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Simplified quantitative risk assessment of rainfall-induced landslidesmodelled by infinite slopes
Abid Ali a,⁎, Jinsong Huang a, A.V. Lyamin a, S.W. Sloan a, D.V. Griffiths a,b, M.J. Cassidy a,c, J.H. Li c
a ARC Centre of Excellence for Geotechnical Science and Engineering, The University of Newcastle, Callaghan, NSW 2308, Australiab Department of Civil and Environmental Engineering, Colorado School of Mines, Golden, CO 80401, USAc ARC Centre of Excellence for Geotechnical Science and Engineering, The University of Western Australia, Crawley, WA 6009, Australia
Article history:Received 12 February 2014Received in revised form 20 June 2014Accepted 30 June 2014Available online 8 July 2014
Keywords:ConsequenceTriggering mechanismLandslideRainfallRiskRandom field
Rainfall induced landslides vary in depth and the deeper the landslide, the greater the damage it causes. Thispaper investigates, quantitatively, the risk of rainfall induced landslides by assessing the consequence of each fail-ure. The influence of the spatial variability of the saturated hydraulic conductivity and the nature of triggeringmechanisms on the risk of rainfall-induced landslides (for an infinite slope) are studied. It is shown that a criticalspatial correlation length exists atwhich the risk is amaximum and the risk is higherwhen the failure occurs dueto a generation of positive pore water pressure.
Landslides cause damage to buildings, infrastructure, agriculturalland and crops. In the majority of cases the main trigger for landslidesis heavy or prolonged rainfall (Brand, 1984; Fourie, 1996). Rainfall-induced landslides are common in tropical and subtropical regionswhere residual soils exist in slopes and there are negative pore waterpressures in the unsaturated zone above the water table (Rahardjoet al., 1995). In an unsaturated soil, these negative pore water pressurescontribute towards its shear strength and thus help tomaintain stability(Fredlund and Rahardjo, 1993). The infiltration of rainwater causes areduction in this negative pore water pressure and an increase in thesoil unit weight (due to an increased saturation), both of which have adestabilizing influence.
Research on rainfall-induced slope failure indicates that severalfactors affect the stability of a slope subjected to rainfall infiltration.Published research in the area (Zhang et al., 2011; Zhan et al., 2012; Liet al., 2013) shows that the rainfall characteristics (duration, intensityand pattern), the saturated hydraulic conductivity of the soil, the slopegeometry, the initial conditions, and the boundary conditions are thefactors that influence the stability of a slope subjected to rainfall.Among these factors, the hydraulic conductivity is a very important
parameter in seepage and stability problems involving unsaturatedsoils (Tsaparas et al., 2002; Rahardjo et al., 2007; Rahimi et al., 2010).
Most studies involving rainfall-induced landslides are deterministicin nature, where the soil is assumed to be homogeneous and averaged(or design) soil properties are considered in the analysis (Gui et al.,2000). The uncertainties associated with the soil parameters are usuallydealtwith by adopting “reasonably averaged” parameters, coupledwithpractical experience (Duncan, 1996). In reality, soil is inherently hetero-geneous with its properties varying from point to point due to differentdepositional and post-depositional processes (DeGroot and Baecher,1993; Lacasse and Nadim, 1996). A few studies focused on the effectsof the spatial variability of the hydraulic conductivity on rainfall infiltra-tion and subsequent slope stability by using random field theory(e.g. Santoso et al., 2011; Zhu et al., 2013; Cho, 2014), but those studiesdid not investigate the nature of the triggeringmechanismor quantifiedthe risk associated with a rainfall-induced landslide when the saturatedhydraulic conductivity varies spatially.
It is now commonly believed that there are two mechanisms thattrigger failure in slopes subject to rainfall infiltration (Li et al., 2013);loss of suction during propagation of the wetting front and the rise ofthe water table (which generates a positive pore water pressure).
Generally, a loss of suction (i.e. reduction in negative pore waterpressure) causes a shallow failure while a rise in the water table(i.e. generation of a positive pore water pressure) causes a deep failure.However, this may not be truewhen the saturated hydraulic conductiv-ity varies spatially, as the water may accumulate at shallow depths
103A. Ali et al. / Engineering Geology 179 (2014) 102–116
(Huang et al., 2010) leading to a positive pore water pressure and ashallow failure. To the authors' knowledge, this important effect hasnot been studied systematically. Another key aspect of the risk assess-ment of rainfall-induced landslides is the assessment of consequence.Rainfall-induced landslides can be shallow or deep. It is clear that adeep-seated landslide will tend to cause more damage and thus has amore severe consequence. Therefore, the consequence associated witha shallow or deep failure should be assessed individually.
The changes in the near-surface pore water pressures caused byrainfall may be determined using field-observations, analytical solu-tions or numerical methods. This steady-state pore-pressure field isthen used to determine the slope stability either analytically or numer-ically. Among these uncoupled approaches, the infinite slope modelcombined with a one-dimensional hydrological model is popular(e.g. Collins and Znidarcic, 2004; Tsai and Chen, 2010; Tsai, 2011;White and Singham, 2012; Zhan et al., 2012; Zhang et al., 2012; Liet al., 2013; Zhang et al., 2014) and will be adopted in this study. Inthe infinite slope model, the landslide is characterized as a slope failureoccurring along a plane parallel to the ground surface. It assumes thateach slice of an infinitely long slope receives the same amount and in-tensity of rainfall (Collins and Znidarcic, 2004); that the time requiredfor infiltration normal to the slope is much less than the infiltrationtime required for flow parallel to the slope; that thewetting front prop-agates in a direction normal to the slope1 (White and Singham, 2012);and that the depth of failure is small compared to the length of the fail-ing soil mass. The validity of these assumptions has been checkedagainst the predictions of two-dimensional numerical models, withthe conclusion that an infinite slope approximation may be adoptedas a simplified framework to assess failures due to the infiltration ofrainfall (Zhan et al., 2012; Li et al., 2013).
