Soil-Based Irrigation and Salinity Management Model:I. Plant Water Uptake Calculations

G. E. Cardon* and J. Letey

ABSTRACTA new soil-based model for irrigation and soil salinity management

has been formulated to rectify deficiencies in existing soil-based modelswith respect to mass balance, numerical stability, and the treatmentof plant growth and water uptake dynamics. The advantages of thenew model are: (i) the treatment of temporal variation in potentialtranspiration (TP) and rooting depth and distribution, (ii) the ad-justment of TP based on feedback from simulated crop water uptake,(iii) the provision for treating growth-stage-specific crop tolerance tosalinity and water stress, and (iv) the provision for multiseasonal sim-ulation through the treatment of noncropped periods. A sensitivitytest and a comparison of model calculations to experimental data wereconducted. The model tended to overpredict relative crop yield froma field experiment based on relative crop water uptake calculations.The mean simulated relative yield was 7% higher than observed witha root squared error (RMSE) of 8.8% relative yield. Willmott's dindex (a statistical index of agreement between measured and observeddata) result was high 0.79 (perfect agreement = 1.0).

GROWERS maintain productivity through well-timedirrigation and soil salinity control in the arid to

semiarid western USA. Historically, irrigations withnonsaline water were used to recharge dry soil profilesand flush salt from the root zone. Currently, somegrowers must alter their irrigation practices in the faceof decreasing quantities and quality of water. In ad-dition, environmental and economic concerns influ-ence irrigation management decisions through controlson solute migration to groundwater and surface dis-posal of agricultural drainage waters. Irrigation man-agement, therefore, has become a more complexproblem.

Researchers have investigated many aspects of ir-rigation and salinity management techniques, includ-ing efficient irrigation technologies (Phene et al., 1985;Lyle and Bordovsky, 1981), irrigation with salinedrainage water (Grattan et al., 1987; Rains et al.,1987; Rhoades, 1984), crop salt tolerance (Maas, 1984,1986), and subirrigation from shallow saline watertables (Ayars and Schoneman, 1986; Namken et al.,1969; Wallender et al., 1979). However, field re-search of all possible combinations of managementpractices and environmental conditions and their long-term impact on soil and crop productivity is not prac-tical. There is a need, therefore, for an accurate methodof predicting the influence of various practices forindividual sets of soil and climatic conditions on along-term basis.

Mathematical models are suited for application toirrigation scheduling and soil salinity management and,potentially, are valuable tools as decision aides. Models

G.E. Cardon, Agronomy Dep., Colorado State Univ., Ft. Collins,CO 80523; and J. Letey, Soil and Environmental Sciences Dep.,Univ. of California, Riverside, CA 92521. Contribution of theUniv. of California, Riverside. Sponsored by the Univ. of Cali-fornia's Salinity/Drainage Task Force. Received 2 Apr. 1991.* Corresponding author.

Published in Soil Sci. Soc. Am. J. 56:1881-1887 (1992).

found in the literature range from very simple to so-phisticated, from crop specific to general, and fromprimarily crop based to soil based. Reviews of exist-ing models are given by Molz (1981) for soil-basedmodels, and by Jones and Ritchie (1990) for crop-based models.

The purpose of this study was to test existing modelsunder arid to semiarid conditions for long-term pre-dictions of soil profile conditions, crop yield, and deeppercolation under various combinations of irrigationscheduling, water quality, and soil salinity. In theprocess of selecting candidate models for testing, mostcrop-based simulation models were found to employvery simple methods of calculating water and solutemovement in the soil profile and are thus unsuitablefor the detailed predictions necessary for soil salinitymanagement (Ritchie et al., 1989; Acock et al., 1983;Baker et al., 1983; Wilkerson et al., 1983). More-over, salinity effects on crop water use are generallynot treated. Soil-based models, on the other hand,generally use sophisticated numerical solutions of waterand solute movement that provide the detail necessaryfor the long-term predictions of soil profile conditionswe were interested in. However, in soil-based models,crop water use is generally calculated by simple sinkterms in the water flux equations, and plant growthdynamics are generally not considered. Given thesetrade-offs, we determined that a soil-based simulationmodel, if modified to treat basic crop growth dynam-ics, would be the most appropriate for the applicationswe were interested in.

