0o3E-075C/00/ 1 6508-62,H31
Soil Science
Copyright O 2000 by Lippincott Williams & Wilkins, Inc.

August 2000
Vol. 165, No.8
Printed in U.S.A.

WATER AND CHLORIDE TRANSPORT IN A FINE.T$fiURED

FIEID EXPERIMENTS AND MODELTNG
D. Ventrellal, B. P. Mohantf, J. Sim$neke, N. Losaviol, dnd M. T.

SOI[:

van Genuchtene

Numerical models are being used increasingly to simulate water andsolute movement in the subsrirface for a variEty;;pt;;l";s in research and soil/water management. Although a iatge"f ,,-"lrr,b"i"?m"autd,eerees of complJxity have U""r, t"o"rcp;; ;;;;;h;,"t_^"I::.tytlg
auvety tew have been tested under field conditions. ve testedll"rr,
the performance of the FrYf)lqs.-1p. computer model t" ti*ri"i. variabry saturated water flow and chloride trairsport in a fine-textured ii"ti"o ,oit
subj-ec-t to a fluctuating saline groundwater table. The modJ was arso
solute tranip-ort pararneters using an inverse optiTi_d.{". estirnating
rnization
scheme. our tesults indicate that including the effects of irnmobile water produced better predictions of chlorilde transport compar-ed- with the traditional convection-dispersion transport^approach.
Including anion exclusion as well did not irriprove the m6del priiiction,
appreciably. occasional deviations between model predictioir and fieldobservation were attribut-"4 j" uffepresented latera-i groundwater flow
processe-s a1{t9 preferential flow through macropores 6r ot}rer structural
voids. The FTYDRUS-1D model was found to be very useful for analyzin.g the.relatively co-rnplex!o,w and solute transporr processes at our {ieldsiteand for estirnating_rnodel pararneters using inve-rse procedures. (Soil
Science 2000; 165 :624-63 1)

words:'water flow, chloride transport, numerical modeling, shal.lowrGy
groundwater table.

Jl 7;'OST coastal zones of Italy are subject to rel_IVlatively high phreatic groundwarer tables, often occurring within 2 m of the soil suface. Suchshallow water tables can have considerable impact
on both the growth of agriculrural crops, anjthe
preservation of soil and groundwater environmentd,resources. Consequendy, optimal agronomic

techniques (irrigation, ferrilization, and tillage)must be based on a precise understanding of thewater and salt balances in the vadose zone and shal_

low groundwater.
The ability to predicr warer flow and solutetransport into and rhrough the vadose zone accu_
rately is essential for effecrive management of soil
and groundwater. A large number of analyticalmodels, and especially numerical models, haverlstituto Sperinentale Ag'onomico,

MiPAF, Vo Celso Ulpiani, 5. 70lg5 &i, lhly.
Dr. Ventrella is conerponding author. E-mail: agronba@interbusinesr.it

rU.S. Salinity Lrboratory, USDA-ARS,

450 \(. 86 Sp;nsr Rdu Rtvcride, Gli

lornio 9?507.
Received Dec. 7, 1999; accepted March gl, g@.

been developed during the past several decades forpredicting water flow in the vadose zone at thefield scale (e.g., Belmans et al., 1983; Hopmans,
1988; Hopmans and Stricker, 1989; deJong andKabat, 1990;Johnsen et al., 1995; Ould Mohamed
et al., 1997). By comparison, relatively few fieldscale model application srudies have been carriedout- to date by soil scientists and hydrologiss.
Mohanry er al. (1997 and 1998) recently used a
two-dimensional numerical model with bimodal
hydraulic funcdons to predict field-scale preGrential flow and nitrate transport in a structured agricultural soil in New Mexico.Although they werereasonably successful in simulating flow andtransport at their experimental field site, based on
several site-specific findings, they recommended
further tesring of the model under different hydrologic, soil, and climatic conditions.
In this paper we evaluare the utiliry of a nu-

merical model, HYDRUS-1D (Simtnek et al.,1998), for simulating long-term (3 year) waterflow and chloride rransporr in an Italian soil sub-tl.'!
,-j -'tvol. 165 - No. 8 WATER AND CHLoRIDE TTRANSpoRT: ExpERTMENTS AND MoDELTNc 625
ject to a shallow, saline groundwater table. As asecondary objective we analyzed a limited data set

from the field to obtain solute transporr parameters in the governing convection-dispersion
solute transport equarion (CDE) with and without accounting for the presence of immobile liquid zones and anion exclusion.

