7. A Model for the Formation of Ice Lenses in an Unconfined, Water-saturated,
Porous Medium consisting of Spherical Particles

7.1 Model
Based on the experimental results obtained in Chapter 4-6, in this chapter, amodel for simulating the formation of ice lenses during freezing of unconfined uniform
porous media is presented.

7.1.1

System and conditionsFirst, we consider an X × Y × Z mm3 rectangle of water-saturated

homogeneous porous medium consisting of monodispersive, rigid-spherical, uniformsized fine particles with N0 particles per unit volume.
the rectangle is constant.

The initial temperature, T0, of

Then, the rectangle is frozen along the Z direction byholding the opposite ends at different temperatures: TH and T L (Figure 32a).
situation, enough water is supplied from the warmer end.

In thisThe rectangle is insulated

except at the ends and does not have any overburden pressures.

Once the thermal

conditions of the rectangle reach equilibrium, the isotherm is advanced at a rate of Vs bycooling both ends of the rectangle while maintaining the temperature difference (Figure
32b).

7.1.2

Generation of an ice lens
In Exp. 3, ice formed while excluding the particles ahead of the growth surface

when the freezing rate, Vf, was lower than the critical freezing rate, Vc.
freezing rate Vc was given by equation (28).

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The critical

TH

Temperature

TH
T0

Tf

Vs
freezing period (a )TLfreezing period (b )ZFreezin gdN0

Fig. 32. Schematic description model system.

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Let us then suppose that the growth of an ice lens in a water-saturated porous
medium consisting of fine particles (e.g. Exp. 1 and Exp. 2) is essentially the same
phenomenon as the ice growth expressed by equation (28).

We would then expect thatthe generation of an ice lens in the porous medium would depend on the critical freezing
rate, V c (Chalmers, 1964; Jackson et al., 1966).

In a porous medium, the particles

adjacent to the ice lens are pushed ahead by the growth surface (Exp. 3) in the samemanner as an isolated particle on an ice surface.

On the other hand, the force comingfrom viscous drag will act not only on the particles adjacent to the ice lens, but also on
the other particles.

Accordingly, the particles adjacent to the ice lens are pushed to thegrowth surface by the surrounding particles as the ice lens grows.

In fact, it isobserved in Exp. 3 that ice had formed while pushing particles ahead of it suddenly
entrap all the particles when the particles accumulated on the surface to a certain
thickness.

This observation supports the influences of surrounding particles on theparticles adjacent to the ice lens.

Therefore, we assume that the critical freezing rate,

Vc', for the generation of an ice lens in a porous medium depends on the number ofparticles per unit volume N at the freezing front. The critical freezing rate is given by
Vc' = AN,

(30)

where, A is a coefficient that includes the surface energy and viscosity in equation (28).

7.1.3

Growth of the ice lensAccording to equation (21), the growth rate of an ice lens, Vil, in saturated soil

is proportional to the supercooling degree at the growth surface, ∆Til.

Ice growth insome porous media also follows this relationship (Ozawa and Kinosita, 1989).
Considering the difference in the chemical potential of water between the growthsurface of the ice lens and pores between particles, Kuroda (1985) presented the growth- 105 -

rate of ice lens Vil as:

Vil =Qm∆Til / Tm (vi ∆pi - vw∆pw)(ak Til / D) + (d/K)

,

(31)

where, Q m is the molecular heat of melting of ice, vi and vw are the molecular volumesof ice and water, ∆pi and ∆pw are the pressure differences of ice- and water-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 water molecule, and K is the hydraulicconductivity near the ice lens.

For the freezing of porous media without overburdenpressure, an ice lens may grow at atmospheric pressure. Then, equation (31) becomesVil =Qm / T m∆Til(ak Til / D) + (d/K)

.

(32)

Vil = B ∆Til .

(33)

We then consider that the growth of an ice lens follows equation (33).

Fromequation (32), B in equation (33) is a coefficient that depends on the heat of melting andthe hydraulic conductivity near, and B is corresponding to 1.0 in equation (21)7.1.4

Heat transfer
Heat energy may be transferred through the porous medium by a number ofdifferent mechanisms, including conduction and convection of latent heat.

