Effects of Hole Irrigation Device Parameters on Soil Water Characteristics Under Different Biogas Slurry Ratios

Abstract

This study investigates the impact of biogas slurry ratio, hole diameter and depth under hole irrigation on the soil wetting front migration distance and cumulative infiltration. In this study, a model describing the water transport characteristics of biogas slurry hole irrigation was developed based on the HYDRUS model. Results demonstrated that the HYDRUS model can be used for biogas slurry hole irrigation (NSE > 0.952, PBIAS ≤ ±0.34). Furthermore, the study revealed that the soil cumulative infiltration and soil wetting front migration distance decreased gradually with an increase in the biogas slurry ratio, while they increased gradually with an increase in the hole diameter and depth. The lateral and vertical wetting front migration distances exhibited a well-defined power function relationship with the soil’s stable infiltration rate and infiltration time (R² ≥ 0.977). The soil wetting front migration distance curve can be represented by an elliptic curve equation (R² ≥ 0.957). Additionally, there was a linear relationship between the cumulative infiltration and soil wetted body area (R² ≥ 0.972). Soil wetting front migration distance model ($X=4.442f_0^{0.375}{t}^{0.24}, Z=11.988f_0^{0.287}{t}^{0.124}, f_0=96.947K_s^{1.151}{D}^{0.236}{H}^{1.042}$, NSE > 0.976, PBIAS ≤ ±0.13) and cumulative infiltration model ($I=0.3365S$, NSE > 0.982, PBIAS ≤ ±0.10) established under biogas slurry hole irrigation exhibited good reliability. This study aims to determine optimal hole diameter, depth, and irrigation volume for biogas slurry hole irrigation by establishing a model for soil wetting front migration distance and cumulative infiltration based on crop root growth patterns, thereby providing a scientific basis for its practical application.

1. Introduction

At present, the world is experiencing extreme water shortage and extreme drought, which have caused a huge crisis and challenge to agricultural production. We must overcome the waste of water resources and shortcomings of low efficiency in traditional irrigation practices and innovate new irrigation technology to improve the efficiency of water and fertilizer utilization [1].

Biogas slurry, a compound liquid organic fertilizer, has high water content and low fertilizer content. Biogas slurry utilization not only protects water resources but also promotes the development of a resource-based and conservation-oriented society [2]. Previous studies have demonstrated that biogas slurry application improved the soil structure of fluvo–aquic soil, enhanced soil aggregate stability [3], increased soil porosity and aggregate structure, reduced soil debris [4], strengthened the binding force of soil particles [5], enhanced soil medium and micro-porosity, and reduced soil macro-porosity, the infiltration rate and hydraulic conductivity of sandy soil [6]. Additionally, biogas slurry improved soil water-holding capacity [7,8]. Field experiments have indicated that the biogas slurry application increased crop yields and crop quality, while reducing irrigation water requirements [9,10,11,12]. However, the traditional irrigation methods with biogas slurry were susceptible to leaching loss of water and nutrients, resulting in a decrease in the rhizosphere soil’s available nutrient retention, which causes a waste of resources and may even pose a pollution threat to groundwater [13]. Consequently, new irrigation methods are urgently needed to improve these conditions.

Hole irrigation, a cost-effective, user-friendly, water-saving and fertilizer-saving technique, widely adopted by farmers in facility agricultural production, is highly practical for promoting water-saving irrigation in arid regions of China [14]. Zheng et al. [15] presented laboratory and simulated experiments on biogas slurry hole irrigation, combining biogas slurry application through hole irrigation technology, and found that the shape of wetting gradually changed from ellipsoid with a larger horizontal axis to one with a smaller horizontal axis with the increase in the hole diameter. The Philip infiltration model was found to accurately describe the variations in the cumulative infiltration amount of biogas slurry hole irrigation over time [16]. Additionally, a generalized Ross infiltration model was established, which more accurately simulates the soil water movement process of clear water under hole irrigation conditions (R² > 0.9) [17]. Previous simulation experiments have accurately simulated the soil wetting front migration distance and soil water content under hole irrigation with clear water. However, research on the relationship between the cumulative infiltration or soil wetting front migration distance and infiltration time, and simulation of the soil wetting front migration distance and cumulative infiltration under biogas slurry hole irrigation is still scarce.