FIELD SITE AND

HYDROLOGIC MONITORING

The experimental field (15 m X 15 m) is
located at the experimenral farm of the Istituto
Sperimentale Agronomico, situated in the coastal
area of the Basilicata region in Southern Italy(Metaponto: lat.40o 24' N and long. 16o 48, E).

The climate of this region is classified as "accennrated thermomediterranean" (according to

the UNESCO-FAO classificarion), with mean

monthly temperatures of 5 oC in the winter and 26
in the surruner. The mean annual precipitarion(1981-1998) at the experimenral farm is 476 mm,with more then 77% of the rainfall occurring
during the winter months (September throughMarch). Typically, a substantial proporrion of theoCannual rain comes in a few days, thereby creatinga rapid rise in the groundwater table during rhe
winter period (Fig. 1).Another fearure of thi climate at our field site is the high annual potential
evaporation rate, with a mean annud pan evaporation rate of 1535 mm. Evaporation rates are
maximum during the months of June,July, andAugust, with mean daily rates of 7.4,8.4, and 7.3mm, respectively. The soil is classified as a Typic
Epiaquerts, similar ro those formed on the silwclay and clay lagoon sediments of the Oleocenethat are present over large portions ofthe alluvial
basin between the Basento and Bradano rivers.
The original sediment affected its evolution andcurrent functional characteristics strongly since itconsists mosdy of swelling clays. The soil, poorly
drained with strong medium angular blocky structure, is relatively low in nitrogen and organic
matter and is classified as alkaline (pH : 8.4).
The clay and silt contenrs, as well as the electricalconductivity, increase with depth, whereas theorganic matter content and the bulk densiry decrease with depth (Table 1).

In November 1993, five sets of nested

15

flo
E
U)
g
()E

ul

100

E

o
-c

100
E
c)

'lc

50:

o

o

E
E

.E

S zoo

.s
(5

tr

o
(u

= 300
O1JAN94 O1JUL94 O1JAN95 O1JUL95 O1JAN96 O1JUL96 O1JAN97!ig. 1' water table depth, daily rainfall (needle), and groundwater salinitr at 3 m during the entire period of stuoy.
The bars indicate

t standard error of the field means.

626'ENTR'LLA,

MonaNrv, srr"r0Nrx, AND vAN GENU.HTEN soL scrENcE
TASLE 1
Selected soil properries at thc cxperimental site

Layer

cm

HorizonParticle size disribution

Clay (%Q Silt (7o)
(F5U

50-70
70-100100-170t70-270s8.8
60.0

Sand ("/d35.8

Dry bulk

Elecnical
conductivity

density

(1:2)

cm-3

dS m-l

1.38

Cationexchange
capaciry

Organic

mttter /o

0.s3

J5-6

+.zJ

IJ.J I

1.48

1.21

0.77

60.2

36.4

J-f, I

71.47

0.83

1.19

I

67.7
77.2

30.8

9.68
10.28

0.61

16.31

u.5 /

A1

11{
0.84

piezometers wsls in5tellsd, each set consisting

1.10

2.38

0.72

of tration, which was found to be strongly corre_
lated (r : 0.95) with the elecnical .oijo.tirrir,,five- piezometen (120-mm i.d.) located at de;rhs

of 0.5, I,7.5,2, and 3 m. The piezometen wereused for measuring hydrostatic pressure heads andcollecting warer samples for chimical analysis. To
prevent clogging, the lower parts of the piezome_ters were lined with glass wool and enveloped by alayer of sand and siliceous fine gravel. Before in_stallation of the piezometers, a piiwas excavated toa depth of 3 m in order to detirmine variations in

the unsarurated soil hydraulic property versus

depth. Three replicates of S-cm-&ametei and 5_

cm-long undisturbed soil cores were taken fromeach layer. Soil water retention data were obtainedby using a sand and kao[n box apparatus (from sat_urarion to -300 cm of water) ind p.essu.e plates(berween -300 cm and-15000 cm).
7994
and December 1996,..BerweenJanuarysoil water pressure heads were measured on a
weekly basis. Ground water samples were taken
every 2 weeks to derermine rhe bulk electricalconductiviry and concentrations of chlorides. sul_phates, carbonates, bicarbonates, sodium, magne_
sium, and calcium ions. Figure 1 shows the Jendof the measured water table depth and the elec_

trical conducriviry at 3-m depth. During thesummer months the water table remained nearlv
constanr at a depth of 1.8 to 2 m, whereas durini
the winter monrhs the water table rose to d.pth
of 0.2 to 0.4 m below the soil surface as "a .es.rlt
of heavy rains.An increase in the evaporative de_ttr:lq of the atmosphere following ih. ,,r-*..
of 1995 .caused the groundwat.i t.bl. to go
down and its saliniry ro increase up to 10 dS m--1.