Two types of plant water uptake terms are com-monly employed in soil-based simulation models. Oneis a mechanistic equation based on Darcy's law (TypeI) and the other is an empirical relationship betweensoil water pressure and relative water uptake (TypeII). Cardon and Letey (1992a) found that the Type IIuptake term provided more physically and biologicallyreasonable values than the Type I term.

We present the formulation of a new model, basedon combination and modification of existing models,for irrigation and salinity management in arid andsemiarid areas. The model is comprised of compo-nents from previously existing models (including aType II root water uptake term), with several modi-fications with respect to the need to model seasonalplant dynamics. We also present a summary of (i) asensitivity analysis of the model to crop-dynamics var-iables, and (ii) a comparison of model-predicted andobserved root water uptake.

MODEL DESCRIPTIONGeneral Model Features

The proposed model is formulated, in part, by acombination of routines adapted from van Genuchten(1987) and Hanks and colleagues (Nimah and Hanks,Abbreviations: TP, potential transpiration; RMSE, root meansquared error; EC, electrical conductivity; EQ, irrigation watersalinity; ET, evapotranspiration; T, actual transpiration.

1881

1882 SOIL SCI. SOC. AM. J., VOL. 56, NOVEMBER-DECEMBER 1992

1973a,b; Childs and Hanks, 1975; Torres and Hanks,1989). The basic concept of how the models havebeen combined is illustrated in Fig. 1. The bulk of acrop-season simulation in which the movement of waterand salt and plant water uptake are calculated is per-formed by routines adapted from the WORM modelof van Genuchten (1987). These routines employ aone-dimensional finite-element solution to the Darcy-Richards equation for water flow, and the convection-dispersion equation for solute movement. Plant wateruptake is included as a sink term, S(z,t) (d-1), in theDarcy-Richards equation as follows:

- K(e}] - s(z-t}

where K(ff) is the soil hydraulic conductivity (cm d-1),C is the soil water capacity (cm-1), h is the soil matricpressure head (cm), t is time (d), and z is depth (cm,positive downwards). Transpiration and water and sol-ute redistribution are calculated for a user-specifiedlength of time until an irrigation or rainfall event oradjustment in crop-dynamic variables (discussed be-low) is desired.

For irrigation or rainfall events, the model uses rou-tines adapted from Hanks, since the Hanks water flowroutines are stable and mass conservative when largehydraulic gradients exist, as occur during infiltrationinto a relatively dry soil profile. The van Genuchtenroutines were not stable and did not provide massconservation for infiltration into a dry soil. The Hanksroutines employ the same governing equations for waterand solute movement as previously described for thevan Genuchten-adapted routines. However, Hanks,

Transpiration / Redistribution(van Genuchten adapted routines)

Infiltration(Hanks adapted routines)

originally used a Type I water extraction term, whichis not appropriate for saline conditions (Cardon andLetey, 1992a). Once the irrigation or rainfall is com-pleted, the modified van Genuchten water uptake- re-distribution routines are used again. The twocomponents of the model exchange a file containingwater and solute distribution data to be used as initialconditions for each respective section. Model execu-tion continues until a user-specified number of cycleshas been completed. During infiltration, root wateruptake is assumed to be zero.

Multiseasonal simulations are facilitated by allow-ing calculations of water and solute movement, infil-tration, and bare soil evaporation during noncroppedperiods. These are calculated entirely by Hanks-adaptedroutines. This is done to facilitate the use of the evap-oration equation of Nimah and Hanks (1973a) and themass balance control and numerical stability of theHanks-adapted infiltration routines. The model, as de-picted in Fig. 1, will be referred to as the modifiedvan Genuchten-Hanks model or, simply, the V-Hmodel.

Root Water Uptake and GrowthThe equation relating water extraction to soil matric

and osmotic pressure heads used in these routines wasdeveloped from work by van Genuchten and Hoffman(1984) and has the general form:

S =/ah + ir\\ TSO J

[2]1 +

where 5 is water uptake (d-1), 5max is maximum wateruptake for no-stress conditions (d-1), h is the soilmatric head (cm), IT is the osmotic head (cm), ir50(cm) is the osmotic head that results in a 50% reduc-tion of 5max, and a is a coefficient equal to TT50lh50,where hso is the matric head that results in a 50%reduction of 5max.

