# A Simplified Infiltration Model for Predicting Cumulative Infiltration during Vertical Line Source Irrigation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. VLSI Modeling

#### 2.1.1. Governing Equation

^{3}∙cm

^{−3}), t is the time (min), r is the radial (horizontal) coordinate (cm), K(h) is the unsaturated hydraulic conductivity (cm∙min

^{−1}), h is the soil water pressure head (cm), and z is the vertical coordinate that is positive downward (cm).

_{e}is the effective degree of saturation, θ

_{s}and θ

_{r}are respectively the saturated and residual water contents (cm

^{3}∙cm

^{−3}), α is an empirical parameter (cm

^{−1}) that is inversely related to the air entry value, and m and n are empirical constants affecting the shape of the retention curve. The value of m is restricted by m = 1−1/n.

_{e}was described using the closed form equation of van Genuchten, which combines the analytical expression (2) with the pore size distribution model of Mualem [27]:

_{s}is the saturated hydraulic conductivity (cm∙min

^{−1}), and l is the pore connectivity parameter estimated by Mualem to be about 0.5 as an average for many soils.

#### 2.1.2. Modeled Scenarios

#### 2.1.3. Initial and Boundary Conditions

#### 2.2. Analytic Method

^{−0.5}), and A is the steady infiltration rate (mL∙min

^{−1}).

#### 2.3. Error Analysis

## 3. Results and Discussion

#### 3.1. Different Factors Affecting Cumulative Infiltration of Vertical Line Source Irrigation

#### 3.1.1. Effect of Initial SWC on Cumulative Infiltration

^{3}·cm

^{−3}, respectively [30,31]. The cumulative infiltration curves at different initial SWC levels are shown in Figure 2. From the figure, it is clear that the initial SWC had little effect on the cumulative infiltration dynamics of vertical line source irrigation. With increasing SWC, the water potential gradient only slightly decreased, leading to a slight decrease in cumulative infiltration. Therefore, the impacts of initial SWC could be ignored in vertical line source irrigation research.

#### 3.1.2. Effect of Tube Burial Depth on Cumulative Infiltration

#### 3.1.3. Effect of Tube Seepage Area on Cumulative Infiltration

_{a}is the seepage area (cm

^{2}), D is the line source diameter (cm), and L is the line source length (cm).

_{a}, D and L values was simulated at a burial depth of 35 cm, initial SWC of 50% field water capacity, and irrigation quota of 40 L. The influence of several selected soil texture classes, S

_{a}, D and L on the cumulative infiltration is shown in Figure 4. Water moves faster through coarse-grained (sandy) soil with larger pores, compared to its movement through fine-grained (clayey) soil with smaller pores. For all treatments, the cumulative infiltration increased with an increase in S

_{a}. From the above analyses, the effects of S

_{a}should be taken into account in vertical line source irrigation research.

#### 3.2. Establishment of a Simplified Model

^{2}) were all larger than 0.95, indicating that the Philip model can adequately describe the relationship between cumulative infiltration and duration.

_{a}, as shown in Equation (9). The cumulative infiltration increased as S

_{a}increased. Further analyses of the relationship of S

_{a}to S and A are shown in Figure 5.

_{a}in an approximately linear way. Thus, Equations (10) and (11) are proposed to describe these relationships:

^{−0.5}), A is the steady infiltration rate (mL∙min

^{−1}), S

_{a}is seepage area (cm

^{2}), and a, b, c, and d are the fitting parameters.

