Hydraulic Conductivity in Frozen Unsaturated Soil
Kunio Watanabe
Graduate school of Bioresources, Mie University, Japan
Tomomi Wake
Graduate school of Bioresources, Mie University, Japan

Abstract
Freezing experiments of silt and sand columns were carried out, and water and heat flows were observed. To estimate
unsaturated hydraulic conductivity of soils, K(h), at above-freezing and subzero temperatures, Darcy’s law was solved
under a non-isothermal condition with ice formation. K(h) steeply decreased with decreasing soil water pressure, h,and more gradually decreased with soil freezing. The results show that the hydraulic model, in which water content
became constant under h< -105 cm, underestimated the K(h) of frozen soil, and suggest that the impedance factor,
which reduced K(h) for frozen soil, is not necessary when accurate soil water and soil freezing characteristics areavailable. The hydraulic model, which can express two types of soil water flow, such as capillary and film flows,
appears to be useful for expressing the hydraulic properties of soils under the freezing process.
Keywords: soil freezing characteristic; soil water characteristic; TDR; unsaturated hydraulic conductivity; water andheat flows.

Introduction
Knowledge of water flow in frozen and thawing groundis important to investigating water and solute redistributions
in soil during winter (Baker & Spaans 1997) and in studying
the mechanism of frost heaving (Wettlaufer & Worster 2006).
Changes in soil properties, such as hydraulic conductivity,
have also received research attention. For example, changes
caused by ground freezing have been examined by applying
an artificial soil-freezing technique to stabilize soil andform a barrier against hazardous waste (McCauley et al.
2002). Moreover, a major concern of hydrological andclimate modeling is how to express change in soil hydraulicproperties.
Burt & Williams (1976) and Horiguchi & Miller (1983)
measured the steep decrease in hydraulic conductivity withsoil freezing, although within a small temperature range.
Using oil as a fluid, McCauley et al. (2002) measured
saturated hydraulic conductivity of frozen soil at various
temperatures. However, few experimental studies haveexamined the unsaturated hydraulic conductivity of frozen
soil, Kf(h). The unsaturated hydraulic conductivity ofunfrozen soil, K(h), is usually expressed by the formula
proposed by Brooks & Corey (1964), Clapp & Hornberger(1979), or van Genuchten (1980). For frozen soil, Harlan(1973) used K(h) instead of Kf(h), assuming the samefilm-water geometry between frozen and unfrozen soils.
However, numerical simulations have suggested that thisassumption overestimates water flow near the freezing front(Harlan 1973, Taylor & Luthin 1978, Jame & Norum 1980).
Guymon & Luthin (1974) and Tao & Gray (1994) expressed
Kf(h) from K(h) by subtracting ice content from saturated
water content. When the soil is frozen, the presence of ice
in some pores may block water flow. To account for thisblocking, several impedance factors have been introduced(James & Norum 1980, Lundin 1990, Smirnova et al. 2000).

However, Black & Hardenberg (1991) criticized the use of
an impedance factor, stating that it is a potent and wholly
arbitrary correction function for determining Kf(h). Newman& Wilson (1997) also concluded that an impedance factor is
unnecessary when an accurate soil water characteristic curveand relationship between K(h) and soil water pressure aredefined.
Measuring unsaturated hydraulic conductivity for frozen
soil remains difficult, and a complete expression for Kf(h) is
still not available. In addition, the model for Kf(h) should becorrelated with the soil water characteristics and soil freezing
characteristics for ease of use in numerical simulations(Watanabe et al. 2007). In this experiment, we estimatedKf(h) from water and heat flow measurements in a frozen
soil column and discuss a model for Kf(h).

