validation of a predictive form of horton infiltration for simulating furrow irrigation

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Validation of a Predictive Form of Horton Infiltration forSimulating Furrow Irrigation

Jean-Claude Mailhol1

Abstract: Due to spatially varying conditions the improvement of furrow irrigation efficiency should be sought not just for a limnumber of furrows or for one specific irrigation event. A simplified predictive modeling approach of the averaged advance-infiltrprocess is proposed in this paper. Horton’s equation, derived from the asymptotic form of the Talsma-Parlange infiltration equation,us to use a predictive approach for the advance infiltration process by means of the exact solution of the Lewis and Milne waterequation. The references to the works of White and Sully, for a surface point source, result in the use of parameters which characthydraulic properties of the soil:Du ~saturated water content minus initial water content!; Ks ~saturated conductivity!; andlc ~macroscopiccapillary length!. The physical meaning of parameters involved in the proposed modeling is attested using field experiments carriin a loamy soil plot context. Assuming a sameDu measured value before irrigation for the whole of a 30 furrow sample, the averagvalues oflc and Ks obtained from calibration on the advance trajectory are comparable to those derived from local infiltration tests~diskpermeameter and double ring methods!. The applicability of the model is then extended to heavy clay soil where the parameterslc andKs still agree with the values proposed in the literature. This paper can be considered as a contribution to the development of aevaluating the impact of irrigation practices on the efficiency at the plot and cropping season scale.

DOI: 10.1061/~ASCE!0733-9437~2003!129:6~412!

CE Database subject headings: Furrow irrigation; Infiltration; Calibration; Predictions; Soil water; Simulation.

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Introduction

Furrow irrigation performance is strongly dependent on soil prerties which govern the infiltration rates. The spatial variabilityinfiltration conditions along a given furrow and between furrowcombined with the discharge inflow variability, characteristicthe water distribution system, may induce a high variability inadvance process~Mailhol and Gonzalez 1993; Or and Silva 199Mailhol et al. 1999; Schwankl et al. 2000!. Consequently, methodology based on a single furrow for assessment of a givengation event, means there is little hope of representing the mbehavior at field level and in improving surface irrigation sytems.

Amongst the different sources of variability affecting the irgation performance, some of them are deterministic and ostochastic~Mailhol et al. 1999; Schwankl et al. 2000!. Soil waterdepletion, governed by water plant consumption, is an exampa deterministic source of temporal variability affecting infiltraticonditions. But, the sources of variability are mainly stochastia modernized surface irrigation context when the plot slopeassumed to be uniform. With regards to impact on irrigation pformance criterion~Schwankl et al. 2000!, inlet discharge is veryimportant. The coefficient of variation@Cv(Q)# of the latter var-ies from 5 to 25% depending on the distribution system~Trout

1Irrigation Reseach Unit Cemagref BP 5095, 34033 MontpelCedex 1 France. E-mail: [email protected]

Note. Discussion open until May 1, 2004. Separate discussionsbe submitted for individual papers. To extend the closing date bymonth, a written request must be filed with the ASCE Managing EdThe manuscript for this paper was submitted for review and posspublication on November 6, 2001; approved on March 24, 2003. Tpaper is part of theJournal of Irrigation and Drainage Engineering,Vol. 129, No. 6, December 1, 2003. ©ASCE, ISSN 0733-9437/200412–421/$18.00.

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1990; Mailhol et al. 1999!. The infiltration rate, which is highlyvariable, is influenced by furrow geometry and, in particular, bysoil preparation~tillage, ridging furrows operation! as well as bythe presence of cracks.

Lastly, soil hydrodynamic parameters such as Ks~saturatedconductivity! noticeably vary depending on soil structure fromthe first to the second irrigation treatment as mentioned by O~1996! and Or et al.~2000! and as shown in Mailhol et al.~1999!for the irrigation of sugar cane in heavy clay soil conditionsInfiltration rates are not only influenced by soil structure differ-ences, as infiltration generally varies along the irrigation seaso~Childs et al. 1993! depending on soil moisture conditions~Mail-hol et al. 1999!. So, proposing an operative approach to establisa link between initial moisture conditions and infiltration param-eters would appear to be useful when attempting to predict infitration rates prior to irrigation events.

Modeling continues to be considered an interesting tool foresearchers seeking to improve surface irrigation systems astested by the number of models published during these last 2years. Most of them are based on numerical solutions of thSaint-Venant equations~full hydrodynamic, zero-inertia, kine-matic wave solutions! coupled with the extended Kostiakov infil-tration equation. TheSIRMODmodel~Walker 1989! is one of themost famous numerical codes. Shayya et al.~1993! proposed areview of available models, and the Schwankl and Wallendemodel ~1988! would appear to be the most useful because it accounts for spatially varying conditions~infiltration, geometry,slope, roughness! along a given furrow by means of the fixed-space method. Analytical solutions of the advance-infiltration process have been proposed by Yu and Singh~1990!, Renault andWallender~1992!, and Mailhol and Gonzalez~1993! as an alter-native to numerical approaches. This eliminates the numericproblems inherent in previous modeling approaches. These i

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volve numerous parameters that are not easy to determine in ficontexts~Or and Silva 1996!.

