# An Empirical Model for Aeolian Sandy Soil Wetting Front Estimation with Subsurface Drip Irrigation

^{1}

^{2}

^{*}

## Abstract

**:**

_{s}, irrigation flow Q, and total irrigation volume V, was proposed. The model’s accuracy was statistically evaluated with the observed data and verified by a numerical simulation using HYDRUS-2D/3D. The mean absolute error (MAE) and root mean square error (RMSE) of the proposed model in the horizontal and downward directions were 0.80 and 0.95 cm, respectively, with a percentage bias (PBIAS) of −3.47 ≤ ±10 and a Nash–Sutcliffe efficiency (NSE) of 0.98, which is close to 1. Thus, this model can contribute to the selection of the appropriate depth and spacing of subsurface laterals.

## 1. Introduction

Category | Applied Soil | Irrigation Method | Equation | Basis for Establishment |
---|---|---|---|---|

Mathematical Model | Silt | Surface point source [27] | $D\left(t\right)/H\left(t\right)=1+\frac{{K}_{s}t}{2\left({\theta}_{f}-{\theta}_{i}\right)}$ | Derivation of Richards’ equation |

Numerical Model | All soil types [28,29,30,31] | - | $\theta \left(h\right)={\theta}_{r}+\frac{{\theta}_{s}+{\theta}_{r}}{{\left(1+{\left|\alpha h\right|}^{n}\right)}^{m}}$ $K\left({S}_{e}\right)={K}_{s}{S}_{e}^{l}{\left[1-{\left(1-{S}_{e}^{l/m}\right)}^{m}\right]}^{2}$ | Inversion of soil parameters |

Empirical Model | Sandy clay loam | Surface point source [32] | $H={Q}^{0.2975}{K}_{s}{}^{3.1679}{t}^{0.3490}{\theta}_{i}{}^{0.0903}{P}_{b}{}^{7.1565}O{M}^{-2.8332}$ $D={Q}^{0.2858}{K}_{s}{}^{1.0710}{t}^{0.4786}{\theta}_{i}{}^{0.2771}{P}_{b}{}^{8.5567}O{M}^{-0.3984}$ | Test fitting |

All soil types | Surface point source [33] | $H=4.01{K}_{s}^{0.242}{d}^{0.105}{t}^{0.313}+0.5D$ $D=3.77{K}_{s}^{0.138}{d}^{0.089}{t}^{0.48{K}_{s}^{0.135}{D}^{0.117}}$ | ||

Loam, sandy loam | Point source [34] | $H=2.3675{t}^{-0.00691}{V}_{m}{}^{0.2623}$ $D=1.2460{t}^{-0.05076}{V}_{m}{}^{0.3124}$ | ||

All soil types | Surface [35] | $H=0.0625\text{}{t}^{0.2562}{Q}^{0.2716}{p}_{b}{}^{-0.0255}{\theta}_{i}{}^{0.1112}$ ${K}_{s}{}^{0.335}{S}^{0.6303}{S}_{i}{}^{0.1222}{C}^{0.6028}$ $D=6.3555\text{}{t}^{0.3903}{Q}^{0.324}{p}_{b}{}^{1.8315}{\theta}_{i}{}^{0.0198}$ ${K}_{s}{}^{-0.084}{S}^{-0.1917}{S}_{i}{}^{0.1105}{C}^{-0.4265}$ | ||

All soil types | Surface [38] | $d=1.32{D}^{0.35}{Q}^{0.33}{K}_{s}{}^{-0.33}$ | ||

Loam, silty and sandy loam | Subsurface line source [22] | $H=3.9{K}_{s}{}^{0.25}{\left({\theta}_{s}-{\theta}_{i}\right)}^{0.14}{L}^{0.14}{D}^{0.08}{t}^{0.34}$ $D=8.1{K}_{s}{}^{0.52}{\left({\theta}_{s}-{\theta}_{i}\right)}^{0.35}{L}^{0.08}{D}^{0.18}{t}^{0.43}+L/2$ |

_{s}is the saturated hydraulic conductivity (cm/s); t is the irrigation time (h); θ

