Continuum modeling and simulation of multiphase di usion ... · Continuum modeling and simulation...

Continuum modeling and simulation

of multiphase diffusion through a porous solid

Qiangsheng ZHAO and Panayiotis PAPADOPOULOS∗

Department of Mechanical EngineeringUniversity of California, Berkeley, CA 94720-1740, USA

To K. Rajagopal, esteemed colleague and friend

Table of contents

1 Introduction 2

2 Definitions 32.1 General definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Balance laws 43.1 Balances of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Balance of linear momentum . . . . . . . . . . . . . . . . . . . . . 53.3 Balance of angular momentum . . . . . . . . . . . . . . . . . . . . 6

4 Constitutive assumptions 7

5 Finite element approximation 95.1 Weak forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.2 Space and time discretization . . . . . . . . . . . . . . . . . . . . . 10

6 Numerical Simulations 116.1 The Leverett experiment . . . . . . . . . . . . . . . . . . . . . . . . 116.2 Squeezing of a saturated cube . . . . . . . . . . . . . . . . . . . . . 126.3 Flexure of a Nafion R© film due to water absorption . . . . . . . . . 13

7 Conclusions 14

Appendix: Consistent Linearization of the Balance Laws 15

AbstractA theory is proposed for the continuum modeling of liquid and gas diffusion through

a deformable porous solid. The solid and the voids are locally homogenized to yield amacroscopic solid phase. Liquid saturation of the voids and gas pressure are includedas state variables to model diffusion. A model based on the proposed theory is imple-mented numerically using the finite element method, and tested with three simulationson sand and Nafion R©.

Keywords: Porous media; multiphase flow; diffusion; capillarity; elasticity; finite element method;Nafion

∗Corresponding author

1

Multiphase diffusion through porous elastic solid

1 Introduction

The diffusion of liquid and gas through porous solids is of considerable technological inter-

est and has been investigated for decades in a wide spectrum of disciplines encompassing

chemical, civil, mechanical, and petroleum engineering. Porous solids of interest are made

of either natural materials (e.g., soil, sand) or man-made materials (e.g., industrial filters,

membranes). In both cases, liquids (e.g., water, crude oil) and gases (e.g., air, oxygen,

natural gas) are driven through the voids in the porous solid by naturally or artificially in-

duced pressure. Nafion R© is an important example of a well-characterized man-made porous

medium due to its extensive use in proton exchange membrane fuel cells. Here, while the

fuel cell is in operation, a mixture of air and water diffuses through the pores of a Nafion R©

membrane. The efficiency of the fuel cell is affected by the variation in the membrane’s

water concentration [1, 2]. In addition, high water concentration has been experimentally

shown to cause substantial volumetric deformation (swelling) of the membrane [3,4], which

may compromise the integrity of the device.

Various approaches have been explored to model liquid and gas diffusion through porous

media and their interaction with the solid phase. These are typically based on mixture [5–9]

or homogenization theory [10–13]. The former, while conceptually appealing, is handi-

capped by the need to impose separate boundary and initial conditions for each phase, as

well as by the absence of liquid volume fraction as a state variable. The latter, while in-

clusive of the microstructural aspects of the flow, requires fine resolution in the microscale,

and hence becomes challenging in its computational implementation. An alternative con-

tinuum approach proposed in [14] admits porosity as a state variable, and includes separate

balance laws for the solid and liquid phases. This approach requires separate time deriva-

tives for each phase and includes the velocities of both phases in the balance laws, which,

again, leads to a computationally intensive solution method.

In an earlier work [15], a continuum approach alternative to that in [14] was proposed

for the modeling of liquid diffusion through a porous elastic solid. The principal novelty

here is that all balance laws are formulated relative to the frame of a macroscopic solid

resulting from the homogenization of the dry solid and the voids. In this approach, the

state of the porous medium is fully characterized by the densities of the solid and liquid

phases, the motion of this homogenized solid and the liquid volume fraction. On the other

hand, the velocity of the liquid phase is not a state variable, but rather is constitutively

related to the velocity of the macroscopic solid by Fick’s law. In the present work, the

2

Q. Zhao and P. Papadopoulos

approach in [15] is further refined by introducing multiphase diffusion, which now includes

the effect of gas pressure (which acts as driving force for diffusion) and capillarity for the

interface between the liquid and gas phases. These effects are of importance for diffusion in

partially saturated porous media. Multiphase diffusion necessitates a reformulation of the

balance laws and introduction of constitutive assumptions for the relative liquid and gas

velocities according to Darcy’s law, as well as for the dependence of the capillary pressure

on liquid saturation. For the latter, a qualitative argument is set forth to establish the

nature of this dependence and, subsequently, a polynomial law is postulated in the spirit

of earlier work on air-water diffusion through soils [16] and Nafion R© membranes [17,18].

The remainder of this article is organized as follows: Section 2 contains definitions of

kinematic and kinetic quantities that enter the proposed theory, while Section 3 includes

the formulation of the balance laws for the multiphase system. All constitutive assumptions

necessary to provide closure in the theory are introduced in Section 4. This is followed

in Section 5 by a brief exposition of the finite element solution of the weak forms of

the balance laws. Section 6 documents three numerical simulations involving sand and

Nafion R©. Closing comments are provided in Section 7.

2 Definitions

2.1 General definitions

Consider a porous body B consisting of solid matter and voids of different sizes and shapes.

When the body is dry, all the pores are occupied by gas (typically, air). Otherwise, the

pores may be either partially or fully occupied by liquid matter, as shown in Figure 1.

Next, assume that the characteristic size of the voids is much smaller than the overall size

of the body. Now, the porous solid can be locally homogenized to yield a macroscopic

continuous solid endowed with a density field ρs. This homogenization altogether excludes

the liquid and gas phases.

