Border Paper (17March10) B

8/8/2019 Border Paper (17March10) B

http://slidepdf.com/reader/full/border-paper-17march10-b 1/28

OPTIMAL IRRIGATION MANAGEMENT FOR SLOPING, BLOCKED-END BORDERS

Jorge Escurra1 and Gary Merkley2

Abstract

An optimization algorithm was implemented in a one-dimensional simulation model for border

irrigation. The model has the capability of successfully simulating all surface irrigation phases for a range

of inflow rates (0.01 - 0.05 m3/s per m), longitudinal field slopes (0.05% - 1.00%), and border lengths (100

- 500 m). The downhill simplex optimization method algorithm was used to determine the recommended

inflow rate and irrigation cut-off time, maximizing a combination of water requirement efficiency and

application efficiency. Optimum values of inflow rate and irrigation cut-off time for a range of longitudinal

slopes, border lengths, and soil types were generated. Most of the calculated optimum values are for

relatively high inflow rate and rapid cut-off time. In addition, exponential relations were developed based

on the simulation results to determine the best irrigation time for maximization of the composite irrigation

efficiency for specified non-optimal inflow rates. The exponential relations are particularly useful in

practice when it is not feasible to use the optimum inflow rate due to constraints at the water source, or

because of irrigation scheduling considerations.

Keywords: optimal irrigation management; border irrigation; irrigation efficiency

___________________________________________

1Engineering Consultant,. [email protected].

1

mailto:[email protected]





Introduction

There are several ways to improve on-farm irrigation management, including: (1) improving

irrigation practices by avoiding water loss due to surface runoff and deep percolation; (2) changing the

irrigation method; and, (3) automating irrigations based on crop water needs. Of these approaches,

improving irrigation practices is usually less expensive and more readily implemented than the other two.

Thus, a research for improved on-farm irrigation management through the application of inexpensive

management changes is useful to irrigators. Surface irrigation accounts for more than 90% of the total

irrigated area in the world, and border irrigation covers more than 20% of the surface-irrigated area in

agricultural regions (FAO 2002) world-wide. Border irrigation is one of the main types of surface

irrigation.

The optimization of surface irrigation system design has been studied by many researchers. For

example, Reddy and Clyma (1980) presented a procedure to optimize furrow irrigation system design

based on design variables, minimum costs, performance parameters, and system constraints. Reddy and

Clyma (1981) proposed another procedure to optimize border irrigation system design using a similar

approach, but adding the maximization of profit after deriving a relationship between the design variables

and the quality parameters (water requirement efficiency). However, the effects of runoff and deep

percolation were not considered in the analysis. Holzapfel et al. (1986) developed a furrow and border

optimization design model for maize under field conditions existing at Chillán, Chile, and Davis, California.

The criterion used in the optimization was the critical required application depth, which was determined on

the basis of the maximum required depth for one irrigation during the season and the peak crop

evapotranspiration. The model generates optimum value of the design variables (inflow rate, field length,

irrigation time, and border width). The model also uses existing linear programming and has been applied

to perform sensitivity analyses.

Esfandiari and Maheshwari (1997) developed an optimization method based on the volume-

balance approach, originally developed for estimating infiltration parameters in border irrigation, using

multiple observations of wetting-front arrival time to adapt the optimization method for use with furrows.

Khanjani and Barani (1999) studied an optimization model for a border irrigation system using the Hook-

Jeev pattern search optimization method in conjunction with a general mathematical model of border

2



irrigation used to maximize irrigation application efficiency. In the analysis, the border irrigation storage

and distribution efficiencies, border slope and length, inflow rate, cut-off time, and the Manning roughness

coefficient are used as constraints. The model was compared to field data and it was shown that a proper

choice of system parameters (length, inflow rate, and cut-off time) can lead to a maximum value of

application efficiency.

Brown et al. (2006) developed an optimal stochastic multi-crop irrigation algorithm which allows

the inclusion of detailed crop-soil models and limits on seasonal water use and system capacity. The

optimal schedulers generally use stochastic dynamic programming. This stochastic dynamic

programming has a limitation due to the requirement of time independence of all parameters except soil

moisture; consequently, the scheduler uses a simulation model with several decision variables which are

used to control irrigation management. These decision variables are optimized using the downhill simplex

method with a custom population-based heuristic method.

Methodology

The present research involves the implementation of the downhill simplex optimization method

(Nelder and Mead 1965) with a one-dimensional unsteady hydraulic simulation model for border irrigation.

