Choosing optimal design depth for surface irrigation systems

Agricultural Water Management, 6 (1983) 335--349

335

Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

CHOOSING OPTIMAL DESIGN D E P T H F O R SU RFA CE I R R I G A T I O N SYSTEMS J . MOHAN REDDY and WAYNE CLYMA Department of Agricultural and Chemical Engineering, Colorado State University, Fort Collins, CO 80523 (U.S.A.)

(Accepted 15 March 1983)

ABSTRACT Reddy, J.M. and Clyma, W., 1983. Choosing optimal design depth for surface irrigation systems. Agric. Water Manage., 6: 335--349.

The surface irrigation system design was formulated as a mathematical programming problem. The minimum cost of a furrow irrigation system for a hypothetical case was calculated for different design depths (25, 51, 76,102 and 127 ram). The crop yields and net returns were simulated for the given design depths. A design (depletion) depth of 51 mm was found optimal under the given conditions.

INTRODUCTION T h e d ep th o f soil water depletion affects crop yields. As the depletion d e p t h increases, crop yields are reduced in p r o p o r t i o n to the a m o u n t o f depletion. Because water flows over the soil in surface irrigation, application efficiency is strongly related to the r e q u i r e m e n t depths. Peri et al. (1979) presented a m e t h o d to evaluate the optimal r e q u i r e m e n t (depletion) d e p t h but did n o t deal with th e question o f how the desired dept h could be applied to the field. In addition, t h e y assumed a linear relationship between cost and the volume o f excess and deficit water. The objective o f this paper is to present a m e t h o d to find t he optimal ( m a x i m u m n e t returns) design dept h along with th e optimal values for the o t h e r design parameters -- flow rate, time o f inflow, length o f run, etc. -- considering t he costs o f t he system and returns f r o m crop p r o d u c t i o n o f a f ur r ow irrigation system. SOIL-WATER DEPLETION AND CROP YIELD Irrigation always incurs water, labor and energy costs which must be balanced against p r o d u c t i o n returns. Higher crop yields can be e x p e c t e d with m o re f r e q u e n t irrigations; however, m or e f r e q u e n t irrigations increase the seasonal cost o f irrigation. On t he o t h e r hand, bot h crop yields and costs are

0378-3774/83/$03.00

© 1983 Elsevier Science Publishers B.V.

336

lower when irrigations are fewer. In addition, from a hydraulics point of view, greater depths of depletion require more time to infiltrate a given depth of water. Therefore, more r u n o f f losses usually occur at the downstream end of the field. Neither of the situations, higher yields with higher costs and lower yields with lower costs, need to be optimal. In optimization, quantitative relationships must be established. First actual and potential evapotranspiration values are needed to quantify plant stress. They can be estimated using the Jensen-Haise m e t h o d (Jensen, 1973). The relationship between the reference crop evapotranspiration, ETr, and the potential evapotranspiration of a given crop, ETp, is expressed as: ETp = Kco ETr

(1)

in which Kco is the crop coefficient (specific to a given crop). The crop experiences a certain degree of stress under field conditions. Therefore, the actual evapotranspiration, ETa, is usually less than the potential evapotranspiration, ETp, and is given as: ET a = K s EWp = g c o K s EWr

(2)-

in which K s is the stress coefficient which is dependent upon the degree of soil saturation. There are several models (linear and power functions} that relate the stress coefficient with the degree of soil saturation. The difference between these models is not significant when the moisture content is close to field capacity. As the moisture content decreases, the difference among these models becomes obvious (Boonyatharokul, 1979}. The linear and power function models have some empirical coefficients which are soil, climate and crop dependent. In lieu of this information for the given situation, the equation developed by Jensen {1973), which is more general, was used to estimate K s and is given as: K s = log(100(0/0s) + 1}/log(101)

