# Applicability of the Modified Green-Ampt Model Based on Suction Head Calculation in Water-Repellent Soil

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## Abstract

**:**

_{f VG}, S

_{f BC}, S

_{f GP}) and the average saturated hydraulic conductivity. Therefore, we obtained three modified models: the Green-Ampt-VG, Green-Ampt-BC, and Green-Ampt-GP models. Indoor one-dimensional water infiltration experiments were conducted to simulate the cumulative infiltration (CI), the distance of the wetting front (Z

_{f}), and the infiltration rate of a hydrophilic treatment and repellent treatments. The results showed that as the degree of repellency increased, the soil suction head decreased, and the relationship between the value of the soil suction head and the degree of WRS was exponential. In addition, the simulated values of the modified CI formula highly fit the measured values of all treatments in the three models (RMSE: 1.696, 1.812, and 0.694). The modified Green-Ampt-VG model had the best simulation effect on the infiltration rate (RMSE: 0.036) and Z

_{f}(RMSE: 3.976). The results indicated that the suction head values obtained from the parameters of the VG model were closest to the actual values compared the other models. These results can provide a reference for the solution of problems involving the suction head and water infiltration into WRS in the future.

## 1. Introduction

## 2. Model Establishment and Validation

#### 2.1. The Traditional Green-Ampt Model

_{s}is the hydraulic conductivity of saturated soil (cm·min

^{−1}); Z

_{f}is the distance of the wetting front (cm); S

_{f}is the suction head at the wetting front (cm); and H is the pressure head at the surface (cm).

#### 2.2. Modified Green-Ampt Model

#### 2.2.1. Assumptions of the Modified Model

_{s}= $\overline{{K}_{s}}$. According to [29,30], the soil hydraulic conductivity should be 0.5 times the saturated hydraulic conductivity, so we considered 0.5 times the saturated hydraulic conductivity as the final hydraulic conductivity.

#### 2.2.2. Calculation of Suction Head

^{3}·cm

^{−3}); ${\theta}_{s}$is the saturated water content (cm

^{3}·cm

^{−3}); H is the pressure head (cm); and α, m, and n are empirical parameters.

^{3}·cm

^{−3}), and the other parameters have the same meanings as above.

_{r}is the relative hydraulic conductivity.

_{a}(1/cm), and λ is the distribution parameter of soil pore size.

_{a}is air entry value (cm).

#### 2.2.3. Cumulative Infiltration and Wetting Fronts of the Modified Green-Ampt Model

^{−1}); S

_{f}corresponds to the suction head solution Formulas (5), (10), and (11) (cm); and Z

_{f}is the migration depth of the wetting front (cm).

#### 2.2.4. The Average Saturated Hydraulic Conductivity

^{−1}); D is the thickness of the soil layer (cm); i is the number of soil layers and N is the number of saturated layers in the soil; and ${K}_{s,i}$ is the saturated hydraulic conductivity of soil layer i.

#### 2.3. Validation of the Model

_{k}is the simulated value, and M

_{i}is the measured value.

^{−3}. The experiments were conducted under ponding conditions with a constant head of 3 cm. The experiments were terminated when the wetting front reached the bottom of the column. Cumulative infiltration and infiltration rate were calculated based on variation in the water level of a Mariotte bottle. A ruler was attached to a column made of clear plastic to determine the wetting front depth. Three replicates were set for each experimental treatment, and the average value of the experimental data was taken as the final value.

## 3. Results

#### 3.1. Relationship between Suction Head and Quantity of Octadecylamine

#### 3.2. Computational Results and Verification of Cumulative Infiltration

#### 3.3. Computational Results and Verification of the Infiltration Rate

#### 3.4. Computational Results and Verification of Wetting Front

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

_{f}, suction head; S

_{i}, suction head at initial water content; VG, van Genuchten; BC, Brooks–Corey; GP, combination of Green-Ampt and the Philip model; ${\theta}_{r}$, residual water content; ${\theta}_{s}$, saturated water content; ${\theta}_{i},$initial water content; S, soil sorptivity in Philip model; CI, cumulative infiltration; Z

_{f}, distance of wetting front; WDPT, water droplet penetration time; K

_{s}, soil saturated hydraulic conductivity; $\overline{{K}_{s}}$, effective saturated hydraulic conductivity; RMSE, root-mean-square error; RETC, retention curve.

