Development of a soil wetting pattern estimation model for drip irrigation

Drip irrigation is one of the most common irrigation technologies, which is widely used for its high efficiency in water-saving. It helps transport water to the soil near plant roots, hence reducing water losses (from evaporation and deep drainage) and improving plant growth and water conservation (Sharmasarkar et al. 2001; Al-Ghobari & Dewidar 2018). The water available in the soil has a wetting pattern, the size and shape of which limit the growth of root systems and crop yields. The similarity between the soil wetting pattern and the distribution area of the crop root system is one of the critical factors used to improve irrigation performance (Wang et al. 2006). Access to the dimensions of the soil wetting pattern will help better determine the cover plant roots and emitter spacing, reduce the cost, and improve water use efficiency. Therefore, accurate estimation of the wetting pattern is a crucial factor in the design and management of drip irrigation systems (Subbaiah 2013). To estimate the soil wetting patterns under drip irrigation, the common models can be mainly divided into analytical models (Philip 1984; Revol et al. 1997; Kilic 2020), numerical models (Cote et al. 2003), and empirical models (Schwartzman & Zur 1986; Amin & Ekhmaj 2006; Naglič et al. 2014).

Analytical models have been developed on the basis of mathematical methods, physical laws, and specific assumptions (Al-Ogaidi et al. 2016). Moncef & Khemaies (2016) proposed an analytical method based on an assumption by Green & Ampt (1911), who assumed a piston-type flow and a semiellipsoid wetting pattern. However, according to Philip (1984), Rooij (2000), and Hill & Parlange (1972), unless the soil has a small volume, the assumption of complete water saturation in the soil profile is inconsistent with reality. Kilic (2020) developed an analytical method to describe three-dimensional (3D) wetting patterns in drip irrigation. They assumed that the composition (horizontal wetting front on the soil surface, horizontal and vertical wetting front in the soil profile) of the wetting pattern is a function of time. They further indicated that with the increase of infiltration time, the infiltration rate gradually decreases and approaches 0 cm min⁻¹, which is inconsistent with reality and does not apply to long-term irrigation. Cook et al. (2003) developed a user-friendly software tool called WetUp, which uses the analytical solution of Philip (1984) to estimate the wetting pattern of homogenous soils from a surface or subsurface point source and an elliptical plotting function to approximate the expected wetting patterns. However, the data used in the model as basic information from the analytical method of Philip (1984) may not be adequate to cover a wide range of soil textures and initial conditions. In addition, the tool uses the hydraulic conductivity of the wetting front as a constant value (1 mm h⁻¹), which differs from the actual field situation (Cook et al. 2006). Therefore, these analytical models are not widely used in the design and practice of drip irrigation systems because of their unreasonable assumptions.

Numerical models use the Richards equation as the governing equation for water flow, which mainly use the finite difference or finite element method to solve variably saturated/unsaturated flow problems (Šimůnek et al. 2006; Kandelous & Šimůnek 2010; Yu & Zheng 2010). Hydrus-2D/3D is the most commonly used numerical model for soil water movement processes (Skaggs et al. 2004; Yu & Zheng 2010). Surendran & Chandran (2022) verified that the soil water content simulated by the HYDRUS model showed agreement with observed values at different the horizontal and vertical wetting fronts from the soil profile. Hydrus-2D/3D has been used to analyze the effects of soil hydraulic properties and emitter discharge rate on the wetting pattern of drip irrigation, and it provides a reference for analyzing soil water movement with different soil textures and initial conditions (Cote et al. 2003; Cai et al. 2017; Shiri et al. 2020; Fan et al. 2021). Therefore, the simulated wetting front was obtained from the Hydrus-2D/3D model.

