note that the matrix is not symmetrical, which

contrasts with the almost universal observation that the permeability tensor is symmetric for a

majority of porous media. At this moment, we do not have any explanation for this divergence

from the conventional wisdom we merely state our results as it is. (Though there is some

anecdotal evidence from co-workers in this area, which suggests that this lack of symmetry is not

uncommon when one tries to estimate the permeability tensor using similar upscaling methods.)

In porous media literature, it is well established that any volume averaged quantity, such

as the permeability, should be independent of the size of the REV such that any changes in this

size should not lead to significant changes in that quantity. We applied the same criterion to our

unit-cell in order to find the proper size of the unit cell in our permeability calculations. Four

concentric square-shaped unit cells with different areas were chosen and the K11 components of

the permeability tensors obtained from these unit-cells were compared to each other.

17

Fig. 4. Four different unit-cells considered to check the insensitivity of the permeability tensor to

unit-cell size.

Fig. 5. Permeability component K11 for different unit cells of varying sizes

It is clear from Fig. 5 that though K11 does increase constantly with the size of the considered

unit cell, it seems to have converged to the value of 2.61e-11

m2. This effect could be explained

by the periodic boundary conditions applied on the boundaries of unit cells, as stated through

Eqn. (18). The influence of the periodic boundary condition, which is caused due to the violation

of the actual non-periodic nature of the porous medium, is known to be large in small regions

near the unit-cell boundaries [1]. However, this effect wanes as one move away from the

0

0.5

1

1.5

2

2.5

3

0.2 0.4 0.6 0.8 1

Per

mea

bil

ity

[m

2]

Unit cell size/total geometry size

Unit cell size study

18

boundaries in larger unit cells. Therefore, the permeability results obtained from the smaller unit-

cells is less accurate compared to the values obtained from the larger unit-cells. As a result, the

convergence of the permeability results for larger unit cells represents an asymptotic approach to

a true permeability, and hence using the entire geometry as a unit cell is appropriate for solving

the closure formulation.

After estimating the size of the final REV, we also wanted to ascertain if the

permeability values do not vary much as the REV is moved around in Fig. 4. This can be

construed as a condition for establishing the homogeneity of the CNF sample. As shown in Fig.

5, the permeability converged to 2.5e-11

m2 approximately, which allows us to conclude that the

unit-cell size with area 60% (or 0.6) of the entire rectangular micrograph would be a good choice

for the REV in the present geometry. After thus establishing the size of the smallest acceptable

REV, we proceeded to calculate the permeability at four different locations in the micrograph by

translating this reduced REV to those locations. These different selected locations of the reduced

REV yield the values of the porosity, a macroscopically averaged quantity estimated through the

image processing capability of COMSOL as the ratio of the void area within the REV to the total

REV area, to be between 0.57 and 0.63. This is a fairly converged result and the overall porosity,

equated to the average of all these different values, was found to be 0.6. By using this porosity

value in our closure-formulation calculations, the average permeability was numerically

estimated to be 2.5e-11

m2. The scatter in this numerical permeability, as reported in Table (3), is

obtained from the maximum and minimum of the set of permeabilities obtained from the

translation of REVs.

After measuring the CNF permeability using the Closure formulation, the falling-head

permeameter was used to estimate the permeability of the CNF sample. In experiments, the

19

height of the liquids was measured as a function of time and the permeability was calculated

using Eqn. (21) by fitting a linear curve on the data points (Section 3.2). Experiments were done

using Decane and Hexane. (These two alkanes, unlike water, were used since the cellulose does

not dissolve in them.) Some of the physical properties of the working fluids and CNF sample are

presented in Table (2).

Table (2): Physical properties of the CNF and the working fluid.