TheoryAssuming that vapor and ice flows are negligible, variably
saturated water flow in above-freezing and subzero soil is
described using a modified Richards equation as follows(Noborio et al. 1996, Hansson et al. 2004).
wT h
wt



Ui wTi T
Ul wt

w §
wh
wT ·
 K h  K h Jh
¨K h
¸
wz ©
wz
wz ¹

(1)

where θ is volumetric liquid water content, θi is volumetricice content, t is time,J z is a spatial coordinate, ρi is density
K (ice,h) ρ is densityw of water, h is the water pressure head,ofw § 'T'h · γ is1theJ h andT is temperature,¨¸ surface tension of soil water.
'z'z ¹
The terms©in parentheses on
the right-hand side of equation(1) represent the water flux, Jw, obtained from the change intheL f of liquid water and ice. Thus, if we measure thedP amountpressuredT vl Tand temperature gradient, K(h) [or Kf(h) at subzero
temperature] would be derived as
1927
Tm  T

TmV
Ui L f r

1928   Ninth International Conference on Permafrost
Table 1. Experimental conditions and physical properties of soil.
g cm–3m3m–3 θm3m–3 θW m 1 -1Kθθθθcm h–1

TDS

1.18
0.40θ
0.569θ

1.45
0.15θ
0.36θ

0.20(0.20)
θ

0.25(0.25)
0.96(1.50)

0.52(0.55)
θ
0.66(0.76)
θ
0.25θ

1.06(1.05)
50.6θ

van Genuchten parameterθr
m3m–3θ 0.03
0.015
α
m–1
0.16
3.36
n
1.38
7
l
0.552
-0.5
Durner parameterθr
m3m–3 θ 0.06
0
α1
m–1
0.35
3.466
n1
3.10
6.40
l
–0.08
-0.5
α2
m–1
0.011
0.027
n2
1.70
1.40
wT h
wT T
w §
wh 0.461
wT ·w2  Ui i
0.105
K
h

K
h

K
h
J
h
¨
¸
wt
w
w
w
wz ¹
U
t
z
z
©
*
l thermal conductivity is average of 2 to 20°C
The value forfor
unfrozen soil and -5 to -20°C for frozen soil.

K ( h)

Jw



(2)
'h ·
§ 'T
 1¸
Jh
¨
'
'
z
z
©
¹
wT h
U wT T
w §
wh
wT ·
 i i
 K h  K h Jh¨K h¸
wt valuewz ©
wz can be obtained fromwz ¹ theUl ofwth in unfrozenThesoil
L f characteristics (relationship between θ and h)
dP watersoil
when
the profile of θ, while h in frozen soil candT measuring
vl T
be calculated from the temperature profile derived from theJgeneralized
K ( h)  form ofw the Clausius–Clapeyron equation by'T'hbetween·
assuming a§¨ differences
ice and water. Specifically,T© m'
Vz J h  'z  1¸¹theTm iceT pressure is sometimes assumed to equal zero gauge
Ui L f rpressure (Williams& Smith 1989, Hansson et al. 2004):

dP L f(3)n mdT
Se 1vlT D hwhere P is the pressure ( = ρwgh), g is the acceleration due togravity, Lf is the latent heat of freezing, and vl is the specificmvolume of Twater.
thesoil freezing
V Thus,2n  m1 equation (3) givesª1  D 2 h n2 ºTSme  Tw1 ª1 m D(relationshiph 1 º  w2between
characteristics
θ and¬ Ui L f r1¼¬
¼ T) from the soil
water characteristics and vice versa.

Se 1  D h Material and2 MethodsmKThe( h) samples
K s Sel ª1 used1  Sine1/ mthisº study were Fujinomori silt«¬»¼
(FSi) and Tottori dune sand (TDS).
Figure 1 shows the soil
 m1
m2
water characteristics
byseveralphysicalmethods
n1 measuredn 
Se w1 ª1  D1h º  w2 ª1  D 2 h 2 º(Jury & Horton
2004)¬¼ for both¬ soils as well
¼ as soil freezing
T i /(T t T r )K ( h ) frozen 10 :K (byh ) pulsed nuclear magnetic
characteristics
measuredresonance (NMR) measurement and depicted by equationn mm
K ( h) K s Sel ª1  1  Se1/ m l º
«
»
¬
K Se
K s w1Se1  w2 Se 2 ¼m2

Liquid water content, θ (cm3cm -3)

Bulk densityθ when packed
θ saturation
Thermal conductivity*at θ = 0.00 (frozen)
at θ = 0.17 (frozen)
at θ = 0.24 (frozen)
at θ = 0.29 (frozen)
Saturated hydraulic cond.