#### 3.3. Evaluation of the Simplified Model

## 4. Conclusions

_{a}) significantly affect the cumulative infiltration and increase with S

_{a}. Furthermore, we proposed a simplified method for predicting the cumulative infiltration for vertical line source irrigation based on the Philip model. Finally, we conducted a comparative analysis of simulations and experiments using the following four statistical measures: mean absolute error (MAE), root mean squared error (RMSE), percent bias (PBIAS), and Nash-Sutcliffe efficiency (NSE). With a low MAE of 0.028–0.480 L, a low RMSE of 0.043–0.908 L, a good PBIAS range (PBIAS < ±1.0) and a great Nash-Sutcliffe coefficient close to 1.0 (NSE ≥ 0.995). This suggests that the predicted cumulative infiltration with simplified method was in a very good agreement with the observed values. For relatively homogeneous soil conditions, the model can be used by irrigation systems designers to estimate cumulative infiltration with irrigation emitter parameters of diameter (D) and length (L). It has to be noted that further research is needed to evaluate such empirical models under in field conditions, where other important factors, such as soil layering, may significantly affect water flow and distribution. In addition, only five soil types were tested in this study, the relationship between model fitting parameters and other soil textures still needs to be explored.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Deng, X.P.; Shan, L.; Zhang, H.; Turner, N.C. Improving agricultural water use efficiency in arid and semiarid areas of China. Agric. Water Manag.
**2006**, 80, 23–40. [Google Scholar] [CrossRef] - Du, S.; Kang, S.; Li, F.; Du, T. Water use efficiency is improved by alternate partial root-zone irrigation of apple in arid Northwest China. Agric. Water Manag.
**2017**, 179, 184–192. [Google Scholar] [CrossRef] - Zeng, C.; Wang, Q.; Fan, J. Effect of initial water content on vertical line-source infiltration characteristics of soil. Trans. Chin. Soc. Agric. Eng.
**2010**, 26, 24–30. (In Chinese) [Google Scholar] - Cheng, H.J.; Wang, Q.J.; Bai, Y.G.; Cai, J.S.; Zeng, C. Influence of line source length of vertical line source irrigation on wetted soil change characteristics. Trans. Chin. Soc. Agric. Eng.
**2010**, 26, 32–37. (In Chinese) [Google Scholar] - Khatri, K.L.; Smith, R.J. Real-time prediction of soil infiltration characteristics for the management of furrow irrigation. Irrig. Sci.
**2006**, 25, 33–43. [Google Scholar] [CrossRef] - Bamutaze, Y.; Tenywa, M.M.; Majaliwa, M.J.G.; Vanacker, V.; Bagoora, F.; Magunda, M.; Obandog, J.; Wasigeh, J.E. Infiltration characteristics of volcanic sloping soils on Mt. Elgon, Eastern Uganda. Catena
**2010**, 80, 122–130. [Google Scholar] [CrossRef] - Bai, D.; He, J.; Guo, L.; Wang, X.; Liang, Z. Infiltration characteristics of vertical tube sub-irrigation as affected by various factors. Trans. Chin. Soc. Agric. Eng.
**2016**, 32, 101–105. (In Chinese) [Google Scholar] - Li, Y.B.; Fan, Y.W.; Liu, Y.; Ma, X.Y. Influencing Factors and Simplified Model of Film Hole Irrigation. Water
**2017**, 9, 543. [Google Scholar] [CrossRef] - El-Nesr, M.N.; Alazba, A.A.; Šimůnek, J. HYDRUS simulations of the effects of dual-drip subsurface irrigation and a physical barrier on water movement and solute transport in soils. Irrig. Sci.
**2014**, 32, 111–125. [Google Scholar] [CrossRef] - Boštjan, N.; Kechavarzi, C.; Coulon, F.; Pintar, M. Numerical investigation of the influence of texture, surface drip emitter discharge rate and initial soil moisture condition on wetting pattern size. Irrig. Sci.
**2014**, 32, 421–436. [Google Scholar] - Huang, M.; Barbour, S.L.; Elshorbagy, A.; Zettl, J.D.; Si, B.C. Infiltration and drainage processes in multi-layered coarse soils. Can. J. Soil Sci.
**2011**, 91, 185–197. [Google Scholar] [CrossRef] - Khoshravesh-Miangoleh, M.; Kiani, R.A. Effect of magnetized water on infiltration capacity of different soil textures. Soil Use Manag.
**2014**, 30, 588–594. [Google Scholar] [CrossRef] - Li, Z.; Wu, P.; Feng, H.; Zhao, X.; Huang, J.; Zhuang, W. Simulated experiment on effect of soil bulk density on soil infiltration capacity. Trans. Chin. Soc. Agric. Eng.
