Plant Water Uptake Terms Evaluated for Soil Waterand Solute Movement Models

G. E. Cardon* and J. Letey

ABSTRACTTwo root water uptake terms, representative of the two major types

commonly employed in soil-based models of water and solute move-ment, were evaluated. Simulations were run to test the sensitivity ofthe two terms to salinity and water content and to investigate thedetails of their respective form and function. The two root wateruptake terms tested were: (i) a mechanistic equation based on Darcy'slaw (Type I), and (ii) an empirical equation relating soil water poten-tial to relative water uptake (Type II). The Type I term was insensitiveto salinity where no reduction in transpiration was shown for increas-ing irrigation water salinity from 0.0 to 6.0 dS/m and applying waterequal to potential transpiration. The Type II term was sensitive tosalinity and showed a 35% reduction in water uptake by increasingwater salinity from 0.0 to 6.0 dS m~' and applying water equal topotential transpiration. Predicted reduction in water uptake due tomatric potential was of the same magnitude as that due to salinity.The Type I term resulted in abrupt shifts in water uptake betweenfull and zero transpiration, occasionally resulting in long periods ofcomputed zero transpiration, uncharacteristic of conditions in the field.It was concluded that the Type I water uptake term may not be ap-propriate for models incorporating root water uptake, particularlyunder saline conditions.

IN ARID AND SEMiARio AREAS of the world, increas-ing demand and decreasing quality of water re-

quires continual assessment of the management of thisvaluable resource. Increasing salinity of water and soilis of particular concern for irrigated agriculture in theseareas. Growers are faced with balancing the benefitsof leaching and drainage to reduce soil salinity againstthe damages from pesticide and other chemical resi-dues that can be transported in drainage water and thecosts of disposing of drainage water. This balance isbeing shifted as irrigation water quality changes or,in some cases, when regulations concerning ground-water pollution impact irrigation management prac-tices.

Numerous management options including the amountand timing of irrigation, using waters of various sal-inities, and crop choice make thorough field researchthat includes all the possible variables impractical.Mathematical simulation models that account for thesevariables are thus useful to guide the development ofwater management practices. Mathematical models havebeen developed to simulate water and solute move-ment through soil and have been coupled with simul-taneous water uptake by plant roots. These modelshave been reviewed by Molz (1981) and Wenkert(1983). Nearly all models calculate water flow by nu-merical solution of the Darcy-Richards equation in-

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. 32:1876-1880 (1992).

eluding a water uptake term, which, written for flowin the vertical dimension, is:

dt[i]L J

where K is the hydraulic conductivity (cm d-1), C isthe soil water capacity (cm"1), h is soil water pressurehead (cm), t is time (d), z is soil depth (cm) takenpositive downwards, and S is the root water uptaketerm (d-1).

The uniqueness of each model is found in the for-mulation of the root water uptake term, 5, of Eq. [1].Molz (1981) listed the plant water uptake equationsof 14 transient-state soil water flow models. The modelsgenerally fall into two groups based on their formu-lation of 5. The most common formulations (whichwill be referred to as Type I) are based on the workof Gardner (1960, 1964) and describe the microscalephysics of water flow from the soil to, and through,the plant roots. In general, these equations take theform of a typical flow equation:

S = B K' G [2]where K' (cm d"1) is a conductivity term (generallythe soil hydraulic conductivity), G (cm) is the waterpressure head gradient from the soil to the root (usu-ally expressed as the difference in root and soil waterpressure head), and B (cm-2) is a water flow geometryterm. In the literature there are many examples of soilwater flow models that employ a Type I plant wateruptake term (Gardner, 1964; Whisler et al., 1968;Nimah and Hanks, 1973; Feddes et al., 1974; Hillelet al., 1976; Herkelrath et al., 1977; Rowse et al.,1978; Wagenet and Hutson, 1987).

