ANALYSIS AND DESIGN OF BORDER IRRIGATION · PDF fileTransactions of the ASAE Vol. 48(5): 1751−1764 2005 American Society of Agricultural Engineers ISSN 0001−2351 1751 ANALYSIS

Transactions of the ASAE

Vol. 48(5): 1751−1764 � 2005 American Society of Agricultural Engineers ISSN 0001−2351 1751

ANALYSIS AND DESIGN OF BORDER IRRIGATION SYSTEMS

D. Zerihun, C. A. Sanchez, K. L. Farrell-Poe, M. Yitayew

ABSTRACT. Application efficiency (Ea) is the primary criterion for border irrigation design and management. The objectiveof this study is to analyze the behavior of the application efficiency function of border irrigation with respect to border length(L) and unit inlet flow rate (qo), given a target minimum application depth. The results show that the application efficiencyfunction is unimodal with respect to L and qo. Optimality conditions are derived for both the Ea(L) and Ea(qo) functions, basedon which simple rules that reduce the design and management procedure into a series of one-dimensional optimizationproblems with respect to qo are developed. The proposed procedure has a variable bounding step in which the feasible rangesof L and qo are determined. This is then followed by a step wherein alternative approximate optimum values of Ea(qo) arecalculated for each of the feasible values of L. Finally, the optimal Ea(qo) is selected from the available alternatives basedon sensitivity analysis and other locally pertinent practical criteria. In addition, the advantages and limitations ofadvance-phase and post-advance-phase inflow cutoff options and their effects on system design and management arediscussed. The distance-based (advance-phase) inflow cutoff option offers two main advantages over post-advance-phasecutoff: operational convenience, and a lower degree of sensitivity of design and management prescriptions to inaccuraciesin inflow measurements and to non-uniformities in the distribution of inlet flow over the width of the border. However, theresults of the study also show that, depending on the parameter set, there exist limiting conditions that preclude theapplicability of the distance-based cutoff criterion in border irrigation management. Even when the distance-based inflowcutoff criterion is feasible, the corresponding design and management scenario can be sub-optimal, in which case anear-optimum operation scenario can be realized only with post-advance-phase inflow cutoff.

Keywords. Border irrigation, Design, Management, Optimum application efficiency.

order irrigation is widely used to irrigate close-growing crops that are susceptible to stem and/orcrown injuries when exposed to prolonged inunda-tion. Properly designed and managed border strips

can apply irrigation water at high levels of efficiency and uni-formity and with minimal adverse effects to the environment.The objective of border irrigation design is to maximize ameasure of merit (performance criterion) while minimizingsome undesirable consequences. The performance criterioncould be economic or physical. In either case, mathematicalmodels are used as design and management tools to relate theselected performance criterion with the decision variables.

Widely used and relatively well tested surface irrigationmathematical models, such as SRFR (Strelkoff et al., 1998),

Article was submitted for review in October 2004; approved forpublication by the Soil & Water Division of ASABE in July 2005. Presentedat the 2002 ASAE Annual Meeting as Paper No. 022177.

The authors are Dawit Zerihun, Former Assistant Research Scientist,Department of Soil, Water, and Environmental Sciences, University ofArizona, Tucson, Arizona; Charles A. Sanchez, Professor and Director,Department of Soil, Water, and Environmental Sciences and YumaAgricultural Center, University of Arizona, Yuma, Arizona; Kathryn L.Farrell-Poe, ASABE Member Engineer, Associate Professor and WaterResources Extension Specialist, Department of Agricultural andBiosystems Engineering and Yuma Agricultural Center, University ofArizona, Yuma, Arizona; and Muluneh Yitayew, ASABE MemberEngineer, Professor, Department of Agricultural and BiosystemsEngineering University of Arizona, Tucson, Arizona. Correspondingauthor: Charles A. Sanchez, Professor and Director, Department of Soil,Water, and Environmental Sciences and Yuma Agricultural Center,University of Arizona, 2149 27th Way, Yuma, AZ 85364; phone:520-782-3836 ; fax: 520-782-1940; e-mail: [email protected].

can accurately simulate processes in irrigation borders byusing either the zero-inertia or the kinematic-wave models,depending on the border bed slope. While SRFR is wellsuited to solving problems that involve system evaluation, itslack of optimal search capability limits its utility as a designand management tool. Simplified solutions that relate borderirrigation performance indices with dimensionless variableswere developed based on the zero-inertia model (Yitayewand Fangmeier, 1984; Strelkoff and Shatanawi, 1985; ElHakim et al., 1988). Site and irrigation specific charts andequations that relate performance indices with pertinentindependent variables were proposed by Reddy (1980),Shatanawi and Strelkoff (1984), and Holzapfel et al. (1986).Yitayew and Fangmeier (1985) used dimensionless curves(Yitayew and Fangmeier, 1984) to develop a procedure forthe design of the reuse system of border strip irrigation. Thedimensionless solutions of Strelkoff and Shatanawi (1985)form the basis for a border irrigation system design andmanagement program, called BORDER, developed byStrelkoff et al. (1996). For infiltration events that canadequately be modeled using the single-term Kostiakov andthe NRCS equations, BORDER can be a useful design andmanagement tool. However, in cases where more generalinfiltration functions are most appropriate, in order toaccurately describe the infiltration process, BORDER cannotbe used as a design and management aid.

Optimal design approaches that use economic cost/benefitcriteria as the objective function were proposed for borderirrigation systems by Reddy and Clyma (1981) and Holzapfeland Marino (1987). However, these approaches are notwidely used, mainly because they are data intensive and theyinvolve relatively complex solution techniques. Design and

B

1752 TRANSACTIONS OF THE ASAE

management procedures based on physical performancecriterion have a minimal data requirement and are amenableto simpler solution techniques. Both economic and environ-mental rationales suggest that, among the physical perfor-mance indices, application efficiency is the primary surfaceirrigation systems design and management criterion (Zerihunet al., 2001). Widely used border irrigation system designapproaches, such as those of Hart et al. (1980) and Walker andSkogerboe (1987), use irrigation performance as the designcriterion. Nonetheless, these procedures, generally, empha-size the development of a feasible-yet-satisfactory, instead ofoptimal, design.

This article presents analyses of the application efficiencyfunction of border irrigation systems. The type of borderconsidered here is a graded and free-draining border withoutcross-slope and with no furrows. Soil and surface roughnessare assumed homogeneous throughout the border, and inletflow rate is considered to be uniformly distributed over theborder width. The analyses show that the applicationefficiency (Ea) of a border irrigation system is unimodal withrespect to length and unit inlet flow rate. Based on theseresults, optimality conditions are derived for the Ea(L) andEa(qo) functions. The advantages and limitations of advance-phase and post-advance-phase inflow cutoff options and theireffects on design and management are discussed. Finally, thearticle proposes a simple design and management procedurefor graded, free-draining border irrigation systems.