In this study, the saturated hydraulic conductivity is modelledas a random field and coupled with Monte-Carlo simulations forthe determination of failure probability, consequence and risk. Therainfall-induced landslide risks of two slopes having different triggeringmechanism are studied by adopting the quantitative risk assessmentframework proposed by Huang et al. (2013). To obtain the pore waterdistributions, the modified form of one-dimensional Richards equation(Richards, 1931) is solved numerically by the HYDRUS 1D software(Simunek et al., 2013).
2. Seepage analysis
Assuming that the effect of pore-air pressure is insignificant and thatwater flow due to thermal gradients is negligible, one-dimensional uni-form flow in a variably saturated soil can be described by a modifiedform of Richards equation (Richards, 1931). Therefore, the flow in anunsaturated infinite soil slope can be described by the 1D equation(e.g. Zhan et al., 2012):
dθdt
¼ ddz
Kdudz
þ cosα� �� �
ð1Þ
where θ is the volumetric water content, t is time, u is the pore waterpressure head, α is the inclination of the slope to the horizontal, K isthe hydraulic conductivity and z is the spatial coordinate as shown inFig. 1. To solve the above equation numerically, the water content θ is
1 The use of an infinite slope model implies that the pore water pressure at a certaindepth is same along all lateral extents of the slope i.e. the pore pressure contours are par-allel to the ground surface when the slope is subjected to rainfall (e.g. Zhan et al., 2012).Pore pressure contours parallel to the ground surface also imply that any variability inthe hydraulic conductivity (parallel to the slope surface) is neglected. If flow is not strictlyone-dimensional, then the pore water pressures will vary along the lateral extent of theslope, even at the same depth. The problem in such a case will no longer be one-dimensional in nature and the use of an infinite slope model would be inappropriate.
assumed to vary with the pore water pressure head u according to thevan Genuchten (1980) model as:
Se ¼θ−θrθs−θr
¼ 11þ auð ÞN� �m
ð2Þ
where Se is the effective degree of saturation, θs and θr are the saturatedand residual water content respectively, a is the suction scaling param-eter and N, m are the parameters of the van Genuchten model. Notingthat the volumetric water content is related to the degree of saturationS and the porosity n (by the relation θ= nS), the effective degree of sat-uration can also be expressed in terms of the degree of saturation S inthe following form:
Se ¼S−Sr1−Sr
ð3Þ
where Sr is the residual degree of saturation. To complete the descrip-tion, the hydraulic conductivity K can be estimated as:
K ¼ KsKr ð4Þ
where Ks is the saturated hydraulic conductivity and Kr is the relativehydraulic conductivity given by van Genuchten (1980):
Kr ¼ S1=2e 1− 1−S1=me
� �mh i2: ð5Þ
In this study, the saturated hydraulic conductivity is modelled as arandom field and Eq. (1) is solved by HYDRUS 1D. The distribution ofpore water pressure and the degree of saturation are then used in theinfinite slope model to assess the slope stability.
2.1. Slope stability assessment
Once the pore water pressure distribution is obtained throughseepage analysis, the factor of safety FS at any given time t can then bedetermined by limit-equilibrium techniques. The stability of an infiniteslope is estimated by using a closed form solution similar to that pro-posed by White and Singham (2012), where the failure is consideredto occur along a plane parallel to the ground surface. A soil column ofa unit width is considered, where the self-weight W is used to obtainthe normal force FN and tangential force FT at any depth. The expression
104 A. Ali et al. / Engineering Geology 179 (2014) 102–116
for the factor of safety is derived along the same lines as White andSingham (2012).
Referring to Fig. 1, resolving equilibrium of the normal forces on theslip plane gives:
FN ¼ W cos α ð6Þ
where W is the self-weight of the soil column above the failure plane.The unit weight of soil during rainfall infiltration will increase due toan increase in water content; neglecting this effect (i.e. assuming aconstant unit weight) could lead to a higher (and thus false) estimateof stability (Tsai and Chen, 2010; Zhan et al., 2012). Therefore, toaccount for the variation in unit weight due to the variation in thewater content with depth, W is determined as:
W ¼Z D f = cosα
0γ
dzcosα
¼ 1cosα
Z D f = cosα
01−nð Þγs þ nSγw½ �dz
¼ 1cosα
Z D f = cosα
01−nð Þγs þ θγw½ �dz
ð7Þ
whereDf is the failure depth,γ is the bulk unitweight of soil,γs is the unitweight of soil solids, and γw is the unit weight of water and the watercontent θ is defined in Eq. (2). Note that the integration takes placealong the vertical direction as illustrated in Fig. 1. From simple statics,the total normal stress σ on the failure surface can be computed as:
σ ¼ FN cos α ð8Þ
σ ¼ W cos2 α ð9Þ
while the corresponding shear force is given by:
FT ¼ W sin α: ð10Þ
Hence the shear stress on the failure surface is:
τ ¼ FT cos α ð11Þ
or
τ ¼ W cosα sin α: ð12Þ
In this study, the shear strength of soil τf (in terms of effective stress)is assumed to be described by the Mohr–Coulomb model:
τ f ¼ c0 þ σ 0 tan ϕ0� �
ð13Þ
where c′ is the effective cohesion,σ′ is the effective normal stress, andϕ′is the effective friction angle. To consider the influence of pore waterpressure on the shear strength of a variably saturated soil, Terzaghi'seffective stress principle is modified according to the formulation ofBishop (1959):
σ0 ¼ σ−uað Þ þ χ ua−uwð Þ ð14Þ
where ua is the pore-air pressure, uw = γwu is the pore water pressure,(ua − uw) is known as matric suction and χ is called the coefficient ofeffective stress and is a constitutive property of the soil that dependson the degree of saturation. For a variably saturated soil, χ denotesthe proportion of matric suction that contributes to the effective stressand generally varies between 0 (for a perfectly dry soil) and 1 (for acompletely saturated soil). Though many mathematical forms of χhave been proposed in the past, in the present study χ is consideredequal to the effective degree of saturation, Se (Vanapalli et al., 1996):
χ ¼ θ−θrθs−θr
¼ S−Sr1−Sr
¼ Se: ð15Þ
Substituting ua = 0 in Eq. (14) and substituting Eq. (15) in Eq. (14)gives:
σ 0 ¼ σ−Seuw: ð16Þ
Slope failure occurs when the applied shear stress τ exceeds themobilized soil shear strength τf. The factor of safety FS can then becomputed as:
FS ¼ τ f
τ¼ tanϕ0
tanαþ c0−Seuw tanϕ0
W cos α sin αð17Þ
where FS = 1 corresponds to a limiting condition for equilibrium, andfailure occurs when FS is less than 1.