(3)

in which 0 is the water content of the soil in the root zone and 0s is the water content of the soil at saturation in the root zone. According to Eq. (3) the actual evapotranspiration never reaches the potential evapotranspiration. However, this was considered not to be a serious problem for this hypothetical case. Under actual conditions, a more appropriate functional relationship may be used. The water content, 0 i, on any day is dependent upon the moisture content on the previous day, 0 i-1 ; rainfall, Ri; irrigation, Di; and evapotranspiration, ETi; on that given day. The relationship is expressed as: 0 i = 0 i- 1 +

Ri - ETi + Di

(4)

The total available water, TAW (mm), in the root zone is: TAW = ( F C - PWP)Tb Y

(5)

in which FC is the field capacity, PWP is the permanent wilting point, 7b is the apparent specific density of the soil and Y is the depth of the root zone

337 (mm). A field is irrigated when only a fraction, say ¢, of the total available water is depleted; so the readily available water, RAW, is given as (Hart, 1975): RAW = ¢ TAW

(6)

in which the availability factor ~b is constrained by 0 < ~ ~< 1

(7)

The optimal value of ¢ is crop, soft and weather dependent. Because TAW varies with r o o t development, different RAW amounts result for a seasonally constant value of ¢. In the present analysis, a constant a m o u n t of water was depleted before each irrigation throughout the season and therefore the value o f ~b was changed with the value of TAW (according to Eq. (6)). The effect of changing root depth on TAW was incorporated (through Eq. (3)) into the crop growth model. A crop production function is needed to relate stress to yield. Any production function that relates actual and potential evapotranspiration to crop yield would suffice. The multiplicative production function developed by Jensen (1968) was used here: N YR = II

(ETa/ETp)

k i

(8)

i=1

in which YR is the relative crop yield, N is the n u m b e r of crop growth stages and ki are the crop senstivity coefficients. Since ki /> 0, a higher ratio of ETa/ET p gives higher yields. A high value of ki indicates that the crop is very sensitive to water stress in stage i, whereas ~.i = 0 implies that the crop is not sensitive at all. The level of soil-water depletion has some effect on the infiltration: the infiltration rate of any given soil is lower at higher water contents. This aspect was n o t incorporated into the model since it makes the analysis more complicated. In addition, the following assumptions were made: (1) The infiltration function was the same for all levels of depletion. (2) The level of depletion was uniform throughout the length of run. (3) A b o u t 90% of the field length was irrigated to the requirement depth, and consequently the effect of nonuniformity on yield was not considered. (4) The soil was well drained so there was no effect from groundwater on crop yield. IRRIGATION SYSTEM DESIGN EQUATIONS

Mathematical equations that express the relationships between the irrigation quality parameters and the design (decision) variables are an integral part of any simulation or optimization study. Although the developed m e t h o d is applicable for border and furrow irrigation systems, only furrow

338

irrigation was considered here. The design equations presented by Ley and Clyma (1980) as developed by the USDA (1979) were used in the present analysis. To derive the equations, a power advance function of the following type was utilized: (9)

tx = gx h

in which t x is the advance time to distance x (min), x is the advance distance (m), h is a constant for a given infiltration family and g is a constant that is defined as follows: g = (3.281 k d q d s d / 2 / C 2

(10)

)

in which So is the slope of the furrow, q is the inflow rate (1 s-1), k is the units conversion factor of 4.831 and C2 and d are constants for a given intake family. Parameters h and d are related by: (11)

h = l -- d

The constants for each infiltration family can be found in Ley and Clyma {1980). Assuming recession time was negligible, the intake oppurtunity time to any given point along the length of run is: (12)

to = ti - - t x or

(13)

to = ti - g x h

where ti is the time of inflow into the furrow (min). TABLE

I

Data used for example

Field variables and c o n s t a n t s S O = 0 . 0 0 1 ; L F = 8 0 5 m ; W F -- 4 0 2 m ; n = 0 . 0 4 ; Q ffi 1 5 8 1 s-~; W = 0 . 7 6 m ; n i = 6 ; I f = 1 . 5 ; a = 2 . 2 8 3 ; b ffi 0 . 7 9 9 ; c -- 7 . 0 ; h = 2 . 2 2 7 ; d = -1.227 a n d C 2 -- 1 3 7 6 7 The soils at the site were assumed of 120 mm per m of soil. Rooting

depth

to be sandy

of the corn crop was assumed

loams with a water

holding

capacity

to be 1.20 m.

