Crop-Water Production Function Model for Saline Irrigation Waters1

J. LETEY, ARIEL DINAR, AND KEITH C. KNAPP2

ABSTRACTThe relationship between crop yield and seasonal amount of ap-

plied water (crop-water production function) is required to determineoptimum irrigation management. A model is developed for the com-putation of crop-water production functions with saline irrigationwaters. The model combines three relationships: yield and evapo-transpiration, yield and average root zone salinity, and average rootzone salinity and leaching fraction. The model allows plant growthadjustment, and therefore evapotranspiration adjustment, to root zonesalinity. Crop-water production functions were computed for tall fes-cue (Festuca elatior arundinacea L.) for various levels of salinity inthe irrigation water. A comparison was made between calculated andpublished experimentally measured values of leaching fractions andyields of tall fescue grown under conditions of various irrigationwater salinities, water application quantities and applied water fre-quencies. Calculated and measured yields were in good agreementconsidering the usual degree of variability of field data. Agreementbetween calculated and measured leaching fractions was not as goodas for yields.

Additional Index Words: leaching fraction, salinity, optimum ir-rigation, tall fescue.

Letey, J., A. Dinar, and K.C. Knapp. 1985. Crop-water productionfunction model for saline irrigation waters. Soil Sci. Soc. Am. J.49:1005-1009.

THE RELATIONSHIP between crop yield and seasonalamount of applied water (crop-water production

function) is required to determine optimum irrigationmanagement. This information is particularly impor-tant when the irrigation water contains significantconcentrations of soluble salts which can affect cropproduction if not properly managed. Sufficientamounts of saline waters must be applied to providefor leaching in addition to evapotranspiration (ET).The leaching requirement has been defined as theminimum fraction of the total amount of applied waterthat must pass through the soil root /one to preventa reduction in crop yield from excessive accumulationof salts (USSL Staff, 1954). Information on the leach-ing requirement as defined, however, is not sufficientto provide information on optimum irrigation. Opti-mum irrigation is considered to be that amount ofwater that maximizes profit to the grower and maxi-mum profit may not coincide with maximum yield.

Feinerman et al. (1984) calculated the yield of corn(Zea mays L.) as a function of seasonal quantities ofapplied irrigation water of various salinities. The cal-culations were made assuming steady-state soil sal-inity conditions and also using a transient state modelpresented by Bresler (1967). Their steady-state anal-ysis assumed that when water is applied in quantitiesless than potential ET occurring with the use of non-

1 Contribution of the Dep. of Soil and Environmental Sciences,Univ. of California, Riverside, CA 92521. This study was supportedby the Univ. of California Kearney Foundation of Soil Science.Received 13 Aug. 1984. Approved 5 Feb. 1985.2 Professor of Soil Physics, Postgraduate Research AgriculturalEconomist, and Assistant Professor of Resource Economics, re-spectively.

saline waters, that no leaching occurs and conse-quently salt concentrations in the root zone increaseto steady-state levels resulting in zero yield. This as-sumption is restrictive in that it does not allow forthe fact that the accumulation of salts in the root zoneleads to a smaller plant, and therefore a lower ET withan increase in leaching.

The model developed in this report allows plant ad-justment to root zone salinity. The model combinesthree separate yet related relationships. These are re-lationships between yield and ET, yield and averageroot zone salinity, and average root zone salinity andleaching fraction for irrigation waters of various sal-inities. The relationship between yield and ET is as-sumed to be independent of whether the yield is pri-marily affected by soil-water content or salinity. Thisassumption is supported by experimental data re-ported by Hanks et al. (1978) in which the ET versusyield curve was not affected by soil salinity levels. Themodel does not take into account any interaction ofthe chemistry of the applied water and the soil.

This paper details the model development, dis-cusses some of the limitations of the model, presentscalculated crop-water production functions for tall fes-cue (Festuca elatior arundinacea L.) with irrigationwater of various salinities, and compares predicted tomeasured response of tall fescue to the salinity andquantity of irrigation water.

