
J. Jpn. Soc. Soil Phys.
¼ T
No. +*1, p. 0- 11 ,**1

 *ῌ*Soil Water Hydraulic Functions for a Sandy Soil and an Aggregated Soil
Masaru SAKAI* and Nobuo TORIDE*
Abstract
The Fayer and Durner models, which modiﬁed the water retention curve of the van Genuchten(VG) model, combined with the statistical pore-size distribution model of Mualem were used to
predict unsaturated hydraulic conductivity functions of Tottori dune sand and aggregatedKumamoto Andisol. Parameters of the Fayer and Durner models, as well as the VG model, wereinversely estimated with the evaporation method. The Fayer model well described unsaturated
hydraulic conductivity for the dune sand reﬂecting pore water retention for higher water contents and absorbed water retention on the surface of soil particles for lower water contents. The
Durner model was suitable to the Andisol whose hydraulic properties were strongly a#ected withthe distinct the aggregated structure.

The Mualem model can predict unsaturated hydraulicconductivity for a wide range of soil water pressure when an appropriate model is used for thewater retention curve and the pore-connectivity coe$cient, l is estimated. Furthermore, it wasalso suggested that we need to use an appropriate soil water retention function to predict unsaturated hydraulic conductivity for much smaller pressure heads than the minimum pressure
head observed during the evaporation experiment.
Key words : soil water hydraulic function, Mualem model, Fayer model, Durner model, evaporation method

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S

7B

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-

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Set al., +332 ; JKῌLM ,**1,-!

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#[2

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Mualem ST4]O >?

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 ^_> )closed-form, W`#O%- C van, 

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OVO)*+ ,**1,-

Mualem )+310, D

VG ST4OB0K)Se,±KsKr῍ ῎῏ + dS ῒ ῐ ῍ + dS῏±KsSle ῐe
῎ῑῌ h)Se, eΐ ῑῌ h)Se, ῐ

,

q&rUs

' qr tO u()c* qr v.



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Mualem,+\ #[2 - qr D!  O w

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 ) , Se D

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2

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u(

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'|})cO qr v.

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D +yA qr v.

#O l±*./ O - ῌ _ Mualem ST4
O  h )Se,    K  h I
D Se O

 D0 u('+*1 cm $/' *  WG

q&rUs' qr opO%!

C'$ - C῎"aST4W` "

VG ST4"

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C'

$ -

O%g%  ' S \^,,  ῌ 

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D 1PPQ

van Genuchten῎

,3

"aST4 kB/0[2 B

3  )451 ,**/,  C'' van Genuchten )+32*, 'NO% van Genuchten

12PQA  "' %&- Durner )+33., D 1 WG
1[2

"V

 6 VG ST4O)VG, ST4 I - VG ST4D°_ CW` 3  #q qr±Se±)+¸¹ah¹n, m
qs qr

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AW Durner S

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

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 $ῌ

Mualem6N_`a $῏

K$Se%&KsSle'Gs$h%ῌGmax),'+\]ah]n)(m\c$h%qaῌqs

῏^^ Gs $h%  Gmaxῑ^^ qa *6N_`a b :cd


,ῌ,ῌ-

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

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῏RSU῍12 hm&(+*1 cm

f12< mno*pqrsr tuvRwxy OP1
2 hm&(+*0 cm~

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
6N_`a* S ¡¢

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6N_`a*34;< Durner $+33.% * «VG a x^ ¬© 6N_`a341? mno* £I ­® ¯ , N° ±>1 , 

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

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 R² ^ 6N_`a* 

qῌqs&Se&'+\]ah]n)(m\+('+\]ah]n)(mc$h%qaῌqs

ῑf5#6N_`aΐῒ῎e c$h%&+(ln$(h%ῌln$(hm%  hm * q&*;῏QRSHTU῍ f%%  ῒ !"#

1?
qῌqs&Se&$+(c$h%qaῌqs%



VG

ῐ

35#

ῐῌ+$a% Fayer $῏ %  $b% Durner $ῒ %
6N_`a 3
,  N° : ³a*  +  ³´a*  ,  ta* ,  ῒ $µ%¶

Fig. +

Two di#erent regions for water retention of (a) the Fayer model (eq. (.)) and (b)the Durner model (eq. (1)) : dashed and chain lines indicate the ﬁrst and second
terms of the right-hand side of the equations, respectively, and solid lines
indicate the sum of the two terms (left-hand side).

