Numerical Investigation of Rainfall Infiltration-Induced Slope Stability Considering Water-Air Two-Phase Flow

Abstract

For insights into rainfall infiltration on soil slopes and coupled transmission mechanisms, two-phase flow and finite element analysis were employed to examine water and air movement during the Shuping landslide. The results indicated a division of the landslide surface into two zones: an upper inflow area and a lower overflow area, driven by contrasting inflow and outflow directions. The total water and air flux remained stable, minimally affected by external factors such as rainfall attributes, surface runoff, and air temperature variations. In the inflow area of the slope surface, when rainfall intensity was greater than the total rate of the infiltration of water and air, the magnitude of infiltration equalled to the total rate infiltration of water and air, and runoff generation occurred in this area. Conversely, when infiltration matched rainfall intensity, runoff was absent in this area. In addition, water pressure in the saturated area of the slope surface can be transferred to the groundwater of the slope by pore air pressure, which could also increase the pressure head of the groundwater, and this was also detrimental to slope stability. Regarding uniform rainfall, it significantly reduces the safety factor, potentially making it the most hazardous pattern for slope failure.

1. Introduction

Rainfall infiltration of a slope is a complex unsaturated seepage process, which is primarily driven by the coupling effect of water and air fluid in the soil [1, 2, 3]. The infiltration intensity is influenced by various factors, such as soil permeability, water-air viscosity, and slope boundary conditions [4, 5]. Understanding these factors is essential to predict the rainfall infiltration process and its impact on slope stability [6, 7, 8].

Most previous studies [9, 10, 11, 12] have used the Richard’s rainfall infiltration equation to describe the infiltration process. However, this equation assumes that the pore air pressure is constant and equal everywhere (usually set to atmospheric pressure), which essentially ignores the coupling and cooperation between water and air in the soil. This simplification may be valid under certain circumstances, but it may not fully capture the complex interactions between water and air in real-world engineering scenarios. Numerous research findings [13, 14, 15, 16, 17] indicated that, in the case of a closed boundary condition, the effect of pore air on the flow of the liquid phase (water) cannot be neglected. During the rainfall infiltration process, the liquid phase (water) exerts pressure on the air in the soil, resulting in a jacking force that opposes the direction of water movement. This phenomenon can cause a decrease in the rainfall infiltration rate because of the reduction of the water pressure gradient. This effect of air on the infiltration process has been observed and reported in numerous studies [18, 19]. Therefore, the consideration of water-air coupling effect is important for accurate modelling and prediction of rainfall infiltration in soil.

Some researchers have demonstrated that the water and air infiltration process of slope rainfall can be accurately captured by considering the influence of the air phase. Based on their findings, it is recommended to adopt the two-phase flow theory of water and air for a more practical exploration of the problem of slope rainfall infiltration. This chapter employs the basic theory and methodology of water- air two-phase flow to simulate and analyse the rainfall infiltration process of Shuping landslide in the Three Gorges Reservoir area, with the aim of investigating the general law of slope rainfall infiltration. While most studies on two-phase flow primarily focus on exploring changes in pressure and other parameters during infiltration and verifying the impact of the air phase, there is a lack of research on the overall induction of rainfall infiltration laws. By studying the general law of slope rainfall infiltration, this research serves as a complement to existing laws of rainfall infiltration and provides a more comprehensive basis for engineering applications. In addition, the effect of rainfall pattern on slope failure is also analysed in this research.

2. Governing differential equation for water-air two-phase flow

Liquid phase (water) flow equation and air phase flow equation are included in the governing differential equations for water-air two-phase flow. These two equations can realize the coupling process by correlation of some variables including saturation, pore air pressure, matric suction as well as porosity. The flow of water and air not only conforms with mass conservation equation driven by pressure but also considers some variation factors including soil deformation, compressibility, viscosity as well as saturation. The governing differential equations for water-air two-phase flow can be described as follows [20]:

$$\mathrm{nS}\frac{\partial \mathit{ln}{\rho }_l}{\partial t}+S\frac{\partial n}{\partial t}+n\frac{\partial S}{\partial t}+\nabla \cdot \left[-\frac{k_r^{l}k}{{\mu }_l}\cdot \left(\nabla p_l+{\rho }_{l}g\right)\right]-\frac{1}{{\rho }_l}{Q}_l=0$$E1

