no. +*0,p.,+-, ,**1 - pc-progress · * * * jiri s‘imu‡nek** numerical modeling of water, heat,...

12
腏腊腅腁腄腂腀腎腍腃腌腒腓腋腉腐腇腆腈 腠膂臱臹*臡臤臯*臒膗 *Jiri S imu nek** Numerical Modeling of Water, Heat, and Solute Transport during Soil Freezing Kunio WATANABE*, Nobuo TORIDE*, Masaru SAKAI* and Jiri S IMU ÷NEK** +腇腅腈腆 膞腐腥腏腌腠腢腔腦致腾腺 腷臎腎腴臑腋腴腌腧膴腦腐腷腐腲腦致臮膵膉膌膃腽 膁腷臎腎腴臑膩腥腢腯腤腍腢腔腦致膍腇腘腴腕腢腑臣腡腋腴腫腜腐腴腰膠腁腽臞腦自臬膤腤腣腥腗腠臄腑膔膒腖腵腠腏腳臄腥腢腯腤腍臞膖腦膋 膄膎膤腦臩腧膭膞腐腢自腥腫腲腤腌腵腥膢膣腦致臮膵腷膲腭腸膅膡腢腔腦致Philip and de Vries +3/1腥膳腟腒臛臜腡腖腵 臒膗臡臤,**0腄腀 (ql (t + r+ (rnqa(t ( (z Klh (h (z Klh KlT (T (z Knh (h (z KnT (T (z 腕腕腡ql qa 腧膝臭致腢致臮膵腦m - m - 腄腁 rl rn 腧膝臭致腢致臮膵腦膈腡 kg m - 腄腁 h 腧膑m腄腁 T 腧膢腃腉腄腁 t 腧臚膱 s腄腁 z 腧膔m腡腋腴Klh ms + 腄腁 KlT m , s + + 腄腁 Knh ms + 腄腁 KnT m , s + + 腧腚腵腛腵膑腢膢腥腱腴 膝臭致膖腢致臮膵膖腥腏腔腴致膿臵腡腋腴膤腷腢腯腤腍腢腔腦致臞腦臚膖臜腦致臜腥膥腎腠臛腦腢腔臞膖 腦膖膘腾臔臜腷膟膗腗腠臁臕腘腴 臡臤腲,**0腄腀 Cp (T (t Ls (rnqa(t ( (z l (T (z Cl ql (T (z Cn qn (T (z Ls (rnqn (z R (c (t D ( , c (z , n (c (z 腕腕腡Cp, Cl, Cn 腧腚腵腛腵膝臭致致臮膵腦腭膏膜 Jm - K + 腄腁 l 腭腟腪膖 Wm + K + 腄腁 ql, qn 腧膝臭致腢致臮膵腦膘膜 ms + 腄腁 Ls 腧致腦臮腱腂腭 ,./*++* 0 ,-03.,T Jkg + 腄腁 D 臔膿臵 m , s + 腄腁 n 膽膱臃膘腈 ms + 腄腁 c 腯腡 molL + 腄腁 R 臞腦腢膚臖腩腦膷腥腱腴膞膙臖腡腋腴*膓膣腦臭臄腘腴膨腩膐腵腥膣腦腢腔臞膖臋臩腡腋腴腑腢腔腥腧腢膚臭致- 腥膥腎臰腜腥腑臋腵腴膢腥腏腔腴膖腢膕腤腴腢膚臖膱臃腥腏腔腴腦臺臤 腸腤腢膚腷膉膯膼腦膛膻腡*膓膣腥 腏腌腠腯臄腗腤腌致膑腦臹腰致腬腝臀臭腦膤腷腢腯腤腌腢腔臞膖腉腍腤膛膻腷膶腪腘腫腜膣腘腴腢腤腢腔腸腤腒腓膤腘腴腜腮 Watanabe and Mizoguchi, ,**,腄腁 腤腢致臶腰腭腝臶臞腑臂腗腓膤腘腴腚腦腜腮臄腷腢腯腤腍腢腔臞膖膍腇腥腏腌腠腧ῌ῏ 臜腦腸膅膡腢腔腦致臞膖臜腥腗腠腸腤致腦膹腫腜腸膅膡腦致膿臵腤腣 腦膖腨腽膤腷腌腐腥膋膄膎膤腘腴腐腑膧腢腤 Harlan +31-膤腽膌腺膀腺膁腽膌膊腹膏膐 腦臜 Generalized form of ClausiusClapeyron equa- tion, GCC 腌腠腤腢腔腦致臜腷臄臏腗臇腐腔腦腭膏膜腥腱腳腢腔致腦腩腦腆膁膤腷腘膋膄 膎腷臧腗腜腕腍腗腜膋膄膎腦臵臁臕腰腤腢膿臵腦腥腞腌腠腧Fukuda et al. +32*腄腁 New- man and Wilson +331腤腣腥腱腳臅臨腑至腮 腲腵腠腒腜Hansson et al. ,**.腤腢腭腟腪膖 腵腃臀臶腷腘膋膄膎腷膒腗GCC 臜腤腣腦腦膋膄膎腷腬臐腭腕腢腡腸膅膡腢腔腲膒臕膂膉膅 HYDRUS-+D S imu nek et al., ,**/腷臢臸腗 腚腗腠膘臛臉腦臝腫腤臄臟臈腏腱腨腥腶腜腴膫腦膩膰腷膩臺腗腜膇膃腡腧腫腙GCC 臜腥膳腟腓臄膨腏腱腨致臮 膵膖腷膲腭腸膅膡腢腔腦致臞膖臜腷膩臼 * 臓臣膬膚臹臙臊膬臆膸膦 /+.2/*1 臗膾臲+/11 ** 腼膍膉腻膎膇腸膬膍膈腂腾腹膆臍膮膺膦膬Department of Environmental Sciences University of California, Riverside Riverside, CA 3,/,+, USA 腉腁腎腁腋 : 腤腢臞膖腸腤膤腽膌腺膀腺膁腽膌膊腹膏膐臜 J. Jpn. Soc. Soil Phys. 自腦腽膕No. +*0, p. ,+-, ,**1腌腊腍腒腐