»IJH66

ῑῌ++*1 ¼ ,**1()*+uEN(¶FH !"#GSE ;=·%(
Mualem +310 =7:hS  l H¸E¹

Table +

 Ks uE0

Soil water hydraulic parameters of the VG, Fayer, and Durner models ﬁt to the observed soil
water retention curves, the observed saturated hydraulic conductivity Ks, and the poreconnectivity coe$cient l presented by Mualem (+310).

Tottori dune sand

qr

qa

qs

a cm+nw,a, cm+n,Ks cm d+l

VG modelFayer model
Durner model

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VG modelFayer model
Durner model

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fg ῎ '0 ῏ 'CK()*+ῐ$%(

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qsqr

(h-./$ Durner -./0w++ahnmw,+a,hn,m,

 w


 w+w,+  ῐ+ bῐ+  !"#$%& ῎ '()*+$ ,&
 Fayer -./0 ῎ '1

ῐ+ a+ 20,2

3%456& 1+2

;<7:($6;=> h+,*** cm ,

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:7

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($6 DEFG@ h+,*** cm

(H$ ῍ ' Fayer -./;%4GI=J4>KLK2@MN6
4 Durner ()*+O

($ ,P6, 3Q9(HR6STU064V&W, 32()*+NP >K

LK-./Q9 6X6-ῌ ῏ ῐ ῒ῎

-./05Q9 ,3(YZ064(H$V6&-./
 [ C\  !"#]%O-ῌ+ ῏ῐ῎῍i) *jklk0$%& ῐ,mnop

qr+ sG,+ tEG+u-6&()*+ klk()*+h῍/* cm (@vw g6

yzX g.$,*῍hx/* cm 

/(G!b;%

4zz{0_Y_@|\K@}9~h῍+,*** cmsG,+ /1\ +33/ u-6 hx

+,*** cm

tEG+ 2ῌ/1 +32. u-6&&W u-93_Z9XJ&`H5 W
4u-+%0\K A  459 , 0()*+$ ;%4hx+/* cm I

?4sG,+u-6

&
t6

klk 7 rb+.0, g cm- 0 rb*..2 g cm- $ >KLK -.2 cm  Durner -./ ^9()*+Y_$

C 2 cm,  1.2 cm C +2.0 cm / 

0@`a Fayer -./ (!b$"86 op$%49g($ ^ 

+V6&Y_$058 - :7X<4;6& ; op

Durner ()*+O ῎ ' $ Mualem -.

$a<64i)g=$&6 i)>B/ ῌ ' #0 c$%( O&¡$%4A-?$49@\t6$¢£CU& k

Y'd@'\K Priesack and Durner, ,**0lk ¤C *./ + , - / cm,  ¤C /./+*./ +/./ cm @¥¡¦!"#$(3§¨

KSeKsw+ Se+w, Se,l
+ῌm m
+ῌm m
w+ a++Se+ w, a,++Se, ,
w+ aw, a,,,e

6 GA©ª$%4/(G$u-6& ,῏ «$B¬CD­@6

A®X\¯°t6$±W& GA©ª B¬CD

Campbell ²

."#n³" CR+*** ´µ6 i)/(G+·,- : ©¸u¹.:qV,F+º»l ¼/

ῑῌ,XaY ¨h©ª© XbY «./0 ,*¬­® : ¯x°4!{ !® ±®
t±²®4¤U³U Fayer  Durner  VG 5V´µ­®¶Fig. ,

Water retention curves for (a) Tottori dune sand and (b) Kumamoto Andisol :open circles are the observed data, and solid, dashed, and chain lines describe
the ﬁtted curves described with the Fayer, Durner, and VG model, respectively.