Constraints D u = 7 6 m m ; E r >I 8 0 % ; L m a x -- 4 0 2 m ; L m i n = 9 1 . 5 m ; q m i n q m a x = 3 . 2 1 1 s-~; T m a x = 8 0 0 0 m i n ; K s -- 0 . 9

Cost c o e f f i c i e n t s Water = $0.004/m s ; Labor = $3/h Ditch construction -- $ 3 . 2 5 p e r m l e n g t h ;

and ~ = 1.0

= 0.63 l s-';

(TAW)

339 The USDA (1979) uses an infiltration function of the following form:

z=atbo +c

(14)

in which z is the cumulative depth of infiltration (mm), and a and b are constants for a given intake family (Table I). The depth Dx infiltrated at any point x is proportional to the infiltration oppurtunity time at that point, so:

Dx = [ a ( t i - tx) b + c ] P/W

(15)

in which W is furrow spacing (m) and P is an empirical coefficient. In furrow irrigation, unlike border irrigation, the infiltration is radial at least initially. The total infiltration per unit length of furrow depends u p o n the wetted perimeter, which is a function of the flow rate, slope of the furrow and the field roughness. To take into account the effect of wetted perimeter on infiltration rate, Ley and Clyma (1980) developed the following relationship between the above parameters and the furrow wetted perimeter:

/ qn \o.31s

v= 0.3304

)

+ 0.21

(16)

in which n is the Manning roughness coefficient. Equations for depth of deep percolation and runoff are given by Ley and Clyma (1980). The reader is referred to the above article for details. OBJECTIVE FUNCTIONANDCONSTRAINTS The most common objective functions are either maximization of profit or minimization of costs. In the present study, minimization of costs, while still providing an adequate irrigation over a specified field length, was chosen. The costs for a furrow irrigation system include: (1) the cost of water, C,; (2) the cost of labor, C2; and (3) the cost of constructing the head ditch, C3. The following equation was used as a cost function for optimal design: min Go = ninlnw

(60c,qtinfs ÷ 1/60c2ati) + c3nlWF

(17)

C~ C2 C3 where Go is the cost function for the system ($ per field), c, is the cost coefficient of water used from either a canal or a well (S/l), c2 is the cost of labor (S/h), ~ is the fraction of the time labor is utilized during irrigation, c3 is the annual cost for ditch construction (S/m), nl is the number of lengths of run in the field, WF is the field width (m), ni is the number of irrigations per season, nfs is the number of furrows irrigated per set and n w is the number of sets in the width direction. In addition to the cost function, constraints must be defined. Constraints -

340 limit the design variables to feasible values. Maximum length of furrow based on physical or legal boundaries, rainfall erosion, and furrow based on pysical or legal boundaries, rainfall erosion, and furrow spacings based on type of crop, machinery or farmer's preference are all examples of constraints. All constraints must be met. The constraints were specified as follows: q/> qmin;

G1 = qmin/q ~< 1

(18)

q ~< qmax;

G2 = q/qmax ~< 1

(19)

in which qmin is the minimum flow rate into the furrow !(ls-1) and qmax is the maximum non~rosive stream (1 s-l). Marr (1967) defined qmax as follows: qmax = 0.63/S;

