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

 *ῌ*Optimum Conditions for Predicting Unsaturated Hydraulic
Properties Using the Evaporation Method
Masaru SAKAI* and Nobuo TORIDE*
Abstract
Optimum conditions for predicting unsaturated hydraulic properties using the evaporation
method were investigated. Numerical simulations for evaporation from a saturated silt soilcolumn were carried out. Sensitivities and correlations for two selected parameters of the van
Genuchten-Mualem model were evaluated with the response surface in terms of the pore-waterpressure heads inside the soil column. The pore-connectivity coe$cient, l, which describes the
tortuosity factor for the unsaturated hydraulic conductivity, had a strong negative correlation
with the shape factor, a, and a positive correlation with the saturated hydraulic conductivity, Ks.
It is therefore necessary to carefully select initial values for these correlated parameters.

Higherpotential evaporation rate, longer column, and pressure heads measured closer to the soil surface
were more valuable for the parameter estimation based on the response surface analysis. The
hydraulic parameters were optimized for a Fujinomori silty soil. The optimized hydraulic parameters were more reliable for longer column and pressure heads measured closer to the soil
surface.

Furthermore, using pressure heads at two di#erent depths in the objective function

resulted in smaller standard deviations for Ks and l than those optimized with single depth data.
Key words : evaporation method, unsaturated hydraulic property, van Genuchten-Mualem
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 ,.0 :_S&,** cm [%'1sij
5C ,, :_S78"#4TUA3 E*.+ cm
d&+ ()1 "#t9su +../ :[%'1vwxyzX{5|}C~Kaw4 IGC
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ῌ,3 L 6 !"#$% E _`3,"#$%

Fig. ,

Actual evaporation rates from a silty
soil with di#erent column length L andthe potential evaporation rate E.

&-/* cm 6 E+ cm d&+ VW ij4kH
 |ῌ- QbR  E*./ cm d&+ *+ L+*
cm 6 ,* cm vwxJKLMab^A cd

eJKLM4&,** cm N-.6IG *./ cm
JKLM4 &2** cm N-.6 L+*ῌ-3 L 6 !"#$% E _`3JKLMab : QaR L+* cm _`3 E
+ *.+ cm d&+ QbR E*./ cm d&+ _`3 L+* ,* cm

Fig. -

Pressure head proﬁles in a silty soil with di#erent column length L and the potential
evaporation rate E : (a) E+, *.+ cm d+ for L+* cm, (b) L+*, ,* cm for E *./ cm d+.

³´µ¶ · +*0 ¸ O,**1P

38cm  ,..  .  L,* cm  -.1  0., g R| 3** IJKLMN}URVW~

89 JKLMN&a:b:H} x*  L,* cm    +* cm 

 L+* cm  . Oqr*.*-. qs*..0 a*.*+0 cm + n+.. Ks0 cm

 ,** cm !  L,*d +P l**./ s (  ῌ rnopq f  * cm  0., "# +* cm 

$ defg'}  k῍.  l 

cm $%&'#( )*+& , ,-

/*

./

*.,l*.2   Ox*, l*P &   J

0123456& +./ ,789 7: L

KLMNnpq ;!<^ { x ,* cm ;!<=>?@A&BC9: D K= Ox* x, l*P (Y f+*/E[` x* xxx*x /ῌ, ῏ῒῐΐ
VG FGH / IJKLMN x Oa, Ks, n, qr, qsP 4nopq f  JKLMN} X
H& ( a Y (QDRST l URVW Ox, lP X(Y 12Z[:8JKLMN \]!^JKLMN_` a:b:

LMN f !^_`g ¡¢£oJK^C¤ l**./ ¥

URVWJKLMN4;cX(Ydefg¦n g f   §JKLMN} h(Yij  k῍.  L+* cm, E+ cm d + lm

H}

"# *./ cm \]!^ / IUR   xx* ¨¦©ª g f   l 

VW Ox, lP X(Ynopq f Oῌ rP stuk_` f &«¬­ X JKLMN'

OdefgP  vJKLMN a:b:s4w

#\]&|®DE ¯( zC-* IxyGMN YzC  a { + Xdef

7 X$JKLMN_`&t(7 JKLMN x _`

^

ῑῌ.