There have been very few studies about the use of multidimensional optimization algorithms to improve

surface irrigation efficiency. On the other hand, multidimensional optimization algorithms have been

widely used for agricultural production studies.

Optimization Model

The optimization algorithms applied in the authors’ study were implemented using a robust

simulation model for border irrigation as described by Escurra and Merkley (2009). The input parameters

to the model are: inflow (m3/s per m); required water application depth, Zreq (m); longitudinal ground slope

(m/m); border length (m); Kostiakov infiltration parameters (a, k, f o); end berm height (m); and, the Chezy

coefficient. The model calculates the initial surface water depths on an irrigated border using the Newton-

Raphson method for the specified input parameters of the model. Then, the model calculates the water

depth and flow rate profiles along the border in which distances, advance times, and recession times are

calculated for different spatial nodes at each time step. During this process, the model handles the

3





reorganized in ascendant order as a function of distance, and linear interpolation is used to generate

these values for 100 equidistant locations along the border, with a uniform spacing of ∆ d. The ∆ d value

is calculated by dividing the last recorded distance by the total number of points, minus one (100 – 1 =

99).

V D

V D P

u pL

o w n

V S R

Z r e q

Fig. 1. Side view of a surface irrigation event

The advance time, recession time, and distance at each of the 100 points are used to calculate

the total volume of water stored in the root zone and the total volume of infiltrated water into the soil. Both

are used to calculate WRE and Ea. The intake opportunity time is the difference between the calculated

recession and advance times at each spatial point. The total volume of water stored in the root zone is

calculated by adding only the volume of infiltrated water remaining in the root zone between each pair of

adjacent points along the length of the border. After WRE and Ea are calculated, the objective function is

formulated using weighting factors for WRE and Ea.

Objective Function

An objective function (ξ ) was used in the model to obtain the highest composite water

requirement efficiency and application efficiency for a given set of dimensional and hydraulic parameters.

The objective function is based on a set of parabolic equations which were formulated by combining

weighting factors and water requirement and application efficiencies, as follows:

2 2

a a1 2

E E WRE WRE1 1

10,000 50 10,000 50

ξ = β − + + β − +

(1)

5



where Ea and WRE are in percentage (0 to 100%); and, β 1 and β 2 are the weighting factors for Ea and

WRE, respectively. The second-degree polynomials in parentheses provide an increasing penalty for

decreasing Ea and WRE, respectively, thereby pushing the solution toward the optimal combination of the

two efficiencies. Also, the slope of the polynomials is zero at Ea = 100% and WRE = 100%, respectively.

The minimization of ξ provides the optimal solution.

After some testing, it was decided to used β 1 = 125 and β 2 = 375, whereby the range of ξ is

from 0 to 500. More weight was given to WRE than to Ea because high values of Ea can be achieved by

using very short cut-off times at the cost of inadequate irrigation, resulting in large water deficits in the

crop root zone and an incomplete irrigation.

Grid Generation

A grid generation procedure was developed to better define the search domain for the

optimization process. Before the downhill simplex method algorithm is activated, the grid generation

procedure is applied. The procedure consists of the objective function, the water requirement efficiency,

and the application efficiency calculations for a range of inflow rates (q in) and cut-off times (tco). This

range starts from the minimum permissible inflow rate (0.01 m3/s per m) which secures robustness, and a

minimum cut-off time (1,000 s). These minimum values are taken as the starting point for the grid

generation. Then, a grid is generated using the increments as described in Eqs. 2 and 3:

( ) ( )

co

co comax minco

t

t tt =

n - 2

−∆ (2)

( ) ( )in inmax minin

qin

q qq

n - 2

−∆ = (3)

where (tco)i+1 and (qin)i+1 are the cut-off time and cut-off time which populate the range, (tco)i and (qin)i are

the initial cut-off time and inflow rate, (tco)max and (tco)min are the calculated maximum water advance time to

the border-end for the minimum inflow rate and the minimum cut-off time, respectively; and, (q in)max and

(qin)min are the respective maximum (0.05 m3/s per m) and minimum permissible inflow rates.

6



In addition, nqin and ntco are the grid sizes which have a direct impact on the number of grids

presented in the generated surface. For different optimizations, the grid size changes according to the

soil type because of the numerical oscillations of small magnitude present in the robust simulation model.