Let the macroscopic solid occupy regions R0 and R at times t0 and t, respectively. If X

is the position vector of a material point in R0 at time t0, then it is mapped by the motion

χ to x = χ(X, t) in R at time t, as illustrated in Figure 2, where the motion is assumed

invertible for any fixed time t. The velocity of a material point is subsequently defined as

v(X, t) = ∂χ(X,t)∂t , where X is the position of the same material point at t0. Likewise, the

deformation gradient at R relative to R0 is defined at any material point as F = ∂χ(X,t)∂X .

The bulk porosity ε (0 ≤ ε ≤ 1) may now be defined at any point x of the macroscopic

3


solid as the ratio of the volume of voids to the volume of the macroscopic solid in a

representative element V centered at the point. Such a definition is meaningful provided

that the porosity field is well-defined for a representative element volume which is much

smaller than the total volume of the body, that is, if vol(V) vol(R).

The preceding homogenization procedure may also be employed to define the satura-

tion φ at a point as the ratio of the volume of liquid to the volume of the voids in the

representative element volume. Therefore, φ = 0 corresponds to dry pores and φ = 1 to

fully saturated ones. In the partially saturated case (0 < φ < 1), gas occupies any portion

on the pores that is not filled with liquid. Therefore, the total density ρ of the porous

medium is

ρ = ρs + ρlεφ+ ρgε(1− φ) , (1)

where ρl and ρg are densities of the liquid and gas phases. Both of these densities are

taken to be constant: for the liquid, this is an immediate implication of homogeneity and

incompressibility, while for the gas it is due to the assumption that the gas is allowed to

move freely through the pores without being subject to compression.

The diffusion of liquid and gas matter in the porous medium is characterized by the

fluxes of liquid ql and gas qg in the macroscopic solid, defined as

ql = ρlεφvrl , qg = ρgε(1− φ)vrg . (2)

Here, vrl and vrg are the liquid and gas velocities relative to the macroscopic solid. These

are related to the respective absolute velocities vl and vg as vrl = vl − v and vrg = vg − v.

3 Balance laws

Consider a part of the macroscopic solid, which occupies an arbitrary closed and bounded

region P ⊂ R with smooth boundary ∂P having outward normal n at time t.

Balances of mass and linear momentum are formulated below by examining the material

in region P. Here, all material time derivatives of integrals over P are defined by keeping

material particles of the macroscopic solid fixed. Therefore, the material time derivatives

of arbitrary scalar and vector functions ψ(x, t) and w(x, t) are given by

dψ

dt=

∂ψ

∂t+ gradψ · v ,

dw

dt=

∂w

∂t+ (grad w)v . (3)

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3.1 Balances of mass

Since all balance laws are formulated on a material domain P relative to the macroscopic

solid material, mass conservation for the macroscopic solid is automatically satisfied. Ad-

ditional mass balance laws are required for the liquid and gas phases.

For the liquid phase, the rate of change of mass equals the flux of liquid leaving the

domain from the boundary ∂P, hence it takes the form

d

dt

∫Pρlεφ dv = −

∫∂P

ql · n da . (4)

Upon using the Reynolds’ transport theorem and the divergence theorem, the local form

of (4) is derived asd(ρlεφ)

dt+ ρlεφ div v = −div ql . (5)

Likewise, the mass balance for the gas phase is written in integral form as

d

dt

∫Pρgε(1− φ) dv = −

∫∂P

qg · n da , (6)

with its local counterpart expressed as

d(ρgε(1− φ)

)dt

+ ρgε(1− φ) div v = −div qg . (7)

3.2 Balance of linear momentum

The proposed theory relies on a single linear momentum balance law, rather than on

individual balances for each phase. This circumvents the need to explicitly resolve all force

interactions between the constituent phases.

The rate of change of the total linear momentum in the material region P is equal

to the sum of the total external force acting on the material and the linear momentum

flux across the boundary ∂P of the liquid and gas phases. This means that the integral

statement of linear momentum balance for the domain P is given as

d

dt

∫P

(ρsv + ρlεφvl + ρgε(1− φ)vg

)dv =∫

Pρb dv +

∫∂P

t da−∫∂P

(ql · n)vl da−∫∂P

(qg · n)vg da . (8)

Taking into account (1) and (2), the left-hand side of (8) may be rewritten as

d

dt

∫P

(ρsv + ρlεφvl + ρgε(1− φ)vg

)dv =

d

dt

∫P

(ρv + ql + qg) dv . (9)

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Using the mass balance equations (5) and (7) and the Reynolds’ transport theorem,

the right-hand side of the preceding equation becomes

d

dt

∫P

(ρv + ql + qg) dv =∫P

(ρdv

dt− v(div qg + div ql) +

dqgdt

+ qg div v +dqldt

+ ql div v)dv . (10)

Combining (10) and (8) and invoking the divergence theorem, the local form of the

linear momentum balance equation is deduced in the form

ρdv

dt− v(div qg + div ql) +

dqgdt

+ qg div v +dqldt

+ ql div v =

ρb + div T− div qgvg − (grad vg)qg − div qlvl − (grad vl)ql , (11)

where T denotes the Cauchy stress, which, by Cauchy’s stress theorem, is defined such

that t = Tn.