Decane Hexane

Viscosity 0.000859 Pa.sec 0.000297 Pa.sec

Density 730 kg/m3 655 kg/m

3

CNF sample radius 0.004 m 0.004 m

Syringe radius 0.013 m 0.013 m

CNF sample length 0.03 m 0.03 m

The 0ln( )x

xvs t plots for these two different working fluids are presented in Fig. 6. The slope of

the linear part of the diagram is used in the estimation of the permeability. The permeability

obtained from the experiments is compared with the numerical estimation in Table (3).

y = 0.001x + 0.0688 R = 0.9957

0

0.5

1

1.5

0 500 1000 1500

Ln

(x

0/x

)

Time (sec)

Decane

y = 0.001x - 0.0128 R = 0.9477

0

0.5

1

1.5

0 500 1000 1500 2000

Ln

(x0/x

)

Time (sec)

Hexane

20

(a) (b)

Fig. 6. Relationship between the ratio of CNF liquid-column heights and time for (a) Decane (b) Hexane.

In order to examine the accuracy of the permeability found using the closure formulation and

experimental method, the empirical correlation by Rumpf and Gupte [29] was used to estimate a

theoretical, true value of permeability. As stated in Eqn. (22), such a permeability is expressed

as a function of the porosity and the mean particle diameter. In this study, the porosity and mean

particle diameter were estimated by using COMSOL multiphysics software in the entire

rectangular micrograph of Fig. 3b. The porosity was calculated through its in-built image

processing software by dividing the void/empty area of the micrograph by the total area of the

rectangular micrograph. In order to find the mean particle diameter, we assumed the solid

particles to be circular and through dividing the total solid area by total number of particle, the

average area, and consequently the mean diameter, of one particle could be measured. The

permeability results obtained using the empirical formulations are presented in Table (3).

Table (3): Comparison of the permeability values as obtained from the closure-based numerical

simulations, experiments, and a theoretical model. Units are 10-11

m2

Simulation Experiment with Decane Experiment with Hexane Theory (Rumpf and Gupte)

K11= 2.61+/-0.23

K22= 3.28+/-0.11

K=3.8+/-0.24 K=1.8+/-0.05 K=3.23

21

As listed in the table (3), there is a good agreement between the permeability values obtained

from the experiments with Decane as the working fluid on the one hand, and the values of the

diagonal elements of the permeability tensor as obtained from the closure formulation on the

other. In addition, the permeability value obtained using the Rumpf and Gupte empirical

correlation are on the same order of magnitude as the permeability from the rest of the methods,

which confirms the accuracy of our simulations.

4. 3 Permeability of particulate porous medium made from sintered polymer beads

In order to examine the validity of the closure formulation for estimation of the permeability

tensor for particle-based media, porous cylindrical wicks sintered from micron size polymer

beads were selected, and SEM micrographs were obtained from thin slices taken from different

cross sections of the wicks. After some manipulations, the micrographs were imported into

COMSOL Multiphysics software, and the governing equations (Eqns. 15 through 18) were

solved in the void region in order to calculate different elements of the permeability tensor. A

sample micrograph and the corresponding binary figure are shown in Fig. 7. The numerical

simulations were carried out for three different samples and as before, the following form of the

permeability tensor was obtained after using Eqns. 20 and 24:

11 3 45 0 52 0 19 0 0910

0 07 0 04 3 8 0 02

. . . .

. . . .

K (m2) (26)

We note that compared to the CNF material, the diagonal terms of the permeability matrix are

much closer to each other. Hence one can conclude that industrially produced wick through the

sintering of polymer beads is much more isotropic that the CNF sample produced in a lab. Also

note that the off-diagonal terms are much smaller than those observed for the CNF material

considered earlier, which indicates that the considered x (horizontal) and y (vertical) directions in

22

Fig. 7(b) are much closer to the principal or material direction of the polymer wick. But once

again, we notice a lack of symmetry in the permeability tensor, which we report without any

explanation.

(a) (b)

Fig. 7. (a) A sample micrograph for permeability calculation, and (b) the

corresponding transformed binary picture obtained using COMSOL.

