Forecasting the seepage loss for lined and un-lined canals using artificial neural network and gene expression programming

Abstract

Canal lining is customarily used to raise water-use effectiveness and reduce seepage loss. The major water losses in an irrigation channel are due to leakage and evaporation. The Egyptian General Integrated Management for Water Resources and Irrigation introduced a proposal for lining the Al-Hagar canal based on these losses. This study investigates the effect of lining in the Al-Hagar canal on flow characteristics, and compares the canal before and after introducing the lining. Additionally, it discusses the most common type of water loss, namely, losses due to seepage. Fieldwork was conducted on the Al-Hagar canal, Al-Saff Center, South of Helwan city, Egypt. The result revealed that the discharge of the canal after the lining is approximately 1.362–1.573 times greater than that of the un-lined section. Water losses in the Al-Hagar canal were 38.736% when un-lined but decreased to 29.253% when lined. The conveyance effectiveness in the un-lined canal, which is approximately 61.26%, increased to 70.75% when the entire canal is lined, which means a 9.483% improvement of conveyance. New relations were introduced using Artificial Neural Network and Gene Expression Programming to forecast the seepage loss in the lined and the un-lined canal as a function of Manning’s coefficient, Froude number and hydraulic radius. The consequences were better using the GEP program than using ANN for the lined and the un-lined canals. The value of the determination coefficient was 0.98, Correlation factor was 0.99, and the RMSE was 0.0017 for lined canals and the value of determination coefficient was 1, Correlation factor was 1, and the RMSE was 0.0003 for un-lined canals.

Keywords:

• Conveyance efficiency
• gene expression programming
• lined canal
• seepage
• un-lined canal
• losses

1. Introduction

Seepage losses via the channel sides and bed, evaporation losses from the free surface of the channel top width, and transpiration losses from weeds and grass are the three primary types of losses in irrigation canals. These losses need to be reduced to ensure the efficient performance of the irrigation system and effective water use. Lined canals are majorly used to prevent water seepage through the soil, however, the un-lined canal is used to recharge the groundwater through the seepage. A tremendous number of authors have studied the lined and the un-lined canals.

