Introduction

In recent years, landslide disasters induced by severe drying-wetting alternation under frequent extreme rainfall due to the El Niño phenomenon have become a major geological hazard in China. Research into the mechanisms of these disasters has been challenging. The serve drying-wetting alternation in soil slopes is characterized by the alternating transformation between the residual and saturated water content of unsaturated soil. During undirectional drying-wetting alternation under extreme rainstorms, unsaturated soil undergoes a strong hygroscopic process, transitioning from residual water content to saturated water content at a high hydraulic gradient.

The mechanism by which regular rainfall affects slope stability can be attributed to the multiphysical coupling effects triggered by water infiltration, primarily involving the interaction between dynamic changes in pore water pressure and the mechanical properties of the soil1,2. Jian et al. investigated the seepage-displacement characteristics of highway slopes under rainfall infiltration by homogenizing fractured soil3. Li et al. developed a trapezoidal wetting front based on the Van Genuchten model and proposed a Green-Ampt infiltration model for slopes that accounts for unsaturated soil characteristics4. Kenta Tozato et al. developed a coupled analysis method integrating infiltration, surface runoff, and slope stability to assess slope failure risks induced by intense rainfall, with field data validating the method’s effectiveness5. These studies have addressed issues such as the homogenization of fractured soil under rainfall, improvements to unsaturated flow models, and coupled analysis methods for rainfall-infiltration-stability. However, the more severe degradation effects caused by wetting–drying cycles remains insufficiently studied. The damage mechanism can be specifically described as follows: Continuous rainwater infiltrates into the unsaturated soil, gradually sealing the interconnected pathways between the interior of the slope and the external soil matrix. This leads to the accumulation of gas within confined cavities, forming closed bubbles. When the infiltration depth of rainwater reaches a critical threshold (i.e., the escape pathways are completely sealed), the entrapped gas pressure and the external maximum infiltration-induced water pressure reach a dynamic equilibrium. As infiltration depth continues to increase, the water-encapsulated closed bubbles are further compressed, causing the pore gas pressure within the confined bubbles to rise continuously. When the gas pressure is less than or equal to the overlying saturated soil stress, air cushions form in the unsaturated soil, weakening the soil structure. As the gas pressure further increases, the pore gas in the unsaturated slope soil escapes, forming an interconnected channel, thereby reducing the soil’s shear strength and severely damaging its internal structure, ultimately triggering slope instability. This process establishes the key hypothesis of a “gas sealing—pressure accumulation—gas migration—slope instability” chain mechanism, which provides a basis for subsequent analysis of pore gas migration behavior.

Wetting–drying cycles are a key environmental factor affecting the long-term stability of slopes. These alternations lead to strength degradation, structural damage, and changes in permeability of geotechnical materials through coupled physical, chemical, and hydraulic processes, which can trigger landslides and other geological hazards6,7. Rosenbalm et al. evaluated the impact of wetting–drying cycles on the volumetric behavior of unsaturated soils8. Bang et al. demonstrated through experimental and numerical analysis that wetting–drying cycles significantly affect the physical–mechanical parameters and permeability of slopes9. Ding et al. found through laboratory tests that repeated changes in water content caused by wetting–drying cycles significantly affect the shear strength of soil10. Jing et al. conducted soil–water characteristic curve (SWCC) tests on unsaturated loess and found that wetting–drying cycles altered both the volumetric water content and the permeability coefficient of the unsaturated loess11. Although the above studies focused on the effects of wetting–drying cycles on slope stability—particularly through changes in permeability, water content, and unsaturated soil volume—they primarily analyzed mechanisms involving key factors such as cycle number and wet-dry amplitude. However, current research has not sufficiently considered the cumulative structural damage to slope soils caused by the repeated transition between residual and saturated water contents during long-term alternating. Dong et al. investigated the stability of waste dump slopes under severe drying-wetting alternation. The results showed that as the intensity of drying-wetting alternation increased, the surface soil of the waste dump transitioned from an unsaturated to a saturated state. This transition led to a reduction in shear strength and a shrinkage of the unsaturated zone12. Further studies revealed that severe drying-wetting alternation can also induce significant gas migration, which further undermines slope stability.

Previous studies have confirmed the significant impact of pore gas on rainfall-induced landslides. Liu et al. revealed the dynamic variation of pore gas pressure and the air-blocking effect using a two-phase water–gas flow model13. Sun et al. demonstrated through numerical simulations that neglecting gas pressure can lead to an overestimation of the slope safety factor14. Zhang et al. further confirmed through infiltration analytical solutions that air entrapment delays the infiltration process, resulting in a lagged landslide response15. However, existing research has several limitations. First, the role of gas is often simplified as a static variable in constitutive models, overlooking its dynamic migration and spatial distribution characteristics of air cushions. Second, the mechanical mechanisms by which locally enriched air cushions reduce the shear strength of weak slip zones and induce non-uniform seepage lack quantitative characterization. Third, studies on the coupled gas–water phase transition process and its cumulative damage effects under under severe drying-wetting alternation remain insufficient.