G2 = q S / 0 . 6 3 ~< 1

(20)

where S is the furrow slope (%). Since qmax for 0.1% slope, using the above equation, was twice the carrying capacity of most furrows in use, the actual qmax used iwas half of that calculated using Eq. 20 (qmax used at 0.1%, slope was 3.2 1 s-I). In this study, the design (depletion) depth was assumed given. The SCS furrow irrigation design procedure requires that the length of the field over which the design depth is to be met by a given irrigation is specified. Under certain circumstances, it may not be beneficial to irrigate the entire root zone to the design (requirement) depth ( R e d d y and Clyma, 1982) and the field may be slightly underirrigated without any economic loss due to reduced crop yields. The fraction of the length of run that is to be irrigated to the design depth is called the design depth, L d = K3L, and is specified in the design. K3 is constrained by:

O
G3 = D u / D x = 0.gL ~< 1

(21)

in which Du is the root zone water requirement depth (mm). The constraint can be expanded, using Eqs. 9 and 15, as: G3 =

Du

~< 1

(22)

{a[ti - g(O.9L) h ] b + c } P / W

or by substituting the relationship for P from Eq. 16 DuW

< 1 G3 = { a [ t i - - g ( O . 9 L ) h ] b + c } [0.3304 ( ~qn . s ) 0.31s + 0 . 2 1 ]

(23)

341 The run length cannot be greater than the actual field length. A shorter run length increases the cost of operation and a greater length may result in a less uniform distribution o f water along the length of the field. Therefore, the field length is restricted at both extremes: nl L = L F -* G4 = L F /nl L ~ 1

(24)

L ~ Lmax ~ Gs = L / L m a x ~< 1

(25)

L ~< Lmin -* G6 = Lmin/L ~ 1

(26)

where LF is the length of the field (m), Lmax is the maximum length of the run (m), and Lmi n is the minimum length of the run (m). Similar constraints were used for time, n u m b e r of furrows per set, and n u m b e r of sets with a given number of furrows and furrow spacing for the width of the field. They were as follows: nfsq = Q ~ G7 = n f s q / Q ~< 1

(27)

nlnw~ <~ Tmax ~ Gs = n l n w t i / T m a x ~< 1

(28)

nwnfs = nf -* G9 = n f / ( n w n f s ) ~< 1 (nf = W F / W )

(29)

where nf is the n u m b e r of furrows in the field, Tmax is the maximum time available per irrigation for the entire field (rain), and Q is the total flow rate available at the field (1 s-l). LEVEL OF DEPLETION AND OPTIMAL DESIGN An example field was analyzed and an optimal design for an irrigation was developed. The data for the field, the constraints and cost efficients are given in Table I. The values chosen for the constraints (based upon farmer's preferences) and some o f the cost coefficients are typical for some parts of Colorado. To evaluate the optimal depth of depletion, the crop yield was estimated b y simulation for different given depths of depletion. Next, the minimal cost design that satisfied the specified constraints (including the one on design depth) was developed using an optimization technique. Five different depths were introduced: 25, 51, 76, 102 and 127 mm. The potential evapotranspiration values from Stewart et al. (1977) for Fort Collins, CO, presented in Table II, were used. The soil water-holding properties are presented in Table I. Using the equations developed earlier, the actual ET was c o m p u t e d for the different levels of depletion. The beginning soil-water c o n t e n t was assumed to be at field capacity. Irrigations were applied whenever the depletion reached the specified depth. The relative yield o f corn was c o m p u t e d using the ki values o f Table III (see Eq. (8)) developed b y Hanks' as presented b y Stewart et al. (1977). The distribution of irrigations through the season is shown in Fig. 1. The number of irrigations per

~'~.