The results from greenhouse tomato planting experiments have shown that hole irrigation is effective in providing water and nutrients to the crop root zone, improving crop yield, and increasing comprehensive nutritional quality and irrigation water utilization efficiency compared to traditional irrigation methods [18,19]. Therefore, it is important to fit different crops with suitable biogas slurry hole irrigation because of various root distribution ranges [20]. Also, it is necessary to determine appropriate biogas slurry hole irrigation parameters (such as hole diameter, depth and biogas slurry ratio) based on the specific root distribution characteristics of the crop. Accurately estimating the irrigation wetted area [21,22,23] and establishing a suitable relationship model between the soil wetted area and crop roots are crucial for improving water and fertilizer use efficiency. Conducting extensive and long-term experimental studies is essential to determine suitable biogas slurry hole irrigation parameters for different crops. However, experimental studies are not an ideal model for the promotion and application of biogas slurry hole irrigation technology. The HYDRUS-2D software has been extensively utilized in various irrigation and fertilization studies to assess the water efficiency of irrigation systems [24,25,26]. In comparison to laboratory and field experiments, using HYDRUS for simulating biogas slurry hole irrigation provides a more comprehensive understanding of the infiltration process. HYDRUS enables researchers to gain insights into the impact of different biogas slurry hole irrigation parameters on the soil wetting front’s shape, soil water content distribution within the wetting body, infiltration rate and variations in cumulative infiltration over time.

Therefore, this study aims to validate the reliability of the HYDRUS-2D simulation results by conducting laboratory experiments. The simulation results were then used to analyze the distribution characteristics of the cumulative soil infiltration amount and the soil wetting front migration distance under different biogas slurry ratios, hole diameters and depths. A soil wetting front migration distance model was established for analyzing the quantitative relationship between the cumulative infiltration amount and the wetted soil body, and the model was verified using laboratory experiment results. These results were expected to provide the theoretical support for the popularization and application of biogas slurry hole irrigation. It is of vital significance to the conservation of water resources and water use efficiency in the agricultural field.

2. Materials and Methods

2.1. Laboratory Experiments

2.1.1. Experimental Materials

The experimental soil was obtained from the soil surface layer (0–40 cm) in the greenhouses near Lanzhou City (36°03′21″ N, 103°46′34″ E), Gansu Province. After naturally air drying and pulverization, the soil was sieved using a 2 mm sieve. The initial water content of the soil was measured to be 0.072 cm³ cm⁻³. The soil particle gradation is presented in

Table 1. Soil particle size distribution.

Particle size/mm	<2.000	<1.000	<0.500	<0.250	<0.100	<0.075	<0.005
Mass fraction/%	100.000	99.995	99.930	99.885	99.789	99.764	3.000

. Based on the soil classification standards, the soil type was identified as silty loam. The bulk density of the experimental soil was determined as 1.35 g cm⁻³, according to the bulk density of local farmland soil.

The biogas slurry used in the experiment was obtained from the biogas project at the Gansu Holstein Dairy Breeding Center in Huazhuang Town, Lanzhou City, China.

2.1.2. Experimental and Simulation Design

Based on the previous experimental results, four biogas slurry ratios (biogas slurry: water, volume ratio) were set: 0 (B₀), 1:8 (B_1:8), 1:6 (B_1:6), and 1:4 (B_1:4). Three hole diameters were conducted: 3 cm (D₃), 5 cm (D₅) and 7 cm (D₇). Two hole depths were conducted: 5 cm (H₅) and 10 cm (H₁₀). A total of 24 treatments were conducted; each treatment had 3 repetitions. A randomized block design was used to arrange the experiments.

In order to use the HYDRUS-2D model to further investigate the effects of hole diameter and depth on the cumulative infiltration amount and the soil wetting front migration distance under hole irrigation with various biogas slurry amounts, this study established 64 simulation scenarios under various biogas slurry ratio conditions, including four biogas slurry ratios (B₀, B_1:8, B_1:6 and B_1:4), four hole diameters (D₃, D₅, D₇ and D₉), and four hole depths (H₈, H₁₀, H₁₂ and H₁₄).

2.1.3. Measurement

The soil water characteristic curve was measured by high-speed constant temperature freezing centrifuge (CR21GⅡ, Hitachi) [27]. Additionally, the soil saturated hydraulic conductivity (K_s) was measured by the variable head permeability test [28].

The biogas slurry hole irrigation system consisted of two main components, a soil tank and a water supply device (Figure 1). To ensure the symmetry of the hole infiltration section, only 1/12 of the actual soil wetting body under hole irrigation was considered as the object of study [27]. The soil box was constructed using a 30° fan-shaped plexiglass device with a height of 50 cm and a radial length of 40 cm. A 520 mL Mariotte bottle was employed to maintain a constant head water supply (Figure 1). When the cumulative infiltration volume reached 520 mL or the infiltration time reached 12 h, the water inlet and exhaust hole valves of the Mariotte bottle were closed. The experiment ended when the biogas slurry in the hole was completely infiltrated.

