continuum modeling and simulation of multiphase di usion ... · continuum modeling and simulation...
TRANSCRIPT
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
Multiphase diffusion through porous elastic solid
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)
4
Q. Zhao and P. Papadopoulos
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)
5
Multiphase diffusion through porous elastic solid
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)
6
Q. Zhao and P. Papadopoulos
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)
7
Multiphase diffusion through porous elastic solid
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
Q. Zhao and P. Papadopoulos
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
Multiphase diffusion through porous elastic solid
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
Q. Zhao and P. Papadopoulos
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
Multiphase diffusion through porous elastic solid
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
Q. Zhao and P. Papadopoulos
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
Multiphase diffusion through porous elastic solid
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
Q. Zhao and P. Papadopoulos
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
Multiphase diffusion through porous elastic solid
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
Q. Zhao and P. Papadopoulos
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
kgµgρgε(1− φ)N e
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
kgµgρgε(1− φ)N e
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
Multiphase diffusion through porous elastic solid
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
Q. Zhao and P. Papadopoulos
[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
Multiphase diffusion through porous elastic solid
[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
Q. Zhao and P. Papadopoulos
Liquid AirSolid
Figure 1: Schematic depiction of partially saturated porous solid
21
Multiphase diffusion through porous elastic solid
χ
X
R
∂R
R0
∂R0x
Figure 2: Motion of homogenized body
22
Q. Zhao and P. Papadopoulos
(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
Multiphase diffusion through porous elastic solid
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
Q. Zhao and P. Papadopoulos
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
liquid saturation φ
hei
ght
(m)
numerical simulationLeverett experiment [20]
Figure 5: Leverett experiment: height of water as a function of the saturation in sand
25
Multiphase diffusion through porous elastic solid
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
liquid saturation φ
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
Q. Zhao and P. Papadopoulos
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
Multiphase diffusion through porous elastic solid
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
Q. Zhao and P. Papadopoulos
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