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Presenting and evaluating a new empirical relationship for estimating the rate of infiltration in trenches
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Abstract: Empirical formulas are indispensable tools in water engineering and hydraulic structure design. Derived from meticulous field observations, experiments, and diverse datasets, these formulas help to estimate water leakage in structures such as dams, tunnels, canals, and pipelines. By utilizing a few easily measurable parameters, engineers can employ these formulas to generate preliminary leakage rate estimates before proceeding with more detailed analyses. In this study, a physical model was developed, and a series of experiments were conducted, considering variables such as inflow rate, materials constituting the unsaturated medium, and variations in infiltration trench depth and width. As a result, a novel artificial recharge method was introduced, and an empirical equation, $ {\text{Q}}_{\text{out}} $=
0.0066 ×$ {{\mathrm{D}}_{50}}^{0.64} $× L ×$ {\mathrm{P}}^{\;0.36} $, was proposed to estimate the infiltration capacity of the trench. This equation incorporates factors such as the wetted perimeter, mean soil particle diameter, trench length, and a coefficient. A comparative analysis between the observed data from nine Iranian earthen canals and the values calculated using the proposed equation revealed an average relative error of 15% between the two datasets. In addition, the Pearson correlation coefficient was determined to be 0.981 and the Root Mean Square Error (RMSE) was 0.381, demonstrating the strong predictive performance of the equation. The parameters considered in the proposed equation allow for its application across diverse regions. Given its accurate performance, this equation provides a reliable initial estimate of the leakage rate, thereby helping to reduce costs and save time.-
Key words:
- Groundwater /
- Artificial recharge /
- Desert area /
- Infiltration rate /
- Physical model
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Figure 1. A simple schematic of the laboratory model.
(a) Side view from the right; (b) Front view
Figure 2. How the moisture front moves with 5-minute intervals on the model wall with a trench width of 8 cm and a trench depth of 10 cm (using medium sand materials)
Figure 3. Illustration of the moisture front movement in the model wall with 5-minute intervals and a trench width of 8 cm and a trench depth of 10 cm (using fine sand materials)
Figure 4. The effect of changes in channel dimensions on flow output in the physical model
Notes: (a) dimensionless graph with an input flow rate of 2.2 liters per minute, a fixed trench width of 8 cm, and different depths of 5, 7.5, 10, 12.5, and 15 cm in the physical model using medium sand materials; (b) dimensionless graph with an input flow rate of 2.2 liters per minute, a fixed trench depth of 10 cm, and different widths of 4, 6, 8, 10, and 12 cm in the physical model using medium sand materials; (c) dimensionless graph with an input flow rate of 0.82 liters per minute, a fixed trench width of 8 cm, and different depths of 5, 7.5, 10, 12.5, and 15 cm in the physical model using fine sand materials;(d) dimensionless graph with an input flow rate of 0.82 liters per minute, a fixed trench depth of 10 cm, and different widths of 4, 6, 8, 10, and 12 cm in the physical model using fine sand materials.
Figure 5. Examining the changes in the volume of the trench and the changes in the output flow in medium sand and fine sand
(a) according to the changes in the depth of the trench; (b) according to the changes in the width of the trench
Figure 6. Examining the changes in the wetted surface of the trench and the changes in the output flow in medium sand and fine sand
(a) according to the changes in trench depth; (b) according to the changes in trench width
Figure 7. Comparison of observed values of $ \mathit{\text{Q}_{\text{out}}} $ and calculated values of $ \mathit{\text{Q}_{\text{out}}} $ from the provided equation
(a) for medium sand materials against the 45-degree line; (b) for fine sand materials against the 45-degree line.
Figure 8. Comparison of the observed values of leakage from the channel with its calculated values form the equation against the 45-degree line
Table 1. Data obtained from the experiments conducted on the physical model with medium sand and fine sand materials
Row Trench width
/cmTrench depth
/cmTrench volume
/cm3The ratio of
the depth to
the
width of
the trenchQoutmax
(Lit/Min) (Medium sand)Qoutmax
(Lit/Min) (Fine sand)Performance
(Medium sand)
(Qout to Qin)Performance
(Fine sand)
(Qout to Qin)1 8 10 6,400 1.25 1.990 0.740 0.905 0.902 2 8 5 3,200 0.63 1.820 0.640 0.827 0.780 3 8 7.5 4,800 0.94 1.930 0.688 0.877 0.839 4 8 12.5 8,000 1.56 2.050 0.775 0.932 0.945 5 8 15 9,600 1.88 2.140 0.810 0.973 0.988 6 4 10 3,200 2.5 1.960 0.690 0.891 0.841 7 6 10 4,800 1.67 1.980 0.720 0.900 0.878 8 10 10 8,000 1 2.010 0.752 0.914 0.917 9 12 10 9,600 0.83 2.030 0.770 0.923 0.939 
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Table 2. Comparison of observed values of $ {{Q}}_{{out}} $ and calculated values of $ {{Q}}_{{out}} $ using the provided equation for different dimensions of the trench in the physical model with medium sand materials
Row Trench width
/cmTrench depth
/cmTrench
length
/cm$ \boldsymbol{Q_{out}} $
(Observational)
/L/Min$ \boldsymbol{Q_{out}} $
(Computational) /L/MinPearson
correlation
(r)RMSE 1 8 10 80 1.990 1.969 0.983 0.073 2 4 10 80 1.960 1.863 3 6 10 80 1.980 1.917 4 10 10 80 2.010 2.019 5 12 10 80 2.030 2.066 6 8 5 80 1.820 1.680 7 8 7.5 80 1.930 1.835 8 8 12.5 80 2.050 2.089 9 8 15 80 2.140 2.198
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Table 3. Comparison of observed values of $ {{Q}}_{{out}} $ and calculated values of $ {{Q}}_{{out}} $ using the provided equation for different dimensions of the trench in the physical model with fine sand materials
Row Trench width
/cmTrench depth
/cmTrench length
/cm$ \boldsymbol{Q_{out}} $ (Observational) /L/Min $ \boldsymbol{Q_{out}} $ (Computational) /L/Min Pearson correlation (r) RMSE 1 8 10 80 0.740 0.745 0.992 0.012 2 4 10 80 0.690 0.705 3 6 10 80 0.720 0.726 4 10 10 80 0.752 0.764 5 12 10 80 0.770 0.782 6 8 5 80 0.640 0.636 7 8 7.5 80 0.688 0.694 8 8 12.5 80 0.775 0.791 9 8 15 80 0.810 0.832
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Table 4. Compares the observed values of leakage from the channel with its calculated values from the given equation
Row Channel name L/m P/m $ {{D}}{_{50}} $/m Q/m3/d/m2 Observational Q/m3/d/m2 Computational Relative error of the main equation/% Pearson correlation
(r)RMSE 1 Sharifabad 0.35 2.85 0.0003 1.89 1.62 16 0.981 0.381 2 Sirian 0.32 3.14 0.0002 1.78 1.36 30 3 Cichi 0.96 1.04 0.00015 1.96 1.99 −1.5 4 Nahr Lulham 0.38 2.6 0.0016 5.58 5.02 11 5 Nahr Sarmast 0.26 3.88 0.0012 3.87 3.24 19 6 Garkan 1 0.39 2.59 0.00005 0.55 0.55 0 7 Garkan 2 0.38 2.61 0.00005 0.59 0.55 7 8 Najafabad 1 0.76 1.32 0.00015 1.12 1.70 −34 9 Najafabad 2 0.43 2.34 0.00015 0.95 1.18 −19
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