FSi

0.6

0.001

|T | (°C)

0.01

1

10
Hanging water
Pressure plate
Vapor pressure
Dew point
NMR
Eq .(5)Eq. (6)

0.4

FSi
0.2

TDS
0

10

2

3

10

4

10

|h | (cm)Figure 1. Soil water characteristics measured by hanging water,pressure plate, vapor pressure, and chilled mirror dew point
measurement methods, and soil freezing characteristics measured
by pulsed NMR measurement for FSi and TDS (in thawing
process).

(1)(3). FSi is highly susceptible to frost and retains much liquidwater even when |h| > 104 cm (T < -1°C),(2)while for TDS, θ =0.03 when |h| > 102 cm (T < -0.01°C).
The TDS were preliminarily washed in deionized water,
and the FSi was passed through a 2 mm (1)screen. Each sample
was mixed with distilled and deionized water and packed(3) diameter of 7.8
into an acrylic column with an internal
cm and a height of 35 cm. Table 1 lists the experimental
conditions and physical properties of the samples. Fifteen(2)copper-constantan thermocouples and seven time domainreflectometry (TDR) probes were inserted into each column,(4)
and the side wall of the column was insulated. The TDR
was preliminarily calibrated for measuring unfrozen watercontent by comparison to the pulsed NMR(3) measurement.
The column was settled at an ambient temperatureof 2°C for(5)
24 h to establish initial water and temperature profiles andthen frozen from the upper end by controlling temperature atboth ends of the column (TL = -8°C and TH = 2°C). During
the experiment, no water flux was allowedfrom either(4)(6)
end, and profiles of temperature, water content, and soluteconcentration (ECa) were monitored using thermocouples
and TDRs. A series of experiments with different durations(5)freezing conditionof freezing were then performed for each(7) gradient). At the(i.e., same freezing rate and temperature
end of the experimental series, the sample was cut into 2.5
cm intervals to measure the total water content by the dry(6)TDR readings, itoven method. From thermocouples and(8)
was confirmed that each column had the same temperature
and water profiles during the series of experiments.

(7)
m

2

5

10

Watanabe and Wake   1929  

0

0h
6h

0h
6h

10

10

28h

28h
20

FSi

Depth (cm)Depth (cm)

20

30

50h

50h

0

0h
10

FSi

30

0h
6h

10

6h

48h

48h
72h

20

30
-8

20

TDS
-6

72h

TDS

30
-4

-2

0

Temperature (°C)

2

4

0

0.1

0.2

0.3

0.4

Water content, θ (cm3 cm-3)

0.5

Figure 2. Temperature profiles for FSi (0, 6, 28, 50 h after freezing
started) and TDS (0, 6, 48, 72 h after freezing started). Arrows
represent the freezing front.

Figure 3. Moisture profiles for FSi and TDS at the same freezingtime as shown in Figure 2. The solid line and dashed line represent
total water and unfrozen water contents, respectively.

Results

θ in the frozen area, and decrease of θ in the unfrozen areawith the advancing freezing front were observed, implyingthat the soil water flowed not only through the unfrozen area,
but also the frozen area. The gradient of the initial θ profile
of TDS corresponded to its soil water characteristics (Fig.
1). In TDS, the soil water in the unfrozen area flowed to andaccumulated near the freezing front because of suction at
the freezing front caused by ice formation. Water flow in theunfrozen area continued 48 h or later after freezing began,although there was no apparent advance of the freezing
front. Meanwhile, much less water flow was observed in thefrozen area.
The profile of water flux, Jw, was then calculated bydeveloping θ profiles (Fig. 3) with the boundary conditionof no water flux. In early stage of freezing (0–6 h), soil
water in almost the entire column moved upward at about
Jw = 0.04 cm h–1 for FSi and Jw = 0.01 cm h–1 for TDS. The
progression of the peak observed in the Jw profile coincidedwith the freezing front. In the frozen area, Jw in SFi was ≥10times larger than that in TDS and exponentially decreased
as the temperature decreased (Jw = 0.007|T|–0.69 for FSi and
0.0012|T|–1.45 for TDS).