**2009**, 25, 40–45. (In Chinese) [Google Scholar] - Yang, J.L.; Zhang, G.L. Water infiltration in urban soils and its effects on the quantity and quality of runoff. J. Soils Sediments
**2011**, 11, 751–761. [Google Scholar] [CrossRef] - León, J.; Echeverría, M.T.; Martí, C.; Badía, D. Can ash control infiltration rate after burning? An example in burned calcareous and gypseous soils in the Ebro Basin (NE Spain). Catena
**2015**, 135, 377–382. [Google Scholar] [CrossRef] - Liu, H.; Lei, T.W.; Zhao, J.; Yuan, C.P.; Fan, Y.T.; Qu, L.Q. Effects of rainfall intensity and antecedent soil water content on soil infiltrability under rainfall conditions using the run off-on-out method. J. Hydrol.
**2011**, 396, 24–32. [Google Scholar] [CrossRef] - Stewart, R.D.; Rupp, D.E.; Najm, M.R.A.; Selker, J.S. Modeling effect of initial soil moisture on sorptivity and infiltration. Water Resour. Res.
**2013**, 49, 7037–7047. [Google Scholar] [CrossRef] - Cheng, H. Experimental Studies on the Characteristics Soil of Water Moisture Movement and Growth Grapevine under Vertical Line Source Irrigation. Master’s Thesis, Xi’an University of Technology, Xi’an, China, 2010. (In Chinese). [Google Scholar]
- Patel, N.; Rajput, T.B.S. Effect of drip tape placement depth and irrigation level on yield of potato. Agric. Water Manag.
**2007**, 88, 209–223. [Google Scholar] [CrossRef] - Šimůnek, J.; Genuchten, M.T.; Šejna, M. Development and applications of the HYDRUS and STANMOD software packages and related codes. Vadose Zone J.
**2008**, 7, 587–600. [Google Scholar] [CrossRef] - Šimůnek, J.; Van Genuchten, M.T.; Šejna, M. Recent developments and applications of the HYDRUS computer software packages. Vadose Zone J.
**2016**, 15, 1–25. [Google Scholar] [CrossRef] - Saito, H.; Simunek, J.; Scanlon, B.R.; Reedy, R.C. Numerical Analysis of Coupled Water, Vapor and Heat Transport in the Vadose Zone using HYDRUS. Vadose Zone J.
**2006**, 5, 784–800. [Google Scholar] [CrossRef] - Skaggs, T.H.; Trout, T.J.; Šimůnek, J.; Shouse, P.J. Comparison of HYDRUS-2D Simulations of Drip Irrigation with Experimental Observations. J. Irrig. Drain. Eng.
**2004**, 130, 304–310. [Google Scholar] [CrossRef] - Li, S.; Wang, Q. Simulation of Soil Water Distribution under Vertical Line Source Infiltration. Trans. Chin. Soc. Agric. Mach.
**2011**, 42, 51–57. (In Chinese) [Google Scholar] - Šimůnek, J.; Van Genuchten, M.T.; Šejna, M. The HYDRUS Software Package for Simulating the Two-and Three-Dimensional Movement of Water, Heat, and Multiple Solutes in Variably Saturated Media; Technical Manual, Version 1.0; PC Progress: Prague, Czech Republic, 2006; p. 241. [Google Scholar]
- Van Genuchten, M.T. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J.
**1980**, 44, 892–898. [Google Scholar] [CrossRef] - Mualem, Y. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res.
**1976**, 12, 513–522. [Google Scholar] [CrossRef] - Zhang, L.J.; Wang, Y.K.; Xin, X.G.; Li, H.Y. Appropriate surge spring root irrigation layout and irrigation norm of jujube on mountain land. J. Northwest A F Univ.
**2010**, 38, 211–217. (In Chinese) [Google Scholar] - Carsel, R.F.; Parrish, R.S. Developing joint probability distributions of soil water retention characteristics. Water Resour. Res.
**1988**, 24, 755–769. [Google Scholar] [CrossRef] - Jiang, P.; Lei, T.; Liu, X.; Yang, W.; Xin, L.; Wang, Q. Principles and experimental verification of capillary suction method for fast measurement of field capacity. Trans. Chin. Soc. Agric. Eng.
**2006**, 22, 1–5. (In Chinese) [Google Scholar] - Rab, M.A.; Chandra, S.; Fisher, P.D.; Robinson, N.J.; Kitching, M.; Aumann, C.D.; Imhof, M. Modelling and prediction of soil water contents at field capacity and permanent wilting point of dryland cropping soils. Soil Res.
**2011**, 49, 389–407. [Google Scholar] [CrossRef] - Philip, J.R. The Theory of Infiltration: 4. Sorptivity and Algebraic Infiltration Equations. Soil Sci.
**1957**, 84, 257–264. [Google Scholar] [CrossRef] - Singh, D.K.; Rajput, T.B.S.; Singh, D.K.; Sikarwar, H.S.; Sahoo, R.N.; Ahmad, T. Simulation of soil wetting pattern with subsurface drip irrigation from line source. Agric. Water Manag.
**2006**, 83, 130–134. [Google Scholar] [CrossRef] - Moriasi, D.; Arnold, J.; Van Liew, M.W.; Veith, T.L. Model evaluation guidelines for systematic quantification of accuracy in watershed simulations. Trans. ASABE
**2007**, 50, 885–900. [Google Scholar] [CrossRef]