In thismodel, the conductive heat flux at the freezing front during freezing with ice lens
growth and without the ice lens are given by

kil dT +LρVl = k u dT ,
dz
dz

(34)

kf dT
dz

(35)

= ku dTdz,- 106 -

where, T is temperature, L is the latent heat of ice, ρ is the density of ice, and kil, ku, andkf are the thermal conductivity of the ice lens, unfrozen region, and frozen region,
respectively.

When an ice lens is growing in porous media, the freezing front

coincides with the growth surface of the ice lens (Exp. 4).

7.1.5 Number of particles near the growth surface
As water with dispersed glass particles freezes unidirectionally, particles
accumulate near the ice-water interface as the ice grows (Exp. 3).

It is expected thatthe particles accumulate near the growth surface of an ice lens in freezing porous media
consisting of unconfined fine particles (e.g. Exp. 1 and Exp. 2).

Therefore, thenumber of particles per unit volume N adjacent to the growing surface of the ice lenswould increase locally, as shown in Figure 33.

We assume that the number of particles

in the shaded portion of Figure 33 equals the number of particles excluded from thegrowing 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 number ofparticles per unit volume near the growing surface is given by3

tp

N = 3C z -Vil dt < z <03

Vil dt +
0

+ N0

C

0

tp

Vil dt0

Vil dt +

tp

2

tp

N0

,

tp

N0

Vil dt
0

C

,

(36)

where, tp is the elapsed time counted from the generation of the ice lens, z is thecoordinate distance from the point where the ice lens was generated, and C is a fitting
parameter. Since no particles 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.

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Ice
Lens

2

N = 3C X' +N0

Generation

Ice
Lens

Initialgrowthsurfacex=0Fig. 33.

X
X'=0

Schematic description change in number of particles near ice lens as it

growth.

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If the change in the number of particles due to the growth of the ice lens isregarded 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.

7.1.6

Ice lensing
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 shown in Figure 34.
First, we separate the freezing period of the porous medium into two periods for thedescription.

One freezing period (a) is the time from when the different temperatures

are first applied to the ends of the rectangle until a stable temperature gradient isestablished, and the other freezing period (b) is the time during which a constant Vs isapplied while maintaining the temperature gradient.

The rate of the advancing

freezing front, V f, is defined as the rate of the advancing isothermal line for the freezing
point of water, Tf.

If freezing progresses through the region where the number ofparticles per unit volume equals the initial number, such as in situ freezing, Tf will beheld at the initial value Tf0.

During ice lens growth, Vf is defined as V il, since thefreezing front corresponds to the surface of the ice lens in such a system (Exp. 4).

(1)

Initial period until a temperature gradient is established (a)Initially, the porous medium in the rectangle considered is held at a constanttemperature above the freezing point of water.

When different temperatures, which

sandwich the freezing point, are applied to opposite ends of the rectangle, the freezing
front within it advances at the rate of Vf.

At the early stage, the critical freezing rate

Vc' remains constant, since the number of particles, N, near the freezing front does notchange.

At this time, the freezing front advances at a rate Vf that is sufficiently faster- 109 -

than V c' and no ice lens forms.
rectangle stabilizes.

Then, V f decreases as the temperature gradient in the

If V f falls below Vc', an ice lens will form at the freezing front.

Growth of the ice lens follows equation (33) (Figure 34-i).
equivalent to the growth rate of the ice lens, Vil.

In this case, Vf is

During ice lens growth, the number ofparticles N adjacent to the growth surface increases following equation (36). Vc' at thegrowth surface decreases as a result of equation (30) (Figure 34-ii).

On the other hand,since the temperature at the growth surface of an ice lens increases with its growth, thesupercooling degree at the growth surface decreases and causes Vil to decay. Therefore,
the ice lens keeps growing while V il is less than V c'; then it stops growing when V ilreaches V c' (Figure 34-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, N0,

as the in situ freezing advances and causes 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 34-iv).