3. Probabilistic analysis
3.1. Random field theory
Random fields are characterized by a distribution (e.g. log-normaltype) and a spatial correlation structure. The present study considersthe saturated hydraulic conductivity, Ks, to be log-normally distributedwhich is consistent with field measurements (Hoeksema andKitanidis, 1985; Sudicky, 1986). A log-normal distribution can be easilyarrived at by a non-linear transformation of the normal (Gaussian) dis-tribution and it ensures that the random variable is always positive(Griffiths et al., 2011). Such a distribution has also been used by severalinvestigators formodelling saturated hydraulic conductivity statistically(Fenton and Griffiths, 1993; Griffiths and Fenton, 1993; Gui et al., 2000;Srivastava et al., 2010; Santoso et al., 2011; Cho, 2012). A log-normallydistributed Ks is defined by two parameters, a mean ( μKs
) and coeffi-cient of variation (νKs) which are related by:
νKs¼ σKs
μKs
ð18Þ
where σKsis the standard deviation for the log-normally distributed Ks.
The equivalent parameters of the normally distributed lnKs — μ lnKsand
σ lnKs(i.e. the mean and standard deviation of lnKs) are:
σ2lnKs
¼ ln 1þ ν2Ks
� �ð19Þ
μ lnKs¼ ln μKs
� �−1
2σ2
lnKs: ð20Þ
In addition to the mean and the coefficient of variation, a third pa-rameter, the spatial correlation length θ lnKs
, is required to completelydefine a randomfield. The spatial correlation length defines the distanceover which the soil properties are significantly correlated; with proper-ties separated by a distance greater than θ lnKs
being generally uncorre-lated. A large spatial correlation length means that the soil propertiesare highly correlated over a large distance, implying less spatial variabil-ity and more uniformity in soil properties. Conversely, a small correla-tion length implies a higher spatial variability and less uniformity inthe soil properties. In the context of random fields, the spatial correla-tion lengths are generally incorporated through a correlation function.The correlation function ρ assumed for the present study is an exponen-tial one of the form:
ρ zð Þ ¼ exp − zj jθ lnKs
!: ð21Þ
Based on the log-normal distribution and the correlation functiondefined above, one-dimensional random fields for saturated hydraulicconductivity Ks can be generated. In the present study, the KarhunenLoève (KL) expansion method (Zhang and Lu, 2004) is used for thispurpose. Note that the required number of terms in the KL expansion
105A. Ali et al. / Engineering Geology 179 (2014) 102–116
increases when the spatial correlation length decreases and, for thesmallest θ lnKs
considered in this study, more than 5000 terms wereused. The Karhunen Loève expansion method generates a Gaussian(normal) random field and Ks, being log-normal, requires a log-normalrandom field. This is obtained through the transformation:
Ksj ¼ exp μ lnKsþ σ lnKs
g j
� �ð22Þ
where Ksj is the saturated hydraulic conductivity assigned to the jthnode (of the 1D finite element mesh) and gj is the Gaussian equivalentof Ksj obtained from a zero mean and unit standard deviation. Thedimensionless form of spatial correlation length Θ is defined as:
Θ ¼ θ lnKs
Dð23Þ
where D is the length of the random field.The exponential correlation function producesmany small scale var-
iations. In order to capture these small-scale variations, the element sizeneeds to be sufficiently small. In the present study the smallest correla-tion length was 0.125 m and the element size was 0.01 m. This meansthat element size is 8 times smaller than the smallest spatial correlationlength. It was observed that stable results can be obtained by this meshdensity.