In summary, current research primarily focuses on conventional rainfall and wetting–drying cycles, while the mechanisms by which severe drying-wetting alternation—such as those induced by extreme climate events—affect slope stability through gas migration remain unclear and require further investigation. This study focuses on severe drying-wetting alternation under extreme climate conditions and performs real-time dynamic calculations of the unsaturated permeability coefficient in unsaturated zones, overcoming the limitations of static parameters in traditional rainfall infiltration studies. A Weibull distribution criterion for pore gas escape and air cushions formation is proposed, which quantitatively reveals the variation patterns of gas migration. Comparative analysis under different conditions—no pore gas migration, gas escape, and air cushions formation—demonstrates that gas escape has a stronger weakening effect on the safety factor than air cushions formation. The study evaluates and analyzes the catastrophic mechanism by which pore gas migration in unsaturated soils under severe drying-wetting alternation induces slope instability (Fig. 1). The findings provide theoretical support for slope stability early warning under extreme climatic conditions.

Fig. 1
Fig. 1
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Research workflow for slope stability analysis incorporating pore gas migration under severe drying-wetting alternation.

Numerical calculation principle under severe drying-wetting alternation

Unsaturated soil shear strength theory

Vanapalli et al16. suggested two independent stress state variables for the shear strength equation of unsaturated soil:

$$\tau_{f} = \left[ {c^{\prime } + (\sigma_{n} - u_{a} )\tan \varphi^{\prime } } \right] + (u_{a} - u_{{\text{w}}} )\left[ {\tan \varphi^{\prime } (S - S_{r} )/(1 - S_{r} )} \right]$$
(1)

where \(\tau_{f}\) is the shear stress on the failure surface of the soil , kPa; \(c^{\prime }\) is the effective cohesion, kPa; \(\sigma_{n}\) is the normal stress, kPa; \(u_{a}\) is the pore gas pressure, kPa; \(\varphi^{\prime }\) is the effective internal friction angle (°); \(u_{{\text{w}}}\) is the pore water pressure, kPa; \(S\) is saturation; \(S_{r}\) is residual saturation.

The effective saturation \(S_{e} = (S - S_{r} )/(1 - S_{r} )\), then17,18:

$$\tau_{f} = \left[ {c^{\prime } + (\sigma_{n} - u_{a} )\tan \phi^{\prime } } \right] + S_{e} (u_{a} - u_{{\text{w}}} )\tan \phi^{\prime }$$
(2)

The expression for total cohesion is as follows19,20:

$$c_{t} = c^{\prime } + S_{e} \left( {u_{a} - u_{{\text{w}}} } \right)\tan \phi^{\prime }$$
(3)

where \(c_{t}\) is the total cohesion, kPa.

Unsaturated soil seepage theory

The unsaturated–saturated soil seepage equation19,20 is as follows:

$${\varvec{q}}_{i} = - k_{r} (S)K_{ij} h_{j} = k_{r} (S)K_{ij} \left[ {\psi + \psi_{z} } \right]_{j}$$
(4)

where \({\varvec{q}}_{i}\) is the unit flow vector; \(k_{r} (S)\) is the relative permeability coefficient, with 0 < \(k_{r} (S)\) < 1 for unsaturated soil and \(k_{r} (S)\) = 1 for saturated soil; \(K_{ij}\) is the permeability coefficient tensor; \(h_{j}\) is the hydraulic gradient; \(\psi\) is the pressure head, \(\psi { = }u_{{\text{w}}} {/}\gamma_{{\text{w}}}\), kPa; \(\gamma_{{\text{w}}}\) is the unit weight of water, kN/m3 and \(\psi_{z}\) is the position head, kPa.

The equation for unsaturated soil permeability coefficient is as follows19,20:

$$k_{u} = kS_{e}^{0.5} [1 - (1 - S_{e}^{1/m} )^{m} ]^{2}$$
(5)

where \(k_{u}\) is the unsaturated hydraulic conductivity, m/s; \(k\) is the saturated permeability coefficient, m/s; \(m\) is the fitting parameter.

Matric suction distribution law

The matric suction-specific discharge equation for natural unsaturated slopes is21:

$$q = - k\exp [ - \beta \psi_{h} \gamma_{w} ][d\psi_{h} /dz + 1]$$
(6)

where \(\beta\) is the variation rate of the soil permeability coefficient, which is dependent on matric suction, kPa−1, \(\psi_{h}\) is the matric head, \(\psi_{h} = (u_{a} - u_{{\text{w}}} )/\gamma_{{\text{w}}}\).

