optimization of furrow irrigation system design parameters considering drainage and runoff water...

Irrig Sci (1994) 15:123-136 �9 Springer-Verlag 1994

J. Mohan Reddy

Optimization of furrow irrigation system design parameters considering drainage and runoff water quality constraints

Received: 5 January 1993

Abstract. The design problem of furrow irrigation systems considering runoff and drainage water quality was formulated as an optimization problem, with maximization of net benefits as the objective. A power advance function with an empirically derived relationship between the furrow irrigation design variables and relative crop yield were used in the formulation. The generalized geometric programming technique was used to solve for the optimal values for the design variables that maximized the net benefits from a furrow irrigation system. The optimal efficiency for which the system must be designed under a given set of soil, crop, and economic conditions is not known in advance. In the design, the application efficiency was not specified a pr ior i . It was an output from the optimal design. The analysis suggested that it might not be economical to design surface irrigation systems to achieve a high application efficiency that is specified a pr ior i . In the ab- sence of environmental degradation problems from irrigation, it may sometimes be profitable to design surface irrigation systems to operate at less than the standard application efficiency (55%-90%) that is routinely used in the design. Formulation of the design problem as an optimization problem would yield the optimal application efficiency that would maximize the net benefits to the farmer under any given set of conditions.

In recent years there has been a growing concern about the increasing presence of agricultural chemicals in groundwater. Among the wide range of agricultural chemicals found in groundwater samples taken in the United States, the largest single contaminant is nitrate nitrogen (NO3-N) (Watts 1989). Nitrates are carried by the water that percolates below the crop root zone during irrigation. It is estimated that as much as 177-220 kg of nitrogen is carried with every hectare-m of water that percolates below the root zone (USDA 1983). Surface irrigation is the applica-

J. M. Reddy Department of Civil Engineering, University of Wyoming, Laramie, WY 82071, USA

tion system with most deep percolation and, therefore, the more likely to produce groundwater contamination. The runoff and deep percolation volumes from surface (furrow) irrigation systems is a function of the design variables: length of run, furrow spacing, furrow flow rate, and time of application. They also depend upon management prac- tices such as irrigation scheduling, and soil intake characteristics, field slope, and furrow geometry and size.

The commonly used objective in the design of surface irrigation systems is the maximization of the application efficiency of the system. This practice does not take into account the farmer who, as the owner of the farm, is inter- ested in maximum profits from crop production. Usually the values of the system performance parameters are specified a p r i o r i and then the set of values for the design variables which satisfy performance parameters are chosen. This design approach may not guarantee that the set of values chosen for the design variables will represent the optimum design because there are several combinations of values for the design variables that yield the same level of system performance but with different net benefits. In the past groundwater and runoff quality has not been explicitly considered in the design of furrow irrigation systems.

The design problem should be formulated so that maximum profits as well as minimum undesirable effects are achieved from the system. This goal can be accomplished by iterating the system design performance with different sets of design variables. This procedure will probably yield the optimum set of design variables; however, it is tedious. Formulation of the design problem as an optimization problem and application of one of the available optimization techniques would eliminate the trial and error solution, and would yield a direct solution for the optimal values of the design variables. The objectives of this paper are to extend the optimal design procedure developed by Reddy and Clyma (1981 a, b, 1983), and Reddy and Apo- layo ( 1991 ) for the design of furrow irrigation systems considering runoff and deep percolation water quality constraints, and to compare the performance of a furrow irrigation system obtained using this procedure with that obtained using the traditional design procedure.

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Mathematical model

Before optimizing a system, the mathematical relation- ships between the system design variables and the performance parameters must be known. In surface irrigation, these are usually obtained by using a hydraulic simulation model. Here, a simple volume-balance model of furrow irrigation is used after neglecting the recession phase.

Advance equation

To compute advance time in a furrow irrigation system a power function relationship (Ley and Clyma 1980) of the following form was used:

t x = r (K 1 L) ~ (1)

where K 1 is fraction of furrow length up to which the design depth is satisfied, L is length of run (m), t x is time of advance to distance x (min), and Icand ]3 are empirical parameters that are defined as follows:

x" = 3.289 K~ qrl so-5. (2)

A

fl= 1 - 7/ ( 3 )

where q is furrow inflow rate (lps), A and 77 are constants dependent on the intake family, So is furrow slope (m/m), and K 2 is units conversion coefficient (4.831 for metric units). Any other advance equation can be used here in place of Eq. (1). The values for A and 77, as given by Ley and Clyma (1980), are presented in Table 1.

Depth of water requirement

The water requirement depth is one of the most important constraints to be satisfied by the system. It is defined as the depth of water needed to bring the soil to field capac-

ity after depletion to the management-allowed deficiency level. The design depth is an input to the model. A procedure for selecting an optimal design depth was presented by Reddy and Clyma (1983 b). In the design, the fraction of the field length that must be irrigated to the requirement depth should be specified when designing a furrow irrigation system. In this study, the fraction of the furrow length up to which the water requirement depth would be satisfied was assumed to be 0.85 L. This would result in a water requirement efficiency of more than 85%. The actual value of water requirement efficiency obtained is calculated after solving for the optimal values for the design variables. In the case of deficit irrigation, where ER could be less than 85%, the mathematical expression for ER can be added as an additional design variable in the problem. This aspect is not considered here. The depth of water infiltrated as defined by the SCS (USDA 1983) infiltration function was set equal to the depth required, and is given as follows:

Dn = ( K V a + 7) -~ (4)

where Dn is water requirement depth (mm), Vis intake opportunity time (min), K and a are coefficients of the intake family, W is furrow spacing (m), and P is the wetted perimeter (m) and was calculated as follows (USDA 1983, after converting to metric units):

( q n / 0 ' 4 2 5

e = 0.265 700 ) + 0.227 (5)

where n is Manning roughness coefficient (assumed to be 0.04 usually). The infiltration opportunity time at a distance of K1 L was written as follows:

V = ( t c o - ~ (K1 L ) ~) (6)

where tco is time of cutoff (min). The other variables have been defined earlier.

Runoff volume

Table 1 Furrow irrigation design parameters

Intake family no A 7/

0.05 129 -0.340 0.10 361 -0.535 0.15 680 -0.650 0.20 1 220 -0.755 0.25 1 851 -0.830 0.30 2 621 -0.890 0.35 3 614 -0.950 0.40 4 772 -1.001 0.45 5 873 -1.038 0.50 7 040 -1.070 0.60 9 603 -1.125 0.70 12 006 -1.166 0.80 14 318 -1.200 0.90 16 178 -1.225 1.00 17 365 -1.240 1.50 13 767 -1.227 2.00 7 660 -1.167

Surface runoff was defined as the depth (or volume) of out- flow from the tail-end of a graded furrow. It is very difficult to design a graded furrow irrigation system without surface runoff. It can be stated that runoff is necessary, in the case of graded furrows, to accomplish the depth of water requirement at the end of the furrow. In some cases the runoff water can be reused by constructing an appropriate surface runoff recovery system. This implies an additional cost to the system. However, this cost can be reduced by subtracting the value of the water recovered from the system.