Reliable and efficient analytical models were derived from thexact solution of the water balance equation~WBE! initiated byLewis and Milne~1938!. Philip and Farell~1964! were the first topropose an exact analytical solutions of the WBE for the advantrajectory when the Kostiakov, Philip two terms~1957!, and Hor-ton equations are chosen, respectively. But these writers hanever validated their theoretical results from field experimen~Renault 1991!. Mailhol and Gonzalez~1993! when justifyingtheir choice of the linear infiltration equation@ I (t)5B1Cst# forcracking soils added another solution to the three proposedPhilip and Farell. Renault and Wallender~1992! with the Horton~1940! infiltration used in the ‘‘ALIVE’’ method, then Mailholand Gonzalez~1993! and Mailhol et al.~1999! for the linear in-filtration and the Horton equation in Mailhol et al.~1997!, and Orand Silva~1996! with the Philip equation, have shown that analytical solutions can satisfactorily simulate the advance infiltratioprocess. The infiltration parameters, are more easily calibratedmonitoring advance trajectory than by performing local infiltration tests in the furrow as explained in Shepard et al.~1993!.

Renault and Wallender~1992! and Mailhol et al. ~1997!showed the generic character of the advance solution derivfrom Horton’s equation in contrast with the advance solution drived from the Philip equation@ I (t)5St1/21At#. Indeed, the lat-ter does not permit satisfactory simulations of the advance trajetory in the case of cracking soils~due to an exponent value of 0.5,not compatible with the linear tendency of the advance trajetory!. Moreover, Philip’s equation in spite of the physical significance ofS: the sorptivity parameter~Philip 1957!, does not al-ways provide realistic solutions as attested by negative valuesA when calibrating the infiltration parameters from the advanctrajectory ~Berthome1985; Mailhol et al. 1997!. For instance,with advance data collected on a loamy soil plot in the Durancvalley ~southeast of France! and used in Mailhol et al.~1997!, weobtain S51.359 L/m/min0.5 and A520.005 L/m/min with Sd51.7 m as the standard error of calibration on the advance trajtory ~with R250.992). When imposing a constraint onA, such asA>0.001 for example, the calibration method~see below! yieldsS51.312 andA50.001, which is the value of the lowA con-straint. Obviously, this artifice significantly affects the quality othe calibration as is shown by Sd53.37 m. In comparison, cali-brating Horton’s equation on the same advance data~with R2

50.999, Sd50.40 m! gives a stabilized infiltration rate offering avery similar value to that obtained from (Q2Qrstab!/FL, whereQ5inflow rate, Qrstab5measured stabilized runoff~or outflow!rate, and FL5furrow length.

One of the major problems in surface irrigation is to proposefficient and operative predictive methods for simulating the avance infiltration process. This is due to the reasons mentionabove, i.e., the soil properties, and consequently, infiltration prameters, vary with time and space. The modeling of the advaninfiltration process has resulted in the development of empiricmodeling approaches with poor prediction levels have been reognized~Haverkamp et al. 1988!. The identification of the infil-tration parameters based on real time advance monitoring~Mail-hol and Gonzalez 1993! allows us to overcome some of thedifficulties previously evoked. This is especially true when aefficient calibration method offers some guaranties regarding tuniqueness of the solution~Mailhol et al. 1997!. Nevertheless, thereliability of the method is greatly affected by varying spatiaconditions, when used as an irrigation management tool. As w

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temporal evolution of the infiltration parameters this elemenmust be taken into consideration.

The objective of this paper is to propose a generic predictivanalytical model which simulates the advance infiltration procesalong an irrigation season. Our work is based on a predictive forof the Horton equation. Therefore, it will be worthwhile assessinthe utility of an adapted simulation method when estimating thimpact of the different sources of variability on the advanceinfiltration process at the plot and season scale.

Theoretical Aspects

The water balance equation~WBE! initiated by Lewis and Milne~1938!

Qt5sA0x~ t !1E0

x

I ~ t2ta~x!!dx (1)

where Qt5inflow volume ~L!; sA0 x(t)5water stored insidethe furrow at time t; *0

x I (t2ta(x))dx5water infiltrated;A05cross-sectional flow area at upstream head~m2! multiplied bya unit coefficient~1,000 L/m3!; s5water line slope coefficient~generally estimated as 0.75–0.80!; t5time ~min!; x(t)5positionof the front at timet; ta(x)5the advance time tox, Q5inlet dis-charge~L/min!; and I cumulative infiltration~L/m! as a functionof opportunity timet2ta(x) , has been widely used for dealingwith the practical aspects of surface irrigation modeling. The solution ~using the Laplace’s transform method! of Eq. ~1! for theadvance trajectoryx(t) proposed by Philip and Farell~1964! withHorton’s equation

I ~ t !5B$12exp~2rt !%1Cs t (2)

~whereB is expressed in L/mr in min21 and Cs the basic infil-tration rate in L/m/min! is given by

x~ t !5A1~12ea1t!1B1~12ea2t! (3)

where

A15Q

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CsJ(4)

a1 , a25~Cs1Br1sA0r!