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 0- '( *.*03 cm- cm)- *+, ,*2 ')-( *..0 cm cm - ".M

 Ecum $ti- %&% *+, , >?@? Fayer / $qa ,/01 2345 $6ῑ,-  7

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F  mt67  cD5 nj D5 jF/01

5 $'i .. + h ῑ+- (Qe URS

F". (G wp , wq ,T01A £¤O l  234¥Ecum $ti- E ti

wE, wq$h- >?@?  h῍)-* cm-   :;  67< Ks

a<¢ k!"# ".A=>?67< ¦HI" 234/01 HIJK(G L

MRST01(Gc URST01A _M ,  /01NJOPQRST01U

VG / § t:; RST01 $qr , qs , a, n, Ks ,<A( RST01VW= $XYῌl- UMZ[ ,**1- ><(( =A\ , kRST01U( ¨©NJO

5 $Z%&%(]^ *./ cm A / cm, *+,PQ@2( (Mrª }«(]^ /./ cm A +/./ cm-   _ qave  U

<AfG <c 5¬M¨©H<A^?234 q `=abcd(Gf?{­(G $Rassam et al., ,**.- c

RST01¨©  2345 Àimůnek ete !? "fG $S

al., +332- _ gh(ijkl  B.m#

$

h $* t- fn%

hA

( U

234GcARST01& 

 $'i .. + h ῑ+- _ 67< Ks ¨©

oB.f5 pqro E $t- kst

^rMs uvwSxv'QMrM<AfGd)+ *+, Ks8,** cm d)+- £¤O l ¨ <()y c ". ©  Mualem $+310- U® $l8*./-   Ecum $ti- /01 zH {*Mys῍¢BC D5]^ADrH ῌa<¢(A $%&% Ks8//* cm

o|kl  ( wp  + A¨©2345AF}~+῎ =F¯°±² , 5". ῌV(/01AIc  wq$h-, wq , wE ]^54EH  pqo E $t- H M  

>?@? +* A _ v ,/01f-³/B.Arl LM./01 IJ(Àimůnek et al., ,**/-  
S· HYDRUS-+D $S

G AH %&%]^ *./ cm   *Mh)1** cm (  , 2345 *q*.-/ (ck>/01 n  (7^?
v8

+
ns,

.

/-^fLM v /01 /-^0 s,῏ 12 -  #_? vE  ".
AH ῌ.+ 

6ῑ, $aῐῌ ?67³/G´µ¶Z%&% 6ῑ, $b-2345¸*+,, Fayer / $

4- Durner / $4- VG / $¹º4- 
5O34C 5 ,/5O34

/01º : L » =O56¼­ ,  

69

K5 J =s *./ cm  |}=I6

 ! "# ,=XI- .* $%& ' ()*#*+!, 

1** cm {=W&

'& -. Durner /012345674kzY

89:;<= qr>*, a,?+*@-, w,>* ABCD Iy= Js56L

2EFG 3H2 34=IG+!,8

|}tT6!wgVk& hq@e  / cm N

T' hq@0* cm {&

k&



9:;<2J῍+ =K5 LML= Fayer /01234.,+ 

&74 N q  h OP=O&GQR W|2pW2

=S56#*T6. UNIV

G~2 -,,/012&34&

W Durner /01 N=XG

YY()*2K5Z[\]^_U=`a& ῍- = Fayer /01 Durner /01 VG /012-,,G~&g|}tN H= Fayer /01234&74 bN ¡¢£2K5 W ῍. =+!,

ch῍@,** cmd IVef56 N¤¥¦2K& -,,89:;<§ ch?@,** cmd QR=S56.

