Abstract

Numerical simulations were carried out with HYDRUS-2D to investigate the influence of soil texture, initial water content, film hole diameter and water depth on cumulative infiltration from a film hole. Soil texture, film hole diameter and water depth are the dominant influencing factors. Philip infiltration model was used to fit the simulated results of the dominant influencing factors. For the same soil, the sorptivity (s) is a power function relationship with film hole diameter, while the steady infiltration rate (a) is a power function relationship with film hole diameter and water depth. On that basis, the calculation formulas for predicting s and a were established, and a simplified film hole infiltration model including the film hole diameter and water depth was proposed. The effectiveness of the model was verified by laboratory experiments and literature data. The predicted values of the model were in good agreement with the experimental observations. The model parameters can be determined only by a set of film hole infiltration experiment, which simplifies the experimental design and can be used as a tool for irrigation engineers or farmers to estimate cumulative infiltration.

cumulative infiltration, film hole infiltration, Philip model, sufficient water supply

INTRODUCTION

Plastic film mulching (PFM) is an important agricultural practice in arid and semi-arid areas of northwest China (Li et al. 2004a; Zhou et al. 2009). Film hole irrigation (FHI) has been also developed on the basis of PFM. FHI is a relatively new irrigation method that involves completely covering a bordered field with plastic film with uniform diameter holes (Saeed & Mahmood 2013; Li et al. 2017). When irrigating, water flows over the plastic film on the soil surface. There is a thin layer of water depth above the film hole, and water infiltrates into the soil through these holes on plastic film. Additionally, these infiltration holes are also seedling holes after germination of the crop. In this regard, FHI belongs to surface irrigation technology. Compared with traditional surface irrigation methods, FHI can effectively save water resources, improve irrigation uniformity and extend the length of surface irrigation (Wu et al. 2001; Li et al. 2005).

For crops with wide row and plant spacing, such as watermelon and melon, the wetting patterns of adjacent film holes do not overlap each other, which is macroscopically a single-point source infiltration. Several field and laboratory experiments have been carried out to study the influence of irrigation parameters (e.g. soil texture, initial water content, film hole diameter and water depth) on the single-point source infiltration under FHI (Jiao & Wang 1999; Wu et al. 2001). Wang et al. (2001) and Wu et al. (2003) used experimental data to establish an infiltration model based on Philip and kostiakov model, respectively. However, these empirical models contained multiple undetermined coefficients, and the experimental design was more complicated.

Numerical simulation is a fast and inexpensive approach to studying optimal irrigation management practices. The numerical model HYDRUS-2D developed by Šimůnek et al. (2016) has been widely used in soil water movement for various irrigation methods. Several investigators have used this model to evaluate either field or laboratory experiments, or other mathematical models (e.g. Skaggs et al. 2004; Provenzano 2007; Naglič et al. 2014; El-Nesr et al. 2014; Karandish & Šimůnek 2017). The HYDRUS model enables its users to trace the movement of water and solutes and the wetting patterns in both simple and complex geometries for homogeneous or heterogeneous soils and for different combinations of initial and boundary conditions. Ma et al. (2015) showed that the HYDRUS-2D model can be used to simulate soil water movement under film hole infiltration with good accuracy.

In this study, the main objectives are to simulate and analyze the influence of soil texture, initial water content, film hole diameter, and water depth on cumulative infiltration and identify the dominant influence factors, and to develop an empirical model for predicting the cumulative infiltration of FHI.

MATERIALS AND METHODS

Laboratory experiments

Figure 1 illustrates the equipment for the laboratory experiments. The soil container was made of 10 mm thick Plexiglas sheets. The bottom of the container had many holes with a diameter of 2 mm for ventilation. Plastic film covered the soil surface and left a ¼ film hole at one corner of container. Water on the plastic film can only penetrated into the soil through the film hole. Holes with a diameter of 2 cm and a spacing of 5 cm were located on the side of soil container near the film hole in order to extract soil and measure its water content after irrigation. Mariotte bottle was fixed in the height adjustable stand and provides a constant head for the film hole through a rubber hose.

Figure 1
Schematic diagram of experimental equipment. h represents water depth above the plastic film, and d represents film hole diameter.
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Schematic diagram of experimental equipment. h represents water depth above the plastic film, and d represents film hole diameter.