Frequently, runoff water contains colloidal materials and other dangerous substances that make it impossible for reuse. In this case, the system design must provide appropriate facilities for its safe disposal. This will increase the cost of runoff. This increase will depend upon the amount of runoff expected in the system, which will dictate the capacity of the facility.

The volume of runoff was estimated using the SCS design equations, and is given as:

I alqtc~ lave]Area (7) V o= L

where VRo is volume of runoff (m 3) as a function of q, tco, L, and Dn, and other parameters such as field slope, soil infiltration characteristics, furrow cross-section and Man- ning roughness factor, tz 1 is units conversion (minutes to seconds) coefficient (60), and lav e is average infiltration depth over a specified length of the furrow (mm). The average infiltration depth, as given by the SCS' design procedure, is a function of the advance relationship which will define the intake opportunity time. Another factor to be taken into consideration in the calculation of the average infiltration is the infiltration function.

The average infiltration depth over a specified furrow length using a linear approximation was written as:

l a v e = (K tcao +7) ~ + (K (tco - ~ (K 1 L) # )a +7) P (8)

o r

~ W P (9) Iave =(K taco + 7) +(K Va + 7) 2 W.

Therefore, the volume of runoff was expressed as:

[ ~/~ (Ktcao+7) P (10)

a4 Nfs gt gwg LW

where N s is number of sets in the length direction, N w is number of sets in the width direction, Nfs is number of furrows irrigated per set, N i is number of irrigations per season, and a4 is units conversion (millimeters to meters) coefficient (0.001).

The results obtained using the linear approximation (Eq. 8) were compared with the average infiltrated depth calculated by integrating the advance function over the length of run. The difference between the exact and the approxi- mated infiltrated depth never exceeded 10% of the volume estimated using the exact method.

Deep percolation volume

Deep percolation losses are usually unavoidable in surface (furrow) irrigation systems. It can be reduced but is very difficult to eliminate, without a concurrent reduction in the water requirement efficiency. This would result in deficit irrigation which may be desirable under some situations. Incorporation of E R as a design variable in the optimal design formulation would accommodate the deficit irrigation case.

Water losses through deep percolation may have several detrimental effects on groundwater quality as well as on the crop root zone. In addition to the loss of water, deep percolation water carries away plant nutrients. These

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chemicals eventually reach the water table and can cause contamination problems. Deep percolation water may also contribute to an increase in the water table level which eventually might cause waterlogging and, depending upon the quality of the water, salinity problems. As a conse- quence of this, 20% of the world's irrigated area suffers from waterlogging and salinity problems (ICID 1980).

Deep percolation loss is a function of the depth of water requirement, time of application, furrow length, type of soil, and flow rate. It is defined as the total water infiltrated less the depth of water requirement, and was estimated as:

V@= (Ktaco+7)~-~+(KV + 7 ) ~ - D n (11)

"a 4 W L N w NI Nfs Ni

where Vdv is volume of deep percolation (m3).

Formulation of optimal design problem

The design of irrigation systems with the purpose of max- imizing profit from crop production involves an optimization problem. The optimization process involves the combination of the physical system itself, the crop response to water and fertilizer, and the on-farm management prac- tices.

In this study, the maximization of profit from crop production was chosen as the objective function. The main purpose of the analysis was to find a particular value of the water requirement efficiency, application efficiency, and the corresponding values of the system design variables which yield the maximum profit from the system while considering the leaching of fertilizers and other chemicals which affect the quality of groundwater. The water requirement efficiency-yield function developed by Reddy and Clyma (1981 a) was used in this study. This function relates the system design variables to crop yield through a two step-process.

Objective function

The objective function includes the system cost function and the crop production function, and is formulated as follows:

max P~=Pv-T~s (12)

where P~ is seasonal net benefit from the system ($), Pv is total value of the produce ($), and Tcs is total cost of the system ($).

Total value of produce

The total value of the produce in Eq. (12) can be expressed a s :

Pv=as Pc YR L NtNwNus W (13)

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where Pc is profit coefficient (S/hectare), YR is relative crop yield, and a 5 is a units conversion coefficient (0.0001).

Reddy and Clyma (1981 a) developed an equation which relates the relative yield of the crop and the irrigation system performance at each irrigation during the season. The model estimates crop yield as a function of the water requirement efficiency. Using a multiplicative crop function, the relative crop yield was simulated as a function of the water requirement efficiency. The statistical relationship obtained between relative yield and water requirement efficiency was as follows:

YR=-0.279 +0.0249 Er-0.0001212 Ez~ (14)

where E r is water requirement efficiency (%), defined as the ratio (in percent) of the volume of water stored in the root zone to the volume of water required in the root zone. Then a surface irrigation hydraulic model simulated the effect of the design variables - length of run, inflow rate, time of cutoff, and the design depth on water requirement efficiency. The statistical relationship obtained between the water requirement efficiency and the system design variables was as follows:

Er=g3 qa tbcotC Da (15)

where K 3 , a, b, c, and d are empirical constants that are site dependent. Although this equation was derived for a border irrigation system, for the purpose of demonstration of optimal design problem formulation, it was also assumed valid for a furrow irrigation system. In addition, crop yields are strongly influenced by the interaction of water and fertilizers. Therefore, if available, an equation that relates crop yield to E R and the fertilizers should be used in place of Eq. (14).

Costs involved in irrigation systems

The irrigation system design costs considered in this study are: the cost of water, the cost of labor, the cost of water delivery facilities, the cost of runoff, the cost of deep percolation, and the cost of producing the crop. In the cost of deep percolation, in addition to the cost of the water itself, the cost of the nutrients carried away by this water and any other negative effects caused by it were included. The total cost was expressed as:

Tc~=Cw + Cl+ Cd+ fro "k" Cdp'St" Cp (16)

where Cw is cost of water ($), Cl is cost of labor ($), Cd is cost of water delivery system construction ($), Cro is cost of runoff ($), Cdp is cost of deep percolation ($), and Cp is cost of production ($).

Cost of water

The cost of water varies depending upon its source. If the water is pumped from a well or a supply stream it will be a function of flow rate, total dynamic head, total irrigation time over the irrigation season, pump efficiency, and unit

cost of energy. This was calculated as follows:

60 / (17)

where h is total pumping head (m), Q is pumping flow rate (lps), Cbh is power cost ($/Kw-h), Ep is pump efficiency, and Ns is number of sets irrigated.