Assuming that the matric head at z = 0 is \(\psi_{h0}\), that is, \(u_{a} - u_{{\text{w}}} { = }\psi_{h0} \,\gamma_{{\text{w}}}\), the integral of Eq. (6) can be obtained:

$$u_{a} - u_{{\text{w}}} = \psi_{h0} \gamma_{{\text{w}}} - 1/\beta \ln [(1 + q/k)\exp ( - \beta z\gamma_{w} ) - q/k]$$
(7)

Under static pressure conditions with q = 0, the matric suction exhibits a linear distribution:

$$u_{a} - u_{{\text{w}}} { = }z\gamma_{{\text{w}}}$$
(8)

Strength reduction method

Using the strength reduction method to analyze and solve the safety factor of unsaturated soil slopes under severe wetting and drying alternation, the expression is22,23:

$$c_{F} = c_{t} /F_{r} = \left[ {c^{\prime } + s_{e} (u_{a} - u_{{\text{w}}} )\tan \varphi^{\prime } } \right]/F_{r}$$
(9)
$$\varphi_{F} = \tan^{ - 1} \left( {\tan \varphi^{\prime } /F_{r} } \right)$$
(10)

where \(c_{F}\) is the reduced cohesion, kPa; \(\varphi_{F}\) is the reduced internal friction angle (°); \(F_{r}\) is the reduction coefficient.

The expression for the safety factor is as follows:

$$K{ = }\int\limits_{{0}}^{{1}} {\left( {c^{\prime } + \left[ {\sigma_{n} - u_{a} + s_{e} (u_{a} - u_{{\text{w}}} )\tan \varphi^{\prime } } \right]} \right)} dl/\int\limits_{0}^{1} {\tau dl}$$
(11)

Drucker–Prager elastic–plastic stiffness matrix

Drucker–Prager criterion expression is frequently used in the analysis of geotechnical materials:

$$f = \alpha I_{i} + \sqrt {J_{2} } = K$$
(12)

Where \(\alpha\) and \(K\) are the constants related to the cohesion \(c\) and friction Angle \(\phi\) of the strength parameters of geotechnical materials.

By adopting a non-associated flow rule matching the Mohr–Coulomb circle and introducing suction components with equivalent cohesion:

$$c_{{\text{e}}} = c^{\prime } + S_{e} \left( {u_{a} - u_{w} } \right)\tan \varphi^{\prime }$$
(13)
$$\alpha = \sin \varphi^{\prime } /3$$
(14)
$$K = \left[ {c^{\prime } + S_{e} \left( {u_{a} - u_{w} } \right)\tan \varphi^{\prime } } \right]\cos \varphi^{\prime }$$
(15)

The elasto-plastic stiffness matrix for the numerical model of unsaturated soil slopes under saturated–unsaturated transition is given by:

$$d\varepsilon = d\varepsilon^{e} + d\varepsilon^{p}$$
(16)

where: \(\varepsilon\) denotes the total strain, \(\varepsilon^{e}\) represents the elastic strain component, and \(\varepsilon^{p}\) signifies the plastic strain component.

The expression of stress increment and strain increment is:

$$d\sigma = \left[ D \right]_{ep} d\varepsilon = \left( {\left[ D \right]_{e} - \left[ D \right]_{p} } \right)d\varepsilon$$
(17)

where: \(\left[ D \right]_{ep}\) is the elastic–plastic stiffness matrix; \(\left[ D \right]_{e}\) is an elastic matrix; \(\left[ D \right]_{p}\) is the plastic stiffness matrix.

The elastoplastic stiffness matrix expression is derived from Eq. (17):

$$\left[ D \right]_{ep} = \left[ D \right]_{e} - \left[ D \right]_{e} \frac{\partial f}{{\partial \sigma }}\left( {\frac{\partial f}{{\partial \sigma }}} \right)^{T} \left[ D \right]_{e} /\left\{ {\left( {\frac{\partial f}{{\partial \sigma }}} \right)^{T} \left[ D \right]_{e} \frac{\partial g}{{\partial \sigma }}} \right\}$$
(18)

Based on the aforementioned numerical calculation principle under severe drying-wetting alternation, an unsaturated–saturated seepage calculation method can be established by coupling the unsaturated soil effective stress equation with seepage flow and shear strength equations. Integrating field tests, the FISH scripting language is employed to dynamically compute the unsaturated permeability coefficient in unsaturated zones during rainfall infiltration. Furthermore, accounting for pore gas escape/cushions formation after severe drying-wetting alternation, the Weibull distribution is adopted to statistically characterize the spatial distribution of gas escape/cushions formation. This framework enables quantitative assessment of the catastrophic mechanism of slope instability induced by gas migration in unsaturated soils under severe drying-wetting alternation. Figure 2 illustrates the computational framework.

