For the unconfined water-saturated uniform
porous media consisting of fine particles, thefollowing results were obtained from unidirectional
freezing experiments.
The freezing rate andtemperature at the growth surface of the warmest ice
lens play important roles in the formation of ice
lenses (Watanabe et al., 1997).
The particle
number changes drastically near the growth surface
of an ice lens during its growth (Muto et al., 1999).
Watanabe (1999) compared ice lenses observed insoil, and in uniform and non-uniform porous media(Figure 1).
They concluded that using suchhomogeneous samples facilitates observation of ice
lenses, and will help to explain the formation of ice
lenses in actual soils. In this paper, we report on amodel that simulates the formation of intermittent
ice lenses in water-saturated porous media
consisting of uniform-sized fine particles, using thefreezing rate, the temperature at the growth surfaceof the ice lens, and the change in particle number
near the growth surface.

2 MODEL
2.1 System and conditionsFirst, we consider an X × Y × Z mm 3 rectangle ofwater-saturated homogeneous porous medium
consisting of monodispersive, rigid-spherical,
uniform-sized fine particles with N0 particles perunit volume. The initial temperature, T0, of therectangle is constant. Then, the rectangle is frozen
along the Z direction by holding the opposite ends at
different temperatures: TH and TL (Figure 2a). In
this situation, enough water is supplied from thewarmer end. The rectangle is insulated except at
the ends and does not have any overburdenpressures. Once the thermal conditions of therectangle reach equilibrium, the isotherm isadvanced at a rate of Vs by cooling both ends of therectangle while maintaining the temperature
difference (Figure 2b).
2.2

Figure 1. Intermittent layers of ice lenses in water-saturated
Fujinomori soil (a; Watanabe et al., 1997), porous medium
consisting of uniform fine glass particles (b; Mutou et al.,1998) and porous medium consisting of non-uniform glass
particles (c; Watanabe 1999). The black portions are ice
lenses. The scale bar indicates 1.0 mm. The ice lenses grewin the direction of heat flow (from left to right in figure).
TH

Temperature

TH
T0

Tf

Vs
freezing period (a)TLfreezing period (b)ZFreezingdFigure 2. Schematic description model system.

N0

Y

X

Generation of an ice lensMutou et al. (1998) performed a unidirectional
freezing experiment of water with isolated fine glass
particles and observed the exclusion and entrapmentof the particles from and into the growth surface ofice crystals. They found that ice forms while
excluding the particles ahead of the growth surface
when the freezing rate, Vf, is lower than the criticalfreezing rate, Vc.
This relationship was also
reported for other kinds of particles by Uhlmann etal. (1964), Chernov and Temkin (1977), and Köber
et al. (1992). For a particle near an ice crystal,
Köber et al. (1992) considered the balance of two
counteracting forces: the attractive force from theviscous drag due to fluid flow around the particle
that favors entrapment and the repulsive force thatoriginates from van der Waals forces. Then, theyexpressed the critical freezing rate Vc as

Vc =

∆γ a
6η d

(1)

where, ∆γ is the difference in the surface energy
between particle-liquid and liquid-ice, a is theaverage molecular distance in the liquid, η is thecoefficient of viscosity, and d is the diameter of theparticle. This equation shows that the criticalfreezing rate decreases with increasing diameter.
Let us then suppose that the growth of an ice lensin a water-saturated porous medium consisting offine particles is essentially the same phenomenon as
the ice growth expressed by equation (1). We
would then expect that the generation of an ice lensin the porous medium would depend on the criticalfreezing rate, Vc (Chalmers, 1964; Jackson et al.,1966). In a porous medium, the particles adjacentto the ice lens are pushed ahead by the growthsurface (Muto et al., 1999) in the same manner as an
isolated particle on an ice surface (Mutou et al.,1998). On the other hand, the force coming fromviscous drag will act not only on the particles
adjacent to the ice lens, but also on the otherparticles. Accordingly, the particles adjacent to theice lens are pushed to the growth surface by thesurrounding particles as the ice lens grows. In fact,
Mutou et al. (1998) observed that ice suddenly
entrapped all the particles when the particles
accumulated on the ice surface to a certain thickness.
This observation supports the influences ofsurrounding particles on the particles adjacent to theice lens. Therefore, we assume that the criticalfreezing rate, V c', for the generation of an ice lens ina porous medium depends on the number of particles
per unit volume N at the freezing front. The
critical freezing rate is given by
Vc' = AN