The factor 5max is related to TP (cm d-1) of a cropby:

[3]

where L is the rooting depth (cm). Potential transpir-ation of a crop is a combination of climatic referencesevapotranspiration (ET0, cm d-1) and an associatedcrop coefficient (Kcr):

TP = [4]

Fig. 1. Flow-chart illustration of the V-H model.

The data for ET0 and K<.r are available for most cli-mates and crops in arid and semiarid irrigated agri-cultural areas. However, ET0 values are obtained bydifferent means. If ET0 values are obtained from freewater evaporation pans, K^ values for most crops typ-ically range between 0.0 and 1.0. If ET0 is measuredusing a well-watered, grass-covered lysimeter, or cal-culated using predictive equations (such as the Pen-man equation), Kcr values may often exceed 1.0.Therefore, the K^ function used must correspond tothe appropriate ET0 value for the specific crop.

CARDON & LETEY: IRRIGATION AND SALINITY MANAGEMENT MODEL: I 1883

Van Genuchten (1987) proposed the followingequation for water uptake at various depths in the soil:

S(z) =

IT-50 0A(z) [5]

where 5(z) is the crop water uptake at depth z, andA(z) is a depth-dependent root distribution coefficientof the form

A(z) =

3L

——f 1 - -12L\ L

;z < 0.2 L

;0.2 L < z < L

;z>L

[6]

As written, Eq. [5] is only appropriate for short-time modeling of root water uptake where 5max and Lcan be considered constants. For season-length sim-ulations, 5max and L are time-dependent functions dic-tated by climatic and soil profile conditions. Moreover,many crops exhibit differential tolerance to soil mois-ture deficit and salinity stress at various growth stages.Three modifications were made to accommodate thesetime-dependent functions.

First, temporal variations in TP are accounted foras follows:

TP(*) = [7]

where time-dependent ET0(t) and K<.r(0 values are pro-vided as tabular input pairs corresponding to user-specified times.

Second, the rooting depth was converted to a time-dependent form, L(t), which is the maximum rootingdepth at an input-specified time. By substituting L(t)for L in Eq. [6], a time- and space-dependent rootdistribution function, A(z,f), was obtained. Note that,for any value of L(t), A(z,f) integrates to unity overthe root zone.

Third, growth-stage-specific stress tolerances of thecrop are handled by allowing the values of hso and•77.50 to vary with tmie by specifying times and valuesfor each parameter.

Moisture and salinity stress cause reduced plantgrowth and transpiration. In effect, the value of Ka(t)depends on the amount of stress previously experi-enced by the plant. To account for this, we introducedan additional variable in the model, Ka(t), where

CT \

CTPJ ;t ta

CTP =

[8a]

[8b]

and where K^t) is the stress-adjusted crop coefficient,CT is the cumulative transpiration calculated by themodel, CTPa is the cumulative stress-adjusted poten-tial transpiration, and t-0 tc, and tn are the initial, cur-rent,and next times during model execution,respectively. Equations [8a] and [8b] represent an in-ternal feedback function that scales the transpirationaldemand if the plant has experienced water or salinitystress. Equations [8a] and [8b] are based on the gen-erally accepted concept that relative transpiration is agood estimator of relative dry-matter production and,hence, relative plant size, which we assume is directlyrelated to the ability of the crop to transpire at theunstressed reference potential level given as input tothe model.

The modified time- and space-dependent water up-take equation, which accounts for basic plant growthdynamics, growth-stage-specific stress tolerance, andstress-induced growth reductions, can be written:

S(z,t) = [9]

where 5^ (t) is the stress-adjusted value of Smax(f)calculated by substituting Ka(t) for K^t) in Eq. [7]and combining the result with Eq. [3]. The variablea(t) is the time-variable analog of the variable a inEq. [2].

The rooting depth of a crop may also be reduceddue to water or salinity stress. In the V-H model, acomplex rooting routine capable of handling the dy-namics of this specific plant organ is not warranted,given the empirical nature of the water uptake equa-tion. Rather, a feedback adjustment of the maximumrooting depth analogous to that proposed for the ad-justment of Ka(t) (Eq. [8a], and [8b]) seems moreappropriate. In the proposed model, however, stress-related rooting depth adjustments are not included basedon the insensitivity of Eq. [9] to changes in L(t), asdiscussed below.

