Optimal and Postirrigation Volume Balance InfiltrationParameter Estimates for Basin Irrigation

E. Bautista, A.M.ASCE1; A. J. Clemmens, M.ASCE2

; and T. S. Strelkoff, M.ASCE3

Abstract: Engineering analysis of surface irrigation systems is predicated on reasonably accurate estimates of a field's infiltrationproperties. Optimal estimation methods pose multiple volume balance equations at various stages of an irrigation event and are assumedto produce the most accurate results among volume balance based procedures. They have the disadvantage of requiring surface volumedeterminations, which may be difficult to obtain in practice under many field conditions. This study contrasts infiltration solutions fromoptimal and a simpler postirrigation volume balance method and examines the implications of those solutions on the performance ofmanagement strategies with zero-slope and low-gradient basins. With those types of systems, there is little benefit in using optimizationover postirrigation volume balance due to the nonuniqueness of solutions and uncertainties of inputs required by the estimation proce­dures. In addition, system hydraulic characteristics mitigate the insensitivity of the distribution uniformity to reasonable variations ininfiltration characteristics from those assumed in the analysis. For the type of systems considered here, management can be optimizedbased on time needed to infiltrate a target depth, even if the infiltration function parameters are uncertain.

001: 10.1061/(ASCE)IR.1943-4774.0000018

CE Database subject headings: Surface irrigation; Irrigation practices; Efficiency; Infiltration; Parameters; Estimation; Flowsimulation.

Introduction

Numerous procedures have been developed for the estimation ofinfiltration properties needed for surface irrigation engineeringanalyses (Strelkoff et al. 2009). Most proposed procedures rely onthe volume balance relationship

in which ti = time at which the balance is computed; VQ = inflowvolume; Vy=surface storage volume; VRo=runoff volume; Az=functional relationship for the infiltration volume per unitlength; T = intake opportunity time (total time t i minus the advancetime to distance x, tx )' and; x(t), the wetted field length at time t.The left-hand side of the expression represents the "measured"infiltration volume Vz. The predicted infiltration V~ is obtained byintegrating the infiltrated profile Az(s) between the limits s=O and

s=x(t), and is represented by the right-hand side of Eq. (1). Gen­erally, Az is calculated using empirical functions, which assumethat infiltration is solely a function of T and the parameters ofthe function, represented by the set 8. The estimation problemconsists of finding an infiltration parameter set 8 that will satisfyEq. (1).

Among the proposed procedures, optimal estimation methodshave the most significant data requirements and are the most com­putationally intensive. Optimal procedures apply the volume bal­ance at multiple times and to different phases of the irrigationevent, and then best fit the infiltration parameters to the resultingseries of volume balance relationships. Included in this categoryof procedures are methods that find the parameters by eye(graphically) (e.g., T. W. Ley, "Sensitivity of furrow irrigationperformance to field and operation variables," unpublished MSthesis, Department of Agricultural and Chemical Engineering,Colorado State University, Fort Collins, Colorado, 1978; Strelkoffet. al. 1999) as well as methods that minimize a nonlinear least­squares objective function (Esfandiari and Maheshwari 1997a;Gillies and Smith 2005). Optimal estimation procedures are con­sidered more accurate than methods that apply Eq. (1) at one or afew number of times and to only one phase of the irrigation event;first, because they include data from all phases of the event, andsecond because they use more equations than unknown param­eters (Gillies and Smith 2005; Khatri and Smith 2005). Thismathematical overconditioning of the estimation problem can av­erage out the effect of field-data nonuniformities (soil texture,hydraulic resistance, field elevation, field geometry, inflow rates,etc.). Also, infiltration formulas vary in their ability to fit infiltra­tion data, and simple estimation methods generally are restrictedto specific functional forms that mayor may not adequately de­scribe a particular soil. In contrast, optimal approaches can beformulated to handle different infiltration functional forms and,therefore, can fit the observations with greater flexibility.

The question to be addressed here is whether optimal methods

(1)1<- - - - Vz - - - - >11<- - ~- - >1

lResearch Hydraulic Engineer, USDA-ARS Arid Land AgriculturalResearch Center, 21881 N. Cardon Ln., Maricopa, AZ 85238 (corre­sponding author). E-mail: eduardo.bautista@ars.usda.gov

2Laboratory Director, USDA-ARS Arid Land Agricultural ResearchCenter, 21881 N. Cardon Ln., Maricopa, AZ 85238. E-mail: bert.clemmens @ars.usda.gov

3Research Hydraulic Engineer, USDA-ARS Arid Land AgriculturalResearch Center, 21881 N. Cardon Ln., Maricopa, AZ 85238. E-mail:theodor.strelkoff@ars.usda.gov

Note. This manuscript was submitted on July 11, 2008; approved onNovember 20, 2008; published online on January 27, 2009. Discussionperiod open until March 1, 2010; separate discussions must be submittedfor individual papers. This paper is part of the Journal of Irrigation andDrainage Engineering, Vol. 135, No.5, October 1,2009. ©ASCE, ISSN0733-9437/2009/5-579-587/$25.00.