2sA0

6@~Cs1Br1sA0r!224CssA0r#0.5

2sA0

TheA0 value is estimated by solving the Manning-Strickler equation for a standard trapezoidal furrow section with a bottom widthof 10 cm and a side slope of 1/2~average values usually observedin France!, the Manning coefficient being fixed at 0.05 for the firstirrigation and at 0.04 for subsequent irrigation treatments~Mail-hol and Gonzalez 1993!. As a result, theB, r, and Cs are the soleparameters to be estimated by calibration.

Predictive Form of Hortons Equation

Parlange et al.~1982! have proposed a general expression for thecumulative infiltration equation involving a weighted factorh:0,h,1. The analysis ofh variations in the vicinity of its limitsresults in finding, once more, either the Green and Ampt~1911! orTalsma and Parlange~1972! equations~Fuentes 1992!. The analy-sis of the asymptotic behavior~for large time values! of Parlangeet al. ~1982! equation gives the following equation:

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I 5Kst1S2

2KsF12expS 2x

2Ks2

S2t D G (5)

This equation has the structure of Horton’s equation in whiccalibration parameterx is proposed in order to increase its fleibility as we shall see below. So, a physical meaning can be gto parameters of Horton’s Eq.~2! as to those of the linear equatio@ I 5B1Cst#

I 5S2

2Ks1Kst (6)

resulting from even greater asymptotic behavior than which gEq. ~5!.

Using the Gardner equation~1958!, relating hydraulic conductivity K to soil pressureh,

K~h!5Ks exp~h//lc! (7)

the macroscopic capillary lengthlc is expressed. The capillarlengthlc is considered a soil characteristic~Bouwer 1966; White1988; Revol 1994; Revol et al. 1996! representing the relativeimportance between capillary and gravity. After Elrick and Renolds ~1989!, it ranges from 3 for coarse sand to 25 cm for thmaterials~clay with no cracks! and much higher~until to 100 cm!according to Kutilek and Nielsen~1994!. Using the integral vari-able known as the Kirchoff transform or the matrix flux potentfunction ~Gardner 1958!:

f~h!5Ehi

h

K~h!dh, h>hi (8)

where hi5initial pressure@u i5u(hi)#, and replacingK(h) byEq. ~7! yields K5f/lc whenKi is neglected and in particular

lc5fs /Ks (9)

According to the simplified relation proposed by White and Su~1987! between matrix flux potentialfs and sorptivityS

fs5bS2

us2u i(10)

and replacingfs in Eq. ~9! yields

lc5bS2/Ks~us2u i ! (11)

With values ranging from 1/2~in the Green and Ampt modeling approach! to p/4 ~Philip 1969!, b is linked to the parametegoverning the exponential variation of diffusivityD(u) Thonyet al. ~1991!. A value of 0.55 as proposed by White and Su~1987! can be used for a soil in place or stabilised~disappearanceof the tillage effect!. The previous equations give a predicticharacter to Eq.~2!. Replacing Eq.~11! in Eq. ~5! the Hortonequation becomes

I ~t!50.9lcDu~12exp@2xKs~lcDu!21t#!1Ks•t (12)

wheret5opportunity time; andDu(cm3/cm3)5us2u i ~saturatedsoil water content minus initial soil water content!. In order toexpress the cumulative infiltration in L/m in compliance with tapplication of Eq.~1!, Eq. ~12! is multiplied by the interfurrowspacing value Fs. Inverting Eq.~3! for opportunity time calcula-tion, Eq.~12! enables the infiltration profile along the furrow to bcalculated.

The fact that parameters of a conceptual modeling approcan be considered as parameters having a physical meaning~es-pecially those governing water transfer in the porous media! is

414 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE


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obviously an interesting ascertainment for prediction purposNevertheless, the interest of the modeling approach involving E~12! has to be confirmed using field tests.

Model Calibration Using the Inverse Method

The three infiltration parameters B,r, and Cs, in Eq.~2! can bederived from the calibration of Eq.~3! according to the optimiza-tion process described in Mailhol et al.~1997! where the ALIVEmethod~Renault and Wallender 1992! is also discussed. With thecalibration method the sum of the square deviation betweenserved and simulated advance trajectory is minimized, under cstraints, as an objective function. The Rosenbrock meth~Rosenbrock 1960! is used to optimize the parameters under costraints. Those imposed to Cs agree with the order of magnituof saturated conductivity Ks for the considered soil type. Mostthe solutions proposed by the optimization algorithms at the eof an iterative process~at least 100 to 150 iterations for eachattempt! are sensitive to initial parameter values. Our searmethod is replicated ten times, from the lower to the uppboundary constraint by increasing by VR/10 the initial paramet(Cs0) at each attempt, VR being a possible variation rang@23Cs0 to 3Cs0] of Cs. The calibration effort is mainly focusedon Cs because the reliability of the infiltration prediction, thfurther along the advance phase, is governed by this paramparticularly for significant differences between cut-off time anadvance time~Renault and Wallender 1992; Mailhol et al. 1997!.