The electrical conductivity during the previous
winter months, when the water table wasihallow,

ranged berween 1 and 4 dS m-t. Befween Tanu_
ary 7994 and October 1995, a slowly incrJasing
trend in salinity was observed until a lirg""rrrorrrriof rainfall in 1996 decreased the electiicalcon_ducdviry to its background value (3 dS m-t). In
thls study we consider only the chloride concen_(Losavio et al., 1997).

-The

experimental field was kept weed freeand no crop was grown.

NUMERICAL MODELINGThe HYDRUS-1D computer model (Sirnf_
nek et al., 1998) was used to simulate o.r._ji*.rr_
sional vertical isothermal variably-saturated flow at
the experimental site. Simulations were performed
for the period from January 11,lgg4,to D...*_
ber 31, 1996. The one-dimensional flow domain
extended to a depth of 3 m and was divided
rnto five separate soil layers, as shown in Table 1.
comprising a total of 100 finite elemenrs.A finei
discretization was used near the soi_l suface to ac_
commodate relatively steep gradients in the pres_sure head.

On the basis of available profile informationand hydraulic properry data, the five soil lavers
were characterized in terms of their field_averaied
hydraulic functions. For each layer, we optimiZedth; hydnulic parameters in the ietenrion ^equarion-ofvan Genuchten (1980) using the R-ETd non_li19ar optimizarion code (van-Genuchten et al.,1991). As the soil suface boundary condirion(x=L), we used a sysrem-dependent atmosphericcondition in accordance with the approach of
Feddes et al. (1974) and Simtnek .iA.
lt9ley.
Class A pan evaporation rates, recorded bv'an au_

tomated data-logger locared at the experimental
farm, were used to calculate potential evaporation
rates on a daily basis. These data. combined withdaily precipitation data from the same weather

station, were used to define the transient top
boundary condition. The bortom boundarv con^-dition (rc:0) consisted of daily prescribed valuesof the presure head [/r,(r)] obiained by linear in_

terpolation from weekly measured daia. The wa_
ter table elevarion measured onJanuary 11,1994,
was used to define the initial condition[ft.(r)] forvor. 165 - No. 8 WareR AND CHLoRIDE TTRANSpoRT: ExpERTMENTS AND MoDELTNG 627
our simularions. Thus, the boundary and initialconditions for our calculations arelanll-K;-Kl =f
ldxl

ot

arr=L

hA<h<hs acx:Lh(x,t):h,(t) atx=0h(x,t):ho@) au:0

(1)

(2)

(4)

filtration and evaporation under the prevailing at*mospheric condirions (LT-t), and h^and h, are,respectively, the minimum and maximum pressure heads allowed at the soil sudace (L). The
value of ft5 was set to 0,whereas ft, was calculatedusing relationships given by Feddes et^1.(1974).
HYDRUS-ID solves numerically theRichards equation (Richards, 1931) for variably
saturated water flow and the convection-dispersion transport equation (CDE) of a single ion.
The latter equation is given by:(e",9\+{:90t dx \-- / 1-u(s)in which\ lql + 0q,,(6)

0Du, =

0x",,,

where c is the solute concenftarion (ML-3), 0 is
the warer conrent (LrL-r), pw is the efFective dispersion coefficient (Lzf-t;,4 is the Darcian fluid

flux densiry (LT-t), }' is the longitudinal dispersivity (L), D* is the molecular difi:sion coeffi-

cient (LzT-t; and r* is rhe tortuosiry factor.
An option in HYDRUS-1D allows users ro
implement the concept of two-region or mobileimmobile solute rransport (MIM: van Genuchren
and Wierenga, 1976). The code for this purpose
partitions the liquid phase into distinct mobile,

9- (L3L-3), and immobile, 0,_ (L3L-3), regions,with solute exchange berween the rwo liquid regions modeled as an apparent first-order exchange process, i.e.,oor,r,,,dt* dO,r,Attu, :a /, p,,, ot^ \_ Oqt^
dr
dt
dx
\"''"

)

(7)