Heat and water flow
Figure 2 shows temperature profiles of the freezing
soils. When both ends of the column were set at different
temperatures, the soil near the column ends quickly reached
the required temperatures. In FSi, the advancing rate of
the 0°C isotherm was 1.57, 0.34, and 0.16 cm h–1 for 0–6,
6–24, and 24–48 h, respectively, and approximately 0.25°C
lowering of the freezing point was observed. The freezing
rate of TDS was similar to that of FSi, although TDS hadlarger thermal conductivity (Table 1). Even after 72 h, thetemperature profile of the subzero area in TDS did not reacha linear shape as in FSi and in the unfrozen area in TDS.
Figure 3 presents water profiles in FSi and TDS at thesame freezing time as shown in Figure 2. The solid line
indicates total water content, θt, measured by the dry-ovenmethod, and the dashed line indicates unfrozen water content
measured by TDR. The ice content, θi, was obtained bysubtracting the unfrozen water from the total water content.
FSi had a relatively vertical initial θ profile, having similar θvalues for h <100 cm (Fig. 1). An increase of θt, decrease of1930   Ninth International Conference on Permafrost

0

Depth (cm)

10

0h

0h

|T|–1.75 and K(h) = 8.2 × 10–14 θ–6.96 for TDS. This power law
relationship was consistent with the relationship between Jw
and |T| mentioned above. The θI–T relationship is sometimes
expressed by power law (Anderson & Tice 1972) and can beconverted through equation (3) to a θ -h relationship, which
may also be expressed by power law (Brooks & Corey
1964). The shape of the formula indicating K(h) may arise
from the soil water characteristics (θ-h) and soil freezingwT h
U wT Tw §whwT ·characteristics
 i i (θi-T). ¨ K h
 K h  K h Jh
¸
wt
wz ©
wz
wz ¹
Ul wt

6h
28h

6h

50h

48h
20

72h
30
10 1

10 2

Discussion

FSi
TDS
10 3

|h| (cm)

10 4

10 5

Figure 4. Profile of soil water pressure estimated from Figures 2
and 3.

Change in hydraulic conductivity
Figure 4 shows the profile of soil water pressure h estimatedby the soil water characteristics (Fig. 1) with the θ profile(Fig. 3) and equation (3) with the temperature profile (Fig.
2). Solute effect was negligible during the experiments, sincevery low solute concentration was confirmed by TDR (ECa)readings. Note that for equation (3) there was discontinuity
in the h profiles near 0°C, especially for unsaturated soil;
therefore, we used linear interpolation to connect the closest
calculated h values between the unfrozen and frozen areas.
With freezing, |h| steeply increased (h < -103 cm). A larger
difference in h between unfrozen and frozen areas wasobserved in TDS than in FSi.
The non-isothermal version of Darcy’s law (Eq. 2) could
consequently by solved with the Jw, T, and h profiles,
obtaining the relationship between the unsaturated hydraulicconductivity, K(h), at above-zero and subzero temperatureswith the soil water pressure, as shown in Figure 5. In therange h > -103 cm (unfrozen), K(h) decreased steeply withincreasing |h|, while it decreased gradually in the range h <-103 cm (frozen). Similar K(h) was observed when different
freezing conditions were applied (TL = -5°C and TH = 5°C
in Fig. 5 for TDS). These changes were clearer for K(h) of
TDS, which agreed well with the value obtained using theevaporation method (Sakai & Toride 2007). In the frozen
area, K(h) in FSi was about 10 times larger than that in TDS,
which is why more water flow was observed in frozen FSi.
Figure 6 shows the relationship between K(h) and liquid(unfrozen) water content measured by TDR. In frozen
TDS, K(h) decreased steeply from 10–5–10–8 cm h–1 withdecreasing θ from 0.04–0.01 cm3 cm–3, while K(h) of frozen
FSi decreased more gradually.
The K(h) of frozen soil also correlated with temperature T
and with ice content θi. The decrease of K(h) with lowering T
and θi was well fitted by power law as K(h) = 3 × 10–6 |T|–1.49
and K(h) = 2.3 × 10–8 θi–2.42 for FSi, and K(h) = 0.7 × 10–6