Validation of the Predictive Modeling Approach

In this section, we focus on the robustness of the predictive aspof the model. Here we have to answer to the following questioCan we find a set of parameters~Ks, lc, andDu! allowing us tocorrectly simulate the advance-infiltration process while beingagreement with that characterizing the hydraulic properties of tsoil on which the model is applied? We will address the questiby first dealing with the case of a loamy soil plot where differentypes of field experiments were carried out. Then, we shall shthat the predictive approach can be extended to a heavy claywith cracks.

On a Loamy Soil Plot

The experimental plot, is located in the Cemagref InstituteMontpellier, France~43°N,3°508E!. The data collection proceduredescribed in Mailhol et al.~2001!, is reproduced here for thewater criteria only and in the case of the Ta treatment~whereusual nitrogen doses were applied! furrow irrigated on FL5130 mlength with corn~Samasara variety! sown in 1999 on May the26th in rows 0.8 m apart. This treatment area was composedaround 35 furrows. A soil analysis performed at its center is prsented in Table 1. The whole of the plot furrow irrigated was lasleveled in February to a slopeS050.25%. Ridge-furrow tillagewas used with a spacing between ridges of 0.8 m and a mevalue of 0.16 m for the difference in elevation between the furrobottom and the ridge. A gated pipe water supply system was usAt the pipe entrance, a tank of 1 m3 enabled constant hydraulichead and discharges close to 20 L/s.

The surface irrigated plots were only watered three timesthere were some rainfall events. A few days after furrow ridginTa received 76 mm~averagedQ51.2 L/s, tco5110 min for the

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first ~8th of July! and 67 mm ~averaged Q50.73 L/s, tco
5160 min) for the second~25th of July! and 53 mm (Q50.77 L/s; tco5120 min) for the third irrigation~25th of August!,respectively. Generally the water distribution was cut off whethe water front advance had reached the end of the furrows. Tclosed-end furrow~CEF! irrigation technique was used. Duringan irrigation event the inlet discharge of each furrow was mesured using a portable flume and the advance front of the irrigafurrows was monitored for the stationsx520, 30, 50, 70, 90, 110,and 130 m. Thirty furrows were monitored at a time except fthe first irrigation event~only 15 furrows!. Three sites equippedwith neutron access tubes for soil water content estimations wpositioned upstream at (x530 m) central~or mid at x565 m),and at downstream sites (x5115 m) along the central furrow.

The monitoring of the infiltration profile was made possiblwith a series of 30 cm length CS6115 Campbell TDR~time do-main reflectrometry! probes inserted in the soil at four differendepths~30, 60, 90, and 120 cm soil depth! on the midsite of thecentral furrow. Lastly, bulk density measurements were carriout using the Gamma densimeter~both the surface and the depthapparatus!.

Supplementary Field Experiments

During the summer of 2000 and a few days after the wheat hvest~sown in December 1999! infiltration tests for hydraulic char-acterization were performed using the double ring method nthe midsite of Ta. Cumulative infiltration was monitored in eacring by maintaining a water depth of 5 cm at the surface ofinitially dry soil.

In 2001 similar fields experiments to those of 1999 were caried out on the same plot. One of the objectives in 2001 wasstudy the impact of nitrate leaching on crop yield due to ovirrigation. Five furrows~F75 to F79! of Ta were irrigated sixtimes whereas the other furrows received only three irrigatitreatments as in 1999. Bearing in mind the objectives of tpresent study we have used data from the previous study concing the advance of F75 to F79 and the initial water content mesured at upstream and downstream~no midsite exits on this sub-treatment!.

Parameters Estimation From the Advance Monitoring

Advance variability is somewhat high as attested by the resuobtained after the third irrigation treatment presented in Tablewith the coefficient of variation of the advance time, Cv~TL!,being, 20, 22, and 19%, for the first, second, and third irrigatitreatments, respectively. Most variability is due to soil preparati~furrow ridging particularly! because discharge inlet variability islow ~6 and 5%, for the first and third! except for the second

Table 1. Soil Distribution Particles on Central Neutron Probe Site

Layer~cm!

Clay ~%!0–2 mm

Loam ~%!2–50mm

Sand~%!50–200mm

Sand~%!200 mm–2 mm

0–38 16.4 42.0 33.4 5.938–68 18.6 43.6 31.7 7.168–98 20.2 47.4 23.7 8.798–128 20.2 54.6 15.0 7.2

128–158 26.7 48.0 8.8 16.4

Note: An averaged value of 0.72% has been taken for organic carcontent.

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~14%!. But, the analysis of the advance variability does not pemit the impact of the engine traffic~when ridging the furrows! tobe clearly shown.

The results of the calibration method from the advance montoring are presented in Table 3 for the third irrigation event. Wcan observe that fixing ther parameter to its average value doesnot affect the quality of the calibration the average value of theR2

being always greater than 99%. This leads us to adopt a value15 for the empirical parameterx in Eq. ~5!, this parameter reflect-ing probably the importance of the transient infiltration processthe furrow scale. As a result, we can consider three paramete

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Table 2. Advance Monitoring~in min! of 30 Furrows for IrrigationNO 3 with Measured Inlet Discharge on Ta~1999!