3.3 Balance of angular momentum

The rate of change of angular momentum in the material region P equals the sum of the

moment of all external forces acting on the material and the angular momentum flux across

∂P of the liquid and gas phases. This means that

d

dt

∫P

x×(ρsv + ρlεφvl + ρgε(1− φ)vg

)dv =∫

Px× ρb dv +

∫∂P

x× t da−∫∂P

x× (ql · n)vl da−∫∂P

x× (qg · n)vg da . (12)

Recalling (1) and (2), and invoking the Reynolds’ transport theorem, the left-hand side of

(12) may be rewritten as

d

dt

∫P

x×(ρsv + ρlεφvl + ρgε(1− φ)vg

)dv =

d

dt

∫P

x×(ρv + ql + qg

)dv

=

∫P

[v ×

(ql + qg

)+ x×

(ρdv

dt+dqldt

+dqgdt

)+ x×

(ρv + ql + qg

)div v

]dv . (13)

Appealing to Cauchy’s stress theorem and the divergence theorem, the second term on the

right-hand side of (12) may be rewritten as∫∂P

x× t da =

∫P

x× div T dv +

∫P

x,i × ti dv . (14)

Likewise, the last two terms on the right-hand side of (12) are recast by way of the diver-

gence theorem as∫∂P

x× (ql · n)vl da =

∫P

vl × ql dv +

∫P

x× vl div ql dv +

∫P

x× (grad vlql) dv (15)

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and∫∂P

x× (qg · n)vg da =

∫P

vg × qg dv +

∫P

x× vg div qg dv +

∫P

x× (grad vgqg) dv .

(16)

Taking into account (13–16) and the local form of linear momentum balance (11), the

angular momentum balance equation (12) reduces to∫P

(v − vl)× ql dv +

∫P

(v − vg)× qg dv +

∫P

x,i × ti dv = 0 . (17)

However, upon recalling the definitions of ql and qg in (2), the first two terms in (17) vanish

identically, and angular momentum balance equation is further reduced to the standard

form ∫P

x,i × ti dv = 0 , (18)

which implies symmetry of the Cauchy stress tensor T. Hence, in the proposed theory,

angular momentum balance does not introduce any further restrictions on the stress.

4 Constitutive assumptions

The balance laws furnish a total of five equations, two from mass and three from linear mo-

mentum balance to determine twelve unknowns, that is, the macroscopic solid displacement

u, the liquid saturation φ, the liquid pressure Pl, the gas pressure Pg, the relative liquid

velocity vrl and the relative gas velocity vrg. Therefore, closure of the system necessitates

the introduction of seven constitutive equations.

The diffusion of liquid and gas matter in the macroscopic solid is assumed to be governed

by Darcy’s law [19], which states that diffusion is driven by (and is proportional to) the

gradient of pressure. Therefore, the constitutive assumptions for the relative velocities of

liquid and gas take the form

vrl = −klµl

gradPl , vrg = −kgµg

gradPg , (19)

where kl, kg, µl, and µg are liquid and gas permeabilities and viscosities, respectively.

The final equation needed to close the system concerns the capillary pressure, which is

the difference in pressure across the interface between gas and liquid in equilibrium, that

is

Pc = Pl − Pg . (20)

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Leverett [20] introduced a dimensionless function J(φ) to which the capillary pressure Pc

is assumed to be proportional, that is

Pc = σJ(φ) , (21)

where σ is a constant with dimension of pressure. Several analytical expressions of the

Leverett J-function have been proposed in the literature for different applications [16, 21,

22].

Here, a qualitative argument is offered for the overall form of the J-function. To this

end, one may consider porous media as comprising many micro-channels formed by pores.

Liquid may diffuse through such micro-channels as in the idealized porous medium whose

cross-section is locally characterized by the regular structure of the representative area

element depicted in Figure 3. Denoting the (average) capillary pressure in a non-saturated

micro-channel pc and letting the (average) cross-section of a micro-channel be A, the local

homogenized capillary pressure Pc is

Pc =NpcA

A, (22)

where N is total number of non-saturated micro-channels and A is the total area of the

representative area element of the cross-section. When the porous medium is essentially

dry as in Figure 3a, (that is, φ approaches zero), all the micro channels are non-saturated,

therefore N is large, which yields a high capillary pressure Pc. As the medium becomes pro-

gressively more saturated with liquid (hence, φ increases), as in Figure 3b,c, N decreases,

and so does the capillary pressure Pc. In the limiting case of full saturation (φ = 1) de-

picted in Figure 3d, N approaches zero, as does Pc. In conclusion, the preceding argument

implies that the J-function is monotonically decreasing to zero with increasing values of

φ. While the argument suggests linearity of J in N , this does not necessarily translate to

linearity of J is φ, since the geometric structure of the micro-channels and the distribution

of pore sizes are far more complex than those of the straight parallel tubes assumed here.

In this article, it is assumed that the J-function depends polynomially on φ, where all

polynomial coefficients have the dimension of pressure and are obtained by curve-fitting of

experimental measurements of the capillary pressure for different values of liquid saturation.

In particular, a simple cubic polynomial approximation is considered in the form

Pc = c1φ3 + c2φ

2 + c3φ+ c4 , (23)

where the coefficients ci, i = 1 − 4, are determined for each porous medium from experi-

mental data. It will be shown in Section 6 that this approximation captures with accuracy

8


the salient features of the dependency of Pc on φ for different types of porous media.

The macroscopic solid, gas and liquid phases contribute to the homogenized stress in

the body, and the resulting Cauchy stress is assumed to be of the form

T = Ts − (Pl + Pg)i , (24)

where Ts is the stress for the macroscopic solid, and i is spatial second-order identity

tensor.

A special case of the theory arises when the capillary pressure is assumed to be linear

in φ, according to

Pc = σ(1− φ) . (25)

It follows, upon recalling (2), (19), and (20), that the liquid mass flux is reduced to

ql = −ρlεφklµl

(gradPg − σ gradφ) . (26)

In addition, if the gas pressure is taken to be uniform inside the porous medium, the

equation for liquid flux is further reduced to be

ql = Kdρlεφ gradφ , (27)

where liquid diffusivity constant Kd is given by klσµl

. This equation coincides to the one

derived in an earlier work [15]. Also, gas mass flux is here zero through out the porous

medium due to uniform gas pressure, therefore mass balance for gas is automatically sat-

isfied.