$$\mathrm{n}\left(1-\mathrm{S}\right)\frac{\partial \mathit{ln}{\rho }_g}{\partial t}+\left(1-S\right)\frac{\partial n}{\partial t}-n\frac{\partial S}{\partial t}+\nabla \cdot \left[-\frac{k_r^{g}k}{{\mu }_g}\cdot \left(\nabla p_g+{\rho }_{g}g\right)\right]-\frac{1}{{\rho }_g}{Q}_g=0$$E2

Where n represents the porosity; $S_r$ represents the degree of saturation; ${\rho }_{\beta }$ represents the density of the phase ($\mathrm{\beta }=1$ represents the water phase, $\mathrm{\beta }=\mathrm{g}$represents the air phase); $Q_{\beta }$ represents the source of phase (kg/m); k represents the intrinsic permeability of soil determined by pore characteristics ($m^2$); $k_r^{\beta }$ represents the relative permeability coefficient of the phase; ${\mu }_{\beta }$ represents the viscosity coefficient of the phase $\left(\mathrm{N}\cdot \mathrm{s}/{\mathrm{m}}^2\right)$; $p_{\beta }$ is the pore pressure of the β phase $\left(\mathrm{N}/{\mathrm{m}}^2\right)$; g is the gravitational acceleration (N/kg).

The dynamics of water-air two-phase flow are described by a complex system of partial differential equations that are nonlinear and depend on both space and time variables. To tackle the spatial domain, the Galerkin finite element method is utilized, while the time domain is discretized using a difference method. This approach enables a robust and accurate calculation of the flow dynamics by considering two unknowns, namely the pore pressure (${\mathrm{\rho }}_g$) and the saturation (${\mathrm{S}}_r$), which are solved iteratively through a cyclic calculation of Eq. (1) and Eq. (2). This iterative process ensures that the solution converges to the correct values. Overall, this methodology provides an efficient and reliable approach for investigating water-air two-phase flow dynamics, which can help to understand the behaviour of fluid flow in porous media, and provide insights into the optimization of industrial processes.

3. Computational model

In pursuit of a more profound understanding of the underlying principles governing rainfall infiltration, this section employs the illustrative framework of the Shuping landslide case study. Through the application of advanced simulation techniques, the dynamic interplay between water and air within the slope during the process of rainfall infiltration is meticulously recreated. This simulation endeavour is strategically designed to shed light on the fundamental tenets that dictate the behaviour of rainfall infiltration on slopes. It takes into account the distinctive attributes of this process, the variations in its intensity, and its propensity for generating runoff.

Subsequent to this, the discussion takes a comprehensive turn by delving into the intricate mechanisms that underpin the intermingling of water and air within the environmental context of the slope. By deciphering these mechanisms, the study aims to offer deeper insights into the multifaceted factors influencing the behaviour of water and air during rainfall infiltration. The discourse seeks to unravel the dynamics that govern their coexistence within the slope’s environment, ultimately contributing to a more holistic and accurate comprehension of this intricate natural process.

3.1 Study area and geometric model

This study presents a case study of the Shuping landslide, which is located on the north bank of the Yangtze River, 61 km away from the dam site of the Three Gorges Project. The landslide body comprises a loose rock clastic layer and a loess soil layer, which have good permeability and are conducive to groundwater infiltration. The bottom of the landslide body is composed of phyllite, which has relatively weak permeability and can form a relative waterproof layer.

The Shuping landslide is characterized as a relict landslide, characterized by a historical chronicle of recurring sliding occurrences. Since the initiation of the Three Gorges Reservoir impoundment in June 2003, a persistent sequence of deformations has been discerned. Consequently, there has been a progressively escalating manifestation of surface deformations within the sliding mass. Particularly notable is the observable trend of continuous propagation observed in the emergence of cracks within the slope. Furthermore, instances of localized significant collapses have been identified. Presently, the landslide remains subjected to an ongoing process of deformation.