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Table 1. Basin Evaluation Data

Test id

B1 B3 B7 B9

Length (L) (m)

Bottom slope (So)

Manning n

Unit inflow rate (q) (m3/s/mX 1,000)

Cutoff time (teo) (min)

Final advance time (tL)(min)

Average applied depth (Dapp) (m XI, 000)

Average intake opportunity time (TAVO) (min)

185.9

1.00 X 10-6

0.183

6.27

53

90

107

433

185.9

o0.125

9.44

33

76

101

345

189.0

4.64 X 10-4

0.177

6.29

43

77

86

229

189.0

5.00 X 10-4

0.145

9.44

26

89

78

186

Note: id = identity.

can generate better estimates of an irrigated field's infiltrationproperties than the simpler postirrigation volume balance (PIVB)methods. PIVB methods use a single volume balance equationcalculated at the end of the irrigation. This is not a trivial questionbecause the optimal method requires values of surface storage foreach time at which Eq. (1) is applied while the surface storagedoes not require such values. Reasonable surface volume esti­mates can be generated under a variety of hydraulic conditionsusing the relationship

Table 1. In the table, the test identifiers (Bl, B3, B7, B9) corre­spond to the identifiers provided by Clemmens and Dedrick.These tests are characterized by high inflow rates and short cutofftimes (tco) relative to tv the advance time to the end of the field.Two of the systems are level (near zero slope), while two are lowgradient. These are the conditions under which it would be diffi­cult to accurately estimate Vy using Eq. (2).

Optimal parameters were computed using the sum-of-squared­errors objective function SSVz

Methodology

Optimal and PIVB calculations were applied to four basin(closed-ended border) irrigation data sets that were reported byClemmens and Dedrick (1986). Reported data include the fieldlength, inflow rate per unit width (measured with a broad-crestedweir), cutoff time, bottom elevations measured at regularly spacedstations (field elevations measured with a standard surveyorlevel), and for each of these stations, advance times, recessiontimes, and depth hydrographs. Manning n parameters neededto describe the field hydraulic roughness were calculated byClemmens and Dedrick (1986) from the depth hydrograph data.A summary of inputs required for this analysis is provided in

However, there are other conditions under which Vy can be de­termined with reasonable certainty only from field measurements,an approach that is too labor intensive and costly for routine use.In Eq. (2), Ao is the upstream flow sectional area, generally cal­culated with the Manning formula by assuming normal depth,while U y is a surface shape factor, which in theory is less than1.0 and, in practice, is a generally assumed constant and in therange of 0.7-0.8 (Scaloppi et al. 1995). Even in cases wherethe use of Eq. (2) may seem reasonable, such as with slopingopen-ended furrow irrigation systems, Vy estimates can be sus­pect and lead to inaccurate infiltration function determinations(Walker and Kasilingam 2004). Inaccurate determinations arisefirst, because of the uncertainty in the determination of the Man­ning roughness coefficient n, and second because Uy is time de­pendent and not necessarily less than unity, as suggested by fieldstudies (Esfandiari and Maheshwari 1997b). The analysis con­ducted examines the characteristics of optimal and PIVB infiltra­tion parameter estimates derived from four basin irrigation datasets, and the implications of using alternative infiltration solutionsfor assessing the performance of the observed irrigation event.The development of recommendations for optimizing the irriga­tion system's operation is also examined.

NX

Vy(tJ = w2: (<!>Yj-l,i + (1 - <!»Yj,J . (Xj -Xj-l) (4)j=2

(3)

NT

SSVz = 2: [VZi(tJ - ~i(tJ]2i=l

NX

~z(t.) = W2: (k( <!>T~-l .+ (1 - <!>)T~ .)l J ,l J,lj=2

in which Vz and V;=left- and right-hand side of Eg. (1) and NT= number of times at which the mass balance is calculated.

The Vy(tJ values needed to compute Vz(tJ were determinedby calculating surface profiles from the measured depth hydro­graphs and integrating the profiles numerically using a modifiedtrapezoidal rule

+ b(<!>Tj-l,i + (1 - <!»Tj,i) + c) . (Xj - Xj-I) (6)

For each t i and station Xj' intake opportunity time was calculatedas

In Eq. (4), W=field width (unit width, for this analysis); Yj,i= flow depth at location Xj and time ti; Nx= number of stationsused in the calculations; and <!>==weighting factor, equal to 0.5except when dealing with the advancing wave tip cell (the incre­ment XNx-XNx-I)' For that tip cell, a shape factor applies instead(Strelkoff et al. 1999). A constant value of <t>=0.7 (Clemmens1991) was assumed for the tip cell in this analysis.