The cost of water withdrawn from an irrigation canal was estimated as

Cw=C 1 ot 1 q tcoNiNfsNtN w (18)

where C1 is water cost coefficient ($/1). In this study, it was assumed that the water used by the

system was withdrawn from an irrigation canal, and that the marginal cost of the water was constant.

Cost of labor

The cost of irrigation labor is the cost of operating the system. Labor is required to operate, check and maintain irrigation equipment during the irrigation. This cost may be high for systems with low labor efficiency. The labor efficiency can be improved by increasing the irrigation water discharge rate applied by an irrigator. This increase is opposed by two basic factors: permissible non-erosive flow rate in irrigation furrows and width of the wetting front. The maximum non-erosive flow rate in the irrigation furrow can be increased by increasing the length and the wetted perimeter of the furrow with the constraint that the wetted perimeter should be less than the furrow spacing, and the length of the furrow should be less than or equal to the length of the field.

Another way of increasing labor efficiency is by means of minor mechanization such as tubes and siphons to con- vey water into the furrow, irrigation pipe lines used instead of temporary sub-laterals, etc. The cost of labor was given by

Ct=Cz tr,2 ~3 tcoNssNwNtNi (19)

where C 2 is labor cost coefficient (S/h), a2 is fraction of the time the irrigator spends in irrigating the field (if the irrigator spends all the time irrigating the field, then a 2 is equal to 1), and th is a conversion factor to convert minutes to hours, i.e. 1/60.

Cost of head ditch

The cost of construction of the delivery system is another important cost of the system. Delivery structures to con- vey the water into the furrows are required. In this study it is assumed that earthen supply ditches are used. Other types of water delivery facilities such as wood or concrete flume, and metal, plastic or concrete pipelines can be used. The cost of the head ditch can be expressed as:

C~ = C3 Nz W s (20)

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where C3 is head ditch construction cost coefficient (S/linear-m), and Wf is field width (m). This is an annualized cost.

Cost of runoff

reduce the differences in intake opportunity time between head and tail end of the furrow. Then the initial flow rate is reduced to an amount which will depend upon the intake characteristics of the soil. This flow rate cannot be so large that it causes erosion, but must be sufficient to reach the end of the furrow. These constraints can be written as:

The cost of runoff is the cost involved in the construction of the runoff recovery system. In general the cost of runoff can be expressed as follows:

f r o = C4 VRO (21)

where Ca is runoff water cost coefficient ($/m3). The volume of runoff (VRo) was estimated using Eq. (10).

Cost of deep percolation

The cost of deep percolation includes the cost of construction of the drainage system as well as the cost of the nutrients leached from the crop root zone. The cost of deep percolation was expressed as:

Cap = C5 Vap (22)

where C 5 is deep percolation water cost coefficient ($/m3). The volume of deep percolation was estimated using Eq. (11).

Cost of production

G2 - q < 1 (24) qmax

G3 = qmin < 1 (25) q

where qmax is maximum non-erosive flow rate (lps), and qmin is minimum flow rate (lps).

The USDA (1983) recommended the use of an empirical relationship to estimate the qmax" This relationship is written as: qmax = IO/S for erodible soils; qmax = 15]S for erosion-resistant soils; qmax=12.5/S for average soils, where S is the slope of the furrow in percent. Although this relationship has been widely used, it does not included a term to account for the furrow capacity or the erodibility of the soil.

Another limitation placed in the system is that the number of furrows per set in the field times the furrow flow rate must be less than or equal to the total available flow rate (Qs). This can be expressed as:

G 4 - q Nfs < 1. (26) Os

This cost includes all the costs necessary for crop production, and is assumed to be a constant for a given area.

If the whole field is irrigated, the following constraint must be satisfied:

System constraints G5 - Nfs Nw < 1 (27) Nf

Constraints are the feasible values at which system design variables should be limited. In this study the depth of water requirement, the minimum and maximum furrow inflow rates, furrow length, the number of sets in the length and width directions and some equality constraints due to runoff, and deep percolation, and water requirement efficiency were considered as the system constraints.

Depth of water requirement

The water requirement is one of the most important constraints to be satisfied. This constraint was written as:

Do W < 1. (23) Gl = (KVa +7) P -

where Nfis the number of furrows in the field in the width direction.

Furrow length

Furrow length is another constraint to be satisfied. It cannot be greater than the field length (Lf). It should be large enough to allow the use of machinery. These constraints can be written as:

G6 = L < 1 (28) tmax

G7 = Lmin <_1 (29) L

Flow rate

The flow rate into the furrow should be large enough to reach the tail end of the furrow as quickly as possible to

G8 = Lf <1 (30) N l L

where Lmi n = minimum allowable furrow length (m).

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Runoff volume, deep percolation volume, and water requirement efficiency

Runoff and deep percolation are other limitations that should be taken into account in the design. The relation- ships defined by Eqs. (10) and (11) must be satisfied as strict equalities in the problem. To guarantee the equality, each equality constraint was replaced by two inequality constraints. For runoff, these constraints may be written as:

tZ1 a4 q Nw Nl Nfs < 1 (31) G9a = VRo + iave a 4 ZWNl Nw Nfs

69 b VR~ + Iave O~4 t W NI Nw Nfs - < 1 ( 3 2 )

al a4 q Nw NI Nfs and for deep percolation, these inequality constraints were written as:

l a v e a 4 L W Nt Nw Nfs GI0 a = < 1 (33)

Vdp + Dn a4 L W NI Nw Nfs

Vdp + Dn a4 L W Nl Nw Nf~ G10 b = < 1. (34)

lave 0~4 L W N l Nw Nfs

The water requirement efficiency is defined as a percent of the amount of water made available for plant use to the water requirement at the time of irrigation (Reddy

and Clyma 1981). It cannot exceed 100%. This was given by

K4 qa b d Gll - tc~ Lc Dn <1. (35) 100

itive and negative coefficients, respectively. They have the following form:

lk N Qk(X)=]~CkiI-Ixan k~" for k = 0, 1, 2 . . . . . K (38)

i=1 n=l

Jk N Pk (X) Ckj x~ k~" for k = 0, 1, 2 . . . . . K (39)

where I k is number of positive terms in constraint k, Jk is number of negative terms in constraint k, Cki and Ckj are positive and negative coefficients in the objective and constraint functions, respectively, and aki n and o~kj n are exponents of the system variables which may be either positive or negative numbers.