The second major type of water uptake term (TypeII) is comprised of macroscopic, empirical functionsthat describe plant water uptake based on observedresponse to water potential. The general form of sucha term is:

5 = a(h,7T)Sa [3]where a(h,Tr) is a dimensionless stress response func-tion equivalent to the ratio between actual, 5, andmaximum uptake, S,^. There are several models usingthis type of term reported in the literature (Feddes etal., 1976; Molz and Remson, 1970, 1971; van Gen-uchten, 1987).

Little systematic comparative testing for sensitivityand applicability of the two types of water uptaketerms across expected ranges in soil and water con-ditions has been done. Testing is critical in selectinga model for resource management decisions. Twomodels were chosen for this study that had the twotypes of water uptake terms. The model by Hanks andcolleagues (Nimah and Hanks, 1973; Childs and Hanks,1975; Torres and Hanks, (1989) uses a Type I termand the model by van Genuchten (1987) uses a Type

1876

CARDON & LETEY: EVALUATION OF PLANT WATER UPTAKE TERMS IN MODELS 1877

II water uptake term. We briefly describe the featuresof the two models and report the results of a compar-ative sensitivity analysis of the two types of wateruptake terms across a range of soil water and salinityconditions.

MODELSBoth models employ the Darcy-Richards equation

(Eq. [2]) to calculate water flow. Salt transport is cal-culated using the convection-dispersion equation fora nonreactive, noninteracting solute:

where c is the concentration of salt (g cm-3), 6 is thevolumetric soil water content (cm3 cm~3), D is thedispersion coefficient (cm2 d-1), and q is the volu-metric flux density (cm d-1). As written, Eq. [4] doesnot account for salt uptake by plant roots.

The nonlinearity of Eq. [1] and [4] requires solutionby numerical methods. Van Genuchten (1987) used amass-lumped finite element method. Nimah and Hanks(1973) utilized the Crank-Nicolson finite differencemethod. It can be shown, for certain combinations ofthe schemes for mass lumping and temporal weighingof the soil water pressure head across a time step, thatthe finite element and finite difference methods areequivalent. The combinations that result in this equiv-alence are often appropriate for conditions such asthose simulated in this study. Therefore, the modelsdo not differ greatly in the handling of water and sol-ute movement and, in fact, operate similarly for manyapplications. This is an important consideration for thecomparisons made in this study, which are partiallydependent on the calculated movement of water andsalt by each respective model. Thus the differences inresults from the two models are associated almostcompletely with differences caused by the two typesof water uptake terms.

The water uptake term in the Hanks model is:

S(z,t) = [Hloot - h(z,t) - Tr(z,t)] RD¥(z)K(0)Ax Az [5]

where Htool is the root water pressure head at the soilsurface (cm), h(z,f) is the pressure head (cm), TT(Z, t)is the osmotic head (cm), K(0) is the soil hydraulicconductivity, RDF(z) is the proportion of total activeroots in the depth increment Az (cm), and Ax is thedistance between the plant root surface and the pointin the soil where h(z,t) and ir(z,i) are measured (ar-bitrarily set at 1 cm in the model).

Salinity effects are incorporated by adding the os-motic to the matric potential to establish the waterpotential gradient from the soil to the root. The valueof //r0ot is iteratively determined until it is such thatcalculated uptake (5) equals potential uptake (Smax),provided //root is above a preestablished limiting lowervalue. After the lower limit on //root is reached, wateruptake (as calculated by Eq. [5]) becomes less thanpotential transpiration and decreases as h or TT de-creases. This reduced water uptake proceeds until the

summation of h and rr equals the limiting value of//root* whereupon water uptake ceases.

The van Genuchten model uses a Type II wateruptake term. Van Genuchten and Hoffman (1984) re-ported that measured salt tolerance data for crops couldbe described by an S-shaped curve relating relativeyield (relative to that under nonsaline conditions) tothe average salt concentration in the root zone. Con-verting salt concentration to osmotic pressure, theirequation was:

7 =1 + —rTso /

[6]

where Y is yield, Ym is maximum yield without os-motic stress, ir50 is the osmotic pressure at which Yis reduced by 50%, and p is an empirical constantfound to be =3 for many crop plants (van Genuchten,1987).