DESIGN AND MANAGEMENT CRITERIA AND

VARIABLESConsidering the type of border described above, the

performance of a border irrigation event can be evaluatedusing three different indices: efficiency (application efficien-cy, Ea [%]), adequacy (water requirement efficiency, Er [%]),and uniformity (distribution uniformity, Du [−]). Ea, Er, andDu can be expressed as (Zerihun et al., 1997):

100

0

0 0

∫

∫ ∫ +−=

co

ov

tod

Lovr

L

adtqC

LZZdxZdxE (1)

1000 0

LZ

LZZdxZdxE

r

Lovr

L

r

ov∫ ∫ +−= (2)

∫

==Lmin

av

min

Zdx

LZ

Z

ZDU

0

(3)

where L = border strip length (m), Lov = length of the borderreach over which the infiltrated amount equals or exceeds Zr(m), Cd = unit conversion factor (10−3 m3/L), qo = unit inletflow rate (L/min/m), tco = cutoff time (min), Z = infiltratedamount (m3/m), Zr = net irrigation requirement (m3/m),Zmin = minimum infiltrated amount (m3/m), and Zav = aver-age infiltrated amount (m3/m). Economic and environmentalrationales suggest that application efficiency is the primaryperformance criterion in the design of surface irrigation sys-tems (e.g., Zerihun et al., 2001). With Ea as the performancecriterion, the border irrigation design problem can be posedas:

( )( )

( )( )

≤

≤−

=−

0and

,0

,0St.

Max

cooi

coomin

coominr

cooa

,tL,qC

,tL,qDUDU

,tL,qZZ

,tL,qE

(4)

where DUmin = minimum acceptable level of distributionuniformity (−), and Ci represents a set of constraints that canbe categorized as variable bounds, conservation-like, andmanagement related. A complete list of these constraints isgiven by Zerihun et al. (1999). Note that the first constraintimposes a restriction on the minimum cumulative infiltrationand target Er. The constraints can be implicitly embeddedwithin the hydraulic simulation model or explicitly enforcedby the optimization algorithm, depending on whether aphysically based model or explicit empirical functions areused to evaluate the terms in the constraint functions. In thisstudy, a simulation model is used to evaluate the terms in theconstraint functions; hence, most of the constraints need notbe enforced explicitly. For reasons of simplicity, the onlyconstraint that is explicitly considered in the current analysisis the requirement on Zmin. Note that the above formulationconsiders unit inlet flow rate (qo), border length (L), and cut-off time (tco) as design variables. While distance-based cutoffcriterion is widely used in border irrigation management, cut-off distance can always be expressed in terms of an equivalentcutoff time. Thus, a time-based inflow cutoff criterion is themore general of the two and is used here.

The determination of border width is an importantelement of the physical design of irrigation borders. Howev-er, the study presented here is based on a one-dimensionalflow analysis; hence, border width is selected as a functionof available flow rate at the field supply channel, field width,width of available machinery, topography, top soil depth, andpreferred aspect ratio. To the extent that width is determinedon the basis of considerations that are not explicitly relatedto performance, it is not considered as a design variable here.For practical design and management purposes, the solutionof equation 4 can be reduced to the solution of a series ofone-dimensional problems (Zerihun et al., 2001), simplify-ing the problem significantly. In subsequent sections, theEa(L) and Ea(qo) functions are analyzed separately toestablish the existence/absence of convexity and unimodal-ity. Based on the results of the analyses, simple equations thatcan be used to calculate approximate optimal length and unitinlet flow rate are developed.

APPLICATION EFFICIENCY AS A FUNCTION

OF BORDER LENGTHGiven the net irrigation requirement (Zr), target water

requirement efficiency (Ert), and unit inlet flow rate (qo), theapplication efficiency, Ea(L), can be given as:

)(

)(Lt

LCLE

coLa = (5)

where CL = Ert Zr/qo. At a stationary point, wheredEa(L)/dL = 0, the following holds:

1753Vol. 48(5): 1751−1764

L

Lt

dL

Ldt coco )()( = (6)

At a stationary point:

Lco

La yLt

LC

dL

LEd ′−=22

2

)]([

)( (7)

where yL ′ = d2 tco(L)/dL2 (Zerihun et al., 2001). Since CL, L,and tco(L) are all positive quantities over the entire range ofL, the Ea(L) function is concave at a stationary point, and thestationary point represents a maximum for:

0>′Ly (8)

Given a parameter set and qo combination and arequirement that Zmin = Zr, intuitive reasoning and experiencewith simulation results show that tco(L) is an increasingconvex function of length. A power function of the followingform can be used to relate tco and L (figs. 1a through 1e):

312 ψ+ψ= ψLtco (9)

where �1 (min/m�2), �2 (−), and �3 (min) are empiricalcurve-fitting parameters. Note that if �2 > 1, then equation 8holds and Ea(L) is concave at a stationary point. In order todetermine the domain of �2, simulation experiments wereperformed using SRFR (Strelkoff et al., 1998). The combina-tions of unit inlet flow rate and the parameter set (i.e., bedslope, surface roughness, and infiltration) used were selectedsuch that a broad range of irrigation conditions was taken intoaccount (table 1). Equation 9 was then fitted to the tco(L) dataobtained using simulation experiments, and the regressionresults are summarized in table 2 and figure 1. Figure 1a rep-resents an irrigation scenario that occurs in a border strip with

a low bed slope and on a high intake rate soil with a very highsurface roughness. Figure 1f, on the other hand, represents anirrigation scenario at the opposite end of the spectrum, whereinfiltration rate and surface roughness are very low and bedslope is steep. Figures 1b through 1e represent irrigation sce-narios that could be described as physically realistic. Notethat most physically realistic irrigation scenarios fall be-tween the two extreme bounds represented by data sets 1 and6 (table 1, figs. 1a and 1f). Moreover, figures 1a, 1b, and 1frepresent irrigation management scenarios where inflow cut-off occurred after completion of the advance phase, and fig-ures 1c through 1e represent conditions in which the inflowis cutoff in the course of the advance phase. The results sum-marized in figure 1 show that in all the cases considered, re-gardless of the inflow cutoff option used, cutoff time remainsa monotonic increasing power function of border length:�2 > 1 (table 2). It then follows that the right side of equa-tion 7 is less than zero at a stationary point. Hence, a station-ary point on Ea(L) represents a maximum point. The absenceof a local minimum automatically precludes the existence ofmultiple local maxima. Therefore, the stationary point on theEa(L) function is a global maximum, and the Ea(L) functionis unimodal.

MAXIMUM APPLICATION EFFICIENCY AS A FUNCTION OFBORDER LENGTH

Combining the first-order optimality condition (eq. 6) andthe power-law expression for tco(L) (eq. 9) yields anexpression for the approximate optimal length (Lopt ′):

2

1

21

3

)1(

ψ

−ψψ

ψ=′top

L (10)

(1a)

Border length (m)

0 40 80 120 160

Tim

e (m

in)

0

60

120

180

240

(1f)

Border length (m)

0 200 400 600

Tim

e (m

in)

0

350

700

1050

1400

Border length (m)

0 200 400 600 800

Tim

e (m

in)

0

400

800

1200

1600

Border length (m)0 200 400 600 800

Tim

e (m

in)

0

300

600

900

(1c)

Border length (m)

0 200 400 600 800

Tim

e (m

in)

0

300

600

900

(1e)

Border length (m)

0 100 200 300 400

Tim

e (m

in)

0

200

400

600

(1d)

(1b)

tco − SRFRtco − Eq. 9ta − SRFR






Figure 1. Cutoff time (tco) and advance time to the downstream end (ta) as a function of border length: (a) data set 1, (b) data set 2, (c) data set 3, (d)data set 4, (e) data set 5, and (f) data set 6.