)*+ L+* cm, °±12²` E+ cm d + H"# *./ cm \]
!^nopq f defg : OaP l a, ObP l Ks, OcP l n, OdP l qr, OeP l qs

Fig. .

Response surfaces of square residual f for pressure heads at *./ cm depth for a silty soil
with column length L῎+* cm silt and the potential evaporation rate E῎+ cm d῍+ : (a) l῍a,(b) l῍Ks, (c) l῍n, (d) l῍qr, (e) l῍qs.

kl`I¼KL : ¦B={'G#$}³MIUM KNvwZ 



 Ks, qr

l  ῎. b  d!" a, n, qs

l # ῎. a  c 

 y!z"+ 3 cm E  f

39

,+.



 FG, HI" ῎, a :"1242+y!45() V:  }) !"  $%&'() *+

+ %6x' *+ +42+y!45

f ,+-. $+ a  l  *+ })+'()f '" /< 7+M 8&' :@: He῎. a  Ks  l  *+ /' ῎. b  012&*+

3'

 x, l 4&'56789 :+ f  ;'"< f,+ -.=+> ?

:+ $@%ABC'῎. a  aDl ABE FG, HI>})*+ +42,_":>! 'W7 _+ "z"& 9@ *./+ cm : K42+y!45&';I'+' FG, 3  JK, , L1'

Fy!z"l+"W7AB

M N OP QR  ῍ 12 ῎. e  ῎/ a  b S 'TU +0 ῎. a E  l 

:"y!45 j ; fj <:789" fuῌfj

M V&2W7 a QR:>XY&Z 

E2I' g"< f / + %&  E/'0:>+'f :>OP:m+vw&'[ l  n, qr, qs  \%  ] f 

0I  1 =% > ;' ¡¢£7

,+-.  (^_`ab x*, l*q'0¤?' g"< [@¥+y!z"

 ]:",&cd&e>+' ῎. c ;> Fy!z"l;>¡¢£V 012f % !" \+Wd  e&20 A!:+;I'

7 gIhI :>OP'TU/ῌ,ῌ, ῏῔ῑΐ% ?

῎0 a  b

+332

S
et al.
iE/' Àimůnek

 l j% VG kl` a, Ks, n, Lu+* cm Eu*.+ cm dD+ vwy!z" *./ cm ={' qsDl  KsDl ABE/

' T¦BC+ + Eu+ cm dD+ DA'A

 \% ? :m+E/'

B  gIhI῎. e  b E/' T¦BC

0 !" n  qr  *+ /'0,+ Eu*.+ cm dD+ E  §&¦BR ¨E: ῎

qr, qs AB+>: qs

:" !" Ks noAB  Ks pq+,  ©ª«% }) 6x:" ῎- a  qs&'0: Ks QRYUr

Dl AB  Eu+ cm dD+  l :>f+s>+' ῎. e D¬- Eu*.+ cm dD+ E /ῌ,ῌ+ ῎ῐ῍ῒ; f ,+(  qsu*..0 ,=+> l p῎/ a  b

q% ῎. e  t Lu+* cm, Eu+ cm

 l D:>f

6+0

dD+ vwxE y!z" + 3 cm ={'#$%|n:" ῎0 a  Eu+ cm dD+  KsDl AB }& qs  l ABE/' '~"< 

*+ f

( ῎. :%:"  })* + +~-,+ ῎. b  [ ῎0 b  Eu*.+ cm dD+

 *./ cm EE  f῎. e +z" + cm, 3 cm 's> lu*./ qsu*..0 G,,=+> f  -.& &8 f&/# qs

u*..0 ={' l f01'  *./ cm ={' f+,*** (*..2ῌlῌ*./- E/

  +,+-. ',+-. ­®<q+ %

 FG, HI" 012 Ks  l  *% FG, l M

3'"< Ks¯ E/'