_{i}is the initial water content (cm

^{3}/cm

^{3}); θ

_{f}is the water content near the wetting front (cm

^{3}/cm

^{3}); θ

_{s}is the saturated water content (cm

^{3}/cm

^{3}); P

_{b}is the soil bulk weight (g/cm

^{3}); OM is the percentage of organic matter (%); d is the infiltration source diameter; L is the line source length; S, S

_{i}, and C are the percentages of sand, silt, and clay in the soil (%), respectively; S

_{e}is the relative saturation of soil (cm

^{3}/cm

^{3}); α and n denote fitting parameters, m = 1 − 1/n, n > 1; α denotes the parameter related to the physical properties of the soil (cm

^{−1}); h denotes the substrate potential (cm); l denotes the connectivity parameter, usually taken as 0.5.

## 2. Materials and Methods

#### 2.1. Infiltration Experiment of Aeolian Sand

#### 2.1.1. Experimental Soil Characteristics

#### 2.1.2. Experimental Setup

#### 2.1.3. Experimental Trials

#### 2.2. Model Establishment Method

#### 2.2.1. Wetting Front Distance Model

_{s}of aeolian sandy soil; (5) irrigation flow Q; and (6) total irrigation volume V. The derivation of the dimensionless formulas for each of the point and line source infiltration sources are also provided:

_{i}and n

_{j}(i,j = 1,…,6) are the equation coefficients and power exponents, respectively.

_{i}and n

_{j}(i,j = 1,…,6) in Equation (3). Finally, the parameters A

_{i}and n

_{j}(i,j = 1,…,6) were substituted into Equations (4) and (5) to derive the wetting front transport distance model.

#### 2.2.2. Wetting Body Elliptical Model

#### 2.2.3. Model Evaluation

_{i}denotes the ith experimental observation, S

_{i}denotes the ith model prediction, ${\overline{O}}_{i}$ denotes the average of experimental observations, and n is the number of individuals.

#### 2.2.4. Numerical Model

## 3. Results

#### 3.1. Wetting Front Distribution

#### 3.2. Proposed Model

#### 3.2.1. Wetting Front Distance Model

_{i}and power exponent n

_{j}(i,j = 1,…,6) values for each dimensionless transport distance, ${U}^{*}$, ${D}^{*}$, ${H}^{*}$, ${\overline{U}}^{*}$, ${\overline{D}}^{*},\text{}\mathrm{and}\text{}{\overline{H}}^{*}$, and total irrigation volume ${V}^{*}\text{}\mathrm{and}\text{}{\overline{V}}^{*}$ are presented in Figure 5 and Figure 6 and Table 3. The equation coefficients A

_{i}and power exponents n

_{j}(i,j = 1,…,6) are imported into Equations (4) and (5) to yield the complete transport distance, as presented in Equations (15) and (16).

#### 3.2.2. Wetting Body Elliptical Model

_{3}, A

_{5}, n

_{3}, and n

_{5}of models H and $\overline{D}$ into Equation (7), the horizontal and vertical coordinates of a wetted front profile point at any time can be obtained.

## 4. Discussion

#### 4.1. Wetting Front Distribution

_{s}(Equation (18)). In this relationship, the wetting body’s maximum width increases with the increase in the flow rate. For this, we substituted the test parameters into Zur’s model to calculate and draw a comparison. Table 6 shows the comparison results. At the same maximum depth, the test value shows a significant gap compared to Zur’s model. The maximum width in the test was only 36–55% of Zur’s model. Such differences were increased further after setting the interval irrigation, which may be caused by differences in surface and subsurface drip irrigation activities and soil sample parameters. Therefore, we proved that the irrigation flow affects the water movement in the aeolian sand. When the flow is slower, the time to reach the same irrigation volume is longer, resulting in a longer water potential difference duration between the outflow boundary and wetting front.

#### 4.2. Comparison with a Numerical Model

_{s}, subsurface irrigation flow Q, and total subsurface irrigation V. To discuss the differences between the present model and the numerical method, HYDRUS-2D/3D numerical simulation software was selected for the validation analysis conducted in this study. The hydraulic characteristic parameters of the test aeolian sandy soil samples presented in Table 7 were calculated using the transfer function method (Rosetta model) in HYDRUS-2D/3D based on the particle gradation of the soil. The transport of the wetting front of the wind-deposited sand under subsurface irrigation conditions was simulated by the software, and the numerical simulation results were compared with the predicted values of the constructed model and experimental measured values.

