1 Introduction

Surface irrigation is the application of water over the land for the purpose of crop production and it is the most common method of irrigation and accounts for 95% of irrigation in the world. It is also well suited for use on both small and large schemes [17, 21]. As an alternative, surface irrigation method is widely used due to low capital investment involved, low operating costs and ease of operation and maintenance [21]. Also surface irrigation systems are less affected by climatic and water quality characteristics. However, surface irrigation systems are typically less efficient in applying water than pressurized irrigation systems because of extreme difficult to achieve good control over the highly variable nature of the movement of water across a soil surface and its infiltration into the soil over a season. Therefore, the land under surface irrigation tends to be more affected by water-logging and soil salinity if adequate drainage is not provided [12].

Even though, it is unrealistic to apply irrigation without loss. These losses appears in the application of irrigation water too less or too much and too short or too large of furrow length. Deep percolation results when water is applied too long to the field and/or the variation of intake opportunity time is too large (inflow rates are too small). These two problems can be remedied by adjusting the time of cutoff (tco) and inflow rate (Qo) [42].

Availability of irrigation water itself is becoming a constraint in some sites, possibly because more farmers have started irrigating than schemes were designed to support [34]. From two selected scheme, [14] stated as the efficiency of the system depends on the strength of institutional set up of the scheme. Adequate water management for irrigated agriculture holds a considerable significance for the future of the Ethiopian agriculture. Irrigation water for Wonji-Shoa Sugar Estate scheme is supplied via a pumping station located on Awash River [9]. In Awash Basin, many industries and major cities abstract water from Awash River. This creates a risk as growing volumes of industrial effluent and urban wastewater are contaminating the water and causing scarcity by reducing the quality of surface water available for downstream users [19]. Further, [19] stated there was a high water application rate compared to irrigation water requirement of crops in most irrigated areas. Poor water management and excess of irrigation water application in turn created the problems of environmental issues (water-logging and salinization), escalating multi-sectoral water demands in the basin and conflict between upstream user (Wonji) and downstream user (Merti) [5, 16]. For this reason, water management at Wonji-Shoa Large-scale Irrigation Scheme needs critical attention which failed under low efficiency of the system because of failure in management and design [9]. Significant water savings can be achieved through improving the application of irrigation water at the field level by reducing the amount of water lost to the crop through deep percolation and surface runoff [8]. This can be achieved only by efficient irrigation systems designed at farm level combined with optimization of management decision variables like: the furrow discharge, cut of time and furrow length as decision support variables [13, 20].

Decision variables are parameters an irrigation designer uses. The variables are modified until acceptable irrigation performance is reached. These are normally the field dimensions (length and width), the flow rate and the cutoff time [21]. The main design and managing irrigation events in surface irrigation considers choosing of decision variables [21, 28]. These variables are unknown and should be specified by the optimization [23]. The optimized variables are used to modify future irrigations in order to achieve the desired level of performance [26]. Based on principal of cost minimization and irrigation efficiency maximization, [24] described the result of the calculations related to inflow rate, length of furrow, irrigation time and irrigation efficiency. The authors stated that with the minimization of cost results of minimized inflow rate and time cutoff in respect to the irrigation efficiency is maximized [32]. Also optimized the inflow rate using simulated advance and recession curves and gets 0.03m3/min/m. The authors stated that the optimized inflow rate gave satisfactory water distribution uniformity and application efficiency for three growth stages of maize: 87% and 89% for emergency stage, 75% and 60% for development stage, and 95% and 89% for maturing stage respectively. In addition, for the existing furrow length (100 m) with uniform inflow rate, [38] obtained optimum inflow rate (5 l/sec) and cut-off times (45 min) that showed better application efficiency (81.16%) and better distribution uniformity (93.43%) of Irrigation System at Tendaho Sugar Estate.