~

g

N

0

0

343 I

2

1441 I ~? ' ' ~~-~' I ~ -~ I "--~ ~ O=

L

T

~

I

1

4

2

4

0 I

3

4

~

Ou =102mm

i

i I

144 ~

Du=127ram

2

i

:3

4

i 5

6

Ou = 76mm a

0

I

I

1

4 0

4

I

y

I

2

3

~

~

45

~

I

I

144

I ~ 0 I

7

~

~

I

8

I

9

I0 ,,Du= 51ram

I

I

2 3 456 7 89t0111213141516171819 ~ ~ v ~ ' ~ I

0

I

6

30

v'-'''v'~v

I

I

60 90 Doys Since Plenting

'J Ou= 25ram 1

120

150

Fig. 1. Number of seasonal irrigations as a function of readily available water in the crop root zone. TABLE

IV

Relationship between depth of depletion, and number of irrigationsper season and relative crop yield Depth of depletion

N u m b e r of irrigations per

(mm)

season

25 51 76 102 127

19 10 6 4 2

Relative crop yield

1.00 0.98 0.92 0.80 0.60

season and the relative crop yield for different levels of depletion are presented in Table IV. The objective o f the design was to minimize the costs. The cost function

344

(Eq. 17), after substituting the cost coefficients and the field width from Table I, was as follows: min Go = ~1 q t i n l n f s n w ~2tinln w + 1306n 1

(17a)

in which ~'1 and ~2 are variables dependent upon the number of irrigations and defined (by comparing Eqs. (17) and (17a) as follows: ~1 = 60Cl

ni

(30) (31)

~2 = 1 / 6 0 c 2 n i

This example problem was formulated for a depletion depth of 76 mm (ni = 6). Therefore, Eq. (17a) t o o k the form: min Go = O . O 0 1 4 4 q t i n l n f s n w + 0 . 3 0 t i n l n w + 1306nl

(17b)

The constraints, after substituting the values of the constants from Table I, were: G, = 0.63q-' ~< 1

(18a)

G: = 0.31q ~< 1

(20a)

~0 G3 =

~1 0.04q 0.31s + 0.21] (2.283[ti -- g(0.gL)2"22T] °'799 + 7.0 }[0.3304 (O.OOD~I (23a)

in which S'0 is a parameter dependent upon the furrow spacing and depletion depth, and given as: ~0 = W Du

(32)

For a depletion depth of 76 mm and furrow spacing o f 0.76 m, ~0 = 58: 64 = 805/(nlL) ~< 1

(24a)

Gs = L/402 ~< 1

(25a)

G 6 = 91.5/L ~< 1

(26a)

G7 = n f s q / 1 5 8 ~< 1

(27a)

G8 = n l n w t i / 8 0 0 0 <~ 1

(28a)

G9 = 5 2 8 / ( n w n f s ) ~< 1

(29a)

In constraint G3 (Eq. 23a), a new variable, v, was substituted to simplify the complexity of the constraint which was given as:

345 v = ti - g ( 0 . 9 L ) 2"227~ G10 = v + g ( 0 . 9 L ) 2 " 2 2 7 < 1 ti

(33)

This s u b s t i t u t i o n t r a n s f o r m e d c o n s t r a i n t G3 t o : 58 G3 =

i

0.04q

[ 2 . 2 8 3 v 0"799 "{" 7.0] [ 0 . 3 3 0 4 I\ ( 0 . 0 ~

~ 1

~0.315 )

(23b)

+ 0.21]

The problem, consisting of Eqs. (17b), (18a), (20a), (23b), (24a), (25a), ( 2 6 a ) , ( 2 7 a ) , ( 2 8 a ) , ( 2 9 a ) a n d ( 3 3 ) , w a s s o l v e d u s i n g t h e g e n e r a l i z e d geometric programming technique (Reddy and Clyma, 1981). In the solution, t h e o b j e c t i v e f u n c t i o n a n d t h e c o n s t r a i n t s w e r e l i n e a r i z e d . I n t h e linearizat i o n p r o c e d u r e t h e c o s t f u n c t i o n w a s a d d e d as a n a d d i t i o n a l c o n s t r a i n t t o t h e p r o b l e m . T h e c o s t w a s f o r m u l a t e d as a n u p p e r b o u n d i n g c o n s t r a i n t b y t h e i n t r o d u c t i o n o f a n a d d i t i o n a l variable, u, i n t o t h e p r o b l e m : rain u

(34)