Figure 1
Schematic diagram of experimental equipment. h represents water depth above the plastic film, and d represents film hole diameter.
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Schematic diagram of experimental equipment. h represents water depth above the plastic film, and d represents film hole diameter.

Numerical modeling

Soil water movement under FHI is an axisymmetric two-dimensional infiltration process. We simulated water infiltration using HYDRUS-2D (Šimůnek et al. 2006). Assuming a homogeneous and isotropic soil, the governing equation for water flow is the 2D Richards equation, which can be written in axisymmetric coordinates as follows:  
formula
(1)
where θ is volumetric water content (cm3/cm3), t is time (min), r is radial coordinate (cm), K(φ) is unsaturated hydraulic conductivity (cm/min), φ is soil water pressure head (cm), and z is vertical coordinate that is positive downward (cm).
The soil hydraulic properties were modeled using the van Genuchten-Mualem (Mualem 1976; van Genuchten 1980) constitutive relationships,  
formula
(2)
 
formula
(3)
where  
formula
and where θs is saturated water contents (cm3/cm3), θr is residual water contents (cm3/cm3), Ks is saturated hydraulic conductivity (cm/min), Se is the effective degree of saturation, α is an empirical parameter (1/cm), and m and n are empirical constants (–).

Figure 2 shows the initial and boundary conditions in this study.

Figure 2
Initial and boundary conditions for film hole infiltration.
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Initial and boundary conditions for film hole infiltration.

Figure 2
Initial and boundary conditions for film hole infiltration.
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Initial and boundary conditions for film hole infiltration.

The initial conditions are as follows:  
formula
(4)
where θ0 is the initial water potential before irrigation (cm).
The boundary conditions are as follows:  
formula
(5)

Modeled scenarios

Given the realities that constrain film hole irrigation in the field, the film hole diameters is typically 3–8 cm, the water depth on film mulch is kept constant within a range of 2–10 cm, and the crops need to be irrigated when soil water content is about 50% available water (AW) (Fan et al. 2008).

To address our main objectives, we have chosen to evaluate the following alternative scenarios:

  • (1)

    Twelve soil textures. Soil hydraulic parameters for twelve textural classes were taken from the soil catalog provided by the HYDRUS software (Carsel & Parrish 1988), presented in Table 1.

  • (2)

    Three soil initial water contents (30%, 50%, and 70%AW). The influence of initial water content was investigated by varying the %AW. AW corresponds to the amount of water between field capacity (FC) and permanent wilting point (PWP). FC corresponds to water content at a soil matric potential of −10 kPa, and θr was used for the PWP (Li et al. 2004b; Naglič et al. 2014).

  • (3)

    Three irrigation water depths (2, 4 and 6 cm).

  • (4)

    Three film hole diameters (3, 5 and 7 cm).

Table 1

The van Genuchten–Mualem model parameters for simulated soils

Soil texture θr (cm3·cm−3) θs (cm3·cm−3) α (cm−1) n (−) Ks (cm·min−1) 
Silty clay 0.070 0.360 0.005 1.09 0.0003 
Silty clay loam 0.089 0.430 0.010 1.23 0.0012 
Sandy clay 0.100 0.380 0.027 1.23 0.0020 
Clay 0.068 0.380 0.008 1.09 0.0033 
Clay loam 0.095 0.410 0.019 1.31 0.0043 
Silt 0.034 0.460 0.016 1.37 0.0042 
Silt loam 0.067 0.450 0.020 1.41 0.0075 
Loam 0.078 0.430 0.036 1.56 0.0173 
Sandy clay loam 0.100 0.390 0.059 1.48 0.0218 
Sandy loam 0.065 0.410 0.075 1.89 0.0737 
Loamy sand 0.057 0.410 0.124 2.28 0.2432 
Sand 0.045 0.430 0.145 2.68 0.4950 
Soil texture θr (cm3·cm−3) θs (cm3·cm−3) α (cm−1) n (−) Ks (cm·min−1) 
Silty clay 0.070 0.360 0.005 1.09 0.0003 
Silty clay loam 0.089 0.430 0.010 1.23 0.0012 
Sandy clay 0.100 0.380 0.027 1.23 0.0020 
Clay 0.068 0.380 0.008 1.09 0.0033 
Clay loam 0.095 0.410 0.019 1.31 0.0043 
Silt 0.034 0.460 0.016 1.37 0.0042 
Silt loam 0.067 0.450 0.020 1.41 0.0075 
Loam 0.078 0.430 0.036 1.56 0.0173 
Sandy clay loam 0.100 0.390 0.059 1.48 0.0218 
Sandy loam 0.065 0.410 0.075 1.89 0.0737 
Loamy sand 0.057 0.410 0.124 2.28 0.2432 
Sand 0.045 0.430 0.145 2.68 0.4950 