Empirical models are based on a regression analysis of measured or simulated values. These models help predict wetting patterns as a function of soil hydraulic properties and emitter discharge rate. Although empirical models may lack a theoretical basis, they are widely used because they are simple and closely resemble the actual situation (Karimi et al. 2020). The currently typical empirical models (Schwartzman & Zur 1986; Amin & Ekhmaj 2006; Naglič et al. 2014) are functional equations of factors that affect the wetting pattern (e.g., saturated hydraulic conductivity, total volume of applied water, and emitter discharge rate). Schwartzman & Zur (1986) developed a model for predicting the soil wetting pattern on the basis of dimensional analysis theory, including three parameters (the soil saturated hydraulic conductivity, emitter discharge rate, and total volume of applied water). However, the model lacked generality (based on the measured values of loamy and sandy soil) and did not consider the effect of the initial soil water content on the wetting front advance. Naglič et al. (2014) used the Hydrus-2D/3D to simulate an infiltration process with different initial conditions (11 soil textural classes, different emitter discharge rates, and varying initial soil water content), redefining the parameters of the model proposed by Schwartzman & Zur (1986) and improving the accuracy of the model. Although the initial soil water content was considered in the definition of the parameters, it was not included in the model used to estimate the wetting pattern. Amin & Ekhmaj (2006) proposed that the wetting front advance is related to the water content and added the average soil water content to the model. However, determining the average soil water content in soils with varying initial soil water content is a difficult process, and replacing half of the saturated soil water content leads to inaccurate results. In addition, these empirical models do not consider the effect of the initial soil water content on the wetting front advance or consider the average soil water content as a parameter. However, according to Witelski (2005), Subbaiah (2013), and Rooij (2000), the initial soil water content has a substantial effect on the wetting front advance. So the improved model needs to include the initial water content as a critical parameter.

Therefore, the development of an empirical model including initial soil water content that can predict the soil wetting patterns under drip irrigation would provide a convenient and accurate method to quantify the horizontal and vertical wetting front (Cristóbal-Muñoz et al. 2022). Moazenzadeh et al. (2022) believed the soil water content was helpful in determining irrigation depth and frequency, and evaluated the performance of optimization algorithms in estimating soil water content. Shiri et al. (2020) simulating wetting front advance in different soil types for surface and sub-surface irrigation systems through the gene expressions programming (GEP) and random forest (RF) techniques, and presented the soil water content has a higher influence on modeling vertical wetting front. Skaggs et al. (2010) used the Hydrus-2D/3D to simulate the effect of the initial soil water content on soil water infiltration in a drip irrigation system. The results indicated that when the initial soil water content increases in drip irrigation systems, the wetting front advance also increases, with the increase being greater in the vertical than in the horizontal direction. Jung et al. (2012) used X-ray photography to investigate the dynamic movement of wetting fronts and changes in the water content. The results indicated that with an increase in the initial soil water content, the wetting front changes from a bulb type to a trapezoidal type. These studies further indicated that the initial soil water content affected the wetting front advance. Meanwhile, the initial soil water content is easy to obtain experimentally, it should be regarded as a critical parameter in wetting pattern estimation models for drip irrigation.

According to previous results, the soil wetting pattern are affected by several factors including soil hydraulic properties and irrigation factors, such as soil saturated hydraulic conductivity, initial soil water content, soil textures, and emitter discharge rate (Kanda et al. 2020; Vishwakarma et al. 2022). Therefore, the objectives of this study were (1) to propose an empirical model which estimate the soil wetting pattern of drip irrigation taking the initial soil water content into account; and (2) to validate and evaluate the model proposed in this study as well as the models of Schwartzman & Zur (SZ model), Naglič et al. (N model), and Amin & Ekhmaj (AE model) by using laboratory experimental data and Hydrus-2D/3D simulations.

Therefore, by determining a₁, a₂, n¹, and n² in Equation (5), the horizontal and vertical wetting front advance can be determined under different initial conditions (V, K_s, q, and θ₀) of the drip irrigation system.