The modified Kostiakov equation

z=kTa + bT + C (5)

was used to represent infiltration in this analysis. In Eq. (5),z=infiltration volume/length/width [L3 / L 2]; T=intake opportunitytime [T]; and k[L / ra]; a [dimensionless]; b[L / T]; and c[L]=empirical parameters. Eq. (5) times the basin width W yields Azin Eq. (1).

Trapezoidal rule integration was used to integrate the infil­trated profile as a function of distance [the right hand side ofEq. (1)]

(2)Vy =Ao ' Uy

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(7)

Each family is associated with a basic intake rate, expressed inin.lh (which is also the family label). The parameters k and aare unique to each family but c is constant (c=7 mm=0.28 in.).A search procedure was used to solve the PIVB problem with theNRCS families: different families were tested in combinationwith Eq. (6) until finding the function that most closely matchedthe predicted and field-measured Vz.

The Merriam-Clemmens families define infiltration usingthe Kostiakov equation but define the k and a parameters as afunction of t IOO' the time (in hours) needed to infiltrate 100 mm(0.1 m)

used to adjust the predicted recession profile. While the simulatedadvance trajectory can be adjusted by modifying a, it provedeasier to hold a constant (=0.5) and use c to make the advanceadjustments. A similar strategy was successfully followed by EI­Haddad et al. (2001) when analyzing basin irrigation data fromEgypt.

Another way of solving for the parameters using PIVB is byfitting Vz to fixed infiltration relationships, i.e., the Natural Re­sources Conservation Service (NRCS) infiltration families (for­merly known as SCS families) [U.S. Dept. of Agriculture-SoilConservation Service (USDA-SCS) 1974] or the Merriam­Clemmens time-rated families (Merriam and Clemmens 1985).The NRCS families use the infiltration equation

(12)

(

0, ti < tx(X)

Tj,i = ti - tx(X) , ti < tr(X)

tr-tx(X), ti;?:tr(x)

In this expression tx(x) and tr(x) are, respectively, the advanceand recession times measured at station Xj' Similar to Eq. (4),<p =0.5 except when dealing with the advancing tip cell. For thatcase, 4>= 11(1 +a) (Strelkoff et al. 1999). Volume balance equa­tions were applied at each advance time and at regular time inter­vals thereafter, until final recession time.

Calculations were implemented in a spreadsheet and thespreadsheet's built-in optimizer was used to minimize [Eq. (3)].Computational difficulties were experienced in the form of lackof convergence, convergence to a local minimum, or convergenceto negative parameters. As a result, solutions were developedfor k and b only (with c=O) at specified values of a. These solu­tions allowed us to examine the behavior of the objective function[Eq. (3)] as a function of the parameters, to contrast their perfor­mance (using unsteady flow simulation), and ultimately to select a"near-optimal" solution, as will be explained later in the Resultssection.

The PIVB (Merriam and Keller 1978; USDA-NRCS 1997)uses the measured advance and recession times to calculate thefinal intake opportunity time at each measurement station. Theunknown parameters can then be found from the infiltration inte­gral [Eq. (6)]. The method was originally developed to solve forthe k parameter of the Kostiakov equation, Z= kTa, with the expo­nent a independently determined (from ring infiltrometer mea­surements or experience). Based on Eq. (6) (but with b=c=O), kis found as

Eq. (13) can be used in combination with Eqs. (3) and (6) toformulate a nonlinear root-finding problemwhere

Vzk=-­SUM I

(8)k = 1001t~00 (13)

NX 1SUM! =L l(T) +T)_I)' (x;-Xj_l)

j=2

(9)NX

o= Vz - L (k[t 100](Tj~?OO] + Tj[t lOO])/2) . (Xj - Xj_l)j=2

(14)

(10)

Here, T and r = observed and predicted values and N the numberof observations, respectively. An additional measure of goodness­of-fit of the advance and recession data are the final average op­portunity time T AVG

Here the notation [t100] is used to refer to the parameters com­puted with Eq. (13).