5 Finite element approximation

5.1 Weak forms

In this section, weak forms of the local balance equations (5), (7) and (11) are constructed

by way of background to the finite element approximation.

For liquid and gas mass balance, equations (5) and (7) are first weighted by arbitrary

scalar test functions η and ζ, then integrated over P. After using integration by parts and

invoking the divergence theorem, the weak forms of liquid and gas mass balance become∫Pηρlε

dφ

dtdv +

∫Pηρlεφ div v dv −

∫P

grad η · ql dv +

∫∂Pηql · n da = 0 (28)

and∫Pζρgε

d(1− φ)

dtdv+

∫Pζρgε(1−φ) div v dv−

∫P

grad ζ ·qg dv+

∫∂Pζqg ·n da = 0 , (29)

9


respectively.

Likewise for linear momentum balance, equation (11) is first weighted by an arbitrary

vector test function ξ, and integrated over P. Again, after the straightforward application

of integration by parts and the divergence theorem, the weak form of linear momentum

balance is expressed as

∫Pξ · ρdv

dtdv −

∫∂P

[(ξ · v)qg

]· n da+

∫P

[(grad ξ)qg

]· v dv +

∫Pξ ·[(grad v)qg

]dv

+

∫Pξ·dqgdt

dv+

∫Pξ·qg div v dv−

∫∂P

[(ξ · v)ql]·n da+

∫P

[(grad ξ)ql

]·v dv+

∫Pξ·[(grad v)ql

]dv

+

∫Pξ·dqldt

dv+

∫Pξ·ql div v dv−

∫Pξ·ρb dv+

∫P

grad ξ·T dv−∫∂P

ξ·t da+

∫∂P

[(ξ·vg)qg

]·n da

−∫P

[(grad ξ)qg] · vg dv +

∫∂P

[(ξ · vl)ql

]· n da−

∫P

[(grad ξ)ql

]· vl dv = 0 . (30)

The preceding equation may be simplified by combining the second and third term with

the fifteenth and sixteenth term, and also the seventh and eighth with the seventeenth and

eighteenth term, leading to

∫Pξ · ρdv

dtdv +

∫∂P

[(ξ · vrg)qg

]· n da−

∫P

[(grad ξ)qg

]· vrg dv +

∫Pξ · [(grad v)qg] dv

+

∫Pξ · dqg

dtdv +

∫Pξ · qg div v dv +

∫∂P

[(ξ · vrl )ql

]· n da−

∫P

[(grad ξ)ql

]· vrl dv

+

∫Pξ ·[(grad v)ql

]dv +

∫Pξ · dql

dtdv +

∫Pξ · ql div v dv

−∫Pξ · ρb dv +

∫P

grad ξ ·T dv −∫∂P

ξ · t da = 0 . (31)

5.2 Space and time discretization

A finite element approximation is effected on the weak forms (28), (29) and (31). Owing

to the arbitrariness of the test functions η, ζ and ξ, the weak forms give rise to a coupled

system of integro-differential equations with the macroscopic solid displacement u, liquid

saturation φ and gas pressure pg as unknown variables.

Following standard procedure, semi-discretization is employed for the solution of (28),

(29) and (31) with displacement-like finite element approximation used for the spatial

discretization of u, φ and Pg. This results in a coupled system of first- and second-order

ordinary partial differential equations in time. These, in turn, are integrated in a typical

10


time interval (tn, tn+1] using an implicit Newmark method [23], according to which

un+1 = un + vn∆tn +1

2

[(1− 2β)

dv

dt

∣∣∣n

+ 2βdv

dt

∣∣∣n+1

]∆t2n ,

vn+1 = vn +

[(1− γ)

dv

dt

∣∣∣n

+ γdv

dt

∣∣∣n+1

]∆tn ,

φn+1 = φn +

[(1− γ)

dφ

dt

∣∣∣n

+ γdφ

dt

∣∣∣n+1

]∆tn ,

Pg,n+1 = Pg,n +

[(1− γ)

dPgdt

∣∣∣n

+ γdPgdt

∣∣∣n+1

]∆tn .

(32)

Here, (·)n = (·) |n= (·)(tn), ∆tn = tn+1 − tn, and β, γ are the Newmark parameters with

0 < β ≤ 0.5 and 0 < γ ≤ 1.

The resulting nonlinear algebraic equations are solved for un+1, φn+1 and Pg,n+1 using

the Newton-Raphson method. This requires consistent linearization of the weak forms with

respect to u, φ and Pg, which is derived in the Appendix.

6 Numerical Simulations

The finite element formulation of the model described in Section 5 was implemented in the

general-purpose nonlinear program FEAP, which is partially documented in [24, 25]. The

model and its numerical implementation were tested on three representative simulations

discussed below. All simulations employed 8-node isoparametric brick elements with full

2 × 2 × 2 Gaussian quadrature. In addition, the time integration parameters defined in

Section 5 were set to β = 0.25 and γ = 0.5.

6.1 The Leverett experiment

The well-known experiment of Leverett [20] is simulated here. In this experiment, a 0.05×0.05×6 m tube is initially filled with dry soil (sand or clay). Next, water is supplied to the

soil from the bottom of the tube at constant pressure, while the free surface of the soil is

subjected to atmospheric pressure. Hence, water is imbibed into the soil due to capillary

pressure. As noted in [20], after several weeks the water distribution reaches steady-state

as the gravitational forces come to equilibrium with capillary pressure. At that time, the

saturation of the soil is measured at different levels in the tube and the corresponding

capillary pressure is deduced from the elevations relative to the bottom of the tube.