The Shuping landslide boasts a historical record marked by recurrent events of collapse and sliding. Its stability is relatively compromised, thereby rendering it susceptible to deformations and failures instigated by external variables such as precipitation and fluctuations in reservoir water levels. If a substantial decrease in the Three Gorges Reservoir water level, reducing it from 175 meters to 145 meters, coincides with intense rainfall, the sliding mass would be susceptible to incurring substantial deformations, thereby possibly precipitating a precarious state of instability.

The thickness of the sliding body provides the material foundation for the landslide. The front edge of the Shuping landslide is 1100 m wide, the back edge is 800 m wide, and the longitudinal length is 550 m. The back edge is distributed in the line of the elevation of 280–338 m. For numerical modelling, the main sliding section is considered, with a horizontal length of 600 m and a vertical height of 350 m. To accurately simulate rainfall infiltration, the surface layer of the landslide body is discretized into elements with varying thicknesses, ranging from 0.25 to 0.75 m along the slope depth direction. The finite element calculation grid obtained by subdivision consists of 3499 elements and 3427 nodes, as shown in Figure 1 (geometric model for numerical simulation). This approach provides a comprehensive understanding of the landslide characteristics and the factors that influence its behaviour, which can contribute to the development of effective mitigation strategies for similar geological hazards.

Figure 1.

Finite element grid diagram of the slope cross-section.

3.2 Initial and boundary conditions

To reduce the uncertainty impact of initial state on this research, this study incorporates measured rainfall data from the Three Gorges Reservoir area spanning several years (provided by the Headquarters of Geological Disaster Prevention in the Three Gorges Reservoir Area of the Ministry of Land and Resources). The seepage field of the slope is calculated and simulated in the fully saturated state until it reaches to a relatively stable state, which is used as the initial condition for subsequent calculations. By adopting this approach, the study aims to minimize potential errors or biases that may arise from using arbitrary initial conditions. The use of measured data and stable conditions provides a more accurate and realistic representation of the initial state of the system, which is critical for achieving meaningful and reliable results. This study conducts a thorough examination of the real-world system under investigation. It investigates various aspects, including the behaviour of materials, the system’s boundaries, its geometries, and significant variables such as the impact of rainfall. The interplay of these diverse elements collectively influences the system’s behaviour. Through a detailed analysis of these factors, the study aims to unravel the intricate dynamics that govern the system’s behaviour within its environmental context.

In this research, the side and bottom of the rear edge of the model are treated as impervious boundaries, and the water level and its variation at the boundary are calculated. The slope surface and leading edge below the reservoir water level are considered as known water pressure boundaries, and their values are dependent on the reservoir water level elevation (145 m). On the other hand, the slope surface above the reservoir water level is treated as the known air pressure boundary. As the runoff water head on the slope is relatively small compared with the atmospheric pressure, it is assumed to be negligible. In the present analysis, the air pressure on the air pressure boundary is set to atmospheric pressure. The proposed approach enables the determination of infiltration rates of both water and air from calculation results, without the need for their a priori specification as flow boundaries. Compared to the traditional single-phase flow calculations, this approach provides a more comprehensive understanding of the dynamics of the water-air two-phase flow, which is important for achieving more accurate predictions and more effective risk assessments in geotechnical engineering applications.

3.3 Constitutive relations and parameters

For the given slope, the mathematical description of the rainfall infiltration process is determined by the governing differential equation, initial and boundary conditions, constitutive relation, and corresponding parameters of water and air two-phase flow. Eq. (1) and Eq. (2) involve five unknowns: S, $k_r^l$, $k_r^g$, $p_l$ and $p_g$, which are solved concurrently by incorporating the relationship between soil-water characteristics determined by soil properties and the relative permeability function of water and air.

In this research, the commonly used Van Genuchten model is adopted to represent the soil-water characteristic curve. The mathematical relationship between matric suction $\left(p_c\equiv p_g-p_l\right)$ and saturation is expressed as follows [21]:

$$p_c=-p_0{\left(S_e^{-1/\lambda }-1\right)}^{1-\lambda }$$E3

Where ${\mathrm{S}}_e$ represents the effective water saturation $\left(S_e=\frac{S-S_{\mathit{rl}}}{1-S_{\mathit{rl}}}\right)$; $S_{\mathit{rl}}$ represents the residual water saturation; $p_0$ and λ are parameters of this model. According to some literature data [22], this calculation adopts the value of $p_0=1.33$, $\lambda =0.41$, $S_{\mathit{rl}}=0.15$ as three parameters.

