Optimizing Water Extraction in Agricultural Systems: A Novel Approach Using a Jet Pump and Multiple Wells

doi:10.21203/rs.3.rs-4353881/v1

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Optimizing Water Extraction in Agricultural Systems: A Novel Approach Using a Jet Pump and Multiple Wells

https://doi.org/10.21203/rs.3.rs-4353881/v1

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This paper proposes a novel approach that utilizes a high-performance jet pump to extract water from multiple wells to a tank, and presents an optimal design of the system in view of the reliability criteria and economic aspects. In this way loss of load probability (LLP) as reliability criteria and the hydraulic network cost as economic criteria are used to simulate the performance of the system. By adopting AFT Fathom simulation software, two designs that exhibit water withdrawal from the wells to the tank were devised. By adjusting the pump's position along the vertical (y) and horizontal (x) axis and implementing different uniform pipe diameters (d) resulted to inconsistent flow rates. The simulation data obtained from these experiments was utilized to construct a Multiple Linear Regression (MLR) model, enabling the prediction of a multi well flow rate through the tank. The MLR model, validated at a 95% confidence level with results showing that R² index and root mean square error (RMSE) values were 0.7987 and 0.03385 respectively. The proposed algorithm selects the optimal pipe sizes, and jet pump position (x, y) relative to tank satisfying both criteria.

Groundwater

Optimal design

Loss Load Probability

Multiple Linear Regression

AFT Fathom

Using one jet pump to draw water from two wells successfully.
Groundwater extraction was examined in Zelabougou, Mali to improve water supply and management.
Minimizing equipment requirements and reduce power consumption, making the system cost-effective.

Developing countries of sub-Saharan Africa, including rural communities, face a myriad of challenges that impede agricultural productivity. Climate change, land degradation, population growth, and heavy reliance on rainfed agriculture are among the social and biophysical constraints that hinder progress (Ngigi 2009). In particular, sub-Saharan African soils are characterized by inherent fertility deficiency, low productivity, and the compounding effects of nutrient scarcity, land degradation, and water stress, which significantly hamper agricultural development in the region (Lahmar et al. 2012). These obstacles pose potential setbacks to agricultural efforts in numerous countries within sub-Saharan Africa, Mali being one of them. Situated in West Africa and landlocked, Mali is among the least developed nations, heavily reliant on rainfed agriculture and livestock for its economy (Ebi et al. 2011). The smallholder agricultural sector primarily focuses on rainfed crops such as millet, sorghum, maize, and cotton for both domestic consumption and market demands. However, agricultural production falls short of meeting Mali's population needs, which are growing at an alarming rate of nearly 3% per year. Currently, over a quarter of Mali's population suffers from malnourishment (IMF 2013). The Malian economy heavily depends on cotton exports, contributing 10% to the national GDP and serving as the primary source of income for 2.5 million farmers in the southern region (Valenghi D 2001).

Mali is traversed by two major rivers, the Niger and the Senegal. Farmers in Mali, particularly in the southern part, confront numerous challenges, including excessive rainwater during the rainy season, water scarcity in the dry season, soil acidity, and nutrient depletion (Yayé et al. 2013). Insufficient land, water, and crop management practices have been identified as key factors contributing to yield gaps in the semi-arid regions of West Africa (Rockstrom et al. 2007). In southern Mali, water runoff was estimated to be between 20–40% of annual rainfall and up to 70% during individual rainstorms (Roose 1987). Cultivated soils experienced significant soil losses, with 25 kg of nitrogen and 20 kg of potassium per hectare per year. These losses in soil fertility accounted for 44% of farmers' income depletion. Southern Mali receives an average annual rainfall ranging from 500 to 1000 mm, characterized by strong variations and irregular rainfall patterns (Akinseye et al. 2016). The scarcity of water due to annual rainfall instability poses a major obstacle in achieving food security and reducing poverty in the region. Recent reports (Serigne et al. 2006; Lobell et al. 2011; Vgen & Gumbricht 2012) indicate that rainfall has become less reliable, and growing seasons have shortened in several areas of West African countries, creating a shift in farming and natural resource management practices.