Table 1. Data sets used in figures 1, 2, 5, and 6.Parameter Units Data Set 1 Data Set 2 Data Set 3[a] Data Set 4[a] Data Set 5 Data Set 6

qo L/min/m 90.0[b] 180.0[b] 150.0[b] 150.0[b] 40.0[b] 30.0[b]

L m 400.0[c] 250.0[c] 200.0[c] 200.0[c] 150.0[c] 100.0[c]

So -- 0.0001 0.0004 0.0008 0.0008 0.0015 0.005n m1/6 0.24 0.19 0.15 0.15 0.12 0.08k mm/ha 42.0 30.0 20.0 20.0 14.0 7.0a -- 0.67 0.5 0.42 0.42 0.25 0.1fo mm/h 15.0 10.0 8.0 8.0 6.0 4.0W m 1.0 1.0 1.0 1.0 1.0 1.0Zr mm 60.0 75.0 40.0 75.0 35.0 40.0

[a] Data sets 3 and 4 are the same except that the required application depths (Zr) are different. Due to differences in Zr, applicable cutoff options are alsodifferent: for data set 3, inflow cutoff occurred during the advance phase (figs. 1c and 2c); for data set 4, inflow cutoff occurred in the post-advance phase(figs. 1d and 2d).

[b] Unit inlet flow rates used to generate figures 2a through 2f.[c] Border lengths used to generate figures 1a through 1f.

The parameters of equation 9 (�1, �2, and �3) can beestimated using the three-point method (Zerihun et al.,2001):

2

22

1113

2

1

23

12

2

1

121

)(

ln

ln

)1(

ψ

ψψ

ψ−=ψ

∆∆

=ψ

−

∆=ψ

LLt

L

L

t

t

Lr

t

co

co

co

co

L

(11)

where

LL

L

cococo

cococo

rLLrL

LLr

LtLtt

LtLtt

/

/

)()(

)()(

312

13

3223

2112

==

=

−=∆

−=∆

(12)

and L3 < L2 < L1. Given a parameter set and unit inlet flow rate(qo), the following procedure can be used to determine theparameters of equation 10: (1) determine L1 as the length ofa block that is irrigated as a unit (discussion on how todetermine L1 is presented in the design section of this article);(2) select the minimum acceptable length (L3) based onoperational and economic considerations; (3) determine rLusing equation 12; (4) determine L2; (5) determine tco(L1),

tco(L2), and tco(L3) using a simulation model, such that Zmin =Zr in each case; and (6) calculate �1, �2, and �3 usingequation 11.

APPLICATION EFFICIENCY AS A FUNCTION

OF UNIT INLET FLOW RATEGiven a parameter set, the net irrigation requirement (Zr),

target water requirement efficiency (Ert), and border length(L), the application efficiency, Ea(qo), can be expressed as:

)(

)(ocoo

qoa qtq

CqE = (13)

where Cq = Ert Zr L. At a stationary point, where dEa(qo)/dqo = 0, the following holds:

o

oco

o

oco

q

qt

dq

qdt )()( −= (14)

At a stationary point:

′−

=

qoco

o

oco

oco

q

o

oa

yqt

q

qt

qt

C

dq

qEd

2

)()]([

)]([

2

)(

2

2

3

2

2

(15)

where yq ′ = d2 tco(qo)/dqo2 (Zerihun et al., 2001). Since Cq,

qo, and tco(qo) are all positive quantities over the entire rangeof qo, the Ea(qo) function is concave at a stationary point, andthe stationary point represents a maximum for:

Table 2. Summary of regression results.Parameter Units Data Set 1 Data Set 2 Data Set 3 Data Set 4 Data Set 5 Data Set 6

Cutoff Time as a Function of Lengthψ1 min/mψ2 8.35 × 10−4 7.24 × 10−6 3.63 × 10−5 2.26 × 10−6 7.24 × 10−6 5.786 × 10−15

ψ2 -- 2.565 2.868 2.508 2.926 2.868 6.481ψ3 min 42.1 98.3 54.0 212.6 98.3 496.1r2 -- 0.99 0.99 0.99 0.99 0.99 0.98

Cutoff Time as a Function of Unit Inlet Flow Rateβ1 minβ2+1/Lβ2 90,297,635 4,747,426.2 2,209,932.8 5,593,698.4 23,917,565 10,423,560β2 -- −2.226 −2.090 −2.276 −2.525 −3.680 −5.071β3 min 55.7 69.4 46.1 206.7 167.5 478.8r2 -- 0.99 0.99 0.99 0.99 0.99 0.99

1755Vol. 48(5): 1751−1764

2

)(2

o

ocoq

q

qty >′ (16)

Given a parameter set and L combination and a require-ment that Zmin = Zr, intuitive reasoning and experience withthe results of surface irrigation simulations suggest thattco(qo) is a decreasing convex function of flow rate (fig. 2).A power function of the following form can be used to relatetco with unit inlet flow rate (qo):

312)( β+β= β

ooco qqt (17)

where �1 (min�2+1/L �2), �2 (−), and �3 (min) are empiricalcurve-fitting parameters. Using equation 17 and the first-or-der optimality condition (eq. 14), it can be shown that at a sta-tionary point:

22

)()1(

o

ocoq

q

qty β−=′ (18)

Comparing equations 18 and 16 shows that a stationarypoint on the Ea(qo) function represents a maximum for �2 <−1. In order to determine the domain of �2, simulationexperiments were performed using five data sets that covera wide range of irrigation conditions using SRFR (Strelkoffet al., 1998). The combinations of border length and theparameter set (i.e., bed slope, surface roughness, andinfiltration) used were selected such that a broad range ofirrigation conditions was taken into account (table 1).Equation 17 was then fitted to the tco(qo) data obtained usingsimulation experiments, and the regression results aresummarized in table 2 and figure 2. Figure 2a represents anirrigation scenario that occurs in a border strip with a low bedslope and on a high intake rate soil with a very high surfaceroughness. Figure 2f, on the other hand, represents an

irrigation scenario at the opposite end of the spectrum, whereinfiltration rate and surface roughness are very low and bedslope is steep. Figures 2b through 2e represent irrigationscenarios that can be described as realistic. Note that mostphysically realistic irrigation scenarios fall between the twoextreme bounds represented by data sets 1 and 6 (table 1,figs. 2a and 2f). Moreover, figures 2a, 2c, and 2e representirrigation management scenarios where inflow cutoff oc-curred during the advance phase, and figures 2b, 2d, and 2frepresent conditions in which inflow is cutoff in thepost-advance phase. The results summarized in figure 2 showthat in all the irrigation scenarios considered, regardless ofthe inflow cutoff option used, cutoff time remains amonotonic decreasing power function of unit inlet flow rate:�2 < −1 (table 2). This shows that equation 16 holds at astationary point on the Ea(qo) function. Consequently, astationary point on Ea(qo) represents a maximum point. Theabsence of a local minimum automatically precludes theexistence of multiple local maxima. Therefore, the stationarypoint on the Ea(qo) function is a global maximum, and theEa(qo) function is unimodal.