Fig. 2
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Development of a numerical framework for unsaturated soil slope analysis.

Numerical model of stability under severe drying-wetting alternation

The establishment of numerical calculation model

Geometric model

The model is a homogeneous unsaturated soil slope, with dimensions of 105 m in length, 3 m in width, and 40 m in height. The slope height is 20 m, with a boundary distance of 55 m from the slope crest and a toe angle of 45°. The model dimensions and grid division are illustrated in Fig. 3.

Fig. 3
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Numerical model of unsaturated soil slope.

Calculation of soil water characteristic curve

Volumetric misturer content versus soil depth curve can be obtained from field exploratory well geotechnical tests. (Fig. 4).

Fig. 4
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Relationship curve between volumetric content and depth.

The VG model was used to fit the soil–water characteristic curve.

$$\theta = (\theta_{s} - \theta_{r} )[1/(1 + a^{n} s^{n} )]^{m} + \theta^{r}$$
(19)

where \(\theta_{s}\) is the saturated volume water content of unsaturated soil; \(\theta_{r}\) is the residual volume water content, \(s\) is the matrix suction.

The fitting parameters can be obtained from Eq. (19), as shown in Fig. 5.

Fig. 5
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Fitting curve of SWCC.

After converting the specific values of unsaturated soil water content test into volumetric soil water content, further mathematical calculations can yield \(\theta_{s} = 0.291\), \(\theta_{r} = 0.026\), then:

$$\theta = 0.265\left[ {1/\left( {1 + 0.0078s^{2.2} } \right)} \right]^{0.32} + 0.026$$
(20)

Constitutive model parameters

The numerical simulation of unsaturated soil slopes employs the Drucker–Prager elastoplastic constitutive model alongside an isotropic seepage model.

The mechanical parameters of slope are determined through water content tests and the direct shear test (Table 1).

Table 1 Mechanical parameters of the slope.
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The specific forms of the total cohesion and uniform shear strength equations can be obtained:

$$c_{t} = 16.1 + \left[ {1/\left( {1 + 0.0078s^{2.2} } \right)} \right]^{0.32} s\tan 22.4^{ \circ }$$
(21)
$$\tau_{f} = 16.1 + \left[ {1/\left( {1 + 0.0078s^{2.2} } \right)} \right]^{0.32} s\tan 22.4^{ \circ } + (\sigma_{t} - u_{a} )\tan 22.4^{ \circ }$$
(22)

Seepage parameters of unsaturated soil slopes

From Eq. (22), the specific equation of the permeability coefficient is as fllows:

$$k_{u} = kS_{e}^{0.5} \left[ {1 - \left( {1 - S_{e}^{3.125} } \right)^{0.32} } \right]^{2}$$
(23)

where \({\text{S}}_{e} = \left[ {1/\left( {1 + 0.0078s^{2.2} } \right)} \right]^{0.32}\).

The results of fluid seepage parameters calculated by the unsaturated–saturated seepage are shown in Table 2. where \(K_{f}\) is the fluid modulus and \(\sigma^{{_{t} }}\) is tensile strength:

Table 2 Fluid seepage parameters.
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Boundary conditions and initial pore pressure

Boundary conditions

Mechanical boundary conditions: soil on the upper part of the unsaturated soil slope model is set as a free boundary, the rest of the side surfaces and bottom of the lower part of slope are set as normal fixed constraints.

Seepage calculation boundary: The soil on the upper portion of slope was set to as an permeable boundary, the other surfaces were set to be impermeable.

Initial conditions

Figure 6 shows the initial pore pressure distribution of unsaturated soil slopes. As depth increases above the groundwater level, there is a linear decrease in pore pressure. Taking the water table as the dividing line, above level is the unsaturated region of negative pore pressure and below is the saturated region of positive pore pressure.

Fig. 6
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Initial pore pressure contour map.

Numerical simulation of slope stability without pore gas migration

Serve drying-wetting alternation intensity

According to the criteria in literature24, rainfall between 30.0 and 69.9 mm in 12 h or 50.0 and 99.9 mm in 24 h is considered as heavy rainfall, and the serve drying-wetting alternation intensity in the soil of the slope is the process of transforming the two limiting states of the unsaturated soil, residual and saturated, the transformation process occurs alternately inside the slope under the conditions of extreme rainfall. Therefore, the intensity of rainfall can characterize serve drying-wetting alternation intensity of the slope.