Groundwater plays a pivotal role as an essential resource in sub-Saharan Africa, where approximately 300 million people depend on aquifers for domestic water supply (Foster & Garduño 2013). Groundwater is generally abundant, of good quality, sufficient in quantity, and affordable (Adelana & MacDonald 2008). Aquifers serve as a dependable source of freshwater during dry spells, particularly in regions with semi-arid climates where rainfall and surface water may be absent for extended periods. This is precisely the case in Mali, where groundwater constitutes the primary source of drinking water supply. Birhanu Zemadim and Ramadjita have conducted extensive studies, mapping and geo-referencing shallow wells in the southern districts of Bougouni and Koutiala in Mali. Out of the 484 wells studied, water availability was observed at an average depth ranging from 5.5 to 15.5 meters in most cases (Birhanu & Tabo 2016).

Water well pumps play a vital role in bringing water from wells and springs to the surface, ensuring a reliable water supply. These pumps can generally be categorized into two main types: shallow well pumps and deep well pumps. Shallow well pumps are typically situated above ground level and operate by creating suction to lift water out of the ground through a suction pipe (Sivasamy et al. 2018). Generally utilized when the water source is close to the surface or in an artesian well, these pumps, often in the form of jet pumps, effectively meet the demand for water extraction. At sea level, the typical lift for suction pumps ranges from 8 to 9 meters, showcasing their efficiency in accessing water from relatively shallow depths.

Deep well pumps are specifically designed to tackle the challenge of pumping water uphill from wells with greater depths. They come in two main variations: submersible pumps and deep well jet pumps. Submersible pumps are designed to be submerged in the water, with the inlet below the water level, enabling them to efficiently lift water to the surface (Zhou et al. 2020). On the other hand, deep well jet pumps utilize a different configuration. They are not limited by suction lift restrictions and have the advantage of a significantly increased depth limit for water well applications, up to four times that of shallow well pumps. Deep well jet pumps are commonly employed in situations where the water source is located at considerable depths. Submersible pumps are impeller-style, ensure effective water extraction and delivery (Sivasamy et al. 2018).

In the context of this study, a deep well jet pump was utilized to investigate its performance and suitability for the specific application. Deep well jet pumps are known for their recirculating-type systems, primarily employed to draw commercial, industrial, and residual water from deep wells. These pumps incorporate deep well ejectors (ecological), which play a crucial role in lifting water and maintaining a net positive suction head (NPSH) (Schiavello & Visser 2009) at the surface mechanical pump. The deep well jet pump, equipped with hose connections, is carefully lowered into the well casing, with its discharge outlet situated at the surface. To initiate the pumping process, an initial priming is required and sustained by a foot valve positioned at the suction inlet of the pump. Once the jet pump is in operation, the flow through the ejector effectively entrains and lifts water from the well, ensuring a continuous supply. The capacity of this type of system is dependent on the depth of the well and the specific capacity of the jet pump, highlighting the importance of selecting the appropriate equipment for optimal performance. In addition, polyethene pipes were selected for water transportation due to their numerous advantages (Wang et al. 2012). Listing some of them: Corrosion resistance — PE pipes are no electrochemical corrosion and no anti-corrosion layer and can be used up to 50 years; PE pipes are excellent bending performance use fewer accessories; strong resistance to low temperature performance; good impact resistance; light weight and economical.

In this study, the conventional use of jet pumps, was pushed further. A groundbreaking approach was introduced, a single, high-performance jet pump to extract water from multiple wells simultaneously. This innovative method ensures an uninterrupted water supply, even if one of the wells becomes depleted. In contrast, the traditional approach would require a separate pump for each well, resulting in higher costs (Billington et al. 1994). The need for multiple pumps and associated equipment was eliminated, leading to cost reduction. Moreover, the utilization of a single pump across multiple wells guarantees a continuous water supply, meeting various demands such as domestic, irrigation, and industrial. Another advantage is the reduction in power consumption. By maximizing energy efficiency through the use of a single high-performance pump, energy consumption was minimized compared to operating multiple individual pumps. This not only contributes to cost-effectiveness but also aligns with sustainable practices, reducing environmental impact.