Unsteady flow simulations needed to fine tune the PIVB modi­fied Kostiakov parameters and to validate the results. These re­sults, generated with other estimation methods, were carried outwith the simulation module of the WinSRFR software package(Bautista et al. 2009) using the inputs of Table 1, but withmeasured field elevations instead of the reported average slope.WinSRFR provides a module for solving the PIVB problem and,therefore, the software was used also for implementing the trial­and-error solution. In addition to calculating k with Eqs. (10) and(11), the software automatically tests the solution with simulation,plots the observations and predictions, and computes the RMSerror for the advance (RMSETA) and recession (RMSETR) times

(15)RMSE=

k= VZ -SUM2

SUM I

where SUM I is as previously defined and SUM2 is given by

SUM2 =~ (~(Tj + Tj_l) + c) .(x) - Xj-I) (II)J=2

The PIVB can also be used in combination with the modifiedKostiakov infiltration in Eq. (5), but requires guesses for the val­ues of a, b, and c. In such case, k is given by

A parameter set representative of the infiltration characteristics ofthe evaluated field can be found with the aid of hydraulic simu­lation and trial-and-error. Based on reasonable initial guesses fora, b, and c, an initial estimate for k is calculated with Eq. (10);the parameter set is then fed into a simulation model. Adjustmentsare then made to the guessed parameters based on differencesbetween measured and predicted irrigation outputs. For theexamples presented here, calculations started with a = 0.5 andb=c=O. Gross adjustments to the parameters were made firstbased on graphical comparisons of the observed versus predictedadvance and recession times; parameter values were then finetuned based on goodness-of-fit indicators computed by theWinSRFR software (RMS error). The parameter b was mainly

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Fig. 1. Average infiltration depth (Dz) as a function of time calcu­lated from volume balance for all basins

Comparison of Optimal and Postirrigation VolumeBalance Solutions

Fig. 1 summarizes the measured Dz [Vz(t)lx(t)] data as a functionof time for all the tests. The data exhibit a large initial scatter,especially for Tests B3 and B9, but varies smoothly for longertimes. For very short times, the mass balance can yield anomalousresults, including negative values. Several factors contribute tothe erratic initial data, including inflow variations that were notcaptured by the flow measurements, large initial transient effectsthat are not captured by the water depth measurements, spatialand temporal variations in surface roughness, and measurementerrors (field elevations, water depths). Because of the initial scat­ter of the data, measurements for times less than 20 min were notincluded in the development of the optimal solutions. The data ofFig. 1 also show significant similarities in infiltration characteris­tics among the tests. Although less water was applied in Tests B7and B9 than to Bland B3, the final slope of the B3 and B7 datasuggests a long-term infiltration behavior similar to that of othertwo tests.

For all the examples, the objective function SSVz as a functionof the exponent a exhibited a well-defined inflection point and,

120

100..

vVv .

v •

0' 80 ~v.·Il:;·aa ° •"V~ 60 ~..-Ie 815 fJ • 83N <tJ0

•a 40'VV •

87° - 8900. °20 .....

00 100 200 300 400 500

Time (min)

'TAVO= Nx

L (xi - Xi-I)i=2

which was calculated for both the field and simulated data.

Results

(16)

consequently, a near-optimal combination of parameters was eas­ily found based on solutions generated for only a few values of a.Because infiltration solutions near this inflection point producedsimilar validation results, no effort was made to find the absoluteminimum to SSVz. As an example, Table 2 summarizes the esti­mation and validation results generated for Test B1. The optimalsolution (minimum SSVz) for B1 is near a=0.5, and that solutionreproduced adequately the observed advance and recession insimulation (RMSETA, RMSETR, and 'TAVO)' Solutions computedfor other values of a produced comparable validation results. Forexample, the solution computed for a =0.4 best matched the ob­served advance times while the observed recession and 'T A VG werebest replicated by the solution computed with a = 0.2. For othertests, the near-optimal solution also yielded the best overall vali­dation results.

The trial-and-error solutions to the PIVB-modified Kostiakovproblem were easy to find with the strategy described earlier.Multiple solutions can be generated with this approach, and thosesolutions differ slightly in their ability to replicate the advance,runoff (for open-ended systems), and recession phases of the irri­gation event. Hence, the choice of a final solution is somewhatarbitrary.

A characteristic of the estimation results generated for all thedata sets presented here is the large prediction errors for recessiontimes in comparison with advance errors. For Test B1, the near­optimal solutions produced RMSETA values that ranged from 2 tonearly 5 min, while RMSETR values varied from about 19 to 26min (Table 2). Optimal and PIVB estimation results for othertests, which will be discussed later, produced even largerRMSETR values. While determination of recession is subjectiveand therefore subject to potentially large errors, these large errorssuggest field undulations that were not measured by the surveyand/or survey errors. Fig. 2 illustrates the large variation in reces­sion times for Test B1, a supposedly zero-slope field. For this test,the survey data show elevation differences of no more than 20mm along the field, yet the measured recession times differed byover 100 min. Contributions to the computed RMSETR value aremostly from the recession times measured in the field segmentbetween 50 and 100 m. Similarly, measurements at a few stationscontribute a large portion of the recession prediction errors forother data sets.

