# A Novel Physically Based Distributed Model for Irrigation Districts’ Water Movement

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Soil Water Movement

^{−1}], and S

_{soil}is the source in unit depth [T

^{−1}] (negative for outflow).

_{surf}is the surface ponding depth [L], which can also be interpreted as the pressure head on the top grid, t is the time [T], Prec is the precipitation rate [L·T

^{−1}], Irri is the irrigation rate [L·T

^{−1}], Evap is the evaporation rate [L·T

^{−1}], Drain is the surface drainage rate [L·T

^{−1}], intch is the downward interchange rate through the land surface [L·T

^{−1}] (positive for infiltration), and S

_{surf}is the source rate [L·T

^{−1}] (positive for incoming source). The interchange rate intch is exactly the upper boundary flux of soil water, which means Equation (2) can be integrated into the upper boundary of Equation (1) through substituting the upper boundary flux with Equation (3), a variation of Equation (2):

_{surf}. The ponding depth (or surface pressure head) H

_{surf}will be solved with soil pressure heads at other grids simultaneously. If the surface is dry, the ponding depth becomes zero, and it is no longer the surface pressure head, then Equation (3) becomes

_{a}is the actual transpiration rate of the root zone [L·T

^{−1}], TD

_{a}is the actual transpiration rate distribution [T

^{−1}], which is a function of elevation z [L] and the corresponding pressure head h [L], TD

_{p}is the potential transpiration rate distribution [T

^{−1}], T

_{p}is the potential transpiration rate of the root zone [L·T

^{−1}], α(h) is the water stress function [-], and β(z) is the distribution function relating to the root density [L

^{−1}].

_{50}is the pressure head at which the transpiration rate is reduced by 50%.

_{top}is the land surface elevation [L], and RD is the root depth [L].

_{p}is calculated through Equation (8):

_{T}is the transpiration coefficient [-], ETp is the crop potential evapotranspiration rate [L·T

^{−1}], K

_{C}is the crop coefficient [-], and ET

_{0}is the potential evapotranspiration rate of reference crop [L·T

^{−1}]. The ET

_{0}is calculated through meteorological information with the Penman–Monteith equation [42]. The K

_{C}is related to the crop and given as a parameter. The K

_{T}is calculated from the Leaf Area Index (LAI) [43]:

#### 2.2. Groundwater Movement

_{grnd}is the groundwater level [L], t is the time [T], $\nabla $ is the gradient operator [L

^{−1}], Ks is the saturated hydraulic conductivity [L·T

^{−1}], H

_{grnd}is the thickness of the groundwater [L], and S

_{grnd}is the source in unit horizontal area [L·T

^{−1}] (positive for inflow). Considering a cell of any shape in unstructured grids, the governing equation can be rewritten as Equation (11):

_{j}is the outer normal direction of the j-th side [L], $\frac{\partial {Z}_{grnd}}{\partial {r}_{j}}$ is the outer normal gradient of the groundwater level at the j-th side [-], A

_{j}is the vertical cross area at the j-th side [L

^{2}], and A

_{cell}is the horizontal area of the cell [L

^{2}]. $\left({K}_{s}\frac{\partial {Z}_{grnd}}{\partial {r}_{j}}{A}_{j}\right)$ is the inflowing flux through the j-th side. The groundwater level change is a result of the side flow and other sources such as soil water recharge, canal seepage, and drainage.

_{grnd}at the interpolated point (say Z

_{gi’}) is a weighted average of Z

_{grnd}at cell centroids nearby (say Z

_{gi}, Z

_{gj}, and Z

_{gk}). The weighting factors are given additionally, and the number of weighting factors remains the same for all ghost nodes in MODFLOW-USG, while it is automatically analyzed from grid information in this study. Cell centroids within the local range of the ghost node will be used as interpolating points, and the weighting factors are calculated with an improved Inverse Distance Weighted (IDW) method, a spatial interpolating method that takes into account the azimuth of the interpolating centroid as well as its distance, which is better than the traditional IDW method that only considers the distance.

#### 2.3. Channel Flow

_{ch}is the water surface width in the channel [L], Z

_{ch}is the channel water level [L], t is the time [T], x is the longitudinal direction [L], q

_{ch}is the source in unit length [L

^{2}·T

^{−1}] (positive for inflow), and D

_{ch}is the diffusive coefficient [L

^{3}·T

^{−1}], as given in Equation (13):

_{ch}is the cross area [L

^{2}], R

_{ch}is the hydraulic radius [L], and N

_{m}is Manning’s roughness coefficient [L

^{−1/3}·T].

#### 2.4. Model Coupling

#### 2.5. Model Inputs and Outputs

_{C}), the leaf area index (LAI), the root depth (RD), and the water stress parameters (p, h

_{50}), which may vary with time to represent the crop growth. The irrigation information is given by round, and each irrigation round requires the round beginning and ending time, crops to be irrigated, the total irrigation water amount, and the seepage and discard ratio of canal chains. The drainage parameters mainly include the maximum allowed ponding depth for surface drainage and averaged lower-level ditches spacing for groundwater drainage.

## 3. Test Case

^{2}, 66.5% of which is irrigated land, and the soil type is mainly silty loam. It is a typical arid region with approximately 150 mm of annual precipitation and 2000 mm of annual evaporation. The main crops here are sunflower (grown from late May to early October), maize (grown from late April to early October), and wheat (grown from early April to mid-July). There are five irrigation rounds during the crop growth period (from April to October) in a year, and the average water table depth is about 1.8 m during this period.

#### 3.1. Input Data

#### 3.2. Simulation Results

^{4}m

^{3}in 2018 but decreases 169.6 × 10

^{4}m

^{3}in 2019. The net recharge from a 1 m depth below is 196.5 × 10

^{4}m

^{3}and 309.5 × 10

^{4}m

^{3}in 2018 and 2019, respectively.