Equations (36) and (37) are called signomials. A signo- mial is the difference of two or more posynomials. In the process of formulating the problem into a generalized geometric programming format, these signomials are transformed into posynomials by moving all the terms with negative coefficients to the right-hand-side of the equation. This is done by rewriting Eq. (37) as follows:

Gk (X) = Qk (X) < 1 + Pk (X) (40)

or

Gk(X)- Ok (X) <1. (41) l + e k (X)

The posynomial (Eq. 41) was then transformed into a monomial, a function with a single term, by the condensation process, which is explained below.

Generalized geometric programming

The problem defined above is a nonlinear programming problem. There are several techniques, such as the quasi- Newton method, the reduced gradient method, the pro- jected Lagrangian method (Gill et al. 1981), and the geometric programming - to solve the above problem. Here, the geometric programming technique is used because of its convergence properties. The generalized geometric programming (Beightler and Phillips 1976; Passy and Wilde 1967; Dembo 1972) is a powerful technique for optimization of engineering problems involving non-linear functions. Let a function, to be minimized with respect to X, be given as follows:

Go (X)= Qo (x)-Po (X) (36)

where G O (X) is objective function formulated in terms of the system variables and the cost coefficients, X represents the number of independent positive variables (x 1 . . . . . XN), and N is the number of variables in the problem. A problem with K constraints is written as:

Gk(X)=Qk(X)-Pk(X)<I for k = l , 2 . . . . . K (37)

where Qk and Pk are called posynomial functions. They are the terms in the objective and constraint functions with pos-

Condensation

Condensation is a mathematical technique used exten- sively in geometric programming. The purpose of this technique is to reduce the number of terms of a polynomial by using primal weights, thereby reducing the degree of difficulty of the problem. Degree of difficulty (or degrees of freedom) is defined as the difference between the number of independent equations and the number of variables in the system minus one. The degree of difficulty is reduced by condensing terms whose exponents are not very different, that is, condensation of terms whose exponents have opposite sign should be avoided. Otherwise, a monotonic approximation to a non-monotonic function will be obtained.

Linearization

After condensation, Eq. (41) resulted in a nonlinear equation with a single term (monomial). This nonlinear function was linearized using a natural log-transformation as follows:

G~ (X, X ~ = Olk In (xl) + 02k In ( X 2 ) "t- . . , -t- ONk In (XN) <-In (~k) (42)

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in which 0ik are constants obtained through the condensation procedure.

In the solution procedure, the objective function was transformed into a constraint of the fol lowing form:

G O (X) = Qo ( X ) - P o (X) <x o . (43)

This added an additional constraint, which was condensed by using the procedure presented above. Finally, the problem was formulated as follows:

min Go (x)= In (xo) (44)

subject to: N

Gt, (X, Xo)=~--1 0~k In ( x , ) < - I n (gtk)

for k=0 , 1 ,2 . . . . . K (45)

Equations (44) and (45), being linear, were solved using a linear programming technique. Because of the inte- ger nature of some of the variables (N z and N w) in the problem, the linear programming technique was augmented by a branch-and-bound technique. For the original nonlinear problem, in general, a global opt imum cannot be guaran- teed. The global solution was approached by solving the linearized problem with several initial solutions.

Application to an example problem

Table 2 Values for the constants used in this study

Parameter Value

Length of field, m 805.0 Width of field, m 402.0 Roughness of field 0.04 Furrow spacing, m 0.76 Slope of field, m/m 0.001 Requirement depth, mm 76.0 Infiltration constants, D n = K tc' + 7 K, mm/min b 2.283 a 0.799 Advance constants, t x = ~r ( K 1 L) fl 77 2.227 fl -1.227 A 13 767 Constants in the relationship Er= K 4 qa tbco L c D d a 0.07678 b 0.28299 c -0.04839 d -0.99380 g 4 2 ,045 Number of irrigations/season 5

Cost coefficients Value

An example problem was considered to demonstrate the application of the technique presented above. The cost coefficients for runoff and deep percolation volumes were obtained f rom Wallender et al. (1987). After substituting the data presented in Table 2, the objective function and the constraints were rewritten as follows:

qmax, lps m a x x 8 = - 2 8 4 4 X l 1 x 4 --I- 5 0 8 8 x 0"923 x O'28 x O'048 x 4 qmin, lps

Lma x , m A,-,- 0 846 0 566 xO.097 X4 - 0.202 x 2 x 3 x 4 Lmi n , m -- q-~q- X 1 " X2.

- 0 . 2 5 x 2 x 3 x 4 - 4 1 6 x I_ x 3 x 4 Ti . . . . . min Er, % - 0 . 0 6 x 6 - 0 . 1 x 7 - 1 5 985 (46) a,.lps

where

q = x l ;

V = x 5 ; tco =x2 ; Nz =x3 ; N w = x 4 ; V k o = x6 ; Vap = x7 ; and P = x8 �9

Transforming the objective function into a geometric programming format resulted in the following:

Go = -x8 (47)

and the system constraints were given as:

Value of produce, S/ha 975.0 Value of produce, S/ha 1,950.0 Cost of production, S/ha 494.0 Cost of water, $/ha-m 40.00 Cost of labor, $/h 3.00 Cost of ditch construction, $/lin. m 13.50 Life of project= 10 years; interest rate= 15% Annual cost, $/lin - m 3.25 Cost of runoff, $/m 3 0.06 Cost of deep percolation, $/m 3 0.10

System constraints Value

3.20 0.63

402.50 92.00

8,000.00 > 80.00 169.00

- - ' , - - 0 8 4 6 0566 0097 G l = 2 8 4 4 x ~ l x a + ~ , 4 x l . x2. x 3" ~ 4 + 0 . 2 0 2 x 2 x 3 x 4 + O . 2 5 x 2 x 3 x 4 + 4 1 6 x ~ l x 3 x 4 + O . O 6 x 6 + O . l x T + x s + 1 5 9 8 5

5 0 8 8 X1-0"923 X 0"283 X 0"097

(48) Ge=0 .63 xi 1 < 1 (49)

G3 = 0.313 x l < 1 (50)

G4 = - 57.76 < 1 G 5 = 2 x31 < 1 (52) 0 .7x 0'425 x 0"799 + 0 .52x 0"799 + 2.1x 0'425 + 1.6

(51) G 6 = 0 . 1 1 3 7 x3< 1 �9 (53)

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Results and discussion

G 7 = 0.000125 x 2 x 3 X4--< 1 (54)

G8 = x5 + 4942 xi -1"227 x~ -2'227 _< 1 (55)

x2

6 9 a = 10.1 x2 x3 x4 < 1 . . . . --0575 0799 +35.3xllxO'799X4 + 278XI0"575X4 +45 .5X10 '575X4 X 0'799 +35.3 Xl I x 4 X 0'799 +216xi-lx4 X 6 -1- "q-D.9 X 1 " X 2 ' X 4