lJ῍, =K&()*2Jag  Durner /012

Fayer /0174 N H ¡ Js74 UN=XG()*

¨wgVk56Z[2IV`a& c

="h=IV34& e LML= VG /012

῍-d W §l\2IV`a&34&74 bNijk2IV

 c῍. cadd ¤¥¦ h OP=O&GQ

Ja56 elT6m nop

qr cq*.*-d

R=©ªS56 VG /01=O& Fayer /01=rs56#*. @2*῍h῍@+*,*** cm  h?@2* cm «,' SzY =6#*

2tuv& hῌ@+*,*** cm twuv& W

2K5 c῍. cbdd Mualem /01 ¬{ Se

=O&G ()*2K52 xH

­&G¤¥¦® ,6 cῌ dm56IyzY ' 34{  nopJ῍+ 89:;<2 |}~=XF6

+*0 cm  qq* 6 Fayer /01 

&G¬{ Se wg . ¤¥¦k¯4V VG /01=°Gwg., 

῍- =

qr =rs56 VG /01=O&G hm q@

LML|}tN

H2' ¤¥¦2tuv56 VG /01±²ῌ-|}t=XF6 LML ³N H : ´µ¶·  ² e²¸-,, Fayer /01 Durner /01 VG /01=I6 ¡¹Fig. -

Pressure heads change during the evaporation process for Tottori dune sand :plots are the observed data, and solid, dashed, and chain lines indicate predicted pressure head changes using the Fayer, Durner, and VG model, respectively.

'§> ¨ +*1 © G,**1H

70

ῌῌ.

lm"[\E  GaH NO P GbH .'9:;z : vwx vP P Py Fayer IJK Durner
IJK van Genuchten IJKA[\x

Fig. .

(a) Water retention curve and (b) unsaturated hydraulic conductivity as afunction of the pressure head for Tottori dune sand estimated with the evaporation method : open circles are the observed retention data, and solid,
dashed, and chain lines are those predicted with the Fayer, Durner, and VG
model, respectively.

῍ῌ, lmA  ¡'¢£f>[\Rd
Table ,

Soil water hydraulic parameters for Tottori dune sand and Kumamoto Andisol obtained inversely with the evaporation method.

Tottori dune sand

qr

qa

qs

a Gcm¤+Hnw,a, Gcm¤+Hn,

Ks Gcm d¤+Hl

VG modelFayer model
Durner model
Fayer (with vapor)
Fayer (with K data)*4*,3*¥*4--*¥*4*23 *4-+1
*
¥
*4--¥
*4*23 *4-+3¥*4*23 *4-+-

*4*-.
*4*-/
*4*-/
*4*-/
*4*-/

.41,
14..
04.,
14-.
140*

¥
¥
¥
¥
¥
¥
*4+1 041-¦+*¤. +4.,
¥
¥
¥
¥
¥
¥

1.422
32422
0/4,.
+**4.012+43.

,4/3¦+*¤,42-¦+*¤.
-4-*¦+*¤*4+/+4+2

Andisol
VG modelDurner model*4***.
*4*+*2

¥
¥

*4*20
*4*-*

+4+.
,4,3

¥
¥
¥
*40- +4++¦+*¤. +4.,

2.-40
.34/

¤*4*3,*421



 

*4020
*4012



f>Cgh*iE 7#$Rj l C[\k   !"#$% & !". lmno:;`aCAWpik& !'()*+,[\"qB.CDE-%./01%2

34.567 8 '9:r  Durner IJKCgXE sjt '9:;;<=>:;?@A 34B.CDE.

u#`a Geῌ-H NO Pvwx Geῌ. GaHH

F6 GTuller and Or, ,**+H Fayer IJK" L:

CAWpiE

Durner IJK

;?@MENO PῌQRIJ

yz{_7 , |}|CDk Fayer IJK.

K.ESTN#$%UVW Mualem IJKC~E& !"&_4CDEXYZ [\"%2]Geῌ. GbHH Durner IJK ,  VG IJKCRC5^E4 E_E B:;`abcRdjE}|NO PC*ikIJK"  Geῌ-H Fayer IJKAZ& !NOW 8 Fayer IJKC

123)¹./ : [º!»0!M5i),|# ¼1

῏.

71

   !"#$%&'
()*

)

+,-./,0

Fayer 1234

VG 123 -56 78)9:

 !"#;.%<  0 .=VG 1238%>?@A8B.C&'F

DE *./ cm

GHI4 0* JKLM=%>NO1* cm P

-RST UVI8WX
Y

῏-Z 4Q.C

VG 123[\[ -56

5]#8^X.%_` 0.,, 

[\[ abP 4 cd efgB5ahijk  l 5ahi8 mno%.