Here, the Leverett experiment is considered for the case of sand saturated by water.

Material properties for this experiment are taken from Table I in [20], and are kl = 3.60×

11


10−13 m2, µl = 1.0 × 10−3 Ns/m2, kg = 3.63 × 10−13 m2, µg = 1.5 × 10−5 Ns/m2, and

ε = 0.40. The coefficients in the constitutive equation (23) are estimated from curve-fitting

the experiment data in [20] for sand and water, such that

Pc = −3.9φ3 + 5.7φ2 − 3.6φ+ 1.8 , (33)

where Pc is calculated in MPa. The polynomial equation (33) and experiment data from [20]

are plotted in Figure 4. A uniform 2× 2× 60 block mesh is used to simulate the Leverett

experiment with constant step-size ∆t = 1000 s. The duration of imbibition is taken to be

10 weeks. In this simulation, the solid is assumed fixed in space.

Figure 5 shows the water saturation at different heights in the tube at the end of the

simulation. These results compare very well with the data in [20].

6.2 Squeezing of a saturated cube

This simulation involves Nafion R©, a polymeric porous solid which is suitable for modeling

within the proposed theory owing to the small size of its pores (diameter of 20−30 nm [26])

compared to the typical thickness of its industrial membranes (0.175 mm [27]).

The material properties for the simulation are set to ρs = 2 × 103 kgr/m3 [28], ρl =

103 kgr/m3, ε = 0.4 [29], kl = 1.0 × 10−13 m2/s, µl = 1.0 × 10−3 Ns/m2 for water [30],

and kg = 0.75 × 10−10 m2/s, µg = 2.0 × 10−5 Ns/m2 for oxygen [31]. Also, Nafion R© is

assumed to be hyperelastic and taken to obey the Kirchhoff-Saint Venant stress-strain law,

according to which the second Piola-Kirchhoff stress Ss = JF−1TsF−T is given by

Ss = λ tr(E)I + 2µE . (34)

Here, E is the Lagrangian strain and λ, µ are elastic constants for the macroscopic solid,

which are set to (λ, µ) = (4.0, 2.7) MPa [28]. For constitutive equation (23), coefficients

for Nafion R© are obtained by curve fitting the experiment data in [32], which leads to the

expression

Pc = −7.8φ3 + 14.4φ2 − 10.2φ+ 3.6 , (35)

where Pc is calculated in MPa. Figure 6 illustrates that the proposed cubic polynomial

equation (35) matches well with the experiment data.

In earlier work [15], a computational experiment was proposed in which water is repeat-

edly squeezed out of a 0.5 mm cube of saturated Nafion R©. The porous solid is compressed

uniformly on one of its faces using displacement control in the direction of compression,

12


while the opposite face remains fixed and each of the four lateral faces is free to slide on

its own plane only. Liquid is allowed to exit from the compressed face, while all the other

five faces are assumed impermeable.

The block is initially squeezed to 93% of its initial length while keeping the liquid

saturation value of φ = 1 on the compressed face. Then, it is stretched back to its original

shape, where it is kept in rest until the liquid saturation reaches steady state. During the

stretching and until steady state is attained, the compressed face is subject to zero liquid

flux conditions. Each of three loading stages is imposed proportionally over a period of

0.1 s. The block is squeezed and stretched repeatedly, until enough liquid is pushed out of

the block and a steady state of liquid saturation is reached. The simulation is carried out

using a uniform 5× 5× 5 mesh with constant step-size ∆t = 1 ms.

Figure 7 illustrates that the water is progressively drained with each squeeze and that

it remains constant in volume during the stretching and resting stage. The figure also

demonstrates that the total volume is reduced faster when compared to the corresponding

result in [15], where the effect of gas is not modeled. Indeed, it is reasonable to assume that

the presence of gas contributes to the exit of the liquid from the block, and increasingly

so as the saturation decreases. It is also interesting to note that during the third squeeze

the liquid saturation φ increases briefly before dropping to its final value. This is due to

the fact that by this time the liquid content is already relatively low and the early stage

of this squeeze generates reduction of the overall volume without removing any liquid.

Liquid diffusion appears to be relatively insensitive to the value of the gas permeability,

as illustrated in Figure 8. Here, a full cycle of squeezing, stretching and resting of the same

saturated Nafion R© cube is simulated using the original mesh and two different values of

gas permeability, kg = 0.75 × 10−10 m2/s (the original value) and kg = 0.75 × 10−9 m2/s.

A comparison of the total liquid volume during this cycle reveals negligible differences

between the two simulations. This implies that the liquid diffusion is not very sensitive to

the diffusivity of the gas.

6.3 Flexure of a Nafion R© film due to water absorption

The flexure of an initially flat Nafion R© film due to the absorption of a water droplet

placed on its top has been experimentally studied in [4]. To simulate this experiment, a

water droplet is idealized by means of a prescribed liquid saturation boundary condition

on a square region of 3 mm side at the center of the top surface of a 10 × 10 × 0.125 mm

Nafion R© block. Zero liquid saturation is prescribed at the four lateral faces and the bottom

13


face to simulate contact with dry air. Also, the block is fixed in space on all of its four

lateral faces. The supply of water is terminated at t = 450 s, at which time the Dirichlet

boundary condition φ = 1 on the droplet region is replaced by a corresponding zero liquid

flux condition.

Following earlier work [15], a uniform 20 × 20 × 5 mesh is used for the analysis, and

symmetry is exploited in modeling only a quarter of the domain. Also, the time step-size

is set to ∆t = 0.2 s.