These small numerical oscillations produce local minimum values of the objective function, particularly

when the soil has low infiltration rate (e.g. clay or clay loam). Consequently, the model uses a 24-by-22

grid size for calculation of cut-off time and inflow rate range in soils with low infiltration rate, and a 10-by-

10 grid size in soils with high or medium infiltration rate (sandy, sandy loam, and silty loam). After the grid

generation is finished, the values of cut-off time and inflow rate which produce the minimum ξ are

recorded. Then, this value is used to ensure a valid starting point in the downhill simplex optimization

algorithm and restrict calculations to the specified parameter ranges.

Downhill Simplex Optimization Implementation

The model uses the downhill simplex method as an optimization algorithm to determine the cut-

off time (tco) and inflow rate (qin) which produce the minimum value of the objective function (i.e. the

highest composite efficiency). The number of starting points is equal to the number of dimensions plus

one (N + 1). For this particular case, there are two parameters to optimize: tco and qin. Therefore, the

algorithm takes three vertices which are produced by the grid generation procedure as starting points.

These vertices are used by the algorithm to bracket the solution of optimum values. The first vertex is the

minimum qin and tco, the second vertex is the q in and tco from the grid generation procedure that produced

the minimum objective function value, and the third vertex is equal to the second vertex after incrementing

by 40%.

The algorithm starts by making its way downhill through the three-dimensional topography until it

encounters a minimum. Thus, the iterations stop when a movement in any direction would result in an

increased objective function value. The steps of the downhill simplex method to find the minimum ξ in

the model are:

• The algorithm reorganizes ξ values in descendent order from the three starting points (qin 1,2,3, tco

1,2,3). Then, the point which gives the highest ξ is moved through the point which gives the

lowest ξ , producing a trial point. This step is called “reflection” (Nelder and Mead 1965), used to

7



conserve the volume of the simplex. The trial point (Xr ) is generated by Eq. 4 where Xg is the

centroid of the two vertex points which have the lowest ξ ; XN+1 is the vertex point with the lowest

ξ .

( )r g g N 1X X X - X += + α (4)

where α , the reflection coefficient, was equal to 1 in this study.

• If the ξ of the trial point is between the second highest and lowest ξ of the simplex vertices,

then the point with the highest ξ is replaced by the trial point (Xr ). If the value of ξ of Xr is lower

than or equal to the point with the lowest ξ , then the trial point is recorded as Xrr to determine if

the value of ξ drops further in the direction of this Xr . The lower ξ of these two points (Xrr and

Xr ) replaces the point with the highest ξ of the simplex; if the ξ of Xr is greater, the simplex

expands in the Xrr direction. In this expansion a new trial point (Xe) is created using Eq. 5 (Nelder

and Mead 1965):

( )e r r gX X X -X= + β (5)

where β , the expansion coefficient, was equal to 2 in this study.

• If the ξ of Xr is higher than or equal to the point with the highest value of ξ , the value Xr is

checked as Xrr ’ to see if ξ is lower between the point with the highest ξ and the average of the

two lowest ξ (Xb). If the ξ of Xr is higher than or equal to the point with the second highest ξ

and lower than the point with the highest ξ the value is recorded as Xrr ” to determine if ξ is

lower between Xb and Xr.

• If the value of ξ of Xrr ’ or Xrr ” is lower than the point with the highest ξ of the simplex considering

Xr , then Xrr ’ or Xrr ” replaces the point with the highest ξ .

• If the value of ξ of Xrr ’ or Xrr ” is higher than the point with the highest ξ of the simplex,

considering Xr, the simplex contracts itself around the point with the lowest ξ . This contraction is

applied using Eq. 6:

( )c g g N 1X X X -X += + γ (6)

8



where γ , the contraction coefficient, was equal to 0.5 in this study.

The downhill simplex method can contract in all directions, and when it does, the method pulls

itself around the point with the lowest value of ξ . To provide robustness, if the generated values (trial

points), after expansion or contraction, have a tco lower than 5 s and qin lower than 0.01 m3/s per m, the

model will automatically start over by ordering the N+1 points (starting) and increase the third vertex

equal to the second vertex incremented by 70%.

In addition, the optimization algorithm applies a fractional convergence tolerance for the function

(f tol = 0.01), and a maximum allowable number of iterations (itmax = 500). If the fractional range from the

highest to the lowest values of ξ is less than the fractional convergence tolerance, the minimum ξ has

been reached and the optimization procedure has found the combination of q in and tco that gives the

highest composite Ea and WRE. Otherwise, computations continue toward an optimum solution.

