Abstract
Heavy rainfall in extreme climates often causes natural disasters such as floods, landslides, and debris flows. Rainfall-induced slope instabilities are major geological natural disasters (Glade in Environ Geol 35:160–174, 1998; Dai et al. in Eng Geol 51:279–290, 1999; Iverson in Water Resour Res 36:1897–1910, 2000; Lee and Pradhan in Landslides 4:33–41, 2007; Li et al. in Landslides 13:1109–1123, 2016a; Li et al. in Ecol Eng 91:477–486, 2016b; Wu et al. in Hydro-mechanical analysis of rainfall-induced landslides. Springer, 2020) that can result in considerable loss of life and damage to infrastructure. Extreme events such as storms, which are becoming more severe because of climate change, can trigger fatal landslides.
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Heavy rainfall in extreme climates often causes natural disasters such as floods, landslides, and debris flows. Rainfall-induced slope instabilities are major geological natural disasters (Glade 1998; Dai et al. 1999; Iverson 2000; Lee and Pradhan 2007; Li et al. 2016a, b; Wu et al. 2020) that can result in considerable loss of life and damage to infrastructure. Extreme events such as storms, which are becoming more severe because of climate change, can trigger fatal landslides. Storm-induced slope failures frequently occur because of rainfall infiltration, particularly in tropical areas (Fourie 1996; Cevasco et al. 2014). Global climate change in many mountainous areas could lead to more severe fluctuations in rainfall, and trigger of soil slope deformations and even slope instability because of the alteration of intensity, frequency, and quantity of rainfall (Dixon and Brook 2007; Jeong et al. 2008). The influence of climate change on rainfall characteristics has the potential to alter the stability of unsaturated soil slopes. Rainfall infiltration causes a decrease in matric suction and an increase in moisture content and hydraulic conductivity in unsaturated soils. The rainfall intensity and duration, initial water table, and hydraulic conductivity are the parameters that significantly affect slope stability (Ng and Shi 1998). An increase in pore-water pressure can reduce the effective stress and thereby weaken the shear strength of slopes. Complex geological environment and human engineering activities are also significant factors of slope instability under rainfall conditions.
Rainfall-induced slope failures have been examined based on experimental modeling, analytical and numerical methods (Ng and Shi 1998; Iverson 2000; Chen et al. 2005; Wu et al. 2009, 2016, 2020; Zhu et al. 2019). Laboratory and field experiments have been carried out to examine the infiltration mechanisms associated with rainfall-induced slope failures (e.g., Lee et al. 2011; Wu et al. 2015, 2017, 2018). Many numerical and analytical studies have investigated the hydraulic responses of slopes to rainfall infiltration and the stability of slopes under such conditions (Iverson 2000; Cai and Ugai 2004; Wiles 2006; Ali et al. 2014; Zhu et al. 2022). Numerical analysis solves for the matric suction or pressure head distribution in a soil slope with varying permeability, and considering different surface conditions of a soil. Many cases demonstrate that change in rainfall patterns may lead to slope failure due to infiltration. The results can provide an indication of the potential influence of climate change on shallow landslides in many mountainous areas (Kim et al. 2012).
Slope stability problems are commonly encountered in engineering projects. Many slope failures are attributed to water infiltration (Cho and Lee 2002; Cai and Ugai 2004). Matric suction is crucial to the stability of soil slopes because dissipation of matric suction leads to decrease in shear strength of unsaturated soils. Slope failure is closely related to the rainfall-induced transient infiltration of slopes in unsaturated soils (Fredlund and Rahardjo 1993; Wu et al. 2020). Shallow landslides are related to periods of intense rainfall. Engineering activities can result in severe geological and environmental issues. Slope failures induced by engineering activities may occur and may progress into landslides. The internal mechanics of slope movements are stress redistribution and the consequent changes in engineering–geological conditions (Marschalko et al. 2012).
Landslide forecasting should take into account rainfall infiltration into soil slopes. The models in predicting the timing and location of landslides are related to dynamic water infiltration in soil slope. The coupled process in an unsaturated soil is of major interest because of its implications for disaster prevention and environmental issues. Analytical approaches have been developed to provide a basic understanding of unsaturated infiltration in terms of the coupling effect (Wu et al. 2020). Meanwhile, numerical approaches provide a powerful tool for solving complex, nonlinear infiltration into unsaturated soils. These numerical models effectively investigate the coupled hydro-mechanical problem involved in unsaturated rooted slope stability issues (Oka et al. 2010; Wu et al. 2020; Zhu et al. 2022).