(2) With a constant Vs (b)Once a constant temperature gradient has been established (a), a constant Vs isapplied to the rectangle.
(Figure 34-v).

If Vf is less than Vc', an ice lens will form at the freezing front

This ice lens grows at the rate Vil.

as the ice lens grows (Figure 34-vi).

Vc' at the growth surface decreases

On the other hand, the mean ice lens growth rate,Vil, becomes constant under Vs (Exp. 1 and Exp. 2).

Therefore, V c' falls below Vil atsome point and the ice lens stop growing (Figure 34-vii).

Even if the ice lens stops

growing, the freezing front does not stop advancing due to Vs.

Then, if the freezing

front reaches a point where Vf falls below Vc' again, a new ice lens will form (Figure 34viii).

With this ice lens, the number of particles adjacent to the growth surface alsochanges as it grows; then the next ice lens forms.

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Freezing rate

Vf

Vf

(a)

V c'
(i)

(iv)

(ii)
V il

(iii)

V il

Elapsed time
Freezing rate

(v)

Vf
V c'

Vf

(b)

(viii)(vi)

V il

Fig. 34.

V il
(vii)
Elapsed time

Schematic description freezing rate.

V ilfreezing period (a) is the time fromwhen the different temperatures are first applied to the ends of the rectangle until astable temperature gradient is established, and the other freezing period (b) is the time
during which a constant Vs is applied while maintaining the temperature gradient.

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7.2

Simulation
The model was solved for a 32 × 1 × 1 (X × Y × Z) mm 3 rectangle filled with awater-saturated porous medium with the following thermophysical properties: latent
heat of melting ρL = 3.34 × 108 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 theinitial value of the freezing point Tf0 = -0.1 °C.

The porous medium consisted ofwater-saturated uniform-sized rigid-spherical particles with diameter d = 10 µm.

Theinitial particle number per unit volume, N0 = 7.0 × 105 mm-3, was found from thespecific gravity of the glass particles, G = 2.19, and the water content by weight,
w0 = 80%.

The coefficients A = 700 mm-2s-1 and C = 1.0 × 10 6 mm -5 were estimatedfrom the initial number of particles per unit volume and the relationship between the
thickness of and intervals between ice lenses shown in the Exp. 2.

The coefficient B =1.0 × 10-6 m s -1°C-1 in equation (33) was estimated from equation (21).

Here, the valueof A corresponds to the value calculated by equation (32) with QmTm-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 m2s-1, and K = 2.7 × 108 m2J-1K-1.
The initial temperature, T 0 = 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 constantVs = 0.6 µm s -1 was applied for a period of 200 min (b).
Figure 35a compares the numerical results for the growth of the warmest ice
lens with the experimental data measured in Exp. 2.

In the measurement, uniform-

sized glass beads with diameter of 9.7 µm and specific gravity of 2.19 were saturated
with pure water at a water content of 80%, then packed into a sample cell with a volumeof 70 × 20 × 3 mm 3.

The sample cell, in which the initial temperature was 2 °C, wasunidirectionally 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 thetemperature gradient was established in the sample cell, a constant Vs = 0.6 µm was- 112 -

applied.
In Figure 35a, the solid line and circles indicate the numerical results andexperimental data (Mutou et al., 1998), respectively.

The elapsed time in Figure 35a

was counted from the time when different temperatures were applied to the sample.
The freezing front that advanced in the sample with in situ freezing was computed in theearly stage.

The warmest ice lens was generated at an elapsed time of 28 min, andthen grew to a thickness of 1.7 mm over 400 min.

The computed results for the final

growth amount and the generation time and location of the warmest ice lens are in good
agreement with the experimental data, as shown in the figure.
Figure 35b shows the growth of each ice lens while applying a constant V s.
The solid line and circles indicate the numerical results and experimental data (Mutou etal., 1998), respectively.

The elapsed time was counted from the time when a constantVs was applied to the sample.
computed.

The rhythmical repetition of ice lens growth was

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 the freezing front again reached a pointthat had a suitable Vc'.

The computation expressed the tendency for the generation andgrowth of each ice lens well, as shown in the figure.

