# A Comprehensive Model for Hydraulic Analysis and Wetting Patterns Simulation under Subsurface Drip Laterals

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Model Development

_{o}is the discharged flow rate, H is the pressure head at the orifice or emitter inlet, C

_{d}is the discharge coefficient, and x is the emitter exponent. When the emitter is buried in the soil, the coefficient C

_{d}also includes the orifice’s resistance and the soil around the pipe. Figure 1 shows an infinitesimal section of a drip lateral with length d

_{x}. The water movement is regulated by the continuity and the motion equations.

_{x}and Q

_{x+dx}are the flow discharge at section x and x + d

_{x}, q

_{ox}is the emitter discharge at section x, A is the cross-sectional area of the lateral pipe, and V

_{x}and V

_{x+dx}are the flow velocity corresponding to Q

_{x}and Q

_{x+dx}, respectively.

_{x}and H

_{x+dx}are the pressure head at the sections x and x + dx, dE is the elevation difference that is due to the lateral slope S (positive for uphill and negative for downhill), h

_{f}is the total head loss (friction and local losses), and g is the gravitational acceleration. The value of dE is given by:

_{x}, and the flow velocity at the section x + dx can be written as:

_{f}can be evaluated with the Darcy–Weisbach equation:

_{x}is the length of the pipe section, and D is the inner diameter of the lateral. As known, the friction factor f depends on the Reynolds number, R, and the pipe roughness. In smooth plastic pipes, the friction factor can be expressed with the Blasius or similar empirical equations [20].

_{x}was applied. Because d

_{x}length is so small, the V

_{x}and f variation are very small and negligible. In addition, the lateral diameter is constant and its variation is equal to zero. Therefore, the partial differential of the hf with respect to length is considered as $\frac{\partial {h}_{f}}{\partial x}$. By substituting from Equation (6) into Equation (3) and then into Equation (8) along with Equation (9), the continuity and the equations of motion result in:

_{ox}, Equation (11) can be written as:

_{d}. Therefore, the water discharged by buried emitters can be estimated from the pressure variations along the lateral. Second, when the distance between the emitters is small, the flow rate discharged by the lateral can be assumed as a linear source [21]. Moreover, when the pipe is horizontal or its slope is small, the term ωd

_{x}can be neglected. In addition, because all the water discharged by the lateral reaches the soil, the rate of the water infiltration in the soil from the emitters has to be considered equal to the sum of the emitters discharge, q

_{ox}. On the other hand, to identify the comprehensive model, Equation (12) has to be integrated into the soil moisture characteristic equation. Therefore, based on water mass volume change and considering that the water flowing in the soil from an area “a” around the buried emitters, we can write:

## 3. Materials and Methods

## 4. Statistical Analysis

_{i}is the observed value, P

_{i}is the predicted value, n is the number of observations, and Ō is the mean of the observed data. The dr index ranges from −1 to 1 and quantifies the prediction errors of the model relative to the observed deviations from the observed mean. Thus, a perfect agreement between data sets would result in dr = 1. In addition, the values of RMSE and MAE indices close to zero indicate an excellent performance for the model. In this study, RMSE, MAE, and dr values were applied to compare separately the soil wetted width and depth. For this purpose, the observed and predicted dimensions of the wetting front at the selected points have been determined. These points were intended at the first and last cross-sections of the lateral, as well as at 10 cm intervals along the longitudinal direction of the lateral.

## 5. Results and Discussion

_{m}), and estimated (H

_{e}) pressure heads, their relative difference (RE), as well as the mean discharge of the buried emitters corresponding to the emitters installed at 0, L/4, 3/4L, and L from the lateral inlet, are reported. To give an example, for the drip lines with 20 cm emitter spacing, the head loss between the emitter placed at a distance of 3/4L (46.4 m) from the lateral inlet and the upstream end of the lateral resulted in 2.428 m, 6.019 m, and 8.734 m under an applied pressure of 50, 100, and 150 kPa, respectively. A similar trend was also observed for the other examined cases. Moreover, for any fixed operating pressure, as a consequence of the pressure head reductions along the lateral even the flow rates discharged by the emitters decreased and the wetting profiles developed non-uniform patterns in the different sections of the laterals.

^{2}, respectively. These values were larger than the corresponding observed values in the experimental cases. On the contrary, the minimum wetted areas were 0.497, 0.021, and 0.018 m

^{2}after 1 h beneath the drip lateral with 50 cm emitter spacing at the operating pressure of 50 kPa.

^{3}, respectively. The mean flow rates of the emitters in the mentioned cases varied from 2.1 to 4.18 L/h. Accordingly, at decreasing emitter spacing and increasing observation time and operating pressure, the wetted zone area and the ratio between the discharged water and the wetted area mostly increased. The results showed how the width and depth of the wetted zone increased with the emitter discharge rate. These results are in agreement with those presented by [5,26]. This phenomenon was observed even after the redistribution phase in the wetted longitudinal profiles, as well as in both cross-sections. The higher operating pressures, similar to the reductions of the emitter spacing, not only determined the increase of the discharged volume but also caused the expansion of the wetted bulb during both the distribution and redistribution processes.