The relative permeability coefficients of water ($k_r^l$) and air ($k_r^g$) are using the Van Genuchten-Mualem [22] model and Corey model, respectively. In this study, the relative permeability coefficients that contain effective saturation variable can be expressed as follows [23, 24]:

$$k_r^l=\sqrt{S_e}{\left\{1-{\left[1-{\left(S_e\right)}^{1/\lambda }\right]}^{\lambda }\right\}}^2$$E4

$$k_r^g={\left(1-S_e\right)}^2\left[1-S_e^2\right]$$E5

The values of some other parameters are as follows [22]: $\mathrm{n}=0.15$, $\mathrm{k}=4.0\ast {10}^{-12}{m}^2$, $\mathrm{g}=9.8\mathrm{N}/\mathrm{kg}$, ${\rho }_g=1.29\mathit{kg}/m^3$, ${\rho }_l=1\ast {10}^3\mathit{kg}/m^3$, ${\mu }_l=1\ast {10}^{3}N\cdot s/m^2$, ${\mu }_g=1\ast {10}^{-5}N\cdot s/m^2$.

3.4 Slope stability analysis method considering pore air pressure

In this study, the slope stability analysis is conducted to use the residual thrust method, which incorporates the consideration of matric suction and pore air pressure. The calculation procedure involves several steps: an initial assumption of the safety factor, subsequent calculation of the thrust starting from the first slide at the top of the slope and progressing towards the last slide, and finally determining the safety factor value at which the thrust reached zero. This zero-thrust condition represents the equilibrium state, thus yielding the final safety factor value [25, 26, 27, 28].

$$\begin{array}{l}{P}_i=W_i\mathit{sin}{\alpha }_i-\frac{c_i^{\prime }{l}_i+\left(W_i\mathit{cos}{\alpha }_i-p_{\alpha }^i{l}_i\right)\mathit{tan}{\phi }_i^{\prime }+l_i\left(p_a^i-p_w^i\right)\mathit{tan}{\phi }^b}{F_s}+P_{i-1}{\psi }_i\\ {\psi }_i=\left[cos\left({\alpha }_{i-1}-{\alpha }_i\right)-\frac{\mathit{tan}{\phi }_i^{\prime }}{F_s}sin\left({\alpha }_{i-1}-{\alpha }_i\right)\right.\end{array}$$E6

Where $F_s$ is the safety factor of the slope, $P_i$ is the sliding force of soil slice, $c_i^{\prime }$ is the effective cohesive force of slice, ${\phi }_i^{\prime }$ is the effective internal friction angle of slice, $l_i$ is the width of soil slice, $W_i$ is the weight of soil slice, ${\alpha }_i$ is the angle of bottom soil slice, $p_a^i$ and $p_w^i$ are pore air pressure and pore water pressure of slice, $\mathit{tan}{\phi }^b$ is the rate of shear strength that increases with an increase in matric suction, and ${\psi }_i$ is the transfer coefficient of slice i. The model recommended by Vanapalli [29] in this research can be expressed as a function of effective saturation. The equation is as follows:

$$\mathit{tan}{\phi }^b=S^{\ast }\cdot \mathit{tan}{\phi }^{\prime }$$E7

Where the effective saturation is expressed as follows:

$$S^{\ast }=\left(S_r-S_{\mathit{rw}}\right)/\left(1-S_{\mathit{rw}}\right)$$E8

4. Rainfall infiltration analysis of slope

In this research, we present the results of finite element calculations of two-phase flow (water and air) in Shuping landslide. Our findings reveal the general rule of rainfall infiltration in slopes, examining characteristics such as infiltration intensity, flow-producing strips, and infiltration patterns. Additionally, we elucidate the coupling force transfer mechanism between water and air in slopes. Through our analysis, we aim to contribute to a deeper understanding of the complex processes that govern the behaviour of landslides under the influence of rainfall infiltration.

