Abstract

Water infiltration in the vadose zone plays a crucial role in the hydrological cycle. Understanding the relationships among rainfall, ponded water, and soil infiltration processes is essential for assessing soil moisture recharge and precipitation redistribution in desert environments. This study focused on interdune lowlands in natural sand dunes, conducting in-situ profile infiltration experiments. The Kostiakov-Lewis, Green-Ampt, Philip, Hydrus-1D, and Hydrus-2D/3D infiltration models were used to simulate vertical soil water infiltration processes, aiming to identify acceptable methods for ponded infiltration in sandy soils. Comparison of simulated and measured results indicated that, considering validation indices such as sum of squared errors and root mean square error, the Philip model could predict infiltration rate, cumulative infiltration, and wetting front advancement in sandy soil, with mean RMSE and R² values reaching 0.018 cm and 0.93, respectively. The Hydrus-3D model simulated soil water content more effectively than Hydrus-2D throughout the wetted zone. The combination of the Philip model and Hydrus-3D can effectively estimate ponded infiltration parameters and simulate water transport processes, thereby improving research on soil water infiltration.

Keywords: vadose zone; ponded infiltration; numerical simulation; sandy soil

Introduction

Infiltration, the process of water entering soil, serves as a critical link between surface water resources and rainfall, playing a vital role in regulating water distribution in the hydrological cycle. Numerous scholars have conducted in-depth studies across different scales, soil types, and climatic regions. The Hydrus-1D software, developed in laboratory settings, can be widely used to simulate one-dimensional water movement in unsaturated media, while Hydrus-2D/3D software is applied to simulate two- or three-dimensional soil water movement and spatial distribution, having received extensive validation and application.

Dune interdune lowlands represent many typical landscape units in arid oasis-desert regions, such as deserts, sand dunes, and farmlands. Among these, sand dunes and their interdune lowlands play important roles in hydraulic connectivity and water cycling in desert ecosystems, affecting the structural and functional evolution of desert ecosystems and oasis stability. In areas with annual precipitation less than 200 mm, rainfall is characterized by scarcity, high variability, and short duration. Although precipitation is minimal, it constitutes the sole water source in arid ecosystems. Particularly, shallow soil moisture storage regulates surface runoff and rainfall infiltration and redistribution processes.

Ponded infiltration occurs when infiltration intensity is less than rainfall intensity, resulting in pressure infiltration. This phenomenon is particularly common in arid regions with thick soil crusts, where prolonged water residence time reduces infiltration capacity. In interdune lowlands, decreased infiltration rates lead to increased soil moisture content in the underlying surface.

Various infiltration models have been proposed for heterogeneous soils to characterize parameters such as infiltration rate and cumulative infiltration, including approximate physical models like the Kostiakov, Horton, Green-Ampt, and Philip equations, as well as the Richards equation algorithm. However, the applicability of different models varies across soil conditions. The Kostiakov equation is simple and can fit measured infiltration data well, but its parameters lack physical meaning and cannot provide detailed information about the infiltration process. The Philip model is commonly used to estimate final infiltration rates and has been validated in sandy loam farmland soils. The Green-Ampt model is primarily used for ponded infiltration in homogeneous soil media, and its simplicity and physical basis have been extended to simulate infiltration, rainfall, and runoff processes under various conditions.

Unsaturated water transport processes are mainly simulated using the Richards equation, which is based on mass conservation and Darcy's laws. The Richards equation is typically nonlinear and cannot be solved analytically, leading to complex calculations, particularly under complicated initial and boundary conditions. Developed by the USDA Salinity Laboratory, the HYDRUS series of software packages numerically solves the Richards equation for variably saturated water flow and solute transport.

Various methods exist for studying soil infiltration processes, such as soil column repacking, single- and double-ring infiltrometers, and soil infiltration instruments. Among these, the single-ring constant-head method enables in-situ measurement of water infiltration and can be used to estimate soil infiltration characteristics. However, field experiments require substantial time and expense, making simple and reliable data processing methods for infiltration tests particularly necessary. Due to soil heterogeneity, many infiltration models still have significant limitations in simulating soil moisture changes. Therefore, identification and selection of soil water transport models under different spatiotemporal conditions require continuous experimentation. Based on this, our study conducted field in-situ infiltration experiments to analyze water infiltration processes in sandy soils and evaluate the performance of different models in simulating soil water infiltration, aiming to identify suitable models for rainfall conditions to reasonably predict dune ponded infiltration processes and enhance understanding of local soil vadose zone water movement and exchange processes in desert dunes.