3.2. Risk assessment
During rainfall, water infiltrates into the soil from the top and cancause shallow or deep failures. It is clear that a deep failure will tendto causemore damage and thus has a more severe consequence. There-fore, the consequence associated with shallow and deep failures shouldbe assessed individually. In the present study, consequence is assumedto be directly related to the failure depth Df. The risk R is defined as:
R ¼Xn f
i¼1
pfi � Ci ð24Þ
where pfi and Ci are the probability and consequence of failure mode i,and nf is the number of failures.2 In applications, an additional vulnera-bility component would be added to Eq. (24). However, it has beenassumed to be one here for simplicity and to concentrate the paper onthe novel developments (e.g. Cassidy et al., 2008). Eq. (24) can berewritten in the traditional form as:
R ¼Xn f
i¼1
pfi � Ci ¼Xn f
i¼1
1nsim
� Ci ¼1
nsim
Xn f
i¼1
Ci ¼nf
nsim
Xn f
i¼1
Ci
nf¼ pf C
ð25Þ
where C is the consequence and nsim is the number of Monte-Carlosimulations.
pf ¼nf
nsimð26Þ
and
C ¼
Xn f
i¼1
Ci
nf: ð27Þ
2 For the Monte-Carlo simulations conducted in this study the rainfall duration wasfixed. The factor of safety (FS) was calculated for each time step and at all depths. When-ever FS b 1, failure occurs and simulation stops. The time and depth of failure are then re-corded. If FS is greater than 1 when rainfall stops, slope is said to be safe. The number offailures (nf) is the sum of failures. Each failure is treated as an independent failure mode,so the probability of one single failuremode is constant, i.e. pfi=1 / nsim. As the simulationstopswhenever FSb 1, it is impossible for another failure to occur after one failure has hap-pened. This means that failures are disjoint.
It can be seen from Eq. (27) that consequence C in the traditionalrisk definition should be redefined as the average consequence of allfailures.
4. Examples
The risks associatedwith the failure of two slopes are assessed in thissection. “Example 1” considers a purely frictional soil slope while“Example 2” considers a cohesive-frictional soil slope. It is shown thatshallow failures are more likely to occur in the first example.
A hypothetical slope, at an inclination of α = 36° to the horizontal,is considered as shown in Fig. 1. A 1 m thick homogeneous soil layeris underlain by rock (i.e. D = 1 m). The water table is initially 2 m(all distances measured normal to the slope) below the ground surfaceand is assumed to be constant throughout the analysis. The initial porewater pressure profile is hydrostatic and, as rainfall occurs, the wateris assumed to infiltrate in a wetting front normal to the slope. The rain-fall intensity is I, and the flux q infiltrating the soil at the top is given byI cosα. No ponding ofwater is allowed at the ground surface at any time.The mechanical and hydraulic properties of the soil are given in Table 1(White and Singham, 2012). In this section, the landslide risk of theslope described above is assessed for the two examples.
4.1. Example 1
In this example it is assumed that c′ = 0. Based on the expressionfor the factor of safety (Eq. (17)) with the parameters described inTable 1 and considering α = 36°, the factor of the safety before rainfall(considering a hydrostatic distribution) is 1.285.
4.1.1. Estimation of pore water pressure distribution at failureAssuming that soil is fully saturated (i.e. Se = 1) at failure, the pore
water pressure required to cause failure at different depths can be esti-mated analytically. Substituting FS = 1 and Se = 1 into Eq. (17) gives:
uw ¼c0− 1− tan ϕ0
tanα
!W cosα sin α
tanϕ0 : ð28Þ
The initial and failure profiles of the pore water pressure are shownin Fig. 2. In this figure, “Initial condition” represents the initial hydro-static pore water pressure distribution and “FS = 1.00” representsthe pore water pressure (corresponding to different depths of failure)estimated from Eq. (28). The pore water pressure at failure is shownon a magnified scale in the inset. It can be observed that failures occurdue to loss of suction only. The residual negative pore water pressure(at failure) increases with an increase in depth. This implies that thefailure depth will depend on the reduction in the negative pore waterpressure and will be the deepest when the reduction in the negativepore water pressure is the least. This is because with an increase indepth, the destabilizing force due to the weight component increasesand, therefore in case of a deep failure, the negative porewater pressureneed not be reduced to the same extent as that required for a shallowfailure.
4.1.2. Preliminary deterministic analysisA series of deterministic analyses is performed where I is kept
constant (I ¼ 0:25μKs) and Ks is varied. Fig. 3 shows the relationship
between Df and Ks. The first passing time of failure is shown in Fig. 4. Ifthe mean value of saturated hydraulic conductivity μKs
is considered,the depth of failure Df = 1.00 m and the first passing time of failuretf = 0.0855 days. If the rainfall duration is restricted to 0.0855 days,then failures would occur only for some values of Ks which are denotedby the (red) solid lines in Figs. 3 and 4. Initially,when the failure depth isthe same (i.e. Df = 0.01 m), an increase in Ks will allow the flux to
Table 1Material properties of the soil (White and Singham, 2012) considered in the present study.
Parameter Symbol Value Units
Porosity n 0.4Unit weight of soil solids γs 20 kN/m3
Unit weight of water γw 10 kN/m3
Mean saturated hydraulic conductivity μKs8.64 m/day
Residual water content θr 0.128Saturated water content θs 0.4Scaling suction a 5 1/mVan Genuchten model parameter N 1.5Van Genuchten model parameter m (N − 1)/NEffective cohesion (Example 1/Example 2) c′ 0/0.15 kPaEffective friction angle ϕ ′ 35 °
106 A. Ali et al. / Engineering Geology 179 (2014) 102–116
infiltrate quickly and this explainswhywith an increase in Ks, the failuretime decreases. However, around Ks = 2.35 m/day, the failure depthstarts increasing with an increase in Ks. Therefore the failure time alsoincreases until Ks ≈ 6 m/day after which the failure time decreasesagain. This is because, when Ks ≈ 6 m/day, the failure occurs at the im-permeable boundary. Increasing Ks any further will not affect the failuredepth (i.e. the failure will still occur at the boundary) but, will reducethe time required for the flux to infiltrate to the bottom and causefailure.
i. From Figs. 3 and 4, it can be seen that there exists a range of saturatedhydraulic conductivity values between which the failure is alwaysshallow. Ksnf represents the lower bound and Ksmin represents theupper bound of this range. The depth of shallow failure is minimumfor Ksnf and maximum for Ksmin. In this range of Ks,Df increaseswhen Ks increases. If Ks is less than Ksnf, there is no failure.
ii. If Ks is in between Ksmin and μKs, there will be no failures. In other
words, no failure is observed in the soil profile between 0.77 m and1.00 m and this shall be referred to as the “region of no failure”later on.
iii. If Ks is greater than μKs, the failure will always be deep (Df=1.00m).