AB°e"Z@ T¦BC E+*../ῌlῌ*./2 ῎/ a   3 cm W7% FRE§&¦B@±&¦B²³V' + %&   ] FG,})*   OP:m+vwE/'cm W7HI" ῎/ b  g"< f&/#2"< M I'G#$H}´F7 +;

+y!z"() l f % OP:m+v

I' ¦B=+> µ¶°e>¦B/

wE/'':·'0῎/ c  dtvwy!z" 3 cm ={'aDl, KsDl ABE/' qsDl AB

3 ¸F¹'6&º 1 7U+G#$H}´F I
l M '"<E/' "»: /! GJ+,-. / +*0 0 &,**1'

40

῎ῌ/

Fig. /

῎ῌ0
Fig. 0 L+* cm, E+ cm d +f "#$% : &a' l qs &() + cm'* &b' l&() 3 cm'  !
qs &() 3 cm'* &c' l a &() 3 cm'* &d' l Ks

Response surfaces of square residual f for pressure heads for a silty soil with column
length L῍+* cm silt and the potential evaporation rate E῍+ cm dῌ+ : (a) lῌqs at + cm depth,(b) lῌqs at 3 cm depth, (c) lῌa at 3 cm depth, (d) lῌKs at 3 cm depth.

 L+* cm, E*.+ cm d + ! f "#$% : &a' l qs, &b' l() *./ cm 
Ks

Response surfaces of square residual f for pressure heads at *./ cm depth for a silty soil
with column length L῍+* cm silt and the potential evaporation rate E῍*.+ cm dῌ+ : (a) lῌqs,(b) lῌKs.

y%º9: : ; !"$%¥(

 n»`a ¼<

41

 B6+..C &i3!4 2 o- n* !"#$ %&} VG y Z' "rs(' ( )*+,-.!/01!2 -LIn9!)i3!

3!4al., ,**.C4  VG y/,- BSÀimůnek et al., +332C4 2 

5ῌ1  E6*./ cm d:- *./ cm

7+

89! L6+* cm, ,* cm;"< 89!= > ?!4 L6+* cm@A B5ῌ1 BaCC qs6*..0  ! DE f

F+*,***

G *.-2ῌlῌ*.0, ?! 8 L6,*

cm

@A B5ῌ1 BbCC *..,ῌlῌ*.0* HIJ =>KLM fN*O9P4 23

BRassam etn  qr  bT*}

;"<G B72**FhF* cmC   n 8 qr *cd-.!2^J ! A
.8S*"}> *9!2+?!4
5ῌ2 BaC,  5ῌ+ A-. VG

y B C 8  n-.

 QR L RS' l T9!UV2* n6, 8 qr 6*.,/ cd-.

O94 E6*./ cm d7+}> ?!4 -@A E6+ cm d7+ WX

RI B5ῌ,C Y Z;"< +,[\4 ] QR L *RI9! Q,""}> {#|$<Z] q -* . S/"}> *O94  5ῌ2 BbC O9Z ;"< VI3!J Y ,

"rs n6+.. Z +,01< EI! B5ῌ- BbCC4 2 ^_`a! 8 n6, 2 )¡0-3!4  2lUb8Lcdefg!hi

3!4 ] jk  lmW= >no-2]i QR*RI9!^J^_ qr "rslm*¢9?! +"1£ "$Ti"2 34L¤5^J[ ( 

*pq9! "rs ( ) tuvhi3!4,**1C4 !wxyz « </- 
/-+



¥¦§2[¨©-3! BoªhF* cm  5ῌ2 BaC

qr 

 !wxyz 5ῌ+ O9^_K

G 72**F

 ,

"}

> ¬6I­9!2  qr*(  4

L"#{#|$<*}~ 

7+** cm 

/-, o-%*}4 VG y*5ῌ+"}> 

 ®¯  HYDRUS  °±² ³ 9! Ni]*} o-

&i3!(  ´µ¶7; 8·¸?

"ῌ1Fig. 1 ! 