MODEL DEVELOPMENTA linear relationship between yield of forages, or total top

weight of nonforages, and ET has been found in several ex-periments such as those reported by Davis (1983), Hanks etal. (1969), Hanks and Retta (1980), Sammis et al. (1979),and Power et al. (1973). A linear relationship between themarketable part of the crop and ET has been reported forcorn grain, chili pepper (Capsicum annuum L.), wheat (Tri-ticium aestivum L.), sugar beets (Beta vulgaris L.), and po-tatoes (Solarium tuberosum L.) (Beese et al., 1982; Stewartet al., 1977; Miller and Hang, 1982; Hanks, 1982; Shalhevetet al., 1983). Cotton (Gossypium hirsutum L.) lint yield wasfound to be curvilinearly related to ET (Grimes et al., 1969;Davis, 1983), however, total cotton plant dry matter weightwas linearly related to ET (Davis, 1983). The following anal-ysis assumes a linear relationship between yield and ET socotton or other crops where the marketable yield componentis not linearly related to ET must be considered as separatecases.

The relationship between yield (Ym) and seasonal appliedwater (AW) for nonsaline irrigation water is illustrated inFig. 1. The seasonal applied water includes preplan! irriga-tion and precipitation which contributes to the availablewater supply to the crop. The yield-ET and yield-AW rela-tionships are assumed to be identical for AW values lessthan ETma,. Applying water at less than ETmax results in def-icit irrigation so the assumption is valid as long as excesswater resulting in deep percolation is not applied at anyirrigation. Maximum ET (ETmax) is associated with maxi-mum attainable yield (Y^) when water is not limiting. Fer-tilization and drainage are assumed to be adequate so thatyields remain constant for AW greater than ETmax.

Consideration is now given to applying saline irrigationwater to the crop which has a production function for non-

1005

1006 SOIL SCI. SOC. AM. J., VOL. 49, 1985

UJV

SLOPE (S)

AW, ET, AW, ETmax

APPLIED WATER

Fig. 1. Relationship between yield (¥„) and seasonal applied non-saline water (AW). Evapotranspiration (ET) is equal to AW forAW values equal to or less than ET .̂ Other symbols are de-scribed in the text.

saline water as depicted in Fig. 1. Assume AW is less thanETmaj, and equal to AW, (Fig. 1). Initially, no leaching occursand salts accumulate during the transient state until the rootzone salinity is sufficient to cause a yield decrement (YD).The yield decrement results in smaller plants and a conse-quent decrease in ET. The YD depicted in Fig. 1 results inET!. The difference between AWt and ET[ is deep perco-lation (DP) which contributes to leaching of salts from theroot zone. Ultimately YD is large enough so that sufficientleaching occurs to mitigate further yield decrements and asteady-state condition may be established. The following de-velopment allows the calculation of YD (and therefore yield)for various values of AW when irrigation is done with watersof various salinities.

The relationship between YD and DP isYD = (DP)5 for AWt < AW < ETmax [1]

where 5 is the slope of the production function for nonsalineirrigation water. Maas and Hoffman (1977) proposed a re-lationship between relative yield and average root zone sal-inity (expressed in electrical conductivity of saturated soilextract, EC,.) as

relative Yield = 100 - fi(ECe - C) [2]where C is threshold salinity and B is the slope of the yield-salinity curve at ECe values greater than C. ECC is com-monly considered to be one half the EC of the soil solution.