Fig. 5 illustrates situations of deep and shallow failures, respectively,as observed in the deterministic study.
4.1.3. Analytical estimation of probability of failureBased on the observations presented in the previous subsection,
the probability of failure can be estimated analytically. Fig. 6 showsthe saturated hydraulic conductivity, Ks, distributed log-normally
Fig. 2. Profiles of porewater pressure— initially and at failure (Example 1); inset, profile ofpore water pressure at failure on a magnified scale.
with mean μKs¼ 8:64 m=day and νKs ¼ 1. The critical values of Ksnf,
Ksmin and μKsare also illustrated when the rainfall duration, t =
0.0855 days and the rainfall intensity I ¼ 0:25μKs.3 For the time t, and
rainfall intensity I, the probability of the saturated hydraulic conductiv-ityKs being below a certain value K, can then be determined analytically,by transforming the log-normal to normal as:
P Ks b Kð Þ ¼ P lnKsb lnKð Þ ¼ P Z blnK−μ lnKs
σ lnKs
!
¼ ΦlnK−μ lnKs
σ lnKs
! ð29Þ
where Φ is the cumulative standard normal distribution and Z is thestandard normal variate. The log-normal parameters μ lnKs
and σ lnKs
are obtained from the parametersμKsandσKs
as explained in Section 3.1.The probability of failure pf can be determined analytically as a sum
of the probabilities of shallow failure pfs (Df b 1.00 m) and deep failurepfd (Df = 1.00 m). This can be written as:
8:64 m=day in the above formulae (Eqs. (30), (31) and (32)), the prob-ability of failure pf can be determined analytically for a range of νKs .
3 It is should be noted that the zoning in Fig. 6 (‘No failure’, ‘Only shallow failure’, ‘Onlydeep failure’) has been done from the results of preliminary deterministic analysis (Figs. 3and 4), where the rainfall duration and intensity were fixed. For a given soil slope, thewidth of the zones will vary if the rainfall duration/intensity is changed.
4 Themaximumfailuredepth in thepresent studywas only 1.0mwhich is referred to asa deep failure. This notion of a “deep failure” having failure depth as 1.0 m should be usedin a restricted sense for the present study only. In a practical sense, such a failure would beregarded as a shallow failure.
Fig. 4. Variation of first passing time of failure tf with saturated hydraulic conductivity Ks for I ¼ 0:25μKs(Example 1).
Fig. 3. Variation of failure depths Df with saturated hydraulic conductivity Ks for I ¼ 0:25μKs(Example 1).
107A. Ali et al. / Engineering Geology 179 (2014) 102–116
4.1.4. Single-random-variable approachTo validate the analytical prediction presented in the previous sub-
section, a single-random-variable (SRV) analysis is conducted. Thesingle-random-variable approach implies a very large spatial correla-tion length θ lnKs
≈∞ð Þ i.e. the saturated hydraulic conductivity is a con-stant in the soil profile. The saturated hydraulic conductivity Ks is drawnat random from a log-normal distribution and assigned to the slope. Forthe SRV analysis, two thousand Monte Carlo simulations were per-formed. For each realization of Ks, seepage analysis is performed for arainfall duration t = 0.0855 days. The factor of safety (FS) for everytime step and depth was recorded. Whenever FS b 1, failure occursand the corresponding time and depth of failure are recorded as tfand Df respectively. If FS is greater than 1 when rainfall stops,the slope is said to be safe, and the failure depth is recorded as zero(i.e. Df = 0) in such cases. The probability of deep failure pfd in thiscase is defined as:
pfd ¼ nfd
nsimð33Þ
where nfd is the number of failures withDf=1.00m.4 The probability ofshallow failure pfs is defined as:
pfs ¼nfs
nsim¼ pf−pfd ð34Þ
where nfs is the number of failures with Df b 1.00 m.The probabilities of failures pf, pfs and pfdwith variousνKs (0.5, 1, 1.5,
2 and 4) are compared with analytical predictions in Fig. 7. A very goodagreement can be observed between the SRV results and the analyticalpredictions. The probability of shallow failure increaseswith an increaseinνKs . This is reasonable, as with an increase inνKs , there is a shift in thelog-normal distribution towards zero, thus increasing the probability ofKs being in between Ksnf and Ksmin. Consequently, there is a decrease inthe probability of deep failures with an increase in νKs . Thus with an in-crease inνKs, the contribution of pfs to pf increaseswhile the contributionof pfd to pf decreases. This clearly shows that the consequence should be
-20 -15 -10 -5 0 5-1
-0.8
-0.6
-0.4
-0.2
0
Pore-water pressure (kPa)
Dep
th (
m)
0.5 1 1.5 2 2.5 3-1
-0.8
-0.6
-0.4
-0.2
0
Factor of safety
Dep
th (
m)
Initial profileK
s = 8.64 m/day
tf = 0.0855 days
Ks = 3.00 m/day
tf = 0.0295 days
Shallow failure
Deep failure Deep failure
Shallow failure
Fig. 5. Illustration of deep and shallow failures for I ¼ 0:25μKsand different Ks (Example 1).