MAXIMUM APPLICATION EFFICIENCY AS A FUNCTION OFUNIT INLET FLOW RATE

Combining the first-order optimality condition (eq. 14)and the power-law expression for tco(qo) (eq. 17) yields anexpression for an approximate optimal unit inlet flow rate(qopt ′):

2

1

21

3’ )1(

β

+ββ

β−=optq (19)

The parameters of equation 17 (�1, �2, and �3) can beestimated using the three-point method:

Unit inlet flow rate (L/min/m)

0 40 80 120

Tim

e (m

in)

0

400

800

1200

(2d)


0 300 600 900 1200

Tim

e (m

in)

0

200

400

600

800

(2e)


0 200 400

Tim

e (m

in)

0

200

400

600

800

(2c)(2a)(2b)


0 40 80 120

Tim

e (m

in)

0

200

400

600

800


0 100 200 300

Tim

e (m

in)

0

200

400

600

(2f)


0 100 200 300 400

Tim

e (m

in)

0

200

400

600







Figure 2. Cutoff time (tco) and advance time to the downstream end (ta) as a function of unit inlet flow rate: (a) data set 1, (b) data set 2, (c) data set 3,(d) data set 4, (e) data set 5, and (f) data set 6.


2

22

1113

2

1

23

12

2

1

121

)(

ln

ln

)1(

β

ββ

β−=β

∆∆

=β

−

∆=β

ooco

o

o

co

co

oq

co

qqt

q

q

t

t

qr

t

(20)

where

q

ooqo

ooq

ocoococo

ocoococo

r

qqrq

qqr

qtqtt

qtqtt

312

13

3223

2112

/

)()(

)()(

==

=

−=∆

−=∆

(21)

and qo3 < qo2 < qo1. Given a parameter set and border length,the following procedure can be used to determine �1, �2, and�3: (1) qo1 can be taken as the maximum non-erosive unitinlet flow rate; (2) determine the minimum unit inlet flow rate(qo3) as the minimum unit inlet flow rate that can reach thedownstream end of the border or the minimum unit inlet flowrate required for adequate spread, whichever is greater;(3) determine rq using equation 21; (4) determine qo2 usingequation 21; (5) determine t(qo1), t(qo2), and tco(qo3) using asimulation model, such that Zmin = Zr in each case; and(6) determine �1, �2, and �3 using equation 20.

EVALUATION OF OPTIMUM LENGTH AND

FLOW RATE EQUATIONSSix test problems (table 3) were used in the evaluation of

the approximate optimality conditions (eqs. 10 and 19). Thesurface irrigation simulation model, SRFR (Strelkoff et al.,1998), was used in the analysis. Data sets 7 through 9 (table 3)were used to test the optimality condition derived for L, (eq.10) and data sets 10 through 12 (table 3) were used to test theoptimality conditions derived for qo (eq. 19). The approxi-mate optimum solutions, [Lopt ′, Ea(Lopt ′)] and [qopt′,Ea(qopt′)], were calculated using the procedures outlinedabove (fig. 3, table 4). The actual optimum solutions, [Lopt,Ea(Lopt)] and [qopt, Ea(qopt)], were determined based on

repeated runs of SRFR (fig. 3). Note that all the approximateoptimum solutions, Ea(Lopt′) and Ea(qopt′), are within threepercentage points of the actual optimum Ea values (fig. 3,table 4). Given the imprecision involved in the determinationof the system parameters and numerical errors, the results aresatisfactory for practical design purposes. Note that fig-ures 3a, 3b, 3d, and 3e represent irrigation managementscenarios in which inflow cutoff occurred during the advancephase, and figures 3c and 3f represent conditions where theinflow is cutoff in the post-advance phase.

INFLOW CUTOFF OPTIONS: ADVANTAGES,

LIMITATIONS, AND EFFECTS ON DESIGN

AND MANAGEMENTINFLOW CUTOFF OPTIONS AND ADVANTAGES

In irrigation borders, inflow cutoff can occur in the courseof the advance phase or at the end of the wetting phase.Advance-phase cutoff offers some practical advantages overpost-advance cutoff. Wattenburger and Clyma (1989a,1989b) and Clemmens (1998) observed that level basindesigns that use distance-based inflow cutoff criterion(i.e., advance-phase inflow cutoff) are less sensitive to widevariations in decision variables and system parameters.Clemmens (1998) stated that design decisions based ondistance-based cutoff criterion are more transferable toirrigators and allow basin designs to be adapted to localpractices. Experience with simulation experiments showsthat similar observations can be made with regard to thesensitivity of the Ea(qo) function of border irrigation systemswhen inflow cutoff occurs during the advance phase(e.g., fig. 4a). Advance-phase cutoff has the effect of damp-ening the influence that changes in qo can have on the runofffraction (Rf) over a large interval of qo (fig. 4a). As a result,Ea becomes nearly insensitive to changes in qo over arelatively wide range (a 300% increase in qo resulted only ina 4.5% change in Ea, fig. 4a).

As can be seen from figures 3a through 3c, the inflowcutoff option used does not have a significant effect on thesensitivity of Ea to changes in L. In general, the Ea(L)function is distinctly unimodal and attains its peak valuewhere Rf approximately equals Df, regardless of the cutoffoption used (figs. 3a through 3c, 4c, and 4d). On the otherhand, Ea(qo) may not necessarily attain its maximum valuewithin physically realistic ranges of qo, if inflow cutoff is tooccur in the course of the advance phase (figs. 3e, 4a, and 4b).Even in the cases where inflow cutoff occurs during theadvance phase, the preceding theoretical observation on the

Table 3. Data sets used in figures 3 through 6.Data Sets Used to Test Equation 10 Data Sets Used to Test Equation 19

Parameter Units Data Set 7 Data Set 8 Data Set 9 Data Set 10 Data Set 11 Data Set 12

L m -- -- -- 340 300 200qo L/min/m 180 38.4 150 -- -- --So -- 0.00038 0.0005 0.00027 0.00065 0.0001 0.00029n m1/6 0.18 0.2 0.235 0.14 0.2 0.199k mm/ha 33.3 10 42 21 18 10.1a -- 0.55 0.25 0.6 0.42 0.3 0.45fo mm/h 9.6 3 8 5 8 12W m 1 1 1 1 1 1Zr mm 75 36 120 61.5 67 97.5

1757Vol. 48(5): 1751−1764


0 40 80 120 160 200

30

40

50

60

70

(3a)

Border length (m)

0 100 200 300 400

35

40

45

50

55

60

65

(3b)

Border length (m)

0 150 300 450 600

35

42

49

56

63

70

Lopt’ = 201 mLopt = 180 m

Ea(Lopt) = 62.2 %

Ea(Lopt’) = 61.7 %

Lopt = 305 m Lopt’ = 307 m

Ea(Lopt) = 67.1 % and E a(Lopt1’) = 67 %


0 100 200 300

30

40

50

60

70

(3d)

qopt = 120 L/min/m

qopt’ = 110 L/min/m

Ea(qopt’) = 66.5 %

Ea(qopt) = 67.0 %

(3e)Unit inlet flow rate (L/min/m)

0 150 300 450 600 750

40

45

50

55

60

65

70

qopt’

= 396 L/min/m qopt

= 420 L/min/m

Ea(qopt) = 66.5 %

Ea(qopt’) = 66.2 % Ea(qopt) = 60.5 %

Ea(qopt’) = 58.9 %

(3c)