2.2. Numerical Simulation Experiments

2.2.1. Governing Equation of Soil Water Movement

Hole irrigation is a form of line source infiltration, characterized by axially symmetrical three-dimensional infiltration. Assuming uniform and isotropic soil, the movement of water can be simplified as a two-dimensional soil infiltration under axially symmetrical conditions. This simplification reduced the experiment duration and disregarded the effects of soil surface evaporation, temperature and crops planting on soil water. The corresponding Richards [28] equation was as follows:

$${\displaystyle \frac{\partial \theta }{\partial t}}={\displaystyle \frac{\partial }{\partial x}}\left(D\left(\theta \right){\displaystyle \frac{\partial \theta }{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial z}}\left(D\left(\theta \right){\displaystyle \frac{\partial \theta }{\partial z}}\right)-{\displaystyle \frac{\partial K\left(\theta \right)}{\partial z}}$$

(1)

where θ is the soil volumetric water content, cm³ cm⁻³; D(θ) is the soil water diffusivity, cm min⁻¹; K(θ) is the soil unsaturated hydraulic conductivity, cm min⁻¹; t is the infiltration time, min; z is the vertical coordinate, specifying that the positive z direction is toward the bottom, cm; and x is the lateral coordinate, cm.

2.2.2. Initial and Boundary Conditions

The actual situation of biogas slurry hole irrigation was fully considered, and the initial and boundary conditions for different modeling scenarios were set (Figure 2). It was assumed that the initial soil water content was evenly distributed in the experimental area, and the initial conditions were expressed as

$$h(x,z,t)=h_0 (0\ge z\ge Z, 0\le x\le X, t=0)$$

(2)

where h₀ is the initial negative pressure head of soil, cm; and Z and X refer to the maximum vertical and lateral distances of the experimental area, respectively, in cm.

The upper boundary AB, set as the no-flux boundary, was not affected by evaporation or precipitation during the experiment. The lower boundary CD, set as the free drainage boundary, was not affected by irrigation. The left boundary OD, set as the no-flux boundary, was the center line of the hole with no water exchange. The right boundary BC was set as the no-flux boundary due to no irrigation water. The boundary ab and Ab were the hole infiltration surface, set as the constant head boundary [29].

In short, the boundary conditions were expressed as follows:

$$\left\{\begin{array}{l}-K(h)\left({\displaystyle \frac{\partial h}{\partial z}}-1\right)=0 \left(t>0, \mathrm{The} \mathrm{boundary}AB \right)\hfill \\ {\displaystyle \frac{\partial h}{\partial z}}=0 \left(t>0, \mathrm{The} \mathrm{boundary} CD \right) \hfill \\ -K(h){\displaystyle \frac{\partial h}{\partial r}}=0 \left(t>0, \mathrm{The} \mathrm{boundary} BC \mathrm{and}OD\right)\hfill \\ \begin{array}{l}h(x,z,t)=H \left(t>0, \mathrm{The} \mathrm{boundary} ab \right)\\ h(D,z,t)=h_i \left(t>0, \mathrm{The} \mathrm{boundary} Ab \right)\end{array}\hfill \end{array}\right.$$

(3)

where h_i is the water head at each point on the hole wall, cm, and the value was equal to the distance from each point on the hole wall to the liquid surface.

2.2.3. Model Parameters

The mixed liquid of biogas slurry and water was considered as muddy water with particulate matter, due to the density and viscosity of biogas slurry limiting the diffusion process in soil [30]. Compared with clear water, muddy water had different the saturated hydraulic conductivity [31,32]. Additionally, the infiltration of biogas slurry affected the soil water characteristic parameters. In this study, the centrifuge method and variable water head method were employed to determine the soil water characteristic curve and soil saturated hydraulic conductivity under different biogas slurry ratio conditions. The soil water characteristic curve parameters were fitted using the van Genuchten–Mualem model [33], as required by the HYDRUS-2D model simulation. The soil parameters obtained for different biogas slurry ratios are presented in

Table 2. Soil characteristic parameters of different treatments.

Ratio of Biogas Slurry	θs (cm3 cm−3)	θr (cm3 cm−3)	ɑ (1 cm−1)	n (-)	Ks (cm min−1)
B0	0.3722	0.0510	0.0112	1.6688	0.00360
B1:8	0.3886	0.0549	0.0097	1.7223	0.00143
B1:6	0.3894	0.0554	0.0095	1.7469	0.00105
B1:4	0.3909	0.0580	0.0093	1.7551	0.00080

.

2.2.4. Statistical Analysis

Five indicators—coefficient of determination (R²), mean absolute error (MAE), root mean square error (RMSE), percent deviation (PBIAS) and Nash efficiency coefficient (NSE)—were used to evaluate the accuracy of numerical simulation and the established model [34]. The MAE and RMSE were closer to 0, −10 ≤ PBIAS ≤ 10, and the NSE was closer to 1. The better the agreement between the simulated and measured values, the smaller the difference.