The variances in water losses using the lined and the un-lined waterways were assessed in a specific area of the Indus Basin in Pakistan (Arshad et al. 2009). It was determined that the lining decreased water loss by 22.5%. The effects of lining on four waterways lined were evaluated by linking the average water loss of 43.5% from the lined to the average water loss of 66% from un-lined waterways (Javaid et al. 2012). It was found that the transportation effectiveness of lined waterway sections ranged from 83% to 90%, whereas un-lined waterway sections ranged from 36% to 69%. A comparison of the performance and cost of various watercourse improvement alternatives was studied (Chatha et al. 2014), and the average water loss rate was calculated to be 1.91, 3.08, and 2.51 lps per 100-m length of the watercourse. The findings showed that lining is superior to earthen renovation and cleaning of watercourses for lasting water savings. Under the existing conditions, the transportation losses from lined and un-lined sections of the canal irrigation network were established (Jadhav et al. 2014). The overall effectiveness and total loss from the lined, un-lined section of the canal, and un-lined field channel were obtained as 75%, 52%, and 35%, and 0.184, 0.61, and 0.183 Mm³, respectively. The water balance or inflow-outflow approach was used to assess the losses across a large length of a canal (Martin and Gates 2014). With calculated probability distributions exhibiting significant uncertainty, the mean- assessed seepage loss rates for the tested canal reached a ranged from around _0.005 (gain) to 0.110 m³ s⁻¹ per hectare of the canal wetted perimeter (or 0.043–0.95 m d⁻¹). The impact of lined and un-lined canals on groundwater recharge was studied in the lower Bhavan basin (Mirudhula 2014). The analysis revealed that the un-lined canal recharges the groundwater about 20% more than the lined canal. The annual rainfall in this region also changed rapidly. The conveyance loss evaluation was studied in the lined and un-lined tertiary channel systems in South Asia (Sultan et al. 2014). The findings showed that in Pakistan, lined watercourses account for around 43.5% of water losses while unlined watercourses account for 66%. However, in India, lined watercourses account for 11% of water losses whereas unlined watercourses account for 20–25%. The ideal design of concrete canal sections for reducing water loss and earthworks expenditures was investigated (Tabari et al. 2014). The optimization process was run and demonstrated using the MATLAB programming language. The final findings were presented as dimensionless graphs, which simplified the optimal design of canal dimensions at the lowest cost per metre length. The water conveyance efficiency, yearly water savings, and cropping intensities on three lined watercourses chosen on Pureed minor in the Jamrao canal command were investigated (Mangrio et al. 2015). It was found that using 30% lining of the initial portion of the watercourses saved 6.5 ha-m, which could be used to cultivate an additional land of 7 ha. The cropping intensity increased by about 29% in the Rabi and 12% in the Kharif seasons. Under Egyptian circumstances, the enhancement of irrigation water conveyance efficiency using lined canals and spvc-buried pipes was investigated (Osman et al. 2016). The results showed that the earthen canal, lined canal, and buried pipes had conveyance efficiencies of 65%, 92.2%, and 98.7% in winter, and 59.6%, 87.1%, and 89.7% in summer, respectively. The lined canal and spvc-buried pipes reduced conveyance losses by 68.1% and 96.3% in summer and 77.7% and 96.3% in winter, respectively, compared to the earthen canal. A comparative study of lining watercourse techniques in Bahawalnagar district in Pakistan was conducted (Zubair et al. 2016). It was noted that the irrigation water losses for the precast concrete parabolic lining watercourse ranged from 35% to 52%, while those for the rectangular lining ranged from 64% to 68%. A comparison study was undertaken comparing watercourses of different forms employing rectangular brick masonry and circular precast parabolic pieces. Using these techniques, it was predicted that the watercourse is hydraulically more efficient, has minimum structure cost, is more durable, has fewer conveyance losses and is most suitable for community farmers. To calculate the percentage of water saved, the losses from lined and unlined watercourses in a nearby geographic area were analysed and compared (Awan et al. 2017). Polynomial regression was used to estimate the percentage of water savings versus the increase in percentage lining, and the optimal lining length for un-lined water channels was calculated to be 50%. Total wetted area, which was determined to be 0.22 and 0.496 for lined and un-lined watercourses, respectively, was estimated and predicted to be impacted by watercourse lining on water distribution (warabandi) (Bhatti et al. 2017). Hence, the effectiveness of warabandi was found to be 78% and 50.4% for lined and un-lined water courses, respectively. This showed that the effectiveness of warabandi was 85% in a lined section of the watercourse, 68% in its tail section, and 40% in the earthen w/c tail section. The conveyance efficiencies in the lined and un-lined sections of the lined watercourse were 98.76% and 90.60%, respectively, and 70.59% for the earthen watercourse, with cropping intensity increased by up to 8% after lining. A research was provided to evaluate the relative influence of watercourse lining on the possibility of seepage reduction (Solangi et al. 2018). According to the research, lining 30% of the early segment of watercourses resulted in an average annual water savings of 10.32 ha-m. Similarly, in the Rabi and Kharif seasons, cropping intensity increased by 15% and 14%, respectively. The efficacy of watercourse lining in Sindh was studied (Soomro et al. 2018), and a research on different types of lining materials for watercourses was conducted. It was found that a fiberglass type plastic material with semi-circular or U-shaped, precast fabricated reinforced cement concrete trapezoidal or parabolic shapes could be used for watercourse lining instead of brick mortar or concrete. With the aid of a hydraulic model constructed on a spatial platform, the conveyance loss of the Dudhganga Right Bank Main canal was estimated and verified using flow-monitoring events (Kulkarni and Nagarajan 2019). To analyse the transportation losses and comprehend the behaviour of the canal, the hydro-spatial model was simulated. The results showed that the Dudhganga Right Bank Main canal had an average of 39.96% water conveyance loss. The results of a numerical simulation to analyse the contributing elements to seepage from earthen channels were estimated (Asl et al. 2020). When the seepage was compared to empirical relationships, it was discovered that the empirical connections contain too many errors in the seepage estimation. The linear and nonlinear multivariate regression connections fit the seepage discharge estimates well. Ponding tests were performed on the canal segment with four different lining statuses to explore the canal lining influence on seepage management and its impact factors in arid locations (Han et al. 2020). The findings indicated that the increased seepage loss is caused by cracks in the joints of the two precast concrete slabs and holes in the geomembrane. When compared to the un-lined canals, the combination of new concrete and geomembrane lining reduces seepage by 86%; after three service years, the combination can reduce seepage by 68%. The impact and appraisal of canal lining in most agriculturally productive areas were studied under a lining project (Lakho et al. 2020). According to the study, total agricultural production grew by 11.72–75.38% with the lining of Bilawal Zardari Minor District Shaheed Benazirabad Sindh, while CCA increased by 15.96–73.25% with the lining of Said Khan Distributary District Matiari. Cropping intensity increased by 23.22% and 17.0%, respectively, on land farmed with the Bilawal Zardari Minor and Said Khan Distributary. A field study in the El-Minia governorate of Middle Egypt was given to examine the degree to which the lining being applied using the suggested approach is compatible with the area’s terrain, soil types, groundwater levels, and other factors (Abu-Zeid 2021). This field study was presented in the hope that it will be integrated with other similar studies to be conducted in North and South Egypt to develop guidelines and a map including the most appropriate rehabilitation methods in each region. The technical and environmental expected impacts that must be achieved in such a national project were discussed (Ashour et al. 2021). The purpose of this field research was to estimate the transmission losses in the El-Sont canal in Middle Egypt (Assuit Governorate) as a representative open channel for a given kind of soil, climate, geography, and beneficiary lifestyle. The created and developed equations offered and advocated by the most prominent writers in this subject were evaluated, summarised, and provided in tabular form for ease of use and comparison. Seepage loss from unlined, lined, and cracked-lined canals was investigated at the Ismailia canal, which stretches from 28.00 to 49.00 Km in Egypt (Elkamhawy et al. 2022). The outcomes revealed that the amount of seepage was greatly influenced by the lining’s hydraulic properties. The concrete liner had the best efficiency, followed by the geo-membrane liner, and then the bentonite liner; in the absence of pumping, these percentages are roughly 99%, 96%, and 54%, respectively via wells, from aquifer.