Van Genuchten (1987) assumed that, where the re-lationship 5/5max = Y/ym holds, then, by substitutioninto Eq. [6],

S = [7]

1750

Van Genuchten further assumed that plant response tomatric pressure could be modeled similarly so that,for a system where matric and salinity effects are bothpresent, a combined expression can be written as:

5 =1 + /ah +

\ Trso

[8]

where the parameter a accounts for the differentialresponse of the crop to matric and osmotic influencesand is equal to the ratio of ir50 to hso, where hso is thematric pressure at which 5max is reduced by 50%.

MODEL TESTING METHODSComparative Testing

A comparative test between the response of the TVpe Iand Type II water uptake terms to irrigation water salinity,irrigation interval, and total applied water, was run usingcrop-specific parameters for corn (Zea mays L.). Values ofh50 and '"so equal to — 0.43 MPa were used for the TypeII equation. The value is equal to the electrical conductivityof a saturated paste extract that causes a 50% reduction incorn yield (Maas, 1986), which was transformed to a soilwater-content-based value by multiplying by 2 and thenconverted to units of megapascals. The A50 value is withinthe range reported by Ehler (1983) and is located midpointbetween values used in the hso sensitivity test discussedbelow. For the Type I equation, the value for //„„„ was setat —15 x 103 cm, as suggested by Nimah and Hanks(1973).

Four water salinity levels of 0.0, 1.0, 3.0, and 6.0 dSm-1, and two irrigation regimes of 1.0 and 0.6 times thepotential transpiration were simulated. The initial soil watercontent was set at a value at equilibrium with a matricpressure of — 0.03 MPa, while the initial salt concentration

1878 SOIL SCI. SOC. AM. J., VOL. 56, NOVEMBER-DECEMBER 1992

was set equal to that of the irrigation water being simulated.A constant potential transpiration rate (5.88 mm d-1) androoting distribution were used for a 119-d simulation. Con-stant values were selected to facilitate computations becausecomparative rather than absolute uptake values were de-sired. The distribution of relative root mass [A(z)J used inall simulations was calculated by the following equationfrom van Genuchten (1987):

Table 1. Hydraulic parameters used in all simulations.Curve-fitting parameters!:

A(z) =

3L25

12 L0

z S 0.2 L

0.2 L <z<L

z > L

[9]

where z (cm) is depth in the soil profile and L (cm) is themaximum rooting depth (set at 200 cm in these simula-tions).

A static root distribution and constant potential transpir-ation were used to isolate the effects of the variables ofinterest on relative water uptake by removing any ambi-guities that could be introduced through the differences inthe handling of the dynamics of .these variables specific toeach individual model. These static conditions are only ap-propriate for this type of study. For modeling field events,plant and climatic dynamics should be more fully treated.

The hydraulic properties for all simulations were for asilt loam soil as calculated by the Hutson and Cass (1987)functions using the parameters listed in Table 1. The curve-fitting parameters are listed with their symbols as used byHutson and Cass (1987) but have no physical meaning.

Type II Term's Sensitivity to Soil MoistureThe Type II water uptake term (Eq. [8]) requires values

of ir50 and hso. Values for irso are available for many crops(van Genuchten and Hoffman, 1984), but values for /iso arescarce and difficult to determine. With uncertainty in hsovalues, a sensitivity analysis was performed to determinethe magnitude of error associated with using erroneous val-ues. The /»50 value for corn was expected to be within therange of -0.25 to -0.65 MPa (Ehler, 1983). The effectof hso is manifest on crop growth as the soil dries. Thus,the effect of hso on yield should depend on the water content(or matric pressure) before irrigation. To test this factor,119 d. of plant water uptake were simulated using 7-, 17-and 24-d intervals between irrigations and two irrigationlevels of 1.0 and 0.6 times potential transpiration for eachirrigation. These simulations were run using a static rootdistribution and nonsaline conditions. In this manner, therelative differences in uptake across several irrigation in-tervals could be ascribed to just those differences arisingfrom variation in the value of hso.