3. Results and Discussion

3.1. Verification for Numerical Simulation

The HYDRUS-2D software was used to simulate the cumulative soil infiltration amount and soil wetting front migration distance with time under the different biogas slurry ratio treatments under D₃H₁₀ conditions. The simulated and measured results are compared in Figure 3a,b. The NSE of the simulated and measured values for the cumulative soil infiltration and soil wetting front migration distance under different biogas slurry ratio treatments was greater than 0.9524, the PBIAS was within ±0.3427. Thus, based on the measured values of soil characteristic parameters, HYDRUS-2D software can be used to simulate the soil water infiltration process under hole irrigation with different biogas slurry ratios, and the boundary conditions established in this study were reliable. The successful application of the HYDRUS model in the hole irrigation with different biogas slurry ratios system in this study expands its applicability under complex water-quality conditions, which contributes significantly to the study of the infiltration of drip irrigation systems using unconventional water sources.

3.2. Effects of Hole Irrigation Parameters on Cumulative Infiltration

Based on the simulated values, a relationship curve was constructed to represent the cumulative soil infiltration with time under different hole irrigation parameters (a, D₅H₁₀; b, B_1:6H₁₀; c, B_1:6D₅) (Figure 4). The soil cumulative infiltration increased with the extension of infiltration time for all hole irrigation parameter combinations. The increase in cumulative infiltration was faster at the initial stage and gradually decreased as the infiltration duration increased. This can be attributed to the fact that at the initial infiltration stage, the surface soil was dry with low water content and large soil matrix potential. The matrix adsorption force and the presence of interconnected large pores and conductive pores in the surface soil contributed to the low gas phase resistance and prominent capillary action of water infiltration, resulting in a rapid increase in the initial cumulative infiltration volume [35]. The low gas phase resistance of water infiltration and the more pronounced capillary action resulted in a rapid increase in the initial cumulative infiltration volume. As the experiment progressed, the water content of the soil gradually increased, leading to the expansion of the water-saturated layer on the soil surface. The increasing thickness of the water-saturated soil layer reduced the suction gradient of the soil matrix and ventilation pores, gradually increasing the gas phase resistance while decreasing the capillary action. Consequently, the infiltration rate gradually decreased. After a certain period of time, the suction force of the soil matrix weakened, approaching zero for both the gradient and capillary action. Under the influence of gravity, the infiltration rate stabilized at a relatively constant level [36,37].

Figure 4a shows the variation of soil cumulative infiltration over time for different biogas slurry ratios (B₀, B_1:8, B_1:6 and B_1:4) under D₅H₁₀ conditions. When the infiltration time reached 140 min, compared to the B₀ treatment, the cumulative soil infiltration under B_1:8, B_1:6, and B_1:4 was reduced by 48.08%, 49.88% and 59.62%, respectively. As the biogas slurry ratio increased, the cumulative soil infiltration amount decreased gradually. The reason was primarily attributed to the gradual increase in tiny organic suspended particles with the increased biogas slurry ratio. The organic suspended particles form a dense layer accumulating on the soil infiltration surface under the biogas slurry hole irrigation [31], which altered the upper boundary condition of hole infiltration and acted as a barrier for infiltration. Figure 4b illustrates the variation curve of the cumulative infiltration amount under different hole diameters over time at B_1:6H₁₀. As the infiltration time was 386 min, the cumulative soil infiltration amounts of D₉, D₇ and D₅ increased by 23.81%, 14.29%, and 10.07%, respectively, compared to D₃. As the hole diameter increased, the cumulative soil infiltration amount gradually increased. Similarly, Figure 4c presents the variation curve of the cumulative infiltration amount under different hole depths over time at B_1:6D₅. At the 296th minute, the cumulative soil infiltration amount of H₁₄, H₁₂, and H₁₀ increased by 52.94%, 34.12% and 17.52%, respectively, compared to H₈. It was suggested that as the hole depth increased, the cumulative soil infiltration amount also gradually increased. That was because during the process of biogas slurry hole irrigation, the hole served as the infiltration interface for water entering the soil. The bigger the hole, including the hole diameter and depth, directly the larger the infiltration area, providing more channels for water infiltration, resulting in an increased infiltration amount. As the hole depth increased, the volume of biogas slurry in the hole also increased, leading to an increase in the gravity field and more pronounced infiltration process.

