Numerical stability analysis of an unsaturated infinite slope under rainfallinduced infiltration conditions was performed using generalized effective stress to unify both saturated and unsaturated conditions The soilwater characteristic curve (SWCC) of sand with a relative density of 75% was initially measured for both drying and wetting processes. The hydraulic conductivity function (HCF) and suction stress characteristic curve (SSCC) were subsequently estimated. Under the rainfallinduced infiltration conditions, transient seepage analysis of an unsaturated infinite slope was performed using the finite element analysis program, SEEP/W. Based on these results, the stability of an unsaturated infinite slope under rainfallinduced infiltration conditions was examined in relation to suction stress. According to the results, the negative porewater pressure and water content within the slope soil changed over time due to the infiltration. In addition, the variation of the negative porewater pressure and water content led to a variation in suction stress and a subsequent change in the slope’s factor of safety during the rainfall period.
Introduction
Most of the recent landslides in Korea occurred via translational slides within 2 m of the ground surface
(Kim et al., 2007)
. In other words, most landslides occurred due to rainfallinduced infiltration and were mainly shallow failures
(Song et al., 2012a
;
Song, 2013)
. The fundamental cause of such slope failure is an increased degree of saturation in the vadose zone and reduced matric suction due to precipitation. Consequently, the suction changes over time and the factor of safety is reduced.
Therefore, this research investigated the effect of suction variation caused by rainfallinduced infiltration on the stability of an unsaturated infinite slope. To do this, we used a new effective stress concept that includes suction stress, as proposed by
Lu and Likos (2004)
. The concept of suction stress is described in greater detail in the following section. The soilwater characteristic curve (SWCC), hydraulic conductivity function (HCF), and suction stress characteristic curve (SSCC) of Jumunjin sand were estimated using data measured by an automated soilwater characteristic curve apparatus. The variation in the matric suction was then predicted for rainfall using the infinite program, SEEP/W, and finally the variations in suction stress and slope stability were analyzed.
Hydraulic characteristics of unsaturated soil
 SoilWater Characteristic Curve (SWCC)
Knowledge of matric suction is essential in understanding flow, stress, and deformation phenomena with respect to unsaturated soil. The SWCC is typically defined as the relationship between the matric suction and the volumetric water content of an unsaturated soil. The SWCC is, therefore, the most important property in unsaturated soil mechanics.
Fig. 1
shows a typical SWCC. Water content can be expressed in terms of volumetric water content or degree of saturation, and soil suction can be expressed in terms of matric suction.
A typical SWCC (modified from Lu and Likos, 2004).
Various models have been proposed to present SWCC in continuous mathematical form. Three models in particular are commonly adopted in the field of geotechnical engineering: the
Brooks and Corey (1964)
model, the
van Genuchten (1980)
model, and the
Fredlund and Xing (1994)
model. According to the analytical results of
Song et al. (2010)
, the
van Genuchten (1980)
model is the most reliable way to fit the relationship between matric suctions and volumetric water content in Jumunjin sand. The
van Genuchten (1980)
model was therefore used in this study, and may be expressed as
where Θ is the normalized volumetric water content,
θ
is the volumetric water content,
θ_{s}
is the saturated volumetric water content,
θ_{r}
is the residual volumetric water content,
S_{e}
is the effective degree of saturation,
S
is the degree of saturation,
S_{r}
is the residual degree of saturation,
α
,
n
, and
m
are the empirical fitting parameters, and
ψ
is the difference between pore air pressure and pore water pressure (i.e.,
u_{a}

u_{w}
).
 Hydraulic Conductivity Function (HCF)
The relative hydraulic conductivity function
k
_{r}
(
ψ
) is a flexible closedform analytical equation proposed by
van Genuchten (1980)
by substituting equation (1) into the statistical conductivity models proposed by
Burdine (1953)
and
Mualem (1978)
, and was adopted as follows
where
k_{r}
=Θ
^{0.5}
[1(1Θ
^{1/m}
)
^{m}
]
^{2}
This allows the hydraulic conductivity function,
k
, of unsaturated soil to be estimated directly from a corresponding model of the SWCC if the saturated hydraulic conductivity,
k_{s}
, is known.