Results

Grid Generation for Different Values of Z req

The model uses a grid generation procedure which produces three vertices to define a starting

region for the downhill simplex optimization method. Using the results from the grid generation

procedure, a three-dimensional graph is generated. The three-dimensional graph considers the values of

ξ (z axis), qin (y axis), and tco (x axis). Three such graphs were generated using three different values

(10, 7, and 4 cm) of net infiltration depth, Zreq. The results from the grid generation procedure were

calculated under the following input parameters: sandy loam soil, bare soil surface (maximum and

minimum Chezy coefficient equal to 60 and 10, respectively), a 0.05%longitudinal field slope, and a

border length of 100 m under a blocked-end downstream boundary condition.

In Figs. 2-4, it is observed that an extended range of minimum ξ values (ξ > 50) is observed

where Zreq = 10 cm. In addition, the region with ξ > 50, where Zreq = 10 and 7 cm, corresponds to an

extended range of qin when tco is high. On the other hand, during the grid generation for Zreq = 4 cm, it was

found that when tco is high, a narrow range of q in produces the ξ > 50 values. In addition, it is recognized

that irrigations with small Zreq values are more susceptible to deep percolation losses.

9



Fig. 2. ξ versus tco and qin for Zreq of 10 cm, sandy loam soil, and longitudinal field slope = 0.05%


10




Also, in the three graphs, there are some inverted spikes (local minimums) located over the

smooth shape of the surface topography. During the grid generation the advance time (time of the water

to reach the border end) was recorded for each qin and tco. The qin and tco, which start to generate the

inverted spikes, had an advance time value close to the time of cut-off. Consequently, it is observed that

when the cut-off time approaches the advance time, the value of ξ suddenly increments. Then, it

reduces again when the cut-off time approaches 130% of the advance time. A backwater condition is

manifested when the water reaches the downhill end of the border until the water spills over the end dike;

consequently, the sudden increment in the objective function magnitude is attributed to this condition.

Finally, crops with small root depth and soils with low water-holding capacity will have a smaller

range of qin and tco that obtain lower ξ values. This range is made up of values of low qin and high tco,

and high qin and low tco. On the other hand, crops with deep roots and soils with high water-holding

capacity will benefit by a greater range of qin and tco that result in low ξ values. This range is comprised

of values of medium and high qin, and medium and high tco.

Presence of Spikes (Local Minimums)

11



An analysis of spike formation during the implementation of the grid generation procedure was

studied. These spikes produce local minimums which can change the results from the optimization

method. Therefore, this is one of the reasons why a grid generation procedure was implemented to find

vertices which are close to the global optimum value.

An analysis was made which involves the calculation of WRE, Ea, and the ξ values using three

different objective function equations. During the analysis, the model was applied under the following

conditions: a border length of 300 m, a q in of 0.01 m3/s per m, a longitudinal ground slope of 0.20 %, a Zreq

of 10 cm, and a silty clay soil type. These conditions were simulated for different values of tco to produce

ξ values based on WRE and Ea.

The reason for working with three objective functions was to determine the impact on ξ values

when Ea and WRE change suddenly due to the numerical oscillations from the solution of the governing

hydraulic equations. Objective functions 2 and 3 are formulas that involve different weighting factors for

Ea and WRE, are shown in Eqs (7) and (8):

2 a500 2E - 3WREξ = − (7)

3 a200 E - WREξ = − (8)

Equations 7 and 8 both have a minimum (best) value of zero, while the maximum values are 500 and

200, respectively.

Figure 5 shows the objective function (ξ ), objective function 2 (ξ 2), and objective function 3 (ξ 3)

as a function of tco for an example data set. During the analysis it was observed that ξ , as used in the

model, amplifies Ea and WRE variations more than ξ 2 and ξ 3. However, the minimum ξ value

corresponds to the highest composite WRE and Ea as proposed for obtaining the best irrigation efficiency.

On the other hand, ξ 2 and ξ 3’s minimum objective value was associated with the highest Ea and

medium WRE. Consequently, the minimum ξ value generated by the model gives a more accurate

representation of the best irrigation efficiency because it ensures that most of Zreq along the border length

is satisfied. Furthermore, it is known that high Ea values do not necessarily correspond to high crop

productivity when Zreq along the length is not completely satisfied. Finally, the problem of amplification of

12



Ea and WRE variations by the objective function is solved by locating the starting vertices close to the

optimum global value using the grid generation procedure. These vertices guarantee convergence at the

global optimum value, avoiding local minimum values during the optimization.