3.3. Effect of Different Factors on the Migration of Wetting Front

3.3.1. Effect of Biogas Slurry Ratio on Soil Wetting Front Migration Distance

Under the conditions of D₅H₁₀ and D₇H₁₂, a numerical simulation analysis was conducted to investigate the migration distance of the soil wetting front under different biogas slurry ratios (Figure 5). The vertical migration distance (Z) and lateral migration distance (X) of the soil wetting front gradually increased over time. Moreover, it was found that the wetting front migration was slower under the higher biogas slurry ratio. Compared to B₀, when the infiltration time was 140th min, the X values under B_1:8, B_1:6 and B_1:4 decreased by 28.26%, 31.37% and 38.49%, respectively (Figure 5a). Similarly, the Z values decreased by 14.76%, 19.58%, and 27.62%, respectively. With the infiltration time of 100 min, the X values of B_1:8, B_1:6 and B_1:4 decreased by 25.81%, 31.10% and 41.71%, respectively, while the Z values decreased by 16.89%, 19.94% and 24.51%, respectively, compared to B₀ (Figure 5b). The migration distance of the soil wetting front decreased as the ratio of biogas slurry increased. This was mainly due to the deposition of organic suspended particles from the biogas slurry, forming a dense layer on the soil surface under biogas slurry hole irrigation. The dense layer acted as a barrier to infiltration, changing the upper boundary of the hole infiltration. As the proportion of biogas slurry increased, the number of organic suspended particles also increased, further strengthening the blocking effect, which led to a decrease in the soil wetting front migration distance. This highlighted the obvious impact of the proportion of biogas slurry on the soil wetting front migration distance; biogas slurry ratio should be considered as an essential factor in practical agricultural irrigation systems.

3.3.2. Effect of Hole Diameter on Soil Wetting Front Migration Distance

Numerical simulation analysis was conducted to investigate the migration distance of the soil moistening front under different hole diameters at both B_1:6H₁₀ and B₀H₈ (Figure 6). The results showed that the larger the hole diameter, the faster the migration of the soil wetting front. In Figure 6a, at the infiltration time of 386 min, X increased by 17.06%, 14.95% and 11.59% under D₉, D₇ and D₅, respectively, compared to D₃. Similarly, Z increased by 10.82%, 6.22% and 4.73% under D₉, D₇ and D₅, respectively. In Figure 6b, at the infiltration time of 144 min, X increased by 25.18%, 11.76% and 5.88% under D₉, D₇ and D₅, respectively, compared to D₃. Similarly, Z increased by 8.72%, 5.91% and 3.64% under D₉, D₇ and D₅, respectively. This indicated that the soil water front migration distance increased with the increase in the hole diameter, and the effect of hole diameter on the lateral migration distance was more obvious than on the vertical migration distance. The reason was that the area at the bottom of the hole increased as the hole diameter increased. This led to an increase in the available channels for water infiltration and the soil wetting front migration distance. Additionally, the lateral wetting front migration distance changed from zero to the hole diameter. As the hole diameter increased, the transverse soil wetting front migration distance also increased.

3.3.3. Effect of Hole Depth on Soil Wetting Front Migration Distance

Numerical simulation analysis was conducted to investigate the soil wetting front migration distance under different hole depths, specifically under B_1:6D₅ and B₀D₇ (Figure 7). In Figure 7a, at the infiltration time of 296 min, X under H₁₄, H₁₂, and H₁₀ increased by 10.11%, 6.25% and 2.52%, respectively, compared to H₈, while Z increased by 35.03%, 25.38% and 12.23%, respectively. In Figure 7b, at the infiltration time of 86 min, X under H₁₄, H₁₂, and H₁₀ increased by 8.37%, 4.65% and 3.48%, respectively, compared to H₈, while Z increased by 34.14%, 21.95% and 12.19%, respectively. The soil wetting front migration distance increased as the hole depth increased, and the influence of hole depth on the soil vertical wetting front migration distance was more significant than on the soil lateral wetting front migration distance. As the hole depth increased, the water head in the hole also increased. This led to an increase in lateral pressure potential energy, gravity potential energy, and the area of the side wall of the hole. Consequently, the channels for water infiltration increased, and the soil wetting front migration distance continued to increase. At the beginning of infiltration, the vertical soil wetting front migration distance increased from zero to the hole depth. The soil vertical wetting front migration distance increased as the hole depth increased. Therefore, both the hole diameter and depth played important roles in determining the soil wetting front migration distance, which were crucial technical factors under the hole irrigation system.