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Page 1: No. +*0,p.,+-, ,**1 - PC-PROGRESS · * * * Jiri S‘imu‡nek** Numerical Modeling of Water, Heat, and Solute Transport during Soil Freezing Kunio WATANABE*, Nobuo TORIDE*, Masaru

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����*�����*��� �*�Jiri S‘imu‡nek**

Numerical Modeling of Water, Heat, and Solute Transport during Soil Freezing

Kunio WATANABE*, Nobuo TORIDE*, Masaru SAKAI* and Jiri S‘IMU÷NEK**

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tion, GCCPK��� ������0,�P�%�>���`�056_A����Ek�D/<� -BCD�R�>42 .*>4BCD�bOef+���.�ab�S�T�� �� Fukuda et al. J+32*K� New-

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**±² ³D´µK¥²¶·¸�¹º»¼«¥`Department of Environmental Sciences University of California, Riverside Riverside, CA 3,/,+, USA

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J. Jpn. Soc. Soil Phys.

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Page 2: No. +*0,p.,+-, ,**1 - PC-PROGRESS · * * * Jiri S‘imu‡nek** Numerical Modeling of Water, Heat, and Solute Transport during Soil Freezing Kunio WATANABE*, Nobuo TORIDE*, Masaru

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l�� h� Z��� ���R� !�_�"� �, !��#����������� e��+ eaff� W�%���w� !����_����$��bR�o������#�"��, !��#��o��������� eDash et al., +33/f� *+ TW� ����W¡¢�����R£¤���h�*+ T()��������R� ���¥¦��a�c�� b��%&'���[V�§a��w� h��������¨©W�ª«�l��!8�%& W�%�l�!R�6��&'�§a��

hh����+ ebf��@P��������W��*()� ���,'�&'()���¬��������­�!�®©�� eWilliams, +30. ; Koopmans

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Fig. + Schematic illustration of liquid water geo-

metry in soil pores : (a) drying with the air-

liquid water interface under room tempera-

ture ; (b) freezing with the ice-liquid water

interface in a saturated soil.