Analytic method

The cumulative infiltration of HYDRUS was determined by the change in water content of each node. The relationship between cumulative infiltration and time were described using the Philip infiltration model (Philip 1957) and expressed as Equation (6).  
formula
(6)
where I is cumulative infiltration (mL), s is sorptivity (mL/min0.5), and a is steady infiltration rate (mL/min).

Statistical analysis

The root mean square error (RMSE) and modelling efficiency (EF) were adopted for further analysis to examine the performance of the simplified infiltration model against observed values. The expressions are:  
formula
(7)
 
formula
(8)
where Mi and Si are the observed and predicted values, respectively. Mmean is the mean of observed data, and n is the total number of observations.

RESULTS AND DISCUSSION

Different factors affecting cumulative infiltration of FHI

The influence of soil texture, initial water content (θ0), film hole diameter (d) and water depth (h) on cumulative infiltrations is shown in Figure 3. Soil texture has a very significant effect on the cumulative infiltration. For a given time, the cumulative infiltration is smaller for fine-textured soil and larger for coarse-textured soil. Film hole diameter and water depth have a great influence on the cumulative infiltration. As the film hole diameter and water depth increase, the infiltration increases. The initial water content has little effect on the cumulative infiltration, and the cumulative infiltration decreases with the increase of the initial water content. Based on the above analysis, we can conclude that the soil texture, film hole diameter, and water depth are dominant influencing factors.

Figure 3
Simulated cumulative infiltration for different influencing factors. (a) θ0 = 50%AW, h = 4 cm, d = 5 cm. (b) Loam, h = 4 cm, d = 5 cm. (c) Loam, θ0 = 50%AW, h = 4 cm. (d) Loam, θ0 = 50%AW, d = 5 cm.
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Simulated cumulative infiltration for different influencing factors. (a) θ0 = 50%AW, h = 4 cm, d = 5 cm. (b) Loam, h = 4 cm, d = 5 cm. (c) Loam, θ0 = 50%AW, h = 4 cm. (d) Loam, θ0 = 50%AW, d = 5 cm.

Figure 3
Simulated cumulative infiltration for different influencing factors. (a) θ0 = 50%AW, h = 4 cm, d = 5 cm. (b) Loam, h = 4 cm, d = 5 cm. (c) Loam, θ0 = 50%AW, h = 4 cm. (d) Loam, θ0 = 50%AW, d = 5 cm.
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Simulated cumulative infiltration for different influencing factors. (a) θ0 = 50%AW, h = 4 cm, d = 5 cm. (b) Loam, h = 4 cm, d = 5 cm. (c) Loam, θ0 = 50%AW, h = 4 cm. (d) Loam, θ0 = 50%AW, d = 5 cm.

Establishment of the simplified infiltration model

In this section, soil texture, water depth, and film hole diameter were simulated. The simulated cumulative infiltration data were fitted using Equation (6) to obtain infiltration parameters s and a, as shown in Table 2. For different combinations of dominant influencing factors, the coefficient of determination (R2) were all greater than 0.99, indicating that the relationship between cumulative infiltration and time can be expressed by Philip infiltration model.

Table 2

Fitted infiltration parameter values

Film hole diameter Water depth Soil texture
 
Silty clay
 		Silty clay loam
 		Sandy clay
 		Clay
 		Clay loam
 		Silt
 		Silt loam
 		Loam
 		Sandy clay loam
 		Sandy loam
 		Loamy sand
 		Sand
 