3.4. Establishment and Evaluation of Soil Wetting Front Migration Distance Model for Biogas Slurry Hole Irrigation

3.4.1. Establishment of Soil Wetting Front Migration Distance Model for Biogas Slurry Hole Irrigation

The current research conducted numerical simulation analysis to investigate the factors influencing soil wetting front migration distance under biogas slurry hole irrigation. The results indicated that all the biogas slurry ratios, hole diameter, hole depth and irrigation time exerted an impact on both the soil vertical and lateral wetting front migration distance. Therefore, these factors should be considered in the design of irrigation systems. To further reveal the relationship between the soil wetting front migration distance and the irrigation time under different scenarios, a power function was established [16]:

$$\left\{\begin{array}{l}X=a{t}^b\hfill \\ Z=c{t}^d\hfill \end{array}\right.$$

(4)

The results showed that the coefficients of determination (R²) of soil wetting front migration distances for the 64 groups under biogas slurry hole irrigation were all greater than 0.977, indicating a high level of significance. This suggested that the power function accurately described the relationship between the soil wetting front migration distance and time under the different hole sizes (hole diameters and depths) and biogas slurry ratios. Further analysis revealed that the changes in the fitting parameters b and d varied across different scenarios, with no clear regular patterns. The values of b ranged from 0.2082 to 0.2967, while d ranged from 0.0972 to 0.1688. To simplify the calculation, the average values of b and d were 0.24 and 0.124, respectively. By substituting the values of b and d into Equation (4), the original Equation (4) can be further transformed.

$$\left\{\begin{array}{l}X=a{t}^{0.24}\hfill \\ Z=c{t}^{0.124}\hfill \end{array}\right.$$

(5)

Equation (5) was used to fit the HYDRUS-2D simulation results and obtain parameters a and c. The fitting results showed that R² ≥ 0.947, indicating the feasibility of simplifying the model by averaging the fitting parameters b and d. Further analysis revealed that the values of a and c varied significantly across different scenarios. However, a strong power function relationship (R² ≥ 0.971) with the steady seepage rate f₀ was obtained (Figure 8).

Both a and c increased with the increase in f₀ (Figure 8), and the power function relationship can be expressed as

$$a=4.442f_0^{0.375}$$

(6)

$$c=11.998f_0^{0.287}$$

(7)

The a and c were substituted into Equation (5), respectively, and the equation converted to the following:

$$\left\{\begin{array}{l}X=4.442f_0^{0.375}{t}^{0.24}\hfill \\ Z=11.988f_0^{0.287}{t}^{0.124}\hfill \end{array}\right.$$

(8)

Equation (8) solely incorporated the undetermined parameter f₀. Hence, accurately estimating the migration distance of the soil moist front hinged on determining f₀. The analysis found that f₀ was consistent with hole diameter (D) and depth (H) as follows:

$$f_0=e{D}^f$$

(9)

$$f_0=g{H}^h$$

(10)

The specific fitting results are presented in

Table 3. Fitting parameters of stable infiltration rate and hole diameter.

Ratio of Biogas Slurry	H/cm	e/(-)	f/(-)	R2
0	8	1.2913	0.2547	0.9900
	10	1.5836	0.2474	0.9838
	12	1.9880	0.2283	0.9950
	14	2.3843	0.2266	0.9889
1:8	8	0.4024	0.2595	0.9873
	10	0.5051	0.2447	0.9870
	12	0.6363	0.2321	0.9971
	14	0.7847	0.2205	0.9815
1:6	8	0.3791	0.2576	0.9857
	10	0.4708	0.2440	0.9877
	12	0.5917	0.2309	0.9969
	14	0.7135	0.2270	0.9811
1:4	8	0.3152	0.2092	0.9882
	10	0.3502	0.2196	0.9899
	12	0.3783	0.2334	0.9971
	14	0.4707	0.2183	0.9887

and

Table 4. Fitting parameters of stable infiltration rate and hole depth.

Ratio of Biogas Slurry	D/cm	g/(-)	h/(-)	R2
0	3	0.1947	1.0418	0.9951
	5	0.2223	1.0283	0.9957
	7	0.2738	0.9766	0.9940
	9	0.2855	0.9926	0.9944
1:8	3	0.0531	1.1088	0.9947
	5	0.0591	1.1073	0.9961
	7	0.0751	1.0403	0.9917
	9	0.0808	1.0444	0.9951
1:6	3	0.0546	1.0661	0.9951
	5	0.0601	1.0687	0.9961
	7	0.0756	1.0075	0.9906
	9	0.0804	1.0154	0.9942
1:4	3	0.0544	1.0045	0.9908
	5	0.0572	1.0173	0.9867
	7	0.0639	0.9942	0.9881
	9	0.0644	1.0045	0.9908

. The coefficient of determination (R²) for the fitting equations under each treatment exceeded 0.98, reaching an extremely significant level. The changes in the fitting parameters f and h under different treatments were minimal and not obvious, f ranged from 0.2092 to 0.2594, while h ranged from 0.9926 to 1.1088. To simplify the calculation, the average values of f and h were taken, resulting in f of 0.236 and h of 1.042. Accordingly, it was proposed that there was a functional relationship between f₀ and both hole diameter (D) and depth (H) as follows:

$$f_0=k{D}^{0.236}{H}^{1.042}$$

(11)

The HYDRUS-2D simulation results were fitted using Equation (11) (Figure 9). It was observed that for any proportion of biogas slurry, it was reliable to consider f and h as 0.236 and 1.042, respectively (R² ≥ 0.9928). Additionally, a power function relationship existed between k and the saturated hydraulic conductivity (Ks) for different biogas slurry ratios (Figure 10). The R² for the fitting equation was greater than 0.9928, indicating that this function accurately described the relationship between Ks and the fitting parameter k.