_{1}) and final (r

_{2}) cross-sections of the lateral with various emitter spacings (E.S) were related to the volume of water discharged (V

_{w}) into the soil that is in agreement with the results of Thorburn et al. [27]; the diameter of wetted soil volume increased nonlinearly with applied water volume (Figure 8).

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Planar view of the input, output, return branches, and the emitter positions in the buried part of the three drip laterals at the 0.2 m depth in the soil box.

**Figure 5.**Comparison between observed and simulated wetting fronts for drip lateral with 20 cm emitter spacing at operating pressures of (

**A**) 50 kPa, (

**B**) 100 kPa, and (

**C**) 150 kPa.

**Figure 6.**Comparison between observed and simulated wetting fronts for drip lateral with 40 cm emitter spacing at operating pressures of (

**A**) 50 kPa, (

**B**) 100 kPa, and (

**C**) 150 kPa.

**Figure 7.**Comparison between observed and simulated wetting fronts for drip lateral with 50 cm emitter spacing at operating pressures of (

**A**) 50 kPa, (

**B**) 100 kPa, and (

**C**) 150 kPa.

**Figure 8.**Relationship between the radiuses of the first and last cross-sections (r

_{1}&r

_{2}) and volume of water discharged (Vw) of the drip laterals with emitter spacing of 20, 40, and 50 cm.

**Figure 9.**Simulated vs. observed wetted width at the first and last cross-sections of drip laterals with emitter spacing of 20, 40, and 50 cm.

**Figure 10.**Simulated vs. observed wetted depth obtained at distances of 0, L/4, 3L/4, and L from the upstream section of the laterals with emitter spacing of 20, 40, and 50 cm.

Soil Properties | Sand (%) | Silt (%) | Clay (%) | Soil BD (gr/cm ^{3}) | Residual Water Content θr (cm^{3}/cm^{3}) | Saturated Water Content θs (cm^{3}/cm^{3}) | Shape Parameters | Saturated Hydraulic Conductivity Ks (cm/h) | ||
---|---|---|---|---|---|---|---|---|---|---|

A (1/m) | n | l | ||||||||

Clay loam | 36 | 31 | 33 | 1.57 | 0.08 | 0.41 | 1.34 | 1.37 | 0.5 | 0.3 |

**Table 2.**Summary of the hydraulic analysis results including the head losses, the measured (Hm) and estimated (He) pressure heads, their relative difference (RE), and the mean discharge of the buried emitters (q

_{o}).

Emitter Spacing (cm) | Pressure (kPa) | 50 | 100 | 150 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Selected Section | 0 | L/4 | 3L/4 | L | 0 | L/4 | 3L/4 | L | 0 | L/4 | 3L/4 | L | |

50 | H_{f} (m) | 0.000 | 0.366 | 0.773 | 0.794 | 0.000 | 0.672 | 1.421 | 1.475 | 0.000 | 0.949 | 2.012 | 2.089 |

H_{e} (m) | 5.000 | 4.634 | 4.227 | 4.206 | 10.000 | 9.328 | 8.579 | 8.525 | 15.000 | 14.051 | 12.988 | 12.911 | |

H_{m} (m) | 5.000 | 4.600 | 4.200 | 4.300 | 10.000 | 9.500 | 8.400 | 8.800 | 15.000 | 14.300 | 13.400 | 12.900 | |

RE (%) | 0.000 | 0.739 | 0.643 | 2.186 | 0.000 | 1.811 | 2.131 | 3.125 | 0.000 | 1.741 | 3.075 | 0.085 | |

q_{o} (l/h) | 2.417 | 2.326 | 2.222 | 2.216 | 3.419 | 3.302 | 3.166 | 3.157 | 4.188 | 4.054 | 3.897 | 3.886 | |

40 | H_{f} (m) | 0.000 | 0.547 | 1.146 | 1.186 | 0.000 | 0.792 | 1.907 | 1.983 | 0.000 | 1.436 | 2.994 | 3.100 |

H_{e} (m) | 5.000 | 4.453 | 3.854 | 3.814 | 10.000 | 9.208 | 8.093 | 8.017 | 15.000 | 13.564 | 12.006 | 11.900 | |

H_{m} (m) | 5.000 | 4.500 | 4.050 | 4.000 | 10.000 | 9.500 | 8.200 | 7.900 | 15.000 | 13.700 | 11.900 | 11.700 | |