Soil freezing characteristics
Jw
characteristics
are sometimes interpreted( h) freezing

KSoil
'
'hpore· curvature, when solute effect isT§from the surfaceJ hforce, 1¸¨'z et al.'z1995,¹ Watanabe & Mizoguchi, 2002).
negligible ©(Dash
The surface force accounts for the power law shape of soil
freezing characteristics and the effect of the curvature known
dPthe Lfas
Gibbs–Thomson
effect, which creates a shoulder todTv
Tthe soill freezing characteristics by means of the freezing
temperature depression, Tm – T depending on the soil poreradius r:
Tm  T

TmV
Ui L f r(4)

where Tm is the freezing temperature of bulk water and σ
is the ice-waterinterface free energy. In the soil pore sizen mSe 1  D htwo peaks are presumed: one from pores among
distribution,
the soil particles (r = 5–50 µm) and the other from pores on theparticle surface (r = 3–10 nm). These peaks would yield two
wm1§
wwTT hh UUto
wwthe
TTi TTsoil2
wwhh
ww-0.001
TT ··n freezingn  maroundw §shoulderscharacteristics
toSe w1 ª1ii Di 1h 1 º ¨¨KKwh2h ª1  DKK2 hhh 2º KK hh JJ hh ¸¸¬UUll -2.5
¼wwzz ©© respectively.
¬wwzz
wwtt andwwttto -10°C,
wwzz the-0.1°C
By¼converting¹¹ soilfreezing characteristics to soil water characteristics throughequation (3), the warmer shoulder would correspond to air
entry. In the range from water saturation to the other (colder)J ww
m 2 predominantly as capillary
K ( h)  soill ªwater
shoulder,will flow
K ( h) K§s 'SeT 1  1'hSe1/ m· º«»¼ at h lower than the colder
flow, but will
¬J h  tofilm
1¸ flow¨ change
'
'
z
z
©
¹
shoulder. Soil water characteristic models proposed by
Brooks & Corey (1964) and van Genuchten (1980) (Eq. 5)
:T i /(T t Tunsaturatedaretosoil with moderate h andr)K (intendedh ) Lfrozen
10express
K (h)
dP
ff
give the constant θ (defined as resident water content, θr) at
dT vllT
extremely
low h. These models, therefore, cannot express
the area around the colder shoulder, which is an important
portion for soil freezing characteristics(Fig. 1). Durnerl
K Se combined
w
S

w
S
(1994)
twovan
Genuchten
models,which express
TTK
V
s
e
e
1
1
2
2
m
m
Tdifferent
Tmm  TT soil
2
watercharacteristics,
todescribewater
retention
Lff rr
1 m1 m1 º
m2 m2 º ·§ D ªU1iiL
ª1  1  S 1/(Eq.
1 aSdual
 w2Ddistributionin a¨ w
soil
porosity6):¸
1 1 having
1
2
2
e
e
¬«
¼» ¹
¬«
¼»
u©
2
w1D1  w2D 2
m
(5)
n m
See 1  D h n