_{s}, irrigation flow Q, and volume V, were needed to calculate the wetting front range more accurately.

## 5. Conclusions

_{s}, irrigation flow Q, and total irrigation volume V. It ould estimate the wetting front transport pattern of aeolian sandy soil under subsurface irrigation conditions. The model proposed in this study can guide the design of irrigation systems for desert roads in planted sand control areas.

- (1)
- The aeolian sandy soil wetting front shape resembles a “bowl”. At the same irrigation volume, the wetting front shape is more significant when the flow rate is low, while at the same irrigation time, the wetting front shape is more prominent when the flow rate is significant. As the time increases, the wetting front transport distance increment first increases and then decreases, and the transport direction is mainly horizontal and then vertical.
- (2)
- The point source infiltration model (Equation (12)) is more suitable for calculating the wetting front transport distance in the horizontal direction. The line source infiltration model (Equation (13)) is more suitable for calculating the wetting front transport distances in the vertical direction. The model proposed in this study (Equation (14)) can accurately calculate the wetting front transport pattern of aeolian sandy soil with subsurface drip irrigation.
- (3)
- The prediction degree of the model is consistent with that of HYDRUS-2D/3D and shows a better fit than HYDRUS-2D/3D in the horizontal direction. In practical applications, the model parameters are easy to obtain and use.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Drip irrigation experiment steps. (

**a**) Schematic diagram of the drip irrigation process; (

**b**) grid line division.

**Figure 2.**Wetted body shape under different flow rates. (

**a**) Q = 0.5 L/h; (

**b**) Q = 1.0 L/h; (

**c**) Q = 1.5 L/h.

**Figure 4.**Movement distance in all directions under different flow rates. (

**a**) Downward; (

**b**) horizontal; (

**c**) upward.

**Figure 5.**Dimensionless relationship between each direction and volume during point source infiltration. (

**a**) Upward; (

**b**) downward; (

**c**) horizontal.

**Figure 6.**Dimensionless relationship between each direction and volume during line source infiltration. (

**a**) Upward; (

**b**) downward; (

**c**) horizontal.

**Figure 7.**Comparisons between predicted and observed point source infiltrations. (

**a**) Upward, Q = 1.0 L/h; (

**b**) upward, Q = 1.5 L/h; (

**c**) downward, Q = 1.0 L/h; (

**d**) downward, Q = 1.5 L/h; (

**e**) horizontal, Q = 1.0 L/h; (

**f**) horizontal, Q = 1.5 L/h.

**Figure 8.**Comparisons between the predicted and observed line source infiltration. (

**a**) Upward, Q = 1.0 L/h; (

**b**) upward, Q = 1.5 L/h; (

**c**) downward, Q = 1.0 L/h; (

**d**) downward, Q = 1.5 L/h; (

**e**) horizontal, Q = 1.0 L/h; (

**f**) horizontal, Q = 1.5 L/h.

**Figure 9.**Comparisons of the predicted and observed values of the models at different angles. (

**a**) Q = 1.0 L/h, 30°; (

**b**) Q = 1.0 L/h, 60°; (

**c**) Q = 1.5 L/h, 30°; (

**d**) Q = 1.5 L/h, 60°.

**Figure 10.**Comparison between the simulated and observed vertical, downward and horizontal movement distances at different flows. (

**a**) 1 L/h, vertical downward; (

**b**) 1 L/h, horizontal; (

**c**) 1.5 L/h, vertical downward; (

**d**) 1.5 L/h, horizontal.

Soil | Particle Size (mm) | Saturated Water Content ${\mathsf{\theta}}_{\mathit{s}}$ (%) | Saturated Hydraulic Conductivity K_{s} (cm/s) | Soil Bulk Density (g/cm^{3}) | ||
---|---|---|---|---|---|---|