**Figure 2.**Effect of initial SWC on cumulative infiltration from a vertical line source irrigation (for different initial soil moisture expressed as the percentage of field capacity). (

**a**) clay loam; (

**b**) silt loam; (

**c**) loam; (

**d**) sandy loam; and (

**e**) sand.

**Figure 3.**Effect of tube burial depth on cumulative infiltration from a vertical line source irrigation. (

**a**) clay loam; (

**b**) silt loam; (

**c**) loam; (

**d**) sandy loam; and (

**e**) sand.

**Figure 4.**Effect of tube seepage area, diameter and length on vertical line source irrigation. In the legend, the first values are the seepage area (S

_{a}), cm

^{2}, the second values are the diameter (D), cm, and the third values are the length (L), cm. (

**a**) silt loam; (

**b**) loam; and (

**c**) sandy loam.

**Figure 5.**Effect of tube seepage area on vertical line source irrigation. (

**a**,

**b**) soil water sorptivity; (

**c**,

**d**) steady infiltration rate. S1 = clay loam; S2 = silt loam; S3 = loam; S4 = sandy loam; and S5 = sand.

**Figure 6.**Comparison of simulated and estimated S and A. (

**a**) soil water sorptivity; (

**b**,

**c**) steady infiltration rate. S1 = clay loam; S2 = silt loam; S3 = loam; S4 = sandy loam; S5 = sand; and L0 = 1:1 line.

**Figure 7.**Comparison of measured data and model predictions. (

**a**) Minqin sandy loam; and (

**b**) Minqin aeolian sand. M1 = measured value (D = 4 cm and L = 25 cm); P1 = predicted value (D = 4 cm and L = 25 cm); M2 = measured value (D = 4 cm and L = 35 cm); and P2 = predicted value (D = 4 cm and L = 35 cm);

**Figure 8.**Comparison of the calculated values and observed values of cumulative infiltration of different soils. (

**a**): Minqin sandy loam; (

**b**): Minqin aeolian sand; (

**c**): Shanshan clay loam.

Diameter (cm) | Length (cm) | Seepage Area (cm^{2}) | Clay Loam | Silt Loam | Loam | Sandy Loam | Sand | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

S | A | S | A | S | A | S | A | S | A | |||

2 | 10 | 63 | 24.21 | 1.02 | 33.09 | 2.33 | 34.96 | 4.22 | 43.87 | 13.91 | 51.39 | 81.17 |

15 | 94 | 33.89 | 1.28 | 46.95 | 2.95 | 50.36 | 5.38 | 66.19 | 17.75 | 87.34 | 104.66 | |

20 | 126 | 43.70 | 1.52 | 60.77 | 3.54 | 65.89 | 6.49 | 88.95 | 21.43 | 129.49 | 124.90 | |

25 | 157 | 53.60 | 1.75 | 74.86 | 4.10 | 81.67 | 7.56 | 112.62 | 24.95 | 174.12 | 146.14 | |

30 | 188 | 63.82 | 1.95 | 89.14 | 4.60 | 98.12 | 8.55 | 137.98 | 28.13 | 228.90 | 165.76 | |

3 | 10 | 94 | 28.07 | 1.14 | 39.53 | 2.66 | 43.95 | 4.90 | 57.65 | 16.66 | 66.96 | 100.63 |

15 | 141 | 39.47 | 1.41 | 56.12 | 3.33 | 63.46 | 6.15 | 85.62 | 21.06 | 114.16 | 127.13 | |

20 | 188 | 51.01 | 1.67 | 73.11 | 3.95 | 83.00 | 7.36 | 115.46 | 25.07 | 167.27 | 150.20 | |

25 | 236 | 62.39 | 1.92 | 89.91 | 4.57 | 102.92 | 8.51 | 146.47 | 28.85 | 228.08 | 173.34 | |