1.1 Study Area Overview

The study was conducted at the Linze Inland River Basin Research Station of the Chinese Ecosystem Research Network (CERN), located at the northern edge of the Badain Jaran Desert in Gansu Province (39°24′41″N, 100°9′30″E, elevation 1381 m). The area has a mean annual temperature of 7.5°C, with minimum temperatures reaching -27°C. Annual potential evaporation ranges from 1900 to 2088 mm, while average annual rainfall is 120 mm. Humidity varies from 7.3% to 80.9% throughout the year, peaking in July. Rainfall is the most sensitive meteorological factor in the study region. The groundwater depth is 4.9 m, and the saturated capillary rise of groundwater does not affect the surface soil layer. During the study period, the region received approximately 10 rainfall events.

1.2 In-situ Experimental Design

Single-ring infiltration tests were conducted in the selected experimental area. First, surface residues that might hinder ring insertion were removed. An iron ring (diameter: 20 cm) was then driven into the soil to an insertion depth of approximately 5 cm using a drop hammer. Care was taken to keep the ring edge vertical to the soil surface and minimize soil disturbance. Before the experiment began, filter paper was placed inside the ring to prevent water flow from disrupting the soil surface and causing uneven water heads. Soil moisture observation probes were inserted at the center position below the infiltration ring at depths of 10, 20, 40, 60, and 80 cm. A data collector continuously recorded soil moisture changes at 1-minute intervals.

1.3 Soil Sample Collection and Analysis

Undisturbed soil samples were collected from the 0-80 cm profile using an auger, with all samples taken from the center of the infiltration profile. The oven-drying method was used to determine soil bulk density, the soaking method measured saturated water content, a laser particle size analyzer obtained soil particle size distribution, and the constant-head method determined saturated hydraulic conductivity (Ks). Soil hydraulic parameters are critical factors for water movement simulation. The main characteristic data are presented in Table 1.

Table 1 Soil hydraulic and structural properties in experimental site

Parameter Value Initial water content (g·cm⁻³) Soil particle size distribution (<0.002 mm) (%) Soil particle size distribution (0.002-0.05 mm) (%) Soil particle size distribution (>0.05 mm) (%) Porosity connectivity parameter (dimensionless)

2.1 Hydrus-1D/3D Model Theory

2.1.1 Soil Hydraulic Properties

Three-dimensional water flow can be described by the Richards equation:

$$\frac{\partial \theta}{\partial t} = \frac{\partial}{\partial x}\left[K(h)\frac{\partial h}{\partial x}\right] + \frac{\partial}{\partial y}\left[K(h)\frac{\partial h}{\partial y}\right] + \frac{\partial}{\partial z}\left[K(h)\frac{\partial h}{\partial z} - K(h)\right] + S$$

where θ is volumetric water content (cm³ cm⁻³), h is pressure head (cm), K(h) is the unsaturated hydraulic conductivity function (cm min⁻¹), t is time (min), z is the spatial coordinate positive upward (cm), and S is the sink-source term, typically representing root water uptake (min⁻¹). This experiment was conducted on bare sand dunes without vegetation, so root water uptake was not considered.

The simplified one-dimensional water flow equation is:

$$\frac{\partial \theta}{\partial t} = \frac{\partial}{\partial z}\left[K(h)\frac{\partial h}{\partial z} - K(h)\right]$$

Soil hydraulic properties were described using the Van Genuchten-Mualem model. The soil hydraulic functions are:

$$S_e = \frac{\theta - \theta_r}{\theta_s - \theta_r} = \left[\frac{1}{1 + |\alpha h|^n}\right]^m$$

$$K(h) = K_s S_e^l \left[1 - (1 - S_e^{1/m})^m\right]^2$$

where θ_r and θ_s are residual and saturated water content (cm³ cm⁻³), respectively; S_e is effective saturation (dimensionless); K_s is saturated hydraulic conductivity (cm min⁻¹); n is the pore size distribution index (dimensionless); m = 1 - 1/n is a soil water retention curve parameter; α is an empirical parameter (cm⁻¹); and l is the pore connectivity parameter (dimensionless). Soil hydraulic parameters for different layers are shown in Table 2.