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**Figure 2.**Relationship between the quantity of octadecylamine and the suction head under different computational methods: (

**a**) VG model; (

**b**) BC model; (

**c**) GP model.

**Figure 4.**Measured infiltration rate curves and simulated infiltration rate curves of modified Green-Ampt model.

**Table 1.**Water repellency degree, quantity of octadecylamine (Q

_{oc}) per kg of soil, WDPT, K

_{s}, and soil hydraulic parameters of soil water retention curve model.

VG | BC | ||||||
---|---|---|---|---|---|---|---|

WRS | Q_{oc} | Measured Values | Fitted Values | Fitted Values | |||

K_{s} | θ_{r} | θ_{s} | α | n | α′ | ||

g·kg^{–1} | cm·min^{−1} | cm^{3} m^{−3} | cm^{3}·cm^{−3} | ||||

Wettable | 0 | 0.0305 | 0.13 | 0.48 | 0.013 | 1.232 | 0.0237 |

Slightly | 0.2 | 0.0210 | 0.13 | 0.48 | 0.018 | 1.230 | 0.0283 |

Strongly | 0.4 | 0.0165 | 0.13 | 0.47 | 0.797 | 1.170 | 0.7162 |

Severely | 0.6 | 0.0125 | 0.13 | 0.45 | 12.97 | 1.112 | 10.086 |

Extremely | 1.0 | 0.0090 | 0.13 | 0.40 | 71.31 | 1.107 | 660.76 |

Treatments | T0 | T1 | T2 | T3 | T4 | T5 | |
---|---|---|---|---|---|---|---|

Different depths | (0–5 cm) | W | ST | SE | SE | E | E |

Water repellency degree | (5–10 cm) | W | SL | SL | ST | ST | SE |

Model | Treatment | Cumulative Infiltration | Infiltration Rate | Wetting Front |
---|---|---|---|---|

Green-Ampt-VG | T0 | 1.12 | 0.027 | 0.916 |

T1 | 1.792 | 0.029 | 2.516 | |

T2 | 1.398 | 0.047 | 4.230 | |

T3 | 2.104 | 0.042 | 4.549 | |

T4 | 1.677 | 0.038 | 5.758 | |

T5 | 2.087 | 0.029 | 5.888 | |

Average | 1.696 | 0.036 | 3.976 | |

Green-Ampt-BC | T0 | 1.12 | 0.626 | 8.097 |

T1 | 1.792 | 0.037 | 13.834 | |

T2 | 1.398 | 0.048 | 16.912 | |

T3 | 2.104 | 0.043 | 15.553 | |

T4 | 1.677 | 0.038 | 17.757 | |

T5 | 2.087 | 0.029 | 18.761 | |

Average | 1.812 | 0.137 | 15.152 | |

Green-Ampt-GP | T0 | 0.50 | 0.031 | 2.458 |

T1 | 1.185 | 0.117 | 6.950 | |

T2 | 0.759 | 0.177 | 9.350 | |

T3 | 0.730 | 0.263 | 9.076 | |

T4 | 0.442 | 0.244 | 10.720 | |

T5 | 0.546 | 0.213 | 11.250 | |

Average | 0.694 | 0.174 | 8.301 |

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**MDPI and ACS Style**

Sun, Y.; Yang, Y.; Zhang, B.; Zhang, X.; Xu, Y.; Xiang, Y.; Chen, J.
Applicability of the Modified Green-Ampt Model Based on Suction Head Calculation in Water-Repellent Soil. *Water* **2023**, *15*, 2925.
https://doi.org/10.3390/w15162925

**AMA Style**

Sun Y, Yang Y, Zhang B, Zhang X, Xu Y, Xiang Y, Chen J.
Applicability of the Modified Green-Ampt Model Based on Suction Head Calculation in Water-Repellent Soil. *Water*. 2023; 15(16):2925.
https://doi.org/10.3390/w15162925

**Chicago/Turabian Style**

Sun, Yixiang, Yalong Yang, Bei Zhang, Xing Zhang, Yangyang Xu, Youzhen Xiang, and Junying Chen.
2023. "Applicability of the Modified Green-Ampt Model Based on Suction Head Calculation in Water-Repellent Soil" *Water* 15, no. 16: 2925.
https://doi.org/10.3390/w15162925