* d[0,i,,*(1 -,f)p&]c,i,, _at

A l.
3r \ dqc...
I -^ le"'Dei:r!'dx\"'dxr;:
0[0,,,,1 (1 - f ) P k]c,,,,

At

(3)

where t is the time (T), r is distance (positive upward) (L), E is the maximum potenrial rate of in-dx

6(0,,,*fpk)c,,,(8)_:cr(rr,_c;u)in which;fis the fraction of the solid phase thatequilibrates instanraneously with mobili fluid, (1_/) is the fraction that equilibrates with immobile
Iiquid, p is the soil bulk densiry, and fr is an emoir-

icd distribution coefEcienr 0-i tvt-t). th. vatu! ofk determines if chemical sorpcion (&>0) or anion
exclusion (&<0) takes place. As shown bv van
Genuchten (1981), the above model rese'mbles
closely the anion exclusion model of Krupp et al.
(1972),where anion exclusion is assumed to be restricted only to the immobile phase.
For chloride transport we implemented a noflux boundary condirion at the soil suface (which,in case of rainfall, assumes that the rain water isfree of solute), whereas observed daily concentrations [c,(r)] were used to define a iime-variant
bottom boundary condition (obtained by linear
interpolation from biweekly measured data). Measured chloride concenrrations [co(x)] from thepiezometers on January 11,1994, were used todefine the initial concentration in the transporr
domain. The boundary and initial conditions ofchloride transport can thus be written as:-oD+*qr:odx

atx= L(9)c(x,t) = c,(t)

atx=0

(10)

c(x,t) = co(x)

at

l: 0

(11)

MODEL PARAMETER ESTIMATIONHYDRUS-1D includes a Marquardt-Levenberg parameter oprimization algoiithm for inverse estimation of transport parameters, basedon minimization of an objective function that
express the discrepancy between observed andpredicted values. We used this procedure to estimate field-scale chloride transport paramerers for
the CDE, MIM, and MIM+AE modeling approaches. For our case the objective function, e,
was taken as

where cr is a mass rransfer coefticient (T-1).
_-{ r
O(b,p) =c,(x,trQf Q2)
We also incorporated anion exclusion (AE)
Errlri @,t) concepts into rhe MIM approach, For this purpose
we used the following transport equations (van where n is the number of measurements, cf areG^enuchten and Wierenga,1976; van Genuchten, the measured concenrrations at time r,
anj depth
1981)
x, c(x, ti,b) are the corresponding model pr.di._

628

VENTRELLA, MoHANTv, SnurriNrr, AND vAN GrNucurEN
TABLE 2Fitted van Genuchten retenf,ion parameters and regression
coefficients for different soil horizons at the field site

Layer 0,0,c
R2cm cm3 cm-3 cm3 cm-3 cm-l
0-500.4791 0.0077 1.113s 0.986(0.00s4)s (0.002s) (0.00e4)

0.5461 0.0145 1.1504 0.996

50-70(0.0087) (0.0033) (0.0074)
0.5501 0.0130 1.1531 0.996(0.0101) (0.0034) (0.0088)
0.5593 0.0084 1.1559 0.993(0.010e) (0.0023) (0.0101)
0.5600 0.0034 1.4100 0.900(0.0107) (0.0014) (0.0e10)70-100100-170t7o-2r0$Values in parentheses are Standard

Errors associated withthe parameter.

tions for the optimized parameter vector b, andu.,, is the weights associated with a particular measurement point, i.'We assumed that all ,/:.s wereequal to 1, indicating that the error variances for
all measured concentrations remained the same.

In this study, the vector of optimized parameters
included \ for the CDE approach and 0,^ and afor the MIM and MIM*AE approaches,in'which
we assumed that \ was known with sufEcient ac_
curacy from the CDE approach. Independent

sensitiviry analysis showed that )t had rilativelv
little impact on rhe results.-We utilized-..ror.i
chloride data during the inicial 100 daysof theexperiment for our inverse modeling. The pa_
ramerer(s) esrimated using this procJdure wL."
used later in the forward problem for simulating
the entire 3-year period.
RESULTS AND DISCUSSION
Table 2 shows the field-average values of thedifferent soil hydraulic p"."*et"., by soil hori_zons optimized w"ith R-ETC from water reren_tion measurements using soil cores. Since nomeasurements were made of the saturated hy_

Sorr ScrcNcr

draulic conducriviry (K), we initially calibrated
our_simulation results to a reference K, of 5 cm

oav ..