MODEL TESTING METHODSSensitivity Analysis

The sensitivity of computed root water uptake to varia-tion in the two crop-dynamic variables, KCI(t) and L(t), wastested. Simulations were run for a 10-d period during themiddle of a hypothetical growth season for corn (Lea maysL.) using the reference values for the uptake parameters asfollows: ET0 = 0.75 cm d-1, Ka = 1.29, L = 200 cm,hso and ir50 = -0.43 MPa.

The tests consisted of holding all other variables constantand changing Kc,(t) or L(t) from 100 to 50% of the referencevalue, in 10% increments. The tests were performed foreach variable under four sets of initial soil conditions toevaluate changes in the sensitivity associated with soil pro-file conditions. The four sets of conditions were two initialsoil matric pressure levels ( — 0.03 and —0.08 MPa) andtwo initial soil salinity levels (0.0 and 6.0 dSm-1). Soilhydraulic properties were computed with the Hutson andCass (1987) equations, using the values of the parameterslisted in Table 1.

1884 SOIL SCI. SOC. AM. J., VOL. 56, NOVEMBER-DECEMBER 1992

Computed relative root water uptake was then comparedwith the relative value of Kc,(t) [or L(t)]. Relative root wateruptake was calculated as cumulative S(z, t) divided by cu-mulative S^axW for the 10-d period.

Experimental Data ComparisonThe V-H model was tested using data from a study with

corn (cv. Jubilee) conducted at the Gilat Agricultural Ex-periment Station in the northern Negev of Israel (Shalhevetet al., 1986). This experiment was used because it providesdata of crop response to several levels of both water andsalinity stress, which is not generally available in the lit-erature. The experiment was a split-plot design with variouslevels of irrigation water salinity (EQ) as main treatments,four irrigation intervals as subtreatments (3.5, 7, 14, and21 d), and three replicates. Prior to treatment with the var-ious saline irrigation waters and irrigation intervals, thecrop was established on all plots with a uniform treatmentof 140 mm of nonsaline water (EQ = 1.0 dS m-1) for 37d. After establishment, saline irrigation water was appliedto replenish transpiration losses from the root zone. Thesoil was a silt loam (Calcic Haploxeralf) whose hydraulicproperties are described by the parameters given in Table

The time-depende'nt crop dynamics and climatic inputsused in the simulations, and the length of the time intervalfor which each value was operative, are summarized inTable 2. The rooting depth values correspond to a growthrate of 5.0 cm d-1 and a maximum rooting depth of 2.0 m(Taylor et al., 1970). The ET0(t) function for Israel wasobtained for a well-watered grass surface (A. Marani, 1991,unpublished weather data). The K(t) function used was de-veloped for sweet corn in the San Joaquin Valley of Cali-fornia under climatic conditions similar to those in Israelusing a grass-covered reference lysimeter and a croppedcompanion lysimeter (Phene et al., 1989). The values ofhso and ir50 for sweet corn were taken from Cardon andLetey (1992a), and were held constant for the entire season.The values of these two variables are included in Table 2,both to give an idea of the format in which a data set isconstructed for the V-H model, and to emphasize that thesedata can be changed during the season to model growth-stage variations in stress tolerance.

Measured relative grain yield was then compared withmodel-predicted relative yield. Measured corn grain yieldswere converted to relative yield by dividing by a maximumyield of 3.1 kg ha-1 reported for the study site (Shalhevetet al., 1986). The V-H model does not calculate crop yielddirectly, rather values of water uptake [S(z,t)] are summedfor a season and must then be converted to yield. Doorenbosand Kassam (1979) reported yield-ET relationships for manycrops and found that the following equation held:

RY = 1 - k + ET \[10]

where RY is relative yield, A: is a crop-specific yield re-sponse coefficient, ET is evapotranspiration, and PET ispotential evapotranspiration. Model-predicted relative yieldswere calculated by substituting cumulative S divided bycumulative S^m for ET/PET in Eq. [10], using the k valuefor corn of 1.25 suggested by Doorenbos and Kassam (1979).The model computations are based on transpiration ratherthan evapotranspiration, but evapotranspiration in the fieldis almost never separated into components of evaporationand transpiration. One unavoidable short-coming of the modelis that ET/PET is considered to be the same as T/TP.

