# Modeling of Border Irrigation in Soils with the Presence of a Shallow Water Table. I: The Advance Phase

^{*}

## Abstract

**:**

^{2}> 0.98 for the cases presented. It is also revealed that, when increasing the time step, the precision is maintained, and it is possible to reduce the computation time by up to 99.45%. Finally, the model proposed here is recommended for studying the advance process during surface irrigation in soils with shallow water tables.

## 1. Introduction

_{I}= ∂I(x,t)/∂t is the infiltration flow, that is, the water volume infiltrated per unit width per unit length of the border, I is the infiltrated depth, g is gravitational acceleration, β = U

_{IX}/U is a dimensionless parameter where U

_{IX}is the projection in the direction of the output velocity of the water mass due to the infiltration, J

_{o}is the topographic slope, and J is the friction slope that can be determined by the fractal law of hydraulic resistance [10]:

_{f}(t) is the position of the wave front at time t and q

_{o}is the unitary discharge at the entrance of the border.

_{f}). The solution considers a hydrostatic initial moisture distribution (Figure 2), where the initial moisture is calculated with the expression θ

_{i}(z) = θ

_{o}+ (θ

_{s}− θ

_{o})(z/P

_{f}). Thus, the moisture content at the soil surface is θ

_{i}(0) = θ

_{o}and at the water-table surface is θ

_{i}(P

_{f}) = θ

_{s}. The suction in the wetting front is a linear function of the moisture content at the front, h

_{f}(θ

_{i},θ

_{s}) = h

_{f}(θ

_{s}− θ

_{i})/(θ

_{s}− θ

_{o}), that is:

_{f}[θ

_{i}(0),θ

_{s}] = h

_{f}and h

_{f}[θ

_{i}(P

_{f}),θ

_{s}] = 0. The infiltrated depth is defined by:

_{M}is the maximum infiltrated depth.

_{f}≠ P

_{f}:

_{f}= P

_{f}:

_{o}is the mean water depth and h

_{o}= (ν

^{2}/gJ

_{o})

^{1/3}(q

_{o}/kν)

^{1/3d}is the normal depth [20].

_{f}→ ∞, considering that I

_{M}= 1/2∆θP

_{f}. In this limit, 1 – (1 – I/I

_{M})

^{1/2}≅ I/2I

_{M}= I/∆θP

_{f}holds. The third term is of the order of 1/P

_{f}and tends to zero. In the second term, the argument of the logarithm tends to 1 + I/∆θ($\overline{\mathrm{h}}$ + h

_{f}) and its coefficient to ∆θ($\overline{\mathrm{h}}$ + h

_{f}). Finally, in the first term, the coefficient of I → 1. Using the definition λ = ∆θ ($\overline{\mathrm{h}}$ + h

_{f}), the Green and Ampt equation is deduced.

## 2. Materials and Methods

#### Numerical Solution

_{R}+ φq

_{L}] + (1 − ω)[(1 − φ)q

_{M}+ φq

_{J}], and the coefficient $\overline{\mathrm{h}}$ = ω[(1 − φ)h

_{R}+ φh

_{L}] + (1 − ω)[(1 − φ)h

_{M}+ φh

_{J}], taking into account the extreme values of each calculation cell, and consequently the coefficient $\overline{\mathrm{J}}$ = ν

^{2}($\overline{\mathrm{q}}$/kν)

^{1/d}/g${\overline{\mathrm{h}}}^{3}$. The weight factors for time and space are denoted ω and φ, respectively.

_{i+1}, and J and M represent the values at the current time t

_{i}.

_{N}and δq

_{N}are equal to zero:

_{L}, q

_{L}, h

_{R}, and q

_{R}.

^{−5}.