5]# /῏/ M ῌ
῍ pMm5ahi8

> Fayer 123ῌ/5ahi =¯o56M=°5]# UVI ±²[\[
5]# M& : ³´µ4U
VI <s qrUs ¶Us4w·
5ahi UVI8.%&' 5ahi8&' UVI8&' Fayer 123 I¸

Fig. /

Impacts of accounting for vapor ﬂow
and the observed unsaturated hydraulicconductivity data for higher water content on the estimation of hydraulic conductivity of Tottori dune sand : open
circles are the observed hydraulic conductivity data, and dashed, gray, andblack solid lines describe predicted hydraulic conductivity using the Fayer
model without neither vapor ﬂow nor
the observed data, with vapor ﬂow, andwith the observed data, respectively.

5]# Yqr

UsZ 8 5ahi8.%&'  Y<sZ
S>t

u 4v.C>%l N h῍O,* cm
x+** cm dO+.%o56 4=w K

I w>O,*yhyO1* cm M=%> *.*+ cm dO+ Q-R !"

4῏, Mt 5z{s8qa, qs, a, n

I4



!"

5ahi M|}T

~I.C Q 5ah8F A
GHIN
 

῏-

5  GH

K']# lI4 5ahi8.%&'4 lx,.2-+*O.
h8&'4
I0

 % lx*.+/%Mm hῌO0* cm

8&' M> 5a0C 


5]#4

l

5ahi
E.I

.C w %4E aMm!"8:5ahi 4E%

5i), 5ah M|}T

῏0 M 0- JKL cd 5 8t 5

<4¤*D%  4 5i), 

ahi8GHM=%>4 cDE

M>4 5ahi mN c5¥¦*., cm dMefE>% !u58^& v+,8t>%4 55ah

 Q [\[ abP 4 !u5i

>E 5"#M-%ef 4 5

M5ahi 4§ !,M,¨ %

ahi$M~  5ahi8

w l § $ !,M=%>N aM

.%GHM=%>4 d  5"; .C>% 4
*Z 5i

῎ p  4 YKuhxῌca

8%lM . .F $8cdM&l῎ p ¥¦8%>
5]# M> ῌa 8&© 5ahi8M
ª4E%
.,- 

0 Y' ,**0Z   Md 5F[\[ aU«M=%> DE / cm

 4 5ahi8¡ Mm¢.

UVI4 U«¬­L - JK

 ef R( YDE *./ cm £RZ

F A

 5

!u5F

hxO,, cm Q®­QaU« 4-Rd-º78} » +*1 ¼ F,**1K72

0 551 "~z!!p} 6

RD.51C-+. 0 5E  r. !"L "4, 

"  p#$S 1Y.51%+. 0 u NO"/ &+

῍ [3"_`no NO ῌ+\51"YZ+B p#

,Dg 51r 1fSAE 0.ῌ- ῏ΐῐ῎῍ῒi῎1 "

67N

OC 0 D /./ cm, +/./ cm 7N
O
ῑῌ0

Fig. 0 "Y'"Y @A(+1* u)"@"*-0 +1*   "p.D /./ cm NOQ.** cm ·!p"L 5bcO : 6l l E¸Edef-/.no1f-nobcO¹

u"*- {+D +/./ cm NO

Calculated water content proﬁles near
the soil surface with (solid line) andwithout (dashed line) considering vapor
ﬂow.

i ῎1 " deu/. Durner UVW1VG UVW"YZ+ 7-7bcj

kC 0 u i῎2 ",<l1
,jk ῎, "%&'(),OC 0 /L
Fayer UVW   !"#  Ks



$%&'()" *+,-.%&'()"/ 0 Ks $%&'()1234,-.51   67"8, 9:;< .%&'()

=>?@ABCDE+.

