Establishment and verification of the prediction model of soil wetting pattern size in vertical moistube irrigation

Vertical moistube irrigation is a new type of moistube irrigation technology that uses a polymer semi-permeable membrane to make a moistube, through which water seeps out for continuous underground irrigation (Quinones-Bolanos et al. 2005; Yang 2010). Vertical moistube irrigation has the advantages over underground drip irrigation of deeper wetting depth, free setting spacing, low working head, and low power consumption. It is more suitable for deep-rooted plants and can keep the soil around the root system moist.

The volume of moist soil is an important basis for optimizing the spacing of irrigation devices, horizontal arrangement, and choosing a suitable irrigation system, and it is also one of the most important factors in the design of irrigation systems. Studies have shown that the wetting depth in the soil profile of the surface drip irrigation and the wetting radius of the soil surface are the main components of the soil wetting pattern (Dabral et al. 2012; Al-Ogaidi et al. 2016a), and that the main components of the soil wetting pattern of subsurface drip irrigation are the wetting depth, wetting radius, and upward wetting distance in the soil profile (Yao et al. 2011). Vertical moistube irrigation is a buried irrigation technology, and the main components of the soil wetting pattern are similar to those of underground drip irrigation (Fan et al. 2018a). During irrigation, the size of the components of the wetting pattern must be determined according to the cross section of the wetting pattern and the volume of the wet soil (Fernandez-Galvez & Simmonds 2006). Domestic and foreign scholars have developed some analytical models (Philip 1984; Chu 1994; Cook et al. 2003; Moncef & Khemaies. 2016), HYDRUS-2D models (Brandt et al. 1971; Šimůnek et al. 2008), and empirical models (Malek & Peters 2011; Al-Ogaidi et al. 2016; Fan et al. 2018b) to quantify the wet soil volume. Researchers compared and evaluated the above three types of models (Kandelous & Šimůnek 2010; Subbaiah 2013). The results show that analytical and HYDRUS-2D models are usually employed to solve control flow equations under specific initial and boundary conditions. Analytical models are based on point source assumptions and special forms of homogeneous soil physical properties. The HYDRUS-2D model can realize a wide range of boundary conditions, including irregular boundaries and soil heterogeneity, improving the popularity of its analysis in the design of moistube irrigation systems. Kandelous & Šimůnek (2010) believe that HYDRUS-2D can estimate soil moisture distribution under drip irrigation conditions more accurately than analysis and empirical models. The model has been successfully applied to simulate the distribution of soil moisture under different irrigation methods. Although each empirical model is relatively simple in form, it is obtained by analyzing laboratory and field data using regression and dimensional analysis methods. The empirical formula is only applicable to specific irrigation techniques, such as the developed point source or line source drip irrigation wetted pattern size, and is not applicable to vertical moistube irrigation (Malek & Peters 2011). Therefore, the development of an empirical model that can predict the size of the soil wetting pattern of vertical moistube irrigation would provide a convenient and practical method for quantifying the size of the wetting pattern and solving the problem that the wetting pattern is difficult to observe.

The combination of numerical simulation and experimental verification is a common method for studying the laws of soil water movement. Numerical simulation can simulate the process of soil water movement under different soil characteristics and different technical elements (Saito et al. 2006; Han et al. 2015; Šimůnek et al. 2016). Experiments can verify the laws summarized by numerical simulation and improve the reliability of research results. Kanda et al. (2020) used the HYDRUS-2D model to numerically simulate the soil moisture distribution of two soil textures (ST; loamy sand and sandy clay loam) under horizontal moistube irrigation, and verified them with experiments, with the results showing that the simulated values are very consistent with the experimental observations. Fan et al. (2018c) verified the reliability of the HYDRUS-2D simulation results of horizontal moistube irrigation by using indoor soil box experimental data. On this basis, the dynamic changes of soil wetting pattern under different ST and loams with different initial water content (θ₀), pressure head (H), and depth of burial (D) were studied. The results show that the movement distance of the soil wet front conforms to the rule of vertical downward > horizontal direction > vertical upward, the contours of soil moisture content are all approximately concentric circles, and that the texture of the soil has a significant effect on the characteristics of the wet body. Zhang et al. (2016) explored the infiltration of water in different soil textures under the condition of horizontal moistube irrigation based on laboratory experiments, with the results showing that the cumulative infiltration amount was negatively correlated with the soil clay content, that the wetting front was an approximate circle centered on the moistube, and that the relationship between the distance and time of the wetting front was approximately a power function. Fan et al. (2018a) determined the specific flow calculation formula of the vertical moistube through indoor experiments, and verified the accuracy of HYDRUS-2D simulation. On this basis, using HYDRUS-2D simulation to study the influence of ST, θ₀, H, moistube length (L) and D on the characteristics of vertical moistube irrigation wetting pattern shows that the contour of the wetting pattern and the water content was approximately an ellipsoid around the moistube. ST has a significant impact on the wetting pattern, and with the increase of soil clay content, the volume of wet soil decreases. The θ₀, H, and L have a greater impact on the distance of the wetting front and the volume of the wet soil, and D affects the position of the wetting pattern.