This study mainly aims to study the effect of lining on seepage loss. The main problems with El-Hager canal include increasing the seepage in the canal and thus reduced the efficiency of water used for irrigation as well as the cultivated area under main crops. So, this study introduced the effect of lining on seepage to improve the efficiency of water used for irrigation and thus increasing the cultivated area. The results were obtained for the cases of lined and un-lined canals. Moreover, the current study was compared with previous works.

2. Fieldwork

2.1. Study area

Fieldwork was conducted on the Al-Hager canal, Al-Saff city, Al Jizah (Giza), Upper Egypt, and South of Helwan city, Egypt. The Al-Hager canal has a total length of 23.81 km, with the last 4.51 km filled. It feeds from the Magror hard Al-lithy on the main Nile, Al-Saff city, which is located between 29^°34′53.6″N and 31°17′02.2″E. It served 12100 feddans to cultivate the following crops, as presented in Table 1.

Table 1. Actual crop patterns cultivated within the Al-Saff area.

Season	Crop type	Cultivated land (%)
Winter	Clover	30
	Wheat	30
	Potato	20
	Barley	10
	Feeds	10
Summer	Sesame	30
	Corn	40
	Feeds	10
	Fallow land	20

Figures 1 and 2 show the study area located between 29°34′53″N and 31°17′02″E at intake and 29°43'24″N and 31°18'19′' E at the end of the canal.

Figure 1. Location of Al-Hager canal in Al-Saff city.

Figure 2. Study area, Al-Hager canal in Al-Saff city.

Data were measured in August 2021 before lining, and the velocity was measured using current meters. Figure 3 shows the average temperature in Helwan for 2021, with a maximum temperature of 34.7 °C and a minimum temperature of 22.1 °C, and Figure 4 shows the average temperature in Helwan for August 2021.

Figure 3. Average temperature in Helwan for 2021.

Figure 4. Average temperature in Helwan for August 2021.

The General Integrated Management for Water Resources and Irrigation, Egypt, introduced a proposal for lining the Al-Hagar canal. The main problem of the canal is water loss due to seepage and decreased cultivated area, which led to increased groundwater levels. Thus, the suggested solution to this problem is the lining of the canal. The canal will be lined with 30-cm dachshum and 10 cm of plain concrete. Therefore, this study investigates the seepage, efficiency, roughness coefficient, and flow characteristics of the Al-Hager canal before and after lining.

3. Methodology

Seepage loss is a significant component of canal water loss. Lining irrigation canals to reduce or eliminate seepage losses ensures better irrigation water conveyance and an improved economic scenario. Several methods for estimating irrigation canal seepage have been developed. Those that are most helpful can be categorized as follows:

Formulas based on empirical research,

Analytical approaches to problem solving, and

Solutions derived from electrical analogies.