RESULTS AND DISCUSSIONPredicted relative transpiration using the two models

for the various combinations of irrigation interval, re-gime, anof water quality are illustrated in Fig. 1. Rel-ative transpiration was calculated as the ratio betweenthe simulated cumulative water uptake and the cu-mulative potential uptake as provided to the models.Both models predicted relative transpiration equal to1.0 for the nonsaline irrigation water applied at 1.0times potential transpiration for both irrigation inter-vals. When simulated irrigations were equal to poten-tial transpiration, the Type I term predicted relativetranspiration equal to 1.0 for all irrigation salinities

bB*«*,

Measured parameters:Saturated hydraulicconductivitySaturation watercontent (v/v)

2.949.12

-4.54kPa-7.20kPa

0.42

0.72 cm h-1

0.49

t Values are obtained through a least-squares fit to the Hutson and Cass(1987) hydraulic retentivity functions.

and both irrigation intervals (the 17-d interval curveis masked by the 7-d interval curve in Fig. 1). TheType II term predicted decreasing relative transpira-tion with increasing irrigation water salinity and slightlylower relative transpiration for the 17-d than the 7-dinterval for a given salinity. Experimentally deter-mined sensitivity of corn to salinity indicates that theplant should not have maintained relative transpirationof 1.0 at irrigation water salinities as high as 6.0 dSm-1.

When simulated irrigations were equal to 0.6 timespotential transpiration, the predicted relative transpir-ation for nonsaline water was 3% higher for the TypeII term than the Type I term for both irrigation inter-vals. The effect of increasing irrigation water salinitywas considerably greater for the Type H term man f°r

the Type I term. Clearly, the Type I term is insensitive

1.0 times TP

7-day interval, Type II term

17-day interval, Type II term

7-day interval, Type I term-•- 17-day interval, Type I term

0.6 times TP0.60

0.50

EC of Irrigation Water (dS/m)Fig. 1. Computed relative transpiration vs. electrical

conductivity (EC) of the irrigation water for 0.6 and 1.0times potential transpiration (TP), two irrigation intervals,and Type I and Type II water uptake terms.

CARDON & LETEY: EVALUATION OF PLANT WATER UPTAKE TERMS IN MODELS 1879

24.0 -3180

-3240

Eo- -3300 ~

OO

- -3360

16.0 -34201140 1164 1188 1212 1236

Time (hrs)Fig. 2. Dynamics of root water pressure at the soil surface (#„„,), and potential and calculated transpiration for a 4-d period

during a simulation using the Type I water uptake term.

to water salinity and, based on experimental data, wouldoverpredict water uptake and yield under saline con-ditions.

The insensitivity of the Type I term to salinity iscaused by the manner in which the salinity effects areincorporated in the uptake term (Eq. [5]). The valueof 5 is dominated primarily by the nonlinear changesin h and K(&). As 6 decreases, both K(0) and h de-crease logarithmically. In contrast, -rr decreases onlylinearly (simple concentration-dilution) and adds verylittle to the gradient term of Eq. [5]. Moreover, in-creasing the salinity of the irrigation water whilemaintaining high water contents results in relativelyhigh K(ff) values and plant water extraction proceedsat or near maximum levels.