(56)

G9b = X6 .t_ 4 ~ . 9 X 1-0575. X2.0799X4 +35.3x{lxO.V99x4+278x~O.SVSx4+,+3.3xl,~ ~- -0575. X4X5.0799 + 35.3 xiqx4 x~ + 216 xi-l x4 < 1

10.1 x 2 x 3 x 4

(57)

Gloa -- 4 5 . 5 Xl ~'575 X 0'799 X 4 + 3 5 . 3 Xi -1 X20'799 X 4 + 278xi -~ X 4 + 45.5xi -~ X 4 X50'799 + 3 5 . 3 xi -1 X 4 X50"799 + 216Xl I X 4 < 1

x 7 + 7834 xi -1 x4

(58)

x7 + 7 8 3 4 x l l x 4 x 0"799 x 0'799 -< l a l 0 b - 4 5 . 5 x10'575 x 0"799 x 4 + 3 5 . 3 xi -1 x~ + 278Xl -~ X 4 + 4 5 . 5 X l "0"575 X 4 + 35 .3 xi -l x4 + 2 1 6 x 1 1 x4

Gll = 0.319 x l l x 4 < -- 1 .

The results obtained are presented and discussed in two parts. The first part discusses the sensitivity of benefit/cost ratio and the performance and design parameters to changes in the system cost coefficients, particularly the changes in the cost of deep percolation and runoff water, are discussed. The second part discusses the analysis of the effect of designing irrigation systems by selecting an application efficiency a priori.

Sensitivity of design and performance parameters to the changes in system cost coefficients

In designing furrow irrigation systems, the design parameters - time of application per set, infiltration opportunity time, length of run, furrow inflow rate, number of furrows per set, and the number of sets in the width and length directions - must be specified. These parameters define the overall performance of the system. The performance parameters are the means the designer uses to evaluate the system. To study the effect of the cost coefficients on these design and performance parameters, the supply flow rate was fixed at 169 lps, which was found to be the optimal supply flow rate for the area under study (Reddy and Apolayo 1991).

The cost of water was varied from $ 5/ha-m to $ 600/ ha-m for a low value crop (value of produce was $ 989/ha), and from $ 5/ha-m to $ 760/ha-m for a high value crop (the value of produce was $1971/ha), the cost of runoff recovery system was varied from $ 0.04/m 3 to $ 0.24/m 3 for the low value crop, and from $ 0.04/m 3 to $ 0.60/m 3 for the

(59)

(60)

high value crop, and the cost of drainage was changed .~ 3 from $ 0.04/m to $ 0.40/m for the low value crop and

$ 0.04/m 3 to $ 0.84/m 3 for the high value crop. These same values were used by Wallender et al. (1990) in San Joaquin Valley, California. The performance parameters - the tail- water efficiency, the deep percolation efficiency, and the application efficiency - are plotted as a function of the benefit/cost ratio. This ratio was obtained by dividing the net benefit from crop production, which is the difference between the total value of the crop less the total cost of the system, by the total cost of the system.

Sensitivity to cost of irrigation water

To study the effect of changes in the cost of water on the design and performance parameters, the cost of water was increased as mentioned in the previous section. The other cost coefficients were fixed at: $ 3/h (labor), $ 3.25/linear-m (head ditch), $ 0.06/m 3 (runoff), and $ 0.10/m 3 (drainage). The results obtained are presented in Figs. I a, b.

In general, as the cost of water increased, the furrow inflow rate decreased. Since the supply flow rate was fixed (169 lps), a reduction in the furrow inflow rate represented an increase in the number of furrows irrigated per set, and therefore, a reduction in the total number of sets in the field. In order to satisfy a desired level of water requirement efficiency, as the inflow rate decreased, the duration of application per set increased.

I00

801

o~ 6 0

U.. ,. 4 0 ILl

2O

0 I -0,1

}_ Z~Z~ Z~A Z~ A Z~ u

- -

_ 0

0 0 0 ~

o Eo Z~ DEEP PERC, EFFIC, n RUNOFF EFFIC,

i i i i. i 0 0.1 0 . 2 0 . 3 0 . 4

BENEFIT/COST RATIO

100

8 0

6 0

0 I.L. U. 40 ILl

b

Fig. 1 a, b

20

0 0 0

0

o Eo DEEP PERC. EFFIC.

rn RUNOFF EFFIC,

0 I I I I I 0 , 5 7 0 , 7 7 0 . 9 7 1.17 1.57 1,57

BENEFIT/COST RATIO

Sensitivity of performance parameters to cost of water. a Low value crop. b High value crop

For both the low and the high value crops, the increase in the cost of water was offset by a less expensive runoff recovery and drainage systems. As the cost of water increased, the inflow rate decreased and, consequently, the water loss due to runoff decreased.

The application efficiency was influenced very much by the cost of water. As shown in Figs. 1 a, b, an increase of 26% in the application efficiency was observed when the cost of water changed from $ 5/ha-m to $ 600/ha-m. Conversely, an increase in the cost of water significantly reduced the net benefit from crop production. For instance, when the cost of water was cheap ($ 5/ha-m) a benefit/cost ratio of 0.4 was obtained for the low value crop. But when

131

the cost of water was expensive ($ 600/ha-m), a negative benefit/cost ratio was obtained. The total cost of the system increased as the cost of water increased. For instance, the cost of the system increased from $ 701/ha to $1100/ha as the cost of water increased from $ 5/ha-m to $ 600/ha-m for the low value crop, and from $ 701/ha to $1192/ha when the cost of water increased from $ 5/ha-m to $ 760/ha-m for the high value crop. Conversely, the net benefit decreased from $ 299/ha to - $ 1 4 4 / h a for a low value crop, and from $ 1218/ha to $ 683/ha for the high value crop for the same range of values. The negative benefit observed in the low value crop represented a high application efficiency, but it also represented loss of money from crop production to the farmer. As the cost of water increased, the total cost of the system increased due not only to the cost of water but also to an increase in the number of ditches required to irrigate the field. The same trend was observed in the high value crop with the exception that the high value of the application efficiency did not result in negative net benefit from crop production. In this case, the high value of the crop absorbed the high system cost, and a higher application efficiency was optimal.

For both the low and high value crops, under given site conditions, the deep percolation efficiency was relatively insensitive to the increase in the cost of water. As shown in Figs. 1 a and b, the value of the deep percolation efficiency fluctuated between 79% and 82%.

The results showed that an increase in the cost of water increased the tail-water efficiency, and the application efficiency. This had a positive impact from the water quality standpoint. However, when the value of the crop was low, application efficiencies higher than 52% were not optimal for the farmer because negative net benefits were obtained from the irrigation system. This showed that a high value of application efficiency was obtained at the expense of an increase in the cost of the system. Therefore, designing a furrow irrigation system for high efficiency may not be justifiable when growing a low value crop. I f the cost of contamination (water quality) increases, growing a low value crop may not be economical.