Infiltration rainfall at the top and bottom of the slope

According to the meteorological data in Beijing area where the test site is located, the single maximum rainfall in 2011–2023 was obtained after statistical analysis (Fig. 7).

Fig. 7
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Single maximum rainfall in Beijing from 2011 to 2023.

The single maximum rainfall in the Beijing area where the test site is located from 2011 to 2023 is the extremely heavy rainfall that occurred on August 2, 2023, which had a maximum rainfall of 744.8 mm, with a maximum rainfall intensity of 111.8 mm/h, the rainfall intensity is taken to be 3.11 × 10−5 m/s after conversion. Meanwhile, the infiltration capacity of unsaturated soils must be taken into account: When the rainfall intensity is less than or equal to the soil’s maximum infiltration capacity, the actual infiltration rate equals the rainfall intensity. If the rainfall intensity exceeds the maximum infiltration capacity, the actual infiltration rate is governed by the soil’s infiltration capacity limit.

Infiltration rainfall on slope surface

The average direction outside the slope surface of unsaturated soil slope model is \(\mathop n\limits^{ \to } (1,0,1)\), the intensity of vertical rainfall is 3.11 × 10−5 m/s.

$$q_{t}^{n} = R_{t} n_{z}$$
(24)

Therefore, the rainfall intensity applied to the slope surface is: \(q_{t}^{n} = 3.11 \times 10^{ - 5} \times 1/\sqrt 2 = 2.20 \times 10^{ - 5} {\text{m/s}}\).

Analysis of seepage results

The effect of different intensities of serve drying-wetting alternation on slope stability can be comprehensively studied by analyzing the changes in pore pressure, shear strain increment, and plastic zone of unsaturated soil slopes with rainfall intensities of 3.67 × 10−6 m/s, 5.55 × 10−6 m/s, 1.10 × 10−5 m/s and 2.20 × 10−5 m/s.

Effects of seepage on pore pressure in unsaturated soils

Figure 8 illustrates the pore pressures of the soil with different intensities of serve drying-wetting alternation. It can be found that the pore pressure of the soil varies greatly after serve drying-wetting alternation. Initially, rainwater infiltration into the slope results in a transient saturated zone at the surface, with further infiltration occurring deeper into the slope, this process realigns the boundaries between the unsaturated and saturated zones within the soil. After serve drying-wetting alternation, some soil bodies in the upper part of slope gradually tend to be saturated from unsaturated, at this time the pore pressure is positive, and this part is the infiltration zone. With the intensity of serve drying-wetting alternation increasing, the infiltration zone extends to the slope.

Fig. 8
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Diagrams of pore pressures in soil with different intensities of drying-wetting alternation.

Effects of seepage on shear strain increment of unsaturated soils

Figure 9 illustrates the effects of different intensities of serve drying-wetting alternation on shear strain increment of unsaturated soil. According to the figure, it is evident that shear damage of slopes occurs first on the surface, and the foot of slope is the first location where shear strain increment occurs. As the serve drying-weting alternation intensities increase, the location of the unsaturated soil slope slip develops from shallow to deep, from foot to the top of slope, and the extent of sliding soil body gradually increases.

Fig. 9
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Shear strain increment for slopes with different intensities of drying-wetting alternation.

Effects of seepage on plastic zone of unsaturated soils

Figure 10 presents the effects of different intensities of drying-wetting alternation on the plastic zone of unsaturated soil. As shown in the figures, seepage with varying intensities of drying-wetting alternation can lead to different degrees of plastic deformation of unsaturated soil slope, the plastic deformation zone first appeared in the foot of slope, and the scope of the plastic deformation zone will be expanded with the intensity of drying-wetting alternation increasing, the location of its generation is mainly concentrated on the slope surface, which is also the slip surface, and the two locations correspond to each other. When the intensity of drying-wetting alternation increases, the extent of the plastic zone gradually widens toward the interior and top of slope, and the location of the formation of through-channels is deepened.

Fig. 10
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Plastic zone of soil with different intensities of drying-wetting alternation.

Slope safety and stability under pore gas migration conditions

Migration of pore gas in unsaturated soil under severe drying-wetting alternation

The rainwater continuously infiltrates into the unsaturated soil body during serve drying-wetting alternation, and the infiltrated rainwater displaces the pore gas inside the soil body. With the increasing depth of infiltration, the infiltrated rainwater seals off the penetration channel connecting the inside of the slope with the outside world, which leads to the formation of closed bubbles.

$$u_{ac} = \gamma_{w} h_{\max }$$
(25)

where, \(\gamma_{w}\) is the weight of water, N/m2; \(h_{\max }\) is the maximum infiltration depth, m.