The results presented in Fig. 1 are the cumulativeeffects of a 119-d simulation. The results can also betracked on an hourly basis. Equations [5] and [8] aredesigned to reduce transpiration as the soil water pres-sure head decreases to less than optimum levels. Fig-ure 2, however, illustrates that the Type I term predictstranspiration equal to potential transpiration until thesum of the matric and osmotic pressure head equalsthe programmed Hroot limit, after which the predictedtranspiration drops to zero. As the soil water pressurehead decreases, the value of Hroot is accordingly re-duced by the model to maintain a gradient sufficientto satisfy potential transpiration. As long as the valueof Hroot is above the imposed lower limit ( — 0.34 x103 cm in these simulations), calculated transpirationwill equal potential transpiration. When the lower limitof Hroot is reached, computed transpiration ceases ex-cept when the redistribution of water occurs duringthe night and raises the water pressure head.

At low soil water contents, very large changes inmatric pressure head are associated with very smallchanges in water content. Therefore, once the lowerlimit of H[00l has been reached, very little water canbe extracted before the soil water pressure head alsoreaches the limiting Hroot level. At this point, the result

DaysFig. 3. Potential transpiration (T) vs. calculated transpiration

for a 10-d period during a simulation using the Type II•water uptake term.

of Eq. [5] becomes zero. These factors produce theresults shown in Fig. 2.

This result has implications for the appropriatenessof the Type I term under any condition. On a cumu-lative basis, a relative uptake <1.0 indicates that theType I term has calculated extraction at 100% of po-tential transpiration for some portion of the simulationperiod and 0% for the remainder. For example, a cu-mulative relative transpiration of 60% indicates that100% transpiration was computed for 60% of the sim-ulation period and zero transpiration for 40% of theperiod. This result is inconsistent with plant behavior,since extended periods of zero transpiration would re-sult in the death of the plant. Based on these results,the use of Type I terms may not be appropriate underany conditions and particularly under saline condi-tions.

In contrast to the Type I term, the Type II termreflects crop stress conditions that are more consistentwith observed plant behavior. Figure 3 summarizes

1880 SOIL SCI. SOC. AM. J., VOL. 56, NOVEMBER-DECEMBER 1992

subdaily data from a test simulation with the Type IIterm. Note that there is a continual gradual reductionin calculated transpiration relative to potential tran-spiration as time progresses and soil water pressuredecreases. One weakness of the Type II term is thatthe calculated transpiration rate only asymptoticallyapproaches zero under dry or highly saline conditions.Thus plant death may occur under conditions where alow value of transpiration is still predicted.

The results of the sensitivity test of the Type II termto hso are summarized in Table 2. Calculated relativetranspiration was rather insensitive to the value of /i50for the range tested. The largest difference was 5%between the range extremes for the 24-d irrigationinterval. Therefore, a difference in water uptake of< 5% was introduced into the comparative tests aboveby using a value of /iso equal to —0.43 MPa. Theinsensitivity of the Type II water uptake term to hsois fortunate because experimentally valid h50 valuesare difficult to determine. An error in the assumed hsovalue does not lead to large errors in predicted tran-spiration.

CONCLUSIONSThe Type I water uptake term was shown to be

insensitive to salinity and was generally inconsistentwith plant behavior in its calculations of root wateruptake. The discrete drops and jumps in the calculatedrate of transpiration (between full potential and zero)are not physically reasonable. All models that employa Type I term for water uptake would be expected tohave similar responses. These models may not be ap-propriate for use in plant-soil-water system models,particularly under saline conditions.

The Type II water uptake term tested in this studywas shown to have a broad range of sensitivity tofluctuations in both the matric and osmotic pressure

Table 2. Relative transpiration calculated by the Type II termfor two values of matric pressure at which maximum plantwater uptake is reduced by 50% (hso) (nonsaline conditions).

Irrigation regime -0.25 MPa -0.65 MPa7-d interval

1.0 x potential transpiration0.6 x potential transpiration

1.000.83

1.000.86

17-d interval1.0 x potential transpiration 0.980.6 x potential transpiration 0.76

24-d interval1.0 x potential transpiration0.6 x potential transpiration

0.920.72

1.000.80

0.970.76

of the soil. Though empirical in its derivation, theType II term provided reasonable calculations of tran-spiration rates and mimics the natural system betterthan the Type I term.