By definitionrelative Yield = 100(7ns - YD)/7ni [3]

where 7M is the yield resulting from irrigating with nonsa-line water. Substitution of Eq. [2] into [3] and rearranginggives

ECe = C' + 100 YD/5Fns for yns > YD > 0 [4]Note that Eq. [2] from Maas and Hoffman (1977) is valid

only for the range ECe > C and ECe less than the valueresulting in zero yield. ECC < C indicates that the thresholdsalinity has not been exceeded and thus no reduction in yieldresults. However ECe < C' in Eq. [4] results in a negativevalue of YD which is physically impossible. Thus any cal-culated negative YD value can be interpreted as resultingfrom a condition where ECe < C' and the true result is YD= 0. Similarly it can be demonstrated that if ECe exceedsthe value for zero production, the calculated YD value is >yns which is not possible. When the latter result occurs, theyield is recognized as being zero or YD = Ym.

A relationship between ECC and irrigation water salinity(EQ) is required. Hoffman and van Genuchten (1983) pres-ent relationships between ECe and EQ for three water uptakefunctions and any of the cases could be used. However, theauthors reported that the exponential uptake function as first

proposed by Raats (1974) gave the best agreement with ex-perimental data so this relationship will be used. The equa-tion for exponential water uptake function is2ECe/EQ =

l/L + (5/ZL)ln[L + (1 - L) exp(-Z/5)] [5]where L is the leaching fraction, Z is depth of the root zone,and 5 is a factor in the exponential uptake function. Hoffmanand van Genuchten (1983) suggest that S = 0.2 Z thereforeboth 5 and Z drop out of Eq. [5]. (Note that Hoffman andvan Genuchten (1983) used EC of the soil solution whilehere saturated soil extract (ECe) is used and ECe is assumedto equal EC/2).

By definition and using relationships depicted in Fig. 1L = DP/AW = YD/(AW)5for AWt < AW < ETmax. [6]

The latter relationship of Eq. [6] is substituted into Eq. [5]for L. From Fig. 1

Yas = S(AW-AWt) for AWt < AW < ETmax .[7]Equation [7] is substituted into Eq. [4] for yns. Finally Eq.[4] is substituted into Eq. [5] for ECe resulting in

IQQ(YD)2

BS(AW-AW t) + (YD)C'

ECj(S)(AW)-O.IEC^SXAW)

xln YD(AW)S + 1- -5) =0. [8]

Equation [8] can be used to calculate YD resulting fromusing given values of AW of irrigation water salinity, EQ,for the range AW, < AW < ETmax.

Yas equals ym» for AW =± ETmax so 7mai is substitutedinto Eq. [4] for these conditions. Deep percolation is (AW- ET^ + YD/5) for AW > ET,̂ and

L = 1 - ETmax/AW + YD/(AW)5. [9]Equation [9] is substituted into Eq. [5]. Again substitutingEq. [4] into Eq. [5] results in

C' 100 YD 0.5 ECl-(ETmax/AW) + [YD/(AW)S]

0.1 EC:

- (ETmax/AW) + [YD/(AW)S]

In 1- fclmax _ YDAW (AW)S [ l -exp(-5)]J=0.

[10]Equation [10] can be used to calculate YD resulting fromusing given values of AW of irrigation water salinity, EQ,when AW > ETm?x. Values of C and B to be used in Eq.[8] or [10] for various crops are available from tables pre-sented by Maas and Hoffman (1977). The crop-water pro-duction function must be known for a given crop irrigatedwith nonsaline water to determine values of 5, ET,,̂ andYOMZ. The values of YD calculated from Eq. [8] and [10] canbe used in Eq. [6] and [9], respectively, to calculate L forvarious values of AW. Solutions to Eq. [8] and [10] wereobtained by the Newton-Raphson procedure for numericalsolutions of nonlinear equations (Ralston and Rabinovitch,1978).

LETEY ET AL.: CROP-WATER PRODUCTION FUNCTION MODEL FOR SALINE IRRIGATION WATERS 1007

The model was developed in terms of absolute values ofyield and applied water. Crop-water production functionsare often expressed in relative terms which is particularlyuseful in transferring relationships between geographical areasof differing climates. For example,

20

- ET/ETmax) [11]where F is a constant, and was used by researchers in fourstates involved in a joint effort to evaluate the influence ofdrought-induced water deficits on crop production (Stewartet al., 1977; Hanks, 1982). ET may be expressed relative toa reference evaporation, such as pan evaporation, instead ofETmax as in Eq. [11]. Use of relative rather than absolutevalues of Y and ET in the present model can be done bymaking appropriate substitution of terms.