Different researchers like [6, 15, 25] in Wonji/Shoa plantation estate have conducted researches focusing on performance assessment of pumps, canals and night storage reservoir. But for water use efficiency, an assessment of performance of on-farm irrigation is vital and resulted from optimization of decision variables which obtained by both numerical and simulation models. From different study, the reported result revealed that as the optimized furrow irrigation decision variables leads to increase the efficiency and the uniformity distribution of water in the furrow. The objectives of this paper is to optimize furrow irrigation decision variables (inflow rate, cutoff time and furrow length) using algebraic equation and simulation model based. The study has the potential benefit of improving irrigation efficiency and reducing stress on water resources.

2 Materials and methods

2.1 Experimental site

The field experiments were carried out at Wonji Shoa Sugar Estate, Oromiya, Ethiopia located at latitude of 80°30' to 8°35' N and longitude of 39°10' to 39°20' E at an altitude of 1540 m above mean sea level (Fig. 1). The Estate was established in 1951 G.C.by foreign private investors, Ethiopian government and Netherland’s H.V.A. Company. Wonji sugar factory is one of the first modern sugar industry in Ethiopia with initial production capacity of 140 tons of sugar per day. Then, Shoa Sugar Factory was established in 1962 with 170 tons of sugar production capacity per day. The two factories are known by the name of Wonji Shoa Sugar Factory (WSSF) and administered as one factory.

Fig. 1
figure 1

Map of study area

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2.2 Data collection techniques/procedures

Prior to any activities, the land had sloped by machine to predetermined single (adjusted once) slope level which was adopted by organization and then, sloped furrow back leveled to zero percent by human power for experiment. After all furrows were leveled, change in elevation of each slope was calculated and drilled at downstream end of the furrow.

The primary data was collected from field focusing on furrow irrigation practices and the secondary data also was collected from the Wonji Sugar Estate of planning and plantation office. The primary data used as basic input for SIRMOD and WinSRFR models were collected from the furrows field. They were: field topography; inflow rate; furrow geometries (furrow length, depth, width spacing, bottom slope, cross sectional area and wetted perimeter); soil infiltration parameters, soil roughness, application depth and cut-off time. The required data such as furrow characteristics and soil moisture contents were collected from March to June 2020.

The experiment was prepared with aim of optimizing that furrow length, inflow rate and cut-off time of the existing system in practice. This study is experimental and observational types of research that evaluate the existing system and gives the optional improvement by conducting the experiment. The treatments applied were slopes, furrow lengths and flow rates. Each treatment has three levels with three replications. The treatment levels were 0.05, 0.075, 0.1% furrow slopes (S), 32, 48, 64 m furrow lengths (L) and 3, 4, 5 l/s flow rates (Q) depending on the maximum non erosive stream size with split-split plot design where slopes constituted the main plot factor, furrow lengths the sub-plot factors and flow rates the sub-sub-plot factors. The selection of these parameters depends on the estate background. The average slope used at the estate was 0.05%, the three of furrow length was used at the estate, and however 32 m were dominantly used. The arrangement was shown in Appendix Table 12.

Each furrow set was consisting of four furrows having 5.8 m width and flow rates were assigned randomly. The middle three furrows were used for testing of irrigation event parameters and the outer furrows used as a buffer to control the effects of lateral flow; the field layout was shown in Fig. 2.

Fig. 2
figure 2

Field layout of the experimental plot

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The experiment was conducted under normal field conditions over two irrigation events. Amount of water to be applied during each event (Zreq) was determined based on soil moisture deficit level at root depth assuming equal to required depth of application during each irrigation event. The irrigation schedule is varying depends on the soil type of the field. The soil type of the study area is heavy soil type and the irrigation schedule for heavy soil at the estate is from 14 to 15 days. Furrow cross sections were determined using profile-meter and the data were used to compute infiltration parameters by volume balance analysis following procedure stated by [37]. The computed parameters values were shown in Table 1.