See

m
m11

w11 ª1  D11h n11 º
¬
¼
n

m
m22

 w22 ª1  D 22h n22 º
¬
¼
n

(6)

where Se = (θ – θr)( θs – θr) –1, m = 1 – n–1, w1 = 1 – w2, θs issaturated water content, α and2 n are empirical parameters,mm 2
K ( h) K ssSelel ª1  1  Se1/e1/mm º
¬«
¼»

K (h)

:TTii /(/(TTtt TTrr ))
10 :
K (h)

Watanabe and Wake   1931  

Eq. (7)(Ω = 4)(Ω = 20)
Eq. (9)

-2

10

FSi

-4

10

0-6h
6-28h
28-50hEvaporation
methodK(h) (cm h-1)

-8

-6

10
K(h) (cm h-1)

-6

10
10

0-6h
6-48h
TDS
48-72hw §whT·T =– 5°C , T w=5°C
 K hL  K h J h H ¸
¨K h
wz ©
wz
w
Evaporation z ¹
method
w §
wh
wT ·

-2

wT h
U wT T i i
wt -4 U
wt
wt

l



Ui wTi T
Ul wt

K ( h)-6 

10

wT h
wt

K ( h)

10 4

10 5

water characteristics, including soil freezing characteristics,
for FSi and TDS(Fig. 1).
n m
Se 1  D hn  m1n  m2
Unsaturated
hydraulicsoilsºSe w1 ª1  D
h 1 º conductivity
h 2frozen
 w2 ª1  D 2for1¬¼
¬conductivity¼ for unfrozen soil,
The unsaturated
hydraulic m2
n  m1 from equationn (5)K(h), is often derived(van Genuchten
Se w1 ª1  D1h 1 º  w2 ª1  D 2 h 2 º¬¼
1980) as¬ follows: ¼(7)m 2
where l is the pore-connectivity
coefficient. For frozen
K ( h) K s Sel ª1  1  Se1/ m º«¬»¼ by an impedance factor Ωsoil, K(h) is sometimes
reduced:T i /(T t T r et
) al. 2004):(Lundin1990,
K ( h ) frozen
10Hansson
K (h)K ( h ) frozen

10 :Ti /(Tt T r ) K ( h )
l



10

Ui L f r
TmV
Tand
T 2 is theweighting factor. Using the parameters listed inm w
Ui L f r  mTable 1, equation(6) was well fitted to a wide range of soiln
Se 1  D h

2

w §
wh
wT ·
 K h  K h Jh
¨K h
¸
wz ©
wz
wz ¹

Frozen Unfrozen
(1)
0-6h

Jw

(8)

KApplyingSe
K s equations
w1Se1  w2(7)Se 2 and (8) to our measured K(h)2m1 could not fit the gradientverifiedchangeof
1 m1 (7)1/ m2 m2 º ·§ w D thatª1  equation
º l
ª
S
1

–3 w S  w2D 2 1  1  S e 2
¨
¸
K
S
K
w
S

1
1
1
e
K(h)
around
h
=
10
cm
for
FSi
and
underestimated
K(h)
at
e
s
e
e
1
1
2
2
«
»
«
»
¬
¼
¬
¼
¹
u©
2
2
1/ m2 m2 º ·§ w D ª1  1  S 1 m1 wm11Dº1wwD
2D 2ª¨ 1 1«¸
2 2 «1  1  S e 2
e1
¬
¼» ¹
¬
¼»
©
u
2
w1D1  w2D 2

dP L f
dT 10
vl-8
T

6-48h
48-72h

(1)

'h ·
§ 'T
 1¸
Jh
¨
'z ¹

Eq. (7)-6 © 'z[K(h)] and soil water pressure at above-zero and subzero
TmV
temperature.
Table 1 lists the parameters for equations (7) and (9).
Tm  Tº¼»

TDS

Ui wTi T
Ul wt-4

Figurev5.
Relationship between unsaturated hydraulic conductivity
dT
lT

m



10

Jw

K s Sel ª1  1  Se1/ m
¬«

-2

10

 K h  K h Jh
¨K h
¸
wz ©
wz
wz ¹

'h ·
§ 'T
 1¸
Jh
¨
'z J w ' z ¹
K ( h)  ©
'h ·
§ 'T
-8
 1¸
Jh
10 L ¨© 'z
'z ¹
dP
f
10 1
10 2
10 3
dT vl T|h| (cm)
dP L f