The results suggested that in areas where the cost of runoff and/or deep percolation are high, which is the case in the San Joaquin Valley, California, and many other irrigated areas in the world, high values of application efficiencies must be achieved. This implies that high value crops should be grown in those areas.

Sensitivity to cost of runoff water

The sensitivity of the system to the cost of runoff recovery system was analyzed for values of runoff water varying from $ 0.04/m 3 to $ 0.24/m 3 for a low value crop, and $ 0.04/m 3 to $ 0.60/m 3 for a high value crop. The cost of irrigation water was fixed at $ 40/ha-m, and the remaining cost coefficients were held constant as shown in the previous section. The results are presented in Figs. 2 a, b.

In the case of a low value crop, as the cost of runoff water increased, the furrow inflow rate decreased with a con-

132

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8O

6C ~.~ Z

" 4 0 I.L

2 0

0 . 2 3

- - o o c - ~ - - o - - - _ . o _ _ ~ ~

o Eo Z~ DEEP PERC, EFFIC. n RUNOFF EFFIC,

I I 0 . 3 3 0 . 4 3

BENEFIT /COST RATIO

I 0 0

I

BO

) . 6 0 r Z W f j u.. ,, 4 0 UJ

2 0 o Eo

DEEP PERC, EFFIC, n RUNOFF EFFIC.

0 I I I I I 1,1:3 1 .23 h33 643 1.53 h63

b B E N E F I T / C O S T R A T I O

Fig. 2a, b Sensitivity of performance parameters to cost of runoff. a Low value crop. b High value crop

comitant reduction in runoff volume while keeping the length of run constant. This resulted in reduced total system cost by reducing the capacity of the reuse system required. The increase in the inflow time per set did not significantly affect the runoff losses from the system although it did affect the deep percolation losses. Tail-water efficiency increased by 20% when the cost of the runoff recovery system was changed from $ 0.04/m 3 to $ 0.12/m 3 (Fig. 2 a). But it was relatively insensitive to further increase in the cost of the runoff water. This can be attributed to the fact that an increase in the tail-water efficiency increased the deep percolation losses, but once a certain level of the deep percolation loss was reached, the system did not loose more wa-

ter by deep percolation. Conversely, the deep percolation efficiency dropped from 82% to 70% when the cost of runoff increased from $ 0.04/m 3 to $ 0.12/m 3. A further increase in the runoff water did not affect this performance parameter. The application efficiency was relatively insensitive to the cost of runoff water. The cost of the system increased from $ 709/ha to $ 809/ha as the cost of runoff water increased from $ 0.04/m 3 to $ 0.24/m 3, whereas the net benefit decreased from $ 306/ha to $ 230/ha for the same range of cost of runoff water. The results suggested that the increase in the cost of the runoff recovery system resulted in an increase in the tail-water efficiency. How- ever, the increase in the tail-water efficiency was absorbed by a decrease in the deep percolation efficiency without a significant change in the application efficiency.

For the high value crop, the tail-water efficiency increased and the deep percolation efficiency decreased as the cost of the runoff recovery system increased. For instance, an increase of 33% in the tail water-efficiency was observed when the cost of runoff water increased from $ 0.04/m 3 to $ 0.60/m 3. Conversely, the deep percolation efficiency dropped from 85% to 66% for the same range of values of the runoff water. The application efficiency increased from 45% to 58% when the cost of runoff water was varied from $ 0.04/m 3 to $ 0.60/m 3. In this case, the high value of the crop could sustain the drainage losses from the system and still give a higher application efficiency (Fig. 2b). The total cost of the system increased from $ 746/ha to $ 911/ha when the cost of runoff changed from $ 0.04/m 3 to $ 0.60/m 3. This increase was due to the cost of runoff water. The net benefits decreased from $1228/ha to $ 952/ha for the same range of cost of runoff water.

Sensitivity to cost of drainage water

To study the effect of the cost of the drainage water on the design and performance parameters of the irrigation system, the cost of the deep percolation water was increased from $ 0.04/m 3 to $ 0.40/m 3 for a low value crop, and from $ 0.04/m 3 to $ 0.84/m 3 for a high value crop. The cost ofrunoffwater was fixed at $ 0.06/m 3, and the remaining cost coefficients (water, labor, and ditch) were held constant at the values shown in the previous section. The results obtained are presented in Figs. 3 a, b.

For the low value crop, the change in the application efficiency in response to the change in the cost of drainage water was not significant because the increase in the deep percolation efficiency was offset by a decrease in the tail- water efficiency. Usually an increase in the deep percolation efficiency would result in a decrease in the cost of the system. In the case of the low value crop, since a high system cost was not optimal, the application efficiency did not increase as the cost of drainage water increased.

In the case of the high value crop, the deep percolation efficiency increased from 62% to 88% as the cost of drainage water increased (Fig. 3 b). The value of tail-water efficiency fluctuated around 70% and 78%. The application

133

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0 Z hi

LL ,. 40 LIJ

20

o hi8

a

0 0 0 ,"~ J ~ 13 ('t

o Ea Z~ DEEP PERC. EFFIC, o RUNOFF EFFIC,

I I I 0.2:5 0.28 0.3:5

BENEFIT/COST RATIO

>- 0 Z LIJ

tl. h LIJ

I00

d

8O

60

40

20

o o

o o o o o o

o Ea

& DEEP PERC, EFFIC. m RUNOFF EFFIC,

I I I I I I 1,02 h 2 2 1,42 1.62

b B E N E F I T / C O S T RATIO

Fig. 3a, b Sensitivity of performance parameters to cost of drainage. a Low value crop. b High value crop

efficiency increased by 39%, which is a significant increase, as the cost of drainage water increased from $ 0.04/m 3 to 0.84/m 3. This increase in the application efficiency was due to the increase in the deep percolation efficiency. In this case, a high value of application efficiency, which usually represents high system cost, was optimal because of the high net return from the crop. This result suggested that high value crops should be grown if high level of net return from crop production is to be achieved in the presence of high cost of drainage water.

In general, deep percolation losses decreased with an increase in the cost of drainage water. A reduction in the deep percolation losses is beneficial not only to crop pro-

duction (less nitrogen is carried away from the crop root zone), but also to the quality of groundwater because of the reduction in the amount of chemicals that would leach out of the crop root zone.