In previous research (Fan et al. 2018a), the authors of this article conducted a qualitative study on the soil characteristic parameters and irrigation technical elements affecting the soil wetting pattern of vertical moistube irrigation. Based on the previous qualitative research on the influencing factors of the vertical moistube irrigation, this study increases the simulation volume and uses the HYDRUS-2D software to obtain the wetting pattern size data with different factors; the laws of the simulation data are analyzed and summarized to construct a simplified empirical model for predicting the size of the soil wetting pattern; and experimental data is used to verify the reliability of the model in order to provide a scientific basis for the design, operation, and management of the moistube irrigation engineering.

The silt loam in Qilihe District, Lanzhou, and the sandy loam and sandy clay loam in Minqin County, Wuwei, Gansu Province, are used for vertical moistube irrigation experiments, with the sampling locations shown in Figure 1. The saturated water content θ_s and saturated hydraulic conductivity K_s of the experimental soil are measured by the ring knife method and the constant head method, respectively. The experimental soil characteristic parameters and irrigation technical parameters are shown in Table 1.

The experiment device consists of five parts: a height adjustable stand, a mariotte bottle, a hydraulic hose, a moistube, and a soil box (Figure 2). The soil box is made of 10 mm thick plexiglass, and the size is 60 cm (length) × 60 cm (width) × 100 cm (height). There are multiple ventilation holes (diameter 2 mm) at the bottom of the soil box to prevent air resistance in the soil. In the test, the mariotte bottle provides a constant head, and the contour of the wetted body at different times is drawn with a marker.

Figure 3 shows the initial and boundary conditions considered when simulating different modeling scenarios in this study. Considering the axial symmetry of the vertical moistube, the area shown in Figure 3 is selected as the simulation calculation domain. The level of soil moisture content determines the rate of water movement in the soil. Therefore, in all simulation scenarios, the soil moisture content is set at a given initial moisture content.

In order to construct an empirical model of the size of the soil wetting pattern of vertical moistube irrigation, seven STs are selected. For each ST, the settings are as follows: three initial water content (60%FC, 70%FC, and 80%FC), three lengths of moistube (10, 15, and 20 cm), three buried depths of moistube (20, 30, and 40 cm) and three pressure heads (100, 150, and 200 cm), a total of 63 groups. During the simulation, the irrigation time is set to 192 h. The field water holding capacity of the soil is calculated using Equation (6), and Q is obtained using Equation (7). The parameters of VG–M model of ST were taken from Carsel and Parrish's data (Carsel & Parrish 1988), and the soil bulk density γ was taken from Pachepsky and Park's data (Pachepsky & Park 2015).

Based on the HYDRUS-2D simulation results, using Equation (10), the fitting parameters a₁ and b₁ in different influencing factor combinations are obtained through Origin 9.0 fitting. The fitting results are shown in Table 2. It can be seen from Table 2 that using Equation (10) to fit the relationship between the vertical upward wetting pattern size of vertical moistube irrigation and the irrigation time, ⁠. It shows that Equation (10) demonstrates the relationship between the vertical upward wetting pattern size of vertical moistube irrigation and irrigation time.

A further analysis of Table 2 finds that the value of the fitting parameter a₁ varies in a small range. However, previous studies have found that the size of the soil wetting pattern of vertical moistube irrigation is mainly affected by ST, θ₀, H, L, and D, while H, L, and D affect Q, which in turn affects the size of the wetting pattern (Fan et al. 2018a). Therefore, taking the parameter a₁ as the average value alone does not meet the actual situation, nor can it meet the requirements for model universality. It is worth considering that Q at different buried depths is different, and due to the influence of gravity, the discharge flow of the adjacent points above and below the moistube will have a superposition effect. Therefore, Q alone cannot fully reflect the water permeability of the moistube, and the product of the average specific flow rate (q) of the moistube and L can be used to characterize the water permeability of the moistube. The θ₀ has a certain influence on the size of the wetting pattern; the influence mechanism is that during the irrigation process, the irrigation water moves around in the soil pores, and gradually fills the soil pores until the soil is fully saturated. Therefore, the final determination of the size of the wetting pattern is the amount of water required to fill the pores of the soil to achieve approximate saturation. It can be expressed by the difference between θ_s and θ₀, that is, θ_s–θ₀. In addition, the influence of ST on the size of the wetting pattern can be characterized by K_s. Therefore, the influencing factors of the simplified vertical moistube irrigation soil wetting pattern size mainly include K_s, qL, and (θ_s–θ₀).