Generally, the root zone soil water content should be maintained between the field capacity (FC) and permanent wilting point (PWP) values because irrigation water is wasted above the FC and crops wither below the PWP (Delgoda et al. 2016). Therefore, the initial soil water content can be expressed by changing the percentage of the maximum available water (AW) in the soil, which is the water volume between the permanent FC and PWP (Sreelash et al. 2017). Here, AW, FC, and PWP were derived from the study of Dai et al. (2013), and 12 typical soil types (USDA soil texture classes) were selected, the hydraulic parameters of which are presented in Table 1 (Carsel & Parrish 1988). In addition, the wetting pattern was simulated by Hydrus-2D/3D to determine the parameters of the dimensional model shown in Equation (5) by using a nonlinear regression approach.

As done in Naglič et al. (2014), 30, 50, and 70% of the maximum effective water were selected as the initial soil water content of different soil types (where 30 and 70% represent dry and wet soil conditions, respectively). Three common and universal emitter discharge rates (i.e., 1, 2, and 3 L h⁻¹) were set for nine soil textures (Moncef et al. 2002; Molavi et al. 2014), with smaller emitter discharge rates (i.e., 1, 1.5, and 2 L h⁻¹) selected for three finer textures to avoid surface runoff (Naglič et al. 2014; Moncef & Khemaies 2016). Table 2 presents the input variables for Hydrus-2D/3D.

Laboratory experimental data were collected from three infiltration experiments published in three papers (from published results; see details in Table 3; Moncef et al. 2002; Molavi et al. 2014; Naglič et al. 2014), including 8 sets of the horizontal and vertical wetting front advance data. These infiltration events occurred under various experimental conditions, including soil textures of sandy loam, loam, sand, sandy clay, clay, silty loam, and silt, emitter discharge rates of 1, 2 and 4 L h⁻¹, in the three different locations of the Tabriz suburbs (Molavi et al. 2014), the experiments under 2D soil tank consisted of 40.3 cm long, 30 cm high and 2.5 cm wide (Naglič et al. 2014), and infiltration experiments in semi-cylindrical container consisted of 120 cm high and 150 cm in diameter (Moncef et al. 2002). More details of the experiments can be found in Molavi et al. (2014), Naglič et al. (2014) and Moncef et al. (2002).

Simulation values were used to evaluate the SZ model, the N model, and the AE model and the proposed model to verify its reliability. The initial soil water content was selected as 40 and 60% of the AW, and the four models were verified using the soil hydraulic parameters presented in Table 1. The input variables used in Hydrus-2D/3D for validation are presented in Table 4. A total of 72 groups of experiments were conducted and included 2 initial soil water concentrations, 12 soil textures (USDA soil texture classes), and 3 emitter discharge rates (i.e., 1, 1.5/2, and 3 L h⁻¹).

To ensure the accuracy of Equations (19) and (20), the MAE, RMSE, and MRE of the simulated and estimated values (horizontal and vertical wetting front) were calculated. The obtained results indicated that the MAE, RMSE, and MRE ranged from 2.77 to 4.02 cm, from 3.75 to 6.32 cm, and from 6.91% to 8.46%, respectively. In addition, the high R² and low errors observed (Figure 2) between the estimated and simulated values indicated the high accuracy of Equations (19) and (20).