Table 1 Infiltration parameters
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The procedure begins by determining basic infiltration rate (fo) from inflow outflow hydro graph, by using the following equation stated by [41]:

$${f}_{o}=\frac{{Q}_{in}-{Q}_{out}}{L}$$
(1)

Next volume balance equation is defined as,

$${Q}_{o}t={\delta }_{y}{A}_{o}x+{\delta }_{z}k{t}^{a}x+\frac{{f}_{o}tx}{1+r}$$
(2)

where Ao = cross sectional area of flow at inlet, (m2) Qo = inlet discharge, (m3/min) T = elapsed time since irrigation started, (min) δy =surface storage shape factor (ranged 0.70 to 0.80) δz =subsurface storage shape factor, defined as

$${\delta }_{z}=\frac{a+r\left(1-a\right)+1}{(1+a)(1+r)}$$
(3)

The exponent in power advance equation, defined as

$$r=\frac{{\ln}(0.5)}{{\ln}\left(\frac{{t}_{0.5L}}{{t}_{L}}\right)}$$
(4)

Thirdly, using two point method, the two empirical fitting parameters (a and k) are computed

$$ a = \frac{{{\text{ln}}\left( {\frac{{V_{L} }}{{V_{{0.5L}} }}} \right)}}{{{\text{ln}}\left( {\frac{{t_{L} }}{{t_{{0.5L}} }}} \right)}} $$
(5)
$$K=\frac{{V}_{L}}{{\delta }_{z}{{t}_{L}}^{a}}$$
(6)
$${V}_{L}=\frac{{Q}_{o}{t}_{L}}{L}-{\delta }_{y}{A}_{o}-\frac{{f}_{0}{t}_{L}}{1+r}$$
(7)
$${V}_{0.5L}=\frac{2{Q}_{0}{t}_{0.5L}}{L}-{\delta }_{y}{A}_{o}-\frac{{f}_{o}{t}_{0.5L}}{1+r}$$
(8)

where \({t}_{0.5L}\)=advance time at one-half field length,(min) tL = advance time to the end of field length, (min) L = field length, (m) V0.51 = infiltrated volume at one-half field length, (m3/m) VL = infiltrated volume at end of field length, (m3/m).

Furrow flow rates were measured using 3 inches Parshal-flumes which were placed at the upstream of the experimental plot, 5 m far from inlet furrow. Prior to the test, the PVC pipe having 75 mm, 90 mm and 110 mm diameters buried at inlet of furrow to distribute flow rates over the replication equally. During the test, advance and recession times were measured for each treatment plots combination. Stakes were driven into the soil along the furrows at fixed interval of 16 m before irrigation events. Advance times were recorded at the time when water reached at each stakes, while recession times (trec) were recorded at the times when water fully infiltrated or disappeared from the furrow bed at observation sections. After determining the depth of water retained in the soil profile, performance indicators were calculated.

2.2.1 Soil data collection

Composite of undisturbed soil samples at two soil depths, 0–30 and 30–60 cm were taken from six spots for the field experiment. The collected undisturbed soil samples were analyzed at Wonji Sugar Corporation Research Center Laboratory for bulk density determinations. Bulk soil bulk density was determined using the methodology described in [40]:

$$Bulk\;density\left(\frac{g}{{m}^{3}}\right)=\frac{dry\;weight\;of\;soil\;sample\;(g)}{volume\;of\;core\;({m}^{3})}$$
(9)

Core samplers of known volume were used and the samples weighted and placed in Oven Dry at 105 °C for 24 h. Field capacity and permanent wilting were taken from the estate as secondary data. The percentages of sand, silt and clay of the composite soil sample were determined by hydrometer analysis method. The percentages of sand, silt and clay of the composite soil sample were determined by hydrometer analysis method in the following procedure stated by [7]. The computed parameters values were shown in Table 2.

Table 2 Soil physical properties
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3 Procedures

  1. 1.

    Mix 100 ml of the 5% dispersing solution and 880 ml of deionized water in a 1000 ml cylinder. This mixture is the blank. (Note: 100 ml + 880 ml = 980 ml. This blank is not diluted to 1000 ml; the other 20 ml is the volume occupied by 50 g of soil.).

  2. 2.

    Weigh 25–50 g of soil and transfer to a dispersing cup. Record the weight of soil up to ± 0.01 g.

  3. 3.

    Add 100-ml of 5% dispersing solution.

  4. 4.