Border length (m)

0 100 200 300 400

35

42

49

56

63

Lopt = Loopt’ = 191 m

Ea(Lopt) = Ea(Lopt’) = 60.1 %

(3f)

qopt’ = 79.9 L/min/m

qopt = 63 L/min/m

Ea

(%)

Ea

(%)

Ea

(%)

Ea

(%)

Ea

(%)

Ea

(%)

Figure 3. Application efficiency as a function of border length: (a) data set 7 (advance-phase cutoff), (b) data set 8 (advance-phase cutoff), and (c) dataset 9 (post-advance-phase cutoff); and application efficiency as a function of unit inlet flow rate: (d) data set 10 (both advance-phase and post-advance-phase cutoff are used), (e) data set 11 (advance-phase cutoff), and (f) data set 12 (post-advance-phase cutoff).

unimodality of Ea(qo) is valid. The fact that tco(qo) is a de-creasing convex function (fig. 2, table 2), irrespective of thecutoff option used, confirms the general validity of the op-timality condition derived above (eq. 19). However, the verylow sensitivity of Ea(qo) over a wide range of qo, when inflowcutoff occurs during the advance phase, means that a distinctmaximum could not be attained within realistic ranges of qo.In which case, the optimality condition developed above isstill applicable, but the optimum qo, calculated as such, maynot be the theoretical optimum. It could, instead, be a valueclose to the maximum feasible unit inlet flow rate (fig. 3e).

The very low sensitivity of Ea(qo) over a wide range of qo,when inflow cutoff occurs during the advance phase, is adesirable property because errors in flow measurements ornon-uniform distribution of inlet flow rate across the bordercan have minimal impact on the reliability of design andmanagement prescriptions. In general, whenever it is feasi-

ble, and when near-optimum management scenarios areachievable, border design and management can preferably bebased on distance-based cutoff criterion (i.e., advance-phaseinflow cutoff option). However, advance-phase inflow cutoffis feasible only if the combination of system parameters andvariables is such that the crop root zone reservoir can bereplenished to the extent desired (say Zmin = Zr), even wheninflow cutoff occurs prior to, or at, the completion ofadvance.

INFLOW CUTOFF OPTIONS AND LIMITATIONS

Given a unit inlet flow rate and a parameter set(infiltration parameters, the Manning roughness coefficient,Zr, and bed slope), there exists a minimum threshold borderlength (Lt) below which a border strip becomes too short tobe operated under the distance-based inflow cutoff criterionand still meet the requirement that Zmin = Zr. In other words,

Table 4. Results used in figure 3.Unit Inlet Flow Rate

(qo)ψ1

(min/mψ2)ψ3(−)

ψ3(min)

Lopt ′(m)

tco ′[a]

(min)Ea (Lopt ′)

(%)Lopt(m)

Ea (Lopt)(%)

180 L/min/m 2.7 × 10−4 2.313 75.7 201 136 61.7 180 62.238.4 L/min/m 2.8 × 10−5 2.714 270.2 307 431 67.1 307 67.1150 L/min/m 4.32 × 10−4 2.370 150.5 191 253.8 60.1 191 60.1

Border Length(L)

β1(minβ2+1/L)

β2(−)

β3(min)

qopt ′(L/min/m)

tco ′[a]

(min)Ea (qopt ′)

(%)qopt

(L/min/m)Ea (qopt)

(%)

340 m 6,823,771 −2.316 166.5 110.5 287 66.5 120.0 67.0300 m 225,208 −1.395 21.1 396.7 78 66.2 420.0 66.5200 m 1.18 × 109 −3.712 280.5 79.7 415 58.9 63.0 60.5

[a] tco ′ = cutoff time corresponding to Lopt ′.


(4a)

Unit inlet flow rate (L/min/m)0 60 120 180

0

20

40

60

(4d)

(4c)Border length (m)

0 150 300 4500

20

40

60

Border length (m)0 100 200 300 400

0

20

40

60

(4b)Unit inlet flow rate (L/min/m)

0 200 400 600

20

40

60

(4a)

(4d)

Ea

(%),

Df (

%),

an

d R

f (%

)

Ea

(%),

Df (

%),

an

d R

f (%

)E

a (%

), D

f (%

), a

nd

Rf (

%)

Ea

(%),

Df (

%),

an

d R

f (%

)

Ea

Rf

Df

0

Ea

Rf

Df

Ea

Rf

Df

Ea

Rf

Df

Figure 4. Application efficiency (Ea), deep percolation fraction (Df), and runoff fraction (Rf) expressed as a function of unit inlet flow rate: (a) data set 11(advance-phase cutoff), and (b) data set 12 (post-advance-phase cutoff); and as a function of border length: (c) data set 7 (advance-phase cutoff), and(d) data set 9 (post-advance-phase cutoff).

for L < Lt, the duration of the advance phase becomes tooshort for the surface storage volume to be sufficiently largeto replenish the root zone in full (figs. 5a and 5b). For any giv-en unit inlet flow rate and a parameter set mix, the corre-sponding Lt can be defined as the border length that yields aninfiltration profile in which Zmin = Zr for R = 100%, where R =(inflow) cutoff distance expressed as a percentage of the bor-der length (figs. 5a and 5b). Note that Lt is a dynamic quantitythat changes in the course of an irrigation season withchanges in irrigation parameters. A similar observation canbe made with respect to border unit inlet flow rate (qo). Givena combination of a field parameter set and border length, thecorresponding threshold unit inlet flow rate value (qot) can bedefined as the minimum qo below which the requirement Zmin= Zr cannot be met if the border strip is to be operated underthe distance-based inflow cutoff criterion (fig. 5c). AlthoughZr can be reduced to overcome this problem, reducing Zr hasits own problems. Lowering Zr means opting for a lighter ir-rigation, which in turn leads to more frequent irrigation. Thismay not always be compatible with the high dose, low fre-quency nature of surface-irrigated systems. In addition, it isimportant to recognize that even though Zr can be adjusted toachieve a feasible irrigation scenario with advance-phasecutoff, such a scenario may correspond to a sub-optimal solu-tion that is inferior to the solution that can be obtained if post-advance-phase cutoff is used.

There exist irrigation scenarios that have two thresholdunit inlet flow rates, qot1 and qot2, where the interval qot1 <qo < qot2 represents the range of qo in which distance-basedcutoff is feasible (fig. 5d). However, outside this range (i.e.,in the ranges qo < qot1 and qot2 < qo), only the post-advanceinflow cutoff option is feasible. In the range qo < qot1, thesurface storage volume at the end of the advance phase is notsufficiently large to replenish the root zone in full; hence,inflow cutoff needs to occur after completion of the advancephase. On the other hand, as qo increases, the duration of theadvance phase [ta(qo, L)] progressively shortens, and eventu-ally as qo approaches qot2, ta(qo, L) becomes shorter than theduration of the recession phase. This causes the location ofZmin to shift from the downstream end to the inlet end of theborder, at which point qo passes a threshold with respect to itseffect on the cutoff time. The tco needed to meet therequirement Zmin = Zr at the upstream end of the borderbecomes virtually insensitive to further increases in qo(fig. 6a, eq. A.1 in the Appendix). In contrast to tco(qo), whichremains nearly constant with further increases in qo, advancetime to the downstream end, ta(qo, L), continues to decline ata relatively higher rate (fig. 6a). Eventually, as qo exceedsqot2, ta(qo, L) falls below tco(qo) and continues to do so withfurther increases in qo, making distance-based cutoff criteri-on inapplicable in the range qot2 < qo.