108 A. Ali et al. / Engineering Geology 179 (2014) 102–116
assessed according to the failure mode. Otherwise, the overall risk willbe incorrectly estimated.
4.1.5. Random field studyIn order to investigate the effect of the spatial variability of the
saturated hydraulic conductivity Ks on landslides risk, Ks is modelledas a random field while other parameters are deterministic. Theprobabilities of failure (pf, pfs, and pfd), consequence (C), and risk (R)associated with rainfall-induced failure of the purely frictional soilslope (i.e. Example 1) are investigated. From the literature it is ob-served that the saturated hydraulic conductivity varies in the rangeof νKs ¼ 0:6 to0:9 (Duncan, 2000). Taking the higher end of therange for the random field study, νKs ¼ 1 is considered. Table 2 sum-marizes the parameters for the random field study. Two thousandMonte Carlo simulations were conducted for each spatial correlationlength. For each realization of the random field of Ks, seepage analy-sis is performed for the duration of t = 0.0855 days and the resultsare then used to perform slope stability analysis.
The variation in failure probabilities (pfs, pfd, pf), consequence (C) andrisk (R) with the spatial correlation length (Θ) is shown in Fig. 8. Thesmallest correlation length (Θ= 0.125) has the greatest spatial variabil-ity (in this study) and therefore results in the largest probability ofshallow failure (pfs) or the smallest probability of deep failure (pfd).
0 1 2 3 40
0.02
0.04
0.06
0.08
0.1
0.12
Ks (
p
0 10 20 30 400
0.05
0.1
0.15
0.2
Ks
p
Only shallowfailure
Ks
Ksnf
No failure
Ksnf
and Ksmin
are determined at
Fig. 6. Saturated hydraulic conductivity distributed log-normally with m
Since the proportion of shallow failures among the total number offailures is a maximum when Θ = 0.125, the corresponding conse-quence is a minimum. With an increase in spatial correlation length(the soil becomesmore uniform,) pfs decreases while pfd increases; sub-sequently C also increases as now more deep failures occur. Althoughthe probability of failure pf decreases with an increase in spatial correla-tion length, the risk reaches its maximumwhen Θ is equal to the depthof the slope i.e. Θ = 1.0. This highlights the importance of individualassessment of the failure consequence.
Fig. 9 shows scatter plots of pore water pressure (uw) at failure, forselected spatial correlation lengths Θ. The (red) solid points representfailures due to the generation of positive pore water pressures whilethe (blue) hollow points represent failures due to loss of suction. Thesolid line represents the porewater pressure at failure estimated analyt-ically (from Eq. (28)). Firstly, the greatest scattering can be observed inthe distribution of porewater pressure at failurewhen the spatial corre-lation length Θ is the smallest. Secondly, at smaller correlation lengths,slope failure may also occur due to the generation of positive porewater pressures as shown by the (red) solid points which could beattributed to non-uniformity in the weight of the failing soil mass W.As explained in Section 2.1, the weight W is obtained by integratingthe unit weight γ throughout the depth to account for variationsin the water content. At a smaller Θ, there is more non-uniformity
5 6 7 8 9 10m/day)
50
No failure Only deepfailure
Ksmin Ks
I = 0.25Ks
and t = 0.0855days
ean μKs¼ 8:64m=day and νKs ¼ 1 (Example 1) at t = 0.0855 days.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ks
p f
pf(analyt)
pf(SRV)
pfs(analyt)
pfs(SRV)
pfd(analyt)
pfd(SRV)
Fig. 7. Comparison of analytical estimation of failure probabilities with the SRV approach (Example 1).
109A. Ali et al. / Engineering Geology 179 (2014) 102–116
(or randomness) in the weightW throughout the depth, thus requiringa greater reduction in the pore water pressure (compared to the analyt-ical estimate) to cause failure. Thirdly, as Θ increases, the maximumdepth of shallow failure also decreases. This is simply due to the soilbecoming more uniform, resulting in the “region of no failure” (as ex-plained in Section 4.1.2) becoming more prominent. Lastly, it shouldbe noted that shallow failures can occur due to loss of suction as welland need not be due to generation of positive pore water pressuresonly (Santoso et al., 2011).
A noteworthy point is that the results of pore water pressure at fail-ure start diverging from the analytical predictions as the failure depthincreases. This is because the effective degree of saturation, Se, varieswith depth and is not constant as assumed in the analytical predictionof pore water pressure at failure. This can be observed in Fig. 10 whichshows the scatter plots of Se for selected spatial correlation lengths.The red (solid) points representing the locations, where Se=1, indicatethat the soil was saturated at failure which implies that positive porewater pressure was generated at shallow depths. The blue (hollow)points representing the locations, where Se b 1, indicate that the soilwas not completely saturated or the failure occurred due to loss ofmatric suction.
Fig. 11 shows the histogram of failure depths for different spatialcorrelation lengths. More shallow failures occur for smaller correlationlengths Θ and most of these shallow failures are concentrated near theground surface with almost zero consequence.
4.2. Example 2
In this example it is assumed that c′=0.15 kPa. Based on the expres-sion for the factor of safety (Eq. (17)) with the parameters described inTable 1 and considering α = 36°, the factor of the safety before rainfall(considering a hydrostatic distribution) is 1.303.
Table 2Parameters for the probabilistic study.