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Page 3: No. +*0,p.,+-, ,**1 - PC-PROGRESS · * * * Jiri S‘imu‡nek** Numerical Modeling of Water, Heat, and Solute Transport during Soil Freezing Kunio WATANABE*, Nobuo TORIDE*, Masaru

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Fig. , Unfrozen water pressure head in a water-

saturated frozen soil h, based on the tem-

perature T with the generalized form of

Clausius-Clapeyron equation.

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Table + Parameter values of van Genuchten model

for three di#erent soils.

Ds qs qr a n m l

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Page 4: No. +*0,p.,+-, ,**1 - PC-PROGRESS · * * * Jiri S‘imu‡nek** Numerical Modeling of Water, Heat, and Solute Transport during Soil Freezing Kunio WATANABE*, Nobuo TORIDE*, Masaru

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YZ�� [\)T��� ]3 K�ql^�U_��V�)`a�� bJb� ��N������LM�H��� %&c�debf�� ��gh�iWjk�

fl� ����������Xm�n��)o�5+� p$�� Jame and Norum �+32*� �� qr��s�������tubf

Kflh!+*�sQKlh ��

$$�� vw�x f ����y� s�z{|,-./0�18�>?@AB� 2 =4��� ,*}-* )~+��� �Gosink et al., +322� Kf� Q �����������(�1� ��. �a� �b� �� �+ - �����b�� O�34��\U *Q�v�W_��V�)b� 67�� G!1� qr��s!. )V� ��������� KflhY � KflT�1� �U *Q�v��?~+�)� ����� Kflh�+*�+* cm s�+ K��l�����v�� >C��(� ����UM��fl��'������vU���1�

-�, ����JZ����� ���������'

��)����������H�f��1� �������� ��d�����)p��������a�V� ���(� ��d���~5��UH�� pfl�� ¡��¢���N£S����� �����p+¤+ Cn, Co, Cair, Cl, Ci �J m�-

K�+� )��)� ����� Cp�����5+�

Cp!qnCn¥qoCo¥qaCair¥qlCl¥qiCi ��

$$�� qn�� ¡�� qo��¢�����1� ����34�v� I�LM��U� ���Y���� p¦§��¨�V��©�1� Kf� ª�����������a��$)�«��1�U� � ¬¦§�­Z������)®b�)`a��$)U�V� ��. �c� �� - ����YZ���������� Cp Y ���JZ��� Ca )34T)^��1� $$�� Cn!+.3,P+*0 kg s�, m�+, Co

!,./+P+*0 kg s�, m�+, Cair!+,,/* kg s�,m�+, Cl!..+2

P+*0 kg s�, m�+, Ci!+.3/-P+*0 kg s�, m�+ �18� ql,

qi����- ^�� Kf qn, qo� , ����f �U *Q�v��?~+��'U¯K�)� �JZ��� Ca����� Cp�!�K�°±�� [�>C���� "²³#��N�m$U�'��fl ���-�� Ca°±����1� 2�� ����´�����µ ���� ��¦§��3K�jk�

-�- ���Campbell �+32/� �� ��¶·� l �W m�+ K�+�

� ���\�����f

l!C+¥C, ql��C+�C.�exp"��C- ql�C/# �

��- GCC ������ [\��J5·¸bf - ���\����N�����¹

Fig. - Unfrozen water curves for three di#erent

water-saturated frozen soils, based on the

soil water characteristic curves described

with Eq. � and the GCCE.

�º�%\ » +*0 ¼ �,**1�24

Page 5: No. +*0,p.,+-, ,**1 - PC-PROGRESS · * * * Jiri S‘imu‡nek** Numerical Modeling of Water, Heat, and Solute Transport during Soil Freezing Kunio WATANABE*, Nobuo TORIDE*, Masaru

���� Ci �i�+� …� /� ����� � � ������������������� C/�.�� !�"��#$%�&'(� l�� )*�+,-"./0+123��Kennedy and Sharratt, +332�# Hansson et al. �,**.��� ��45678"$%�&'(�594�:8;#

l�C+<C,�q<Fqi�=�C+=C.�exp

>=?C-�q<Fqi�@C/A ��

F�+<F+ qF,i ��

���� q�BCDEFG�;)B�� F�BH*�&'(��I�+JKLM��# N�. �d� �� -OP�%+Q8"��4��RS;$%�&'(���#�T� q�ql, %+,�U F+�+-.*/� F,�+.*0� qi�N�-�VM4� W�X�YZ[\]�:�,��5^�;# %� *_`a+bc!��H� *E�dG+8;��$%�&'(��eL��# ��f� ghij�%kB�lm�$n3�op%�&'(��� q$B5rLs3�tu%+vw"e�# x;� op%�q$BE� *_`a�yc�+ qr+zi3�;S �N�-�� {|}~+T��&'(������#