When these methods were employed in the conditions for which they are intended, analytical methods had consistently given results that are incredibly exact, yet empirical formulas can only produce rough estimations. These solutions are regarded to be the most accurate and easy ways of estimating seepage values given known hydraulic conductivity of the subsoil, canal shape, and groundwater table location.

The accepted methods currently for measuring the quantity of water lost by seepage from existing canals are limited to ponding, inflow–outflow and seepage meter determination.

The inflow–outflow method measures the water that flows in and out of a section of the irrigation canal, as shown in Figure 5. Seepage is the difference between the quantities of water flowing in and out of the canal section. (1) $$S=Q_{\mathrm{i}}+R-Q_{\mathrm{o}}-D+I-E,$$(1) where S represents the seepage rate, Q_i represents the upstream inflow, R represents the rainfall, Q_o represents the downstream outflow, D represents the flow diverted along the reach, I represents the inflow along the reach, and E represents evaporation. To produce a measured loss using this approach, constant flow conditions and extensive canal stretches must be assumed (Blackwell 1951).

Figure 5. Mass balance for the inflow–outflow method.

3.1. Empirical relationships for calculating the seepage

Over the past few years, various researchers have provided empirical relationships for calculating water seepage from channels. Four of the most well-known are discussed as follows.

The Moritz formula was proposed to calculate seepage loss (Kraatz and Mahajan 1975): (2) $$S=0.0186\times c\times \sqrt{\frac{Q}{V}},$$(2) where S is the seepage rate over 1 km (m³/s); c is a constant, which is 0.41 for clay and clay loam soils and 0.66 for sandy loam soil (0.66); V is the velocity of the water in the channel (m/s); Q is the amount of discharge (m³/s) in the channel. Note that the channel length does not appear in Equation (2)(2) $$S=0.0186\times c\times \sqrt{\frac{Q}{V}},$$(2) because seepage is presented for estimation over a 1 km distance.

For all canals in the Almanna network, the empirical formula of Moleswerth and Yennidunia was employed to estimate seepage losses in different earthen sections. This formula, which is used in Egypt to assess seepage losses (Kraatz and Mahajan 1975), is as follows: (3) $$S=c\times L\times p\sqrt{R},$$(3) where S is the conveyance losses for a particular canal length (m³/sec), L is the canal length in kilometres, P is the wetted perimeter in metres, R is the hydraulic radius in metres, and c is the soil type factor, which is 0.0015 for clay.

Moleswerth and Yennidumia developed the fourth empirical formula and is being used by the Egyptian Irrigation Authority to estimate seepage from channels (Kraatz and Mahajan 1975). (4) $$S=86.4C\text{ R},$$(4) where S is the water seepage flux (m³/m²/day), and C is a coefficient, which is 0.0015 for clay soils and 0.003 for sandy soils.

Davis and Wilson proposed the empirical equation below for determining seepage discharge from channels (Kraatz and Mahajan 1975): (5) $$S=0.45\text{ C }\frac{\text{P L}}{4\times {10}^6+3650\sqrt{V}}{y}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.},$$(5) where C denotes the coefficient, which ranges from 1 to 70 depending on the channel cover and is equal to 1.0 for concrete channels; P represents the wetted perimeter (m); L represents the channel length (m); V represents the average velocity in the channel (m/s); y represents the water depth in the channel (m); and S represents seepage from the channel along the channel length (m³/s).

3.2. Water conveyance efficiency

The system’s ability to transport water is assessed using conveyance efficiency. It evaluates the effectiveness of canals used to transport water from ponds and wells to fields. The percentage difference between the amount of water delivered to farms or fields and the amount of water diverted from sources is known as the water conveyance efficiency. Equations (6)(6) $$\text{Conveyance Loss }\left(Q_L\right)\left(\%\right)=\frac{Q_i-Q_o}{Q_i}\times 100,$$(6) and (7)(7) $$\text{Conveyance efficiency }\left(\%\right)=\frac{Q_o}{Q_i}\times 100,$$(7) were used to calculate the conveyance losses and conveyance efficiency in the watercourse samples after the flow rate was determined using a current metre. (6) $$\text{Conveyance Loss }\left(Q_L\right)\left(\%\right)=\frac{Q_i-Q_o}{Q_i}\times 100,$$(6) (7) $$\text{Conveyance efficiency }\left(\%\right)=\frac{Q_o}{Q_i}\times 100,$$(7) where Q_i and Q_o are the inflow and outflow rates observed in m³/sec, respectively.