RE (%) | 0.000 | 1.044 | 4.840 | 4.650 | 0.000 | 3.074 | 1.305 | 1.481 | 0.000 | 0.993 | 0.891 | 1.709 | |

q_{o} (l/h) | 2.417 | 2.281 | 2.124 | 2.113 | 3.419 | 3.281 | 3.076 | 3.061 | 4.188 | 3.983 | 3.747 | 3.730 | |

20 | H_{f} (m) | 0.000 | 0.550 | 2.428 | 2.519 | 0.000 | 3.335 | 6.019 | 6.151 | 0.000 | 4.805 | 8.734 | 8.929 |

H_{e} (m) | 5.000 | 4.450 | 2.572 | 2.481 | 10.000 | 6.665 | 3.981 | 3.849 | 15.000 | 10.195 | 6.266 | 6.071 | |

H_{m} (m) | 5.000 | 4.500 | 2.900 | 2.800 | 10.000 | 6.500 | 4.100 | 3.700 | 15.000 | 9.900 | 6.300 | 6.100 | |

RE (%) | 0.000 | 1.111 | 11.310 | 11.393 | 0.000 | 2.538 | 2.902 | 4.027 | 0.000 | 2.980 | 0.540 | 0.475 | |

q_{o} (l/h) | 2.417 | 2.280 | 1.733 | 1.702 | 3.419 | 2.791 | 2.156 | 2.110 | 4.188 | 3.452 | 2.706 | 2.663 |

**Table 3.**Volumes of water discharged (V

_{w}), the mean rate of the discharged water per emitter (q

_{sm}), and wetted area measured along the longitudinal direction (A

_{L}), initial (A

_{1C}), and final (A

_{2C}) cross-sections were obtained in all the experiments. The ratios between the discharged water to the wetted area are also indicated.

Operating Pressure (kPa) | Time-T (h) | 1 | 2 | 3 | 24 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Emitter Spacing (cm) | 50 | 40 | 20 | 50 | 40 | 20 | 50 | 40 | 20 | 50 | 40 | 20 | |

50 | V_{W} (m^{3}) | 0.009 | 0.012 | 0.022 | 0.019 | 0.023 | 0.043 | 0.028 | 0.034 | 0.063 | 0.028 | 0.034 | 0.063 |

q_{sm} (l/h) | 2.371 | 2.320 | 2.170 | 2.360 | 2.280 | 2.140 | 2.340 | 2.240 | 2.100 | - | - | - | |

A_{L} (m^{2}) | 0.497 | 0.538 | 0.679 | 0.570 | 0.657 | 0.701 | 0.697 | 0.786 | 0.867 | 0.856 | 0.921 | 0.980 | |

V_{W}/A_{L} (m) | 0.019 | 0.022 | 0.032 | 0.033 | 0.035 | 0.061 | 0.040 | 0.043 | 0.073 | 0.033 | 0.036 | 0.064 | |

A_{1C} (m^{2}) | 0.021 | 0.023 | 0.042 | 0.028 | 0.040 | 0.046 | 0.054 | 0.052 | 0.097 | 0.080 | 0.099 | 0.137 | |

V_{W}/A_{1C} (m) | 0.452 | 0.513 | 0.523 | 0.674 | 0.577 | 0.930 | 0.520 | 0.642 | 0.649 | 0.351 | 0.339 | 0.460 | |

A_{2C} (m^{2}) | 0.018 | 0.018 | 0.030 | 0.029 | 0.030 | 0.040 | 0.047 | 0.052 | 0.060 | 0.086 | 0.095 | 0.107 | |

V_{W}/A_{2C} (m) | 0.527 | 0.630 | 0.723 | 0.651 | 0.760 | 1.070 | 0.597 | 0.646 | 1.047 | 0.327 | 0.354 | 0.589 | |

100 | V_{W} (m^{3}) | 0.013 | 0.017 | 0.034 | 0.026 | 0.033 | 0.066 | 0.039 | 0.048 | 0.095 | 0.039 | 0.048 | 0.095 |

q_{sm} (l/h) | 3.320 | 3.300 | 3.400 | 3.280 | 3.250 | 3.300 | 3.220 | 3.220 | 3.150 | - | - | - | |

A_{L} (m^{2}) | 0.537 | 0.574 | 0.684 | 0.625 | 0.782 | 0.870 | 0.886 | 0.891 | 0.936 | 0.933 | 1.011 | 1.075 | |

V_{W}/A_{L} (m) | 0.025 | 0.029 | 0.050 | 0.042 | 0.042 | 0.076 | 0.044 | 0.054 | 0.101 | 0.041 | 0.048 | 0.088 | |

A_{1C} (m^{2}) | 0.026 | 0.027 | 0.043 | 0.032 | 0.046 | 0.102 | 0.077 | 0.077 | 0.111 | 0.119 | 0.151 | 0.180 | |