4.1 Rainfall infiltration characteristics of slope surfaces

According to the calculation results of Shuping landslide, the direction and total velocity of material (water, air) entering and leaving the slope surface show a certain regularity. According to different directions of material (water and air) inflow and outflow, the slope surface can be divided into inflow area and overflow area. If the fluid tends to flow in the direction of the slope, exhibiting an upward or uphill movement, this area is commonly known as the inflow area. Conversely, the overflow area represents the zone where the flow rate of water or air predominantly moves away from the slope (Figure 2). The distribution of inflow area and overflow area of slope surface has strong regularity. The water and air inflow area is usually located in the middle and upper part of the slope, while the overflow area is generally located in the lower part. The boundary line between the suction zone and the overflow zone is usually not fixed. It is not only related to the geometric characteristics and permeability characteristics of slope, but also affected by rainfall characteristics such as rainfall duration, rainfall interval, and rainfall intensity. With the extension of rainfall duration, the boundary line between inflow area and overflow area will move upward, the inflow area of slope surface will shrink, and the overflow area will increase. With the increase of rainfall interval, the boundary line will gradually move down, and the inflow area of slope surface will increase, while the overflow area will decrease.

Figure 2.

Slope surface (water, air): Inflow and overflow area.

Based on numerical analysis, the total flow rate of water and air through the slope surface exhibits phase-pair stability, and its magnitude is primarily influenced by slope geometry and the transmission characteristics of geotechnical materials, while external factors such as rainfall characteristics, slope flow production, and air temperature variation have relatively little effect. The flow rate of water and air into and out of the slope surface provides a measure of the intensity of rainfall inflow and overflow. Figure 3 depicts the relationship between the intensity of water and air inflow (overflow) and time. The curve indicates that, as the rainfall duration increases, the maximum inflow and overflow intensities of the Shuping landslide slope attain a certain stability, with the former being around 0.4 mm/min and the latter around 0.5 mm/min. The average inflow and overflow intensity values of water and air on the slope exhibit a slight downward trend, with both values being quite similar at approximately 0.1 mm/min.

Figure 3.

Variation of inflow and overflow intensity over time.

4.2 Rainfall infiltration intensity and flow-producing conditions on slope surface

The assessment of rainfall infiltration intensity and runoff production on slope surfaces is closely linked to the relative magnitude of infiltration and the total inflow rate of the slope surface, comprising both water and air. In the inflow area of the slope surface, infiltration of water and air relies primarily on the composition of water and air at the slope’s boundary. In the absence of rainfall on the surface, air is drawn into the inflow area through its boundary. Conversely, when rainfall occurs on the surface, if the rainfall intensity is lower than the total inflow rate of the slope (consisting of water and air), the slope’s inflow area will not produce flow, and the infiltration intensity of rainwater will match the rainfall intensity. The slope’s boundary will absorb a mixture of water and air with varying compositions. However, if the rainfall intensity exceeds the total inflow rate of the slope (comprising both water and air), runoff will occur on the slope, and the infiltration intensity will equal the total inflow intensity of water and air. In such cases, the surface of the slope is covered by air, while water is drawn at the inflow area’s boundary.

Regarding the Shuping landslide, rainfall intensity is much greater than the total inflow intensity of the slope, approximately 0.1 mm/min, which is a prerequisite for runoff to be generated on the slope. The maximum inflow area is located at the top of the slope, when the rainfall intensity exceeds the maximum inflow intensity of the slope surface, which is approximately 0.4 mm/min, and the slope may experience full flow generation. These observations suggest that rainfall intensity and the total inflow rate of the slope surface are essential factors to consider when analysing the potential for runoff generation and slope stability during rainfall events.

In the overflow area of the slope surface, the overflow of water and air is closely linked to the water and air content at the slope’s boundary and the slope saturation. According to a report by the Yangtze River Scientific Research Institute in 2017, when rainfall infiltration intensity was 0, the slope surface produced runoff, and the outflow at the boundary of the overflow area was water. In an unsaturated state, the infiltration intensity of rainfall is equivalent to the intensity of the rainfall itself, resulting in a lack of flow on the slope surface. The outflow from the boundary of the overflow zone consists of a combination of water and air. In scenarios where the entire overflow area is dry, the outflow at the boundary of the overflow area exclusively consists of pore air.