 Suction Stress Characteristic Curve (SSCC)
There are two widely recognized macroscale approaches for describing the state of stress in unsaturated soil: (1) the modified effective stress approach, which is generally attributed to the work of
Bishop (1959)
; and (2) the independent stress state variable approach, which is generally attributed to the work of
Fredlund and Morgenstern (1977)
.
Bishop’s (1959)
effective stress approach involves a modified form of
Terzaghi’s (1943)
classic effective stress equation (3) written as follows:
where the “effective stress parameter”
χ
is generally considered to vary between zero and unity as a function of the degree of porewater saturation. The difference (
σ
–
u_{a}
) is the net normal stress and the difference (
u_{a}
–
u_{w}
) is matric suction. For
χ
equal to zero (corresponding to perfectly dry conditions) and for
χ
equal to unity (corresponding to saturated conditions), equation (4) reduces to Terzaghi’s effective stress. For
χ
between zero and unity, the second term in equation (4),
χ
(
u_{a}
–
u_{w}
), describes the contribution of matric suction to effective stress. Following the approach by
Bishop (1959)
, the macroscopic engineering behavior of unsaturated soil is described using the effective stress defined by equation (4) within the established framework of saturated soil mechanics. Shear strength, for example, may be described by incorporating the modified effective stress expression into the classical MohrCoulomb failure criterion:
where
c
' is the effective cohesion and
Φ
′ is the effective angle of internal friction.
Following the independent stress state variable approach by
Fredlund and Morgenstern (1977)
, net normal stress (
σ
–
u_{a}
) and matric suction (
u_{a}
–
u_{w}
) were treated independently in terms of their roles in the mechanical behavior of unsaturated soil. Macroscopic behavior is described in terms of the independent stress state variables and conjugate material properties. Shear strength, for example, may be described as
where the first two terms comprise the classical MohrCoulomb criterion and the third term introduces
Φ^{b}
as an additional friction angle to capture the contribution of matric suction to shear strength.
The effectiveness, validity, and practicality of these two different approaches for describing the state of stress and corresponding behavior of unsaturated soil remain largely uncertain. Difficulties associated with the experimental or theoretical determination of the effective stress parameter
χ
have limited the general applicability of Bishop’s approach in both research and practice. Experimental studies have suggested the nonuniqueness of
χ
=
f
(
S
). Similar experimental and conceptual difficulties associated with the determination of required material variables such as
Φ^{b}
and uncertainties in their uniqueness over a wide range of saturation have limited the practical applicability of the independent stress variable approach
(Lu and Likos 2006)
.
Lu et al. (2010)
recently conceptualized a closedform equation to represent the state of effective stress in unsaturated soil by introducing the suction stress characteristic curve as equation (7).
where
σ
′ is the effective stress,
σ
is the total stress,
u_{a}
is the pore air pressure, and
σ^{s}
is the suction stress.
The suction stress which includes van der Waals attraction, electrical doublelayer repulsion, cementation forces, surface tension forces, and attractive forces arising from negative porewater pressure
(Lu and Likos 2004
,
2006)
is expressed as equation (8).
where
σ_{pc}
is the physicochemical stress attributable to van der Waals attraction, electrical doublelayer repulsion, and cementation forces, and
σ_{cap}
is the capillary stress arising from surface tension and negative porewater pressure.
The suction stress also graphically represents the area under the normalized soil water characteristic curve and can be expressed as equation (9).
where
u_{a}
is the pore air pressure,
u_{w}
is the pore water pressure, and
S_{e}
is the effective degree of saturation.