    Attach dispersing cup to mixer and mix the sample for 30–60 s.

  5. 5.

    Transfer the suspension quantitatively from the dispersing cup to a 1000 ml cylinder.

  6. 6.

    Fill to the 1000-ml mark with deionized water equilibrated to room temperature, or allow standing overnight to equilibrate.

  7. 7.

    At the beginning of each set, record the temperature, and the hydrometer reading of the blank, using the procedure described below.

  8. 8.

    To determine the density insert plunger into suspension, and carefully mix for 30 s. until a uniform suspension is obtained. Remove plunger (begin 40 s timer) and gently insert the hydrometer into the suspension.

  9. 9.

    Record the hydrometer reading at 40 s. This is the amount of silt plus clay suspended. The sand has settled to the bottom of the cylinder by this time. (Repeat 8 –9 for each sample)

  10. 10.

    Record the hydrometer reading again after 6 h, 52 min. This is the amount of clay in suspension. The silt has settled to the bottom of the cylinder by this time. Then calculate percentage of particles after hydrometer reading was corrected.

% clay = corrected hydrometer reading at 6 h, 52 min. × 100/ wt. of sample.

% silt = corrected hydrometer reading at 40 s. × 100/ wt. of sample—% clay.

% sand = 100%—% silt—% clay.

3.1 Soil physical properties

The soil samples data were taken from the field plot for determination of bulk density, soil texture and moisture content. As per the results obtained, which are presented in Table 2, the soil textural class is heavy clay type with Field capacity (FC) of 52.7% and Permanent Wilting Point (PWP) of 24.3% on weight basis.

The bulk density of the study area sampled from six spot was found to be presented in Table 3. As a result observed the average bulk density of soil was 1.10 g/cm.3

Table 3 Bulk density of the study area
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3.1.1 Soil infiltration and surface roughness

Typical value of roughness (n) was taken as 0.04 and 0.03 for first and later irrigation events, respectively.

As the result indicated the infiltration function over the two events were formed in equation Z1 and equation Z2 respectively.

$${Z}_{1}=0.33567{\tau }^{-0.18801}+0.00542\tau $$
(10)
$${z}_{2}=0.22911{\tau }^{-0.107002}+0.00542\tau $$
(11)

where: Z1 = cumulative infiltrated depth for first irrigation event and Z2 = cumulative infiltrated depth for later irrigation event. The maximum opportunity time of the field plot was observed as 290.3 min and 296.95 min during first and later irrigation event respectively.

3.2 Surface irrigation software package

Mathematical models for the design, operation, and evaluation of various surface irrigation methods have been used in user-friendly computer programs such as the WinSRFR [4], SURDEV [21], and SIRMOD [42]. These software can be used to predict irrigation performance, for example, system uniformity and application efficiency, given certain gradients, soils, field dimensions, in-row or inter-row planting. This prediction capability may facilitate the modification of furrow irrigation operational guidelines and, if necessary, layouts/design, so that performance is comparable to more marketed irrigation systems [18].

3.2.1 SIRMOD software

SIRMOD is a comprehensive software package for simulating the hydraulics of surface irrigation systems at the field level, selecting a combination of sizing and operational parameters that maximize application efficiency and a two-point solution of the “inverse” problem allowing the computation of infiltration parameters from the input of advance data [42]. SIRMOD was developed at Utah State University to simulate both border and furrow irrigation for continuous flow irrigations as well as surge flow and cutback methodologies. It employs a full hydrodynamic model, as well as zero-inertia and kinematic-wave approximations and field experiment used to validate the model. The tool is continually being further developed [29, 33]. [35] used SIRMOD simulation model to optimize design parameters and compare the hydraulic behavior between surge flow and continuous flow of furrow irrigation for the cultivation of black tobacco in Cuba. The authors described that the variable cycles surge flow irrigation can increase the application efficiency by more than six fold and reduce the water volume by more than 80% compared to continuous flow irrigation and also verified that the model has an excellent correlation between simulated and measured advance times both for continuous and surge irrigation.