The stress of saturated soil overlying a closed bubble is

$$\sigma_{t} = \gamma_{sat} h$$
(26)

where, \(\gamma_{sat}\) is the saturation density of the soil, N/m2; \(h\) is the depth of the closed bubble, m.

Based on the above conclusions, under ideal state, assuming that the soil body is of uniform soil quality, when rainwater infiltrates into the deepest part of the slope, a wetting front will be formed at this location, which is regular in shape and is generally a horizontal plane. When the pore water pressure at the wetting front is equal to the internal pressure of pore gas, the seepage process is terminated. As shown in Fig. 11a, the wetting front is horizontal at any position. However, most of the unsaturated soil is not homogeneous in the actual condition, and the seepage process is affected by other factors such as boundary conditions, so the seepage analysis should take into account the unequal seepage velocity of the unsaturated soil at each location, the shape of the wetting front in the actual condition is not regular, and it is generally a folded line (Fig. 11b).

Fig. 11
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Wetting front and compressed pore gas escape.

Although the wetting peak pore water pressures are different, the compressed pore gas pressures are equal as shown in Fig. 11, and their pressures are equal to the maximum water pressure at the wetting peak.

The intake value of the soil at the wetting front formed by seepage is:

$$\left( {u_{a} - u_{w} } \right)_{b} = 2T_{s} /R_{b}$$
(27)

where, \(T_{S}\) is the surface tension of water, N/m; \(R_{b}\) is the maximum radius of soil pores, m.

When the surface tension of the boundary shrink film is considered, the pressure inside the bubble is expressed as:

$$u_{ac} = 2T_{s} /R + \sigma_{t}$$
(28)

where, \(R\) is the radius of the bubble, m.

Combining Eqs. (27) and (28) yields:

$$u_{ac} \ge \left( {u_{a} - u_{w} } \right)_{b} + \sigma_{t}$$
(29)

Thus the criterion for pore gas escape associated with the slope soil stress can be obtained as:

$$\sigma_{t} < u_{ac} - \left( {u_{a} - u_{w} } \right)_{b}$$
(30)

The criterion for the formation of air cushions by pore gas is:

$$\sigma_{t} \ge u_{ac}$$
(31)

In unsaturated soil slopes under different drying-wetting alternation intensities, pore gas escaping from the unsaturated slope soil form a ventilation channel when condition (30) is met, and an air cushion is formed inside the unsaturated soil when condition (31) is met.

Evaluation method of slope failure in the presence of pore gas migration

The location of pore gas migration was statistically analyzed using the Weibull distribution evaluation method based on the criterion of the escape of confined bubbles and the formation of air cushions.

The zone principal stress distribution of unsaturated soil slopes obeys the Weibull distribution, the results of all zone seepage calculations for modeled soils were statistically analyzed using the FISH language and taking 95% as the confidence interval. The probability of pore gas escape/air cushions formation, i.e., slope destruction/weakening probability, is analyzed from the perspective of principal stresses, so as to realize the quantification of the pore gas escape/air cushions formation-induced slope instability problem, which makes it possible to solve the problem of pore gas migration under the complex conditions.

The probability density function using the Weibull distribution statistics is:

$$f(t) = [m/(\eta - t_{0} )][(t - t_{0} )/(\eta - t_{0} )]^{m - 1} e^{{ - [(t - t_{0} )/(\eta - t_{0} )]^{m} }}$$
(32)

where \(t\) is the independent variable; \(m\) is the shape parameter (the shape of the distribution curve); \(\eta\) is a scale parameter (characteristic lifespan); \(t_{0}\) is the position parameter (minimum lifespan parameter), it is safe and stable (\(\le t_{0}\)), it is unstable and damaged (\(> t_{0}\)).

The independent variable \(t\) takes the soil stress \(\sigma_{t}\), the applicable Weibull probability density function is obtained from Eqs. (30) to (32):

$$f(t) = [m/(u_{ac} - \sigma_{t0} )][(\sigma_{t} - \sigma_{t0} )/(u_{ac} - \sigma_{t0} )]^{m - 1} e^{{ - [(\sigma_{t} - \sigma_{t0} )/(u_{ac} - \sigma_{t0} )]^{m} }}$$
(33)

where \(\sigma_{t}\) is the soil stress, \(m\), \(\sigma_{t0}\) and \(u_{ac}\) are Weibull distribution parameters.