s a s a s a s a s a s a s a s a s a s a s a s a 
3 2 0.97 0.01 1.65 0.07 1.39 0.04 2.69 0.04 2.09 0.25 2.56 0.35 2.85 0.59 3.75 1.08 2.50 0.97 4.18 4.50 4.52 14.25 4.47 30.31 
3 4 0.97 0.01 1.66 0.07 1.35 0.06 2.70 0.05 2.07 0.29 2.54 0.39 2.78 0.67 3.65 1.34 2.12 1.51 4.35 6.01 5.62 18.85 5.99 39.52 
3 6 0.97 0.01 1.66 0.08 1.04 0.13 2.27 0.13 1.93 0.35 2.50 0.43 2.71 0.77 3.37 1.73 2.53 1.88 5.55 7.21 6.58 23.15 6.48 48.38 
5 2 1.89 0.01 3.50 0.08 2.93 0.05 5.29 0.02 4.42 0.39 5.50 0.54 6.20 0.95 8.42 1.78 5.26 1.82 10.05 8.00 11.43 25.66 11.53 54.64 
5 4 1.88 0.01 3.51 0.09 2.56 0.14 4.80 0.13 4.22 0.50 5.51 0.62 6.02 1.13 8.18 2.34 5.77 2.57 11.81 10.26 13.77 33.14 13.91 69.83 
5 6 1.85 0.02 3.49 0.11 2.41 0.24 4.10 0.33 3.66 0.72 5.20 0.75 5.33 1.46 7.86 3.01 6.44 3.22 13.46 12.35 15.37 40.29 15.98 84.43 
7 2 3.10 0.01 6.03 0.08 4.93 0.07 8.65 0.02 7.72 0.53 9.73 0.73 10.94 1.32 15.08 2.52 9.18 2.81 19.27 11.89 21.15 38.97 21.01 83.16 
7 4 3.06 0.01 6.03 0.09 4.81 0.12 8.34 0.12 7.53 0.62 9.61 0.79 10.75 1.46 14.99 2.83 9.96 3.12 19.99 12.94 22.85 42.28 23.32 89.85 
7 6 2.97 0.02 5.94 0.11 4.79 0.16 8.31 0.18 7.66 0.68 9.51 0.87 10.89 1.56 14.95 3.12 10.53 3.41 21.14 13.90 24.37 45.48 23.88 96.47 
Film hole diameter Water depth Soil texture
 
Silty clay
 		Silty clay loam
 		Sandy clay
 		Clay
 		Clay loam
 		Silt
 		Silt loam
 		Loam
 		Sandy clay loam
 		Sandy loam
 		Loamy sand
 		Sand
 
s a s a s a s a s a s a s a s a s a s a s a s a 
3 2 0.97 0.01 1.65 0.07 1.39 0.04 2.69 0.04 2.09 0.25 2.56 0.35 2.85 0.59 3.75 1.08 2.50 0.97 4.18 4.50 4.52 14.25 4.47 30.31 
3 4 0.97 0.01 1.66 0.07 1.35 0.06 2.70 0.05 2.07 0.29 2.54 0.39 2.78 0.67 3.65 1.34 2.12 1.51 4.35 6.01 5.62 18.85 5.99 39.52 
3 6 0.97 0.01 1.66 0.08 1.04 0.13 2.27 0.13 1.93 0.35 2.50 0.43 2.71 0.77 3.37 1.73 2.53 1.88 5.55 7.21 6.58 23.15 6.48 48.38 
5 2 1.89 0.01 3.50 0.08 2.93 0.05 5.29 0.02 4.42 0.39 5.50 0.54 6.20 0.95 8.42 1.78 5.26 1.82 10.05 8.00 11.43 25.66 11.53 54.64 
5 4 1.88 0.01 3.51 0.09 2.56 0.14 4.80 0.13 4.22 0.50 5.51 0.62 6.02 1.13 8.18 2.34 5.77 2.57 11.81 10.26 13.77 33.14 13.91 69.83 
5 6 1.85 0.02 3.49 0.11 2.41 0.24 4.10 0.33 3.66 0.72 5.20 0.75 5.33 1.46 7.86 3.01 6.44 3.22 13.46 12.35 15.37 40.29 15.98 84.43 
7 2 3.10 0.01 6.03 0.08 4.93 0.07 8.65 0.02 7.72 0.53 9.73 0.73 10.94 1.32 15.08 2.52 9.18 2.81 19.27 11.89 21.15 38.97 21.01 83.16 
7 4 3.06 0.01 6.03 0.09 4.81 0.12 8.34 0.12 7.53 0.62 9.61 0.79 10.75 1.46 14.99 2.83 9.96 3.12 19.99 12.94 22.85 42.28 23.32 89.85 
7 6 2.97 0.02 5.94 0.11 4.79 0.16 8.31 0.18 7.66 0.68 9.51 0.87 10.89 1.56 14.95 3.12 10.53 3.41 21.14 13.90 24.37 45.48 23.88 96.47 
For the same soil texture, s is mainly affected by d, while h has little effect on it, and a is related to d and h. Based on extensive verification, we found that the relationship between s and d satisfies the power function, while a is a power function relationship with d and h. Predictive model for s and a were proposed as below:  
formula
(9)
 