3.2.2 WinSRFR software

WinSRFR is a software package for the hydraulic analysis of surface irrigation systems. Intended users are irrigation specialists, consultants, extension agents, researchers, university level instructors and students, and farmers with moderate to advanced knowledge of surface irrigation hydraulics. The software offers four analytical functionalities, which are identified as WinSRFR Worlds. These functionalities are: Event Analysis, Simulation, Physical Design and Operations Analysis. With furrows, simulations consider only a single furrow and, therefore, neighboring furrows are assumed identical. Any variation in properties from furrow to furrow must be modeled separately. Outputs include the advance and recession curves, flow and depth hydrographs at specified locations, water surface profiles at specified times, and a variety of performance measures such as application efficiency, distribution uniformity, and adequacy of the irrigation [4, 31] optimized closed-end furrow irrigation design based on field and simulated advance data and described that the WinSRFR software were in agreement with measured values, and the absolute error average values of Ea, Du, and Es of all the irrigation furrows were 6.87%, 7.67%, and 6.15%, respectively.

According to the findings of [39], full hydrodynamic and zero inertia models were very powerful in simulation process. For increasing of field slope until amount of 0.01 full hydrodynamic and zero inertia models had not any difference but for more increasing of slope due to the increasing of velocity, accuracy of zero inertia model dropped. This study was conducted simulation of furrow irrigation by volume balance, Zero Inertia and full hydrodynamic model, and using SIRMOD software compared with WinSRFR software for prediction of irrigation performance.

3.3 Calibration of the model

The model was calibrated by using calibration parameters such as manning’s roughness coefficient’n’ stated by [37]. The calibrated value of SIRMOD model presented in Appendix Table 13.

The most effective way to evaluate these systems is via use of a calibrated hydraulic simulation model. The role of field measurement is to collect enough information in order to calibrate this model so it can be as true as possible to the actual irrigation. As the result viewed in Appendix Table 13, the average surface roughness of the field was of 0.16 and 0.10 at first and later irrigation, respectively.

3.4 Suitability of surface irrigation models

To evaluate the suitability of the surface irrigation models, three criteria were chosen to analyses the degree of the goodness fit. These criteria taken from [10] stated as follows:

  1. 1.

    The coefficient of determination (R2)

    $$ R^{2} = \frac{{\left[ {\left( {\sum_{i = 1}^{n} \left( {o_{i - } \mathop o\limits^{^{\prime}} } \right)*(p_{i} - \mathop p\limits^{^{\prime}} } \right)} \right]^{2} }}{{\sum_{i = 1}^{n} \left( {o_{i - } \mathop o\limits^{^{\prime}} } \right)^{2} *\sum_{i = 1}^{n} \left( {p_{i} - \mathop p\limits^{^{\prime}} } \right)^{2} }} $$
    (12)
  2. 2.

    The value of R2 ranges from 0.0 to 1.0, indicating a better agreement for the values close to 1.0. Root Mean Square Error (RMSE)

    $$RMSE=\sqrt{\frac{{\sum }_{i=1}^{n}{\left({o}_{i-}{p}_{i}\right)}^{2}}{n} }$$
    (13)

    The RMSE has minimum value of 0.0, with a better agreement close to 0.

  3. 3.

    Standard error (SE)

    $$ SE = \sqrt {\frac{{\frac{1}{n}\sum_{i = 1}^{n} \left( {o_{i - } p_{i} } \right)^{2} }}{{\mathop p\limits^{^{\prime}} }}} $$
    (14)

where Ao = cross sectional area of flow at inlet,(m2) Qo = inlet discharge, (m3/min) T = elapsed time since irrigation started, (min) δy =surface storage shape factor (ranged 0.70 to 0.80), δz = subsurface storage shape factor, t0.5L =advance time at one-half field length,(min) tL = advance time to the end of field length, (min) L = field length, (m) V0.51 = infiltrated volume at one-half field length, (m3/m) VL = infiltrated volume at end of field length, (m3/m) Z1 = cumulative infiltrated depth for first irrigation event and Z2 = cumulative infiltrated depth for later irrigation event n—Number of observations Oi—ith value of the observed measurement Pi—ith value of the predicted measurement Ō—Mean of the observed values Ṕ—Mean of the predicted values.

