# Three-Dimensional Model of Soil Water and Heat Transfer in Orchard Root Zone under Water Storage Pit Irrigation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Experimental Design and Test Items

#### 2.2.1. Experimental Design

#### 2.2.2. Test Items

_{0}and atmospheric temperature changes during the experiment are shown in Figure 4. The rainfall and irrigation during the experiment are shown in Table 2.

#### 2.3. Establishment of a Three-Dimensional Model of Soil Water and Heat Transfer in Orchard under Water Storage Pit Irrigation

#### 2.3.1. Determination of Simulation Area

#### 2.3.2. Governing Equation

- (1)
- Governing equations of soil moisture movement

^{3}·cm

^{−3}, $t$ is time in min, x, y, and z are the space coordinates, where z up is positive, in cm, K(h) is the unsaturated hydraulic conductivity in cm/min, and S is the root water uptake rate in $\mathrm{L}/\mathrm{min}$.

^{3}·cm

^{−3}, ${\theta}_{r}$ is the residual moisture content in cm

^{3}·cm

^{3}, $\alpha $, m, and n are the shape parameters of soil water characteristic curve, and $m=1-\frac{1}{n},(n>0)$.

_{0}, h

_{1}, h

_{2}, and h

_{3}are the matrix potential heads corresponding to ${\theta}_{s}$, $80\%{\theta}_{f}$, $60\%{\theta}_{f}$, and ${\theta}_{w}$.

_{m}, Y

_{m}, and Z

_{m}are the maximum extension depths (cm) of root in the x, y and z directions, respectively, A

_{xy}is the area of computational region on the surface in cm

^{2}, and $\beta \left(x,y,z\right)$ is the shape factor that describes the spatial distribution of root water uptake potential.

- (2)
- Governing equations for soil heat transfer

^{3}·°C), K

_{h}is the thermal conductivity in J/(cm·min·°C), and ${c}_{w}$ is the specific heat capacity of water in J/(cm

^{3}·°C).

_{h}is related to moisture content, and the result of Chung et al. (1987) [48] was used for model simulation.

_{h}is the thermal conductivity in W/(m·°C), and b

_{1}, b

_{2}, and b

_{3}are the regression coefficients.

#### 2.3.3. Definite Conditions

#### Initial Conditions

#### Boundary Conditions

_{w}is the temperature of irrigation water in °C, and ${g}_{k}\left(x,y,z,t\right)$ is the heat flux density function in the pit in $\mathrm{J}/\left({\mathrm{cm}}^{2}\xb7\mathrm{min}\right)$.

_{1}is the end time of water infiltration.

_{d}is the z coordinate value corresponding to the boundary of the pit bottom.

_{5}is the area of boundary LKCD.

#### 2.3.4. Model Solving

#### Calculation of Surface Soil Heat Flux

#### Determination of Water and Heat Boundary in Water Storage Pit

- (1)
- Boundary of soil moisture in the pit

^{3}, N is the total number of nodes in the computing region, and n is the node on the boundary of the pit wall. Additional parameters are detailed in Supplementary Materials.

- (2)
- Boundary of soil temperature in the pit

#### 2.3.5. Parameters Determination

#### 2.3.6. Model Evaluation Indices

#### 2.3.7. Programming

## 3. Results and Discussions

#### 3.1. Model Validation

#### 3.1.1. Comparison and Analysis of Simulated and Measured Values

^{3}·cm

^{−3}, and the distribution tended to be uniform. Therefore, the model can be a good simulation of soil moisture movement in the field.

#### 3.1.2. Model Performance Evaluation

#### 3.2. Simulated Interday Dynamic Changes of Soil Water and Heat

#### 3.3. Simulated Intraday Dynamic Changes of Soil Water and Heat

^{3}·cm

^{−3}), and the area with high moisture content gradually spread outward. In this condition, as the water redistribution proceeded, the water content distribution gradually became uniform. The calculation results of statistical indexes are shown in Table 8.

#### 3.4. Simulated the Characteristics of Soil Water and Heat Transfer in Orchard under Different Irrigation Amount

#### 3.5. Simulated the Characteristics of Soil Heat Transfer in Orchard under Different Irrigation Water Temperature

## 4. Conclusions

^{−1}, but the influence of water temperature only lasted for 18–24 h after irrigation.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Distribution of soil moisture content and temperature measurement points. (