formula
(10)
where m1, m2, n1, n2, and n3 are the fitting parameters.
The m1 and n1 were obtained from fitting s using Equation (9) for each soil in Table 2. Similarly, the m2, n2, and n3 were determined by fitting a with Equation (10). We found that n1 changed little (from 1.35 to 1.73), and n2 and n3 varied greatly for fine-textured soils (silty clay, silty clay loam, sandy clay, and clay) and little for coarse-textured soils (n2 from 0.76 to 0.91, and n3 from 0.25 to 0.35). It can be seen from Equation (6) that I is affected by st0.5 and at. For fine-textured soils, s is much larger than a, and the effect of st0.5 on cumulative infiltration is much greater than at during irrigation time. Although the fitting results of a were not good, it had little effect on cumulative infiltration. To simplify the parameters, n1 can take the average value of twelve soils, and n2 and n3 can take the average value of coarse-textured soils, i.e. n1 = 1.6, n2 = 0.8, and n3 = 0.3. Therefore, Equations (9) and (10) can be further varied to:  
formula
(11)
 
formula
(12)
where k1 and k2 are the fitting parameters.
Combining Equations (6), (11) and (12), a simplified infiltration model for predicting the cumulative infiltration of single film hole was proposed:  
formula
(13)

Evaluation of the simplified infiltration model

A soil tank experiment was designed to verify whether the simplified infiltration model is accurate. The soil sample was taken from Yangling area, China. Given the USDA soil taxonomy, the soil texture is silt loam and sandy loam. The cumulative infiltration data with d of 5 cm and h of 5 cm was used to fit and determine the parameters s and a with Equation (6). Then, using the Equations (11) and (12), we get the fitting parameters k1 and k2. The simplified prediction models were established.

For Yangling silt loam:  
formula
(14)
For Yangling sandy loam:  
formula
(15)

Additionally, other experiments were also carried out with three film hole diameters and two water depths to verify the accuracy of Equations (14) and (15). The comparison result between predicted and observed values was illustrated in Figure 4(a).

Figure 4
Comparison of five soil types between predicted and observed values of cumulative infiltration.
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Comparison of five soil types between predicted and observed values of cumulative infiltration.

Figure 4
Comparison of five soil types between predicted and observed values of cumulative infiltration.
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Comparison of five soil types between predicted and observed values of cumulative infiltration.

To further judge whether the proposed model is reliable, the published data were also compared for three different soils, i.e. Dingbian sandy loam from Wu et al. (2003), Xi'an silt from Li et al. (2001) and Yulin loam from Jiao & Wang (1999). Similarly, the s and a were determined from fitting the cumulative infiltration with d of 5 cm and h of 2 cm, the k1 and k2 were obtained by Equations (9) and (10). To this end, the cumulative infiltration prediction models for the other three soils were established and written in Equations (16), (17) and (18) below. The comparison result between predicted and observed values for three soils was presented in Figure 4(b).

For Yulin loam:  
formula
(16)
For X'an silt:  
formula
(17)
For Dingbian sandy loam:  
formula
(18)

The values of RMSE is close to 0 (RMSE = 0.06 L); meanwhile, the values of EF approach 1.0 (EF = 0.99). It was noteworthy that all the results are consistent, indicating that the simplified model can accurately describe the cumulative infiltration from a film hole.

CONCLUSIONS

HYDRUS-2D simulation showed that soil texture, film hole diameter and water depth are the dominant factors affecting the cumulative infiltration from a film hole. For the same soil texture, the sorptivity (s) increases with the film hole diameter, and the relationship between them is a power function, while the steady infiltration rate (a) increases with increases of film hole diameter and water depth; the a is a power function relationship with film hole diameter and water depth. The calculation formulas of s and a were created, and a simplified model of cumulative infiltration of single-hole, including film hole diameter and water depth, was proposed. The effectiveness of the proposed model was verified by infiltration experiments of film hole infiltration and literature data. The calculated values of the model were in good agreement with the measured values. The model has only two undetermined coefficients, which can be determined only by a set of film hole infiltration tests. The simplified model in this study can provide a reference for agricultural irrigation project designers to estimate cumulative infiltration.