0

50

100

150

200

250

300

350

400

450

500

O b j e c t i v e F u n c t i o n s

Time of Cutoff, tco(s)

OBJ

OBJ2

OBJ3

Ea (%)

WRE (%)

ξξ2

ξ3

Fig. 5. Ea and WRE variations using ξ , ξ 2, and ξ 3

Evaluation of Optimization Results for Different Soil Types, Border Lengths, and Slopes

Many simulations were performed to find the optimum qin and tco to obtain the best irrigation

efficiency. These simulations were developed for five soil types. Three border lengths (100, 250, and

500 m) were simulated for four of the five soil types, five border lengths (100, 175, 250, 375, and 500 m)

were simulated for one type of soil (sandy loam), three longitudinal slopes (0.05, 0.5, and 1 %) for four of

the five soil types, five longitudinal slopes (0.05, 0.15, 0.25, 0.5, and 1%) for one of the five soil types

(sandy loam), and a Zreq equal to 10 cm. The results were the qin and tco which generate the minimum ξ .

In addition, the results from the grid generation show the different values of qin and tco, and their ξ , Ea,

and WRE values.

13



Exponential Relation Between qin and t co The grid was relatively dense (24 x 22) for the clay and

clay loam soil types because they exhibit a greater number of downward spikes, or local minimums,

thereby requiring more topographical detail to assist in the search for the global optimum value. On the

other hand, silty loam, sandy loam, and sandy soil types had a 10 x 10 grid size due to a smoother

topography of the solution domain. The grid generation contains a range of qin and tco in which qin and tco

present a calculated ξ , WRE, and Ea for a determined border length, longitudinal field slope, and soil

type. A range of tco belongs to each qin. The lowest ξ generated for each range (qin with several tco) was

grouped and then plotted to observe their mathematical tendency. It was found that some of them,

especially those that have mild field slopes and short to medium border length, have an exponential

mathematical relationship. This means that it is possible to find the best irrigation time to obtain the

highest combined WRE and Ea for non-optimal inflow rates. This is useful, especially if the irrigator is

unable to use the optimum inflow rate or irrigation time due to constraints at the water source, or because

of irrigation scheduling considerations. Figures 6 to 10 show an exponential relationship between qin and

tco which conform to the lowest ξ for a specific qin and a range of tco.

In Figs. 6 to 10 it is observed that the optimum values for all five soil types are located in a range

when qin is high (> 0.03 m3/s per m) and tco is low (> 900 s). Also, low ξ values (< 3) are located in the

same range. After the exponential relation was observed in the sandy, sandy loam and silty loam soils,

the grid generation procedure was repeated using a 24 x 22 solution grid. Based on the results, it was

found in the exponential equations for the five soil types that coefficient values vary from 8.9 to 5.9, in

descendent order from sandy to clay soil types, with the exception of the silty loam soil which has a

coefficient of 9.3, and the lowest coefficient of determination (0.76). In addition, the exponential values

vary from 0.83 to 0.92; however, there is no discernible trend from sandy to clay soil types.

14



0.000

0.010

0.020

0.030

0.040

0.050

0.060

I n f l o w p e r u n i t w i d t h , q

i n ( m

2 / s )

tco(s)

optitren

ξ>3

ξ<3

qin=8.89 tco-0.83

r2=0.89

Fig. 6. Exponential relationship between qin and tco for length of 250 m, slope of 0.05 %, and sandy soil

15



0.000

0.010

0.020

0.030

0.040

0.050

0.060


i n ( m

2 / s )

tco(s)

optitren

ξ>3

ξ<3

qin=6.34 tco-0.92

r2=0.99

Fig. 7. Exponential relationship between qin and tco for length of 100 m, slope of 0.05 %, and sandy loam

soil

16



0.000

0.010

0.020

0.030

0.040

0.050

0.060


i n ( m

2 / s )

tco(s)

optitren

ξ>3

ξ<3

qin=9.30 tco-0.84

r2=0.76

Fig. 8. Exponential relationship between qin and tco for length of 250 m., slope of 0.05 %, and silty loam

soil.