Furrownumber

x ~m!

20 30 50 70 90 110 130 Q (L/s)

52 7 12 23 38 52 67 96 0.7853 6 11 23 34 49 59 70 0.7854 7 12 26 41 61 82 120 0.8355 8 13 24 36 50 64 84 0.7756 6 10 21 34 48 51 72 0.7757 7 13 24 36 51 66 83 0.7858 7 11 21 32 44 60 83 0.7459 6 11 21 31 44 58 84 0.7860 7 11 23 34 46 62 84 0.7861 6 9 19 28 38 49 60 0.7762 8 13 24 37 51 70 95 0.7863 8 13 24 34 47 58 75 0.7564 10 16 29 49 64 92 125 0.7465 7 12 23 34 48 60 80 0.7466 8 14 25 36 51 65 93 0.7867 8 13 23 33 44 54 71 0.7568 7 11 20 30 41 53 71 0.7769 7 11 20 28 39 49 63 0.7870 7 12 23 34 45 59 84 0.871 8 13 22 33 44 55 69 0.8672 8 13 25 36 49 63 80 0.7473 6 11 21 30 41 54 73 0.8374 8 12 25 42 62 83 124 0.8375 8 12 21 32 43 58 91 0.8376 6 13 23 35 49 63 84 0.7577 7 11 21 32 45 58 80 0.7378 7 11 20 30 43 52 68 0.7379 7 12 24 35 49 63 90 0.7380 7 11 22 35 45 60 86 0.7888 7 13 28 40 56 74 97 0.73Average 7 12 23 34 48 62 85 0.77Cv~%! 12 11 10 12 13 16 19 5

Table 3. Calibration on Advance Trajectory~on 30 Furrows! forIrrigation 3 on Ta in 1999

r settled to 0.2 mn21 r calibrated m50.185, Cv515%

B515 L/m Cv514% B516 L/m Cv517%

Cs50.18 L/m/min Cv530% Cs50.17 L/m/min Cv528%Sd52.0 m Cv535% Sd52.3 m Cv536%R250.9965 Cv50.5% R250.9960 Cv50.6%

Note: Average values ofB, and Cs; Cv5coefficient of variation,R25determination coef, expressed as the Nash criterion, Sd5standarderror of calibration on the advance trajectory.

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only: lc Ks andDu. Note that parameterx permits us to use theHorton equation in heavy clay soil with cracks instead of tlinear infiltration @Eq. ~6!#. It is therefore necessary to settlex tohigher values,~see below!, for reducing the transient effect~notcompatible with the macropore effect! of the infiltration due to thelow value of the ratio Ks/lc under heavy clay soils.

From Parameter B of Eq. ( 2) to Macroscopic CapillaryLength lc

Averaged initial soil moisture conditions (u i51/z*0Z51mq(z)dz)

established from the neutron probe measurements performethe central site are plotted in Fig. 1. We can observe that inisoil conditions of the third irrigation are drier than those of thtwo previous irrigation. As attested by Fig. 2 initial soil condtions are not very different along the furrow, the significanthigher value at the downstream site reflecting the impact of Cpractices as shown in Mailhol et al.~1999!. The averaged valuesof u i calculated on the first soil meter depth, where the plamainly uses up the water, are 0.23, 0.21, and 0.17 for first, sond, and third irrigations, respectively. The value ofus before thefirst irrigation is 0.4 cm3/cm3 according to the bulk density value~Bd! of 1.4 Mg m23 ~Revol 1991; Revol et al. 1996!. Soil com-paction due to subsequent surface irrigation events has adocumented impact~Berthome 1991; Or 1996; Mailhol et al.1999! on hydraulic soil properties. Changes in compaction levare more significant between the first and the second irrigatevents~Berthome1991!. Existing empirical formulas allow someparameters, that characterize the water transfer, to be adaptenew soil conditions particularly those involving bulk density. Fo

Fig. 1. Averaged value of initial soil water contentu i before eachirrigation event on different sites along central furrow

Fig. 2. Initial soil moisture conditions of Irrigation 3 along centrafurrow

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instance, the Verrecken pedotransfer functions~Verrecken et al.1989; Verrecken et al. 1990! can be used to updateus and Ks tonew Bd values:

us50.8120.283 Bd10.001 ~Clay! (13)

Ln~Ks)520.6220.96 Ln~Clay)20.66 Ln~Sand)

20.46 Ln~Carb)28.43 Bd (14)

where Carb5organic carbon soil content~as for clay and sand,expressed in %, Ks being expressed in cm/day!. Using an aver-aged value for Bd of 1.5 for the first meter depth, obtained fromeasurements performed at the end of the irrigation season,can reasonably propose a value of 0.38 forus after the first irri-gation. So, using Eq.~12! with Fs50.8 m, Du50.21, and thevalue of 15 L/m forB ~see Table 3!, an averaged macroscopiccharacteristic capillary length value of around 10 cm can be drived. This lc value is very close to that obtained by Revo~1994! and Revol et al.~1996! on the same plot using the diskpermeameter method~Perroux and White 1988!. Two methodswere used during field experiments on the nodes of a 4 m38 mgrid. The first method consists of creating pressure variatiowhereas the second method relies on sorptivity estimation durthe transient infiltration rate and the flux estimation during thsteady infiltration rate~Smetten and Clothier 1989!. The coeffi-cient of variation oflc derived from the previous method~localinfiltration tests! is of 35%. Moreover the average values oflcobtained after the first irrigation agree with those proposedWhite and Sully~1987!, in the 10 cm range for a loamy soil~Table 1!.