(2)

at atmospheric pressure.
becomesVil =Then, equation (4)Qm / Tm∆Til .
(ak Til / D) + (d/K)(5)

We then consider that the growth of an ice lensfollows equation (3). From equation (5), B inequation (3) is a coefficient that depends on the heatof melting of ice and the hydraulic conductivity near
the ice lens.
2.4 Heat transfer
Heat is transferred through the porous medium by anumber of different mechanisms: conduction,
convection and so on.
In this model, theconductive heat flux at the freezing front duringfreezing with ice lens growth and without the ice
lens are given by

where, A is a coefficient that includes the surface
energy and viscosity in equation (1).

kil dT +LρVil = ku dT
dz
dz

(6)

2.3

kf dT
dz

(7)

Growth of the ice lens

Watanabe et al. (1997) reported that the growth rateof an ice lens, Vil, in saturated soil is proportional to
the supercooling degree at the growth surface of theice lens, ∆Til.
Vil = B∆Til(3)

where, B is an empirical constant. Ice growth insome porous media, such us water-saturated fineglass particles, also follows this relationship (Ozawa
and Kinosita, 1989; Watanabe, 1999). Considering
the difference in the chemical potential of water
between the growth surface of the ice lens and pores
between particles, Kuroda (1985) presented thegrowth rate of ice lens Vil as:
Vil =Qm∆Til / Tm (vi ∆pi - vw∆pw)(ak Til / D) + (d/K)(4)

where, Q m is the molecular heat of melting of ice, vi
and v w are the molecular volumes of ice and water,∆pi and ∆pw are the pressure differences of ice- andwater-atmospheres, k is the Boltzmann factor, Til is
the temperature at the growth surface of the ice lens,
D is the self-diffusion coefficient of the watermolecule, and K is the hydraulic conductivity near
the ice lens. For the freezing of porous media
without overburden pressure, an ice lens may grow= ku dT
dz

where, T is temperature, L is the latent heat of ice, ρis the density of ice, and kil, ku, and kf are the thermal
conductivity of the ice lens, unfrozen region, andfrozen region, respectively. When an ice lens is
growing in porous media, the freezing front
coincides with the growth surface of the ice lens
(Watanabe and Mizoguchi, 2000).
2.5

Number of particles near the growth surface

As water with dispersed glass particles freezes
unidirectionally, particles accumulate near the icewater interface as the ice grows (Mutou et al., 1998).
It is expected that the particles accumulate near thegrowth surface of an ice lens in freezing porous
media consisting of unconfined fine particles (e.g.
Muto et al., 1999). Therefore, the number ofparticles per unit volume N adjacent to the growing
surface of the ice lens would increase locally, as
shown in Figure 3. We assume that the number ofparticles in the shaded portion of Figure 3 equals thenumber of particles excluded from the growing ice
lens, and then approximate the curved line, which
defines the shaded portion, with a simple quadratic
equation. In this manner, the change in the numberof particles per unit volume near the growing surface
is given by3

tp

N = 3C z -Vil dt0

Vil dt +3tp

Vil dt < z <tp

N0

Vil dt0

Vil dt +0+ N0 ,

C0tp

2.6 Ice lensing

2

tp

N0

(8)

C

0

where, tp is the elapsed time counted from thegeneration of the ice lens, z is the coordinate
distance from the point where the ice lens was
generated, and C is a fitting parameter. Since noparticles exist in the ice lens, N equals zero in the ice
lens. As freezing progresses without an ice lens, N
at the freezing front keeps the initial value N0.
If the change in the number of particles due to thegrowth of the ice lens is regarded to be equivalent to
the theory of consolidation proposed by Terzaghi
and Peck (1967), equation (8) is an exponential
relationship. In this study, however, we used aquadratic relationship to ease computation.

Ice
Lens

2

N = 3C X' +N0

Generation

growthsurface

Ice
Lens

Initialx=0

Figure 3.
ice lens.

X'=0

Schema of change in number of particles near the

VfFreezing rate

X

Vf

(a)

Vc'
(i)

(iv)

(ii)
Vil

(iii)

Vil

Freezing rate

Elapsed time
(v)

Vf
Vc'

Vf

(b)

(viii)
(vi)

Vil

Vil
(vii)
Elapsed timeVil

Figure 4. Schematic diagram of freezing rate and criticalfreezing rate.