The statistical analysis procedure regarding the escape/formation of pore gases from unsaturated soil slopes is shown below:

  1. (a)

    The seepage calculation of the unsaturated soil slope model was carried out using FLAC3D, and the stress distribution in each zone of the unsaturated soil slope was obtained based on the results;

  2. (b)

    Using the FISH language embedded in the FLAC3D platform for secondary development, Calculations were extracted in each zone of soil that satisfy the conditions for the escape of pore gas/formation of air cushions;

  3. (c)

    According to the statistical method of Weibull’s probability density function, combined with the above obtained stress distribution of the zone soil body and the calculation results of the pore gas escape/formation of air cushion conditions, the stress distribution of the unsaturated soil slope at this time is analyzed, so as to obtain the slope damage probability, i.e., the probability of pore gas escape/formation of air cushions is obtained;

  4. (d)

    A 95% confidence interval is taken as the final judgment, the zones in that interval are searched and the results are recorded.

Numerical simulation of pore gas escape in slope

The formation/escape of pore gas will generate channels inside the soil body. On this basis, Using the statistical method of Weibull distribution, the location of each channel formed by pore gas formation/escape was determined, and the safety and stability of unsaturated soil slopes in the case of pore gas escape was further investigated by numerical simulation.

Location of pore gas escape

Table 3, Figs. 12 and 13 demonstrate the results of Weibull distribution evaluation in case of pore gas escape. The process of pore gas migration is jointly influenced by permeability conditions and stress distribution. As observed in the figures and tables, gas initially escapes from the toe region of the slope, where the higher hydraulic gradient and greater soil saturation facilitate the initial accumulation and subsequent release of gas. With the increasing of severe drying-wetting alternation intensity, the number of gas escape units rapidly increases from 9 to 29, and the gas migration pathways progressively extend from the slope toe toward the crest. This expansion process is closely associated with water infiltration paths and gas pressure distribution. Ultimately, gas migration further develops into the interior of the slope, forming a more complex channel network whose propagation pattern is governed by the heterogeneity of slope materials and the characteristics of the stress field.

Table 3 Evaluation results of Weibull distribution of pore gas escape.
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Fig. 12
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Evaluation results of Weibull distribution of pore gas escape.

Fig. 13
Fig. 13
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Evaluation results of Weibull distribution of pore gas escape.

The effects of pore gas escape on slope stability

Figure 14 illustrates the shear strain increment in the presence of pore gas escape under different intensities of serve drying-wetting alternation. From the figure, it can be observed that when the pore gas escapes from the interior of the slope, the area where shear damage occurs extends from the foot to the slope face, and damage depth in the range of the foot is greater than that in the case where there is no migration of the pore gas.

Fig. 14
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Shear strain increment when pore gas escapes.

Figure 15 illustrates the plastic zone in the presence of pore gas escape under different intensities of drying-wetting alternation. It can be observed that the escape of pore gas leads to plastic deformation in the slope surface area, which corresponds to the location of the slip surface generated by air cushions.With the intensity of drying-wetting alternation increasing from 3.67 × 10−6 to 2.20 × 10−5m/s, the plastic zone first appeared at the foot, and then the range of plastic zone gradually expanded, finally extended to the top. When the drying-wetting alternation is further intensified, the plastic zone extends into the slope, i.e., the locations within the plastic zone that produce channels are further deepened.

Fig. 15
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Plastic zone during gas escape.

Numerical simulation of air cushions formation by pore gas in slope

Based on the criterion that pore gas form air cushions, the location of the formed air cushions were determined using the Weibull distribution method.

Air cushions position

Table 4, Figs. 16 and 17 show the results of the evaluation of the Weibull distribution for the case of pore gas forming air cushions.

Table 4 Evaluation results of Weibull distribution in the case of pore gas forming an air cushion.
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Fig. 16
Fig. 16
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Evaluation results of Weibull distribution in the case of pore gas forming air cushions.

Fig. 17
Fig. 17
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Weibull evaluation results of air cushions.

Figure 16 clearly illustrates the migration and accumulation processes of air cushions formed by pore gas within the slope. With the intensity of drying-wetting alternation increasing from 3.67 × 10−6 to 2.20 × 10−5 m/s, the formation of air cushions exhibits distinct stage-wise characteristics. In the initial stage, gas preferentially accumulates at the slope toe, forming five discrete air cushion units due to the region’s elevated hydraulic gradient and stress concentration effects. With intensified drying-wetting alternation, the number of air cushion units increases significantly to 25, accompanied by a systematic expansion in their spatial distribution. This expansion manifests not only as vertical propagation from the slope toe toward the crest but also as lateral penetration into the slope interior. The phenomenon reveals a dynamic coupling relationship between pore gas migration mechanisms in unsaturated soils and the evolving water infiltration pathways/soil structure modifications. Notably, the slope toe consistently serves as both the initial and primary zone for air cushion formation, which is fundamentally attributed to its unique hydraulic boundary conditions and stress state characteristics.