The inflow and outflow method was used to measure the seepage in the field.

3.3. Determination of roughness coefficients

Roughness coefficients of all selected sections were determined using the following formulas

1. Mannin

2. Chez

3. Darcy–Weisbach

The formulas are as follows: (8) $$\text{Manning}\text{n}=\frac{R^{\frac{2}{3}}{i}^{\frac{1}{2}}}{V},$$(8) (9) $$\text{Cheney}C=\frac{V}{\sqrt{Ri}},$$(9) (10) $$\text{Darcy}\mathrm{–}\text{Weisbach}f=\frac{8gRi}{V^2},$$(10) where i is the hydraulic slope (m/m), and g is the acceleration due to gravity (m/s²).

4. Results and analysis

The seepage loss at different Al-Hager canal sections was investigated using empirical and analytical formulas. It was discovered that the minimum seepage losses happened at sections starting at km (0.00) to km (10.50), and the maximum seepage losses occurred at sections starting at km (10.50) to km (19.30). The consequences showed that the seepage rate values in the canal were higher before lining than after lining. However, the seepage generally shows a slight increase in the downstream direction along this canal. When the average water loss from lined and un-lined canals was compared, it can be concluded that lining significantly decreases water conveyance loss in developing countries.

The conveyance efficiency in the un-lined canal, which is about 61.26%, can be increased to 70.75% when the entire canal is lined, as shown in Table 2, implying a 9.483% improvement of conveyance occurred due to the lining of the watercourse. Therefore, there is significant scope for improving water use efficiency by lining the system when the performance of water distribution improves during the same crop period. It was observed that the discharge of a canal after the lining is approximately 1.362–1.573 times greater than that of the un-lined section.

Table 2. Water efficiency in the lined and un-lined Al-Hager canal.

Water loss in the lined and un-lined watercourses
	Discharge at the inlet (Qi) (m^3/s)	Discharge at the outlet (Qo) (m^3/s)	Water efficiency (%)
Un-lined	7.231	4.43	61.264
Lined	9.852	6.97	70.747

Water losses in the Al-Hager canal were 38.736% when un-lined, but losses decreased to 29.253% when the canal was lined, as presented in Table 3.

Table 3. Water losses in the lined and un-lined Al-Hager canal.

Water losses in lined and un-lined watercourses
	Discharge at the inlet (Qi) (m^3/s)	Discharge at the outlet (Qo) (m^3/s)	Water loss (%)
Un-lined	7.231	4.43	38.736
Lined	9.852	6.97	29.253

Table 3 shows the seepage losses calculated for the various earthen portions of the Al-Hager canal. The largest value of Al-Hager canal seepage losses, according to this table, occurs in portion one, from the input regulator to km 10.50. Part four also has the lowest value of the same losses from kilometre 10.50 to 19.30.

The values of seepage losses determined using the most often used algorithms above vary greatly. The Molesworth equation produces the greatest value, whereas Moritz’s equation gives the lowest values for the Al-Hager canal, as presented in Tables 4 and 5.

Table 4. Calculated values of seepage losses for the un-lined canal using different methods.

Methods for calculating Seepage	Un-lined canal
	Section 1	Section 2	Section 3	Section 4	Section 5
Davis	0.1291	0.1130	0.0968	0.0711	0.0562
Moleswerth and Yennidumia	0.1722	0.1669	0.1543	0.1385	0.1268
Mortiz	0.0405	0.0373	0.0327	0.0257	0.0216
Davis and Wilson	0.0583	0.0510	0.0437	0.0321	0.0253

Table 5. Calculated values of seepage losses for the lined canal using different methods.

Methods for calculating Seepage	Lined canal
	Section 1	Section 2	Section 3	Section 4	Section 5
Davis	0.1180	0.1146	0.0946	0.0707	0.0586
Moleswerth and Yennidumia	0.1671	0.1627	0.1541	0.1380	0.1274
Mortiz	0.0378	0.0363	0.0320	0.0252	0.0214
Davis and Wilson	0.0532	0.0517	0.0427	0.0319	0.0264

Figures 6–11 show the relationship between water depth (y) and seepage (S) from the canal before and after the lining. The polynomial form was the best curve fit for these figures. Seepage is directly proportional to the water depth, and channel leakage rises with the rise in the water level. After lining the canal, seepage loss was significantly decreased, the seepage control effect was obviously improved, and the stable infiltration time was relatively prolonged.