On this basis, we here consider the formation of ice
lenses in an air- and solute-free porous medium
consisting of uniform-sized fine particles, as shownin Figure 4. First, we separate the freezing periodof the porous medium into two periods for thedescription. One freezing period is the time fromwhen the different temperatures are first applied tothe ends of the rectangle until a stable temperature
gradient is established, and the other freezing period
is the time during which a constant Vs is applied
while maintaining the temperature gradient. The
rate of the advancing freezing front, Vf, is defined as
the rate of the advancing isothermal line for thefreezing point of water, Tf. If freezing progresses
through the region where the number of particles perunit volume equals the initial number, such as in situ
freezing, Tf will be held at the initial value Tf0.
During ice lens growth, Vf is defined as Vil, since thefreezing front corresponds to the surface of the ice
lens in the porous medium (Watanabe and
Mizoguchi, 2000).
2.6.1 Initial period until a temperature gradient isestablished (a)Initially, the porous medium in the rectangle
considered is held at a constant temperature above
the freezing point of water.
When different
temperatures, which sandwich the freezing point, areapplied to opposite ends of the rectangle, thefreezing front within it advances at the rate of V f.
At the early stage, the critical freezing rate Vc'
remains constant, since the number of particles, N,
near the freezing front does not change. At thistime, the freezing front advances at a rate Vf that is
sufficiently faster than Vc' and no ice lens forms.
Then, Vf decreases as the temperature gradient in therectangle stabilizes. If Vf falls below Vc', an ice
lens will form at the freezing front. Growth of theice lens follows equation (3) (Figure 4-i). In thiscase, Vf is equivalent to the growth rate of the ice
lens, Vil. During ice lens growth, the number ofparticles N adjacent to the growth surface increases
following equation (8). Vc' at the growth surface
decreases as a result of equation (2) (Figure 4-ii).
On the other hand, since the temperature at thegrowth surface of an ice lens increases with its
growth, the supercooling degree at the growthsurface decreases and causes Vil to decay.
Therefore, the ice lens keeps growing while Vil isless than Vc'; then it stops growing when Vil reaches
Vc' (Figure 4-iii). Once the ice lens has stopped
growing, the freezing front advances with in situ
freezing. N at the freezing front approaches its
initial value, N 0, as the in situ freezing advances andcauses V c' to increase. If the freezing front reaches
a point where Vf falls below Vc' again, a new ice
lens will then form (Figure 4-iv).