As shown in Figure 3, the estimated values of the four models agreed with the measured values and were evenly distributed on both sides of the 1:1 line. Table 5 shows the MAE, MRE, and RMSE values calculated for analyzing the measured and estimated values. The proposed model exhibited the smallest error in the horizontal and vertical wetting front advance distance, with MAE values of 2.77 and 3.56 cm, RMSE values of 6.45 and 10.61 cm, and MRE values of 9.04% and 10.51%, respectively. Following the proposed model, the AE model and the N model demonstrated similar results, with MAE values of 2.91 and 4.99 cm and 2.91 and 5.08 cm, RMSE values of 6.59 and 11.32 cm and 7.78 and 11.34 cm, and MRE values of 9.46% and 13.75% and 10.50% and 15.84%, respectively. The SZ model exhibited the largest error, with MAE values of 4.13 and 5.19 cm, RMSE values of 7.56 and 12.90 cm, and MRE values of 14.01% and 15.78%, respectively. These results indicated that the estimated values of the four models were consistent with the measured values, although some differences were observed. These differences were mainly due to inevitable experimental errors and the simplified assumptions of the models. The SZ model has the lowest consistency of the four models since the effect of initial soil water content was not taken into account and that only two soil textures (loam and sand) were included instead of a variety of soil textures. Because the effect of soil water content was not neglected, the results of the N model and AE model are similar and more consistent than those of the SZ model. Although Naglič et al. (2014) considered the initial soil water content in the parameter definition, the N model did not include the initial soil water content. Amin & Ekhmaj (2006) proposed that the wetting front advance was affected by soil water content and added the average soil water content to the model. However, it was found that the results of the AE model for the soil wetting front with different initial soil water contents were not as accurate as the model proposed in this paper. This indicated that the initial soil water content in the AE model could not be replaced by half of the saturated soil water content.

Overall, the estimated values of the proposed model were more accurate than those of the SZ model, the N model, and the AE model. This is because the proposed model considers the effect of the initial soil water content on wetting front advance, which is more consistent with the actual infiltration process. During the infiltration process, any variations in the initial soil water content result in different advance rates of the wetting front. Soils with high initial soil water content require less water to fill soil pores, whereas soils with low initial soil water content require more water to fill soil pores. Therefore, soils with higher water content have a higher wetting front advance speed, which is why ignoring the initial soil water content leads to inaccurate results. In conclusion, this study, the proposed model based on the dimensional analytical approach with the initial soil water content considered simultaneously, exhibits improved accuracy over existing models, and obtains more consistency between estimated and measured values.

The results indicated that the proposed model exhibited the smallest estimation error in the horizontal and vertical wetting front advance, with MAE values of 4.27 and 6.20 cm, RMSE values of 6.20 and 6.58 cm, and MRE values of 5.61% and 7.15%, respectively. Following the proposed model, the AE model and the N model exhibited similar results. For horizontal and vertical wetting front advance, the AE model exhibited MAE values of 4.33 and 6.85 cm, RMSE values of 6.50 and 8.78 cm, and MRE values of 8.91% and 10.50%, respectively. For horizontal and vertical wetting front advance, N model exhibited MAE values of 5.12 and 4.88 cm, RMSE values of 6.44 and 6.74 cm, and MRE values of 9.50% and 8.55%, respectively. For horizontal and vertical wetting front advance, SZ model exhibited the largest error, with MAE values of 6.11 and 11.42 cm, RMSE values of 6.92 and 13.54 cm, and MRE values of 10.99% and 13.08%, respectively. Although the estimated values of the four models were close to the simulated ones, some differences were observed. The estimation model was based on measured or simulated values, and it helped simplify the complex infiltration process. The improved accuracy observed can be attributed to the initial soil water content, which was regarded as a variable in the estimation of the wetting front advance in the proposed model. Meanwhile, the proposed model was based on 12 soil textures (USDA soil texture classes), and can be estimated the soil wetting patterns of various soil textures, which is applicable to a variety of soil textures.

Overall, the model proposed in this study is more accurate than the other three models discussed (the SZ model, the N model, and the AE model) for estimating the wetting pattern in drip irrigation. This is mainly because the proposed model considers the effect of the initial soil water content on the wetting front advance, which is consistent with the actual situation. In addition, given the infiltration results obtained for 12 different soil textures (USDA soil texture classes), the model has a favorable level of universality.

