
J. Jpn. Soc. Soil Phys.
:a
No. +*0, p. ,+ -, ,**1

ῌῌ

S ‡nek***ῌ*ῌ *ῌJiri ‘imu
Numerical Modeling of Water, Heat, and Solute Transport during Soil Freezing
Kunio WATANABE*, Nobuo TORIDE*, Masaru SAKAI* and Jiri ‘
SIMU÷NEK**

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;hZi2< JmsV+K c  =) JmolLV+K R 

  ῌῌ      !"#

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KnT Jm, sV+ XV+K \@]@W*^-')^- _

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Department of Environmental Sciences University of California, Riverside Riverside, CA 3,/,+, USA :   ῌ0ῌ , " Q<#!$῏#!P»*+ ~ +*0 ¼ e,**1f

22

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kg mk-f l m %O P Z[VR῍ \] nop C @qr`abc
Cp*() ,'&' ()¬
­!®© eWilliams, +30. ; Koopmans
and Miller, +300f hw _¯¬#

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Schematic illustration of liquid water geometry in soil pores : (a) drying with the airliquid water interface under room temperature ; (b) freezing with the ice-liquid waterinterface in a saturated soil.

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23

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(+,!

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z{ dql ῌ dT +VW dqi ῌ dT.-dql ῌ dT Y

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2 T 34 K )* 5# "+ 6 789:
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2<=>?@A+ ῍῎D)EF GH/+ * 5) ῍

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4 a m-+ n, m )* ῒ+  2 - D)* Xῒ
-  - Drswx  GCC

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Table +

ῌ,
Fig. ,GCC

34 T fW2
GDM G/ h.

Unfrozen water pressure head in a watersaturated frozen soil h, based on the temperature T with the generalized form ofClausius-Clapeyron equation.

- Ds van Genuchten ·¸Parameter values of van Genuchten modelfor three di#erent soils.

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24

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῕+O 34\U *Q

vW_V)b 67 G!1 qrs!.

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ῒ
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 Kflh Y  KflT 1 U *Qv?

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bf -  \ N
¹

Fig. -

Unfrozen water curves for three di#erent
water-saturated frozen soils, based on thesoil water characteristic curves describedwith Eq. ῎
῏ and the GCCE.

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UH p fl ¡ ¢ N£S   p+¤+ Cn, Co, Cair, Cl, Ci J m-- 

K+ ))   Cp  5+-+ 

 ῌ

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'  vU1

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  

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 34v ILMU Y

Genuchten, +32*
Klh!KsSel"++S+eῌmm#,ῐ῎+$$ Ks  m s  l %&'()*+,-./01 2 34   p ¦§¨V©1 Kf ª a$)«1U  ¬¦§­Z )®b)`a$)UΐV ῕. c  -  YZ ῎  KlT m, s+ K+   5+ Noborio et al., +330

T ) ^1 $$ Cn!+.3,P+*0 kg s, m+, Coῌ῍

KlT!Klh hG+ dg ῎g* dT῏ῑ῎ Cp Y῍

P+*0 kg s, m+, Ci !+.3/-P+*0 kg s, m+ 18 ql,
qi ῕->?@ABCD/EF GH I 2 J5 +U *Q

KLM Nimmo and Miller, +320 g N

 Ca   Cp

g* ,/Q!1+.23 g s, 1

RS

 Knh, KnT  Scanlon et al. ,**- )T $)UV
K 

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^ Kf qn, qo ,f v?~+'U¯K) JZ !K°±[>C "²³#N m$U'fl ῕- Ca°±1 2´

µ   ¦§3Kjk--W  'X=

 Ca )34!,./+P+*0 kg s, m+, Cair !+,,/* kg s,m+, Cl !..+2$$ G 6718 9-:;/< . =4

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

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ῑῌ.

$%| a +,BBM Kﬂh, b |+,BBM KﬂT, c &E Cp H&E Ca, d &'( l.

Fig. .

Temperature dependence of physical properties of frozen soils : (a) isothermal hydraulicconductivities Kﬂh ; (b) thermal hydraulic conductivities KﬂT ; (c) soil heat capacity Cp andapparent heat capacity Ca ; (d) thermal conductivity l.

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Table ,
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F+<F+ qFi ,Thermal parameter values for three
di#erent soils.

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Tinit ;/<( =123&4 5678

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*HI( ῌ J3

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\ !῏/ RcS( *< 9 RbcS srh3&4 bckl:( *./ sr3 ,mm h?+( / sr3 *./ mm h?+ )sr))dP W9: OC( bckOi4 2* sr{ bckl:OP4)( 9: b3 *.--< cm?+( b
3 +< cm?+ )f¡( b=O¢]P R!῏/ RaSS D ( b£¤ O
῎ JbA'¥¡Qj3&4 R!῏/ RbSS P@( ῍K¦£¤(! §y%( b=3¥)4῏ῌ/

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F#+, /<( E#+, +/<S ºFig. /(a) Temperature and (b) thermal conductivity proﬁles in a freezing loam soil ; (c) Progress of the freezing front (*< isotherm) for
Tinit;/<, TL;?/<, TR;/< and the +*<isotherm for Tinit;+/<, TL;/<, TR;+/<.

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῏0

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e  VWX cl, f YZP[ V\] qcl. ^_41` da_4+] b_4 Wc d ]<e f g4h]i`Proﬁiles of (a) temperature, (b) pressure head, (c) water ﬂux, (d) water contents, (e) soluteconcentration cl and (f) amount of solute in a unit soil qcl in a freezing loam soil. The dashedline indicates the freezing front. In ﬁgure (d), the di#erence between the total water content(solid line) and the liquid (unfrozen) water content (dotted line) gives the ice content.

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29

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 qlT, "78 qnT.

Fig. 2Isothermal liquid (qlh) and vapor (qnh) ﬂuxes
and thermal liquid (qlT) and vapor (qnT) ﬂuxes
in a freezing loam soil after / hours for different initial water pressure heads.

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Proﬁles of (a) water and (b) pressure headand (c) isothermal hydraulic conductivity
in a freezing loam soil after / hours for
di#erent initial water pressure heads. In
ﬁgure (a), the solid and dotted lines indicate total and unfrozen water contents, respectively.

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QR(@h"78234ijk6>l( 01"23 qlh (mn ;ῌ2 ;b<< => ῌ1

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30

man and Wilson +331   \ ^23cdefgG. .*R$ .0    qinit*.-- /  !" hinit#/-.

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/ 78º <5 /5@º
Water content proﬁles for three di#erent
freezing soils after / hours. The solid anddotted lines indicate total and unfrozen
water contents, respectively.

Campbell, G.S. (+32/) : Soil physics with BASIC, Elsevier, New York.
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pqrstuv :w1hxyzfreezing and soil water curve data for Windsor
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ῌῌ{@pqr

31++*/ῌ+++-.
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Two-dimensional model for water, heat and so-

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S ‡nek, J., Mizoguchi, M., Lundin, L.C.

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+32. :

 !"#


$%&'# .3 ; /.ῌ0+.

dients. Trans. AGU, -2 (,) : ,,,ῌ,-,.

() *ῌ+,-. ,**0 : /0123 J.R. Philip and
D.A de Vries 4 56789:;<=>?@A  !"B#

$%&'# +*- : +*/ῌ++,.

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eb%@fghij# klmno$%&bῌῌcdῌ

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 W X  +*0  ,**1

32

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