This simulation was done based on an environment found in Zelabougou village located in western region of Mali where there is insufficiency of water facilities. The only sufficient water source in Zelabougou is underground water. In this work two wells were drilled such that they are 50 m apart and a jet pump was employed to draw water from well A and B simultaneously to the tank. Well A was found to provide enough water at the depth of 13 m, and well B at the depth of 10 m. The water tank was situated at a distance of 70.2 m from well B and 7.25 m high. In this case study the pipes and fittings used were polyethylene. This type of pipes and fittings are suitable for irrigation systems and have a pressure rating that matches or exceeds the working pressure that will exist in the well. The pipes and fittings were employed to fully advance the efficiency and transport of water through the system. The simulation was run by a commercial software AFT fathom. It was used for figuring the possibility of using a single jet pump to draw water from two wells to the tank and obtaining the flow rate the tank receives.

3.1 Pump Selection

Pump Sizing

To choose a pump the first major parameter determined was how much water is required. The pump flow rate was 1.5 m³/h. This was calculated from the area of the field, and how much water must be provided to the crop seeds.

Second major parameter was the total dynamic head, this parameter is crucial because it is a measure of how much pressure the pump must generate in order to draw water from the wells and its delivery to the required level in the tank. Factors to be considered in finding the total dynamic head are as follows (Atoyebi et al. 2015):

Vertical rise
Pumping level
Static water level
Drawn down
Dynamic water level
Pump depth
Well depth
Friction losses

$$\text{T}\text{D}\text{H} = {\text{P}}_{l}+ {\text{V}}_{r} + \left[{\text{L}}_{t}+ {\Sigma }\left({\text{n}}_{f}. {\text{f}}_{e}\right)\right]\times {F}_{h}\times {30.48}^{-1}$$

Here TDH – Total dynamic head; P_l – Pump level; V_r – Vertical rise; L_t – Total length; n_f – Number of fittings; f_e – Fittings equivalent in meters of pipe; F_h – Friction loss;

Figure 1 shows the measurement which were used in computing TDH. The static lift of water is the distance from the inlet of the pump to the surface of water in the borehole before the debut of pumping. When pumping was established the water level drops down until the drawdown level is reached. The difference between the previous level and the current level represents dynamic lift of the borehole. Hence pumping level was defined as the sum of the static and dynamic lift of water. Here we assume the pump level for the two wells combined was 18.25 m with a total depth of 23 m. The vertical rise is the height from the outlet of the pump to the inlet of the tank as shown in Fig. 1. Again, in this study the value for the tank vertical rise was taken to be 6 m and the pump was 1.25 m above the ground. The resistance of water as it flows through a pipe is known as friction loss. A variety of factors determine the value of frictional losses, including flow rate, pipe type, its inner diameter, type and number of fittings. The losses are expressed in terms of equivalent head (meters) and can be calculated using Pipe Sizing Charts or Hazen-William’s equation. Hazen-William’s formula is an empirical equation that estimates the friction loss in water pipes which is expressed as follows:

$${F}_{h}=\frac{7.925\times {10}^{-12}{(1000\times Q)}^{1.85}\times L}{{C}^{1.85}\times {d}^{4.87}}$$

Here F_h – Friction loss (m); Q – Flow rate (m3/hour); L – Length of the pipe (m); C – Roughness coefficient; and d – Inner diameter of the pipe. A typical roughness coefficient C of a PE pipe is 140.

Now employing to Eq. (2) in which total pipe length of 103.7 m, pipe inner diameter of 40 mm, roughness coefficient is 140 (PE pipe), we get the following head loss:

$${F}_{h}=\frac{7.925\times {10}^{-12}{(1000\times 1.5)}^{1.85}\times 103.7}{{140}^{1.85}\times {0.04}^{4.87}}=0.42$$

Valves and fittings on a pipe also contribute to the overall head loss that occurs, but these must be calculated separately from the pipe wall friction loss, using a method of modeling pipe fitting losses with k factors. In addition, the number of similar fittings attained to assembly the model was 7 elbow joints and a single t-joint. Each with a minor loss coefficient of 0.91 and 1 respectively.

Therefore, using Eq. (1) we get TDH as follows:

$$\text{T}\text{D}\text{H} = 18.25+ 6 + \left[103.7+\left(7\times 0.91+1\times 1\right)\right]\times 0.42\times {30.48}^{-1}=26 m$$

Pump Performance

It is crucial to note that the Valco deep well jet pump BCV222-2 is specifically designed for deep well employment and is satisfactory for wells with low capacity. This pump is self-priming and can effectively draw water from deep boreholes.