A closedform expression for suction stress for the full range of saturation can be arrived at by substituting equation (1) into equation (9) and eliminating matric suction:
Similarly, a closedform expression for suction stress for the full range of matric suction can be arrived at by substituting equation (1) into equation (9) and eliminating the degree of saturation:
Substituting equation (11) into equation (7), the closedform equation for effective stress in the entire pore water pressure range (fully saturations) is
Hydraulic characteristics of Jumunjin sand
During this study, physical tests and soilwater characteristic tests were performed to estimate the soilwater characteristic curve, hydraulic conductivity function, and suction stress characteristic curve of Jumunjin sand with a relative density of 75%.
Table 1
lists the results of the tests.
Physical properties of Jumunjin sand.
Physical properties of Jumunjin sand.
 SoilWater Characteristic Curve (SWCC)
The relationship between matric suction and volumetric water content was measured using an automated SWCC apparatus
(Song et al., 2012b)
. The test for measuring SWCC was conducted for both drying and wetting processes in order to examine the hysteresis of SWCC. The fitting parameters of the
van Genuchten (1980)
model were estimated for each test using the data measured from the experiments and a nonlinear leastsquares computer program.
Fig. 2
shows soilwater characteristic curves for drying and wetting processes.
Table 2
lists the parameters used in van Genuchten’s SWCC model. In
Table 2
,
α
is the parameter related to the air entry value,
n
is the parameter related to the slope of SWCC, and
m
is the parameter related to the residual water content.
Soilwater characteristic curves for drying and wetting processes.
Fitting parameters used in thevan Genuchten (1980)model.
Fitting parameters used in the van Genuchten (1980) model.
As shown in
Fig. 2
, the hysteresis phenomenon, exemplified by different matric suctions over the same volumetric water content, was observed between drying and wetting processes and was attributed mainly to inkbottle and contact angle hysteresis. In addition,
Fig. 2
shows that the inclination of the transient region (funicular zone) of the curve is extremely steep and that the matric suction corresponding to residual volumetric water content was around 4 kPa. These results are attributed to the highly uniform particle composition and narrow grain size distribution of Jumunjin sand.
 Hydraulic Conductivity Function (HCF)
HCF for both drying and wetting processes was estimated using equation (2) and SWCC, as displayed in
Fig. 3
. Similar to SWCC, HCF was maintained until the airentry value was obtained, followed by a rapid reduction. Significantly, the hydraulic conductivity also exhibited the hysteresis phenomenon.
Hydraulic conductivity functions for drying and wetting processes.
 Suction Stress Characteristic Curve (SSCC)
The SSCC for Jumunjin sand with a relative density of 75% was estimated for both paths using equation (9) and SWCC, as shown in
Fig. 4
. The SSCC shows that the suction stress ranged from –0.65 kPa (wetting) to approximately –1.8 kPa (drying) at a matric suction of 2 kPa, and it exhibited the hysteresis phenomenon like the SWCC and HCF. Moreover, where matric suction was less than the airentry pressure, the suction stress decreased as the matric suction increased; however, when matric suction was greater than one, the suction stress increased to zero again as the matric suction decreased. The suction stress caused an increase in shear strength and effective stress of unsaturated soil because it had a negative value. Therefore, the suction stress aided slope stability under unsaturated conditions.
Suction stress characteristic curves for drying and wetting processes.
Stability analysis of an unsaturated nfinite slope
The change of matric suction and volumetric water content in soils due to the rainfallinduced infiltration conditions should be considered when analyzing the stability of an unsaturated infinite slope. The transient seepage analysis was conducted using a finite element analysis program, SEEP/W, and we examined the variations in matric suction and volumetric water content within the soil over time.
For seepage analysis, we assumed a slope inclination of 1:1.5 and the geometry of the model shown in
Fig. 5
. In addition, it was assumed that the depth of the slope was 4.05 m, sufficient for seepage to occur, and that the water table was located 3.05 m below ground level (b.g.l.). The hydraulic characteristic curve for the wetting process was also used for reproducing rainfallinduced infiltration and the rainfall condition was 0.07 m/hr in terms of the saturated hydraulic conductivity listed in
Table 3
.