Go = 0 . 0 0 1 4 4 q t i n l n f s n w + 0.30tinlnw + 1 3 0 6 nl ~ u

(35)

T h e r e f o r e , t h e final p r o b l e m c o n s i s t e d o f Eqs. {34), ( 3 5 ) a n d t h e a b o v e c o n s t r a i n t s . T h e d e c i s i o n variables w e r e q, ti, L, nfs, n w , n 1 ( i n d e p e n d e n t ) a n d TABLE V Values of ~0, ~'1, ~2, design parameters, and cost of construction for different depletion depths Depth of depletion

Value of the variable

Values of design parameters

Construction cost (S/ha)

(mm)

~0

~'1

~'2

25

19

0.00456

0.95

q = 29 1 s - l ; n f s ffi 48 t i f f i 2 9 1 m i n ; n w •11 L f 4 0 2 m ; n lffi2

412

51

39

0.0024

0.50

q = 3.29 1 s - z ; nts ffi 48 ti = 3 2 0 m i n ; n w • l l L ffi 402 m; n 1 ffi 2

273

76

58

0.00144

0.30

q ffi 3.29 1 s -1; n f s = 48 t i = 355 min; n w = I I L = 402 m; n I ffi 2

209

102

77

0.00096

0.20

q ffi 3.29 I s - l ; n f s ffi 48 ti f f i 3 5 5 m i n ; n w = 1 1 L ffi 402 m; n I ffi 2

174

127

97

0.00048

0.10

q ffi 3 . 2 9 1 s - ' ; n t s ffi 48

131

ti f f i 4 1 2 m i n ; n w • l l L ffi 402 m; n I ffi 2

346 v (dependent). All these variables were interrelated because of the constraints. Using the optimization technique, the following optimal values were obtained for the design variables: q = 3.29 1 s-l; ti = 355 min, L = 402 m, n 1 = 2,, nw = 11 and nfs = 48. The cost of system construction and operation was $6759 ($209/ha). In the optimal design for the different depletion depths, the cost coefficients in the objective function (which reflect the effect of number of irrigations) and the coefficient in constraint G3 (Eq. 23b), were changed to reflect the effect of requirement (depletion) depth on the cost of irrigation. The values of ~0, ~1 and ~'2 for the different depths of depletion are shown in Table V. The problem was solved four more times and the solutions are presented in Table V. To evaluate the benefits for different depletion levels, Ymax was assumed to be 10 000 kg/ha. With a price of $0.15/kg, the gross return GR from the crop was: GR = 1500 YR S/ha The net NR return was given by: NR = R G - Cp - Cco in which ,Cp represents the production costs per ha, excluding the cost of irrigation system construction and operation, Cco. A production cost of $800/ha was assumed. RESULTS AND DISCUSSION The value of all the design parameters except the flow time, were the same in all optimal solutions for the different design depths (Table V). This does not mean that the solutions were not sensitive to different design depths, but rather that the solutions were constraint bound. Since the soils were sandy loams, the optimal flow rate (to minimize deep percolation) was equal to the m a x i m u m permissible flow rate for all the design depths. The values of the design parameters are interrelated and are heavily dependent upon the flow rate. Hence, when the flow rate did not change, the values of the other design parameters (except time) did not change either. The value of flow time (ti) was different for different design depths in order to satisfy the constraint for the depth of infiltration. The average depth of water applied and the application efficiency for the design depths are presented in Table VI. The application efficiencies were low, 14% to 48%. Too much water was applied to the field, most of which was lost as deep percolation. Even though the application efficiencies were low, the solutions obtained were optimal for the minimum cost. The number of irrigations per season increased with a decrease in the design (depletion) depth. For a design depth of 25 mm, the number of irrigations per season was 19, whereas only two irrigations per season were required for a design

347 TABLE VI Application efficiency of irrigation system for different levels of depletion Case no.

Depth of depletion (mm)

Depth of water applied (mm)

Application efficiency (%)