Effect of designing irrigation systems by selecting an application efficiency a priori

Traditionally, furrow irrigation systems have been designed by arbitrarily specifying a value of application efficiency, and then selecting a combination of design variables which gives this application efficiency. Since there are several combinations of design variables that give the same application efficiency, the design may or may not guarantee an optimal design (maximum net benefits) from the farmer's point of view. The effect of selecting an application efficiency a priori on the net benefit is discussed next.

The preceeding section has shown how the application efficiency of a furrow irrigation system was influenced by the system cost coefficients. For instance, high application efficiencies were required when the cost of water was high. Similarly, the application efficiency increased as the cost of runoff water and drainage water was increased. Gener- ally, as the application efficiency increased the cost of the system also increased. As expected, high values of application efficiency represented low values of runoff and deep percolation losses (Figs. 1-3) . This is a positive aspect from the water quality standpoint. However, this also represented low net benefits from crop production when a low value crop was grown. Therefore, the main objective in the design of a furrow irrigation system should be to maximize the net benefit and allow the mathematical model to select the optimal value of the application efficiency and the corresponding values for the optimal design parameters - furrow inflow rate, length of run, time of inflow, number of furrows per set, and number of sets in the length and width directions.

In this study the optimal cost and the optimal application efficiency were obtained for each combination of the cost coefficients. The results obtained are summarized into two graphs (Figs. 4 a, b), one for the low value crop, and the other for the high value crop. In each case, the cost coefficients of the three optimal designs were selected. Then, the cost coefficients of these optimal designs were used to- gether with the design variables (not the cost coefficients) of several other optimal designs to calculate the net benefits from the irrigation system. Table 3 presents the values of the cost coefficients for the optimal designs selected.

Curve no 1 of Fig. 4 a, obtained with changes in the cost of runoff, shows an optimal application efficiency of 47%. When the application efficiencies were less than the optimal, the net benefit did not change significantly. However, when the application efficiencies were greater than the optimal, the net benefits dropped very rapidly. Curve no 2 of the same figure was obtained with changes in the cost of drainage water. It shows an optimal application efficiency of 51%. In this case the application efficiencies, varying

134

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OPTIMAL

OPTIMAL ==

E # 5

~ET BENEFITS ARE GIVEN IN THOUSANDTHS I I I I

45 50 55 60 65 A P P L I C A T I O N EFFICIENCY, %

Table 3 Cost coefficients for the optimal designs used in Figs. 4a, b

Coefficient Low value crop (high value crop)

Changes in Changes in Changes in cost of cost of cost of water runoff drainage

Cost of water, 400 (120) 40 (40) 40 (40) (S/ha-m)

Cost of labor, 3.00 (3.00) 3.00 (3.00) 3.00 (3.00) (S/h) Cost of ditch, 3.25 (3.25) 3.25 (3.25) 3.25 (3.25) (S/L-m) Cost of runoff, 0.06 (0.06) 0.12 (0.08) 0.06 (0.06) ($/m 3)

Costofdeep 0.10 (0.10) 0.10 (0.10) 0.36 (0.36) percolation, ($/m 3) Optimal application 55 (52) 47 (47) 52 (54) efficiency, (%)

40

:55 OPTIMAL

3 0 OPTIMAL

, 25

U_ W z 2 0 W m

W Z

Io

5

NET BENEFITS ARE GIVEN IN THOUSANDTHS

C I i I I I :55 4 0 45 50 55 60

b A P P L I C A T I O N EFFICIENCY, %

Fig. 4a, b Optimal application efficiencies, a Low value crop. b High value crop

from 47% to 53%, were very close to the optimal. But for application efficiencies greater than 54%, a decrease in the net benefits was observed. Curve no 3 of Fig. 4 a, was obtained with variation in the cost of water. An application efficiency of 55% was considered for this case. However since all the application efficiencies of this curve represented negative benefits, they were not optimal to the farmer. A similar trend was also observed for the high value crop (Fig. 4 b). However, in this case, the net benefits did not drop so rapidly as the application efficiencies decreased or increased from the optimal application efficiencies. Curve no 1 of this Figure, obtained when the cost of run-

off was changed, shows an optimal application efficiency of 50%. Application efficiencies below and above the optimal application efficiency showed a decrease in the net benefit. Curve no 2, which was obtained when the cost of water was changed, shows an optimal application efficiency of 55%. A decrease in the net benefit was observed when the application efficiencies were below or above the optimal application efficiencies. Finally, curve no 3, obtained when the cost of drainage was changed, shows an optimal application efficiency of 52%. Application efficiencies below and above this optimal show a decrease in the net benefit to the farmer.

Finally, data from Figs. 4 a, b were reduced into a single graph. In this graph, the relative application efficiency was plotted against the relative net benefit. The results obtained are presented in Fig. 5. The relative application efficiency and the relative net benefit were calculated as follows:

Relative E a = E , (61)

and

Relative Net Benefit = NB (62) NB *

where E* represents the optimal application efficiency in Figs. 4a, b, Ea represents the application efficiency selected a p r i o r i (the non-optimal cases), NB* represents the net benefit corresponding to the optimal application efficiency, and NB represents the net benefit corresponding to the application efficiency selected a p r i o r i (the non-optimal cases).

This study was carried out to show how the net benefits changed as the application efficiency deviated from the optimal application efficiency. Using statistical methods, a quadratic curve was fitted to the data, and a high corre- lation coefficient (R 2) of 0.853 was obtained between the

135

Fig. 5

I-- ,7 LU Z I.=,1

I'- LU Z

W >

I-- < J w

1.2

hO

0.8

0,6

0,4

0,2

OPTIMAL POINT

�9 X 0 0

0 , ( I I I I I I 0 . 6 0 . 8 1.0 1.2

R E L A T I V E Eo

Relative application efficiency versus relative net benefits

actual and predicted relative net benefits. As shown in Fig. 5, a decrease in the relative net benefit was observed as the relative application efficiency deviated (increased/decreased) from the optimal.

These results showed that the optimal designs obtained were optimal for its corresponding cost coefficients. Once these cost coefficients were changed, a decrease in the net benefits was observed. Furthermore, for a selected application efficiency, several different optimal designs were possible depending upon the relative magnitude of the cost coefficients and constraints. This suggested that the same application efficiency could give different net benefits to the farmer depending upon the combination of values used for the design variables to obtain the specified application efficiency. The role of the designer is, therefore, to obtain the optimal combination of values for the design variables that would yield the maximum net benefit to the farmer. This can be accomplished by formulating the design problem as an optimization problem.

Conclusions

The design problem of a furrow irrigation system was formulated as an optimal design problem in terms of the system variables and system constraints. The design variables considered were the furrow inflow rate, the time of application per set, the length of run, and the number of sets in the width and length directions.