4 Results and discussions

4.1 Hydraulic performance evaluation of existing furrow irrigation system

Existing irrigation system was carefully evaluated to identify that may be effective and feasible in improving the systems’ performance. As early stated, the dominant soil type of the estate was heavy clay soil. Two fields were selected for furrow length of 32 m and 48 m and all input parameters were measured accordingly. The soil physical characteristics of the field like soil texture, field capacity, permanent wilting point, bulk density and soil moisture content, before irrigation event, were presented in Tables 4 and 5, respectively.

Table 4 Soil texture, field capacity and permanent wilting point of field number 108
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Table 5 Soil moisture before irrigation event
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The results revealed that the heavy clay soil had particle size distribution of 10%, 17% and 73% for sand, silt and clay, respectively, with required application depth of 14 mm computed before irrigation events, by assuming required application depth equal to soil moisture deficit. The hydraulic performance of the field was evaluated over the irrigation events by using WinSRFR software package.

The estate applies 75 l/sec of flow rate per six furrows through tube at inlet of field. Each furrow inlet receives 12.5 l/sec according to the estate application. The two existing field number (16 and 108) of application efficiency and distribution uniformity were 43% and 86% for field number 16 and 25% and 44% for field number 108, respectively. The results of evaluation were presented in Table 6. From selected field for evaluation of the existing system, the findings presented in Table 6 demonstrate that the existing furrow systems were inefficient with the value of application efficiency and distribution uniformity of 43% and 86% for field number 16 and 25% and 44% for field number 108, respectively which doesn’t fit the typical values. [30] stated that the typical application efficiency range was between 60 and 80%. So, the system needs improvement to get the required efficient. The water distribution profile from an irrigation event of existing was indicated in Fig. 3. The Evaluation of existing system analyzed by WinSRFR software package in Fig. 3 showed that the recession time delayed because of high application of inflow rate which is 75 l/sec per furrow set (6 furrows). In addition, the water advanced the down end field quickly; the reason is that high flow rate used to increase gravity force that drives flow. The deep percolation of the result became high with heavy clay soil type because of the end condition of the field is closed and there is no surface runoff loss occurred. The only chance for loss was deep percolation.

Table 6 Evaluation of existing furrow system
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Fig. 3
figure 3

Advance/recession curve of existing system

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The results came from insufficient design variables (furrow length and slope) and low level of management on-field practice. Since the irrigators assigned at each field number have different management skills, the irrigation efficiency at each field was varying. There was over irrigation resulting in high deep percolation loss. From less performance systems, maximum yield of production is unexpected. According to [2] findings, the relative yield has direct relationship with storage efficiency of the system. For this reason, the system should be improved to save water use (application efficiency) and to give maximum yield production (storage efficiency). Bautista [3], also stated that low efficiency of the system was resulted from poor design or management.

Different researchers tried to recommend the way to improve the inefficient system [1], found that, optimizing irrigation efficiency can be possible through changes in irrigation operation and field design, which may lead water saving by reducing deep drainage losses and can enhance the efficiency of the system [28, 43], reported that application efficiency of the system can be increased by shortening the field length and decreasing of the inflow rate [38]. Also stated that efficient furrow irrigation can be ensured by selecting proper combination of decision variable. Application efficiency was affected by the rate of supply, bed slope, infiltration rate of soil, storage capacity of the root zone and land leveling.

4.2 Measurement of advance and recession time

Before any measurement was undertaken, the pharshall flume was installed at 5 m from furrow inlet. The measurement was taken at 2 min initially and increased 5 min and 10 min until the required head has reached. The relationship between stream size and head of 3-inch pharshall flume was taken from [22]. After required stream size reached the four tubes installed at inlet furrow were opened at the same time. The time of advance water front was recorded by stopwatch at 16 m interval of distance starting from application of water to the inlet furrow along the length of 32 m, 48 m and 64 m.