Parameter Symbol Value Units
Mean saturated hydraulic conductivity μKs8.64 m/day
2.00, 8.00, 100Length of random field D 1 mNumber of simulations nsim 2000
4.2.1. Estimation of pore water pressure distribution at failureBased on Eq. (28), the pore water pressure required to cause failure
at different depths can be estimated analytically. The initial and failureprofiles of the pore water pressure are shown in Fig. 12. Comparedto Example 1, the pore-water pressure in the top half of the soil profileis positive while the bottom half is negative at failure. This implies thatfailures in the top half will be due to generation of a positive pore waterpressure only, whereas in the bottom half the failures can occur due toloss of suction as well.
4.2.2. Preliminary deterministic analysisLike Example 1, a deterministic analysis is performed for Example 2
with a constant I (¼ 0:25μKs), and a varying Ks. To observe the variation
in failure pattern, Df and tf are recorded for each simulation of Ks.Fig. 13 shows the relationship between Df and Ks. The first passing
time of failure is shown in Fig. 14. If the mean value of saturatedhydraulic conductivityμKs
is considered, the first passingdepth of failureDf =1.00m and the first passing time of failure tf = 0.0860 days. If therainfall duration is restricted to 0.0860 days, then failures would occuronly for some values of Ks which are denoted by the (red) solid linesin Figs. 13 and 14. The only difference compared to Example 1 is inthe values of Ksnf and Ksmin. Compared to Example 1, Ksnf is muchgreater while Ksmin is smaller in Example 2. This means that therange of Ks over which shallow failure can occur is reduced. Fig. 15illustrates examples of deep and shallow failures as observed in thedeterministic study. For Example 2, Ksnf = 1.253 m/day, Ksmin =3.7584 m/day and μKs
¼ 8:64m=day.
4.2.3. Analytical estimation of probability of failureBased on the results obtained in the previous section and using
the expressions presented in Section 4.1.3 the probabilities of failure(pf, pfs, pfd) can be analytically estimated.
4.2.4. Single-random-variable approachSimilar to Example 1, the SRV analysis is performed to determine the
probabilities of failure (pf, pfs, and pfd) for selected values of νKs (0.5, 1,1.5, 2 and 4). Two thousand Monte-Carlo simulations are performedfor each νKs . For each realization of Ks, seepage analysis is performedfor a rainfall duration t = 0.0860 days and then the results of porewater pressure are incorporated in Eq. (17) to check for failure. Theprobabilities of failures pf, pfs and pfd with various νKs (0.5, 1, 1.5, 2and 4) are compared with analytical predictions in Fig. 16. A verygood agreement can be observed between the SRV results and the
10-1
100
101
1020
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Dimensionless spatial correlation length,
p f, C, R
pf
pfd
pfs R C
Fig. 8. Variation of failure probabilities, consequence and risk with spatial correlation lengths (Example 1).
-0.4 -0.2 0 0.2-1
-0.8
-0.6
-0.4
-0.2
0
Pore water pressure (kPa)
Dep
th (
m)
= 0.125
Analyticalu
w < 0
uw
0
-0.4 -0.2 0 0.2-1
-0.8
-0.6
-0.4
-0.2
0
Pore water pressure (kPa)
= 1.00
-0.4 -0.2 0 0.2-1
-0.8
-0.6
-0.4
-0.2
0
Pore water pressure (kPa)
= 8.00
-0.4 -0.2 0 0.2-1
-0.8
-0.6
-0.4
-0.2
0
Pore water pressure (kPa)
= 100.00
Fig. 9. Scatter plots of porewater pressure at failure for selected spatial correlation lengths (Example 1). (For interpretation of the references to colour in this figure, the reader is referred tothe web version of this article.)
0.96 0.98 1-1
-0.8
-0.6
-0.4
-0.2
0
Dep
th (
m)
= 0.125
Se
< 1
Se
= 1
0.96 0.98 1-1
-0.8
-0.6
-0.4
-0.2
0 = 1.00
0.96 0.98 1-1
-0.8
-0.6
-0.4
-0.2
0 = 8.00
0.96 0.98 1-1
-0.8
-0.6
-0.4
-0.2
0
Se
Se
Se
Se
= 100.00
Fig. 10. Scatter plots of effective degree of saturation (Se) at failure, for selected spatial correlation lengths (Example 1). (For interpretation of the references to colour in this figure, thereader is referred to the web version of this article.)
110 A. Ali et al. / Engineering Geology 179 (2014) 102–116
0 200 400 600-1
-0.8
-0.6
-0.4
-0.2
0D
epth
(m
)
Frequency
= 0.125
0 200 400 600-1
-0.8
-0.6
-0.4
-0.2
0
Frequency
= 1.00
0 200 400 600-1
-0.8
-0.6
-0.4
-0.2
0
Frequency
= 8.00
0 200 400 600-1
-0.8
-0.6
-0.4
-0.2
0
Frequency
= 100.00
Fig. 11. Histogram of failure depths for selected spatial correlation lengths (Example 1).
111A. Ali et al. / Engineering Geology 179 (2014) 102–116
analytical predictions. In Example 1, pf increasedwith an increase inνKs,reached a peak (atνKs ¼ 2) and then decreasedmarginally (atνKs ¼ 4);on the other hand, pfs increased continuously while pfd decreased con-tinuously with an increase in νKs . For Example 2, pf increases with anincrease inνKs and reaches a peak (atνKs ¼ 1:5) then decreases sharply;the magnitude of pf being much smaller for each value of νKs comparedto Example 1. Unlike Example 1 where pfs continuously increased withνKs , pfs for Example 2 reaches a peak (at νKs ¼ 1:5) and then decreases.Interestingly, pfd varies exactly in the same manner for both the caseswithin the same range of values as well. This implies that, due to achange in the triggering mechanism, there are fewer shallow failuresfor Example 2, resulting in a reduction in pfs. Consequently, this resultsin a decrease in the overall probability of failure pf as well.