Soils with low water content have a small matrix potential (with negative values). The difference in potential energy increases the flow of water. Hence, the presence of low water content in the soil pores increases the time required to fill the pores with water, which in turn reduces the advance of the wetting front and increases the amount of water required (Liu et al. 2019; Patle et al. 2019; Cheng et al. 2021). When the water content increases, the difference in potential energy decreases, which decreases the flow of water, decreases the time required to fill the pores with water, accelerates the advance of the wetting front, and decreases the volume of applied water (Talsma 1974; Or et al. 2001; Guo & Liu 2019; Roy et al. 2020). This, however, contradicts the concept of dimensionless analysis, in which the advance of the wetting front is proportional to the total volume of applied water [Equation (4)]. This contradiction indicates that the initial soil water content has a major effect on soil water infiltration (Bauters et al. 2000). Therefore, the accuracy of the proposed model is higher than the AE model, the N model, and the SZ model (Figures 3 and 4, Tables 5 and 6).

The results revealed relatively large errors in the estimation of the wetting front advance of coarse-textured soils, which is consistent with previous studies (Al-Ogaidi et al. 2016; Wang et al. 2020). The same results were also reflected in the SZ model, the N model, and the AE model for estimating wetting front advance in coarse-textured soils (not shown in this study). These results may be due to the presence of macropore flow during the infiltration of coarse-textured soils, a process that cannot be completely captured by Hydrus-2D/3D (Cameira et al. 2003; Jarvis et al. 2009). In addition to the matrix potential, the vertical wetting front advance is also affected by gravity potential. During the infiltration process, the water content and matrix potential of coarse-textured soils rapidly increase. When the water content reaches the field capacity, the soil water starts moving rapidly downward under the effect of gravity. Compared with soils of other textures (medium and fine soil textures), the vertical wetting front in coarse-textured soils advances much faster than the horizontal wetting front (Siyal & Skaggs 2009). Therefore, even for the same infiltration time and total volume of applied water, some errors may occur in the regression analysis of the vertical wetting front advance in coarse-textured soils, hence reducing the accuracy of the estimation model. To improve the accuracy of the soil wetting pattern model, the effect of the initial soil water content should be considered when estimating the wetting front advance. Therefore, the proposed model, with the initial soil water content considering simultaneously, provided a good estimation of the soil wetting pattern and was suitable for application in a variety of soil textures.

The proposed model is more accurate in estimating the soil wetting patterns without coarse-textured soils than coarse-textured soils. Further studies are required to deeply explore the mechanism of coarse-textured soil infiltration to improve the accuracy of the model when applied to a variety of soil textures. In recent research, many scholars (Kisi et al. 2021; Abdalrahman et al. 2022; Singh et al. 2022) have applied artificial intelligence (AI)-based approaches and data-driven technologies (e.g., multi-layer perceptron (MLP), artificial neural network models (ANN), generalized regression neural network (GRNN), support vector machines (SVMs), and multivariate adaptive regression splines (MARS)) to irrigation system design and management. Therefore, a combination of these techniques and dimensional analysis theory should be considered in further studies to develop more applicable and accurate models of soil wetting patterns for drip irrigation.

In this study, we used the models proposed by Schwartzman & Zur (1986) and Naglič et al. (2014) as the basis for and added the initial soil water content to the proposed model of this study, which helped improve the estimation accuracy of the horizontal and vertical wetting front advance during drip irrigation. We then verified the reliability and universality of the proposed model by using experimental and simulated values in Hydrus-2D/3D. Compared with the SZ model, the N model, and the AE model, the proposed model of this study exhibited the smallest error in the horizontal and vertical wetting front advance distance, with MAE, RMSE, and MRE values ranging from 2.77 to 4.69 cm, from 6.20 to 10.61 cm, and from 5.61% to 10.51%, respectively. These results indicate that the proposed model is reliable and highly accurate and can be used with a variety of soil textures.

This research was supported by grants from the National Natural Science Foundation of China (52279043), and Scientific Research Program of the Shaanxi Provincial Education Department (20JS099).

The authors declare that they have no conflict of interest.

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

The authors declare there is no conflict.