**a**) Horizontal distribution; (

**b**) Vertical distribution.

**Figure 4.**Variation of ET

_{0}and atmospheric temperature during the experiment. (

**a**) Variation of ET

_{0}during the experiment; (

**b**) Variation of atmospheric temperature during the experiment.

**Figure 9.**Simulation of soil moisture content distribution on different days. (

**a**) Soil moisture content after 1 day of irrigation; (

**b**) Soil moisture content after 5 days of irrigation; (

**c**) Soil moisture content after 11 days of irrigation; (

**d**) Soil moisture content after 17 days of irrigation.

**Figure 10.**Simulation of soil temperature distribution on different days. (

**a**) Soil temperature after 1 day of irrigation; (

**b**) Soil temperature after 5 days of irrigation; (

**c**) Soil temperature after 11 days of irrigation; (

**d**) Soil temperature after 17 days of irrigation.

**Figure 11.**Simulation of soil moisture content distribution at different times after 1 day of irrigation. (

**a**) 6:00 soil moisture content; (

**b**) 12:00 soil moisture content; (

**c**) 18:00 soil moisture content; (

**d**) 24:00 soil moisture content.

**Figure 12.**Simulation of soil temperature distribution at different times after 1 day of irrigation. (

**a**) 6:00 soil temperature; (

**b**) 12:00 soil temperature; (

**c**) 18:00 soil temperature; (

**d**) 24:00 soil temperature.

**Figure 13.**Simulation of soil moisture content under different irrigation amounts. (

**a**) Irrigation amount: 200 L; (

**b**) Irrigation amount: 300 L; (

**c**) Irrigation amount: 400 L.

**Figure 14.**Simulation of soil temperature under different irrigation amounts. (

**a**) Irrigation amount: 200 L; (

**b**) Irrigation amount: 300 L; (

**c**) Irrigation amount: 400 L.

**Figure 15.**Simulation of soil temperature under different irrigation water temperatures. (

**a**) Water temperature: 15 °C; (

**b**) Water temperature: 20 °C; (

**c**) Water temperature: 25 °C.

Soil Depth/cm | ${\mathit{\theta}}_{\mathit{f}}/{\mathbf{cm}}^{3}\xb7{\mathbf{cm}}^{-3}$ | ${\mathit{\theta}}_{\mathit{s}}/{\mathbf{cm}}^{3}\xb7{\mathbf{cm}}^{-3}$ | Dry Bulk Density/g·cm^{−3} | Soil Texture |
---|---|---|---|---|

0–40 | 0.30 | 0.51 | 1.49 | silt loam |

40–70 | 0.28 | 0.52 | 1.44 | silt loam |

70–120 | 0.29 | 0.49 | 1.56 | silt loam |

120–170 | 0.32 | 0.50 | 1.51 | loam |

170–200 | 0.30 | 0.52 | 1.45 | loam |

Date | Rainfall/mm | Irrigation/L |
---|---|---|

24 June | 300 | |

27 June | 0.6 | |

28 June | 0.2 | |

3 July | 0.6 | |

9 July | 2.0 |

Soil Depth/cm | ${K}_{s}$/cm·h^{−1} | ${\theta}_{r}$/cm^{3}·cm^{−3} | ${\theta}_{s}$/cm^{3}·cm^{−3} | α | n |

0–40 | 0.522 | 0.0545 | 0.51 | 0.0071 | 1.5801 |

40–70 | 0.768 | 0.0536 | 0.52 | 0.0080 | 1.5552 |

70–120 | 0.684 | 0.0496 | 0.49 | 0.0085 | 1.5296 |

120–170 | 0.606 | 0.0465 | 0.50 | 0.0094 | 1.5243 |

170–200 | 0.888 | 0.0443 | 0.52 | 0.0091 | 1.5419 |

p_{x} | p_{y} | p_{z} | x* | y* | z* |

1.81 | 1.02 | 1.82 | 103.48 | 52.38 | 82.23 |

RMSE | MAPE | MAD | |
---|---|---|---|

Soil moisture content | 0.0269 | 10.05% | 0.0214 |

Soil temperature | 0.9460 | 3.23% | 0.6984 |

**Table 6.**Calculation of statistical indexes of daily soil moisture content for water storage pit irrigation.

Parameters | Average (cm ^{3}·cm^{−3}) | Standard Deviation (cm ^{3}·cm^{−3}) | Coefficient of Variation | Maximum (cm ^{3}·cm^{−3}) | Minimum (cm ^{3}·cm^{−3}) | |
---|---|---|---|---|---|---|

Date | ||||||

1 day | 0.2166 | 0.0788 | 0.3636 | 0.4655 | 0.0625 | |

5 days | 0.2088 | 0.0429 | 0.2056 | 0.2822 | 0.0538 | |

11 days | 0.1924 | 0.0253 | 0.1315 | 0.2279 | 0.0631 | |

17 days | 0.1795 | 0.0165 | 0.0920 | 0.1990 | 0.0799 |

**Table 7.**Calculation of the statistical indexes of daily soil temperature for water storage pit irrigation.

Parameters | Average (°C) | Standard Deviation (°C) | Coefficient of Variation | Maximum (°C) | |
---|---|---|---|---|---|