Figure 9 illustrates the flexure of the film at the center of the water droplet. It is noted

that maximum flexure in the range of approximately 0.7 mm is achieved at t = 450 s. After

the water supply is terminated, the flexure starts decreasing rapidly with time. When

compared to the results in [15], the flexure here is more pronounced, while the rate of its

decrease after the termination of the water supply is steeper. Both variations are attributed

to the nonlinear dependence of the liquid pressure on saturation, which, in turn, affects the

driving force of the liquid diffusion. These results are both qualitatively and quantitatively

consistent with the experimental findings in [4].

7 Conclusions

This work provides a simple, yet reasonably general theory for modeling multiphase diffu-

sion and capillarity in a porous solid undergoing finite deformation. The three numerical

simulations on sand and Nafion R© film demonstrate both the versatility and predictive

power of the proposed theory. The simulation of water imbibition in sand demonstrates

that Leverett-type constitutive assumptions for capillarity are appropriate for this approach

and yield results that match well with the outcomes of the original experiment. The simu-

lation of squeezing of an initially water-saturated Nafion R© cube highlights the role of gas

pressure as driving force for liquid diffusion. Lastly, the simulation of flexure of a Nafion R©

film shows the superior predictive capacity of the proposed multiphase diffusion approach

over an earlier single phase one [15].

Further work remains to be done in exploring the nature of the nonlinearity in the

constitutive relation between liquid saturation and capillary pressure. Furthermore, the

significance of the gas phase in multiphase diffusion needs to be better understood with

the aid of relevant experiments.

14


Acknowledgment

The authors are grateful to Professor Stephen J.S. Morris of the University of California,

Berkeley for his helpful comments on capillarity.

Appendix: Consistent Linearization of the Balance Laws

Consistent linearization of the weak forms (28), (29) and (31) is required for the solution

of the nonlinear balance laws using the Newton-Raphson method at a given time tn+1.

To this end, and after time discretization by (32), let the interpolation in element e be

φn+1 =∑N

I=1NeI φI n+1, pg n+1 =

∑NI=1N

eI pgI n+1, and un+1 =

∑NI=1N

eI uI n+1 for the

dependent variables, and, likewise, ηn+1 =∑N

I=1NeI ηI n+1, ζn+1 =

∑NI=1N

eI ζI n+1, and

ξn+1 =∑N

I=1NeI ξI n+1 for the test functions. Here, N e

I denotes the I-th element interpo-

lation function of element e and the notation “(·)I” signifies the value of variable (·) at

node I. The subscript n+ 1 is omitted in the rest of this appendix for brevity.

The linearization of the weak forms (28), (29) and (31) gives rise to an element tangent

stiffness matrix[Ke]

of the form

[Ke]

=

[Kφφ

] [Kφp

] [Kφu

][Kpφ

] [Kpp

] [Kpu

][Kuφ

] [Kup

] [Kuu

] . (A.1)

Here, the N ×N submatrix[Kφφ

], N ×N submatrix

[Kφp

]and the N × 3N submatrix[

Kφu]

result from the linearization of the liquid mass balance equation (28) in the directions

of the increments ∆φI , ∆pI and ∆uI , I = 1, 2, . . . , N , of the nodal variables. Likewise, the

N ×N submatrix[Kpφ

], the N ×N submatrix

[Kpp

], and the N × 3N submatrix

[Kpu

]results from the linearization of the gas mass balance equation (29), while the 3N × Nsubmatrix

[Kuφ

], the 3N ×N submatrix

[Kup

]and the 3N × 3N submatrix

[Kuu

]result

from the linearization of the momentum balance equation (31) along the same directions.

Explicit expressions are provided below for all the submatrices. In these expressions, the

variables i, j, k are associated the spatial coordinates xi, xj , xk, respectively.

It can be easily deduced from the linearization of (28) that the components of the first

15


row of submatrices in (A.1) are

KφφIJ =

∫Ωe

ρlεNeIN

eJ dv +

∫Ωe

ρlεNeIN

eJ div v dv

+3∑

k=1

∫Ωe

N eI,kρl

klµlε(N eJ

(p,k +

dPcdφ

φ,k)

+ φ(dPcdφ

N eJ,k +

d2Pcdφ2

N eJφ,k

))dv , (A.2)

KφpIJ =

3∑k=1

∫Ωe

N eI,kρl

klµlεφN e

J,k dv , (A.3)

and

KφuI(jJ) =

γ

β∆t

∫Ωe

ρlεφNeIN

eJ,j dv . (A.4)

Here, N eJ,j is the partial derivative of N e

J with respect to the spatial coordinate xj and Ωe

is the domain of element e.

Similarly, the linearization of (29) yields

KpφIJ = − γ

β∆t

∫Ωe

ρgεNeIN

eJ dv −

∫Ωe

ρgεNeIN

eJ div v dv

+

3∑k=1

∫Ωe

N eI,kρg

kgµgεN e

Jpg,k dv , (A.5)

KppIJ =

3∑k=1

∫Ωe

N eI,kρg

kgµgε(1− φ)N e

J,k dv , (A.6)

and

KpuI(jJ) =

γ

β∆t

∫Ωe

ρgε(1− φ)N eIN

eJ,j dv . (A.7)