## 3. Results

#### 3.1. Soil Characterization

_{o}= 0.00085 m/m, moisture content at saturation θ

_{s}= 0.5245 cm

^{3}/cm

^{3}, and the dimensionless parameter β = 0. The values of the hydraulic conductivity at saturation (K

_{s}) and the suction in the wetting front (h

_{f}) were optimized using the Levenberg–Marquardt algorithm [31]. In the hydraulic resistance law, Equation (3), d = 1 was used. The values of q

_{o}, P

_{f}, I

_{M}, initial moisture (θ

_{o}), $\overline{\mathrm{h}}$, K

_{s}, and h

_{f}are reported in Table 1 for each irrigation test.

_{s}and h

_{f}. Good adjustments were observed for the three irrigation tests.

#### 3.2. Applications

_{s}and h

_{f}.

_{s}and h

_{f}, which is corroborated by the high R² values obtained.

#### 3.3. Comparison of the Theorical Advance Curve with the Measured Data

_{f}= 50 cm) was used for a more exhaustive analysis. Figure 5 shows the advance curve obtained by the model, which shows a good fit between the measured data obtained in the field and those obtained with the model.

_{M}calculated with Equation (8), the soil is no longer capable of storing more water. Therefore, the soil was considered completely saturated 36 min after irrigation had started.

#### 3.4. Time Step Analysis

^{®}Core

^{TM}i7-4710 CPU @ 2.50 GHz and 32 Gb of RAM.

## 4. Discussion

_{s}and h

_{f}, it was possible to represent surface irrigation in soils with shallow water tables, obtaining excellent results for the coefficient of determination for each irrigation test, which confirms that the proposed model adequately reproduces the data measured in the field. In addition, with the values obtained it is possible to calculate, for the next irrigation, the optimal irrigation flow that should be applied in each border, by means of an analytical representation that takes into account the border length, the net irrigation depth, and the characteristic parameters of infiltration obtained through the inverse process shown here [32].

## 5. Conclusions

_{s}and h

_{f}) that are easy to obtain.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 9.**Relationship between measured data and data simulated by the model with optimized parameters at different time steps: (

**a**) δt = 1.5 s; (

**b**) δt = 2.0 s; (

**c**) δt = 5.0 s; (

**d**) δt = 10.0 s.

Test | q_{o} (m³/s/m) | P_{f} (cm) | I_{M} (cm) | θ_{o} (cm³/cm³) | $\overline{\mathbf{h}}$ (cm) | K_{s} (cm/h) | h_{f} (cm) | R² |
---|---|---|---|---|---|---|---|---|

1 | 0.001428 | 152 | 14.54 | 0.3331 | 2.73 | 1.1800 | 23.84 | 0.9983 |

2 | 0.001428 | 50 | 2.15 | 0.4386 | 2.73 | 1.5325 | 44.00 | 0.9814 |

3 | 0.001238 | 52 | 2.32 | 0.4353 | 2.60 | 0.0500 | 10.00 | 0.9967 |

δt | Computation Time (min) | R^{2} |
---|---|---|

1.0 | 55.0 | |

1.5 | 14.5 | 1.0 |

2.0 | 8.2 | 1.0 |

5.0 | 1.3 | 1.0 |

10.0 | 0.3 | 1.0 |

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**MDPI and ACS Style**

Fuentes, S.; Chávez, C.
Modeling of Border Irrigation in Soils with the Presence of a Shallow Water Table. I: The Advance Phase. *Agriculture* **2022**, *12*, 426.
https://doi.org/10.3390/agriculture12030426

**AMA Style**

Fuentes S, Chávez C.
Modeling of Border Irrigation in Soils with the Presence of a Shallow Water Table. I: The Advance Phase. *Agriculture*. 2022; 12(3):426.
https://doi.org/10.3390/agriculture12030426

**Chicago/Turabian Style**

Fuentes, Sebastián, and Carlos Chávez.
2022. "Modeling of Border Irrigation in Soils with the Presence of a Shallow Water Table. I: The Advance Phase" *Agriculture* 12, no. 3: 426.
https://doi.org/10.3390/agriculture12030426