3.2 Flow Design of Water from Wells to Tank

3.2.1 Process Flow of the System

The provided flow chart in Fig. 3 illustrates the comprehensive process flow of the entire system. In order to monitor the water levels in the wells, deep well water level sensors were submerged at the surface of each well. These sensors were responsible for measuring the height of the available water in the wells. Additionally, a tank float valve was employed to determine whether the tank had reached its maximum capacity or not. The water level readings in well A and B were denoted as L1 and L2 respectively, and these readings were updated every second.

The extraction of water from the boreholes was carried out by the jet pump, following a sequential process based on a series of conditions. The primary condition involved prioritizing the well with a greater volume of water. If the two wells had water available, the one with the highest water level would be given priority for extraction. Simultaneously, the system would monitor the tank to determine if it had reached its peak capacity.

Once the well with the highest water volume was identified, a secondary condition would be evaluated before initiating the extraction process. This condition involved checking whether the water level in the selected well had reached its drawdown point, denoted as Ld. The drawdown point signifies the minimum water level that should be maintained in a well to ensure its proper functioning. If the water level in the well had not reached the drawdown point, the pump would be triggered on, allowing water to flow into the tank until either the well reached its threshold or the tank became full.

When the tank reaches its maximum capacity, indicating that it was full, the pump would be automatically triggered off. This mechanism prevents overfilling of the tank and ensures efficient water management within the system.

Figure 3: Process Flow

3.2.2 First Model Proposal

Figure 4 illustrates the first model, it was designed in a way that the pipes (P26 and P27) which drew water from each well were connected to the surface pipe by elbow joints (J9 and J10) where P2 and P5 were attached from, towards elbow joints (J12 and J13) respectively. From joint 12 and 13, the pipes (P19 and P18) were connected towards the t-joint (J14) where both streams intersect. Pipe 13 fitted from the t-joint (J14) was joined to an elbow fitting (J22) and then P14 extends from elbow joint (J22) to the inlet of the jet pump (J4). Pipe (P28) connected from outlet of the pump (J4) to the elbow joint (J20) which is then connected to pipe (P16) that extends to joint 21. Pipe (P15) from joint (J21) was connected to the tank inlet. Table 1 presents the lengths of each pipe shown in Fig. 4.

Table 1: Pipe length

No	Pipe identity	Length (m)
1	P26	13.5
2	P27	10
3	P2	60
4	P5	10
5	P19	0.25
6	P18	0.25
7	P13	59
8	P14	0.5
9	P28	1
10	P16	6
11	P15	0.2

The pump was shifted along the horizontal axis through pipes (P13 and P28) such that P13 with initial length of 59 m reduces at a decrement of 5 m to a final length 14 m. Meanwhile P28 simultaneously increased at an interval of 5 meters from a length of 1 m to 46 m.

The pump was adjusted along the vertical axes between pipe 14 and pipe 16. When the pump inlet pipe (P14) was increased to a length of 1 m from 0.5 m, simultaneously pipe 16 was reduced from 6 m to 5.5 m. This adjustment was done such that the position of the tank was maintained at 7.25 m high. The pump adjustment was done at an interval of 0.1 m from the initial positions of 0.5 m and 6 m respectively to pipe 14 and pipe 16 along the vertical direction. At each position of the pump the pipe diameters were altered at a set of: 40-, 50-, 65-, and 80-mm. Each diameter was uniform throughout all the pipes in the system. All these adjustments were made so that a maximum flow rate towards the tank was attained.

The Fig. 5 illustrates graphs of pump position along the x-axis against flow rate at different pipe diameters. Each graph represents its own pump height with respect to the tank. It was observed that an increment in pipe diameter leads to an increase in the flow rate, and a decrease in pipe 16 length and increase pipe 14 length (suction depth) will slightly increase the flow rate (Nanda et al. 2011). The closer the pump to well B along the horizontal axis results in a slightly larger flow rate.