These observations yield insights into the pivotal roles played by several key factors in the process of water and air overflow on the slope surface. Among these factors, the water and air content existing at the boundary of the overflow region, as well as the saturation state of the slope, emerges as critical determinants. Moreover, these findings indicate that a confluence of variables, namely rainfall intensity, slope saturation, and the water and air content at the slope’s boundary, collectively governs the production of runoff and the characteristics of overflow events.

In essence, these insights underscore the intricate interplay of factors that orchestrate the dynamics of water and air behaviour within the context of slope surface overflow. The water and air content at the boundary of the overflow zone, alongside the slope’s saturation state, serves as foundational considerations that shape the overflow process. Furthermore, the interrelation between these variables and external factors such as rainfall intensity adds layers of complexity to the observed behaviours. These findings contribute to an enhanced understanding of the multifaceted mechanisms governing the overflow dynamics, thereby facilitating more accurate assessments and predictions in geotechnical and hydrological investigations.

4.3 Water-air coupling force transfer mechanism

Rainfall infiltration is a process where water and air interact within soil pores, influencing each other’s movement. Water moves downward due to gravity, causing air to move within the pores. The compression of air generates a counteracting force against the water’s movement. However, due to its low viscosity (about 1% that of water), air’s effect on slope stability is unfavourable. Figure 4 depicts the force transfer mechanism resulting from water-air coupling. This coupling leads to pressure on the subsurface water, highlighting the detrimental influence of air on slope stability.

Figure 4.

Mechanism of water-air coupling in force transfer.

Figures 5–7 show the saturation, pore air pressure, and pore water pressure distribution cloudy map of Shuping landslide at a specific time. It reveals that the saturated water pressure at the top of the slope is transmitted to the saturated area at the slope’s front through closed pore air in the unsaturated area. This phenomenon leads to a partial elevation of groundwater level in the anti-sliding area at the slope’s front, resulting in the formation of external normal thrust that adversely affects slope stability.

Figure 5.

Distribution of water saturation for Shuping landslide at one time.

Figure 6.

Distribution of pore air pressure for Shuping landslide at one time.

Figure 7.

Distribution of pore water pressure for Shuping landslide at one time.

4.4 Effect of rainfall type on infiltration

By analysing pertinent data, the rainfall pattern can be categorized into four distinct conditions, as depicted in Figure 8. These conditions are characterized as four rainfall patterns. Uniform rainfall pattern: during this condition, the rainfall intensity remains relatively constant over the specified time period. Pre-peak rainfall pattern: this condition exhibits a linear decrease in rainfall intensity over time, leading up to a peak point. Post-peak rainfall pattern: in this condition, the rainfall intensity demonstrates a linear increase over time, following a peak point. Mid-peak rainfall pattern: this condition displays a rainfall intensity that initially increases and then subsequently decreases, forming a peak point.

Figure 8.

Four types of rainfall patterns.