2.6.2 With a constant Vs (b)Once a constant temperature gradient has been
established (a), a constant Vs is applied to therectangle. If Vf is less than Vc', an ice lens will
form at the freezing front (Figure 4-v). This ice
lens grows at the rate Vil. Vc' at the growth surface
decreases as the ice lens grows (Figure 4-vi). On
the other hand, the mean ice lens growth rate, V il,
becomes constant under Vs (Watanabe et al., 1997).
Therefore, V c' falls below Vil at some point and theice lens stop growing (Figure 4-vii). Even if the ice
lens stops growing, the freezing front does not stopadvancing due to Vs. Then, if the freezing front
reaches a point where Vf falls below Vc' again, a
new ice lens will form (Figure 4-viii). With this ice
lens, the number of particles adjacent to the growthsurface also changes as it grows; then the next ice
lens forms.
3 RESULTS AND DISCUSSION
The model was computed for a 32 × 1 × 1 (X × Y ×Z) mm3 rectangle filled with a water-saturated
porous medium with the following thermophysicalproperties: latent heat of melting ρL = 3.34 × 10 8 J
m-3, volumetric specific heat C = 4.21 J g-1K-1,
thermal diffusivity αf = 0.64 × 10 -6 m 2s-1, αuf = 0.57
× 10 -6 m 2s-1, αil = 0.53 × 10 -6 m 2s-1, and the initialvalue of the freezing point Tf0 = -0.1 °C. The
porous medium consisted of water-saturated
uniform-sized rigid-spherical particles with diameterd = 10 µm. The initial particle number per unit
volume, N0 = 7.0 × 10 5 mm -3, was found from thespecific gravity of the glass particles, G = 2.19, andthe water content by weight, w0 = 80%. The
coefficients A = 700 mm -2s-1 and C = 1.0 × 106 mm-5
were estimated from the initial number of particles
per unit volume and the relationship between the
thickness of and intervals between ice lenses shownin the experiment performed by Mutou et al. (1998).
The coefficient B = 1.0 × 10 -6 m s -1°C-1 in equation(3) was estimated from observation of ice lenses(Watanabe et al., 1997, Watanabe, 2000). Here,the value of A corresponds to the value calculated byequation (5) with Q mTm-1 = 3.70 × 1023 J K-1, a = 3.8
× 10-10 m, k = 1.38 × 10-23 J K -1, D = 1.0 × 10-10 m2s1
, and K = 2.7 × 108 m2J-1K-1.
The initial temperature, T0 = 2 °C, is set. Then,
TH = 2 °C and TL = -6 °C were applied at opposite
ends of the rectangle for a period of 800 min (a).
Afterwards, a constant Vs = 0.6 µm s -1 was applied
for a period of 200 min (b).
Figure 5a compares the numerical results for thegrowth of the warmest ice lens with theexperimental data measured by Mutou et al. (1998).
In the measurement of Mutou et al. (1998), uniformsized glass particles with diameter of 9.7 µm andspecific gravity of 2.19 were saturated with purewater at a water content of 80 %, then packed into asample cell with a volume of 70 × 20 × 3 mm3.
The sample cell, in which the initial temperature was
2 °C, was unidirectionally frozen with different
temperatures: TH = 4 °C and TL = -4 °C at points 32
mm apart, i.e. the temperature gradient established
was 0.25 °C mm -1. After the temperature gradientwas established in the sample cell, a constant Vs =0.6 µm was applied.
In Figure 5a, the solid line and circles indicate thenumerical result and experimental data (Mutou etal., 1998), respectively. The elapsed time in Figure
5a was counted from the time when different
temperatures were applied to the sample. The
freezing front that advanced in the sample with insitu freezing was computed in the early stage. The
warmest ice lens was generated at an elapsed time of
28 min, and then grew to a thickness of 1.7 mm over
400 min. The computed results for the final growthamount and the generation time and location of thewarmest ice lens are in good agreement with theexperimental data, as shown in the figure.
The growth of an ice lens calculated using themodel obeys equations (3) and (5). The selfdiffusion coefficient and the hydraulic conductivity
shown in equation (5) may vary due to alteration ofsample uniformity, growth of the ice lens, and theredistribution of particles near the ice lens. The
phase transition of water in a small pore depends on
the pore size (Handa et al., 1992). Accordingly, it
is considered that the freezing point is depressed
locally at or near the growth surface of the ice lens,
where particles are accumulating due to the
exclusion of particles from the growth surface. The
growth tendency differs between the computed andexperimental values in Figure 5a because weneglected these effects and gave B a constant value
in equation (3). It seems necessary to adopt somefunction for B in order to calculate the growthtendency accurately.
Figure 5b shows the growth of each ice lens while
applying a constant Vs. The solid line and circles
indicate the numerical results and experimental data
(Mutou et al., 1998), respectively. The elapsed
time was counted from the time when a constant Vs
was applied to the sample.
The rhythmical
repetition of ice lens growth was computed. The
first ice lens grew for 60 min. The second ice lens
was generated in an area warmer than the first one
and grew for 30 min. Once this ice lens stopped
growing, the next ice lens then formed when thefreezing front again reached a point that had asuitable Vc'.
The computation expressed thetendency for the generation and growth of each icelens well, as shown in the figure. Furthermore, thecalculated locations at which ice lenses weregenerated were in good agreement with theexperimentally observed locations. The rhythmical
generation of ice lenses depends on equation (8).

REFERENCE

2 (a)

1
Experimental results
Numerical method

00

400
800
Elapsed time (min )(b)

Experimental results

0.8

Numerical method

0.4

0

60
120
Elapsed time (min )Figure 5. (a) Growth amount of the warmest ice lens.
(b) Growth amount of ice lenses under constant Vf.

As mentioned above, a quadratic relationship wasused for equation (8) instead of an exponential
relationship to ease computation.
Althoughcomputations using equation (8) are appropriate
when the ice lenses are a few millimeters thick, theequation tends to underestimate the value of Vc'
when thicker ice lenses form with a small Vs. The
slightly larger and earlier growth of the ice lenscomputed in Figure 5b may come from the form ofequation (8). The equation will be modified in the
future.
4 CONCLUSIONS
A model was developed for the formation of ice
lenses in air- and solute-free porous media
consisting of unconfined uniform-sized fine particles
under unidirectional freezing conditions.
Thismodel is based on the freezing rate, the temperature
at the growth surface of the ice lens, and the change
in the critical freezing rate due to the change in thenumber of particles near the growing ice. Our
results show that this model can be used to simulate
the location, growth, and formation of intermittent
layers of ice lenses in homogeneous porous media
consisting of fine particles.
This model isappropriate for the analysis of heat and mass transfer
in unconfined, water-soaked porous media underfreezing conditions.

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