Evaluation of predicted vs. measured relative yield wasthen made by calculation of two objective functions. The

Table 1. Hydraulic parameters used in the sensitivity analysisand experimental data comparison simulations.

Curve-fittingparameters!: b

B

Measured parameters:Saturated hydraulicconductivitySaturation watercontent (v/v)

2.949.12

-4.54kPa-7.20kPa

0.42

0.72 cm h-

0.49t Values are obtained through a least-squares fit to the Hutson and Cass

(1987) hydraulic retentivity functions.

first is the RMSE, which is calculated in the followingmanner:

RMSE =N [11]

where Pt and Of are the ith predicted and observed valuesof interest, respectively. The value of RMSE has the sameunits as the corresponding data (fractional relative yield inthis case) and is a measure of the average deviation ofpredicted from observed values. The second objective func-tion is Willmott's index of agreement (d) expressed as:

d = 1 -

- N

2 (Picn

[12]

where P{ = P, — Om, O{ = Ot — Om, and Om is the meanobserved value (Willmott, 1981). The value of d is an index

Table 2. Time-dependent tabular input dataf for the climaticand plant-dynamics variables, with the time interval forwhich each set was operative.

Intervald1.05.06.06.06.06.06.03.53.53.53.53.53.53.53.53.53.53.53.53.53.5

ET0

cmd-1

0.530.590.510.500.620.650.540.600.640.640.660.690.750.700.650.660.710.710.700.770.75

Ac,

0.150.190.230.270.310.350.490.620.760.891.021.161.291.371.341.301.261.221.191.151.11

Lcm5.0

30.060.090.0

120.0150.0180.0198.0200.0200.0200.0200.0200.0200.0200.0200.0200.0200.0200.0200.0200.0

A*,MPi

-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43

"so

-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43-0.43

t ET0 = reference evapotranspiration; K,, = unadjusted crop coefficient;L = maximum rooting depth; AM = 50% transpiration reduction valueof matric pressure; trx = 50% transpiration reduction value of osmoticpressure.

CARDON & LETEY: IRRIGATION AND SALINITY MANAGEMENT MODEL: I 1885

of how well the predicted and observed deviations aboutOm correspond to each other, both in magnitude and sign.It varies between 0.0 and 1.0, with 1.0 representing perfectagreement. The two objective functions (RMSE and the dindex) in conjunction quantify the agreement between sim-ulated and observed data.

RESULTS AND DISCUSSIONSensitivity Analysis

Results from the sensitivity analysis are given inFig. 2 and 3. Calculated root water uptake is relativelyinsensitive to changes in L(t), particularly under moistsoil conditions where S'ma3i can be met even when theroot zone is confined to one-half its potential depth.Sensitivity increases somewhat under dry or salineconditions where root water uptake cannot reach po-tential levels when the rooting depth is reduced.

Water uptake calculations are very sensitive tochanges in the variable Kcr(t)(Pig. 3). Reductions inrelative root water uptake are nearly the same as re-ductions in relative KCT(t), as expected, since KCI(t) isa primary factor in determining uptake demand or TP(Eq. [7]). Results of this sensitivity analysis are thebasis for choosing to not include stress-related feed-back adjustments to L(t). If stress-related adjustmentsof L(t) were made as proposed for Ker(t) (Eq. [8a] and[b]), then both KCI(t) and L(t) would be reduced bythe same relative amount. Since relative uptake is asimultaneous function of both L(t) and Kct(t) (Eq. [9])the dominance of KJf) in the calculation (Fig. 3) masksthe influence of changes in L(t).

J£5Q.

t_*•*CO

OccCD"̂

1

1.1U

1.00

0.90

0.80

0.70

0.60

0.50

0.40

1.00

0.90

0.80

0.70

0.60

0.50

f\ Af\

« — * * * '

:

1 :

:Moist :

:

:

^^^^^'^^

:

——— O.OdS/m ]

Dry •••' 6.0 dS/m ;

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10Relative Value of L(t)

Fig. 2. Sensitivity of the V-H model to rooting depth undermoist and dry soil conditions at two salinities.