1.1 Rainfall Infiltration Equation
According to the modified Green–Ampt model (Mein and Larson 1973), the water infiltration process in soils during uniform precipitation can be divided into two stages: a stage controlled by rainfall intensity and a stage controlled by pressure head. The infiltration rate (fa) determined by rainfall intensity (qr) can be written as:
where qr is the rainfall intensity; and β is the slope angle.
The infiltration rate (fb) determined by the pressure head can be expressed as:
where ks represents the permeability coefficient at saturation; sf represents the suction head of the wetting front; and zf represents the wetting front depth.
If fa = fb, the water ponding time (tp) can be obtained as:
where θs and θi are the saturated moisture content and initial moisture content, respectively.
According to the water balance principle and Darcy law, wetting front movement during rainfall can be calculated as follows:
Equation (1.4) is the GA model suitable for slopes (Chen and Young 2006).
A penetration test of saturated sand layers was conducted and found a quantitative relationship between the infiltration velocity of water in the soil and the head loss, namely Darcy law:
where q is the flux (discharge per unit area, with units of length per time, m/s), H is the total water head, and L is the seepage length.
When soil mass is unsaturated, most scholars believe that Darcy law can also be used for analyzing water movement in unsaturated soils. Richards (1931) extended the Darcy law of saturated soil to the unsaturated infiltration and introduced the continuity equation to derive the equation of motion of the unsaturated soil–water flow, namely the Richards’ equation. The permeability coefficient (k) is expressed as a function of matric suction, and Darcy’s law can be expressed as:
where \(\nabla H\) is a hydraulic gradient, including two components of gravity and suction.
Continuity equation:
Then:
Substituting Eq. (1.6) into Eq. (1.8), one can obtain:
As shown in Fig. 1.1, the Richards’ equation governing one-dimensional vertical infiltration in unsaturated soils can be written as:
1.2 Infiltration Equation for Unsaturated Slopes
The 2D generalized RE for unsaturated infiltration is expressed as (Ku et al. 2017; Wu et al. 2020):
where \(K_{z} (h)\) and \(K_{x} (h)\) are the permeability coefficients along the vertical direction and lateral direction in unsaturated soils, respectively. To study the groundwater flow of the unsaturated slope (Fig. 1.2), the RE needs to be rotated. Total head can be written as:
and elevation head E can be described as:
By substituting Eqs. (1.12) and (1.13) into Eq. (1.11), Eq. (1.11) can be re-expressed as:
According to Iverson’s model (2000), the modified RE only considers infiltration in the vertical direction, which can be given by:
1.3 Linearized Richards’ Equation
Combined with an exponential model, Eq. (1.15) is linearized. Here, a new parameter \(h^{*}\) is defined as:
where \(\lambda\) is the key parameter for solving the linearized Richards’ equation in the conversion method, which can be defined as a constant \(\lambda = {\text{e}}^{{\alpha h_{{\text{d}}} }}\) (Tracy 2006; Liu et al. 2015); hd is the pressure head value when the soil is dry. The relative permeability coefficient is expressed as:
Taking the derivative of Eq. (1.16) with respect to z, one can obtain:
Equation (1.18) can be re-stated as follows:
Furthermore, substituting Eq. (1.16) into Eq. (1.19), one can obtain:
Equation (1.20) can be rewritten as:
Equation (1.17) can be derived from z again:
Substituting Eq. (1.19) into Eq. (1.22), one can have:
An exponential model is employed to describe soil moisture (Gardner 1958):
where \(\theta (h)\) is volumetric water content; \(\theta_{{\text{r}}}\) is the residual volumetric water content.
The derivation of both sides of Eq. (1.24) with respect to time t has the following relationship:
Substituting Eqs. (1.19), (1.23), and (1.24) into Eq. (1.10), the linearized Richards’ equation can be obtained:
where \(c = \alpha (\theta_{{\text{s}}} - \theta_{{\text{r}}} )/k_{{\text{s}}}\).
Equation (1.26) is also expressed as:
where \(K_{\theta } = k_{{\text{s}}} /\left( {\theta_{{\text{s}}} - \theta_{{\text{r}}} } \right)\), \(K_{a} = K_{\theta } /\alpha\).