K ( h)

FSi

-8

10

10

wT h

0-6h
6-28h
28-50h-4

1010Frozen Unfrozen-210(2) (Ω = 4)(Ω = 20)(2)
Eq. (9)

0

0.1

0.2

0.3
(3)

0.4

Water content, θ (cm3 cm-3)

0.5

(3)
TmV
6. Relationshipbetween unsaturated hydraulic conductivity,
TFigurem T
rK(h), and Usoil
content at above-zero and subzero temperature.
i L fwater
Table 1 lists the parameters for equations (7)(4)and (9).
n mShe < -10
1 2 Dh for TDS (Fig. 5). The impedance(4)cm
factor might
be useful for expressing the steep decrease of K(h) near
(5)0°C, if equation (5) were fitted to the whole range of soil

m
 m2
1
n2 However,use of anª1  D1h n1 º (FSiª1Figs.
ºSwaterw1characteristicsh 6).(5)
 win
 D5,
e
2
2
¬ factor requires¼¬
impedance
cautionsince it¼ will underestimate
K(h) as freezing progresses.
(6)
The design of equation (7) was based on the bundle ofcapillary tube model in which water flow decreases with2
(6)
1/ m m(tube)
radius, as Poiseuille flow,ºKsquare( h) decrease
K s Sel ª1 in1the Sporee¼» in K(h) on a log–log scale
and produces¬« a linear reduction
when θ is constant. On the other hand,(7)film water can be
regarded as the flow proportional to the first or less root of
T i /(TK(h)t T r )film
Thesoil(7)consequently has aK( h )thickness.
10 :
Kof( hfrozen)
frozenlower grade than that of unfrozen soil. Thus, the Durnermodel was again applied to K(h), taking(8)care that θ did not
become constant at 10–3< h < 10–6 cm (-0.1 < T < -100°C).
l
(8)
K Se
K s w1Se1  w2 Se 22
(9)
§ w D ª1  1  S 1 m1 m1 º  w D ª1  1  S 1/ m2 m2 º ·¨ 1 1«
¸
2 2 «
e1
e2
»
»
¬
¼¹
¬
¼
u©
2
w1D1  w2D 2(9)(9)

1932   Ninth International Conference on Permafrost
Equation (9) showed good agreement with the K(h) obtained
from the column freezing experiment (Figs. 5, 6). This model
was originally used for explaining water flow containing two
different flow rates, such as among and within soil aggregates.
Our results suggest that this model is also suitable for soils
under freezing-thawing processes, in which soil water flowchanges from capillary flow to film flow.

Conclusion
The sand and silt columns were frozen directionally, and thewater and heat flows during soil freezing were measured. The
flows depended on the soil types. Unsaturated hydraulic conductivity for frozen and unfrozen soils was estimated by solving Darcy’s law under non-isothermal conditions with ice formation, although further consideration of the precision of flux
measurements and limits of the Clausius-Clapeyron equation
may be required. Hydraulic conductivity steeply decreased withdecreasing soil water pressure and water content in unfrozen
soil but more gradually decreased in frozen soil. In both unfrozen and frozen states, the silt had higher hydraulic conductivity
than the sand, resulting in more water flow during silt freezing.
The shapes of soil water characteristics and soil freezing
characteristics were discussed from the viewpoint of the poresize distribution. Use of an impedance factor for calibrating
the hydraulic conductivity of frozen soil, which has sometimes
created unstable numerical simulations, appears to beunnecessary when the hydraulic model can appropriately
express both the soil water and freezing characteristics. Rather,
the results suggest that the Durner model is useful for expressing
the hydraulic properties affected by the change in the type ofwater flow during soil freezing.

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