The stream was cutoff when waterfront reached to the end of the furrows. The recorded advance and recession times of waterfront in experimental plot during both irrigation events were recorded. The advance relationships between the first and later irrigation events were developed by EXCEL software and presented in Fig. 4. The result revealed in Fig. 4 that the advance time of the first irrigation event was delayed than the later irrigation event because of the fact that there was, at first, high infiltration than the later and at later there was moisture of the first irrigation that led water to fast to the end of the field.

Fig. 4
figure 4

Advance and recession curve of first and later irrigation events

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4.3 Optimal hydraulic performance parameters

The hydraulic performance indicators of the field were evaluated by assuming calculated target application equal with soil moisture deficit. The required parameters these are furrow geometry, computed infiltration parameters, furrow slope were inserted in the WinSRFR and SIRMOD soft ware package as per their guidelines. The author selected the two latest software packages to simulate the system.

The simulation model for WinSRFR was restricted to use Zero Inertia model because of the end condition of the furrow is blocked; it is not applicable in Kinematic wave model. According to result indicated in Table 7, the maximum application efficiency of 86% and 34% was obtained by S3L1Q2 treatment in first and later irrigation, respectively. As the Performance indicators result observed in Table 7, the efficiency of the later irrigation has decreased due to less requirement depth which is based on soil moisture deficit with the same amount of applied water. The result agreed with [36] findings which stated that the application efficiency varies between irrigation events [28]. Also found that the later irrigation event had less application efficiency than that of first irrigation event.

Table 7 Performance indicators of WinSRFR output
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After running the SIRMOD model, the simulated values of application efficiency, distribution uniformity, requirement efficiency, runoff and deep percolation loss percentages were obtained by feeding the inputs which were collected from field measurement in SIRMOD software. These predicted hydraulic performance indicators of the full Hydrodynamic and Zero Inertia were summarized in Table 8 with maximum application efficiency of 77.72%. As the result indicated in Table 8, the performance using full hydrodynamic and Zero inertia solution models were almost the same. But as the slope increased, the relation was far each other which agree with [39] findings.

Table 8 Performance indicators of SIRMOD output
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As a result of optimization by SIRMOD viewed in Table 8, the high application efficiency of 77.72% was recorded at furrow length of 48 m, furrow stream of 4 l/s, furrow slope of 0.075% and at 43 min cutoff time (treatment S2L2Q2). The second maximum application efficiency (70.72%) was observed by treatment S3L1Q2, but the first maximum application efficiency in volume balance and in WinSRFR model. Therefore, the treatment S3L1Q2 was taken as optimum treatment comparing to other treatments over the three models (WinSRFR, SIRMOD and Volume balance).

The application efficiency simulated in the SIRMOD software under zero inertia and hydrodynamic models of furrow irrigation with closed end boundary condition was similar in comparison with WinSRFR software. Distribution uniformity and storage efficiency were also good indicating the same trend as that of application efficiency. The treatment S3L1Q2 screen output by SIRMOD model was viewed in Fig. 5.

Fig. 5
figure 5

Window of SIRMOD output

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4.4 Validation of WinSRFR simulation model

The whole computation of the best fit evaluation of experimental based and WinSRFR model based was presented in Appendix Table 14 and the output of the computation summarized in Table 9.

Table 9 The Best fit criteria of WinSRFR model and Experimental
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According to the criteria selected and computed, the coefficient of determination (R2) value (0.95) was closer to unity which fills the standard of best fit. The Root Mean Square Error (RMSE) value of 4.78 far from zero which indicate less agreement to best fit and Standard error (SE) value of 0.11 closer to zero that agree to best fit standard. As the result indicates, the WinSRFR model best fit with Experimental based result by Coefficient of determination (R2) and Standard error (SE) criteria.

4.5 Validation of SIRMOD simulation model

The same evaluation criteria to SIRMOD and Experimental based was taken and the computation was tabulated in Appendix Table 15. The output of computed best fit evaluation criteria were summarized in Table 10.