17



0.000

0.010

0.020

0.030

0.040

0.050

0.060


i n ( m

2 / s )

tco(s)

optitren

ξ>3

ξ<3

qin=6.05 tco-0.91

r2=0.98

Fig. 9. Exponential mathematical relation between qin and tco for length of 100 m, slope of 0.05%, and

clay loam soil

18



0.000

0.010

0.020

0.030

0.040

0.050

0.060


i n ( m

2 / s )

tco(s)

optitren

ξ>3

ξ<3

qin=5.94 tco-0.90

r2=0.98

Fig. 10. Exponential relationship between qin and tco for length of 250 m, slope of 0.05%, and clay soil

Example Use of the Exponential Equations These exponential relationships (Figs. 6 to 10) are

very useful for irrigators to find maximum combine WRE and Ea for a qin. Two examples are given to

explain the use of these exponential curves:

1. Case 1: Sandy loam soil; 0.05% field slope; 100-m border length; and, qin = 0.012 m3/s

per m; and,

2. Case 2: Clay soil; 0.05% field slope; 250-m border length; and, qin = 0.024 m3/s per m.

In these examples, a hypothetical of water supply constraint was considered. it was assumed

that the fields cannot be irrigated with more water than the current inflow because they are the maximum

allowable inflow rates. Consequently, the irrigator can use the exponential relationship to obtain the best

irrigation time to improve the composite irrigation efficiency. For Case 1:

19



1

0.92

co

0.012t 911 s

6.34

− = =

(9)

And, for Case 2:

( )1

0.900.90

in co

0.024q 5.94t 456.52 457

5.9

−− = = = ≈

(10)

1

0.90

co

0.024t 447 s

5.94

− = =

(11)

Finally, the calculated irrigation times in both cases, according to the exponential relations

developed in this research, will give a relatively high combined WRE and Ea for those qin (0.012 and 0.024

m3/s per m).

Evaluation of the Optimum Objective Function Values The minimum ξ was found for different

values of border length, longitudinal slope, and soil type. The optimum qin and tco for sandy soils were

determined, and Table 1 shows nine optimum qin and tco values for the sandy soil type. These

optimizations were performed for three longitudinal field slopes (0.05, 0.50, and 1.00%), and three border

lengths (100, 250, and 500 m) with a Z req of 10 cm. Each optimum value has its calculated ξ , WRE, Ea.

Also, the ratio (tco

/taL

) between the advance time (time of water to reach the end) and cut-off time was

calculated. The tco/taL ratio is close to unity in most cases, but there are a few cases when the tco/taL ratio is

greater than 1.5, especially when the longitudinal field slope is 0.50 or 1.00%, and the border length is

250 or 500 m.

Table 1. Optimum Values of Inflow Rate and Cut-off time for Sandy Soils

20



Cutoff Time Inflow rate Slope Border Ea

(s) (tco)(m3/s/m)

(m/m) length (m) (%)232 0.048 0.0005 100 99

150 0.051 0.005 100 69

150 0.053 0.01 100 99

Optimum Values

The optimum qin and tco values for sandy loam soils were also determined, and Table 2 shows

twenty-three optimum values of qin and tco for sandy loam soils. These optimizations included five

longitudinal field slopes (0.05, 0.15, 0.25, 0.50, and 1.00%) and five border lengths (100, 175, 250, 325,

and 500 m,) with a Zreq of 10 cm. However, for the border length of 500 m only three longitudinal slopes

were used (0.05, 0.50, and 1.00%). From Table 2, it is also seen that the optimum qin values are higher

than the average, and tco values are relatively low. In a few cases, it is observed that qin is relatively low

and the tco is high when the longitudinal field slope is 0.50% for a border length of 500 m, or for slopes of

0.25% and 0.15% with a 250-m border length.

Table 2. Optimum Values of Inflow Rate and Cut-off time for Sandy Loam Soils.

21





found for a slope equal to 0.5 and a border length equal to 250 m. It is also observed that ξ values

increase, and WRE and Ea decrease, as the longitudinal field slope increases.

Table 3. Optimum Values of Inflow Rate and Cut-off time for Silty Loam Soils.


(s) (tco) (m3/s/m) (m/m) length (m) (%)

203 0.049 0.0005 100 97

147 0.06 0.005 100 99

360 0.023 0.01 100 65

Optimum Values

Table 4 shows nine optimum values of q in and tco for clay loam soils. These optimizations were

performed for three longitudinal ground slopes (0.05, 0.50, and 1.00%) and three border lengths (100,

250, and 500 m,) with a Z req of 10 cm. From Table 4, it is again found that the optimum qin values are

relatively high and tco values are relatively low. The tco/taL ratio values are close to 1.0; a few tco/taL ratios

greater than 1.5 are found when the longitudinal field slope is 0.50% for a border length equal to 250 m,

and slope is 1.00% for a border length equal to 100 m.