The application efficiency was sensitive to the cost of water. As the cost of water increased, the application efficiency also increased. For instance, the application efficiency increased from 45% to 61% as the cost of water in-

creased. Tail-water efficiency increased with an increase in the cost of water. The deep percolation efficiency was relatively insensitive to the increase in the cost of water. The cost of irrigation water was offset by a least expensive runoff recovery system. The total cost of the system increased and the net benefit decreased as the cost of water increased. The optimal design was influenced by the value of the produce. For example, application efficiencies greater than 52% were not optimal when a low value crop was used; nevertheless, they were optimal when a high value crop was used.

In the case of a low value crop, the design and performance parameters were found to be sensitive to the increase in the cost of the runoff recovery system up to $ 0.12/m 3. But they were relatively insensitive to a further increase in the cost of runoff. The optimal furrow inflow rate decreased as the cost of runoff water increased. Time of application per set increased as the cost of runoff increased. The optimal length of run was relatively insensitive to the cost of runoff. The reduction in the inflow rate reduced the volume of runoff. The increase in the inflow time did not affect significantly the runoff losses. The tail-water efficiency increased by 20%, and the deep percolation efficiency decreased 12% as the cost of runoff water was increased. The application efficiency was relatively insensitive to the cost of runoff. In this case, the increase in the tail-water efficiency was absolved by the decrease in the deep percolation efficiency: However, the total cost of the system increased and the net benefit decreased as the cost of runoff increased.

In general, tail-water efficiency increased and deep percolation efficiency decreased as the cost of runoff increased. The cost of runoff was offset by a less expensive runoff recovery system but the total cost of the system increased and the net benefit decreased as the cost of the runoff recovery system increased.

The design and performance parameters were sensitive to the cost of drainage water. The optimal inflow rate increased and the time of application per set decreased as the cost of drainage increased in both the low and the high value crops. For the low value crop, the deep percolation efficiency increased by 12% and the tail-water efficiency decreased by 20% as the cost of drainage water increased. The application efficiency was relatively insensitive to the cost of drainage water. The increase in the deep percolation efficiency was offset by the decrease in the tail-water efficiency. However, the cost of the system increased and the net benefit decreased as the cost of drainage increased.

In the case of the high value crop, the deep percolation efficiency increased by 20% and the tail water efficiency fluctuated around 78% and 70% as the cost of drainage increased. The application efficiency increased by 39% with the increase in the cost of drainage water. But the total cost of the system increased and the net benefit decreased as the increase in the cost of drainage.

As the application efficiency deviated (increased) from the optimal, the net benefits decreased very rapidly. How- ever, when the deviation was negative, the net benefit decreased very slowly. Similarly, the relative net benefit de-

136

creased as the re la t ive appl ica t ion ef f ic iency decreased and/or increased f rom the op t imal appl ica t ion eff iciency.

The procedure presented here can be used to obtain the combina t ion of des ign var iables o f a furrow irr igat ion sys tem that max imizes the net benef i ts f rom crop produc- t ion whi le sa t i s fy ing the water qual i ty constraints . The procedure s impl i f ies the des ign p rob lem and reduces the t ime required to obta in the op t imal design.

Severa l combina t ions o f the des ign var iables may g ive the same appl ica t ion eff iciency. Therefore , the des ign o f furrow i r r igat ion sys tems should be formula ted as an op- t imiza t ion p rob lem with maximiza t ion o f profi ts f rom crop product ion as the objec t ive funct ion a long with a set of constraints to be sat isf ied by the design. Selec t ing the ap- p l ica t ion ef f ic iency a priori might y ie ld low or, in some cases, negat ive benefi ts to the farmer.

As expected , high values o f the appl ica t ion ef f ic iency represented low values o f runoff and deep perco la t ion volumes, but also represent h igh cost of the system. There- fore, in areas with high contamina t ion costs, high value crops should be grown in order to avoid negat ive benefi ts f rom crop product ion. I f low value crops are to be grown, then other means to increase the appl ica t ion ef f ic iency whi le keeping the sys tem cost low should be sought.

Abbreviations: The following symbols are used in this paper: a, b, c, d = exponents in water requirement efficiency equation; C a = head ditch construction cost; Cdp=cost of deep percolation; Cki and Ckj= positive and negative coefficients in the objective function and constraints, respectively; C t = seasonal cost of labor; Cp = cost of production; C~o = cost of runoff; Cw = seasonal cost of water; C 1 = water cost coefficient; (?2 = labor cost coefficient; C 3 = head ditch construction cost coefficient; Ca = runoff water cost coefficient; C 5 = deep percolation water coefficient; D n = net depth of application; E R = water requirement efficiency; Gi = constraint function i; G O = objective function;/ave=average infiltration rate along a furrow; Ik=number of positive terms in constraint k; Jk = number of negative terms in constraint k; K=infiltration coefficient; K~ =fraction of field length irrigated to requirement; K 2 = units conversion coefficient; K 3 = coefficient in water requirement equation; L=furrow length; Lf=field length; Lmax=maximum allowable furrow length; Lmin=mmimum allowable furrow length; N= number of design variables; Nf= number of furrows in field; Nfs=number of furrows irrigated per set; Ni=number of irrigations per season; Nl=number of sets in length direction; Ns= number of sets per irrigation in field; Nw = number of sets in width direction; n = Manning's roughness coefficient; P = furrow wetted perimeter; Pn = seasonal net benefit; Pv = total value of

produce; Qs = total available flow rate at field; q = furrow inflow rate; qmax=maximum nonerosive flow rate into furrow; qmin=minimum flow rate into furrow; So=furrow slope; Tcs=total cost of system; too=time of application per set; V= infiltration opportunity time at point x; Vap= volume of deep percolation; VRo = volume of runoff; W= furrow spacing; Wf= field width; xi=design variable i; YR=relative crop yield; a= infiltration exponent; akin and akjn = exponents of system variables of positive and negative terms, respectively, of objective function and constraints; a z , a 3, a4, a5 =units conversion coefficients; tr~ = fraction of set time used for irrigating filed; fl and ~r advance parameters; A and r/=constants dependent upon intake family; 0nk = coefficient of variable k of term n, after linearization; ~k = right-hand-side constant of constraint k, after linearization.

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Gill EG, Murray W, Wright MH (1981) Practical optimization. Academic Press, New York

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Reddy JM, Clyma W (1981 a) Optimal design of border irrigation systems. J Irrig Drain Eng ASCE 107: 289-306

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Reddy JM, Clyma W (1983 a) Optimizing furrow irrigation runoff recovery systems. Trans ASAE 4: 1050-1056, 1063

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Reddy JM, Apolayo H (1991) Sensitivity of furrow irrigation system cost and design variables. J Irrig Drain Eng ASCE 117:201-219

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optimization of furrow irrigation system design parameters considering drainage and runoff water...

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