Zero-Slope BasinTables 3 and 4 summarize the optimal and PIVB estimation re­sults for the zero-slope basins (B 1 and B3, respectively). In eachtable, the column labeled OPT identifies the optimal results, whilethe columns labeled PIVB identify the postirrigation mass bal­ance results (MK-modified Kostiakov equation; NRCS-NRCSinfiltration families; TR-time-rated infiltrated families). The lastcolumn, labeled AF, will be explained later. The row labeled IF

Table 2. Parameter Values, SSVz, and Performance of Optimal Solutions Computed at Discrete Values of Exponent a for Example B 1

a

0.1 0.2 0.3 0.4 0.5 0.509

k (mm/ha) 29.2 30.9 32.7 34.9 37.9 38.3

b (mm/h) 10.03 8.49 6.53 3.96 0.43 0.00

SSVz (m3 / m)2 0.51 0.32 0.20 0.13 0.11 0.11

RMSETA (min) 4.8 3.3 2.3 2.0 2.6 2.6

RMSETR (min) 20.1 19.2 19.8 21.9 25.1 25.8

TAVG (min) 427.75 432.73 438.50 444.38 448.94 450.14

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600

~.../\.

500 -- 0""0_____

""'-- ---- °/0

c 400 w

I 300Q)

Ei= 200

100

00 50 100 150 200

Distance (m)

Advance (Observed)Recession (Observed)Advance (Predicted)Recession (Predicted)

Fig. 2. Observed and simulated advance and recession times forExample B1 (simulated results based on the optimal infiltration esti­mate)

identifies the infiltration family associated with the PIVB-NRCS(infiltration rate, in in./h) and time-rated family (t 100' in hours)solutions. The row SSVz includes the objective function valuecomputed based on the PIVB solutions. The last two rows arethe simulated minimum infiltrated depth D min and the low-quarterdistribution uniformity DU1q, which are included to compare thefinal infiltration distribution predicted with each of the estimatedfunctions.

For both of these examples, the PIVB-MK and optimal solu­tions produced equally accurate advance and recession predic-

tions. For B1 (Table 3), the optimal solution produced slightlybetter advance predictions but slightly worse recession pre­dictions, while the opposite was true for the second example(Table 4). Also, although the PIVB solution produced larger SSVzvalues than the optimal solution, by an order of magnitude forExample B3, both solutions predicted essentially the same finalinfiltration distribution, as indicated by the Dmin and DU1q values.

Because the NRCS and time-rated infiltration families arefixed infiltration functional forms, they were expected to produceworse advance and recession predictions than the PIVB-MK so­lution. Results confirm that the expectation, especially with theB3 test, for which the PIVB-NRCS and PIVB-TR functions pro­duced relatively large advance prediction errors [RMSETA = 12.1and 11.4 min, respectively (Table 4)]. However, like the optimaland PIVB-MK functions, they predicted nearly the same finalinfiltration profile.

The close agreement in infiltrated profiles predicted with thefour estimated functions merits further discussion. Fig. 3 depictsthe infiltration solutions computed for Basin B3, which are pre­sented to illustrate the differences in the shape of the functions.Differences in predicted infiltrated depths at relatively shorttimes, as much as 20 mm in the first 10 min, result in largedifferences in infiltrated depth during advance (Fig. 4), and ulti­mately account for the differences in RMSETA values of Table 4.However, differences in infiltrated depth with distance during ad­vance are compensated during latter phases of the irrigation. Thiscompensation occurs because the hydraulic conditions of the testproduce relatively small differences in opportunity time with dis­tance [a zero-slope basin with no runoff and a short advancephase relative to the duration of the irrigation event (see tLand

Table 3. Optimal and PIVB Results for Example B 1: Parameter Values, Infiltration Family, and Simulation Results

OPT PIVB-MK PIVB-NRCS PIVB-TR AF

k (mm/ha ) 37.9 25.0 23.6 39.5 41.4

a 0.500 0.5 0.742 0.506 0.518b (mm/h) 0.4 5 0 0 0c (mm) 4 7 0 0IF 0.45 6.27 5.48SSVz (m3 / m)2 0.11 0.24 1.35 0.60 2.17

RMSETA (min) 2.6 3.3 7.9 2.1 1.9RMSETR (min) 28.5 24.5 28.9 27.8 66.8

'TAVG (min) 450.1 432.4 421.2 432.6 375.6Dmin (mm) 94 93 92 93 93

DU1q 0.93 0.92 0.92 0.92 0.91

Table 4. Optimal and PIVB Results for Example B3: Parameter Values, Infiltration Family, and Simulation Results