3.2.3 Figure 5: Flow rate of a two well pumping system by adjusting the pump position and pipe diameter

3.2.4 Second Model Proposal

Figure 6 illustrates the second model; the pump was shifted along the horizontal axis through pipes (P3 and P4) such that P3 with initial length of 49 m reduces at a decrement of 5 m to a final length 4 m. Meanwhile Pipes (P4 and P6) simultaneously increased at an interval of 5 meters from a length of 1 m to 46 m and a length of 71 m to 116 m respectively. The diameters were maintained throughout all the pipes in the system for the same set of diameters as mentioned in the first proposal model. Another pump adjustment was made along the vertical axis between pipe 5 and pipe 8. When the pump inlet pipe (P5) was increased to a length of 0.75 m from 0.25 m, simultaneously pipe 8 was reduced from 6 m to 5.5 m. This adjustment was done such that the position of the tank was maintained at 7.25 m above the ground. The pump adjustment was done at an interval of 0.1 m from the initial positions of 0.25 m and 6 m respectively to pipe 5 and pipe 8 along the vertical direction. For each height of the pump, the previous experiments were repeated in this model as stated above. These adjustments were made so that the maximum flow rate towards the tank is achieved. Table 2 presents the lengths of each pipe shown in Fig. 6.

Figure 6: Second Model

Table 2

Pipe lengths
No	Pipe identity	Length (m)
1	P1	13
2	P9	10
3	P3	49
4	P4	1
5	P6	71
6	P8	6
7	P7	0.2

The Fig. 7 represents the pump position along the x-axis against flow rate along y-axis at a range of different pipe diameters. Each graph represents its own pump height with respect to the tank. An increment in pipe diameter led to an increase in the flow rate, and a decrease in pipe 8 length and increase pipe 5(suction depth) length slightly increased the flow rate (Nanda et al. 2011). The closer the pump to well B along the horizontal axis resulted in a slightly larger flow rate.

3.3 Model Development

Many engineering problems involve determining the correlation between two or more variables. A Multiple Linear Regression is a method used to model the linear relationship between a dependent variable and one or more independent variables. The dependent variable is sometimes known as the response, and the independent variables as the explanatory. The term “regression” is often used to convey the concept of returning to the mean or average. MLR evaluates the correlation between two or more input variables by fitting a linear equation to observed data (Khademi & Behfarnia 2016; Sadrmomtazi et al. 2013). It involves summarizing data as well as investigating the relationship between variables.

Table 3

Correlation matrix between variables
	Flow rate (Q)	x	y	d
Flow rate (Q)	1	-0.34	-0.56	0.6
x	-0.34	1	0.00	0.00
y	-0.56	0.00	1	0.6
d	0.6	0.00	0.00	1

Table 4

Model summary
Model	R²	Adjusted R²	Std. error of the estimate
Equation	0.7987	0.7975	0.03385

Table 5

ANOVA table
Model		Sum of squares	Df	Mean square	F	Sig.
Equation	Regression	2.16421	3	2.16421	629.6	0.000
	Residual	0.54537	476	0.00115
	Total	2.70958	479

Table 6

Multiple regression coefficient
Model	coefficient	Std. error	t	P-value	Lower bound	Upper bound
Constant	3.269000	0.052440	62.35	2e-16	3.166452	3.372531
x	-0.000679	-0.000041	-16.63	2e-16	-0.000759	-0.000599
y	-0.248200	-0.009046	-27.43	2e-16	-0.265967	-0.230416
d	2.989000	0.101900	29.32	2e-16	2.788855	3.189479

Through semi-empirical knowledge, the wide knowledge in analyzing the behaviors of flow rate provided by the pump is a functional relationship between the independent variables and the dependent variables. For the system under study, we have flow rate received at the tank Q as the response variable, position of the pump along the horizontal and vertical axis x and y respectively, and diameter of the pipe d as explanatory variables. For the system under study, we have flow rate received at the tank Q as the response variable, position of the pump along the horizontal and vertical axis x and y respectively, and diameter of the pipe d as explanatory variables. The response values ‘Y’ i.e., flow rate Q are obtained from a fluid dynamic simulation software (AFT Fathom), as explained in the previous section. Table 3 shows the correlation between the dependent variable (flow rate) and any other independent variable (x, y, d), and also the correlation among the independent variables. As shown in the correlation matrix table, diameter of the pipe (d) has the highest correlation with the flow rate compared to other variables (r = 0.6) followed by the jet pump position (y) along the vertical axis (r = -0.56). All of the predictor variables except (d) have negative correlation with flow rate. There is a slight relationship among x and Q. The explanatory variables are independent from each other, but indeed have a relationship with the response.