Table 10 The Best fit criteria of SIRMOD model and Experimental
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For the performance evaluation criterion considered, the result obtained from the SIRMOD model were coefficient of determination (R2) 0.68, Root mean Square Error (RMSE) 58.97 and standard error (SE) 1.87. According to the result indicated the SIRMOD model was less agreed with experimental based of application efficiency relative to WinSRFR model because of, the model has a tendency to over predict the volume infiltrated. Even in free drainage end condition Hornbuckle, (2001) found that SIRMOD model poorly correlated (R2 = 0.448) between predicted and measured outflow so, in closed end condition which has no outflow, the model can predict over the amount of water infiltrated. In addition, [27] also reported the WinSRFR model had carried out the simulation of application efficiency with a three-percent error and with higher accuracy than SIRMOD model.

4.6 Water saving

Significant savings can be achieved through improving the application of irrigation water at the field level. The key is to improve the efficiency of irrigation by providing just enough irrigation water to match the available storage in the root zone, thus reducing the amount of water lost to the crop through deep percolation and surface runoff [8]. The result of Table 11 revealed that 42.3% and 60.3% of water can be saved by the optimized decision variables. Water saving observed in Table 11 was differing from field number to field number due to different management level of the irrigators over the field. Thus, the estate can save average of 51.3% of water from two fields and can irrigate 0.51 ha additional land according to [11] reported that from 30.62% of water saving, 0.31 ha of land could be irrigated.

Table 11 Water saving
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5 Conclusion

In this study, to improve the efficiency of the irrigation system for sugarcane fields, optimum decision variables were determined. According to result obtained, the application efficiencies of existing furrow irrigation were in the ranges of 25% to 43% for furrow lengths 48 m and 32 m, respectively, that indicate less performance of the system due to excess application of water and less management skills of irrigators during irrigation events. Even the irrigator management experience were different from one field to another that leads unequal water saving. The more experienced irrigator manages irrigation water effectively and save more water. Thus, the existing furrow irrigation has revealed inefficient performance which needs careful design of the furrow geometry and good management practices. The measured data of experimental site was inserted in the SIRMOD software and WinSRFR software package. Furrow lengths of 32 m, 48 m, and 64 m, slopes of 0.05%, 0.075% and 0.1%, and flow rates of 3 l/sec, 4 l/sec and 5 l/sec were analyzed using volume balance method and simulation model.

It was observed that first irrigation event treatments had better application efficiencies than that of later irrigation event treatments. Maximum values simulated by WinSRFR model were 86% and 34% for first irrigation event and later irrigation event treatments, respectively. This is due to decrease of water required depth in later case with the same application of flow rate which increases deep percolation. The optimum decision variables were observed that gave maximum application efficiency which reflects the overall use of water. From the study, interaction factor that gave performance indicators above the recommended was S3L1Q2 which is third level of slope which is 0.1%, first level of length (32 m), and second level of flow rate (4 l/sec). The interaction gave maximum values during both irrigation events. The optimum design factor of the system was found that furrow length of 32 m which was already abundantly used at the estate and furrow slope of 0.1% which needs improvement. In addition the management factors were also found from this study. The management factors which on-field operators (irrigators) should care were flow rate of 4 l/sec and cutoff time of 15.79 min. The authors’ findings demonstrate that the optimum decision variables are the key recommendation to improve the existing irrigation system. Additionally, many efforts are underway to establish Basin Information Systems (BISs) in major river basins at present. However, these need to be conveyed to a centralized system at federal level that collects water data from different sources including Regional Water Bureaus (RWBs). Data should then be synthesized and translated into useful information for decision making purposes at federal, basin and sub-basin level coupled with investments into capacity building on data management and analysis as well as the necessary software and hardware. Therefore, the outcome of this study may serve as a source of information for use by policy makers and planners during the design and operation of irrigation development programs. In addition, provide motivation for a designer to implement remedial measures if a design is not up to standard.

For this study only two models were used (SIRMOD and WinSRFR models) to evaluated and designed hydraulic performance parameters of furrow irrigation. Future studies should consider comparison of different surface irrigation simulation software packages depending on their inputs and accessibility as that may enhance prediction of the hydraulic performance of furrow irrigation.