In addition to the computational analysis, the Falling head Permeameter was used as before to

find the height of the liquid column within the wick as a function of time, and to eventually

estimate permeability. Some of the physical properties of the wick and the working liquid used in

this experiment are gathered in Table (4).

Table (4): Physical properties of the wick and the working fluid

Liquid substance Dodecane (C12H26)

Density (Kg/m3) 753

Viscosity (mPa.s) 1.33

Wick material Polyethylene

Wick Length (mm) 74.5 0.2

Cross-section Diameter (mm) 6.4 0.1

Particle size (m) (appx.) 17525

23

As shown in Fig. 8, there is a good linear relationship between Ln(x0/x) and time, and therefore,

using Eqn. (21) as before, the permeability was calculated. The experiments were repeated three

times and the scatter bars show the range of Ln(x0/x) values obtained.

Fig. 8. Relationship between the ratio of liquid-column heights and time for a polymer wick

using Dodecane.

Table (5): Comparison of the permeability values as obtained from numerical simulations,

experiments and theory (units are in 10-11

m2)

Simulation Experiment Theory (Kozeny-Carman)

K11=3.45

K~2.7 K~6.28 K22=3.8

As listed in Table (5), there is a good agreement between the permeability value obtained from

the falling head permeameter and the values of the diagonal elements of the permeability tensor

as obtained from the closure formulation. While the theoretical model yields a permeability value

on the same order of magnitude as the experiments, the simulation provides the more accurate

results.

5. Summary and Conclusions

In this study, the closure formulation employed by Whitaker to derive Darcys law using the

volume averaging method [1] was used to estimate the permeability tensor for porous media

24

found in CNF sample and sintered polymer wicks. The main advantage of the proposed closure-

formulation method is its direct estimation of the permeability tensor, unlike the Stoke-Flow

simulation method that requires simulating flow along three different directions in the pore space

to resolve the full tensor. The SEM micrographs of the CNF sample and polymer wicks were

used to create 2D binary pictures representing the solid and pore phases. The permeability tensor

was estimated by solving the partial differential equations associated with the closure

formulation within the pore space of a unit cell. From the closeness of the K11 and K22 terms in

the 2-D permeability tensor, the two considered porous materials were found to be almost

isotropic, the polymer wick being more so than the CNF sample. The presence of non-diagonal

terms indicated that the chosen x and y axis in the micrographs did not match the principal or

material directions. The permeability tensor was observed to be slightly non-symmetric, once

again, less so for the polymer wick than for the CNF sample. The convergence in the estimated

permeability values was established by varying the size of unit cells as well as by translating the

unit cells in the binary domain. The numerical permeability thus estimated was compared with

the experimental permeability obtained using the falling-head permeameter. A good agreement

between the two demonstrated the accuracy of the closure formulation and the resulting

simulation. In addition, the permeability values were also compared with two

theoretical/empirical (Kozeny-Carman and Rumpf and Gupte) permeability relations obtained

from the literature and a reasonable match was observed, thereby establishing the accuracy of the

proposed method again.

6. Future Work

Despite introducing an improvement to the permeability estimation process for realistic, irregular

porous materials, the 2D micrographs cannot perfectly represent the 3D pore structure of the

25

considered porous media. Even though the permeability values from the 2D simulation were

close to the experimental data, it would be more accurate to perform a full 3D analysis by

reconstructing the porous media from a series of 2D micrographs for parallel planes using a

micro-CT scanner. Though such a simulation will be computationally challenging due to high

memory and storage requirements, it will be worth the effort to undertake this step. In addition, it

is recommended to repeat the simulation and experiment for a variety of CNF and wick samples

so as to have a wider range of porosity and particle size variations in the porous media

considered, and hence have a more rigorous validation of Whitakers closure-formulation

approach for estimating permeability.

7. Acknowledgement

The authors gratefully acknowledge the financial support extended by the Research Growth

Initiative (RGI) program of the graduate school at University of Wisconsin-Milwaukee.

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