The insensitivity of the tco(qo) function, in the range whereZmin occurs at the inlet end of the border, can be explained

1759Vol. 48(5): 1751−1764

(5d)


0 100 200 300

50

60

70

80

90

100

(5c)


0 200 400 600

40

50

60

70

80

90

100

(5a)

Border length (m)

0 100 200 300 400

50

60

70

80

90

100

L = Lt

qo = qot

qo = qot1 qo = qot2

(5b)

Border length (m)

0 250 500 750 1000

40

60

80

100

L = Lt

Zmin

Zr

R

Zmin

Zr

R

R

Zr

Zmin

R

Zr

Zmin

R (

%),

Zm

in (

mm

), a

nd

zr

(mm

)R

(%

), Z

min

(m

m),

an

d z

r (m

m)

R (

%),

Zm

in (

mm

), a

nd

zr

(mm

)R

(%

), Z

min

(m

m),

an

d z

r (m

m)

Figure 5. Cutoff ratio (R), minimum infiltrated depth (Zmin), and required depth (Zr) expressed as a function of border length: (a) data set 7, and (b)data set 2; and as a function of unit inlet flow rate: (c) data set 11, and (d) data set 10.

using an equation that relates cutoff time (tco) with the re-quired intake opportunity time, �req(Zr), and the duration ofthe depletion phase (tdep):

depcorreq ttZ +=τ )( (22)

As can be seen from figure 6a, tdep is virtually insensitiveto changes in qo in the range where Zmin occurs at the inlet endof the border. In addition, for a given Zr and infiltrationparameter set, �req(Zr) is a constant. Thus, equation 22 showsthat if the requirement Zmin = Zr is to be met, then tco(qo) alsoneeds to be nearly constant.

Note that tco can also be insensitive to changes in L whenborder lengths are very short (figs. 1c and 1d). Here as well,it is the combined effect of a constant �req(Zr) and a nearlyinsensitive tdep(L) (fig. 6b) that renders tco nearly insensitiveto changes in L for short borders. In addition, it can be seenfrom figures 5a and 5b that if L is increased beyond Lt thecutoff ratio decreases steadily. However, if L becomesexcessively high, then the consequent progressive steepeningof the advance curve and the final infiltration profile near thedownstream end of the border make the cutoff distance verysensitive to changes in L. As a result, the inflow cutoffdistance begins to grow at a faster rate than L; hence, R beginsto back up (fig. 5b). Depending on the range of L consideredin the analysis, R may back up to 100% (fig. 5b). Thissuggests that a second threshold border length may exist. Ingeneral, the question of a second threshold border length ispertinent only when extremely long borders are considered(fig. 5b). Such border lengths are physically unrealistic, andhence the issue of a second threshold border length is of nopractical design and management significance.

INFLOW CUTOFF OPTIONS AND DESIGN AND MANAGEMENTIMPLICATIONS

Based on the preceding discussion, the following infer-ences are drawn: (1) regardless of the cutoff option used theEa(L) and Ea(qo) functions are unimodal; however, wheninflow cutoff occurs during the advance phase, Ea could benearly insensitive to changes in qo, and as a result, themaximum Ea may not be attained within physically realisticranges of qo; (2) there exist limiting conditions, which aredependent on the field parameter set, that preclude theapplicability of the distance-based cutoff criterion in borderirrigation management; and (3) even when distance-basedinflow cutoff criterion is feasible, the corresponding designand management scenario could be sub-optimal, in whichcase, a near-optimal operation scenario can be realized onlywith post-advance-phase cutoff.

DESIGN AND MANAGEMENT APPLICATIONSConsidering tco, L, and qo as the three border irrigation

design and management variables, the border irrigationdesign and management procedure can be simplified sub-stantially. Given a parameter set and the condition Zmin = Zr,tco is a function of qo and L. Consequently, tco cannot betreated as an independent variable in itself. This leaves onlyqo and L as the design variables. In addition, for a given fieldlength and parameter set combination, the set of allpractically realistic values of L is limited to a couple ofknown alternatives. For each feasible value of L, Ea cantherefore be optimized with respect to qo. In which case, thetwo-dimensional problem is reduced to a series of one-dimensional optimization problems. The following optimaldesign procedure is proposed:


(6a)

−60 −40 −20 0 20 40 60 80

0

10

20

30

40

(6b)

−50 0 50 100 150 200 250

−10

0

10

20

30

40

tco

ta

tdep

ta

tco

tdep

Change in unit inlet flow rate from qot2 [see Figure 5d] (%)

Change in border length from Lt [see Figure 5b] (%)

Rel

ativ

e se

nsi

tivity

of t

co, t

a, a

nd

t dep

(−)

Rel

ativ

e se

nsi

tivity

of t

co, t

a, a

nd

t dep

(−)

Figure 6. Relative sensitivity of cutoff time (tco), advance time to the downstream end (ta), and depletion time (tdep) as a function of: (a) unit inlet flowrate (data set 10), and (b) border length (data set 2).

1. Establish the feasible range of L and qo.1a. Determine the maximum non-erosive unit inlet

flow rate (qmax) and the minimum unit inlet flowrate required to ensure adequate spread (qmin). Anyappropriate set of equations can be used todetermine qmax and qmin. For example, using the

empirical equations of Hart et al. (1980), qmax(in L/min/m) for non-sod-forming crops, such asalfalfa and small grains, is:

4/359.10

omax

Sq = (23)

1761Vol. 48(5): 1751−1764

For well-established, dense sod crops, qmax can betwice as large. From Hart et al. (1980), the qmin(in L/min/m) required to ensure adequate spreadis:

n

SLq o

min357.0

= (24)

where n is the Manning roughness coefficient andSo is border bed slope. Note that equations 23 and24 are inapplicable to combinations of L, So, andn that result in qmax < qmin.

1b. Specify the field parameter set, field length (Lf),and minimum acceptable border length (Lmin).Selection of Lmin can be based on operational andeconomic considerations.

1c. Determine qmin and check if the correspondingmaximum possible advance distance (Lmax) is lessthan or greater than Lf.� Calculate qmin as a function of Lf using equation

24.� Using a simulation model (e.g., SRFR), check

if the irrigation stream can advance to thedownstream end of the border (for L = Lf andqo = qmin). If the advance phase is completed,then Lmax > Lf. If, on the other hand, the streamfailed to reach the downstream end of theborder, then Lmax < Lf, and Lmax needs to bedetermined through repeated runs of a simula-tion model.

1d. Determine the maximum possible border length(L1) that can be irrigated as a block: (1) L1 = Lf ifLf < Lmax, or (2):

=

=

>

+

=

max

f

max

f

max

f

f

max

f

max

f

max

f

f

L

L

L

L

L

L

LL

L

L

L

L

L

L

LL

intfor

int

intfor

1int

1

1

(25)

if Lmax < Lf, where int = an operator that truncatesthe fraction part of the quotient of Lf and Lmax.