Furthermore, the calculated

locations at which ice lenses were generated were in good agreement with theexperimentally observed locations.

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Growth amount (mm)Growth amount (mm)

2 (a)

1
Experimental results
Numerical method

00

0.8

400
800
ELAPSED TIME (MIN.)(b)Experimental results
Numerical method

0.4

0

60
120
ELAPSED TIME (min.)Fig. 35. (a) Growth amount of the warmest ice lens.
under constant Vf.

- 114 -(b) Growth amount of ice lenses

7.3

Discussion
The growth of an ice lens calculated using the model obeys equations (30) and(33).

The self-diffusion coefficient and the hydraulic conductivity shown in equation(33) may vary due to alteration of sample 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 thatthe 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 growthsurface.

The growth tendency differs between the computed and experimental values

in Figure 35a because we neglected these effects and gave B a constant value inequation (30).

It seems necessary to adopt some function for B in order to calculate

the growth tendency accurately.
The rhythmical generation of ice lenses depends on equation (36).

As

mentioned above, a quadratic relationship was used for equation (36) instead of an
exponential relationship to ease computation.

Although computations using equation(36) are appropriate when the ice lenses are a few millimeters thick, the equation tends
to underestimate the value of Vc' when thicker ice lenses form with a small Vs.

Theslightly larger and earlier growth of the ice lens computed in Figure 35b may come fromthe form of equation (36). The equation will be modified in the future.
This model can be applied to freezing of porous medium consisting of micro
particles (from clay to silt size) and intermittent formation of ice lenses in it.

In thismodel, the behavior of particles in the vicinity of an ice lens is assumed as shown in
Figure 33. Then the generation and jump of ice lenses are described on the basis of thechanges of number of particles adjacent to the growing surface of the ice lens. In
actual soil, however, there are heterogeneous and non-uniform particles and someparticle contact each other then make framework structure. There would be mixture of- 115 -

large particles that build the framework and small particles that change their number
adjacent to an ice lens and affect the growth near an ice lens in a non-uniform porous
medium (Figure 36).

In this manner, our model can be used to explain the behavior ofsmall particles since the ice lens grows with excluding them.

And it is obvious fromequation (28) that the large particles are easier to encapsulate into the ice lens than thesmall particles.

Therefore, the large particles would not influence the growth of ice

lens very much.

On the other hand, the both sized particles would push andconsolidate each other; i.e. they would interact.

However, it dose not understand how

the interaction should be dealt with. For extending the model to the non-uniform
porous media or actual soil, it is necessary to consider the presence of particles thatbuild framework and the interaction of the particles with micro particles in the future.

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Freezing

Initial number

distance from ice lens x (mm)Fig. 36. Schematic description of the vicinity of the ice lens.

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7.4

Summary
On the basis of Exp. 1-4, a model was developed for the formation of ice

lenses in air- and solute-free porous media consisting of unconfined uniform-sized fineparticles under unidirectional freezing conditions.

This model is based on the freezing

rate, the temperature at the growth surface of the ice lens, and the number of particles
near the growing ice.

An ice lens generates while a porous medium is freezing

unidirectionally when the rate of the advancing freezing front falls below a critical rateof progress. An ice lens grows as a result of supercooling at its growing surface. The
number of particles adjacent to the growing surface of the ice lens increases as particles
are excluded from the growing surface and accumulate.

This change in the number ofparticles per unit volume is assumed to alter the critical rate of progress of the freezing
front, and the ice lens then stops growing.

Once one ice lens has stopped growing, a

new ice lens then forms when the freezing front again reaches a point that has a suitable
number of particles per unit volume.
Our results show that this model can be used to simulate the location, growth,and formation of an intermittent layer of ice lenses in such a system.

This model isappropriate for the analysis of heat and mass transfer in unconfined, water-soaked
porous media under freezing conditions.

To consider a model in such an ideal systemis effective for understanding of the basic mechanism of the ice lensing, although thefreezing of actual soil is indeed complex phenomenon.

Further examination ofcoefficients in equations of this mode and extension of the model to the non-uniform
porous media and actual soil remain to be solved in the future.

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