Table 4. Optimum Values of Inflow Rate and Cut-off time for Clay Loam Soils.



267 0.037 0.0005 100 99

146 0.06 0.005 100 89

336 0.033 0.01 100 63

Optimum Values

23



Table 5 shows nine optimum values of q in and tco for clay loam soils. These optimizations

included three longitudinal slopes (0.05, 0.50, and 1.00%) and three border lengths (100, 250, and 500

m), with Zreq of 10 cm. It was again found that the optimum qin values are relatively high values, and the

best tco values are relatively low. The tco/taL ratio values are close to unity. Also, it is found the highest

value of the tco/taL ratio (equal to 3) was greater than that of the other soil types; and the highest tco/taL ratio

is found when the slope is equal 0.50 and 1.00% for a border length equal of 100 m. It is also observed

that the ξ values increase as the slope becomes steeper; on the other hand, WRE and Ea decreases as

the slope steepens.

Table 5. Optimum Values of Inflow rate and Cut-off time for Clay Soils



264 0.039 0.0005 100 98

660 0.021 0.005 100 58606 0.021 0.01 100 46

Optimum Values

Finally, according to all the evaluations of the tco/taL ratio among the optimum values of qin and tco

for all soil types, it is observed that the average ratio is 1.1 among all the these values for the five soil

types. This is without considering tco/taL ratios higher than 1.5 because those results could be influenced

by small numerical oscillations (spikes) which generate some variations in water depth profile, producing

an impact in the calculation of the WRE and Ea. According to the average tco/taL ratio, it is advisable that

when irrigators have the optimum qin to obtain the best combined efficiency, tco should be between 0.9 to

1.1 times the advance time, which is the time until the water reaches the end of the border to obtain the

best irrigation efficiency. The value of 0.9 is used for short border lengths, soil with low infiltration rate,

and mild slopes. On the other hand, the value of 1.1 is used for long border lengths, soil with high

24



infiltration rate and steep slopes. The main reason that the tco/taL ratio is close to unity is to satisfy Zreq,

especially at the downhill end of the border. Also, the results indicate that a relatively high inflow rate and

a low cut-off time provide the best combined efficiency, in most cases.

In addition, nine optimization results from Tables 1 to 5 produced unacceptably low (less than

70%) WRE and Ea values. These results are observed when the borders present longitudinal slope

higher or equal than 0.50% and border length higher or equal than 250 m. The reason is that steep

slopes and long border lengths present small numerical oscillations which do not affect the convergence,

but do impact the water depth profile; therefore, low variations are observed in the WRE and Ea

calculations.

Summary and Conclusions

A robust hydraulic model which successfully simulates all surface irrigation phases for a range of

inflow rates (0.01 - 0.05 m3/s per m), longitudinal field slopes (0.05% - 1.00%), and border lengths (100 -

500 m) was developed. A downhill simplex optimization algorithm was implemented to determine the

recommended inflow rate and irrigation cut-off time, maximizing a combination of water requirement

efficiency and application efficiency. As part of the downhill simplex optimization implementation, a grid

generation procedure was developed to calculate the three vertices which define the starting search

domain for the optimization process. Based the results from the grid generation and the downhill simplex

implementations, the conclusions for optimal irrigation management of blocked-end, sloping borders are

as follows:

1. Crops with deep roots and soils with high water-holding capacity, which correspond to a relatively

high value of Zreq, will benefit from a large range of qin and tco values that generate relatively high

composite WRE and Ea, including the calculation of the best composite irrigation efficiency (global

optimum value). In addition, surface irrigated fields with relatively steep longitudinal slopes (> 2.5

%) are associated with a decrease in the highest combined WRE and Ea, causing the minimum

ξ to increase.

2. In most of the cases for the five included soil types, the minimum ξ values (global optimum

values) are associated with relatively high q in and low tco. In only a few cases are the optimum

25



values for low qin and high tco. Therefore, in most cases a high qin and a short tco are needed to

obtain the best possible combined irrigation efficiency.

3. If irrigators can use the global optimum qin to obtain the best combined irrigation efficiency, tco

should be close to the advance time. From the results, it is observed that tco is relatively greater

than the advance time because the model has been designed to give more weight to WRE than

to Ea, prioritizing the satisfaction of Zreq.