_{i}is the i-th simulated value, O

_{i}is the i-th observed value, and O

_{ave}is the average of observed data. The relative error is

_{max}is the maximum observed value, and O

_{min}is the minimum observed data. The root mean square ratio is

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**An example of spatial discretization. The red lines stand for canals, the blue lines stand for ditches, and the yellow grids stand for cells of the surface field, soil water, and groundwater.

**Figure 6.**Location of Shahaoqu Sub-Irrigation Area and the distribution of canals, ditches, cells, and observation sites for soil water and groundwater.

**Figure 9.**Process of precipitation and irrigation, and the simulated and observed groundwater level at an observation site.

Item | Value | ||||
---|---|---|---|---|---|

Groundwater | Ksg [m·d^{−1}] | 14.4 | |||

μ [-] | 0.03 | ||||

Soil water | Layer depth [cm] | 0~40 | 40~170 | 170~250 | 250~300 |

θr [-] | 0.050 | 0.043 | 0.073 | 0.043 | |

θs [-] | 0.413 | 0.460 | 0.500 | 0.450 | |

α [cm^{−1}] | 0.010 | 0.012 | 0.007 | 0.012 | |

n [-] | 1.567 | 1.467 | 1.200 | 1.700 | |

Kss [cm·d^{−1}] | 10.333 | 17.467 | 6.000 | 28.100 | |

Canal flow | Nm [m^{−1/3}·s] | 0.022 | |||

Ditch flow | Nm [m^{−1/3}·s] | 0.029 |

Year | Round | Time | Water Amount [10^{4} m^{3}] | Crops to Irrigate |
---|---|---|---|---|

2018 | 1 | 4/23~5/14 | 577.4 | wheat, sunflower, melon |

2 | 5/14~5/25 | 365.6 | wheat, sunflower | |

3 | 6/13~6/26 | 368.3 | wheat, maize, sunflower | |

4 | 7/3~7/19 | 398.9 | maize, sunflower | |

5 | 7/26~8/4 | 173.6 | maize, sunflower | |

2019 | 1 | 4/28~5/16 | 530.4 | wheat, sunflower, melon |

2 | 5/16~5/30 | 500.7 | wheat, sunflower | |

3 | 6/14~6/23 | 194.6 | wheat, maize | |

4 | 7/6~7/20 | 424.8 | maize, sunflower | |

5 | 7/26~8/9 | 259.1 | maize, sunflower |

Item | 2018 | 2019 | |
---|---|---|---|

Precipitation | 827.7 | 408.8 | |

Irrigation | Inlet | 1820.9 | 1834.3 |

To fields | 1197.3 | 1210.7 | |

Seepage | 513.3 | 513.5 | |

Evaporated | 18.7 | 11.4 | |

Discarded | 93.8 | 100.9 | |

Surface evaporation | 883.8 | 856.8 | |

Crop transpiration | 1335.0 | 1241.8 | |

Groundwater drainage | 218.8 | 222.4 | |

Groundwater boundary flow ^{1} | −119.7 | −119.9 | |

Water storage change ^{2} | −71.1 | −337.1 | |

Balance error ^{3} | 52.1 (2.9%) | 29.1 (1.6%) |

^{1}flow through lateral boundaries, negative for flowing out of the region.

^{2}storage for both soil water and groundwater, negative for storage reduction.

^{3}percentage to irrigation water amount form the inlet.

Item | 2018 | 2019 |
---|---|---|

Precipitation | 827.7 | 408.8 |

Irrigation | 1197.3 | 1210.7 |

Evapotranspiration | 2218.8 | 2098.6 |

Net recharge from 1 m depth below | 196.5 | 309.5 |

Water storage change ^{1} | 2.7 | −169.6 |

^{1}storage for the 1m-depth soil layer, negative for storage reduction.

**Table 5.**Relative calculating time consumed with different numbers of threads (all the computing times are relative to the total computing time with one thread).

Number of Threads | 1 | 2 | 4 | 8 |
---|---|---|---|---|

Total time consumed | 1 | 0.75 | 0.50 | 0.35 |

Time for canals | 0.01 | 0.01 | 0.01 | 0.01 |

Time for ditches | 0.01 | 0.01 | 0.01 | 0.01 |

Time for soil water | 0.81 | 0.56 | 0.33 | 0.19 |

Time for groundwater | 0.08 | 0.08 | 0.08 | 0.08 |

Time for output | 0.10 | 0.08 | 0.06 | 0.06 |

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**MDPI and ACS Style**

Mi, B.; Chen, H.; Wang, S.; Jin, Y.; Jia, J.; Chang, X.; Fu, X.; Chai, R.; Wei, M.
A Novel Physically Based Distributed Model for Irrigation Districts’ Water Movement. *Water* **2021**, *13*, 692.
https://doi.org/10.3390/w13050692

**AMA Style**

Mi B, Chen H, Wang S, Jin Y, Jia J, Chang X, Fu X, Chai R, Wei M.
A Novel Physically Based Distributed Model for Irrigation Districts’ Water Movement. *Water*. 2021; 13(5):692.
https://doi.org/10.3390/w13050692

**Chicago/Turabian Style**

Mi, Boyu, Haorui Chen, Shaoli Wang, Yinlong Jin, Jiangdong Jia, Xiaomin Chang, Xiaojun Fu, Ronghua Chai, and Meiling Wei.
2021. "A Novel Physically Based Distributed Model for Irrigation Districts’ Water Movement" *Water* 13, no. 5: 692.
https://doi.org/10.3390/w13050692