Table 3 shows the simulation results based on HYDRUS-2D. Using Equation (11), the values of fitting parameters a₂ and b₂ under different influencing factor combinations and R² are obtained by fitting with Origin 9.0. It can be seen from Table 3 that using Equation (11) to fit the relationship between the horizontal wetting pattern size of vertical moistube irrigation and the irrigation time gives ≧ 0.985, indicating that Equation (11) demonstrates the relationship between the horizontal wetting pattern size of vertical moistube irrigation and the irrigation time.

Using the HYDRUS-2D simulation results and Equation (12), the fitting parameters a₃ and b₃ under different combinations of influencing factors were obtained by Origin 9.0. The values of parameters a₃ and b₃ and R² are shown in Table 4. It can be seen from Table 4 that Equation (12) is used to fit the relationship between wetting pattern dimension of vertical downward of vertical moistube irrigation and irrigation time, and ≧ 0.989, indicating that Equation (12) demonstrates the relationship between wetting pattern dimension of vertical downward of vertical moistube irrigation and irrigation time.

To evaluate the accuracy of the prediction model of the soil wetting pattern size of the vertical moistube irrigation, the indoor experiments is used to verify the model, with the calculated value of the prediction model compared with the measured value (Figure 4). Based on Equations (14)–(16), a statistical analysis is carried out on the calculated and measured values of the model (Table 5).

Figure 4 shows that the measured values of soil wetting pattern of vertical moistube irrigation is consistent with the calculated values of the model, but the horizontal fit is the best. It can be seen from Table 5 that MAE of the statistical indicators of the built model is between 0.30 and 1.42 cm, RMSE is between 0.42 and 1.65 cm, and NSE is not less than 0.93. This shows that the model prediction effect is good, but there are still some errors. The reason may be that when the prediction model of soil wet pattern size for vertical moistube irrigation is established, in different combinations of K_s, qL, and (θ_s–θ₀), the power function exponent of the wetting pattern size and irrigation time varies little in the vertical upward, horizontal, and vertical downward direction. In order to simplify the calculation, the average values are taken, which affects the accuracy of the calculation results to a certain extent. In addition, only using the difference between K_s and (θ_s–θ₀) to characterize the size of different ST wetting patterns is also one of the reasons for the partial error. It should be noted that in the process of irrigation advancement, the wetting boundary will reach the surface and the vertical upward Equation (23) is no longer applicable, but the buried depth value can be taken when the estimated value is greater than the buried depth of the moistube. The arrival of the wetting boundary on the surface has little effect on the horizontal and vertical downward migration of wetting patterns, and Equations (24) and (25) still apply. For soils with large pores between soil particles, such as sand, the model should be used with caution.

On the basis of qualitative research on the influencing factors of wetting pattern of vertical moistube irrigation, this study adds the simulation quantity. The HYDRUS-2D software is used to simulate the size data of the wetting pattern under different factors. By analyzing and summarizing the laws of simulation data, a simplified empirical model for predicting the size of soil wetting patterns has been constructed, with the reliability of the model verified with experimental data. The following conclusions are obtained.

The power function is used to fit the relationship between the size of the soil wetting pattern and the irrigation time. The power function exponent varies slightly in the vertical upward, horizontal, and vertical downward directions of the moistube, with the average values taken as 0.476, 0.406, and 0.428, respectively. The power function coefficient conformed to the power function relationship with K_s, qL, and (θ_s–θ₀).

Based on this, an empirical model of the size of the vertical moistube irrigation wetting pattern was established. The model includes t, K_s, qL, and (θ_s–θ₀). The MAE of the statistical indicators of the built model is between 0.30 and 1.42 cm, RMSE is between 0.42 and 1.65 cm, and NSE is not less than 0.93. The prediction effect of the model is good, and it can provide a scientific basis for the design, operation, and management of the moistube irrigation engineering.

This study was supported by the National Natural Science Foundation of China (No. 51409137 and 51969013), the Natural Science Foundation of Gansu Province, China (No. 18JR3RA144), and Hongliu Supporting Discipline of Lanzhou University of Technology.

All relevant data are included in the paper or its Supplementary Information.