Date | |||||

1 day | 20.56 | 2.03 | 0.0988 | 24.53 | |

5 days | 20.68 | 1.85 | 0.0895 | 25.95 | |

11 days | 21.93 | 2.59 | 0.1183 | 28.32 | |

17 days | 20.12 | 1.02 | 0.0507 | 22.98 |

**Table 8.**Calculation of statistical indexes of soil moisture content for water storage pit irrigation day.

Parameters | Average (cm ^{3}·cm^{−3}) | Standard Deviation (cm ^{3}·cm^{−3}) | Coefficient of Variation | Maximum (cm ^{3}·cm^{−3}) | Minimum (cm ^{3}·cm^{−3}) | |
---|---|---|---|---|---|---|

Time | ||||||

6:00 | 0.2129 | 0.0780 | 0.3664 | 0.5000 | 0.0648 | |

12:00 | 0.2166 | 0.0788 | 0.3636 | 0.4655 | 0.0626 | |

18:00 | 0.2144 | 0.0692 | 0.3229 | 0.3997 | 0.0604 | |

24:00 | 0.2141 | 0.0646 | 0.3016 | 0.3745 | 0.0592 |

**Table 9.**Calculation of the statistical indexes of soil temperature for water storage pit irrigation day.

Parameters | Average (°C) | Standard Deviation (°C) | Coefficient of Variation | Maximum (°C) | |
---|---|---|---|---|---|

Time | |||||

6:00 | 20.12 | 1.75 | 0.0870 | 23.22 | |

12:00 | 20.56 | 2.03 | 0.0988 | 24.53 | |

18:00 | 21.10 | 2.65 | 0.1258 | 26.65 | |

24:00 | 20.72 | 2.05 | 0.0989 | 23.78 |

Parameters | Average (cm ^{3}·cm^{−3}) | Standard Deviation (cm ^{3}·cm^{−3}) | Coefficient of Variation | Maximum (cm ^{3}·cm^{−3}) | |
---|---|---|---|---|---|

Irrigation Amount | |||||

200 L | 0.2062 | 0.0586 | 0.2842 | 0.3693 | |

300 L | 0.2166 | 0.0788 | 0.3636 | 0.4655 | |

400 L | 0.2177 | 0.0816 | 0.3748 | 0.5000 |

**Table 11.**Calculation of statistical indexes for soil temperature under different irrigation amount.

Parameters | Average (°C) | Standard Deviation (°C) | Coefficient of Variation | Maximum (°C) | Minimum (°C) | |
---|---|---|---|---|---|---|

Irrigation Amount | ||||||

200 L | 20.58 | 2.04 | 0.0990 | 24.65 | 17.03 | |

300 L | 20.56 | 2.02 | 0.0988 | 24.53 | 17.19 | |

400 L | 20.52 | 2.00 | 0.0976 | 24.63 | 17.20 |

**Table 12.**Calculation of statistical indexes for 0–80 cm soil temperature under different irrigation water temperature.

Parameters | Average (°C) | Standard Deviation (°C) | Coefficient of Variation | Maximum (°C) | Minimum (°C) | |
---|---|---|---|---|---|---|

Irrigation Water Temperature | ||||||

15 °C | 20.89 | 1.95 | 0.0932 | 25.27 | 17.23 | |

20 °C | 21.61 | 1.74 | 0.0807 | 25.36 | 18.89 | |

25 °C | 22.33 | 1.96 | 0.0878 | 25.45 | 18.95 |

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**MDPI and ACS Style**

Su, Y.; Guo, X.; Lei, T.; Zheng, L.; Ma, J.; Sun, X.; Hao, L.; Hu, F.
Three-Dimensional Model of Soil Water and Heat Transfer in Orchard Root Zone under Water Storage Pit Irrigation. *Water* **2022**, *14*, 1813.
https://doi.org/10.3390/w14111813

**AMA Style**

Su Y, Guo X, Lei T, Zheng L, Ma J, Sun X, Hao L, Hu F.
Three-Dimensional Model of Soil Water and Heat Transfer in Orchard Root Zone under Water Storage Pit Irrigation. *Water*. 2022; 14(11):1813.
https://doi.org/10.3390/w14111813

**Chicago/Turabian Style**

Su, Yuanyuan, Xianghong Guo, Tao Lei, Lijian Zheng, Juanjuan Ma, Xihuan Sun, Linru Hao, and Feipeng Hu.
2022. "Three-Dimensional Model of Soil Water and Heat Transfer in Orchard Root Zone under Water Storage Pit Irrigation" *Water* 14, no. 11: 1813.
https://doi.org/10.3390/w14111813