Table 2 Soil parameters for different soil layers in the experiment site

Depth (cm) Saturated water content θ_s (cm³ cm⁻³) Residual water content θ_r (cm³ cm⁻³) Pore size distribution index n (dimensionless) Saturated hydraulic conductivity K_s (cm·d⁻¹) 0-20 20-40 40-60 60-80

2.1.2 Initial and Boundary Conditions

For Hydrus-1D simulation, the upper boundary was an atmospheric boundary, and the lower boundary was a free drainage boundary. The simulation domain infiltration surface quickly reached saturation after infiltration began, representing a constant head boundary. The boundary and initial conditions were:

Initial condition: h(z,0) = -200 cm (0 ≤ z ≤ 200 cm); θ(z,0) = θ_i (0 ≤ z ≤ 100 cm)

Upper boundary: h(0,t) = 0 cm (0 ≤ t ≤ 100 cm); q(0,t) = -K(h)(∂h/∂z + 1) = i(t), 0 ≤ t ≤ 200 cm

Lower boundary: ∂h/∂z = 0 (z = 200 cm, t = 0-100 cm)

For Hydrus-2D and Hydrus-3D simulations, the water flow boundary conditions under single-ring infiltration were applied (Fig. 3). The simulation domain was discretized into radial grid spacing of 1 cm and vertical grid spacing of 1 cm. The three-dimensional simulation domain had an infiltration surface with a diameter R = 20 cm. The upper boundary was an atmospheric boundary, side boundaries were impermeable, and the lower boundary was free drainage. The infiltration circular surface quickly reached saturation after infiltration began, forming a constant head boundary. The boundary and initial conditions were:

Initial condition: h(x,y,z,0) = -200 cm (0 ≤ x ≤ 200 cm; 0 ≤ y ≤ 100 cm; 0 ≤ z ≤ 200 cm)

Boundary conditions:
- Upper boundary: h(x,y,0,t) = 0 cm (0 ≤ x ≤ 200 cm; 0 ≤ y ≤ 100 cm)
- Side boundaries: ∂h/∂x = 0 (x = 0 or 200 cm); ∂h/∂y = 0 (y = 0 or 100 cm)
- Lower boundary: ∂h/∂z = 0 (z = 200 cm)

The total simulation duration was 200 minutes.

Fig. 3 The Hydrus 2D/3D simulation domain and boundary conditions

2.2 Infiltration Model Theory

Green-Ampt Model

The Green-Ampt model initially analyzed ponded infiltration in soil columns under constant head conditions using Darcy's law. The infiltration equation is:

$$i = K_s \left(1 + \frac{S_f}{Z_f}\right)$$

$$I = K_s t + S_f (\theta_s - \theta_i) \ln\left(1 + \frac{I}{S_f (\theta_s - \theta_i)}\right)$$

where I is cumulative infiltration (cm), i is infiltration rate (cm min⁻¹), K_s is saturated hydraulic conductivity (cm min⁻¹), S_f is wetting front suction (cm), Z_f is the conceptual wetting front depth (cm), and θ_i is initial soil water content (cm³ cm⁻³).

For layered homogeneous soil infiltration, cumulative infiltration and wetting front advancement over time can be derived through integration:

$$Z_f = \frac{I}{\sum_{j=1}^{N} D_j (\theta_{s,j} - \theta_{i,j})}$$

$$t_N = \frac{1}{K_{s,N}} \left[Z_{f,N} - S_{f,N} \ln\left(1 + \frac{Z_{f,N}}{S_{f,N}}\right)\right] - \frac{1}{K_{s,N-1}} \left[Z_{f,N-1} - S_{f,N-1} \ln\left(1 + \frac{Z_{f,N-1}}{S_{f,N-1}}\right)\right]$$

where D is soil layer thickness (cm), and subscripts i, s, j, and N represent initial state, saturated state, layer number, and saturated layer number, respectively.

Philip Model

The Philip model expresses cumulative infiltration and infiltration rate as:

$$I(t) = St^{1/2} + At$$

$$i(t) = \frac{1}{2}St^{-1/2} + A$$

where A is a function of soil properties and hydraulic conductivity (cm min⁻¹), and S is sorptivity, a function of soil matrix potential (cm min⁻¹/²).

Kostiakov-Lewis Model

The Kostiakov model, modified by Lewis for long-term periods, is described as:

$$i(t) = i_f + Bt^{-C}$$

$$I(t) = i_f t + \frac{B}{1-C}t^{1-C}$$

where B and C are equation parameters (C > 0, 0 < C < 1), and i(t) and i_f are infiltration rate and stable infiltration rate, respectively.