The averagelc values was obtained assuming thatDu is thesame for the whole of the 30 furrows sample. That seems a reistic assumption when one considers that, unlike with crackinsoil with a permanent crop~sugar cane! ~Mailhol et al. 1999!,advance velocity decrease is not significantly affected by tchange in soil water content along the last few meters of a furroassociated with CEF practices. Moreover, plant water consumtion tends to establish homogeneous initial soil conditions. Wshould note that what we wanted to show here was that thelcvalues derived from the advance calibration at the furrow scalerelatively close to those obtained from a reduced scale~that of thedisk! according to a mechanistic solution for unsaturated steastate flow initiated by Wooding~1968! for the axisymetric geom-etry. Nevertheless, thelc identification based on advance procesmonitoring, leads us to believe that variability also results fromphenomenon other than those specific to infiltration. These phnomenon~furrow geometry variability for example! have notbeen taken into account in our simplified modeling approach~fur-row geometry variability for instance!. Lastly, the 30lc values fita Gaussian distribution as in Revol~1994! and as Parameter B ofthe linear equation in Mailhol and Gonzalez~1993!.

From Cs Parameter to Ks

The Cs parameter is a calibration parameter. But, it is of obviointerest to link the averaged Cs value of Table 2 to the Ks paraeter of our loamy soil plot. The pedotransfer functions@Eq. ~14!with Ks calculated in cm/day# proposes a realistic Ks value. In-deed, with Bd51.4, we find Ks51.8 cm/h which is of the sameorder of magnitude~Ks52.8 cm/h with Bd51.4! as the valueobtained by Revol~1991! on the same part of the experimentaplot using the internal drainage method~Vachaud et al. 1978!.Although for the 15 first cm depth only, the median value of Kderived from the multipressure method by Revol~1994! is 2.4

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cm/h ~Bd51.4! with a coefficient of variation of 35%. The valuof Ks derived from the classical double ring method performedJuly 2000 in the area of the midneutron probe site, equalscm/h ~with Bd51.5! as attested by the graph plotted in Fig.This Ks value is deduced from the slope~0.225 gives 1.4 cm/h! ofthe line I /sqrt~t)5 f @sqrt(t)# obtained from the weighted~by therespective areas of the rings! cumulative infiltration of each ring~central and lateral ring!. One can also argue in favor of this Kvalue when adapting Eq.~14! to local conditions~by adapting theresidue of the regression equation: 20.62 to 21.17 usingcouple Bd51.4 and Ks52.8!. The result obtained is Ks51.4 cm/hwith Bd51.5.

The value of 2.4 cm/h found above by Revol~1994! is veryclose to the averaged Ks value obtained from the first irrigattreatment by converting Cs in L/m/min into Cs in cm/h forspacing furrow of 0.8 m. Similarly, the averaged value of50.18 L/m/min~Table 3! gives 1.35 cm/h. This value that can btaken as the Ks of the individual irrigated furrow after the fiirrigation treatment. It appears that, at the furrow scale, theverse method results in a Cs value which concurs with thevalue of the soil as suggested in Shepard et al.~1993! for theparameter representing the stabilised infiltration rate in thetended Kostiakov infiltration.

These results raise questions about the importance of thetidimensional nature of the infiltration process that is supposepredominate in the individual furrow scale. The multidimensioninfiltration process which prevails in the disk permeamemethod should concur more fully with the assumed twdimensional~2D! furrow infiltration process than with the theoretical 1D infiltration process of the double ring method. Thseems to be strengthened by the fact that averaged paramderived from the disk permeameter method, are close to thobtained with the inverse method in spite of the differencetween the two analysis scales~few cm for the disk, the furrowlength for the inverse method!. Despite the fact that the 1D chaacter of the infiltration in our double ring experiment is questioable, it is interesting to note that the averaged Ks value derifrom the inverse method is similar to that obtained from tdouble ring method.

The 30lc values fit a Gaussian distribution as the 30 Log~Ks!values. Although the correlationlc versus Ks is weak, its tendency is noticeably negative. It is worth noting this with a viewthe application of a modeling approach to simulate the variabof the advance infiltration process by means of the well kno

Fig. 3. Infiltration test in Summer 2000 using double ring metharound midsite of treatment Ta. Intermediary ridge is relativeweighted cumulative infiltration of each ring.

JOURNAL OF IRRIGATION AND DRA


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Monte Carlo method~Mailhol et al. 1999! assuming that inletdischarge Q fits a Gaussian distribution.