MODEL LIMITATIONSThe valid use of any model is restricted to a set of con-

ditions assumed in the model development. The followingis a discussion of some of the assumptions used in the modeland a guide for achieving approximate results when assumedconditions are not met.

Crop such as cotton which have a nonlinear relationshipbetween marketable yield and ET require a slight modifi-cation in the analysis. Since the total shoot weight of non-forage crops is linearly related to ET (Hanks et al., 1969;Davis, 1983), Eq. [8] and [10] can be used to calculate totalshoot weight for various applied seasonal quantities of watersof different salinities. The production function for market-able yield can then be determined if the relationship betweenmarketable yield and total shoot weight (harvest index) isknown. Note that the appropriate C and B values in thiscase are for shoot growth. The values reported by Maas andHoffman (1977) are for the marketable plant component un-less otherwise specified and therefore may not be valid forthe present case.

Fertilizer application and drainage were assumed to beadequate so that they would not affect yields even underhigh values of AW. In principle, fertilization can be man-aged to be nonlimiting under various values of AW; how-ever, the rate of deep percolation (drainage) is constrainedby soil profile characteristics. The rate of deep percolationis limited by the hydraulic conductivity of the least perme-able soil profile layer. The limit of validity of Eq. [10] canbe estimated if the value of hydraulic conductivity is known.Multiplication of the hydraulic conductivity by the irriga-tion season time period approximates the maximum surfacedepth of water which can be transmitted through the soiland become deep percolation.

The model assumes irrigation with water of a constantEQ. Except for very arid regions, a combination of precip-itation and irrigation water satisfy crop water requirements.Under these conditions, a weighted average of the EC of theprecipitation and irrigation water can be used for EQ in theequations to achieve approximate results. There is experi-mental evidence (Meiri, 1984) that plants respond to theweighted average of the EC of various waters applied duringthe season.

RESULTS FOR TALL FESCUEWe tested the model by using published data from

a multiple year investigation on the response of tallfescue to irrigation water salinity, leaching fraction,and irrigation frequency (Hoffman et al., 1983). Val-ues of ymax equal to 20.6 Mg/ha, ETmax equal to 185cm, S equal to 0.111, and AWt equal to zero weredetermined from these data. Mass and Hoffman (1977)

10

100 200

APPLIED WATER, cm300

Fig. 2. Relationships between calculated yield of tall fescue and sea-sonal applied irrigation waters of various values of EC (dS/m).

reported C = 3.9 dS/m and B = 5.3%/dS per m fortall fescue. These values were used to calculate thecrop-water production functions for EQ = 0, 1.0, 2.5,and 4.0 dS/m.

Crop-water production functions for tall fescue areillustrated in Fig. 2 for various values of EQ. Irriga-tion water of EQ up to 4.0 dS/m can be used to irrigatetall fescue without yield decrement if sufficient irri-gation water is applied. Some yield is achieved whenirrigating with saline waters at AW < 185 cm whichis considered deficit irrigation even with nonsalinewater. Yield decrements increase with increase in EQat the lower values of AW.

After YD has been calculated by Eq. [8] or [10] forgiven values of AW, associated values of leaching frac-tion (L) can be calculated from Eq. [6] or [9]. Resultsof these computations are illustrated in Fig. 3. A con-stant L is calculated for a given value of EQ for allvalues of AW less than ETmax. The value of L increasesas ECi increases. Significantly, some leaching occurseven at low values of AW when saline waters are usedfor irrigation. This result is consistent with observa-tions of Hoffman et al. (1983) where leaching wasmeasured when irrigating tall fescue with saline waterswith quantities considerably below ETmax. L increaseswith an increase in AW for waters of all values of EQwhen AW is greater than ETmax as expected. The val-ues of L for ECi = 0, 1, and 2.5 converge as AWincreases and the AW value at convergence is equalto the value at which YD becomes zero.