Table 3 shows average daily water fluxes
across the top (evaporation and infiltrarion) andbottom (inflow to and ourflow from the domain)boundaries calculated with HyDRUS-1D from

December 10, t994, ro March 16, 1996. This
simulation period was divided into five intervals
characterized by different water table conditions.

During the winter monrhs of both yean (De_cember 70, 1994 to January 13, 7995 and irlo_
vember 26,1995 to February 5, 1996), fluxes
were directed predominantly into the flow do_main because of infiltration through the surface
and inflow through the bottom associated with arising groundwater table. The drop in the ground_qble between January 74, 7995, and May
1vat9r_10, 1995, was caused primarily by outflow fromthe flow domain, but soil evaporation was also
g!lT*.: During the summer monrhs (May 11,

1995 to November 25,1995),when the groundwater table wx deep and relatively stabli (about
200 cm fiom the soil suface), soil evaporation
dominated all other fluxes. Finally, duiing theperiod between February 6,1996,and Marc-h 16.
1996, ner infiltrarion (infilrration-evaporarionj
across rhe top boundary equaled the outflow
across the bottom boundary, thus defining an
qpproximately stationary shallow groundwatertable. Flux dara such as rhose showi in Table 3,,|:dd be helpful for quantifiing the impacr ofornerent agrlcultural management practices on
the leaching of solutes to groundwater.
Table 4 shows the resulis of the parameter op_
timization urilizing measured chloride .orr..rrtorions during the initial 100 davs. For D... we used a
Iiterature value of 1.7 cmz day-l (van h.ees et al.,1991). It is evident that the inclusion of the mo_bile-immobile water concepr improves the correspondence berween measured and simulated chlo_ride concentrations at diferent depths (Fig.2), u
indicated by higher regression coeficient (R2) andTABLE 3Avenge daily water fluxes across the top end bottom flow boundaries during partof simulation (Dec. 10, 1994 - Mar. 16,1996) as calculated with HyDRUS_1D
Average daily fluxes (cm day-1)

Observed

Durationwater table

Top boundary

condition
Dec. 10,94-Jan. 13, 1995

Rising

0.09

Jan. 14,95-May 10, 1995

Falling

0.17

May 11,95-Nov.25,7995
Nov. 26, 95-Feb . 5,7996
Feb. 6, 96-Mar. 16,1996

Bortom boundary

ion In{iltrarion Inflow to domain Outflow from domarn
0.16
0.04
0.09

0.01

Rising

0.16
0.08

0.26
0.05
0.07
0.28

v. rz

0.15

Shallow

0.01

v.Jz

0

0.31

0.40
0

vor' 165 - No. 8 WATER AND CHLoRIDE TTRANSpoRT: ExpERTMENTS ANDMoDELING 629

TABLE 4
Results of transport parameter estimation from experimental concentration
ApproachConstant DarametersS

Initial Esrimate Final Estimates Standard Errcr

D",= t.t,O,,n = 0,g : 0

CDE

MIM
MIM

10

D, = 1.7, )t = 10
Dw= 1.7,1\ = 20

MIM

data

Estimated parametersS

Ds:1.7,\=30

MIM+AE D. = 1.7,X = 10,k = -0.01

R2

0.78

0.001

0.41
0.91

e.

0.1

0.209

c

0.012

0.008

0.016

0.005

0,"
c

0.1

0.210
0.016

0.012

0.008

ou,
a

0.1

0.210

0.008

0.012
0.005

0.91

0.016

A.

0.1

0.203

c

0.012

0.91

0.008

0.015

0.005

SD, (c*zday-r), \ (cm),0,. (cm3 cm-3), c (d"y-r), & (cms g-t1.

lower standard error compared with the CDE.
The CDE actually posed convergence problems.
With the MIM approach, Oim and cr were esti-

mated to be 0.21 and 0.016 day-l, respectively,
independent of the value of )t. In addition, incorporating anion exclusion in the MIM conceor
resulted in a slight decrease in 0,_ with no im_provement in R2 or the standard eiior. This resulr
suggests that anion exclusion is not as importanta factor under transient field conditions as it isunder the steady-state flow condition for which

the concep_t was originally developed. Furrher_
more, our findings reconfirm the importance ofthe mobile-immobile water concep-t, which isused widely to characterize prefereniial transport

in field soils. In addition by using different initialconditions, we were able to identify the (unique)transport parameters of the MIM approach.