OPT PIVB-MK PIVB-NRCS PIVB-TR AF

k (mm/hQ) 38.2 21.6 25.5 40.9 41.4

a 0.1 0.500 0.748 0.515 0.518b (mm/h) 9.6 5 0 0 0c (mm) 20 7 0 0

IF 0.5 5.7 5.48SSVz (m3/m)2 0.02 0.21 3.20 1.04 1.05

RMSETA (min) 2.9 1.9 12.1 11.4 11.3

RMSETR (min) 26.4 28.4 24.2 30.8 31.0

'TAVG (min) 344.2 344.6 376.2 345.3 346.3Dmin (mm) 89 90 89 88 88

DU1q 0.94 0.94 0.94 0.93 0.93

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Table 6. Optimal and PIVB Results for Example B9: Parameter Values, Infiltration Family, and Simulation Results

OPT PIVB-MK PIVB-NRCS PIVB-TR AF

k (mm/hQ) 35.80 25.323 29.3 43.1 41.4

a 0.200 0.5 0.757 0.528 0.518

b (mm/h) 10.9 5 0 0 0

c (mm) 0 18 7 0 0

IF 0.6 5.0 5.48

SSVz (m3/ m)2 0.13 0.25 1.57 0.45 1.03

RMSETA 7.1 2.2 13.8 12.3 12.6

RMSETR 41.7 39.6 46.7 44.0 46.9

TAVG (min) 186.2 186.7 194.1 188.6 209.0

Dmin (mm) 56.0 59.0 48.0 54.0 52

DUlq 0.81 0.83 0.72 0.78 0.76

functions computed for each test are shown in Fig. 5. Thesecurves suggest slightly lower infiltration rates for the zero-slopebasins than for the graded basins, which could be the result of thegrading operation. The similarity in infiltration characteristics isless evident from the optimal parameter values, given in Tables2-5. For example, the optimal parameters for B1 (Table 3) coulddescribe a soil where the steady infiltration rate is very small andis reached onl'] aftet a vet'] lon'b time, while the o\1timal \1atam­eters for B3 (Table 4) suggest a soil where infiltration reaches asteady state rapidly and the steady rate is high. Clearly, the opti­mized parameters are poor indicators of the infiltration propertiesof these fields. Similar differences in parameter values can beobserved between Tests B7 and B9.

The PIVB-MK parameter values are only marginally betterindicators of infiltration properties than the optimal parameters.Because of the strategy used to fit the parameters, the PIVB-MKresults differ only in the k and c parameters, but these parametersvary substantially and give little indication of the similarity ininfiltration predictions among the various tests. In contrast, theNRCS and Merriam-Clemmens infiltration family results suggestsimilarities in infiltration characteristics among the tested basins.The NRCS infiltration family values ranged from 0.45 to 0.6. Thetime-rated family information appears particularly useful, as thecombined information from the two tests indicates that, for thisfield, the time needed to infiltrate 100 mm is between 4.9 and 6.27h. Because of the uncertainty in the measured infiltration proper­ties, the t 100 value can be more useful than specific parameterswhen developing management recommendations.

An average infiltration function was developed by combiningthe results of all four basins. Different approaches can be fol­lowed in developing a combined function from the evaluation

results. For simplicity, the approach used here is to represent in­filtration with the time-rated infiltration families and to averagethe t 100 value of the four tests. Simulation results computed withthe average function for each test are presented in the columnlabeled AF in Tables 3-6. The average function is very close tothe function estimated for Basin B3 (Table 4) and, therefore, re­sults are very similar to those obtained with the PIVB-TR solu­tion. For the other basins., the main effect of the average functionis to shift the recession curve upward or downward, which isindicated by the change in RMSETR and 'TAVG values. Fig. 6 con­trasts the optimal functions for Tests Bland B9 with the averagefunction. The average function predicts more infiltration as afunction of time than the B1 optimal function but predicts smallerdepths than the B9 solution. Hence, for B1 the 'TAVG with theaverage increases relative to the value computed with the averagefunction while for B9 'TAVG decreases. Despite these differences,final infiltrated profile predictions for all of these examples do notchange or change little when using the average function. Theseresults provide confidence in using the estimated function for op­timizing fieldwide operations or for improving the design.

Two optimal management scenarios were developed for the B1and B9 basins based on the average function. Those basins wereselected because their optimal infiltration function differ the mostfrom the average function. The scenarios assume an irrigationtarget Dreq =80 mm (0.8 m). A recommended q-tco combinationwas developed by trial-and-error, with the objective of meetingDreq everywhere (D min~ Dreq ). The optimized q-tco combination isgiven in the AF column of Table 7 (Scenarios 1 and 2). The q-tco

combination was then tested with the estimated functions for eachexample. Those results are summarized in the OPT, PIVB-MK,

175 --....-------.------

Time (min)

Ol..-_"'---...I...-_..J.....-........_....I--.....

o 100 200 300 400 500 600

5lE0J

......;,.,..,.,..,.,..,.,.......,.,..,.,..