+ E6*./ cm d7+ *8h!QR L xyz¹  ! :- *./ cm< n89! DE f
l7qs = > : BaC L6+* cm, BbC L6,* cm;"

Response surfaces of square residual f for pressure heads at *./ cm depth for a silty soil
with di#erent column lengths L and the potential evaporation rate E#*./ cm d$+ : (a) L#+*cm, (b) L#,* cm.

s¹º » +*0 ¼ m,**1o

42

῎ῌ2

¬2 qr  n x19yF VG ·6¸ mqsn*./1. an*.**,-/ cm~+ Ksn/ cm d~+ ln*./o mao uXE{ mbo
u5uFig. 2(a) Volumetric water content vs. pressure head and (b) unsaturated hydraulic conductivity
vs. volumetric water content for di#erent sets of parameter values of qr and n for the VG
model (qs῍*./1., a῍*.**,-/ cmῌ+, Ks῍/ cm dῌ+, l῍*./).

s  s 

3/ῌ  /-*} FG n OP 89 
h42 u5] r&6#$!"#$%&'()*+, /. , -*.

?9*=2!"#$0(123 45!"#$/

6#$5 7289: ; <=> ?@[*+ bc* qr %&[3fgA vuXE{hra<=><*+89A <=>2 /B/ CD/ 2  Z [ %& +>E *./ FG2HI/:J KLMNO

Q*^_ ? mh~2** cmo jJ*P> Q RHIST2KLMN<=Àimůnek et al., +332o+J+ mS

UVWXPY Z [ \] ^_Q  fgF,u qs %&a*`!"#$a/ ` 

 /. , -CDE bc YdJ
῍+  , e fg^_hijJ !"#$%&a

*+ 2.- cm k!l mLn2.- 

m2.- cm k!l* qsn*./22 +0 cm k!l*

qsn*./-*o fgg+0 cm k!l* qs OP  fg MN¡Fu([Y (  u5 Ks
m2.- cm k!l* Ksn+.1- cm d~+

cmo * pP *.1 -.+ /., cm  - q*r&/ st%&aAuv ῌ w45  +0 cm k!l mLn+0+0 cm k!l* Ksn+.,0 cm d~+o 2 u¢uf

cmo * pP + / +- cm  - qx19y4iFu5 Ksn/.3 cm d~+ *+5£¤ uXE{r&6#$45¥!"#$a >uXE{ F,(¦ uXE{%&*= Q §*+ N|fgA J}pPstuvÀimůnek et al.,¨HIA©>2ª«*= mSz῍3  %&/6#$

89*+ 2.- cm k!l +0+332o /B/ r&6#$(¬289 Ks :cm k!lB%&PuXE{ ~,**huXE{hr6#$45¥~2** cm *M./  z῍+ u

A%&(V­/2 q*+XE{j/ nn+..  qr n*.*, ?9a*

Y  ®r&/+  ^_ %&* 4<=2 n  qr `Jhr6#$+89 A hra%&a¯2  °±2a² ³´µ¶Jm2.- cm k!l89* nn+.3* qrn*.,,, +0

cm k!l89* nn-.-- qrn*.-01o  /. -. ++./ 

qs, Ks ~j : {|} 012 j Q

43

ῌ+ {|} VG ~ =6 $&
Table +

Optimized parameter values and standard deviations (lower row) for the VG model based on
the evaporation method.

Lengthtensiometersqr

qs

a (cmB+)nKs (cm dB+)l

L[+0 cm+, /, +- cm

*4-01
*4**0

*4/-*
*4**2

*4**,-,,4-3A+*B/-4-*4*1+4,0*4+-24+,A+*B/+4-.A+*B.

+ cm

*4-23
*4**/

*4/,2
*4**2

*4**,-0,4+.A+*B/

-42,
*4*2

+/43
,4.1

*4**,
*4*+-

/ cm

*4-32
*4**-

*4/-,
*4**.

*4**,/++421A+*B/.4+*4+*,402*40,*4**32
*4*,1

+- cm

*4.*3
*4**2

*4/-3
*4*++

*4**,0-4*3A+*B/

.4-+
*4+.