Q+/* cm u@"*-0.]>?

 bc1/

 %&'(),-1/Z0 5E  *,<l.l

Fayer UVW, C ,<l "

r/.51 ¡ 1fSAE 0
Durner UVWno 7bcO

NO1

Y4{|- Fi῎1K u ,<lNFSÀimůnek et al., +332 ; GOY4¢- Fi῎2 FaKK0 Durner UVW

HῌIJ ,**1K0 >?"L.+,-I M   ,<l"8;-+C- h

Ks B NO FKsP//* cm dQ+K " *+RD/O

PQ,** cm !p£E< ¤C-0 5E  ¥F 1 S T Fayer U V W  ,  O Ks P 32.22 cm/¦1¥/0¦}§¨}©/ 5

dQ+K0 5 X

FA +33/K "YZ+N-NO FhPQ,1.1 cm "L+/,.+0 cm dQ+ hPQ-/./ cm "L

11ª-+. 0/«¬ h­Q+** cm 

KP

/*  ¥/®¦"

KP2.,- cm dQ+K

L ¯°"2±}7C-+L h

῍ [K "\- ]>?"Y Fayer UV3 Fῌ

῍Q+** cm *@W%&'(),^/Z0 /*  ¥/0¦¤²³3"/Z51

NO"@

de

C-+. 1>´r 0 ,fgh"^/Z0 i῎/ " 

<l"Y4 o- ¥/0µ}1ª-NOS+]>?-,jk FlK NO11

/r

B"C- %&'(),jk῎, "C 0 ebcNV()mSno ,3 1¶S 0

jk

_`a

wP+* 1- bc

KsP2.-.0 cm dQ+ 1/ NO"pq.0

Durner UVW   -e4}{+ VG UVW .+ 7^/Zno 

51r NOmS+B  67NO ¢7bcO"s. tAEh u hvQ/./ cm  +1* /* ¢r/. Fi῎.* cm  

1K0 ,<l NO/A@"-¤wx yz{|-+.

B D

./0%¸¹ : Tº5Bn5 "#$%&' »¼

 

73.. 

 ῌ2  ! "#$%&'

Mualem ./01 FGH BFGIJKLMN

()*+,- VG ./0123456789

OPQ 5 &R<./0=S

(:;<=>?@ABCDE TUTV,Q1 W "X5YRZ[A\1 ] "X^_`B1 
a*b?

-cd1 Mualem ./01 e"fghijklQ1R<=>?,BCDE m n5V,Q  nIo=cd
1hij*b?pq&*S @ rst
V,Q , uH avw "Hxyzῌ

{./0=S Fayer ./0B Durner./01 TUT|2345}~R<B*=>

5(:R

<Q,BCDEl Mualem ./0V^_{)l 1

yRBQ*ῌ1Fig. 1}~V«r234556 7
89 : ¬­¡®1¯° ¯z l±²z1
 Durner ./0 VG ./0 ¨³(9´µ¶
Pressure heads change during the evaporation process for Kumamoto Andisol : plots are the observed data, andsolid and chain lines indicate predictedpressure head changes using the Durner and VG models, respectively.

Mualem ./ uBQ l*./ Q,?, Mualem, +310
1  ?&
- l "HxyzB1 Q

B56 7'  D ¡¢ £¤BQ&*¥, ¦,7fg "HxyzQ()&], Fayer ./0| Durner./0V,Q E^_{)l §P

BQ ,B ¨ Mualem ./0  "Hxyzῌ

{./0©&ª!

ῌ2}~·=§P2345 a
"Hxyz b B56 7
' : ¬­¡®1¯° ¯z l±²z1  Durner ./0 VG ./0§P¶Fig. 2(a) Water retention curve and (b) unsaturated hydraulic conductivity as afunction of the pressure head for Kumamoto Andisol estimated with theevaporation method : open circles are the observed retention data, and solidand chain lines indicate those predicted with the Durner and VG model,
respectively.

» WX! Y +*1 ¼ %,**1.

74 

  D*+*+10 Fayer 67821G! " #$ 

$

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75

Durnerb%  W

    X,  M. Th. van Genuchten  

Fayer

 

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!"#$%&'()*+Marquardt, D.W. (+30-) : An algorithm for least-

01232345637 89:;squares estimation of nonlinear parameters.

<=>?@AB C/%

SIAM J. Appl. Math., ++ : .-+ῌ..+.
Millington, R. J. and Quirk, J.M. (+30+) : Permeabilityῌ ῏ ῎ ῍of porous solids. Trans. Faraday Soc., /1 : +,**ῌ

Brooks, R.H. and Corey, A.T. (+30.) : Properties ofporous media a#ecting ﬂuid ﬂow. J. Irrig. Drainage Div., ASCE Proc. 1, (IR,) : 0+ῌ22.