Seasonal depth of water applied (mm)

1 2 3

25 51 76 102

188 207 229 249

14 28 33 41

3572 2070 1374 996

127

266

48

532

4 5

depth of 127 mm (Table IV). Since the difference in the time of irrigation between the t w o design depths (25 mm and 127 mm) was only 122 min (Table V), the cost of labor was very high for 25 mm design depth. The seasonal depth of water applied ranged from 532 mm to 3572 mm (Table V). The seasonal cost of operation increased also with the a m o u n t of water applied. The cost of system construction and operation for different depletion depths were $412, $273, $209, $174 and $131 per ha, respectively, for 25, 51, 76, 102 and 127 mm (Table V). In the solution procedure, no constraint was included on the a m o u n t of deep percolation. Therefore, the higher cost of the system at smaller design depths was primarily due to the increased cost of labor and excess amount of water applied. If serious consequences such as a high water table with a waterlogging condition exist, a constraint can be specified on the maximum allowable deep percolation. The authors realize that the application efficiencies obtained were low because of the nature of the soil and the criteria of minimal cost of construction and operation. The crop yields were highest (10 000 kg/ha) at a design depth of 25 mm because of more frequent irrigations, b u t the design cost was also highest ($412/ha). Conversely, because of more water stress, the yields were lowest (6 000 kg/ha) at a design depth of 127 mm, b u t the cost of construction was also the lowest ($131/ha). The gross returns, the minimal cost of construction and operation, and net returns from crop production for the whole farm (32.32 ha) are presented in Fig. 2. The net returns were $9 308, $12 831, $11 506, $7 304 and $1 002, respectively, for the depletion (requirement) depths of 25, 51, 76, 102 and 127 mm. The optimal depth of depletion was 51 m m with a m a x i m u m net return of $12 831 for the season. The next best solution was 76 mm depth of depletion with maximum net returns of $11 506. Even though 51 mm depth of depletion was economically optimal, the other important factor that influences the farmers decision is the a m o u n t of water available for the whole season. Therefore, the a m o u n t of water available must be superimposed on the optimal depth of depletion. F o r example, if the seasonal a m o u n t of water available per ha is 996 mm, the farm-

348 50

~

,

,

,

,

I00

125

40

×

_o o

30

~

Cost of Production ~ + Irrigotion System

Io

ao

w

ii1

-iio

I0

0

0 0

i

I

25

50

I 75 (ram)

50

Fig. 2. Relationship between depth of depletion and g r o s s r e t u r n s , n e t r e t u r n s a n d c o s t o f design.

er may choose to deplete 102 mm before each irrigation with net returns of $7 304. The results also indicate that the farmer realizes a loss of $1 002 at a depletion depth of 127 mm. SUMMARY AND CONCLUSIONS

The depth of depletion is an important decision variable in the design of irrigation systems. A combination approach of simulation and optimization was utilized in the analysis: a simulation model for crop production and an optimization (geometric programming) model to find the optimal design of the applicable system. The optimal depth of depletion was determined by considering maximum net returns and by evaluating the gross returns and the minimum cost design for different depths of depletion. For the given situation, a depletion depth of 51 mm was found optimal with net returns of $12 831. Since the soils were very light textured, the deep percolation losses were more than 50 percent. If waterlogging problems are anticipated due to heavy deep percolation losses, an optimal design including a constraint on the maximum amount of deep percolation can be developed. The method presented shows the importance of management on net returns from optimizing irrigation system construction, operation and crop production. ACKNOWLEDGEMENTS

The work reported was supported by the United States Agency for International Development through contract AID/NE-C-1351 with the Consor-

349 t i u m f o r I n t e r n a t i o n a l D e v e l o p m e n t f o r t h e E g y p t W a t e r Use a n d Management Project, and the Colorado State University Experiment Station through t h e D e p a r t m e n t o f Civil Engineering. T h e i r h e l p is sincerely a p p r e c i a t e d .

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