In this case study, the MLR model for the flow rate of the two wells pumping system was obtained as follows:

$${R}_{Q}={{\beta }}_{0} + {{\beta }}_{1}\text{x} + {{\beta }}_{2}\text{y}+{{\beta }}_{3}\text{d}$$

The coefficients value of Eq. (3) and their units are as follows:

β₀ = 3.269(m³/h) β₁ = -0.000679(m²/h)

β₂ =-0.2482(m²/h) β₃ = 2.989 (m²/h)

The fitting performance of the derived model is analyzed mainly from the values of two parameters i.e., R² and RMSE. The value of R² lies in the range of 0 to 1 usually. Higher the value of R² and lower the value of RMSE, the better is the constructed model in performing the task of the prediction of the response variable. Here as seen from Tables 4 and 5 the R² = 0.7987 is quite close to 1, and RMSE is quite low; thus, it can be said that the derived model is quite fit to imitate the process of flow rate prediction from the available data. The information regarding the regression coefficients, standard error, and the significance test values are shown in Table 6. The p-values are derived from the t-test values, and they represent the significance of the independent parameter in the prediction of the dependent parameter. The p-values in Table 6 prove that all the parameters are highly significant in the forecast of the tank flow rate Q. The values of the fitted model parameters predicted with a 95% confidence level are shown in Table 6 as well.

Figure 8 shows the normal probability plot for the multiple regression model. The normal probability plot is used for evaluating the assumption that the distribution of the errors (residuals) is normal. The points in Fig. 8 fall reasonably close to a straight line, suggesting that the distribution of the error terms does not depart substantially from a normal distribution.

3.4 Optimization

3.4.1 Algorithm Proposal

A well-designed irrigation system must effectively meet the water demand of crops at a given site. Thus, during the months corresponding to the crops' vegetative cycle, it is necessary to determine the required water volume for irrigating the crops and ensure that the hydraulic system and reservoir volume can adequately supply this volume. In this study, we have adopted the load losses probability (LLP) method for water storage in the tank. The LLP is defined as the ratio between the water deficit and the total water requirement. The primary objective is to calculate values that satisfy the specified threshold LLP. It is essential to emphasize that the chosen sizes of system components must fulfill the irrigation requirements for all months of the crops' vegetative cycle, spanning from July to December.

Therefore, the algorithm's principal aim is to achieve a cost-effective system design that maximizes the flow rate while ensuring the necessary water volume for irrigation. The algorithm relies on the following key factors:

• the reservoir volume,

• the hydraulic system,

• the desired flow rate,

• the threshold LLP.

3.4.2 Algorithm Determination

In order to find the optimal minimum costs, the optimal components sizes which are: the pipe diameter, the jet pump position (x, y) relative to the reservoir along the horizontal and vertical axes, and flow rate are first found. The algorithm used to obtain the result costs is described by the following steps:

Step 1. Define Structures: In this step, the models utilized in the application are defined. Each model consists of the total pipe length, joints prices based on the pipe diameter and the number of joints used, and the initial and final positions of the jet pump along the horizontal axis relative to the tank. The pipe price per meter is determined based on the diameter, and the volume of the reservoir is taken into consideration. Additionally, the flow rate equation is derived from a MLR model, and a maximum LLP is defined.

Step 2. Initialize Models: The models are initialized by assigning the respective total pipe lengths, obtaining joints prices from local contractors, and specifying the initial and final positions along the horizontal axis for each model, as determined in the proposed model.

Step 3. Generate Vectors: A set of vectors is generated, consisting of different diameters used in the project, along with their corresponding pipe prices per meter and joints prices.

Step 4. Calculate Optimal Values: For each generated vector, the flow rate is calculated based on the jet pump's position (x, y) along the vertical and horizontal axes, as well as the pipe diameter. The LLP is computed using the formula:

$$LLP=\frac{{V}_{tank}-{Q}_{i}\times 7}{{V}_{tank}}$$

If the calculated LLP is less than or equal to the predetermined threshold LLP, it is stored as an optimal value for the corresponding model. The process continues to find additional optimal values. If the LLP exceeds the threshold, the jet pump position and pipe diameter are adjusted iteratively to find the next optimal value that satisfies the LLP condition.