+.4**
04-2

*4**23*4*-1+, / cm

*4-1,
*4**/

*4/-*
*4**1

*4**,-.
,4+*A+*B/

-4/2
*4*1

+403
*4,0

-4/0A+*B.
-4.1A+*B-+, +- cm

*4-.3
*4*+*

*4/-,
*4*+*

*4**,,-4/2A+*B/-4**4*2*42.
*4*3+4-+A+*B0+41/A+*B.

/, +- cm

*4.*+
*4*,*

*4/-*4**-*4**,//+401A+*B/

.4+1
*4*3

-4-*4-0-4-,A+*B.
,42*A+*B.

*.1, -.+, /., cm

*4,,,
*4*+3

*4/22
*4**2

*4**-*,
24*.A+*B/

+43*
*4*/

+41+4*+

*4*/*
*430*

L[2.- cm

2.- cm  +0 cm   qr, qs, a, n   ! " # $% ῌ +& ' ( ) L  * +  ! "$Hopmans et al., ,**,&, -./ 0123 45 Ks 6 l  2.- cm /
7898 +.*+ *.30* 6:;<=>?@ +0 cm
/7898 *.+- +.-.A+*B. 6!"<=6<C
D, = le ?@le B,sῌlῌle E,s 6

F"8

3/῍ GHIJ 2.- cm /B+.21ῌlῌ+.31
+0 cm / B+.20A+*B. ῌlῌ-..3A+*B. /KC

D, Lῌ+*   @D0123 45MN  l 
3/῍ GHIJ6@

ONP8Q:=6Q!=R

DST=UV, 2.- cm 0123
ῌ3

Fig. 3{|}/ @D   :p= p +0 cm  =  2.- cm 
=Water retention curves for a Fujinomori silty soil estimated with the evaporation method : plots are observed
data, the solid line is for L+0 cm, anddashed line is for L2.- cm.

45

3/῍ GHIJ?WVXY D6NZ h[B2** cm ,A+*B-ῌKῌ/A+*B, cm dB+ 6\ , -.

+0 cm /L]/^_/;< `abGHIJ

cdefg , b8Phi '() L d) 60123 45 j l  klm]"nb6U@o 5=pqWrs $/. ,. -t& /uP8Dhi6v-wV,/-- 



5=pqxlyzo  /º ! » +*0 ¼ f,**1g

44+*\. *.+- aJ , EG:)j$klmD>&b` neoJ E p_ , EG!"# q r/stK Lde7XuD>
&s NOvJ ,-."#/ wx$yῌ+ z) Ks[/.3 cm d\+ l[*./ {|+ cm (:) /12 qr[*.**+ qs[*./+/ a[*.**+33 cm\+ n[,.+/ Ks[*.**. cm d\+l[*.**+ &J }/^~%&  Ks B & 7 CD> n 
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r : ·/G·1)r l  3/ῌ K ¸ QK ¹

Fig. +*

Unsaturated hydraulic conductivity as
a function of soil water pressure head
for a Fujinomori silty soil estimatedwith the evaporation method : solidlines are optimized conductivity functions and dashed lines represent the
conﬁdence ranges based on the 3/ῌ
conﬁdence limits for the pore-connectivity coe$cient l.

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a / ;<=  >?> s @A B CD> <0$89&`n, l 0$ E ,-."#/&` $  VWªo,-."#  l /u«¬FGHI s B ;&J K LM >­Z , ¡¢QRS

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'()*+$,-."#/ l  a,-."#'()*+TUZ

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l     Eching, S.O. and Hopmans, J.W. (+33-) : Optimization

of hydraulic functions from transient outﬂow

 !"#$% & '()*+$, -. /0  !*1234*5)6

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and soil water pressure data. Soil Sci. Soc. Am. J.,/1 : ++01ῌ++1/.

3ῌGH

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45

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Hopmans, J.W., Àimůnek,
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J., Romano, N. and Durner,

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W. (,**,) : Inverse method-Evaporation method,  %$ >"  G

Methods of soil analysis Part .. pp. 312ῌ32+, ASA

%1 &: 'F$,()

and SSSA., Madison.