Figures 9 and 10 illustrate the variations in pore air pressure and safety factor under four different rainfall patterns. In the case of uniform rainfall pattern, when the rainfall intensity remains constant, the saturation of the soil on the slope surface becomes saturated. Consequently, the slope surface forms a relatively impermeable state, which limits the outflow of air from the slope surface. As a result, the pore air pressure continues to rise. With increasing saturation, the matric suction decreases, leading to an increase in pore water pressure. During this period, the rainfall not only increases the load on the slope but also weakens the strength parameters, causing a decrease in the safety factor of the reservoir bank slope. As the rainfall duration increases, the infiltration intensity on the slope surface gradually decreases, resulting in a reduction in the amount of infiltrated water. Consequently, the pore air pressure, pore water pressure, and safety factor eventually reach a stable state. In the case of pre-peak rainfall, the initial stage shows a similar pattern to uniform rainfall, where the pore air pressure and pore water pressure increase non-linearly, leading to a decrease in the safety factor. During linearly decreasing rainfall, the slope surface gradually transitions from a saturated state to an unsaturated state. Pores in unsaturated soil provide a pathway for the overflow of pore air pressure from the slope body. In comparison with the early stages of rainfall, the pore air pressure decreases rapidly, while the pore water pressure increases in the latter half of the pre-peak rainfall. Consequently, the slope safety factor is gradually recovered. As the rainfall intensity is relatively low in the early period, the pore air pressure begins to increase from 0. As the rainfall continues, the intensity gradually increases. As a result, the surface soil becomes saturated rapidly during the later stage of rainfall, leading to a rapid increase in pore air pressure and pore water pressure, while the safety factor of the slope decreases. In the case of mid- peak rainfall, the rainfall intensity gradually increases during the early stages, and the pore air pressure increases gradually from 0. As a result, the safety factor experiences a slow decrease in the initial stage. With the continuous increase in rainfall intensity, the pore air pressure decreases rapidly, while the pore water pressure increases, leading to a rapid decrease in the safety factor. At the point when the rainfall intensity reaches its maximum, both the pore air pressure and pore water pressure reach their maximum values, resulting in the minimum safety factor. Subsequently, as the rainfall intensity starts to decrease, the pore air pressure and pore water pressure also decrease, causing a slow increase in the safety factor.

Figure 9.

Temporal variation in pore air pressure associated with four rainfall patterns.

Figure 10.

Temporal variation in safety associated with four rainfall patterns.

5. Conclusions

This study initially presents the fundamental patterns governing rainfall infiltration on slopes. This presentation encompasses various aspects, including slope surface characteristics related to rainfall infiltration, the intensity of rainfall infiltration, and the prevailing runoff conditions. Subsequently, a comprehensive explanation is provided regarding the intricate mechanism underlying the coupled transmission forces of water and air within the slope’s environment, which also constitutes the novelty of this work. Furthermore, this research also encompasses an analysis of the impact of rainfall patterns on slope failure.

Based on the principles of water-air two-phase flow, the rainfall infiltration process of Shuping landslide is analysed through finite element method. The following conclusions could be drawn from the analysis:

1. The direction of water and air entering and leaving the slope differs, resulting in the formation of an inflow zone and an overflow zone on the slope surface, with the former typically located at the upper part and the latter at the lower part of the slope.

2. The total flow rate of water and air in and out of the slope surface depends mainly on the material transmission characteristics and geometric characteristics of the slope and is relatively stable, with less influence from the flow production condition and runoff depth.

3. The relative relationship between rainfall intensity and the total inflow rate of water and air can serve as a judgement criterion for rainfall infiltration intensity and runoff production condition on the slope surface. In the inflow area, when the rainfall intensity is less than the total inflow rate of water and air, no flow is produced, and the infiltration intensity is equal to the rainfall intensity. Conversely, the slope produces flow, and the infiltration intensity is equal to the total inflow intensity of water and air.

4. The water pressure in the saturated area of the slope surface can be transferred to the underground saturated area through the pore air in the unsaturated area, which increases the underground water pressure and the normal thrust outside the slope foot, and adversely affects the slope stability. The analysis provides a theoretical basis for engineering examples by simulating and generalizing the law of rainfall infiltration.

5. During period of uniform rainfall, there is a noticeable decline in the safety factor. When the rainfall type transitioned to post-peak rainfall, the safety factor exhibits an overall downward trend. Notably, during the initial stage of rainfall, the reduction rate of the safety factor was significantly lower compared to the subsequent stages of rainfall. In contrast, both mid-peak rainfall and pre-peak rainfall experienced minimal precipitation in the later period, resulting in an increase in the safety factor. However, due to the numerous factors affecting the process, quantitative analysis of the infiltration rule is still limited. Therefore, further studies will focus on analysing and controlling the influencing factors and quantifying the general rule of the infiltration process.

Acknowledgments

The work was supported by the Geotechnical Laboratory at Ghent University and Geosound.be. The first author would like to acknowledge the financial support received from the China Scholarship Council (No.201908420298).

Conflict of interest

The authors declare no conflict of interest.