Figure 6. Relationship between water depth and seepage using the Davis formula.

Figure 7. Relationship between water depth and seepage using Mortiz formula.

Figure 8. Relationship between water depth and seepage using Moleswerth formula.

Figure 9. Relationship between water depth and seepage using the Davis and Wilson formula.

Figure 10. Relationship between water depth and seepage using all methods for the un-lined case.

Figure 11. Relationship between water depth and seepage using all methods for the lined case.

Figures 12–15 show the relationship between discharge (Q) and seepage (S) from the canal before and after lining using different calculation methods. The power form was the best curve fit for these figures. These figures indicate that the seepage loss of a canal is directly proportional to the flow rate, and at the same discharge value, the seepage increases. After lining the canal, the discharge and cultivated area increase. This resulted in an increase in the number of crops cultivated to meet Egyptian demand. The Molesworth equation gives the biggest value, whereas Moritz’s equation gives the lowest values. The resulting values from Molesworth were bigger than those from Mortiz in the 76.446–83.0058% range.

Figure 12. Relationship between flow rate and seepage using the Molethwerth method.

Figure 13. Relationship between flow rate and seepage using the Davis method.

Figure 14. Relationship between flow rate and seepage using the Davis and Wilson method.

Figure 15. Relationship between flow rate and seepage using the Mortiz method.

Figures 16–19 demonstrate the relationship between the Froude number (F_r) and seepage (S) from the canal before and after the lining. The power form was the best curve fit for these figures. These figures indicate that the seepage loss of a canal is inversely proportional to the Froude number and that seepage decreases with a higher Froude number. The Froude number and seepage have high relationships, according to analysis of the hydraulic characteristics, with an average coefficient of determination of 0.8195.

Figure 16. Relationship between Froude number and seepage using the Davis method.

Figure 17. Relationship between Froude number and seepage using the Molethwerth method.

Figure 18. Relationship between Froude number and seepage using the Mortiz method.

Figure 19. Relationship between Froude number and seepage using the Davis and Wilson method.

4.1. Manning’s roughness coefficients (n)

Tables 6 and 7 show that the computed values of Manning’s method before and after lining the canal ranged from 0.0124 to 0.0404 and 0.0078 to 0.02475, respectively.

Table 6. Roughness coefficients for the un-lined canal.

Roughness coefficient	Section 1	Section 2	Section 3	Section 4	Section 5	Average
n	0.0404	0.0286	0.0259	0.0149	0.0124	0.0245
f	0.1058	0.0542	0.0470	0.0168	0.0122	0.0472
C	27.2297	38.0677	40.8593	68.2642	80.1649	50.9172

Table 7. Roughness coefficients for the lined canal.

Roughness coefficient	SEC1	Sec 2	SEC 3	sec4	sec5	Average
n	0.02475	0.0286	0.0201	0.0118	0.0078	0.0186
f	0.04058	0.0541	0.0283	0.0104	0.0048	0.0277
C	43.9783	38.0677	52.6774	86.7512	127.5997	69.8149

4.2. Chezy’s roughness coefficients (C)

Tables 6 and 7 show that the completed values of Chezy’s method before and after lining the canal ranged from 27.2297 to 80.1649 and 43.9783 to 127.5997, respectively.

4.3. Darcy–Weisbach’s roughness coefficients (f)

Tables 6 and 7 show that the completed values of Darcy–Weisbach’s method before and after lining the canal ranged from 0.0122 to 0.1058 and 0.0048 to 0.04058, respectively.

The average values of Mannin’s, Chez’s, and Darcy’s methods after lining the canal are 0.0186, 69.8149, and 0.0277, respectively, whereas the average values before lining the canal are 0.0245, 50.9172, and 0.0472 respectively.

Figures 20–22 show the relationship between roughness coefficients and seepage using the Davis method. The power form was found to be the best fit for these figures. The seepage is directly proportional to Manning’s and Darcy’s coefficients but inversely proportional to Chezy’s.

Figure 20. Relationship between Manning’s coefficient and seepage using the Davis method.

Figure 21. Relationship between Darcy’s coefficient and seepage using the Davis method.

Figure 22. Relationship between Chezy’s coefficient and seepage using the Davis method.

Figure 23 shows the relationship between specific energy (E_s) and seepage (S). The power form was found to be the best fit for this figure. Seepage is directly proportional to the specific energy. The energy percentage losses before and after the lining canal were 41.233% and 25.623%, respectively.