The averaged advance trajectory is correctly simulated usinthe following parameters;lc510 cm, Ks51.4 cm/h, andDu50.21 as shown in Fig. 4, the simulated water application dep~WAD! at the midsite (x565 m) equates to 42 mm using Eq.~12!. This value is close to that obtained from~TDR! ~43 mm!and neutron probes~40 mm! measurements made 24 h after irri-gation at the midsite of the plot. Lastly, we can note that thaveraged WAD delivered to the plot is 53 mm~0.77360831208/130 m/0.8 m!. So, the value of around 40 mminfiltrated to a point located just before the beginning of theflooded part~the CEF impact! is a realistic value.

Under our loamy soil context, the averaged value of Ks obtained from the second irrigation is very similar~Ks51.45 cm/h!to that obtain from the third irrigation treatment. In contrast, thlc value obtained for the second irrigation treatment is significantly higher~14 cm with Sd52.7 cm!. However it is signifi-cantly lower than the value obtained for the first irrigation (lc520 cm, with Ks52.4 cm/h!. Note that the value of 0.55 chosenfor the White and Sully parameterb in Eq. ~10!, corresponds tothe value of a soil ‘‘in place’’~i.e., stabilized!.

Unfortunately, only three irrigation treatments were carried ouin 1999. In order to validate the value of Ks andlc found for thethird irrigation treatment and to assess the role played by initi

Fig. 4. Advance trajectory for Irrigation No. 3 of Ta treatment in1999. Comparison between measurements and simulation (lc510cm, Ks51.4 cm/h, andDu50.21!.

Fig. 5. Initial soil moisture conditions before 4th~IC4! and 5th~IC5!irrigation events on over irrigated treatment of Ta in 2001 at upstreaand downstream sites of plot

INAGE ENGINEERING © ASCE / NOVEMBER/DECEMBER 2003 / 417

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Fig. 6. Validation of parameters (lc510 cm and Ks51.4 cm/h! on advance phase trajectory~a! of 4th ~Du50.26 andQ50.55 L/s! and~b! 5th~Du50.20 andQ50.92 L/s! ~b! irrigation of over irrigated treatment in 2001
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soil conditions we tested the model on the fourth and fifth irrigtion treatments of the over irrigated set of furrows in 2001 on tsame plot. Initial soil conditions~Fig. 5! are averaged on the firsmeter depth and for the two neutron probe sites, givingDu50.26for the first example andDu50.20 for the second one. The valueof inlet dischargeQ are 0.46, 0.54, 0.57, 0.62, and 0.57~averagedQ50.55 L/s) for furrow numbers 75–79 for the fourth and 0.80.94, 0.91, 0.99, 0.87~averaged Q50.92 L/s!, for the fifth irriga-tion respectively. The results of the validation presented in Figattest to the utility of the predictive modeling approach in oloamy soil plot.

Under a Heavy Clay Soil with Cracks

In this section, our objective is to show that the predictivadvance-infiltration modeling approach can be extended toheavy clay soil with cracks. To this end, we shall take the eample of the experimental plot located in the Gharb valley~65–70% of clay, 20–30% of sand! in Morocco near Kenitra. Theexperimental setup, made up three sugar cane subplots~laser lev-eled: S050.2%) furrow irrigated using siphons~FL5175 m!,floppy pipes~FL5230 m! and gated pipes~FL5230 m!, is de-scribed in Mailhol et al.~1999!.



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The value of Ks for this soil after the first irrigation afterplanting is 0.24 cm/h. This value, in agreement with that proposby Eq. ~14! for a Bd value of 1.5 g/cm3, is derived from advancecalibration on samples of 30 furrows collected during differenirrigation events from 1996 to 1998~averaged Cs50.06 L/m/min→0.24 cm/h for a furrow spacing of 1.5 m!. In the paper, itwas shown that advance velocity increases significantly whenfront reaches the part of the plot influenced by flooding resultinfrom the CEF practices in heavy clay soil conditions. Using thsolution proposed by Renault and Wallender~1994! for advancesolutions under heterogeneous soil conditions, due to the variaity of the initial soil water content as illustrated in Fig. 7, theaveraged advance process is well simulated~Fig. 7! with mea-sured values ofDu150.395 (us50.48) for the first 150 furrowlength,Du250.06 for the last 80 m~generally affected by flood-ing!, and for the whole of the furrow length, Ks50.24 cm/h and alc570 cm, the latter resulting from calibration on the averageadvance trajectory ~Fig. 7! of Irrigation 3 with Q51.54 L/s@Cv(Q)515%]. The data concerned is presented iTable 3 of the cited article. Note that in heavy cracking soil thmodel resulting from the choice of the linear infiltration, Eq.~6!correctly simulates the advance trajectory. The condition requir

Fig. 7. Model calibration~b! under closed-end furrow practice in heavy clay soil with cracks,Du1 andDu2 are derived from measurement~a!,~Q51.54 L/s, Cv~Q!515%,lc570 cm, and Ks50.24 cm/h!


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Fig. 8. Model validation under free draining practice in heavy clay soil with cracks~b!, ~Q50.65 L/s, Cv~Q!511.6%,lc570 cm, and Ks50.24cm/h!, measuredDu50.044~a!