.� � � �

.�+ ����$n5HF��%k�B��&��+��"� ��x��8;��4H������45^�"�:�++�8

��, -OP�%��&��YZ[\]�Table , Thermal parameter values for three

di#erent soils.

%� qn qo C+ C, C- C.

�\�����\�op�\�

*4-

*4//

*4/1

*

*

*

*4/-

*41,

+4/2

*42.

*42.

*42.

24-2

24-2

,1

*4*3-

*4*3-

*4*3-

��. $%������|����� �a� ����+,���B����BM Kflh, �b� |���+,���B����BM KflT, �c� &�E CpH����&�E Ca, �d� &'(� l.

Fig. . Temperature dependence of physical properties of frozen soils : (a) isothermal hydraulic

conductivities Kflh ; (b) thermal hydraulic conductivities KflT ; (c) soil heat capacity Cp and

apparent heat capacity Ca ; (d) thermal conductivity l.

������� : $n5HF��%k�B��&� p����� 25

Page 6: No. +*0,p.,+-, ,**1 - PC-PROGRESS · * * * Jiri S‘imu‡nek** Numerical Modeling of Water, Heat, and Solute Transport during Soil Freezing Kunio WATANABE*, Nobuo TORIDE*, Masaru

� -��������� ,* cm���� ������������������� ��������� !�.�"��� �����# �$�%&'( �)�*�+,-�./0 123&4� 5678 ( 9:Tinit;/<( �=�>� hinit;?+**( @AB?+,*** cm,

�*�: cinit;+* m mol L?+3�C)��� DD3( +,9:)�%(E# TL;?/<(F# TR;/<�G���*HI ( �J3GK�( �L�MNOPQ�*RR;+S ����( T L;+ cm, UTV� D*;*.+

cm, day?+)��� W�( �*�A4XYZ[\]�*��L�� ( �>]>��^L�_` �a��� bc�)dPe�HI3 ( bc��f3�=�>�]���Og#��hi4�j( ��bc��k�l:OlQmn( o�pPqr@ABsrtu�vwi4��O&4� ���j( bcOxy4�z�qrtu�*.*, cm, 56�srtu� *.*.�)��� {\( 56>� hinit;?+** cm�2|����c}�W~"�( ����P456�78( ���A4�������

.�, ����!�/ RaS @AB!�0 RaS �( E#��9:�\��2|��9:��"i� �� �9#�K���3&4� �*�A4XYZ[\����%QPQ�j( *<���Obc����i4 R!�0=��S� E#��9:�\�4)( /<3�C3&���9O(E#�K ���\��� !�/ RcS ( *<��9� Rbc�S ����sr�h3&4� bc��k�l: ( *./sr3 ,

mm h?+( /sr3 *./ mm h?+)��sr))d���P��� W�9:�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�b�A'¥¡Q�j3&4 R!�/ RbSS� P@( ��J�K�¦���£¤ ( �!��§y%( b�=3�¥��)4�!�/ RcS � ( bc¨©A'9:O +*<��ª�«Q78 R5678 Tinit;+/<( E#+, TL;/<( F#+, TR;+/<S �@¬4 +*<��9������hd"��� bcOxyPQ�=���%­y9:®�+,78�G4)( ,.sr�� �=��HIO��C����( �����9�O"K�4� ���( bcOxy49:783 ( b�=�@¬4¯��#¦O°��j( 9:��$± �9��²��P4� W�( �£¤ �=��!)�!��h�³´i4 RJ��S��9�@Q% ( 9:���A4��µ�Oxy4O( �£¤ ���¥¡P%' PQ�j( C��

3 +*<��9�O������=¶���i4� ��( bcOxy4)( b���£¤ O�b�� -&f��P4�j R!�/ RbSS( �·ª� ���� -�.W

��/ 2|��bc�¸� RaS 9:�( RbS�£¤ ��( RcS *<� Rbc¹�S ���º RcS �+*<� ( bc�PQ9:78 R569: +/<(F#+, /<( E#+, +/<S ��º

Fig. / (a) Temperature and (b) thermal conductiv-

ity profiles in a freezing loam soil ; (c) Pro-

gress of the freezing front (*< isotherm) for

Tinit;/<, TL;?/<, TR;/< and the +*<isotherm for Tinit;+/<, TL;/<, TR;+/<.