Figure 23. Relationship between the specific energy and seepage using the Davis method.

5. Statistical analysis

5.1. Statistical package for the social sciences software

A software program called Statistical Package for the Social Sciences (SPSS), also called IBM SPSS Statistics, is used to evaluate a variety of statistical data. Survey results, customer databases from businesses, Google Analytics, study findings from academic studies, and server log files are common sources. Almost all structured data formats and a wide variety of data types can be analysed and modified using SPSS. It provides bivariate and descriptive statistics data analysis, numerical outcome projections, and group identification forecasts. The programme also allows for data editing and charting.

Because of their strength, adaptability, and simplicity, Neural Networks are the chosen tools for many predictive data mining applications. In applications where the underlying process is complex, predictive neural networks are very helpful. The ability to compare model predictions to known values of the target variables makes supervised neural networks, such the multilayer perceptron (MLP) and radial basis function (RBF) networks, ideal for predictive applications. MLP and RBF networks can be fitted using forecasting and decision trees, and the resulting models can be saved for scoring. A set of nonlinear data modelling tools called a computational neural network consists of input and output layers as well as one or more hidden layers. Each layer’s connections between neurons have corresponding weights that are iteratively calculated.

The seepage loss is affected by various factors such as the flow velocity, lining material type, Ground water elevation, bed slope, water depth, and canal cross section. Manning’s coefficient depends on the type of lining (Chow 1959). Since the water velocity depends on Manning’s coefficient, there is a relation between the seepage and the Manning’s coefficient and Froude number.

This study used a neural network to predict seepage as a function of Manning’s coefficient, Froude number, hydraulic radius for un-lined canal, as shown in Figure 24, and for lined canal, as shown in Figure 25.

Figure 24. Neural network for predicted seepage as a function of Manning’s coefficient, Froude number and hydraulic radius for un-lined canal.

Figure 25. Neural network for predicted seepage as a function of Manning’s coefficient, Froude number, and hydraulic radius for lined canal.

Using the non-linear regression in the SPSS program, the following empirical formulas were introduced

For un-lined canal (11) $$S=0.05R^{0.835}+0.097n^{0.563}{F}_r{}^{-0.069}{R}^2=0.998,$$(11)

For lined canal (12) $$S=0.116\mathrm{EXP}(n)\times F_r{}^{0.704}+n\mathit{LnR}{R}^2=0.917,$$(12)

5.2. Gene expression programming

An evolutionary method called Gene Expression Programming (GEP) generates computer programs on demand. They can be traditional mathematical models, neural networks, sophisticated nonlinear regression models, logistic regression models, complicated polynomial structures, logic circuits and expressions, and other types of computer programs. However, all GEP programs—regardless of how complicated they are—are encoded in extremely basic linear structures called chromosomes. These straightforward linear chromosomes are a breakthrough since they consistently and correctly encode functional computer programs. We can therefore modify them, pick the best ones to produce, develop stronger programs from them, and so on indefinitely. Of course, this is one of the requirements for having an effective system that is constantly seeking for new and better solutions.

For un-lined canal:

The seepage for un-lined canals can be calculated using the following formula: $$S=\frac{n}{n+\mathrm{Exp}\text{(R)}\text{-}\frac{R}{c_1}}+\frac{\mathrm{R}}{\frac{c_2-R}{c_3}-c_2{F}_r}$$

In which $C_1=1.629{\text{, C}}_2=-8.872{\text{, c}}_3=-0.5655.$ The value of determination coefficient = 1, Correlation factor = 1, and RMSE = 0.0003. Figure 26 shows the relation between observed and predicted values from GEP.

Figure 26. The relationships between observed and predicted values from GEP for un-lined canal.

For lined canal:

The seepage for un-lined canals can be calculated using the following formula: $$S=n\text{ Log }\left[{\text{C}}_{{}_{\text{1}}}+2R-F_r\right]+\frac{1}{{(C_2-Ln{C}_3)}^2}$$

In which $C_1=4.7952{\text{, C}}_2=4.8338{\text{, C}}_3=1.4262$

The value of determination coefficient = 0.98, Correlation factor = 0.99, and RMSE = 0.0017. Figure 27 shows the relation between observed and predicted values from GEP.

Figure 27. The relationships between observed and predicted values from GEP for lined canal.