4. A relationship was found between the best tco to obtain high combined WRE and Ea for non-

optimal qin values. The relation is given by a series of exponential equations for the different soil

types, border lengths, and longitudinal slopes studied in this research. The exponential equations

have coefficients from 8.9 to 5.9, and they are arranged in descendent order from sandy to clay

soils, with the exception of the silty loam soil. The exponents are all close to 0.90. With these

exponential relations, irrigators can obtain a high combined irrigation efficiency even if the

optimum qin cannot be used.

Notation

The following symbols are used in this paper:

Ea = application efficiency (%);

f tol = fractional convergence tolerance;

itmax = maximum allowable number of iterations;

N = number of dimensions;

qin = inflow rate entering to the border (m3/s per unit width);

qin 1,2,3 = starting points/vertices of inflow rate (m3/s per unit width);

taL = advance time (s);

tco = cut-off time (s);

tco 1,2,3 = starting points/vertices of cut-off time (s);

VD = volume of water deficit (m3);

VDP = volume of deep percolation (m3);

VSRO = volume of surface runoff (m3);

26



VZR = volume of water stored in the root zone (m3);

WRE = water requirement efficiency (%);

Xe = trial point due to expansion;

Xg = centroid of two vertex points with lowest ξ ;

XN+1 = vertex point with the lowest ξ ;

Xr = trial point due to reflection;

Xrr , Xrr ’, Xrr ” = checked Xr which is used during the reflection, expansion, and

contraction process;

Zreq = net infiltration depth (cm);

α = reflection coefficient (equal to 1);

β = expansion coefficient (equal to 2);

β max1, β max2 = weighting factors for application and water requirement efficiencies;

γ = contraction coefficient (equal to 0.5);

∆ d = uniform spacing (m), and;

ξ , ξ 2, ξ 3 = objective function; objective function 2; and, objective function 3.

References

Brown, P. D., Cochrane, T. A., Krom, T. D., Painter, D. J., and Bright, J. C. (2006). “Optimal On-Farm

Multicrop Irrigation Scheduling with Limited Water Supply Food and Agriculture.” Computers in

Agriculture and Natural Resources 4th World Congress Conference, Proceedings. Orlando, Florida.

Escurra, J. J. and Merkley, G. P. (2009). “Robust Simulation Model for Sloping, Blocked-end Border

Irrigation.” (submitted for review)

Esfandiari, M., and Maheshwari, B. L. (1997). “Application of the Optimization Method for Estimating

Infiltration Characteristics in Furrow Irrigation and its Comparison with other Methods” Agric. Water

Manag., 34(2): 169-185.

Holzapfel, E. A., Mariño, M. A., and Chavez-Morales, J. (1986). “Surface Irrigation Optimization Models.”

ASCE J. Irrig. Drain. Eng., 112(1): 1-19.

27



Khanjani, M. J., and Barani G. A. (1999). “Optimum Design of Border Irrigation System” J. Amer. Water

Res. Assoc., 35(4): 787-792.

Nelder, J. A., and Mead, R. (1965). “A simplex Method for Function Minimization.” The Computer J., 7(4):

308-313.

Organization of the United Nations (FAO) (2002). Aguas y Cultivo, Logrando el Uso Óptimo en la

Agricultura, Rome, Italy.

Playán, E. (2004). Design, Operation, Performance Evaluation and Surface and Subsurface Irrigation,

Instituto Agronomico Mediterraneo-CIHEAM, Bari, Italy.

Reddy, J. M., and Clyma, W. (1980). “Optimal Design of Furrow Irrigation System.” American Soc. of

Agric. Engineers Summer Meeting, June 15-18, San Antonio, Texas.

Reddy, J. M., and Clyma, W. (1981). “Optimal Design of Border Irrigation System.” J. Irrig. Drain. Eng.,

107(3): 289-306.

Walker, W. R. (2003). SIRMOD III-Surface Irrigation Simulation, Evaluation, and Design, Guide and

Technical Documentation, Biological and Irrigation Engineering Dept., Utah State Univ., Logan, Utah.

Walker, W. R., and Skogerboe, G. V. (1987). Surface Irrigation, Theory and Practice, Prentice-Hall Inc.,

Englewood Cliffs, New Jersey.

Zerihun, D., Sanchez, C. A., and Farrell-Poe, K. L. (2002). “Analysis and Design of Border Irrigation

System.” ASAE, Annual International Meeting , Chicago, Illinois.

28

Documents

Border Paper (17March10) B