3 Results

3.1 Observed Infiltration Characteristics

Results showed that infiltration rate continuously decreased after the experiment began, reaching a stable infiltration rate of approximately 0.57 cm·min⁻¹ after about 200 minutes. Cumulative infiltration increased progressively over time, reaching 398 cm within the observation period. The wetting front advanced rapidly in both vertical and horizontal directions after infiltration began, with the advancement rate slowing thereafter. At the end of infiltration, the horizontal wetting front distance was greater than the vertical distance. Power functions could fit the measured data well (R² = 0.85). However, power functions are simple empirical models for estimating soil water infiltration without specific physical parameters.

Furthermore, results indicated significant differences in soil water content across profiles within the 0-80 cm depth (Fig. 4). Soil water content decreased with depth from the surface to steady-state, but the maximum water content in the 10-80 cm profile was significantly lower than in other profiles and below saturated water content, likely related to soil properties such as bulk density and particle size composition.

Fig. 4 Dynamic of observed infiltration rate, cumulative infiltration, depth of wetting front and soil water content during the study period

3.2.1 Infiltration Rate

Comparisons between simulated and observed infiltration rates are shown in Fig. 5. Results indicated that the R² coefficient for infiltration rate ranged from 0.75 to 0.95 across the five hydrological models. The Philip model showed the highest R² coefficient. The Kostiakov-Lewis model slightly overestimated infiltration rates compared to observed values, while the Green-Ampt model slightly underestimated them, particularly in the later infiltration stage. The Hydrus-1D simulation showed lower agreement with measurements, with R² coefficients below 0.75. Overall, under ponded infiltration conditions in desert sandy soil, the Philip model better described soil water infiltration compared to other models.

Model performance was evaluated using the coefficient of determination (R²) and root mean square error (RMSE):

$$R^2 = 1 - \frac{\sum_{i=1}^{n}(X_i - \hat{X}i)^2}{\sum$$}^{n}(X_i - \bar{X})^2

$$RMSE = \sqrt{\frac{\sum_{i=1}^{n}(X_i - \hat{X}_i)^2}{n}}$$

where X_i represents the ith observed value, \hat{X}_i is the simulated value, \bar{X} is the mean of observed values, and n is the total number of observations. Smaller RMSE values and R² values closer to 1 indicate better simulation accuracy.

Fig. 5 Comparison of simulated infiltration rate from models with observed result

3.2.2 Cumulative Infiltration

The relationship between simulated and observed cumulative infiltration is shown in Fig. 6. Results demonstrated that all five models showed good correlation with measured data. The Philip model had the highest R² coefficient, while the Kostiakov-Lewis model had the lowest. The Green-Ampt model significantly overestimated cumulative infiltration, particularly in the early infiltration stage. However, both Hydrus-1D and Hydrus-3D models had relatively high R² coefficients, indicating they are effective models for predicting cumulative infiltration in sandy soil, especially the Philip model.

Fig. 6 Comparison of simulated cumulative infiltration from models with observed result

3.2.3 Wetting Front Distance

Comparisons between simulated and observed wetting front distances are presented in Fig. 7. Results showed that Hydrus model simulations differed significantly from observed values, particularly the Green-Ampt model, which substantially overestimated wetting front advancement, with R² values less than 0.75. Based on mean R² results, the Philip model predicted wetting front advancement in sandy soil more accurately.

A possible explanation is that Hydrus-1D ignores lateral water flow and air flow. Previous studies have shown that Hydrus-2D overestimates the water storage capacity of the wetted zone. However, considering both R² and RMSE values, Hydrus-3D is an ideal choice for describing and simulating soil water content under ponded conditions in sandy soil.

Fig. 7 Comparison of simulated wetting front depth from models with observed result

Table 3 Goodness-of-fit parameters for simulation results with models

Model Infiltration Rate Cumulative Infiltration Wetting Front Distance RMSE (cm·min⁻¹) R² RMSE (cm) Kostiakov-Lewis Green-Ampt Philip Hydrus-1D

Note: RMSE represents root mean square error, R² represents coefficient of determination. The same below.

3.2.4 Soil Water Content with Hydrus-2D

Hydrus-2D simulated soil water content is shown in Fig. 8. Results indicated low agreement between simulated and observed soil moisture, with mean R² of only 0.75, particularly in the 0-10 cm surface layer where R² was just 0.45. Based on R² and RMSE mean values, the Hydrus-2D model could not effectively predict soil water content changes under ponded infiltration conditions.