The effect of pore gas formation of air cushions on slope stability

Figure 18 shows the incremental shear strain in the soil during the formation of air cushions by pore gas under different intensities of drying-wetting alternation. It is evident that the shear damage occurs at the foot of slope first in the case of air cushions formation, and the location is between the damage depth of the case of no migration of pore gas and the pore gas escape, with the intensity of drying-wetting alternation increasing from 3.67 × 10−6 to 2.20 × 10−5m/s, the scope of air cushions generation escalated to the inside and extends from the initial foot to the surface of the slope, At this point, the destructive tendency also spreads towards the top, finally forming a through shear zone.

Fig. 18
Fig. 18
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Incremental shear strain when pore gas forms air cushions.

Figure 19 illustrates the plastic deformation zone of the soil in the presence of pore gase forming air cushions under different intensities of drying-wetting alternation. From the figure, it can be observed that the plastic zone is located on the surface, corresponding to the position of slip surface generated by the air cushions, and the position is higher than the slip surface position generated by the pore gas escaping. The plastic zone expands toward the interior with the intensity of drying-wetting alternation increasing. Since the formation of the air cushions affects the internal structure of the soil, the strength of the soil decreases sharply and the plastic zone expands further. The range of air cushions is smaller than the range of pore gas escape, the soil structure is not damaged, so under the same drying-wetting alternation intensity, the formation range of the plastic zone in the case of the air cushions is smaller than that in the case of the pore gas escape.

Fig. 19
Fig. 19
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Plastic zone when pore gase forms air cushions.

The influence of pore gas migration on slope stability under serve drying-wetting alternation

Based on the results of numerical calculations of unsaturated–saturated seepage in unsaturated soil slopes under serve drying-wettig alternation, the safety factor was obtained from the strength discount method in the absence and presence of pore gas migration, as shown in Fig. 20.

Fig. 20
Fig. 20
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Curve of safety factor under different drying-wetting alternation intensities.

According to Fig. 20, it can be found that the safety factors under all three conditions decline dramatically as the intensity of drying-wetting alternation increases. The safety factors decrease more before the intensity of drying-wetting alternation is 2.20 × 10−5 m/s, the rate of descent decreases significantly when it is greater than 2.20 × 10−5 m/s. When the intensity of drying-wetting alternation grows up to 2.20 × 10−5 m/s, the safety factor decreases to 0.97 in the case of no pore gas migration, and decreases to 0.89 and 0.92 in the case of gas escape and formation of air cushions, respectively. After the intensity of drying-wetting alternation reaches 2.20 × 10−5 m/s, the safety factors under three conditions are less than 1, which indicates that the intensity of drying-wetting alternation has already led to slope instability at this time. Further observation of the three safety factor curves shows that when the intensity of drying-wetting alternation is the same, the lowest safety factor is obtained in the case of pore gas escape compared to the case of formation of air cushions and the case of no pore gas migration, the rate of decrease of the safety factor in the case of no gas migration is lower than that of the other two cases. At the same time, the escape of pore gases inside the slope will generate a large number of channels, which will seriously damage the soil structure, thus leading to a sharp decline in the safety factor of the slop. The formation of gas cushions only weaken the soil structure, so the damage of gas cushions to slope stability is smaller than that of pore gas escape.

Conclusion

Based on the effective stress and shear strength equations and test parameters, the unsaturated permeability coefficients for the unsaturated–saturated seepage condition are calculated in real time. Based on the criterion of pore gas escape/cushions formation after serve drying-wetting alternation, combined with the statistical method of Weibull distribution, the location of pore gas escape/cushions formation is accurately obtained. The numerical calculation can effectively analyze the problem of instability induced by the migration of unsaturated pore gas in soil slopes under serve drying-wetting alternation. The results of the study show that it has certain reference value for the study of landslide disaster triggered by serve drying-wetting alternation under complex climate. The findings suggest that:

  1. (1)

    Pore gas escape/gas cushions formation is first generated at the foot of the slope, and both are extended towards the interior of the slope with the intensity of the drying-wetting alternation increasing.

  2. (2)

    The infiltration of rainwater into unsaturated soil slopes under the conditions of no pore gas migration, pore gas escape and formation of air cushions follows the unsaturated saturated seepage law. Pore gas migration weakens the soil shear strength, and the slip damage increases, which leads to a drastic decrease in slope stability.

  3. (3)

    From the value of the safety factor, it can be seen that with the intensity of the drying-wetting alternation increasing, safety factors of the case of no pore gas migration, pore gas escape/formation of air cushions all show a decreasing tendency, the degree of damage to the safety and stability of slopes caused by the escape of pore gase is greater than that in the case of the formation of air cushions.