16


Lastly, upon linearization of (31) it follows that

Kuφ(iI)J =

∫Ωe

N eI ρlN

eJ

dvidt

dv +

∫Ωe

kgµgN eI ρgN

eJ

3∑k=1

(vi,kpg,k

)dv

+3∑

k=1

∫Ωe

N eI ρl

klµl

(vi,kεN

eJ(pg,k +

dPcdφ

φ,k) + φd2Pcdφ2

N eJφ,k + φ

dPcdφ

N eJ,k

)dv

−∫

Ωe

kgµgρgN

eI εN

eJpg,i div v dv

+

∫Ωe

klµlρlN

eI ε(N eJ(pg,i +

dPcdφ

φ,i) + φ(d2Pcdφ2

N eJφ,i +

dPcdφ

N eJ,i))

div v dv

− 1

γ∆t

∫Ωe

kgµgρgN

eI εN

eJpg,i dv

−∫

Ωe

klµlρlN

eI

( 1

γ∆tN eJ(pg,i +

dPcdφ

φ,i) +dφ

dt

dPcdφ

N eJ,i +

dφ

dt

d2Pcdφ2

N eJφ,i

)dv

−∫

Ωe

klµlρlεN

eIN

eJ(dp,idt

+dPcdφ

dφ,idt

+d2Pcdφ2

dφ

dtφ,i) dv

−∫

Ωe

klµlρlεN

eIφ(d2Pcdφ2

dφ,idt

N eJ +

1

γ∆t

dPcdφ

N eJ,i +

1

γ∆t

d2Pcdφ2

N eJφ,i +

d2Pcdφ2

dφ

dtN eJ,i

)dv

−∫

Ωe

kgµgφ,iN

eI,k(

kgµgρgεN

eJpg,k) dv +

∫Ωe

klµl

(dPcdφ

N eJ,i +

d2Pcdφ2

)3∑

k=1

(N eI,k(ql)k

)dv

+

∫Ωe

klµl

(p,i +dPcdφ

φ,i)

3∑k=1

N eI,k

(−klµlρlε(N eJ(pg,k +

dPcdφ

φ,k) + φdPcdφ

N eJ,k + φ

d2Pcdφ2

N eJφ,k

))dv ,

(A.8)

Kup(iI)J = −

3∑k=1

∫Ωe

N eI vi,k

kgµgρgε(1− φ)N e

J,k dv −3∑

k=1

∫Ωe

N eI vi,k

klµlρlεφN

eJ,k dv

−∫

Ωe

N eI


J,i div v dv −∫

Ωe

N eI

klµlρlεφN

eJ,i div v dv

−∫

Ωe

N eI

kgµgρgε( 1

γ∆t(1− φ)N e

J,i −dφ

dtN eJ,i

)dv −

∫Ωe

N eI

klµlρlε(

dφ

dtN eJ,i + φN e

J,i

1

γ∆t) dv

+

∫Ωe

kgµg

3∑k=1

(N eJ,iN

eI,k(qg)k − pg,iN e

I,k


J,k

)dv

+

∫Ωe

klµl

3∑k=1

(N eJ,iN

eI,k(ql)k − (pg,i +

dPcdφ

φ,i)NeI,k

klµlρlεφN

eJ,k

)dv , (A.9)

17


and

Kuu(iI)(jJ) =

1

β∆t2

∫Ωe

N eI ρN

eJδij dv +

γ

β∆t

∫Ωe

( 3∑k=1

N eI (qg)kN

eJ,k

)δij dv

+γ

β∆t

∫Ωe

( 3∑k=1

N eI (ql)kN

eJ,k

)δij dv +

γ

β∆t

∫Ωe

N eI (qg)iN

eJ,j dv

+γ

β∆t

∫Ωe

N eI (ql)iN

eJ,j dv +Km

(iI)(jJ) , (A.10)

where Km(iI)(jJ) denotes the material tangent stiffness from the stress-divergence term in

(31) and δij is the Kronecker symbol.

References

[1] T.V. Nguyen and R.E. White. A water and heat management model for proton-exchange-membrane fuel cells. Journal of the Electrochemical Society, 140:2178–2186,1993.

[2] T.V. Nguyen and M.W. Knobbe. A liquid water management strategy for pem fuelcell stacks. Journal of Power Sources, 114:70–79, 2003.

[3] J.A. Elliott, S. Hanna, A.M.S. Elliott, and G.E. Cooley. The swelling behaviour ofperfluorinated ionomer membranes in ethanol/water mixtures. Polymer, 42:2251–2253, 2001.

[4] S. Goswami, S. Klaus, and J.B. Benziger. Wetting and absorption of water drops onNafion films. Langmuir, 24(16):8627–8633, 2008.

[5] A. Fick. On liquid diffusion. Philosophical Magazine, 10:30–39, 1855.

[6] A.E. Green and P.M. Naghdi. A theory of mixtures. Archive for Rational Mechanicsand Analysis, 24:243–263, 1967.

[7] A.E.Green and P.M.Naghdi. On basic equations of mixture. Quarterly Journal ofMechanics and Applied Mathematics, 22:427–438, 1969.

[8] R.J. Atkin and R.E. Craine. Continuum theories of mixtures: Basic theory and histor-ical development. Quarterly Journal of Mechanics and Applied Mathematics, 29:209–244, 1976.

[9] K.R. Rajagopal and L. Tao. Mechanics of Mixture. World Properties, Singapore,1995.

[10] S.M. Hassanizadeh and W.G. Gray. General conservation equations for multiphasesystems: 1. Averaging procedure. Advances in Water Resources, 2(4):131–144, 1979.

[11] S.M. Hassanizadeh and W.G. Gray. General conservation equations for multiphasesystems: 2. Mass, momenta, energy and entropy. Advances in Water Resources,2(4):191–203, 1990.

18


[12] S.M. Hassanizadeh and W.G. Gray. General conservation equations for multiphasesystems:3. Constitutive theory for porous media flow. Advances in Water Resources,3:25–40, 1980.

[13] Y. Bachmat and J. Bear. Macroscopic modelling of transport phenomena in porousmedia. 1: The continuum approach. Transport in Porous Media, 21:213–240, 1986.

[14] O. Coussy. Poromechanics. John Wiley and Sons, Chichester, 2004.

[15] Q. Zhao and P. Papadopoulos. Modeling and simulation of liquid diffusion through aporous finitely elastic solid. Computational Mechanics, 52:553–562, 2013.