30 | 283 | 74.33 | 2.12 | 107.35 | 5.08 | 123.59 | 9.50 | 178.67 | 32.36 | 289.28 | 196.40 | |

4 | 10 | 126 | 33.48 | 1.30 | 48.43 | 3.05 | 55.15 | 5.64 | 73.43 | 19.52 | 89.46 | 119.63 |

15 | 188 | 46.93 | 1.60 | 68.44 | 3.75 | 79.03 | 6.98 | 108.75 | 24.20 | 139.57 | 150.41 | |

20 | 251 | 60.15 | 1.89 | 88.98 | 4.42 | 103.18 | 8.25 | 145.54 | 28.48 | 231.22 | 169.96 | |

25 | 314 | 74.13 | 2.14 | 109.39 | 5.05 | 127.80 | 9.43 | 183.33 | 32.61 | 279.36 | 200.83 | |

30 | 377 | 88.13 | 2.35 | 130.34 | 5.58 | 152.95 | 10.48 | 223.24 | 36.15 | 353.32 | 223.69 | |

5 | 10 | 157 | 37.62 | 1.51 | 54.85 | 3.51 | 64.42 | 6.45 | 89.28 | 22.39 | 108.58 | 139.51 |

15 | 236 | 52.31 | 1.84 | 76.76 | 4.31 | 91.62 | 7.91 | 131.77 | 27.26 | 174.43 | 171.80 | |

20 | 314 | 67.18 | 2.15 | 98.95 | 5.08 | 119.25 | 9.28 | 175.19 | 31.93 | 244.54 | 200.53 | |

25 | 393 | 82.48 | 2.44 | 121.37 | 5.80 | 147.27 | 10.57 | 221.34 | 36.03 | 331.86 | 227.52 | |

30 | 471 | 97.97 | 2.66 | 144.54 | 6.37 | 176.23 | 11.66 | 267.90 | 39.90 | 435.52 | 250.93 | |

6 | 10 | 188 | 48.13 | 1.57 | 69.93 | 3.69 | 73.04 | 7.28 | 105.56 | 25.17 | 132.16 | 159.15 |

15 | 283 | 65.59 | 1.90 | 96.68 | 4.48 | 102.91 | 8.86 | 154.40 | 30.38 | 201.64 | 194.72 | |

20 | 377 | 84.37 | 2.18 | 123.78 | 5.20 | 133.32 | 10.34 | 205.64 | 35.06 | 288.74 | 224.75 | |

25 | 471 | 102.50 | 2.45 | 150.73 | 5.90 | 164.26 | 11.72 | 256.96 | 39.65 | 394.96 | 251.61 | |

30 | 565 | 120.73 | 2.67 | 178.33 | 6.44 | 196.17 | 12.91 | 312.54 | 43.51 | 506.28 | 278.22 |

Soil Texture | a | b | c | d |
---|---|---|---|---|

Minqin sandy loam | 0.521 | 29.91 | 0.017 | 10.48 |

Minqin aeolian sand | 1.181 | 21.38 | 0.183 | 62.65 |

Soil | MAE (L) | RMSE (L) | PBIAS (%) | NSE |
---|---|---|---|---|

Sandy loam from Hexi Corridor | 0.480 | 0.908 | 0.444 | 0.995 |

Aeolian sand from Hexi Corridor | 0.428 | 0.642 | 0.321 | 0.997 |

Clay loam from Turpan Depression | 0.028 | 0.043 | 0.900 | 1.000 |

© 2018 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fan, Y.; Huang, N.; Gong, J.; Shao, X.; Zhang, J.; Zhao, T.
A Simplified Infiltration Model for Predicting Cumulative Infiltration during Vertical Line Source Irrigation. *Water* **2018**, *10*, 89.
https://doi.org/10.3390/w10010089

**AMA Style**

Fan Y, Huang N, Gong J, Shao X, Zhang J, Zhao T.
A Simplified Infiltration Model for Predicting Cumulative Infiltration during Vertical Line Source Irrigation. *Water*. 2018; 10(1):89.
https://doi.org/10.3390/w10010089

**Chicago/Turabian Style**

Fan, Yanwei, Ning Huang, Jiaguo Gong, Xiaoxia Shao, Jie Zhang, and Tong Zhao.
2018. "A Simplified Infiltration Model for Predicting Cumulative Infiltration during Vertical Line Source Irrigation" *Water* 10, no. 1: 89.
https://doi.org/10.3390/w10010089