The finite difference format of the linearized RE (Eq. 1.26) can be expressed as:
where \(i,\Delta z,\Delta t\), and n denote the nodal point, grid size, time step, and time level, respectively.
It can be seen from Eqs. (1.26) and (1.27) that the nonlinear partial differential equation (Eq. 1.26) has been transformed into a linear partial differential equation. Once the linear partial differential equation is solved to obtain a numerical solution, the actual pressure head can be written as:
1.4 Unsaturated Soil Slope Stability Under Rainfall
Shear strength is a fundamental material property that is required to address a variety of engineering problems including bearing capacity, slope stability, lateral earth pressure, pavement design, and foundation design. Recently, many researches have focused on the shear strength of unsaturated soils (Fredlund et al. 1996; Lu et al. 2010).
According to Mohr–Coulomb criterion and effective stress, the shear strength of saturated soils can be expressed as:
where \(\tau_{{\text{f}}}\) is the shear strength (kPa), c is the cohesion (kPa), \(\sigma_{{\text{n}}}\) is the normal stress acting on the failure surface (kPa), and \(\varphi\) is the angle of internal friction (°). Cohesion and cohesive shear strength are due to chemical bonding between soil particles and surface tension within the water films (Lu and Likos 2006). Frictional shear strength (\(\sigma_{{\text{n}}} \tan \varphi\)) is owing to internal friction between soil particles that depends on the normal stress acting on the failure surface.
Engineering practices indicate that the shear strength equation of saturated soils can meet the engineering requirements. The shear strength parameters are also influenced by matric suction. With an increase in matric suction, c and \(\varphi\) increase, which depends on soil texture and structure. Soil shear strength significantly increases with an increase in net normal stress, matric suction, and the parameters of shear strengths.
However, several phases of unsaturated soils make the shear strength equation of saturated soils difficult to apply. Therefore, some studies on the shear strength criteria of unsaturated soils have been carried out. There are main representative shear strength criteria here.
Bishop (1959) developed a shear strength criterion for unsaturated soils:
where \(\tau_{{\text{f}}}\) is the shear strength of unsaturated soils; c′ and \(\varphi^{\prime}\) are the effective cohesion and friction angle, respectively; (ua − uw) is the matric suction; ua is the pore air pressure; uw is the pore-water pressure (h = uw/γw, γw = ρwg); and \(\chi\) is the function of the degree of saturation.
Based on two stress state variables, the following equation was developed to describe shear strength (Fredlund and Rahardjo 1993):
where \(\varphi^{b}\) is the internal friction angle due to the distribution of matric suction.
Lu and Likos (2004) proposed a unified form of shear strength equation:
in which
The first two terms in Eq. (1.33), c′ and c″, represent shear strength due to the so-called apparent cohesion in unsaturated soils. In an unsaturated soil, the third term represents frictional shearing resistance provided by the effective normal force at the grain contacts. The apparent cohesion captured by the first two terms includes the classical cohesion c′ representing shearing resistance arising from interparticle physicochemical forces, and the second term c″ describing shearing resistance arising from capillarity effects. The term c″ is defined as capillary cohesion hereafter. Physically, capillary cohesion describes the mobilization of suction stress \(\chi\)(ua − uw) in terms of shearing resistance. The relationship between capillary cohesion and the maximum suction stress at failure, \(\chi_{{\text{f}}} \left( {u_{{\text{a}}} - u_{{\text{w}}} } \right)_{{\text{f}}}\) is defined as shear strength also affects the water movement of the soils (Eudoxie et al. 2012).
Slope failure in unsaturated soil regions induced by rainfall is due to shear strength of unsaturated soils (Fredlund and Rahardjo 1993; Lu and Likos 2004; Guzzetti et al. 2008; Muntohar and Liao 2009). Both rainfall characteristics (rainfall intensity and duration) and soil permeability may influence failure mechanism.
The soil slope stability was commonly followed by stability analysis according to the pressure head and/or the stress condition within the soil slope profile. Various techniques were employed to compute factor of safety (Fs), and the conventional limit equilibrium methods (Alonso et al. 2010). The limit equilibrium approach is mostly effective for slope failure with a small depth compared with their length and breadth. A slope sliding at a depth happens as the driving stress contributing to failure exceeds the anti-slip stress offered by the soil mass strength. Namely, sliding can occur at a particular depth as follows:
where \(F_{{\text{s}}} \left( {z,t} \right)\) is the safety factor over depth and time; W is the weight of the sliding mass; and γw represents the unit weights of water.