[16] K.S. Udell. Heat transfer in porous media considering phase change and capillarity -the heat pipe effect. International Journal of Heat Mass Transfer, 28:485–495, 1985.

[17] T. Berning and N. Djilali. A 3D, multiphase, multicomponent model of cathode andanode of a pem fuel cell. Journal of The Electrochemical Society, 150:1589–1598, 2003.

[18] U. Pasaogullari and C.Y. Wang. Two phase transport and the role of micro-porouslayer in polymer electolyte fuel cell. Electrochimica Acta, 49:4359–4369, 2004.

[19] H. Darcy. Les Fontaines Publiques de la Ville de Dijon. Dalmont, Paris, 1856.

[20] M.C. Leverett, W.B. Lewis, and M.E. True. Dimensional-model studies of oil-fieldbehavior. Transaction of the AIME, 146:175–193, 1942.

[21] R.H. Brooks and A.T. Corey. Hydraulic properties of porous media. Colorado StateUniversity, Fort Collins, 1964.

[22] M.Th. van Genuchten. A closed-form equation for predicting the hydraulic conductiv-ity of unsaturated soils. Soil Science Society of America Journal, 44:892–898, 1980.

[23] N.M. Newmark. A method of computation in structural dynamics. ASCE Journal ofEngineering Mechanics, 85:67–94, 1959.

[24] R.L. Taylor. FEAP - A Finite Element Analysis Program, User Manual. Universityof California, Berkeley. http://www.ce.berkeley.edu/feap.

[25] R.L. Taylor. FEAP - A Finite Element Analysis Program, Programmer Manual. Uni-versity of California, Berkeley. http://www.ce.berkeley.edu/feap.

[26] D.J. Gargas, D.A. Bussian, and S.K. Buratto. Investigation of connectivity of hy-drophilic domains in Nafion using electrochemical pore-directed nanolithography.Nano Letters, 5:2184–2187, 2005.

[27] K.A. Mauritz and R.B. Moore. State of understanding of Nafion. Chemical Reviews,104:4535–4586, 2004.

[28] M.B. Satterfield, P.W. Mjsztrik, H. Ota, J.B. Benziger, and A.B. Bocarsly. Me-chanical properties of Nafion and Titania/Nafion composite membranes for polymerelectrolyte membrane fuel cells. Journal of Polymer Science Part B: Polymer Physics,44(16):2327–2345, 2006.

19


[29] S. Song, W. Zhou, J. Tian, and R. Cai. Ethanol crossover phenomena and its influenceon performance of DEFC. Journal of Power Sources, 145:266–271, 2005.

[30] S. Motupally, A.J. Becker, and J.W. Weidner. Diffusion of water in Nafion 115 mem-branes. Journal of the Electrochemical Society, 147:3171–3177, 2000.

[31] L. Zhang, C. Ma, and S. Mukerjee. Oxygen permeation studies on alternative protonexchange membranes designed for elevated temperature operation. ElectrochimicaActa, 48:1845–1859, 2003.

[32] J. Divisek, M. Eikerling, V. Mazin, H. Schmitz, U. Stimming, and Y.M. Volfkovich. Astudy of capillary porous structure and sorption properties of nafion proton-exchangemembranes swollen in water. Journal of The Electrochemical Society, 145:2677–2683,1998.

20


Liquid AirSolid

Figure 1: Schematic depiction of partially saturated porous solid

21


χ

X

R

∂R

R0

∂R0x

Figure 2: Motion of homogenized body

22


(a) Dry (b) Low saturation

(c) High saturation (d) Full saturation

Figure 3: Local structure of the representative area element of an idealized porous mediumat four different states of saturation (“wet” micro-channels are shown as filled blue circlesand “dry” ones are shown as unfilled circles)

23


0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

liquid saturation φ

cap

illa

ryp

ress

ure

(MP

a)

Leverett experiment [20]proposed polynomial model

Figure 4: Capillary pressure as a function of liquid saturation for sand: polynomial fit ofexperimental data

24


0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


hei

ght

(m)

numerical simulationLeverett experiment [20]

Figure 5: Leverett experiment: height of water as a function of the saturation in sand

25


0

0.5

1

1.5

2

2.5

3

3.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


cap

illa

ryp

ress

ure

(MP

a)

Nafion experiment [32]proposed polynomial model

Figure 6: Capillary pressure as a function of liquid saturation for Nafion R©: polynomial fitof experimental data

26


44.24.44.64.8

5

0 100 200 300 400 500 600 700 800 900

time (ms)

volu

me

of

liqu

id(m

3) ×10−11

0.8

0.85

0.9

0.95

1

0 100 200 300 400 500 600 700 800 900

time (ms)

liqu

idsa

tura

tion

×10−11

1.15

1.17

1.19

1.21

1.23

1.25

0 100 200 300 400 500 600 700 800 900

time (ms)

tota

lvo

lum

e(m

3)

×10−11

×10−10

multiphase diffusionearlier work [15]

Figure 7: Squeezing of a saturated cube: Liquid and total volume as a function of time forthe case of squeezing

27


4.6

4.7

4.8

4.9

5

0 50 100 150 200 250 300

time (ms)

volu

me

ofli

qu

id(m

3)

×10−11

original oxygen permeabilityincreased oxygen permeability

Figure 8: Comparison of two different values of oxygen permeability kg: Liquid volume asa function of time for the case of squeezing

28


0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700 800 900 1000

time (s)

max

imu

mflex

ure

(mm

)

multiphase diffusionearlier work with C = 1.0 MPaearlier work with C = 1.3 MPaearlier work with C = 1.6 MPa

Figure 9: Flexure of Nafion R© film due to localized water absorption on its top surface

29

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