Table 8 shows a sample of calculated and predicted seepage from GEP and SPSS for un-lined and lined canals, as follows:

Table 8. A sample of calculated and predicted seepage from GEP and SPSS for un-lined and lined canals.

Un-lined canal			Lined canal
Seepage (S)	(S) Predicted from GEP	(S) Predicted from SPSS	Seepage (S)	(S) Predicted from GEP	(S) Predicted from SPSS
0.100	0.0997	0.0997	0.0940	0.0878	0.0912
0.092	0.0923	0.0918	0.0913	0.0935	0.0985
0.0819	0.0820	0.0815	0.0808	0.0794	0.0782
0.0669	0.0668	0.0664	0.0664	0.0661	0.0617

From results of SPSS and GEP, it is observed that the two programs give a good expression for calculation seepage of lined and un-lined canals, but GEP gives more accurate results than SPSS

5.2.1. Comparison between the present study with the previous studies

To verify these results, the data of seepage loss obtained from all previous methods were compared to those of this study, as shown in Figures 28 and 29 for lined and un-lined canals, which apparently share the same trend. R² = 0.9902 (Mortiz), R² = 0.9898 (Cavis and Wilson), R² = 0.9898 (Davis), and R² = 0.9932 (Moleswerth) which agree with the R² = 0.9904 calculated in this study.

Figure 28. Comparison of the overall (y – s) relationship obtained in the present study with the data collected from previous works for lined canals.

Figure 29. Comparison of the overall (y – s) relationship obtained in the present study with the data collected from previous works for un-lined canals.

6. Conclusion

To guarantee efficient operation and optimal water consumption, the losses from canals must be kept to a minimum. One of the main factors in water loss from canals is seepage loss. This study investigates the effect of lining on seepage loss. The result revealed that the discharge of a canal after the lining is about 1.362–1.573 times greater than that of the un-lined section. In addition, the average values of Mannin’s, Chez’s, and Darcy’s coefficients are 0.0186, 69.8149, and 0.0277 for the lined canal and 0.0245, 50.9172, and 0.0472 for the unlined canal, respectively. Comparing the average water loss in the Al-Hager canal of 38.736% from unlined to the average water loss of 29.253% from lined watercourses, it was estimated that the lining reduced water loss by 9.483%. In a comparison of the preceding efforts, it is discovered that the Moleworth equation yields the highest value, while Moritz’s equation generates the lowest result. The investigation of the hydraulic parameters showed that the seepage and flow features, such as roughness coefficients, Froude number, and water depth, have significant connections. The present study is introduced in comparison to earlier investigations. In comparison to the ANN program, the GEP program produced better results. The GEP and ANN programs provide expressions for seepage of lined and unlined canals as a function of Manning’s coefficient, Froude number, and hydraulic radius. It was discovered that utilizing GEP rather than ANN produced superior results in terms of outcomes.

Author contributions

The study’s inception and design contained input from all authors. Waleed Omar prepared the material and collected the data. Hanaa Abdelhaleem wrote the initial draught and Manal Gad prepared the data analysis. The final manuscript was read and accepted by all writers.

Statements and declarations

We hereby state that this thesis is an original account of our investigation. By signing this statement, we affirm that this thesis is our own work and has not been submitted to a journal.

Ethical approval

This paper does not contain any studies with human participants or animals worked by any of the authors.

Notation
A	=	Cross-sectional area
C	=	Chezy coefficient
c	=	Constant
D	=	Flow diverted along the reach
E	=	Evaporation
Es	=	Specific energy
Fr	=	Froude number
f	=	Darcy coefficient
g	=	Acceleration due to gravity
I	=	The inflow along the reach
i	=	Hydraulic slope
L	=	Canal length
n	=	Manning coefficient
P	=	Wetted perimeter
Q	=	Discharge of flow
Qi	=	Upstream inflow
Qo	=	Downstream outflow
R	=	Hydraulic radius
S	=	Bed slope
T	=	Top width
V	=	Mean velocity
y	=	Water depth.

Acknowledgments

We would like to acknowledge and give our warmest thanks to Delta Higher Institute for Engineering and Technology, specially Dr. Mohamed rabie Nasser and Eng. Hatem Khalefa for their assistance and the encouragement them to the scientific research at Delta Higher Institute for Engineering and Technology.

Disclosure statement

No potential conflict of interest was reported by the authors.

Data availability statement

The paper contains all of the data sets that were created or analysed for this investigation.

Additional information

Funding

The authors didn’t receive support from any organization for the submitted work.