Fig. 8 Comparison of simulated soil water content by Hydrus 2D model with observed result

3.2.5 Soil Water Content with Hydrus-3D

Hydrus-3D simulated soil water content is shown in Fig. 9. Results demonstrated that simulated and observed soil water content values were essentially consistent, with mean R² of 0.93 for the 0-80 cm profile. Although soil water content was slightly underpredicted in the later infiltration stage, considering both R² and RMSE values, Hydrus-3D is an ideal choice for describing and simulating soil water content under ponded conditions in sandy soil.

Fig. 9 Comparison of simulated soil water content by Hydrus 3D model with observed result

Table 4 Goodness-of-fit parameters for simulation results between Hydrus-2D and Hydrus-3D

Depth (cm) Hydrus-2D Hydrus-3D RMSE (cm³ cm⁻³) R² 0-10 10-20 20-40 40-60 60-80

Fig. 10 Comparison of observed the velocity of the wetting front with two directions

4 Discussion

4.1 Soil Water Infiltration Characteristics

The single-ring ponded infiltrometer is one of the most commonly used methods for measuring soil water infiltration rates and has been widely applied to assess and monitor rainfall infiltration. Research indicates that infiltration rate decreases with increasing infiltration amount, eventually reaching a stable infiltration rate. This study found that the stable infiltration rate for sandy soil was 0.57 cm·min⁻¹, consistent with previous studies reporting sandy soil infiltration rates ranging from 0.3 to 0.77 cm·min⁻¹. Differences in results may be attributed to soil texture, initial soil water content, and surface cover conditions.

During soil water transport, studying the dynamic characteristics of wetting front advancement in the unsaturated zone is important. Research has shown that single-ring infiltration is affected by lateral water movement beneath the ring, with matrix potential flux dominating at the wetting front. This study found that horizontal wetting front distance was greater than vertical distance. Additionally, within the first 100 minutes after infiltration began, wetting front advancement velocity decreased over time before remaining constant. A possible explanation is that funnel flow induced by sandy soil is driven by matrix potential in the horizontal direction and by gravitational potential in the vertical direction.

4.2 Soil Profile Water Status

Soil moisture distribution is an important factor affecting water uptake effectiveness by plant roots in arid desert regions. Results showed that soil water content was higher in surface layers than in deep layers, with ponded infiltration causing non-uniform, unsaturated distribution throughout the wetted zone. This may be related to the influence of soil structure on water holding capacity. Although the entire profile consisted of sandy soil, differences in particle size composition ratios between layers led to slight differences in water holding capacity, resulting in spatial variability of soil water content.

Previous studies have shown that the Philip model can be used to predict the relationship between infiltration rate and time in sandy soils, with similar reports confirming its suitability for cumulative infiltration prediction. Zolfaghari et al. found that Hydrus-1D can predict infiltration rates in layered soil columns under ponded conditions. However, due to differences in model parameters and performance across soil conditions and times, extensive experimentation is needed to determine the applicability range and conditions of infiltration models.

The Hydrus-3D model yielded the best results for numerical simulation of soil water content, with the advantage of considering three-dimensional variably saturated porous media flow in soil. The combination of numerical simulation and field measurements can improve research efficiency on soil water infiltration in dry environments.

5 Conclusions

Taking interdune lowlands in natural sand dunes as the research object, single-ring infiltration experiments were conducted to determine soil hydraulic parameters and analyze soil water variation responses to infiltration in fixed-head seepage zones. The adaptability of several infiltration models under ponded infiltration conditions was validated, yielding the following main conclusions:

1) Infiltration rate decreased sharply with time in the initial infiltration stage, then stabilized. Throughout the infiltration process, horizontal wetting front advancement was faster than vertical direction. Infiltration rate, cumulative infiltration, and wetting front depth showed power function relationships with time.

2) Affected by boundary and initial conditions and soil hydraulic parameter settings, the Hydrus-1D hydrological model performed poorly in simulating water infiltration in sandy soil. However, under current experimental conditions, comparison of performance evaluation indices indicated that the Philip and Green-Ampt models could reasonably describe sandy soil infiltration parameters and reflect relatively realistic soil water dynamics. The Kostiakov-Lewis model also effectively predicted cumulative infiltration.

3) Single-ring infiltration experiments need to consider both one-dimensional flow inside the ring and three-dimensional flow processes outside the ring. The Hydrus-3D model can basically reflect three-dimensional soil water content changes in different sandy soil profiles. In summary, combining the Philip model with Hydrus-3D hydrological model to determine sandy soil infiltration is feasible. However, this result is based solely on sandy soil, and whether it can be applied to other soil types requires further research.

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