.' /...~/

..;/'/../../

.'/'.·h

.·h25 :U

75

50

:E.. 100Q)"0Co

~'+=.E

'0 150~

k 125

180 _-__-_-..,..--....,.----r-----,

C") 160 ~

~ 140 ,/'iCE /.,?; 120 / .

i 1:~ ·:~·,~#:?~<I··:::········ ~~ I'+= 40 .'/ ---- 87E: 20 IJ _ .. _.. 89

OL.---~--'---..l.--...I..----'---......o 100 200 300 400 500 600

Time (min)

Fig. 5. Optimal infiltration functions for all basins

Fig. 6. Optimal infiltration functions for Basins Bland B9 and av­erage infiltration function from all tests

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Table 7. Optimal Management Scenarios for Basins Bland B9: Inflow, Cutoff Time, and Performance

AF OPT PIVB-MK PIVB-NRCS PIVB-TR

Scenario I-Basin Bl

q (m3 / s/ m XI, 000) 6.56 6.56 6.56 6.56 6.56

teo (min) 45 45 45 45 45

tL (min) 94.4 89.2 85.8 82.2 91.8

D min (mm) 81 82 81 81 81

DUzq 0.89 0.91 0.90 0.90 0.90

AE (%) 84 84 84 84 84

Scenario I-Basin B9

q (m3 / s/ m XI, 000) 1.64 1.64 1.64 1.64 1.64

teo (min) 165 165 165 165 165

tL (min) 203 245.8 258 191.3 232.5

D min (mm) 81 65 64 74 65

DU1q 0.97 0.87 0.89 0.95 0.87

AE (%) 93 92 92 93 92

PIVB-NRCS, and PIVB-TR columns of Table 7. The row labeledAE in Table 7 is the application efficiency (Burt et al. 1997).

For the B1 basin, the predicted performance is relatively in­sensitive to changes in infiltration from the assumed averageinfiltration function (Table 7, Scenario 1). The hydraulic charac­teristics of the level-basin systems, which mask the effect of in­filtration characteristics on distribution uniformity, help explainthese results. Also, the average solution is more restrictive thanany of the estimated functions for design and operation optimiza­tion purposes because it predicts more infiltration for a giventime. Hence, q-tco recommendations generated with the averagefunction can be expected to perform well if actual infiltrationrates are lower than expected.

For the B9 basin, changes in infiltration characteristics relativeto the assumed function result in slight underirrigation at thedownstream end of the field. This happens because more waterinfiltrates as a function of time with the estimated function thanwith the average function. However, since the optimized opera­tion results in AE in excess of 90%) (Table 7, Scenario 2), therecommended q-tco combination can be improved by increasingcutoff time slightly. This will slightly lower AE but will ensurethat the requirement can be met everywhere under the range ofconditions assumed in this analysis.

when both surface volume and infiltration rates are changingrapidly.

• Parameter values are very sensitive to the data and can providefew clues about the similarity or differences in infiltrationcharacteristics among a group of tested basins. Measures suchas t lOO or the average opportunity time associated with the finalaverage depth infiltrated of a test are potentially more usefulthan the parameter values for making comparisons among testsand for generating management recommendations.

• Despite the uncertainty of results, infiltration estimates gener­ated with both the optimal and PIVB methods are useful. Thebasin irrigation tests presented here provide a measure of therange of infiltration characteristics that could be encounteredacross the tested field. An average infiltration was definedfrom the four tests and used to assess current performance andgenerate operational recommendations. Sensitivity testsshowed that predicted irrigation performance will only slightlydegrade if actual infiltration conditions deviate from the as­sumed average conditions within the range of conditionsstudied.

Notation

c

b

Dzk

qRMSETA

RMSETR

The following symbols are used in this paper:

Az functional relationships for infiltration volumeper unit length;

Ao upstream flow sectional area;a - exponent of the Kostiakov infiltration equation;

steady-state infiltration rate term of the extendedKostiakov infiltration equation;storage term of the extended Kostiakov infiltrationequation;average infiltration depth [Vz/x(t)J;transient parameter of the Kostiakov infiltrationequation;number volume balance equations used in theestimation;number of stations used in the calculations ofinfiltrated or surface volume;unit inflow rate;RMS error of advance times;RMS error of recession times;

• Optimal estimation based on multiple volume balance equa­tions and a detailed set of surface depth measurements do notyield more reliable estimates of a field's infiltration character­istics than the one obtained by PIVB under the conditions ofthis analysis. Reasons include the non-uniqueness of solutions,the similarity in predicted infiltrated depth at the average op­portunity time of the test, and small differences in opportunitytime along the field, which mitigate differences in infiltrationbehavior predicted with any of the functions estimated for theexamples presented here.