A soil wetting front migration model for biogas slurry hole irrigation can be constructed as follows:

$$\left\{\begin{array}{l}X=4.442f_0^{0.375}{t}^{0.24}\hfill \\ Z=11.988f_0^{0.287}{t}^{0.124}\hfill \\ f_0=96.947K_s^{1.151}{D}^{0.236}{H}^{1.042}\hfill \end{array}\right.$$

(12)

Formula (12) contained only one undetermined parameter, K_s, which was related to the ratio of biogas slurry and soil texture, which needed to be measured using either the constant head method or the variable head method.

3.4.2. Evaluation of Soil Wetting Front Migration Distance Model for Biogas Slurry Hole Irrigation

The wetting front migration distance model was evaluated and verified using different biogas slurry ratios at H₅D₃, H₅D₅ and H₅D₇ (Figure 11). The simulated statistical indicators for different parameter groups are presented in

Table 5. Comparative evaluation index of calculated and measured values of wetting front migration distance.

D/cm	NSE	RMSE	PBIAS	MAE
3	0.9992	0.0281	−0.2376	0.3196
5	0.9996	0.0010	0.0087	0.4667
7	0.9763	0.1264	−0.2235	0.6611

. The NSE values ranged from 0.9763 to 0.9996, RMSE values ranged from 0.001 to 0.1264, PBIAS values ranged from −0.2235 to 0.0087, and MAE values ranged from 0.3196 to 0.6611. These results demonstrated that the model was highly reliable and can be effectively used for predicting the soil wetting front migration distance under hole irrigation with different ratios of biogas slurry.

3.5. Establishment and Evaluation of Soil Cumulative Infiltration Model for Biogas Slurry Hole Irrigation

3.5.1. Establishment of Soil Cumulative Infiltration Model for Biogas Slurry Hole Irrigation

Figure 12 illustrates the variation process of the soil wetting front migration distance under B₀D₅H₁₀ and B_1:6D₃H₁₂ over time. It clearly showed that the soil wetting front migration distance was well fitted to the elliptic curve at various time points.

In order to further determine the fitting performance between the soil wetting front migration distance and the elliptic curve, correlation analysis was performed on the two sets of measured results (

Table 6. Determination coefficients between wetting front and elliptic curve.

t (min)	B0D5H10	t (min)	B1:6D3H12
7	0.9566	20	0.9926
26	0.9891	74	0.9890
54	0.9950	158	0.9961
90 140	0.9919 0.9961	264 416	0.9880 0.9900

).

The analysis revealed that the R² for the two sets of experiments conducted at different times varied between 0.9566 and 0.9961, which is almost approaching one. This suggests that the fitting was highly accurate and the shape of the soil wetting front migration distance could be effectively described by an elliptical equation. Then, the wetting area can be obtained by subtracting the hole area from the ellipse area, which was expressed as

$$S=\pi XZ-{\displaystyle \frac{1}{2}}DH$$

(13)

By substituting Ks = 0.0036 cm min⁻¹, D = 5 cm, H = 10 cm Ks = 0.00105 cm min⁻¹, D = 3 cm, and H = 12 cm into Equations (12) and (13), respectively, the relationship between the soil wet area and infiltration time was obtained (Figure 13). The analysis revealed that the variation pattern of the soil wetting area was consistent with the variation in cumulative infiltration amount. Consequently, the relationship correlation between cumulative infiltration amount and the soil wetting area under B₀D₅H₁₀ and B_1:6D₃H₁₂ was analyzed, as show in Equation (14).

$$I=\alpha S$$

(14)

Using Equation (14), the cumulative infiltration amount and wetted area of soil under different scenarios of biogas slurry cave irrigation were fitted (Figure 14). The results showed that the R² values were both greater than 0.972, reaching an extremely significant level. Equation (14) accurately described the relationship between the cumulative infiltration amount of biogas slurry hole irrigation and the area of the soil wetting body, taking into account different ratios of hole diameter and depth. Subsequent analysis showed that the fitting parameter α ranged from 0.3291 to 0.3578 across various treatments, with only small numerical variations. To simplify the calculation, the average value (α = 0.3365) was incorporated into Equation (14). The original equation can be further transformed as follows:

$$I=0.3365S$$

(15)

3.5.2. Evaluation of Soil Cumulative Infiltration Model for Biogas Slurry Hole Irrigation

Equation (15) represented the relationship between the cumulative infiltration of soil and the soil wetting area under hole irrigation. By measuring the horizontal and vertical soil wetting front migration distances, the soil cumulative infiltration can be simulated using Equations (13) and (15). To validate the accuracy of Equation (15), the values of soil cumulative infiltration with different biogas slurry ratios were evaluated under H₅ and H₈ conditions (Figure 15). The corresponding evaluation index results are presented in

Table 7. Comparative evaluation index of cumulative infiltration calculated and measured value.