0.40

0.20

50.10

EC;

100 200

APPLIED WATER, cm300

Fig. 3. Relationships between calculated leaching fractions for tallfescue and seasonal applied irrigation water of various values ofEC (dS/m).

1008 SOIL SCI. SOC. AM. J., VOL. 49, 1985

COMPARISON OF CALCULATED ANDMEASURED RESULTS

The experiment (Hoffman et al., 1983) was con-ducted in a rhizotron consisting of 24 fully enclosedsoil plots on which the amounts of applied and drain-age water were carefully controlled and accuratelymeasured. Rainfall was excluded from the plots. Theexperimental variables consisted of three values of EQ(1, 2.5, and 4 dS/m), three target values of L (0.09,0.18, and 0.27) and three irrigation frequencies. Theirrigation frequency treatments were pulse irrigationsdaily to prevent any significant depletion of soil waterbeyond that required to maintain the desired leachingfraction, irrigation when approximately 1/3 of theavailable soil water had been lost, and irrigation whenapproximately 2/3 of the available soil water had beendepleted. The study was conducted for 3 yr followingan initial period to establish the plants and treatments.The fescue was periodically harvested and the resultswere reported for accumulated annual yield as well asannual quantities of irrigation and drainage water.3The reported annual AW and the appropriate EQ wereused in Eq. [6], [8], [9], or [10] to calculate yield andleaching fraction for each experimental plot for eachyear.

The calculated annual yield versus the measuredyield from each experimental plot is plotted in Fig. 4.The data points from 1979 to 1980 (2nd yr of the 3-yr study) are identified because some of the greatestdeviations from perfect correlation occurred duringthat year.

The calculated leaching fraction versus the mea-sured leaching fraction from each experimental plot isplotted in Fig. 5. One feature of the model output isa predicted leaching fraction that is constant at allvalues of AW less than ETmax. The lowest predictedleaching fractions were 0.175, 0.12, and 0.04 for EQvalues of 4, 2.5, and 1 dS/m, respectively. The exper-

3 Appreciation is expressed to Dr. Glenn J. Hoffman for providingdata on annual quantities of irrigation and drainage water for allplots that were not reported in the publication.

12 14 16 18 20MEASURED YIELD (Mg/ho)/yr

22 24

Fig. 4. Comparison between calculated and measured yields of tallfescue. Data points identified with (2) are those obtained in the2nd yr of a 3-yr study. The line represents perfect agreementbetween calculated and measured results. Units of EC, are dS/m.

imental plan was set up to achieve a predeterminedaverage leaching fraction, and irrigation was managedaccordingly. The lowest target leaching fraction was0.09 which is lower than the model predicts for EQvalues of 2.5 and 4.0 dS/m. The target leaching frac-tion was not achieved during every irrigation, but ap-plied water was adjusted so that "on the average" thetarget leaching fraction was achieved.

Three types of analyses were done to provide somequantitative comparison of calculated and measureddata. The correlation coefficients between calculatedand measured yields and leaching fractions were com-puted. The average of the total calculated and averageof the total measured data was compared to determinewhether the model tended to over predict or underpredict results; and the average percent difference be-tween individual observations (D) was computed byEq. [12] as an index on the degree of variation betweenindividual measured and calculated area.

D = \Y . — V . l /y •>I •* mi •* at/ •* mit [12]

where Ym, Yc, i, and n represent measured yield, cal-culated yield, observation index, and number of ob-servations, respectively. Similar computations weredone with leaching fraction data.