�»��'� � +*0¼ R,**1S26

Page 7: No. +*0,p.,+-, ,**1 - PC-PROGRESS · * * * Jiri S‘imu‡nek** Numerical Modeling of Water, Heat, and Solute Transport during Soil Freezing Kunio WATANABE*, Nobuo TORIDE*, Masaru

��������.�- �����0 �a� �����������������0

�b�� ��� !"#$% &�'()*�+��,-./0 ����0 �c� �1� 22�� 3 ��,

-./045�67 8* ��9:;� �<� ��� ����4� �� �=>;���� ��0 �a�� ?)'()*� @+*,*** cmABCDE�FGH ��0

�b�� 2 �I� ������C� J��?)��7:�K ��LM�NO� J��+P ������:�

��0 QRS ��TU �a� ����� �b� ����� �c� ��,-./0��� �d� ������e� ��� VWX� cl, �f� YZP[�� V\] qcl. ^_4����1�` �d� a_4+��]� b_4��� Wc� �d��� ]�<e� f g4h]�i�`

Fig. 0 Profiiles of (a) temperature, (b) pressure head, (c) water flux, (d) water contents, (e) solute

concentration cl and (f) amount of solute in a unit soil qcl in a freezing loam soil. The dashed

line 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.

jklmnop : ���Cqrs�� ���t�V\uvjkl 27

Page 8: No. +*0,p.,+-, ,**1 - PC-PROGRESS · * * * Jiri S‘imu‡nek** Numerical Modeling of Water, Heat, and Solute Transport during Soil Freezing Kunio WATANABE*, Nobuo TORIDE*, Masaru

�������� ����� ������������������������ !"�0 !c## $ ������ ��%&'()��*+ ���,-.�/�� � 0���,1� ���234� � �5��67������� !"�0 !b## ���,-.�8� �8 � ��9�:;�<=>?� @� ���1�� ������� �����AB !"�0 !c## C � ���� ��������5 � D���EF?� $��� � ��GH� �I��J�;B�K@�<=�$82� !"�. !a##���7 ��D� �L��� ����M"�0 !d# N. "� OP�Q���� �P�� �����*3� R 9����*� ���S� � �����AB� R �;�T ����UVB�� �WX���YB C � � ������ Z�6[�\FB���?]�� Q��� !����^��#����23JI_��YB B_B� O` ��]���� ������a�F?>� C ��J��J�bc�� � ������YB?�$8��d=���� ! 8ef� Fukuda et al., +32* ; New-

man and Wilson, +331# $��� : g GCCh8���ijPkl>�� ���jPM� �����D�;B�5!"B���$8�mn�*� .?op� "�-N.2q� *r ����������;s.� @�*� $ ��� ��� ���#Mt$B ����jPM"��?( u%�te���

.�. ����"�0 !e# � "�0 !a# 8&' vwWX][��(xyz* 3 � ){*� clMN. ���.�8�:+|B �_�){�,�}=�� @���� ){*�� ��~�H-�>?��]���� ���2+�����8����?� @ !"�0 !d##� ��� ){*�5�?+ C � ��� ){*� ���� :+|� ��?���.S���� ���,-.�/�� ��� ){*���>?3� C *�����I_?+ $��� �������I:+|���AB� C ��2�){D���� @�*� ��� �]���� ){*��UVB ����23Hmm