>0.075 (%) | 0.075~0.002 (%) | <0.002 (%) | ||||

Aeolian sandy soil | 90 | 2.1 | 7.9 | 41.54 | 2.37 × 10^{−3} | 1.565 |

**Table 3.**Point and line source infiltration characteristic model parameters for wetting body aeolian sand.

Point source infiltration | A_{1} | n_{1} | A_{2} | n_{2} | A_{3} | n_{3} |

0.5096 | 0.2273 | 2.4990 | 0.4922 | 1.5700 | 0.3128 | |

Line source infiltration | A_{4} | n_{4} | A_{5} | n_{5} | A_{6} | n_{6} |

0.1985 | 0.2273 | 1.3440 | 0.4922 | 0.6784 | 0.3128 |

Model | MAE (cm) | RMSE (cm) | PBIAS (%) | NSE (−) |
---|---|---|---|---|

Downward | 0.50 | 0.61 | 1.007 | 0.99 |

Horizontal | 1.16 | 1.39 | −5.89 | 0.94 |

**Table 5.**Statistical characteristics analysis of the predicted and observed values of the models at different angles.

Model | MAE (cm) | RMSE (cm) | PBIAS (%) | NSE (−) | |
---|---|---|---|---|---|

1.0 L/h | 30° | 0.70 | 0.95 | 1.28 | 0.98 |

60° | 0.83 | 0.95 | −3.01 | 0.98 | |

1.5 L/h | 30° | 0.71 | 0.95 | 1.37 | 0.98 |

60° | 0.88 | 0.96 | −3.47 | 0.99 |

Flowrate | Q = 0.5 L/h | Q = 1.0 L/h | Q = 1.5 L/h | |||
---|---|---|---|---|---|---|

Wetted Body Scale | Maximum Width d | Maximum Depth D | Maximum Width d | Maximum Depth D | Maximum Width d | Maximum Depth D |

Test | 47.5 | 34.6 | 46.6 | 32.9 | 44.2 | 32.2 |

Zur’s model | 17.5 | 34.6 | 21.6 | 32.9 | 24.5 | 32.2 |

Soil | Saturated Water Content θ_{s}(cm ^{3}/cm^{3}) | Residual Water Content θ_{r}(cm ^{3}/cm^{3}) | Parameter α | Parameter n | Saturated Hydraulic Conductivity (cm/s) |
---|---|---|---|---|---|

Aeolian sandy soil | 0.415 | 0.025 | 0.292 | 2.428 | 3.02 × 10^{−3} |

**Table 8.**Statistical characteristics analysis of the simulated and observed values of the two models.

Model | Irrigation Flow (L/h) | Wetting Front Direction | MAE (cm) | RMSE (cm) | NSE (−) |
---|---|---|---|---|---|

Hydrus 3D | 1.0 | Horizontal | 2.53 | 2.69 | 0.84 |

Downward | 1.05 | 1.16 | 0.98 | ||

1.5 | Horizontal | 3.06 | 3.10 | 0.68 | |

Downward | 0.54 | 0.62 | 0.99 | ||

Proposed model | 1.0 | Horizontal | 1.00 | 1.20 | 0.95 |

Downward | 0.75 | 0.83 | 0.98 | ||

1.5 | Horizontal | 1.16 | 1.39 | 0.94 | |

Downward | 0.50 | 0.61 | 0.99 |

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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Qiao, W.; Luo, Z.; Lin, D.; Zhang, Z.; Wang, S.
An Empirical Model for Aeolian Sandy Soil Wetting Front Estimation with Subsurface Drip Irrigation. *Water* **2023**, *15*, 1336.
https://doi.org/10.3390/w15071336

**AMA Style**

Qiao W, Luo Z, Lin D, Zhang Z, Wang S.
An Empirical Model for Aeolian Sandy Soil Wetting Front Estimation with Subsurface Drip Irrigation. *Water*. 2023; 15(7):1336.
https://doi.org/10.3390/w15071336

**Chicago/Turabian Style**

Qiao, Wei, Zhihua Luo, Daming Lin, Zhongjian Zhang, and Songjiang Wang.
2023. "An Empirical Model for Aeolian Sandy Soil Wetting Front Estimation with Subsurface Drip Irrigation" *Water* 15, no. 7: 1336.
https://doi.org/10.3390/w15071336