ACKNOWLEDGEMENTS

This research was supported by the National Basic Research Program of China (‘973’ Program) (2015CB452701), the Natural Science Foundation of Gansu Province, China (18JR3RA144), and the Independent Research Project of the State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research (2016ZY01).

REFERENCES

REFERENCES
Carsel
R. F.
&
Parrish
R. S.
1988
Developing joint probability distributions of soil water retention characteristics
.
Water Resources Research
24
(
5
),
755
–
769
.
Google Scholar
Crossref
Search ADS
 
Fan
Y. W.
,
Ma
X. Y.
,
Wang
B. L.
,
Gao
W. Q.
&
Nie
W. B.
2008
Numerical simulation on soil wetting pattern moisture distribution and infiltration characteristics for film hole irrigation
.
Transactions of the Chinese Society for Agricultural Machinery
39
(
11
),
35
–
41
.
Jiao
X. Y.
&
Wang
W. Y.
1999
Experimental research on infiltration from single hole of membrane
.
Journal of Hydraulic Engineering
30
(
7
),
33
–
40
.
Li
Y. N.
,
Fan
X. K.
&
Ma
X. Y.
2001
Experimental research on infiltration through membrane hole irrigation single hole
.
Agricultural Reseach in the Arid Areas
19
(
1
),
54
–
58
.
Li
Y.
,
Liu
Y.
,
Li
F.
&
Fei
L. J.
2005
Ecological environment effect of irrigation with mulching
.
Transactions of the Chinese Society of Agricultural Engineering
21
(
11
),
61
–
63
.
Li
Y. B.
,
Fan
Y. W.
,
Liu
Y.
&
Ma
X. Y.
2017
Influencing factors and simplified model of film hole irrigation
.
Water
9
(
7
),
543
.
Google Scholar
Crossref
Search ADS
 
Ma
H. Y.
,
Wang
X.
,
Zhang
Z. Y.
,
Feng
G. X.
&
Lü
N. J.
2015
Numerical simulation of water-salt distribution under brackish water film hole furrow irrigation based on HYDRUS-3D model
.
Transactions of the Chinese Society of Agricultural Machinery
46
(
2
),
137
–
145
.
Saeed
M.
&
Mahmood
S.
2013
Application of film hole irrigation on borders for water saving and sunflower production
.
Arabian Journal for Science & Engineering
38
(
6
),
1347
–
1358
.
Google Scholar
Crossref
Search ADS
 
Šimůnek
J.
,
Van Genuchten
M. T.
&
Šejna
M.
2006
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
, p.
241
.
Šimůnek
J.
,
Van Genuchten
M. T.
&
Šejna
M.
2016
Recent developments and applications of the HYDRUS computer software packages
.
Vadose Zone Journal
15
(
7
),
1
–
25
.
Google Scholar
Crossref
Search ADS
 
Skaggs
T. H.
,
Trout
T. J.
,
Šimůnek
J.
&
Shouse
P. J.
2004
Comparison of HYDRUS-2D simulations of drip irrigation with experimental observations
.
Journal of Irrigation & Drainage Engineering
130
(
4
),
304
–
310
.
Google Scholar
Crossref
Search ADS
 
Van Genuchten
M. T.
1980
A closed-form equation for predicting the hydraulic conductivity of unsaturated soils
.
Soil Science Society of America Journal
44
(
5
),
892
–
898
.
Google Scholar
Crossref
Search ADS
 
Wang
W. Y.
,
Tan
Q. L.
,
Jiao
X. Y.
,
Zhang
J. H.
&
Ding
X. L.
2001
Establishment and verification of the point source infiltration model for the filmed hole irrigation
.
Advances in Water Science
12
(
3
),
300
–
306
.
Wu
J. H.
,
Fei
L. J.
&
Wang
W. Y.
2001
Study on the infiltration characteristics of single filmed hole and its mathematical model under filmed hole irrigation
.
Advances in Water Science
12
(
3
),
307
–
311
.
Wu
J. H.
,
Fei
L. J.
&
Li
H. E.
2003
A philip model of film hole infiltration under adequate water supply
.
Journal of Irrigation and Drainage
22
(
4
),
18
–
21
.
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