C� /���� $��� ��8��6� ���7 ��D�2�D0���B @�*� $qB �_�,�}=��){ ��/� ��=����J�� ){��� �� :;�?�G��.�

"�0 !f# �� "�0 !d# ����� ql8"�0 !e# ){*� cl z�1e���(x ){� qlcl���*� ){��� �����>�~���>� ���>?+ �]���� ��� ){*� cl���� !"�0 !e##� ��?�'(���� ql�\F.� !"�0 !d## $ @� 2��]���� � ){��WX2���38?+ ��� ���){*� cl�+|����?��� ��2��� 0������ $ @� ){��� ���S>��>� ����%&�S>?+ C ����]��){��������� $qB ){� ������ ,-885<=>?+ "�0 !f# ���L�� O` ��1������� ){ ���?�z! 8efKonrad and Mc-

Cammon, +33*# M2>0�B������ O` 4�5�IO �L]���� ��85?q� ){*� �A8 � ){*� ¡¢�BfBf���� $qB ���L8O` �£ :¤�� ¥¦�§A� C ){ ,�}BJ�?( ¨©.ªLM«6.�¡�� ¬7?­��*�8te���

.�/ ����� �"�08&' ��w8/����?�%&��®¯

hinit°±+,*** cm8 �aM-+ "�1 !a# �� ����² /WX hinit°±+** cm8±+,*** cm �����*� ��� %&��]��5� ���S� �][����A8� ��UV���� �� %&�����8 �_��7 ��D���\FB ��� � �������9®¯�'©:B>?+ "�1 !b#� !c# � $ 8� ����8���� ���D�;.���GH Klh��MN.%&��2���� �����'©:B�$ 2q����� ����%&����B?� �� ��h23� ����M����kl�1e��� @�*� .?op� �����:B�� ����8����:B� ��� � ��GH�� ��h23��s�� �23<=>?� $ @� � ��GH���� ³� hinit

°±+** cm 'q������ � ��GH �A�� �� ³� hinit°±+** cm 'q����"�2�� %&��±+** cm]2´±+,*** cm;.� /WX² .µ� �������� .?op���� ��� qlh, ���� ��� qlT, ���� �¶· qnh, ���� �¶· qnT ���*� ����� 6��� ���6� !"Z6�# �¸�*� ��� %&��]��5� ���S� qlh M¹>�

º ;<i » +*0¼ !,**1#28

Page 9: No. +*0,p.,+-, ,**1 - PC-PROGRESS · * * * Jiri S‘imu‡nek** Numerical Modeling of Water, Heat, and Solute Transport during Soil Freezing Kunio WATANABE*, Nobuo TORIDE*, Masaru

���������� ������2��������� ������������� ! � "���#$ hinit%&+** cm'�� qlh()*+,�������

3/��� qlT( /��-��� ./� *+,'�����"������� 3/�� qnT(-�� 01"23456 "7823(9:�� ;��2 ;a<<� =>�� ?�@4ABC*"�(DE�� Kflh(FG4DE����'H!� "���E�$ hinit%&+,*** cm'�� )*+,'� qlh( 2-� IJ5�B� KLM4 qlT ;++�<N"7823 qnT ;0�< (O�5�!� P�� QRST(O�$*UVWX'�� qlh(YTM'H!� Z[4\]��1^_`'�� *UVa�"�234Lb!"7823�cd��$ $e!� P�� *+,'�� fgQR(�@��h"78234ij�k6>�l��(�01"23 qlh(mn�� ;��2 ;b<<� =>����1;c< 4o��Ap4� q�(IJ5� r"st Kflh(O�$��'H!� Cuv*+�r"st�wxy4z$��� {|st4z$�h}~(HB ;� e�� New-

��1 ��!fg+,"QR�*U��l6 /���� ;a< "���� ;b< QR��� ;c< r"st��� ;a< �����"���� ���C*"��ob�

Fig. 1 Profiles of (a) water and (b) pressure head

and (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 indi-

cate total and unfrozen water contents, re-

spectively.

��2 ��!fg+,"QR�*U��l6 /����QRST4A!01"����� qlh, "78����� qnh, �A�?ST4A!01"����� qlT, "78����� qnT.