In order to tackle the cost optimization problem, a maximum LLP threshold of 20% is established, ensuring that each design must on average meet the water demand 80% of the time. The 20% LLP serves as an upper limit, and certain designs may exhibit greater reliability with lower LLP values if the lowest-cost system can fulfill a greater proportion of the demand. This LLP restriction is based on the guidelines provided by the Food and Agriculture Organization (FAO) for tomato cultivation, which indicates that achieving 80% of the evapotranspiration requirements results in higher crop yields. To carry out the cost optimization for the case study involving a 500 m² land surface in Zelabougou with a tank volume of 18 m³, the proposed algorithm is tested using the parameters outlined in Table 7 and Table 8.

Table 7

Hydraulic Parameters
Model	Total pipe length (m)	X_initial (m)	X_final (m)
Model 1	160.7	1.2	46.2
Model 2	150.3	71.2	116.2

Table 8

Cost of the Hydraulic Parameters
Diameter (mm)	Pipe price per m (USD)	Total joint prices for model 1 (USD)	Total joint prices for model 2 (USD)
40	0.30	0.70	0.56
50	0.34	1.25	0.97
65	0.50	2.09	1.53
80	0.58	3.06	2.23

The optimum values have been highlighted in Table 9. They show that the strategy ensures the needed water volume in 63 solutions. The solutions are categorized in three costs (USD): 82.55, 90, and 97. These costs are gotten from different adjustment in the two models such as diameter of the entire hydraulic network, position of the jet pump relative to the tank. However, with considering two conflicting objectives function the best compromising solution is the solution with the minimum cost and maximum flow rate. The best compromising solution with ranking 1 is shown in the first row of the table. The corresponding cost (USD) is 82.55 and related flow rate (m³) is 2.0937 and loss of load probability (%) is 18.44 this solution is achieved from model 1 at pipe diameter 65 mm, position of the jet pump with respect to the tank (1.2 m, 5.5 m).

Table 9

Summary Results of the Calculation of the Minimum Costs
Ranking No	Model		Diameter (mm)	X (m)	Y (m)	Q (m³)	LLP (%)	Cost (USD)
1	Model 1		65	1.2	5.5	2.09737	18.4356	82.55
2	Model 1		65	6.2	5.5	2.09398	18.5676	82.55
3	Model 1		65	11.2	5.5	2.09058	18.6997	82.55
4	Model 1		65	16.2	5.5	2.08719	18.8317	82.55
5	Model 1		65	21.2	5.5	2.08379	18.9637	82.55
6	Model 1		65	26.2	5.5	2.0804	19.0957	82.55
7	Model 1		65	31.2	5.5	2.077	19.2278	82.55
8	Model 1		65	36.2	5.5	2.07361	19.3598	82.55
9	Model 1		65	1.2	5.6	2.07255	19.4008	82.55
10	Model 1		65	41.2	5.5	2.07021	19.4918	82.55
11	Model 1		65	6.2	5.6	2.06916	19.5329	82.55
12	Model 1		65	46.2	5.5	2.06682	19.6239	82.55
13	Model 1		65	11.2	5.6	2.06576	19.6649	82.55
14	Model 1		65	16.2	5.6	2.06237	19.7969	82.55
15	Model 1		65	21.2	5.6	2.05897	19.9289	82.55
16	Model 2		80	71.2	5.5	2.09468	18.5404	82.55
17	Model 2		80	76.2	5.5	2.09128	18.6724	90.00
18	Model 2		80	81.2	5.5	2.08789	18.8045	90.00
19	Model 2		80	86.2	5.5	2.08449	18.9365	90.00
20	Model 2		80	91.2	5.5	2.0811	19.0685	90.00
21	Model 2		80	96.2	5.5	2.0777	19.2005	90.00
22	Model 2		80	101.2	5.5	2.07431	19.3326	90.00
		...
61	Model 1		80	46.2	5.7	2.06201	19.8107	97.00
62	Model 1		80	11.2	5.8	2.06096	19.8517	97.00
63	Model 1		80	16.2	5.8	2.05756	19.9838	97.00

The Fig. 10 offer a comprehensive overview of the optimum flow rate values achieved by the jet pump when extracting water from either two wells (Q1) or a single well (Q2). Upon careful examination, it becomes evident that Q1 consistently exhibits higher flow rate values compared to Q2. The variance in flow rates between Q1 and Q2 can be primarily attributed to the divergence in water availability. Notwithstanding the similarity in pump performance across both scenarios, the jet pump drawing water from two wells delight in an ample water supply, thereby facilitating higher flow rates. This pivotal distinction underscores the pivotal role that water volume plays in shaping the system's overall potency.