1e. Determine the feasible set that contains whole-number divisors of L1 that are greater than or equalto Lmin. For example, if L1 = 300 m and Lmin =75 m, then the set containing whole-numberdivisors of L1 that are greater than or equal to Lminis given as Lfs = {300, 150, 100, 75}.

2. For each element of Lfs, determine Ea(qopt ′).3. Perform a sensitivity analysis and select the scenario

that provides the highest application efficiency andsatisfies other locally pertinent practical requirements.

4. Determine border ridge height as a function of the flowdepth at the inlet and with an allowance for freeboard.The flow depth at the inlet can be calculated using theManning equation.

5. Determine border width taking into account availablemachinery width, field width, available field supplychannel discharge, top soil depth, cross-slope, andpreferred aspect ratio.

The procedure presented above is for the design of aborder irrigation system. Once an irrigation system isconstructed, the length of the border is known, and hencesystem management is a function of qo only. A managementproblem can therefore be considered as a particular case ofa design problem. Consequently, the design procedureproposed here can be directly applied to solve managementproblems by setting L constant.

EXAMPLE DESIGN PROBLEMIt is required to determine the combination of border

length and flow rate that yields maximum applicationefficiency given specific field conditions. The field parame-ter set used in this example is: So = 0.0008, n = 0.1 m1/6, k =15 mm/ha, a = 0.3, fo = 5 mm/h, and Zr = 75 mm, where k, a,and fo are the coefficients and exponent of the modifiedKostiakov-Lewis infiltration function. The procedure used todetermine the optimal L−qo combination is described below:

1. Establish the feasible range of L and qo.1a. Considering an alfalfa crop, the maximum flow

rate (qmax) calculated using equation 23 is 2226 L/min/m. This is an extremely high value to beconsidered realistic; hence, a lower value of500 L/min/m is used as qmax. Note that qmaxcorresponds to qo1 in equations 20 and 21 (table 5).

1b. The field length (Lf) is 400 m, and the minimumacceptable border length (Lmin) is taken as 100 m.

1c. The minimum unit inlet flow rate (qmin) calculatedusing equation 24 for L = Lf, is 40 L/min/m. UsingSRFR (Strelkoff et al., 1998), it can be shown thatwhen qo = qmin = 40 L/min/m, the irrigation streamcan advance to a distance well beyond Lf. Hence,Lf < Lmax, and the maximum possible border length(L1) is 400 m.

1d. The feasible set for L that contains a whole-num-ber divisor of L1 that is greater than or equal to Lminis: Lfs = {400, 200, 100}.

1e. Determine qmin for each element of Lfs usingequation 24. For L = 400 m, qmin = 40 L/min/m; forL = 200 m, qmin = 20 L/min/m; and for L =100 m,qmin = 10 L/min/m. Note that the qmin values herecorrespond to qo3 in equations 20 and 21 (table 5).

Table 5. Results of design example.

Parameter Units L = 400 m L = 200 m L = 100 m

qo1 L/min/m 500.0 500.0 500.0qo2 L/min/m 141.4 100.0 70.7qo3 L/min/m 40.0 20.0 10.0

tco(qo1) min 475.0 487.0 496.0tco(qo2) min 496.0 491.0 515.0tco(qo3) min 1600 1587.0 1555.0

β1 minβ2+1/L 2.215 × 108 3.704 × 107 1.178 × 105

β2 -- −3.260 −3.483 −2.046β3 min 474.6 474.9 474.6

qopt ′ L/min/m 67.1 32.97 15.14tcopt ′[a] min 711.0 720.0 756.0

Ea(qopt ′) % 62.9 63.2 65.5qopt L/min/m 60.0 30.0 14.0

tcopt[b] min 785.0 760.0 805.0

Ea(qopt) % 63.6 65.7 66.5[a] tcopt ′ = cutoff time corresponding to Lopt ′.[b] tcopt = optimum cutoff time.


1f. For each element of Lfs calculate qo2 (qo1, qo3)using the procedure outlined above (eqs. 20 and21). The results are summarized in table 5.

2. For each element of Lfs, determine Ea(qopt ′):2a. For each element of Lfs, three simulation runs

(corresponding to qo1, qo2, and qo3) are performedusing SRFR, and the resulting cutoff times,tco(qo1), tco(qo2), and tco(qo3), are summarized intable 5.

2b. For each element of Lfs, the parameters ofequation 17 (�1, �2, and �3) are calculated usingequations 20 and 21 (table 5).

2c. The approximate optimum unit inlet flow rate(qopt′) is calculated using equation 19, and theapproximate maximum application efficiency,Ea(qopt′), is calculated using equation 13 (table 5).

2d. For L = 100 m, the approximate maximumapplication efficiency is 65.5%; for L = 200 m, itis 63.2%; and for L = 400 m, it is 62.9% (table 5).On the other hand, the maximum applicationefficiency values estimated through repeated sim-ulation runs using SRFR are 66.5 % for L = 100 m,65.7% for L = 200 m, and 63.6% for L = 400 m(table 5).As can be seen from table 5, qopt′ varies as afunction of length, but the optimum applicationefficiency remains nearly unchanged. This resultconcurs with observations made by Zerihun et al.(1993) that the potential maximum applicationefficiency of surface irrigation systems is afunction of the parameter set only. Consequently,from the point of view of maximization ofapplication efficiency, all three design scenariosare equally valid. This implies that practical andeconomic considerations need to be taken intoaccount in the selection of the best option among

the three border lengths. For instance, in situationswhere flow regulation and measurement devicesare of low accuracy, the design scenario with theleast sensitivity to flow rate variation around theoptimum Ea(qo) is recommended.

3. Analysis of the sensitivity of Ea(qo) around theoptimum (fig. 7) shows that in a close vicinity of theoptimum (between −5% and +15%), Ea shows the leastsensitivity to changes in qo for L = 100 m, followed byL = 200 m, and then L = 400 m. Hence, if flowregulation and measurement devices are of lowaccuracy, then L = 100 m is the best option. However,final selection of the border length needs to take intoaccount other local economic and operational consid-erations.

4. Calculate border ridge height. Considering the optionL = 100 m and using the Manning equation, the normaldepth at the inlet is 13.5 cm. Taking the freeboard as100% of the normal depth, the ridge height can be givenas 27 cm.

5. The actual calculation of border width in itself is verybasic. However, the number of factors that need to betaken into account, as enumerated above, are many andtheir interrelationship is not as simple. This makes thedetermination of border width more of a subjectiveprocess, dominated by intuition rather than by mathe-matical rigor. For instance, if the available flow rate inthe field supply channel is known, then a first estimateof the border width can be calculated as the ratio of thefield supply channel discharge to the optimum unitinlet flow rate. This initial estimate needs to be revisedsuch that the final border width is an integer divisor ofthe field width. Other factors that need to be consideredare width of available farm machinery in relation toborder width and preferred aspect ratio, if any, asrelated to adequate spread of water across the border.

Change in unit inlet flow rate from the optimum (%)

−40 −30 −20 −10 0 10 20 30 40

0.0

0.5

1.0

1.5

2.0

2.5

3.0

L = 400 m

L = 200 m

L = 100 mRel

ativ

e se

nsi

tivi

ty o

f E

a (q

o)

Figure 7. Relative sensitivity plot around the optimum unit inlet flow rate for the three alternative lengths.