+,*1.
Mualem, Y. (+310) : A new model for predicting thehydraulic conductivity of unsaturated porous

Burdine, N.T. (+3/-) : Relative permeability calculations from pore-size distribution data. Trans.
Am. Inst. Min. Metall. Pet. Eng., +32 : 1+ῌ11.
Campbell, G.S. and Shiozawa, S. (+33,) : Prediction ofmedia. Water Resour. Res., +, : /+-ῌ/,,.

HIJῌῌ
]

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ῌpq

r 8+33/; : #Y pp. ++/ῌ+-0 }~23i[}~Y

hydraulic properties of soils using particle size

Philip, J.R. and de Vries, D.A. (+3/1) : Moisture move-

distribution and bulk density data, in Interna-

ment in porous media under temperature gradi-tional workshop on indirect methods for estimating the hydraulic properties of unsaturated soils,
University of California Press, Berkeley, +33,.

ents. Eos Trans. AGU, -2 (,) : ,,,ῌ,-,.
Priesack, E. and Durner, W. (,**0) : Closed-form expression for the multi-modal unsaturated con-Childs, E.C. and Collis-George, N. (+3/*) : The permea-

ductivity function. Vadose zone J., / : 23,ῌ232.

bility of porous materials. Proc. Roy. Soc.

Rassam, D., Àimůnek,S
J. and van Genuchten, M.Th.
8,**.; : HYDRUS-,D 

London, Ser. A, ,*+ : -3,ῌ.*/.
Durner, W. (+33.) : Hydraulic conductivity estima-

[ HYDRUS £(¤ }~Ytion for soils with heterogeneous pore structure.

Rossi, C. and Nimmo, J.R. (+33.) : Modeling of soil

Water Resour. Res., -* : ,++ῌ,,-.
Eching, S.O. and Hopmans, J.W. (+33-) : Optimizationof hydraulic functions from transient outﬂow
and soil water pressure data. Soil Sci. Soc. Am. J.,
Fayer, M.H. and Simmons, C.S. (+33/) : Modiﬁed soil
water retention functions for all matric suctions.

Water Resour. Res., -* : 1*+ῌ1*2.
S¥D¦ῌ§E ¨ῌJiri Àimůnekῌ
hij 8,**0; :] +*. : 0-ῌ1-.

§E ¨ῌhij 8,**0; :   : J.R. Philipand D.A. de Vries  ®¯°±²©ªb

Water Resour. Res., -+ : +,--ῌ+,-2.

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Jury, W.A. and Horton, R. 8,**0; : #]water retention from saturation to oven dryness.

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ij k : Elmῌn oῌpq rῌsÀimůnek,SJ., van Genuchten, M.Th. and Àejna,SM.

tuῌvwx; pp. 23ῌ3- +*1ῌ+*2 yz{(,**/) : The HYDRUS-+D software package for| }~Ysimulating the movement of water, heat, andKosugi, K. (+330) : Lognormal distribution model for

multiple solutes in variably saturated media.

unsaturated soil hydraulic properties. Water

Version -.*. HYDRUS Software Series +, Dep. of

Resour. Res., -, : ,031ῌ,1*-.

Environmental Sciences, Univ. of California Riv- 8,**1; :   : Y. Mualem  

erside, Riverside, CA.

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76

I-b wFmC mDwmBwm ln wmDwln wEÀimůnek,SJ., Wendroth, O. and van Genuchten, M.Th.
(+332) : Parameter estimation analysis of theevaporation method for determining soil hy-

A1

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draulic properties. Soil Sci. Soc. Am. J., 0, : 23.῏

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Tuller, M. and Or, D. (,**+) : Hydraulic conductivity
H wFln wDmwBm mD+of variably saturted porous media. Water Resour.
Res., -1 : +,/1῏+,10.

A3

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tion for predicting the hydraulic conductivity of

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unsaturated soils. Soil Soc. Am. J., .. : 23,῏232.

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