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Inoue, M., Àimůnek,
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nitzer, V. (+332) : In situ estimation of soil hydraulic functions using a multistep soil-water3]^_`*4=5:
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.O pp. ,*,ῌ,*. N P

extraction technique. Water Resour. Res., -. :+*-/ῌ+*/*.
Klute, A. and Dirksen, C. (+320) : Hydraulic con-sductivity and di#usivity-Laboratory methods,+ 6 06  *4

Methods of soil analysis Part +. pp, 021ῌ1-., ASA

'*5)hi

qr, qs, ajkd l.

) n, l ? 06 s

and SSSA., Madison.

gnQ ,**1 : R*S Y. Mualem T +,-J*Km,U ¡@ 2[/*'$,¢)_?"78 jkn:, ; a), , £ ).¤ M. Th. van Genuchten T +,-

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WXO +*0 : .1ῌ0*.

& %rs tsmS Marquardt, D.W. (+30-) : An algorithm for least-&*u\ , 6!*4,

squares estimation of nonlinear parameters.

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SIAM J. Appl. Math, ++ : .-+ῌ..+.
Mualem, Y. (+310) : A new model for predicting the

VW<>*v=,--p, >% (,- (,- 78m@a),\ ?4:,-@Au qs ,-2[/ Ks @'P

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hydraulic conductivity of unsaturated porous
media. Water Resour. Res, +, : /+-ῌ/,,.

GH

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\ +33/ : ¥WX«¬5 pp. ++/ῌ+-0 P ­]P) qs, Ks y$,V !U,hiL

Rassam, D., Àimůnek,S
J. and van Genuchten, M.Th.

@ VD'D 9c*5) (,-z,**. HYDRUS-,D m, +,-^®¯{%|})EF:U, ~VG _t(N./O_y@ =5 *5)6
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X³´a HYDRUS µ_ ¶ PS·3¸bῌ¹ ºῌJiri Àimůnekῌ°­±_ ,**0 :+,- (N._LM,¬»78 ¥ WXO +*. : 0-ῌ1-.

Carsel, R.F. and Parrish, R.S. (+322) : Developing jointÀimůnek,SJ., van Genuchten, M.Th. and Sejna, M.

probability distributions of soil water retention(,**/) The HYDRUS-+D software package forcharacteristics. Water Resour. Res, ,. : 1//ῌ103.

simulating the movement of water, heat, and4 

46

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multiple solutes in variably saturated media.

unsaturated soils. Soil Soc. Am. J., .. : 23,ῌ232.

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

van Genuchten, M.Th., Leij, F. and Yates, S. (+33+) :Environmental Sciences, Univ. of California Riv-

The RETC code for quantifying the hydraulicerside, Riverside, CA.

functions of unsaturated soils. Technical ReportÀimůnek,SJ., Wendroth, O. and van Genuchten, M.Th..
(+332) : Parameter estimation analysis of the evaporation method for determining soil hydraulic

EPA/0**/,ῌ3+/*0/, US Environmental Protection Agency.
Wendroth, O., Ehlers, W., Hopmans, J.W., Kage, H.,properties. Soil Sci. Soc. Am. J., 0, : 23.ῌ3*/.

Halbertsma, J. and Wosten, J.H.M. (+33-) : Re-

van Dam, J.C., Stricker, J.N.M. and Droogers, P. (+33.) :evaluation of the evaporation method for de-Inverse method for determining soil hydraulictermining hydraulic functions in unsaturatedfunctions from multi-step outﬂow experiments.
Soil Soc. Am. J., /2 : 0.1ῌ0/,.
van Genuchten, M.Th. (+32*) : A closed-form equation for predicting the hydraulic conductivity of῍

soils. Soil Sci. Soc. Am. J., /1 : +.-0ῌ+..-.
Wind, G.P. (+302) : Capillary conductivity data estimated by a simple method., Water in the unsaturated zone. pp, +2+ῌ+3+, IASAH, Gentbrugge.

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