1763Vol. 48(5): 1751−1764

In addition, in situations where substantial land gradingand shaping is involved, the border width should notexceed a maximum permissible width, which is a func-tion of the topography and the topsoil depth.

CONCLUSIONS AND SUMMARYApplication efficiency is the primary criterion in border

irrigation system design and management. The applicationefficiency function of border irrigation systems is unimodalwith respect to length and unit inlet flow rate. Optimalityconditions are derived for the Ea(L) and Ea(qo) functions.Differences between the solutions obtained using the approx-imate optimality conditions derived here and the actualoptimal solutions are less than three percentage points. Giventhe imprecision involved in the determination of the systemparameters and numerical errors, the results are satisfactoryfor practical purposes. The advantages and limitations ofadvance-phase and post-advance-phase inflow cutoff optionsand their effects on design and management are discussed.Based on the optimality conditions derived here, simpledesign and management rules are developed.

REFERENCESClemmens, A. J. 1998. Level basin design based on cutoff criteria.

Irrig. and Drain. Systems 12(2): 85-113.El-Hakim, O., W. Clyma, and E. V. Richardson. 1988. Performance

functions of border irrigation systems. J. Irrig. and Drain. Eng., ASCE 114(1):118-129.

Hart, W. E., H. G. Collins, G. Woodard, and A. J. Humphereys.1980. Chapter 13: Design and operation of surface irrigationsystems. In Design and Operation of Farm Irrigation Systems.ASAE Monograph No. 3. M. E. Jensen, ed. St. Joseph, Mich.:ASAE.

Holzapfel, E. A., M. A. Marino, and J. Chevez-Morales. 1986.Surface irrigation optimization models. J. Irrig. and Drain.Eng., ASCE 104(3): 275-281.

Holzapfel, E. A., and M. A. Marino. 1987. Surface-irrigationnonlinear optimization models. J. Irrig. and Drain. Eng., ASCE113(3): 379-391.

Reddy, J. M. 1980. Irrigation system improvement by simulationand optimization. PhD diss. Fort Collins, Colo.: Colorado StateUniversity.

Reddy, J. M., and W. Clyma. 1981. Optimal design of borderirrigation systems. J. Irrig. and Drain. Div., ASCE 107(3):289-306.

Shatanawi, M. R., and T. Strelkoff. 1984. Management contours forborder irrigation. J. Irrig. and Drain. Eng., ASCE 110(4):393-399.

Strelkoff, T., and M. R. Shatanawi. 1985. Normalized graphs ofborder irrigation performance. J. Irrig. and Drain. Eng., ASCE110(4): 359-375.

Strelkoff, T. S., A. J. Clemmens, B. V. Schmidt, and E. J. Solsky.1996. BORDER: A design and management aid for slopingborder irrigation systems. V. 1.0. WCL Report No. 21. Phoenix,Ariz.: USDA-ARS, U.S. Water Conservation Laboratory.

Strelkoff, T. S., A. J. Clemmens, and B. V. Schmidt. 1998. SRFR:Computer program for simulating flow in surface irrigation:Borders-basins-furrows. V. 3.31. Phoenix, Ariz.: USDA-ARS,U.S. Water Conservation Laboratory.

Walker, W. R., and G. V. Skogerboe. 1987. Surface IrrigationTheory and Practice. Englewood Cliffs, N.J.: Prentice-Hall.

Wattenburger, P. L., and W. Clyma. 1989a. Level basin design andmanagement in the absence of water control: Part I. Evaluation

of completion of advance irrigation. Trans. ASAE 32(3):838-843.

Wattenburger, P. L., and W. Clyma. 1989b. Level basin design andmanagement in the absence of water control: Part II. Designmethod for completion of advance irrigation. Trans. ASAE32(3): 844-850.

Yitayew, M., and D. D. Fangmeier. 1984. Dimensionless runoffcurves for border irrigation. J. Irrig. and Drain. Eng., ASCE110(2): 179-191.

Yitayew, M., and D. D. Fangmeier. 1985. Reuse system design forborder irrigation. J. Irrig. and Drain. Eng., ASCE 111(2):160-174.

Zerihun, D., J. Feyen, J. M. Reddy, and G. Breinburg. 1993. Designand management nomograph for furrow irrigation. Irrig. andDrain. Systems 7: 29-41.

Zerihun, D., J. Feyen, and J. M. Reddy. 1996. Analysis of thesensitivity of furrow irrigation performance parameters. J. Irrig.and Drain. Eng., ASCE 122(1): 49-57.

Zerihun, D., Z. Wang, R. Suman, J. Feyen, and J. M. Reddy. 1997.Analysis of surface irrigation performance terms and indices.Agric. Water Mgmt. 34(1): 25-46.

Zerihun, D., J. Feyen, J. M. Reddy, and Z. Wang. 1999. Minimumcost design of furrow irrigation systems. Trans. ASAE 42(4):945-955.

Zerihun, D., C. A. Sanchez, and K. L. Farrell-Poe. 2001. Analysisand design of furrow irrigation systems. J. Irrig. and Drain.Eng., ASCE 127(3): 161-169.

APPENDIX: RELATIVE SENSITIVITY

EQUATIONThe equation used to calculate relative sensitivity in

figures 6 and 7 (e.g., Zerihun et al., 1996) is:

∑ ∆

∆

=m

o

m

o

m

m

x

x

f

f

RS1

(A.1)

where RSm is the relative sensitivity at the mth perturbationof x, x is the independent variable, xo is the reference valueof the independent variable, f is a function whose sensitivityis being analyzed, fo = f(xo), and:

1

11

and

)()(

−

−−

−=∆

−∆+=∆

mmm

mmmm

xxx

xfxxff

(A.2)

NOMENCLATURE

a = exponent of the modified Kostiakov-Lewisinfiltration function

CL = constant expressed as Ert Zr/qoCq = constant expressed as Ert Zr LEa = application efficiencyEr = water requirement efficiencyErt = target water requirement efficiencyDU = distribution uniformityDUmin = minimum acceptable level of distribution

uniformity (−)fo = coefficient of the linear term of the modified

Kostiakov-Lewis infiltration function


k = coefficient of the power term in the modifiedKostiakov-Lewis infiltration function

L = border lengthLf = field lengthLmax = maximum advance distance corresponding to

maximum non-erosive flow rateLopt = optimal border lengthLopt ′ = approximate optimal border lengthn = Manning roughness coefficientqo = inlet flow rateqo1 = upper bound of the feasible set for qoqo2 = flow rate value calculated as the product rq qo1qo3 = lower bound of the feasible set of qo

qmax = maximum non-erosive flow rateqopt = optimal border inlet flow rateqopt ′ = approximate optimal border lengthrq = constant calculated as (qo3/qo1)1/2

So = border bed slopeSt. = subject tota = advance timetco = cutoff timeW = border widthZmin = minimum infiltrated amountZr = required amount of application�req = required intake opportunity time

Documents

ANALYSIS AND DESIGN OF BORDER IRRIGATION · PDF fileTransactions of the ASAE Vol. 48(5): 1751−1764 2005 American Society of Agricultural Engineers ISSN 0001−2351 1751 ANALYSIS