These analyses were computed using all the dataand also using data segregated by individual years andgiven values of EQ, L, and irrigation frequency. Theresults are tabulated in Table 1. Better agreement be-tween calculated and observed results was achievedfor yield than for leaching fraction in almost everycase. This observation is possibly associated with thefact that the experiment imposed one leaching fraction(0.09) which is lower than the model predicts is pos-sible for EQ of 2.5 and 4 dS/m.

With one exception, the total average calculated yieldwas within 5% of total average measured yield. The

.35

2.30

cc.25

5 -20

.15

0 .05 .10 .15 .20 .25 .30 .35MEASURED LEACHING FRACTION

Fig. 5. Comparison between calculated and measured leaching frac-tions. The line represents perfect agreement between calculatedand measured results. Units of EC, are dS/m.

LETEY ET AL.: CROP-WATER PRODUCTION FUNCTION MODEL FOR SALINE IRRIGATION WATERS 1009

Table 1. Various comparisons between measured and calculatedvalues of yield (Y) and leaching fraction (L).

Correlationcoefficientt Rt 0§

All data1978-791979-801980-81EC; = 1EC; = 2.5EC; = 4L = 0.09L = 0.18L = 0.27Daily irrig.1/3 depl. irrig.2/3 depl. irrig.

0.730.730.660.960.690.660.860.600.690.720.680.710.79

0.660.740.560.700.850.610.740.29NS

-0.13NS0.02NS0.730.570.61

1.010.961.090.991.041.030.960.951.041.031.001.031.00

0.890.841.000.940.760.881.101.220.780.920.890.940.94

8.16.6

13.45.58.49.77.1

14.16.95.18.97.78.6

32.323.939.434.740.027.829.465.531.022.232.830.433.7

t All correlation coefficients are significant at the 0.01 level except for thevalues identified by NS which are not significant.

t R is the ratio of the total average calculated to the total average meas-ured values.

§ Average percent difference between individual calculated and measuredvalues.

exception was 1979 to 1980 when the difference was9%. The 3-yr avg of Fmax and ET^^ were used in themodel without adjustment for variations in these val-ues during individual years. The Ymiui for 1979 to 1980was 6% lower and Fmax.s for 1978 to 1979 and 1980to 1981 were 3% higher than average. Standardizingthe model calculations for individual year growth andclimatic conditions would have provided slightly bet-ter agreement between calculated and measured val-ues. The average difference between individual cal-culated and measured yields ranged between 5.1 and14.1%. The correlation coefficients between measuredand calculated yields range between 0.60 and 0.96 andare all significant at the 0.01 level.

Segregating the data by individual years and givenvalues of EQ, L, and irrigation frequency provide re-sults similar to those determined from the aggregateddata and suggests that the model is adequate for allvalues of EQ, L, and irrigation frequency used in theexperiment.

Validation of the model also suggests validity ofcomponents upon which the model was constructed.The Maas and Hoffman (1977) threshold and slopevalues for crop-salinity sensitivity are vital parametersin the model. Many of their reported values wereachieved in greenhouse experiments where largequantities of water were frequently applied resultingin high leaching fractions. The data reported here pro-vides evidence that the Maas and Hoffman coeffi-cients are valid over a range of leaching fractions andirrigation frequencies.

The relationship between average root zone salinityand leaching fraction used in the model was derivedby Raats (1974) under conditions of steady water flow.Our results provide evidence that the assumed rela-tionship between average root zone salinity and leach-ing fraction is not restricted to high frequency irriga-tion.

Since the experiment continuously used irrigationwater at a given EC and rainfall was eliminated fromthe plots, no evidence is provided on the validity ofusing a weighted average EQ when waters of various

EC values (such as periodic rain) are intermittentlyused. Thus, that aspect remains in question. Never-theless, the analysis reported here provides evidencethat the model has utility under a variety of field con-ditions.

ACKNOWLEDGEMENTThe contributions of Dr. Kenneth H. Solomon, Agricul-

tural Engineer, U.S. Salinity Laboratory to this study areacknowledged. Discussions the authors had with Dr. Solo-mon were helpful in formulating the problem.