0.91

0.005

Adopting the MIM approach (N : 30 cm,0t_= 0.21,1, and d. : 0.016 day-t1, we carried oui

another independenr simularion fromJanuary 1 0,
1995, to December 37,1996,in order to tesi theperformance of the model by comparing simu_
lated and measured chloride concentraiions at
50, 100, 150, and 200 cm depths. Resuls in Fig.
3.
thow that the model prediitions agree .".rori_
ably well with the available data lt different
depths. The model underestimated and some-what delayed the arrival of the chloride peak in
October 1995 at the 200 cm depth. A similar

trend was noticed for the other depths (resula
not shown). This finding suggests that several
field-scale processes *... ,roi correctlv represented in the HYDRUS-1D model.
The processes that were most likely not modeled correctly are (i) lareral groundwater flow,(A)

11
CEApfld

COE App@c}l

€:

ot
o
o

MIM Approsdl

o

o
=

o
l+
l{an

Zl{ar

Fig. 9. Comparison of inverse analysis results of CDE (I:9.7g

1+tl.fn) and MIM ()t=30 cm, 0r.=0,21 cm3 cffi-3r c=0.016
day-1) approaches (solid lines) with measured (symbols) soll chloride concentrulons
at depths of 50 cm (A) and
900 cm (B).

630

VENTRELLA, M.HANT', SuutNer, AND vAN

GrNucnteN Son ScmNcr

o
F

m

o
o
o
o

o5

t0

(=J

o
0{_
0rJN

0{_
0tuffiFig. 3. Comparison of the simulated (solid line) andmeasured (symbols) soil chloricte concentration at depthsof50 cm (A) and 900 cm (B).

and (ii) preferential flow through larger pores andstructural cracks in the vadose zone. Ouruesuls

agree with Zurmiihl and Durner (1996) who
showed that the MIM approach was not very ef_
fective, in describing preferential t..rrrpor, u.rd".
natural field conditions, unless bimodJ funcrionsof the hydraulic properries were considered in at_
tempts to account for flow through the structured(macroporous) pars of rhe sofu.1he importanceof using bimodal funcrions was confiried bv

M1h.anry et al. (1998), who simulated the preferential transport of nirrare in a field soil. Zurmiihl
and Durner (1996) atso demorutrated that a betterde.scriprion of preferenrial transport in structured
soi-ls can be achieved when 0* is defined as atran_
sient variable that varies in tiile, depending on rhe
slope of the unsaturated hydnulic cond-ucrivity
curve. The current version of HyDRUS_1D as_
sumes that immobile water content is constant. A

follow-up paper will address some of these issues.

CONCLUSION

The performance and uriliry of the Hy_
DRUS-1D model was assessed bysimulating wa_
ter flow and solute transport in a bare, fine-_tex_
tured soil in ltaly subject ro rransient water_tableboundary condirions. The model, with estimated

MIM transport parameters obtained from inverse

modeling (based in part of available data), repro_
duced the general trends of measured ttt"ria.
concentrafion at the field site fairly well, except
for some underestimation and a delayed arrivd tf
the concentration peaks. possible r."ro* for thisbehavior could be unrepresented lateral ground_water flow and preferential flow through macro_pores and other structural voids. Considering
the complexiry of our field conditiorrs. the

HYDRUS-1D model was found to be a useful
tool for analyzing solute transport data and esti_

mating transPort parameters.

REFERENCES
Belmans, C.,J. G.Wesseling, and R. A. Feddes. 19g3.
Simularion model of the water balance of a cropped
soil: SWATRE. J. Hydrol. 63:27 l-296.
de Jong, R., and P. Kabat. 1990. Modeling waterbal_

ance and grass production. Soil Sci. Soc. Am. T.
54:1725-1732.
Feddes, R. A., E. Bresler, and S. p, Neuman. 1974.
Field
test of a modified numerical model for water
uptake by root systems. Water Resour. Res. 10:'t199-1206.

Hopmans,-J.'f,/. 1988. Treatment of sparially variablegroundwater levels in one-dimenslonal stochastic
unsaturated water-flow modelling. Agric. Water
Manage. 15:19-36.
Hopmans,J.'W., and J. N. M. Stricker. 19g9. Stochastic
1"rly*_"{:"! water regime in a watershed.J. Hy_
drol. 105:57-84.
Johnsen, K. E., H. H. Liu,J. H. Dane, L. R.Ahuja, and
S. R. Workman. 1 995. Simulating flu*uatin{
water
tables and tile drainage with a modified .oo"t zone
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