Equation (1.35) can be re-arranged as:
in which, γsat represents the unit weight of the saturated soil. When Fs approaches 1, the infinite soil slope reaches a limit state. Based on Eq. (1.36), the limit-state pore-water pressure head can be obtained.
Rainfall-induced landslides may occur in unsaturated soils above the groundwater table, usually with shallow sliding surfaces parallel to the slope surface (Lu and Godt 2008), which involves 2D and 3D problems. However, an infinite slope model is usually used as a simplified model of the 2D or 3D issues with simple geometry and ignores the stress concentration, the practice sometime demonstrates its effectiveness for assessing shallow slope stability (Michalowski 2018).
Slope instabilities are often hydrologically initiated by the advancement of the wetting front alone (Muntohar and Liao 2010), a rise in groundwater level (Asch et al.1999; Montgomery et al. 2009), and positive pore-water pressure on the soil–bedrock boundaries (Baum et al. 2010). The most common mechanism for rainfall-induced landslides occurs when the soil slides on a low-conductivity layer. Rainfall infiltration leads to a rise in the pressure head, resulting in positive pore-water pressures (Iverson 2000; Muntohar and Liao 2010).
Generally, unsaturated soil slope failures happen most frequently during or after rain periods (Wu et al. 2020). The characteristics of water flow, change of pore-water pressure, and shear strength of soils are the major parameters related to slope stability analysis involving unsaturated soils that are directly affected by the boundary conditions (i.e., infiltration and evaporation) at the soil–atmosphere interface. The relative importance of soil properties, rainfall intensity, initial water table location, and slope geometry in inducing instability of soil slopes under different rainfall was investigated through a series of studies. Soil properties and rainfall intensity were found to be the primary factors controlling the slope instability due to rainfall, while the initial water table location and slope geometry only played a secondary role (Rahardjo et al. 2007).
The Green–Ampt model is a typical approximate infiltration model. Due to the simplicity and few parameters, the approximate infiltration model has become popular (Grimaldi et al. 2013). The classic GA model is only suitable for infiltration in horizontal soils. Therefore, modified GA models have been developed to describe the water infiltration in layered soils and slopes (e.g., Mein and Larson 1973; Chen and Young 2006; Kale and Sahoo 2011). Some modified infiltration models that account for rainwater redistribution have also been proposed (e.g., Corradini et al. 1997; Dou et al. 2014). These infiltration models have been extended to regional rainfall-runoff models for the hydrological prediction of catchments (Yuan et al. 2019). However, the actual infiltration process is very complicated and affected by many factors such as soil heterogeneity and rainfall conditions, and becomes difficult to be described accurately based on theoretical formulations (Srivastava et al. 2020). These theoretical equations generally tend to overestimate the factor of safety of soil slopes, resulting in slides and geological hazards (Kim et al. 2012). Some intelligent methods have been developed to predict the water infiltration into soils using machine learning techniques (Sihag et al. 2018).
Hydrological responses and slope factor of safety due to rainfall are concerned from a perspective of hydro-mechanical coupling. Coupled and uncoupled hydro-mechanical behaviors in unsaturated soils have been carried out to characterize the physical responses of unsaturated infiltration (i.e., variation of soil moisture, matric suction, effective stress, shear strength, and slope stability) (Casini 2013). The coupled issues are strongly linked in unsaturated soil slopes due to water infiltration, and the coupled poromechanical model actually examines the behavior and stability of rooted soils subjected to rainfall (Kim et al. 2012). Pressure heads generated in the uncoupled analysis are employed to examine deformation or soil slope stability (Cai and Ugai 2004; Yoo and Jung 2006). The accuracy and computational efficiency of the uncoupled analysis highly depend on the selected time increments (Huang and Lo 2013). The soil hydraulic and mechanical responses are calculated simultaneously in the coupled analysis. The coupled analysis produced a reasonably well defined wetting front and a lower critical Fs for unsaturated soil slopes. The coupled investigation could produce more accurate assessment of soil slope stability due to water infiltration and demonstrate a better physical representation of water infiltration and stress variation within unsaturated soil slopes.