Ratio of Biogas Slurry	NSE	RMSE	PBIAS	MAE
B0	0.9990	0.5913	0.0673	1.3087
B1:8	0.9817	0.2293	−0.0262	1.3894
B1:6	0.9993	0.8626	0.0982	1.3816
B1:4	0.9997	0.8825	−0.1482	2.6522

. The NSE values ranged from 0.9817 to 0.9997, RMSE values ranged from 0.2293 to 0.8825, PBIAS values ranged from −0.0262 to 0.0982, and MAE values ranged from 1.3087 to 2.6522, which indicated the performance of the model for predicting the soil cumulative infiltration under hole irrigation with varying proportions of biogas slurry was credible.

The Philip model is currently the most widely used line-source infiltration method [38,39]. However, this model relies on soil infiltration test data, necessitating a substantial number of experiments to ascertain the parameters prior to application. In contrast, as a physical model, Equation (15) requires only the measurement of the saturated hydraulic conductivity of the soil to directly calculate the cumulative infiltration amount, presenting significant advantages over traditional empirical models. Furthermore, the dynamic distribution characteristics of crop roots during the growth period render the water distribution within the soil’s wet body a critical factor influencing growth. The model proposed in this study determines the appropriate soil wetting front migration distance based on the root distribution range, subsequently allowing for the accurate calculation of irrigation water requirements and the determination of the hole diameter and depth for biogas slurry hole irrigation. This approach provides a theoretical basis for the precision irrigation practices associated with biogas slurry hole irrigation.

3.6. Limitations of the Current Study and Future Outlook

Although this study explored the precision irrigation of biogas slurry hole irrigation and provided useful insights, its limitations cannot be ignored. From the perspective of model verification, the feasibility of the model has only been preliminarily validated through indoor infiltration tests. However, the farmland environment is complex and variable; thus, the actual application effectiveness of the model at the farmland scale and its accuracy require further validation. Regarding experimental conditions, this study was conducted under dry soil conditions, whereas the actual initial soil moisture content can vary significantly, which may affect key parameters, such as the soil wetting front migration distance and cumulative infiltration. Therefore, it is essential to further investigate the parameter changes under different initial soil water contents to enhance the model’s practicality. In light of this, future research can advance in the following areas: first, deepen the field-scale verification efforts and comprehensively assess the model’s applicability in complex environments, such as those exhibiting soil heterogeneity; second, conduct field experiments and design hole irrigation devices capable of dynamically adjusting the hole diameter and depth according to crop growth requirements, ensuring that the wetted area closely aligns with the distribution range of crop roots; finally, develop a dynamic irrigation decision model that comprehensively considers multiple factors, including soil moisture and crop growth stages, to achieve precise matching of the wetted area and crop roots, thereby providing robust support for the promotion and application of biogas slurry hole irrigation technology.

4. Conclusions

In this research, based on the HYDRUS-2D method, a soil wetting front migration distance model was established, and the quantitative relationship between the cumulative infiltration amount and the wetted soil body was analyzed. We have optimized both the parameters and the corresponding model simulation, which improves the accurate utilization of water resources and model evaluation in the hole irrigation process. The model was verified with the field scale. We draw the following conclusions:

(1)

The constructed hole irrigation HYDRUS model more accurately describes the water transport characteristics of the biogas slurry hole irrigation, expanding the application scenario of the HYDRUS model.

(2)

The infiltration process of biogas slurry is greatly affected by its ratio, hole diameters and hole depths. Its stable seepage rate is strongly power-law related to pore size and depth (R² ≥ 0.98). Meanwhile, the lateral and vertical migration distance of the soil wetting front exhibits a robust power function relationship with stable infiltration rate and infiltration time.

(3)

The soil wetting front curve under biogas slurry hole irrigation can be described using an elliptic curve equation. Additionally, the cumulative infiltration of the soil irrigated with biogas slurry through holes exhibited a linear relationship with the soil wetting area.

(4)

The established wetting front migration distances model and cumulative infiltration model under biogas slurry hole irrigation demonstrated high reliability and accurately depicted the variations in hole irrigation parameters and biogas slurry ratios for soil moisture dynamics.