Furthermore, the utilization of two wells offers an additional benefit. In the event that one of the wells dries up or experiences a temporary disruption, the second well can still be operational, ensuring a continuous water supply. On the other hand, when relying on a single well, any drying or interruption would result in a momentary shortage of water until the issue is resolved.

Through rigorous investigation and dynamic water flow simulations employing AFT Fathom, this study conclusively demonstrates the feasibility of utilizing a single jet pump to draw water from two wells. The main concerns are to achieve the optimal solution with the minimum design cost and, at the same time, satisfy the required minimum LLP through commercially available pipe sizes. The detailed results are as mentioned below.

• To establish a comprehensive understanding of the relationship between the explanatory parameters—specifically, the pump positions (x and y) and the pipe inner diameter (d)—and the resulting flow rate to the tank, a robust mathematical model based on Multiple Linear Regression (MLR) was meticulously derived with a 95% confidence level. The significance test results of the MLR model affirm the substantial impact exerted by the explanatory parameters on flow rate, as indicated by an adjusted coefficient of determination (R²) of 0.7987. These statistical findings provide invaluable insights that can guide the achievement of two wells flow rate.

• To maximize the system's performance, it is imperative to carefully consider the positioning of the jet pump relative to the well. Placing the pump too high above the well could impede its ability to effectively draw water. It is advisable to adopt a sequential operational mode while using a single jet pump to draw water from two wells, as described before in the process flow chart. However, it is worth noting that simultaneous extraction from both wells is also a viable option, albeit with careful consideration of system dynamics and hydraulic constraints.

• An optimized solution was obtained in the two models for different adjustment such as pipe diameter of the entire hydraulic network and position of the jet pump relative to the tank position while maintaining the constrains. However, with considering two conflicting objectives function the best compromising solution is the solution with the minimum cost and maximum flow rate.

• Comparing the system employing a single jet pump drawing water from two wells on the first model to the alternative configuration utilizing a single well, the results yield a justifiable outcome. The system featuring two wells and a single jet pump offers a distinct advantage—an additional well serves as a reliable backup water source, ensuring an uninterrupted flow in the event of either well running dry. Furthermore, this configuration eliminates the time lost waiting for a single well to replenish its water levels, resulting in enhanced efficiency.

This research highlights that the first model was chosen according to its feasibility and efficiency gain associated with exploring a single jet pump to draw water from multiple wells. The derived mathematical model and the comprehensive analysis of the proposed algorithm provide invaluable conclusions that can guide the achievement of optimized solution. By incorporating these findings into system design and operational strategies, practitioners can enhance the reliability and performance of water extraction processes, offering significant benefits for various applications.

Ethical Approval The authors confirms that this article is original research and has not been published or presented previously in any journal or conference in any language (in whole or in part).

Consent to Participate The authors declares that they are aware and consent with his participation on this paper.

Consent to Publish The authors declares that they are consent with the publication of this paper.

Author Contributions All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Hogen Wu and Doucoure Ibrahim. The first draft of the manuscript was written by Hlomayi Brendon, Paramananthan Mahilan revised the language, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Funding The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Competing Interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Optimizing Water Extraction in Agricultural Systems: A Novel Approach Using a Jet Pump and Multiple Wells

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Abstract

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Highlights

1. INTRODUCTION

2. THE STUDY AREA

3. METHODS

3.1 Pump Selection

3.2 Flow Design of Water from Wells to Tank

3.2.1 Process Flow of the System

3.2.2 First Model Proposal

3.2.4 Second Model Proposal

3.3 Model Development

3.4 Optimization

3.4.1 Algorithm Proposal

3.4.2 Algorithm Determination

4. Result and Analysis

5. Conclusion

Declarations

References

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