• Both the optimal and PIVB methods require inflow, outflow,advance, and recession measurements, which can be subject toerror. Optimal methods add the uncertainty of surface volumedeterminations. When dealing with measured surface volumes,small field surveyor depth measurement errors can lead toanomalous results, particularly early during the advance phase

Conclusions

586/ JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / SEPTEMBER/OCTOBER 2009

SSVz objective function, sum-of-squared-errors ofinfiltrated volume;

SUM l summation term used in computing parameter kwith the PIVB method;

SUM2 summation term used in computing parameter kwith the PIVB method;

t i time at which the balance is computed;tx advance time to distance x;

t100 Merriam-Clemmens infiltration familycharacteristic time, equal to the time (in hours)needed to infiltrate 100 mm;

VQ inflow volume;VRO runoff volume;

Vy surface storage volume;Vz infiltration volume;W basin width;x - distance along the direction of flow;z infiltration depth;e vector of infiltration parameters;

(J" y surface shape factor;'T intake opportunity time;

'TAva average intake opportunity time; and'P shape factor used in surface or infiltrated volume

calculations.

References

Bautista, E., Clemmens, A. J., Strelkoff, T. S., and Schlegel, J. L. (2009)."Modern analysis of surface irrigation systems with WINSRFR."Agric. Water Manage., 96, 1146-1154.

Burt, C. M., et al. (1997). "Irrigation performance measures-Efficiencyand uniformity." J. Irrig. Drain. Eng., 123(6),423-442.

Clemmens, A. J. (1991). "Direct solution to surface irrigation advanceinverse problem." J. Irrig. Drain. Eng., 117(4), 578-594.

Clemmens, A. J., and Dedrick, A. R. (1986). "Surface irrigation model­ing." Annual Rep., USDA-ARS U.S. Water Conservation Laboratory,Phoenix, 23-86.

EI-Haddad, Z., Clemmens, A. J., EI-Ansary, M., and Awad, M. (2001)."Influence of cultural practices on the performance of long level ba-

sins in Egypt." Irrig. Drain. Syst., 15(4), 327-353.Esfandiari, M., and Maheshwari, B. L. (1997a). "Application of the op­

timization method for estimating infiltration characteristics in furrowirrigation and its comparison with other methods." Agric. Water Man­age., 34(2), 169-185.

Esfandiari, M., and Maheshwari, B. L. (1997b). "Field values of theshape factor for estimating surface storage in furrows on a clay soil."Irrig. Sci., 17(4),157-161.

Gillies, M. H., and Smith, R. J. (2005). "Infiltration parameters fromsurface irrigation advance and run-off data." Irrig. Sci., 24(1), 25-35.

Khatri, K. L., and Smith, R. J. (2005). "Evaluation of methods for deter­mining infiltration parameters from irrigation advance data." Irrig.Drain., 54(4), 467-482.

Merriam, J. L., and Clemmens, A. J. (1985). "Time rated infiltrated depth

families." Proc., Development and Management Aspects of Irrigation

and Drainage, Specialty Con!, Irrigation and Drainage Div., ASCE,New York, 67-74.

Merriam, 1. L., and Keller, 1. (1978). Farm irrigation system evaluation:A guide for management, Dept. of Agricultural and Irrigation Engi­neering, Utah State Univ., Logan, Utah.

Scaloppi, E. J., Merkley, G. P., and Willardson, L. S. (1995). "Intakeparameters from advance and wetting phases of surface irrigation."J. Irrig. Drain. Eng., 121(1),57-69.

Strelkoff, T. S., Clemmens, A. J., and Bautista, E.(2009). "Estimation ofsoil and crop hydraulic properties." J. Irrig. Drain. Eng., 135(5),537-555.

Strelkoff, T. S., Clemmens, A. J., EI-Ansary, M., and Awad, M. (1999)."Surface-irrigation evaluation models: Application to level basins inEgypt." Trans. ASAE, 42(4), 1027-1036.

U.S. Dept. of Agriculture-Natural Resources Conservation Service

(USDA-NRCS). (1974). National irrigation handbook, Part 652,

Irrigation guide, National Technical Information Service, Washington,D.C., 754.

U.S. Dept. of Agriculture-Soil Conservation Service (USDA-SCS).

(1974). National engineering handbook, Chap. 4, Border irrigation,

National Technical Information Service, Washington, D.C., 250.Walker, W. R., and Kasilingam, B. (2004). "Correcting the volume bal-

ance equation for shape factors during advance." Proc., 2004 World

Water and Environmental Resources Congress: Critical Transitions inWater and Environmental Resources Management, ASCE, Reston,Va., 1690-1695.

JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / SEPTEMBER/OCTOBER 2009/587