Experimental Data ComparisonResults for the experimental data comparison sim-

ulations are presented in Fig. 4 and Table 3. The V-H model tended to overpredict relative yield as dem-onstrated by the higher mean calculated relative yieldcompared with the observed mean (Table 3) and theposition of the points relative to the 1:1 correspon-dence line in Fig. 4. The overprediction is particularlypronounced for the 21-d irrigation interval simula-tions. Based on the values of the two objective func-tions, however, the correlation between predicted andmeasured relative yield is high. The average deviationbetween measured and predicted relative yield is < 9%and the value of the d index is fairly high (0.79). Theoverall trend in the predicted data also correspondswell to that for the measured data.

The general overprediction may be because the ET0(f)and Ka(t) data specific to the experimental data wasnot available and the values used in the simulationmay have been too low. The 21-d interval simulationsare the most affected by TP due to the greater soildrying during the long transpiration periods. Simula-tions using a higher value of TP produced better agree-ment between observed and simulated values (data notshown).

If the data for the 21-d interval simulations are leftout of the calculation of the data statistics and objec-tive functions, the inherent structure of the data re-mains unchanged (little or no change in the mean orstandard deviation in Table 3), but the correlation be-tween predicted and measured relative yield is greatly

co

1.10

1.00

0.90

0.80

0.70

0.60

CD 0.50*-"co§ 0.40- 1.10O

DC 1.00<D.£ 0.90«J® 0.80DC

0.70

0.60

0.50

0.40

Moist

Dry

—— 0.0 dS/m

-"" 6.0dS/m

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10

Relative Value of KCR(t)Fig. 3. Sensitivity of the V-H model to the crop coefficient

Kcr(t) under moist and dry soil conditions at two salinities.

1886 SOIL SCI. SOC. AM. J., VOL. 56, NOVEMBER-DECEMBER 1992

Table 3. Descriptive statistics and objective function resultsfor the observed data (O), the full experimental data com-parison (w/21), and the experimental data comparison with-out the data for the 21-d irrigation interval (w/o 21).

O w/21 w/o21MeanStandard deviationRoot mean

squared errorWillmott'sd index

0.660.10

0.730.08

0.090.79

0.740.08

0.060.90

1.0

10.8

•0.7

1 0.6^n£0.5

0.4

• 3.5-day interval

• 7-day interval

+ 14-day interval

» 21-day interval

0.4 0.5 0.6 0.7 0.8Measured Relative Yield

0.9 1.0

Fig. 4. Observed vs. predicted sweet corn relative yields forthe experimental data comparison simulations.

enhanced. The average deviation from measured rel-ative yield drops to 6.2% and the d index increases to0.9. This suggests that overall model performance maybe better for shorter irrigation intervals or more salttolerant crops.

CONCLUSIONSA new soil-based model, named the V-H model for

irrigation management in arid and semiarid areas wasdeveloped. The model formulated from existing modelshas the following advantages over previously existingmodels: (i) temporal variation in potential transpira-tion and rooting depth-distribution, (ii) feedback foradjustments in potential transpiration based on simu-lated crop stress, (iii) treatment of growth-stage-spe-cific crop tolerance to salinity or water stress, and (iv)multiseasonal simulation capability through the addi-tion of routines for bare soil evaporation during non-cropped periods. The V-H model has excellent massbalance control and numerical stability across all soilwater contents and employs a sophisticated numericalsolution of water and solute movement, which pro-vides the detail necessary for predicting the long-termimpact of irrigation management on soil salinity, soilwater, and losses of water to deep percolation andevaporation. Moreover, root water uptake is based, inpart, on crop salt-tolerance data, which enhances theapplicability of the V-H model to saline conditions.

The V-H model was shown to reasonably predictexperimental relative crop yields calculated from sim-ulations of root water uptake across a wide range ofboth salinity and irrigation levels. Through the modeltended to overpredict relative yield, the agreement with

observed data is sufficient to make the V-H model auseful tool for irrigation management in areas wheresoil water and salinity levels are the primary deter-minants of crop water uptake.

Further testing of the V-H model with focus on theagreement between predicted and measured soil pro-file distributions of water and salinity is reported inCardon and Letey (1992b).