More and more methods highlight the role of vegetation because of their interception role of the canopy and the root characteristics. Meanwhile, recent studies indicate that vegetation cannot control the rainfall-induced shallow landslide distribution (Emadi-Tafti et al. 2021). Some researches focus on the effect of roots on root–soil composite strength, or saturated hydraulic conductivity (Alessio 2019). The more complex the root architecture is, the stronger the root-composite strength becomes, while the faster the rainfall infiltrates. It has generally been concluded that vegetation roots mechanically and hydrologically affect slope stability. The plant roots seem act as a positive function in root-composite strength, while a negative role in water infiltration. Plant roots have various architectures in different land ecosystems and climatic conditions (Ma et al. 2018). Increasing studies related to soil–root complex focus on the root architectures (Burylo et al. 2011; Li et al. 2016a, b). One major controversy exists, e.g., the plant roots play positive role and enhance slope strength (Arnone et al. 2016). The roots could advance rainfall infiltration, thus contributing an adverse effect on slope stability (Ghestem et al. 2011; Garg et al. 2015). The root–soil composite strength and the hydraulic conductivity are of utmost importance for the rooted soil slope stability.
References
Alessio P (2019) Spatial variability of saturated hydraulic conductivity and measurement based intensity-duration thresholds for slope stability, Santa Ynez Valley, CA. Geomorphology 342:103–116
Ali A, Huang JS, Lyamin AV, Sloan SW, Cassidy MJ (2014) Boundary effects of rainfall-induced landslides. Comput Geotech 61:341–354
Alonso EE, Pereira JM, Vaunat J, Olivella S (2010) A microstructurally based effective stress for unsaturated soils. Géotechnique 60(12):913–925
Arnone E, Caracciolo D, Noto LV, Preti F, Bras RL (2016) Modeling the hydrological and mechanical effect of roots on shallow landslides. Water Resour Res 52(11):8590–8612
Asch TWJV, Buma J, Beek LPHV (1999) A view on some hydrological triggering systems in landslides. Geomorphology 30(1):25–32
Baum RL, Godt JW, Savage WZ (2010) Estimating the timing and location of shallow rainfall-induced landslides using a model for transient, unsaturated infiltration. J Geophys Res Earth Surf 115(F3):1–26
Bishop AW (1959) The principle of effective stress. Tekn Ukebl 39:859–863
Burylo M, Hudek C, Rey F (2011) Soil reinforcement by the roots of six dominant species on eroded mountainous marly slopes (Southern Alps, France). Catena 84(1):70–78
Cai F, Ugai K (2004) Numerical analysis of rainfall effects on slope stability. Int J Geomech 4(2):69–78
Casini F (2013) Coupled processes during rainfall an experimental investigation on a silty sand. In: Poromechanics. ASCE, pp 1542–1549
Cevasco A, Pepe G, Brandolini P (2014) The influences of geological and land use settings on shallow landslides triggered by an intense rainfall event in a coastal terraced environment. Bull Eng Geol Environ 73:859–875
Chen L, Young MH (2006) Green–Ampt infiltration model for sloping surfaces. Water Resour Res 42(W07420):1–9
Chen CY, Chen TC, Yu WH, Lin SC (2005) Analysis of time-varying rainfall infiltration induced landslide. Environ Geol 48:466–479
Cho SE, Lee SR (2002) Evaluation of surficial stability for homogeneous slopes considering rainfall characteristics. J Geotech Geoenviron Eng 128(9):756–763
Corradini C, Melone F, Smith RE (1997) A unified model for infiltration and redistribution during complex rainfall patterns. J Hydrol 192(1–4):104–124
Dai FC, Lee CF, Wang SJ (1999) Analysis of rainstorm-induced slide-debris flows on natural terrain of Lantau Island, Hong Kong. Eng Geol 51:279–290
Dixon N, Brook E (2007) Impact of predicted climate change on landslide reactivation: case study on Mam Tor, UK. Landslides 4:137–147
Dou HQ, Han TC, Gong XN, Zhang J (2014) Probabilistic slope stability analysis considering the variability of hydraulic conductivity under rainfall infiltration-redistribution conditions. Eng Geol 183:1–13