V_{W}/A_{1C} (m) | 0.511 | 0.623 | 0.791 | 0.820 | 0.710 | 0.647 | 0.502 | 0.626 | 0.851 | 0.325 | 0.320 | 0.525 | |

A_{2C} (m^{2}) | 0.026 | 0.026 | 0.043 | 0.042 | 0.043 | 0.069 | 0.072 | 0.075 | 0.082 | 0.094 | 0.135 | 0.134 | |

V_{W}/A_{2C} (m) | 0.511 | 0.635 | 0.789 | 0.625 | 0.756 | 0.952 | 0.540 | 0.642 | 1.148 | 0.411 | 0.358 | 0.705 | |

150 | V_{W} (m^{3}) | 0.017 | 0.020 | 0.038 | 0.031 | 0.038 | 0.066 | 0.042 | 0.047 | 0.105 | 0.042 | 0.047 | 0.105 |

q_{sm} (l/h) | 4.180 | 4.000 | 3.800 | 3.850 | 3.770 | 3.300 | 3.500 | 3.150 | 3.500 | - | - | - | |

A_{L} (m^{2}) | 0.650 | 0.686 | 0.772 | 0.745 | 0.790 | 0.917 | 0.898 | 0.900 | 1.030 | 0.990 | 1.140 | 1.160 | |

V_{W}/A_{L} (m) | 0.026 | 0.029 | 0.050 | 0.042 | 0.048 | 0.072 | 0.047 | 0.053 | 0.102 | 0.042 | 0.041 | 0.091 | |

A_{1C} (m^{2}) | 0.029 | 0.032 | 0.052 | 0.038 | 0.048 | 0.104 | 0.081 | 0.079 | 0.136 | 0.120 | 0.181 | 0.220 | |

V_{W}/A_{1C} (m) | 0.587 | 0.625 | 0.731 | 0.811 | 0.785 | 0.635 | 0.519 | 0.597 | 0.772 | 0.350 | 0.261 | 0.477 | |

A_{2C} (m^{2}) | 0.028 | 0.029 | 0.053 | 0.050 | 0.051 | 0.070 | 0.072 | 0.077 | 0.100 | 0.113 | 0.160 | 0.163 | |

V_{W}/A_{2C} (m) | 0.597 | 0.690 | 0.724 | 0.616 | 0.739 | 0.939 | 0.583 | 0.614 | 1.050 | 0.372 | 0.295 | 0.644 |

**Table 4.**Mean absolute errors (MAE), the root mean square error (RMSE), and the refined index of agreement (dr) indices obtained when comparing the observed and predicted wetted areas in the longitudinal and transversal flow directions.

The Wetted Area around the Buried Lateral | |||||||||
---|---|---|---|---|---|---|---|---|---|

Emitter Spacing | Mean Absolute Errors (m) | Root Mean Square Error (m) | Refined Index of Agreement | ||||||

50 kPa | 100 kPa | 150 kPa | 50 kPa | 100 kPa | 150 kPa | 50 kPa | 100 kPa | 150 kPa | |

20 cm | 0.002 | 0.003 | 0.004 | 0.015 | 0.020 | 0.024 | 0.913 | 0.918 | 0.924 |

40 cm | 0.002 | 0.002 | 0.004 | 0.013 | 0.020 | 0.030 | 0.920 | 0.921 | 0.902 |

50 cm | 0.002 | 0.002 | 0.004 | 0.015 | 0.013 | 0.027 | 0.924 | 0.927 | 0.886 |

Initial and final cross-sections of the lateral | |||||||||

20 cm | 0.003 | 0.004 | 0.002 | 0.020 | 0.020 | 0.020 | 0.912 | 0.814 | 0.914 |

40 cm | 0.001 | 0.003 | 0.002 | 0.011 | 0.035 | 0.014 | 0.937 | 0.876 | 0.939 |

50 cm | 0.001 | 0.003 | 0.001 | 0.014 | 0.021 | 0.012 | 0.942 | 0.919 | 0.921 |

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**MDPI and ACS Style**

Zamani, S.; Fatahi, R.; Provenzano, G.
A Comprehensive Model for Hydraulic Analysis and Wetting Patterns Simulation under Subsurface Drip Laterals. *Water* **2022**, *14*, 1965.
https://doi.org/10.3390/w14121965

**AMA Style**

Zamani S, Fatahi R, Provenzano G.
A Comprehensive Model for Hydraulic Analysis and Wetting Patterns Simulation under Subsurface Drip Laterals. *Water*. 2022; 14(12):1965.
https://doi.org/10.3390/w14121965

**Chicago/Turabian Style**

Zamani, Saeid, Rouhollah Fatahi, and Giuseppe Provenzano.
2022. "A Comprehensive Model for Hydraulic Analysis and Wetting Patterns Simulation under Subsurface Drip Laterals" *Water* 14, no. 12: 1965.
https://doi.org/10.3390/w14121965