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to maintain the quality of the simulation, when using the modderived from Eq.~5! is to adopt a higherx value~x540! than thatsettled for soils subject to low or non cracking phenomenon~x515!. The role played byx in the flexibility increase of Eq.~5! isunderlined here.

A validation of the previous couple~Ks50.24 cm/h and alc570 cm! is proposed using data collected on the beginningJuly 1998 for an irrigation event performed under relatively wsoil conditions~Fig. 8!. During this irrigation season, the freedraining furrow method was used. The advance monitoring offurrows sample of the plot irrigated using the siphons techniquepresented in Table 4. Averaged discharge inlet wasQ50.65 L/swith Cv(Q)512%, and cut-off timetco54508. Runoff wasmonitored from the beginning of the storage phase for 22 furro~one furrow on 2! and for the stabilized phase of runoff only fothe other furrow giving an estimation of Ro545% for the mea-sured runoff losses with a Cv~Ro!527%. As shown in Fig. 8, theaveraged advance trajectory is correctly simulated using a msured valueDu50.044 ~left part of Fig. 8! with the couple Ks50.24 cm/h and alc570 cm.

To give an other example of model applicability one can refto the article of Mailhol and Gonzalez dealing with the caseloamy clay soil~35% clay, 55% loam, and 10% sand! for which alc525 cm @in agreement with the values cited by Elrick anReynolds~1989! and White and Sully~1987! for a clay soil# withKs50.7 cm/h allow satisfactory simulations of the averaged avance trajectory TL in a low variability context@Cv~TL!,9%#according to an estimation on the value ofDu ~Du50.25! basedon the standard soil water balance.

Table 4. Averaged Advance Time and Coefficients of Variation~Cv!for Siphon Plot of Experimental Site in Gharb Valley

x ~m!

Furrow 25 50 75 100 125 150 175Q (L/s) Ro ~%!

Average 23 51 82 111 141 175 221 0.65 43%Cv~%! 29 25 21 19.5 17.7 17 17.5 11.6 27%



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Summary and Conclusion

An operative modeling approach for predicting the advancinfiltration process under furrow irrigation through the irrigatioseason is proposed. The Horton equation, as an expression oasymptotic form of Talsma-Parlange equation, with theS ~sorp-tivity ! and Ks parameters characterizing the unsaturated pormedia, provides an exact WBE solution for the advance traject~Philip and Farell 1964!. Using the works of White and Sully’s~1987! for a surface point source, the advance infiltration proceinvolves the followings parameters:Du ~saturated soil water con-tent minus initial soil water content!; Ks ~saturated hydraulic con-ductivity!; andlc ~capillary length!. These criteria are recognizedto be representative of the hydraulic properties of a given soi

In order to check if the parameters involved in our simplifiemodeling approach can be linked with ‘‘measured’’ paramete~or parameters having a physical meaning!, the inverse method,which consists of deriving the infiltration parameters from thadvance process monitoring, was used in a loamy soil. The aaged parameters values of Ks andlc obtained from a sample of30 furrows are very close to those derived from local infiltratiotests ~disk permeameter method and internal drainage meth!and match the characteristic values proposed by the literaturelcin the range of 10 cm for a loam!. The applicability of the pre-dictive approach highlighted in our loamy soil modeling was thsuccessfully applied to a heavy clay soil with cracks. Althoughhas not been demonstrated in this work, the applicability on tmodeling approach to silty soils is questionable due to the fthat this soil type has a tendency to develop a surface cthereby considerably reducing infiltration during the advance pcess. In that soil type the drier the soil the higher the advanvelocity until the crust is broken inducing a drop in velocity of thadvance front. Proposing threshold values for Ks versusu couldbe a means to further broaden the applicability of the propomodeling approach.

Acknowledgments

The writer is grateful to the assistance of engineers and teccians of the ORMVAG of Kenitra~Marocco! in the data collec-


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tion under the context of the Gharb valley. The writer is alsgrateful to the very constructive criticisms of an anonymoureviewer.

Notations

The following symbols are used in this paper:A 5 basic infiltration rate in Philip’s equation;

A0 5 cross-sectional flow area at upstream head;B 5 1st parameter of Horton’s equation;

Cs 5 basic infiltration rate~2nd parameter in Horton’sequation!;

Fs 5 interfurrow spacing;h 5 soil water pressure head;I 5 cumulative infiltration;

Ks 5 saturated conductivity;Q 5 furrow inflow rate;

Qrsatb5 stabilized runoff discharge;Ro 5 percentage of losses through runoff;

S 5 sorptivity;S0 5 field slope;

t 5 time;Ta 5 treatment having received usual fertilizer

application;tco 5 cut-off time;

x 5 distance in direction of flow~abscissa!;b 5 parameter of White and Sully equation linked to

diffusivity properties of soil;Du 5 us2u i ;u i 5 initial water content;us 5 saturated water content;lc 5 macroscopic characteristic capillary length;

r 5 empirical parameter governing transient phase inHorton’s equation;

s 5 waterline slope coefficient;t 5 intake opportunity time;f 5 matric flux potential;

fs 5 matric flux potential corresponding to saturatedconditions; and

x 5 empirical parameter in Horton’s equation accordingto asymptotic form of Parlange equation.

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