# Moment Analysis for Modeling Soil Water Distribution in Furrow Irrigation: Variable vs. Constant Ponding Depths

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Field Experiment

^{−3}) = 1.5.

^{−1}(30 gpm) per furrow. The experimental and buffer furrows were equipped with a flume to record accurate measurements of the inflow. It took about 10 min for the inflow to reach a constant rate. The inflow reached the downstream end at t

_{L}= 36.58 min and it was cut-off at t

_{co}= 77 min. We used large heads to achieve quick completion of advance in zero slope furrows. The quick advance allows for better application and distribution efficiency. We would have topped the furrow if we did not cut-off when we did [4]. The experiment continued until almost the whole water content had receded (infiltrated) (t

_{r}= 124 min) and was well into the redistribution (t = 100 h).

#### 2.2. Numerical Computations

^{3}L

^{−3}]; ${\theta}_{r}$ is the residual volumetric water content $\left[{\mathrm{L}}^{3}{\mathrm{L}}^{-3}\right]$; ${\theta}_{s}$ is the saturated volumetric water content $\left[{\mathrm{L}}^{3}{\mathrm{L}}^{-3}\right]$; ${K}_{s}$ is the saturated hydraulic conductivity [LT

^{−1}]; ${S}_{e}$ is the relative water content or effective saturation $[-]$; $\alpha $ is the inverse of the air-entry value (or bubbling pressure) $\left[{\mathrm{L}}^{-1}\right]$; $n$ is a pore size distribution index $[-]$; and $m$ is a pore connectivity parameter [–], for which a value of 0.5 is used as an average for many soils. The hydraulic properties of the soil were predicted using Rosetta (Schaap et al. [17]) and are summarized in Table 4.

#### 2.3. Moment Analysis

#### 2.4. Data Processing

## 3. Results and Discussion

#### 3.1. Numerical Computations

#### 3.2. Moment Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Measured flow depth hydrographs at five stations along the test furrow. “x” is the distance from the furrow inlet.

**Figure 3.**Simulated soil wetting patterns with variable (VAR) and constant (CONST) ponding depth at stations 1 (S1), 3 (S3), and 5 (S5). Volumetric soil water contents (cm

^{3}·cm

^{−3}) are denoted by th [-].

**Figure 4.**The location of the center of mass, ${z}_{C}$, and the semi axes of the ellipses, ${\sigma}_{x}$ and ${\sigma}_{z}$, with variable (VAR) ponding depth at stations 1 (S1), 3 (S3), and 5 (S5).

**Figure 5.**The time-evolving ellipses with variable (VAR) head calculations at stations 1 (S1), 3 (S3), and 5 (S5).

Field | Soil Mapping Unit Symbol | Textural Class | Depth (cm) | Sand (%) | Clay (%) | BD (g·cm^{−3}) |
---|---|---|---|---|---|---|

18 | TR | Clay loam | 0–70 | 25–45 | 27–40 | 1.4–1.55 |

Parameter | Unit | Value |
---|---|---|

Furrow length (L) | m | 100 |

Furrow spacing (FS) | cm | 102 |

Bottom width (BW) | cm | 7 |

Top width (TW) | cm | 52 |

Maximum depth (hmax) | cm | 12 |

Side slope (SS) | cm·cm^{−1} | 1.88 |

Average bed slope (S_{0}) | m·m^{−1} | −0.00013 |

Station | Position (m) | Arrival Time (min) | Peak Depth (mm) | Average Depth (mm) |
---|---|---|---|---|

1 | 6.1 | 2.23 | 96 | 72 |

3 | 50 | 11.27 | 82 | 53 |

5 | 93.9 | 32.07 | 84 | 59 |

Parameter | θ_{r} | θ_{s} | α | n | l |
---|---|---|---|---|---|

Unit | cm^{3}·cm^{−3} | cm^{3}·cm^{−3} | cm^{−1} | - | - |

Value | 0.0788 | 0.4142 | 0.0136 | 1.3817 | 0.5 |

**Table 5.**Observed (OBS) and simulated volumetric soil water content with variable (VAR) and constant (CONST) ponding depth at a station.