Emadi-Tafti M, Ataie-Ashtiani B, Hosseini SM (2021) Integrated impacts of vegetation and soil type on slope stability: a case study of Kheyrud Forest, Iran. Ecol Model 446:109498
Eudoxie GD, Phillips D, Springer R (2012) Surface hardness as an indicator of soil strength of agricultural soils. Open J Soil Sci 2:341–346
Fourie AB (1996) Predicting rainfall-induced slope instability. Proc Inst Civ Eng Geotechn Eng 119(4):211–218
Fredlund DG, Rahardjo H (1993) Soil mechanics for unsaturated soil. Wiley, New York
Fredlund DG, Xing A, Fredlund MD, Barbour SL (1996) The relationship of the unsaturated soil shear to the soil-water characteristic curve. Can Geotech J 33(3):440–448
Gardner WR (1958) Some steady stale solutions of the unsaturated moisture low equation with applications to evaporation from a water table. Soil Sci 85(4):228–232
Garg A, Coo JL, Ng CWW (2015) Field study on influence of root characteristics on soil suction distribution in slopes vegetated with Cynodon dactylon and Schefflera heptaphylla. Earth Surf Process Landforms 40(12):1631–1643
Ghestem M, Sidle RC, Stokes A (2011) The influence of plant root systems on subsurface flow: implications for slope stability. BioScience 61(11):869–879
Ghestem M, Veylon G, Bernard A, Vanel Q, Stokes A (2014) Influence of plant root system morphology and architectural traits on soil shear resistance. Plant Soil 377(1–2):43–61
Glade T (1998) Establishing the frequency and magnitude of landslide-triggering rainstorm events in New Zealand. Environ Geol 35(2):160–174
Grimaldi S, Petroselli A, Romano N (2013) Curve-number/Green–Ampt mixed procedure for streamflow predictions in ungauged basins: parameter sensitivity analysis. Hydrol Process 27(8):1265–1275
Guzzetti F, Peruccacci S, Rossi M, Stark CP (2008) The rainfall intensity–duration control of shallow landslides and debris flows: an update. Landslides 5(1):3–17
Huang CC, Lo CL (2013) Simulation of subsurface flows associated with rainfall-induced shallow slope failures. J GeoEng 8(3):101–111
Iverson RM (2000) Landslide triggering by rain infiltration. Water Resour Res 36(7):1897–1910
Jeong S, Kim J, Lee K (2008) Effect of clay content on well-graded sands due to infiltration. Eng Geol 102:74–81
Kale RV, Sahoo B (2011) Green–Ampt infiltration models for varied field conditions: a revisit. Water Resour Manage 25(14):3505–3536
Kim J, Jeong S, Regueiro RA (2012) Instability of partially saturated soil slopes due to alteration of rainfall pattern. Eng Geol 147:28–36
Ku CY, Liu CY, Su Y, Xiao JE, Huang CC (2017) Transient modeling of regional rainfall-triggered shallow landslides. Environ Earth Sci 76(16):570
Lee S, Pradhan B (2007) Landslide hazard mapping at Selangor, Malaysia using frequency ratio and logistic regression models. Landslides 4(1):33–41
Lee LM, Kassim A, Gofar N (2011) Performances of two instrumented laboratory models for the study of rainfall infiltration into unsaturated soils. Eng Geol 117(1–2):78–89
Li WC, Dai FC, Wei YQ, Wang ML, Min H, Lee LM (2016a) Implication of subsurface flow on rainfall-induced landslide: a case study. Landslides 13(5):1109–1123
Li YP, Wang YQ, Ma C, Zhang HL, Wang YJ, Song SS, Zhu JQ (2016b) Influence of the spatial layout of plant roots on slope stability. Ecol Eng 91:477–486
Liu CY, Ku CY, Huang CC (2015) Numerical solutions for groundwater flow in unsaturated layered soil with extreme physical property contrasts. Int J Nonlinear Sci Numer Simul 16(7):325–335
Lu N, Godt J (2008) Infinite slope stability under steady unsaturated seepage conditions. Water Resour Res 44(11):W11404
Lu N, Likos WJ (2006) Suction stress characteristic curve for unsaturated soil. J Geotech Geoenviron Eng 132(2):131–142
Lu N, Godt JW, Wu DT (2010) A closed-form equation for effective stress in unsaturated soil. Water Resour Res 46(5):W05515
Lu N, Likos WJ (2004) Unsaturated soil mechanics. John Wiley and Sons, Inc
Ma C, Deng JY, Wang R (2018) Analysis of the triggering conditions and erosion of a runoff-triggered debris flow in Miyun County, Beijing, China. Landslides 15:2475–2485