Infinite slope model for the seepage analysis.
Parameters for the stability analysis.
Parameters for the stability analysis.
Seepage analysis is greatly affected by the initial state of the soil, such as the antecedent moisture conditions due to preceding rainfall levels. Therefore, in this study, it was assumed that, although the initial matric suction of the soil increased hydrostatically from the water table, the depth with maximum value was limited to 2 m below the ground surface. For simplicity, it was assumed that the matric suction of the region which is uppermost 0.05 m was 3 kPa.
The factor of safety of the unsaturated infinite slope was evaluated by limit equilibrium analysis using the generalized effective stress, which included the suction stress, as proposed by
Lu and Likos (2004
,
2006)
. Equation (13) was used to calculate the factor of safety of the critical sliding surface. The first term on the right hand side of the equation represents the internal frictional resistance of soil, the second term is the cohesion, and the third term is the suction stress.
where
H_{ss}
is the distance from the ground surface to the critical failure surface,
β
is the inclination of the slope, and
c
' and
Φ
′ are the cohesion and the internal friction angle of soil, respectively.
Fig. 6
shows the variation in porewater pressure and the factor of safety over time at different depths within the slope. As shown in
Fig. 6
, the initial distribution of porewater pressure was hydrostatically reduced in proportion to distance from the water table under the initial condition, until the pore water pressure approached approximately –20 kPa, when it became constant. Subsequently, over time, positive porewater pressure was generated as the pores within the soil became saturated from the ground surface due to rainfallinduced infiltration.
Temporal variations in porewater pressure and factor of safety.
The initial factor of safety was 1.412 (the same as the factor of safety of the infinite slope without seepage) at more than 0.4 m above the water table, and slightly higher than 1.412 due to the effect of suction stress in the zone from 0 to approximately 0.4 m above the water table. This result is attributed to the suction stress, which helps to stabilize the slope under unsaturated conditions. However, the slope was saturated gradually due to the rainfall and then, after approximately 404 minutes, the factor of safety of the slope was 0.81 at 1.5 m above the water table.
Conclusions
To investigate the effect of rainfallinduced infiltration on the stability of an unsaturated infinite slope, the soilwater characteristic curves (SWCCs) of sand with a relative density of 75% were estimated for the drying and wetting processes using data measured by an automated soilwater characteristic apparatus. The unsaturated hydraulic conductivity function (HCF) and suction stress characteristic curve (SSCC) were then calculated. Consequently, the stability analysis of an unsaturated infinite slope over time and under rainfallinduced infiltration conditions was conducted by employing the limit equilibrium method, with a focus on the suction stress.
The suction stress slightly increased the factor of safety near the water table, and the maximum safety factor occurred at the location corresponding to the airentry pressure value. Subsequently, the saturated zone was gradually formed below the ground surface over time, and, as a result, the factor of safety was rapidly reduced due to the formation of positive porewater pressure. It was also shown that the effective range of suction stress was in the funicular zone. Therefore, the variability in matric suction and suction stress should be considered in stability analyses of unsaturated slopes.
Acknowledgements
This research was supported by the Basic Research Project (Grant No. 143215) of the Korea Institute of Geoscience and Mineral Resources (KIGAM) funded by the Ministry of Science, ICT, and Future Planning of Korea.
BIO
YoungSuk Song
Korea Institute of Geoscience and Mineral
Resources, Geological Hazards Department
124 Kwahakro, Yuseonggu, Daejeon 305350
Tel: 042) 8683035
Fax: 042) 8683414
Email: yssong@kigam.re.kr
WoongKi Hwang
Korea Maritime and Ocean University, Department of Civil Engineering
727 Taejongro, Yeongdogu, Busan 606791
Tel: 051) 4104945
Fax: 051) 4104460
Email: golda1142@nate.com
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