Depth (cm) | Bed | Station | Ridge | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

12–27 | 27–42 | 42–57 | 57–72 | 72–87 | 87–102 | 0–15 | 15–30 | 30–45 | 45–60 | 60–75 | 75–90 | 90–102 | ||

S1 | ||||||||||||||

OBS | 0.18 | 0.18 | 0.26 | 0.19 | 0.16 | 0.15 | 0.16 | 0.18 | 0.18 | 0.15 | 0.14 | 0.14 | 0.17 | |

VAR | 0.24 | 0.23 | 0.22 | 0.18 | 0.14 | 0.13 | 0.24 | 0.23 | 0.21 | 0.16 | 0.13 | 0.13 | 0.13 | |

CONST | 0.24 | 0.23 | 0.22 | 0.18 | 0.13 | 0.13 | 0.23 | 0.22 | 0.20 | 0.15 | 0.13 | 0.13 | 0.13 | |

S3 | ||||||||||||||

OBS | 0.17 | 0.19 | 0.18 | 0.17 | 0.14 | 0.12 | 0.15 | 0.17 | 0.18 | 0.13 | 0.12 | 0.11 | 0.13 | |

VAR | 0.23 | 0.22 | 0.20 | 0.16 | 0.13 | 0.13 | 0.21 | 0.20 | 0.16 | 0.13 | 0.13 | 0.13 | 0.13 | |

CONST | 0.23 | 0.22 | 0.20 | 0.16 | 0.13 | 0.13 | 0.21 | 0.19 | 0.16 | 0.13 | 0.13 | 0.13 | 0.13 | |

S5 | ||||||||||||||

OBS | 0.15 | 0.19 | 0.17 | 0.13 | 0.19 | 0.15 | 0.13 | 0.15 | 0.16 | 0.13 | 0.11 | 0.12 | 0.11 | |

VAR | 0.22 | 0.22 | 0.20 | 0.15 | 0.13 | 0.13 | 0.20 | 0.19 | 0.15 | 0.13 | 0.13 | 0.13 | 0.13 | |

CONST | 0.22 | 0.22 | 0.20 | 0.15 | 0.13 | 0.13 | 0.19 | 0.17 | 0.14 | 0.13 | 0.13 | 0.13 | 0.13 |

**Table 6.**The location of the center of mass, ${z}_{C}$, and the semi axes of the ellipses, ${\sigma}_{x}$ and ${\sigma}_{z}$, with variable (VAR) ponding depth at stations 1 (S1), 3 (S3), and 5 (S5).

Time (h) | S1-VAR | S3-VAR | S5-VAR | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1.5 | 6 | 60 | 99 | 1 | 1.5 | 6 | 60 | 99.5 | 1 | 1.5 | 6 | 60 | 100 | |

${z}_{C}$ (cm) | −14.1 | −17.2 | −21.6 | −27.5 | −29.3 | −13.4 | −15.2 | −20.2 | −24.8 | −26.1 | −11.3 | −13.8 | −19.2 | −23.9 | −25.2 |

${\sigma}_{x}$ (cm) | 15.5 | 17.6 | 22.0 | 27.2 | 27.9 | 12.4 | 15.0 | 19.1 | 25.2 | 26.3 | 10.9 | 15.0 | 18.8 | 24.8 | 26.0 |

${\sigma}_{z}$ (cm) | 8.2 | 10.5 | 13.2 | 17.1 | 18.1 | 7.8 | 9.2 | 12.4 | 15.4 | 16.4 | 6.4 | 8.0 | 11.9 | 14.9 | 15.7 |

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**MDPI and ACS Style**

Kazemi, H.; Sadraddini, A.A.; Nazemi, A.H.; Sanchez, C.A. Moment Analysis for Modeling Soil Water Distribution in Furrow Irrigation: Variable vs. Constant Ponding Depths. *Water* **2021**, *13*, 1415.
https://doi.org/10.3390/w13101415

**AMA Style**

Kazemi H, Sadraddini AA, Nazemi AH, Sanchez CA. Moment Analysis for Modeling Soil Water Distribution in Furrow Irrigation: Variable vs. Constant Ponding Depths. *Water*. 2021; 13(10):1415.
https://doi.org/10.3390/w13101415

**Chicago/Turabian Style**

Kazemi, Honeyeh, Ali Ashraf Sadraddini, Amir Hossein Nazemi, and Charles A Sanchez. 2021. "Moment Analysis for Modeling Soil Water Distribution in Furrow Irrigation: Variable vs. Constant Ponding Depths" *Water* 13, no. 10: 1415.
https://doi.org/10.3390/w13101415