Marschalko M, Yilmaz I, Bednárik M, Kubečka K (2012) Influence of underground mining activities on the slope deformation genesis: Doubrava Vrchovec, Doubrava Ujala and Staric case studies from Czech Republic. Eng Geol 147–148:37–51
Mein RG, Larson CL (1973) Modeling infiltration during a steady rain. Water Resour Res 9(2):384–394
Michalowski RL (2018) Failure potential of infinite slopes in bonded soils with tensile strength cut-off. Can Geotech J 55(4):477–485
Muntohar AS, Liao HJ (2009) Analysis of rainfall-induced infinite slope failure during typhoon using a hydrological–geotechnical model. Environ Geol 56:1145–1159
Muntohar AS, Liao HJ (2010) Rainfall infiltration: infinite slope model for landslides triggering by rainstorm. Nat Hazards 54:967–984
Montgomery DR, Schmidt KM, Dietrich WE, Mckean J (2009) Instrumental record of debris flow initiation during natural rainfall: Implications for modeling slope stability. J Geophys Res–Earth 114(F1):1–16
Ng CWW, Shi Q (1998) A numerical investigation of the stability of unsaturated soil slopes subjected to transient seepage. Comput Geotech 22:1–28
Oka F, Kimot S, Takada N (2010) A seepage-deformation coupled analysis of an unsaturated river embankment using a multiphase elasto-viscoplastic theory. Soils Found 50(4):483–494
Rahardjo H, Ong TH, Rezaur RB, Leong EC (2007) Factors controlling instability of homogeneous soil slopes under rainfall. J Geotech Geoenviron Eng 133(12):1532–1543
Richards LA (1931) Capillary conduction of liquids through porous mediums. Physics 1(5):318–333
Sihag P, Singh B, Sepah Vand A, Mehdipour V (2018) Modeling the infiltration process with soft computing techniques. ISH J Hydraul Eng 26(2):138–152
Srivastava A, Kumari N, Maza M (2020) Hydrological response to agricultural land use heterogeneity using variable infiltration capacity model. Water Resour Manage 34:3779–3794
Tracy FT (2006) Clean two and three-dimensional analytical solutions of Richards’ equation for testing numerical solvers. Water Resour Res 42(8):8503, 1–11
Wiles TD (2006) Reliability of numerical modelling predictions. Int J Rock Mech Min Sci 43(3):454–472
Wu LZ, Zhang LM (2009) Analytical solution to 1D coupled water infiltration and deformation in unsaturated soils. Int J Numer Anal Met 33(6):773–790
Wu LZ, Huang RQ, Xu Q, Zhang LM, Li HL (2015) Analysis of physical testing of rainfall-induced soil slope failures. Environ Earth Sci 73(12):8519–8531
Wu LZ, Selvadurai APS, Zhang LM, Huang RQ, Huang JS (2016) Poro-mechanical coupling influences on potential for rainfall-induced shallow landslides in unsaturated soils. Adv Water Resour 98:114–121
Wu LZ, Zhou Y, Sun P, Shi JS, Liu GG, Bai LY (2017) Laboratory characterization of rainfall-induced loess slope failure. Catena 150:1–8
Wu LZ, Zhang LM, Zhou Y, Xu Q, Yu B, Liu GG, Bai LY (2018) Theoretical analysis and model test for rainfall-induced shallow landslides in the red-bed area of Sichuan. Bull Eng Geol Environ 77:1343–1353
Wu LZ, Huang RQ, Li X (2020) Hydro-mechanical analysis of rainfall-induced landslides. Springer
Yoo C, Jung HY (2006) Case history of geosynthetic reinforced segmental retaining wall failure. J Geotech Geoenviron Eng ASCE 132(12):1538–1548
Yuan W, Liu M, Wan F (2019) Calculation of critical rainfall for small-watershed flash floods based on the HEC-HMS hydrological model. Water Resour Manage 33:2555–2575
Zhu S R, Wu L Z, Shen Z H, et al (2019) An improved iteration method for the numerical solution of groundwater flow in unsaturated soils. Comput Geotech 114: 103113
Zhu SR, Wu LZ, Huang JS (2022) Application of an improved P(m)-SOR iteration method for flow in partially saturated soils. Comput Geosci 26(1):131–145
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Wu, L., Zhou, J. (2023). Background. In: Rainfall Infiltration in Unsaturated Soil Slope Failure . SpringerBriefs in Applied Sciences and Technology. Springer, Singapore. https://doi.org/10.1007/978-981-19-9737-2_1
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