Fig. 2 Isothermal liquid (qlh) and vapor (qnh) fluxes

and thermal liquid (qlT) and vapor (qnT) fluxes

in a freezing loam soil after / hours for dif-

ferent initial water pressure heads.

wxy���� : *U� h�p+,�"������23wxy 29

Page 10: No. +*0,p.,+-, ,**1 - PC-PROGRESS · * * * Jiri S‘imu‡nek** Numerical Modeling of Water, Heat, and Solute Transport during Soil Freezing Kunio WATANABE*, Nobuo TORIDE*, Masaru

man and Wilson �+331��� �����������.�0 ����� ��� �������������������

�� ����� qinit�*.--��� !" �hinit�#/-.

cm� $%&' !" �hinit�#+** cm�� (�%&' !"$��)�*����� qinit�*.+,�+, !" �hinit�#+** cm� �-�./���01�2�3��4�35� -6 ��� /789�:����;����������� 4�<�5���� ��5�����=� >�?5���@�� !"$%&' !"5���,A���� BC��D����@��� >���� ��E�F�����)��/BC��� 9�������G�.5�����������4�-�����H��=%&' !"�BI)J�� >���� �����GK�)L��M�%&' !"���*�����N����OP ���4�3�� (�� ��������5� ���H� �4�-��=%&' !"�BIQ�N3�� R���� %&' !"�����!S�� qi5 *.,/$ !"� *.,

���"T.�*�3��9�� +, !"5� #����"T.�$%&J���� ��'(*U2��� (�� P V)����Q�*� ���*I��P 5�(=W�X�N3�� +YZ������,[$��\]�01-5�^����8���P ���5"_�`��� RI�����"_.a�2F�/5b001�1/�

�� \�� ^����23cde��fg�G�.� ���������.�*R$�����

/� � �

94hijklkm�ijno p�E�5�.� ���)$�����q��rs&�t6�� 9�:���������������7,P �85��01-�@��� 9u�:!v�w;��rs&���RI��b0rs&x<��01�2IR$5� ������eD�����=>�����?y�@5��������'AA�zj{im��� .-�zj{im|�e� �������������7B}�~� >;N�5CD�EW�01M���5�M�� (��\(85�-�Q�������7,P �,�./� �����85V)��R$��FM�$�F���=�G�t��2I��� ��������-�

.� ��H��,�����|H��?!v��e�IJ�����H�� KtL�eM'N�~����O�0�D��P�� �IJ���Q��b�� ���57,�-�.� ��/��e7,���)Rh� 7,���O�SM��G���=����e7,���|��F����rs&O�T����� U����.���.�*����$�F�X��9�� ���$/�I���O,P ����01

5� �j�!��b�*� (�?�����������`���8�78���q�X������AV�  p¡¢!��£1H�5WX��:¤�.��� >X/�G� ���01578����� >���� 01¥&�:¤�U¦��§¨�©F.� ,[$��ª[eq��«�����rs&e01�¬Yh/­���$�F��\]b001�5�� HYDRUS5� ®�¯°�X.��±!l²p�³Z��/���� \(§¨5�´�X.�������b001µ ¶j"[5��15M� ·A���R$���5�M�$�F�� \�5� �j�!����e<>0$�"_�~�\�.� rs&�³Z$¸]��:¤�U��=;���

� � �

Campbell, G.S. (+32/) : Soil physics with BASIC, Else-

vier, New York.

Dash, J.G., Fu, H. and Wettlaufer. J.S. (+33/) : The pre-

melting of ice and its environmental consequences.

Rep. Prog. Phys., /2 : ++/�+01.

Black, P.B. and Tice, A.R. (+323) : Comparison of soil

��3 -6 ����-�.01����°¹N�/78������º <�5����� /�5�����@�º

Fig. 3 Water content profiles for three di#erent

freezing soils after / hours. The solid and

dotted lines indicate total and unfrozen

water contents, respectively.

�»�O^� _ +*0¼ �,**1�30

Page 11: No. +*0,p.,+-, ,**1 - PC-PROGRESS · * * * Jiri S‘imu‡nek** Numerical Modeling of Water, Heat, and Solute Transport during Soil Freezing Kunio WATANABE*, Nobuo TORIDE*, Masaru

freezing and soil water curve data for Windsor

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3 : +-+.�+-,-.

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and van Genuchten, M.Th. (,**.) : Water flow

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+32. : �� ������������������� !"# �$�%&'# .3 ; /.�0+.

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