4.2.5. Random field studyThe parameters for the random field simulation in Example 2 are the
same as in Example 1, only the soil cohesion c′ is changed to 0.15 kPafor the slope stability assessment. The purpose here is to observe the ef-fect of a change in the triggeringmechanism on the failure probabilities(pf, pfs, and pfd), consequence (C), and risk (R). The seepage analysis isperformed for duration of t = 0.0860 days and the results are thenused to perform slope stability analysis.
-15 -10 -5 0-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
Pore water pressure (kPa)
Dep
th (
m)
Initial conditionFS = 1.00
-1 -0.5 0 0.5 1-1
-0.8
-0.6
-0.4
-0.2
0
Pore-water pressure (kPa)
Dep
th (
m)
Fig. 12. Profiles of pore water pressure— initially and at failure (Example 2); inset, profileof pore water pressure at failure on a magnified scale.
Fig. 17 shows the variation in the failure probabilities (pfs, pfd, pf),consequence (C) and risk (R) with spatial correlation length (Θ). Com-pared to Example 1, although the trend in the variation of the failureprobabilitieswithΘ is somewhat similar, the variation occurs over a dif-ferent range of valueswith a significant reduction in pfs at all values ofΘ.This reduction can be attributed to a different triggeringmechanism (i.e.generation of a positive pore water pressure) which causesfewer shallow failures. The variation in pfd is very similar to Example1. Therefore, the reduction in pf is primarily due to a reduction in pfs.The consequence is a minimum for Θ = 0.125 and increaseswith Θ as the soil becomes more uniform, similar to what was ob-served in Example 1. However, deeper shallow failures cause anincrease in the consequence C compared to Example 1. The riskR depends on pf and C, and pf decreases while C increases withΘ. Subsequently, a maximum risk can be observed at Θ = 0.50,implying that a critical spatial correlation length exists at whichthe risk is greatest. The variation in the risk R with Θ occurs overa very narrow range (in Example 2) with a higher risk being ob-served at smaller spatial correlation lengths (Θ = 0.125, 0.25,0.5) compared to Example 1.
Fig. 18 shows the distribution of pore water pressure (uw) at failureplotted against the corresponding failure depths Df for different spatialcorrelation lengths Θ. The (red) solid points represent failures dueto the development of positive pore water pressure while the (blue)hollow points represent failures due to a loss of suction. The solid linerepresents the pore water pressure at failure estimated analytically(from Eq. (28)). A significant scatter can be observed in the results(of uw) for small correlation lengths (Θ). The scatter decreases with anincrease in Θ, similar to that observed for Example 1. However, in con-trast to Example 1, where most of the surficial failures are due to lossof suction, the shallow failures in the top half of the soil profile areonly due to generation of a positive pore water pressure. Also, fewfailures at the bottom occur due to generation of a positive pore waterpressure, which was not observed in Example 1. The number of failuresoccurring due to positive pore water pressure generation is greatest forthe smallest Θ, and decreases with an increase in Θ. As Θ increases, thenumber of shallow failures occurring near the surface also starts de-creasingwhich causes a decrease in pfs as observed in Fig. 17. The devel-opment of positive pore water pressure is also confirmed in Fig. 19,which shows scatter plots of Se at failure for selected spatial correlationlengths. The (red) solid points representing the locations, where Se=1,indicate that the soil was saturated at failurewhich implies that positivepore water pressure was generated at shallow depths. The (blue)hollow points representing the locations, where Se b 1, indicate thatthe soil was not completely saturated or the failure occurred due toloss of matric suction. Fig. 20 illustrates examples of shallow and deepfailures occurring due to generation of a positive pore water pressurefor the random fields shown in Fig. 21.
10-1
100
101
102
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Ks (m/day)
Df (
m)
tf
0.0860 days
tf > 0.0860 days
Ksnf
Ksmin
tf increasing
tfincreasing
Fig. 13. Variation of failure depths Df with saturated hydraulic conductivity Ks for I ¼ 0:25μKs(Example 2).
10-1
100
101
1020
0.05
0.1
0.15
0.2
0.25
0.3
Ks (m/day)
t f (da
ys)
tf
0.0860 days
tf > 0.0860 days
tf = 0.0860 days
Ksnf
Ksmin
Fig. 14. Variation of first passing time of failure tf with saturated hydraulic conductivity Ks for I ¼ 0:25μKs(Example 2).
0.5 1 1.5 2 2.5 3-1
-0.8
-0.6
-0.4
-0.2
0
Factor of safety
Dep
th (
m)
Initial profileK
s = 8.64 m/day
tf = 0.0860 days
Ks = 3.00 m/day
tf = 0.0700 days
-20 -15 -10 -5 0 5-1
-0.8
-0.6
-0.4
-0.2
0
Pore water pressure (kPa)
Dep
th (
m)
Deep failure
Shallow failure
Deep failure
Shallow failure
Fig. 15. Illustration of deep and shallow failures for I ¼ 0:25μKsand different Ks (Example 2).
112 A. Ali et al. / Engineering Geology 179 (2014) 102–116
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ks
p f
pf(analyt)
pf(SRV)
pfs(analyt)
pfs(SRV)
pfd(analyt)
pfd(SRV)
Fig. 16. Comparison of analytical estimation of failure probabilities with the SRV approach (Example 2).
10-1
100
101
1020
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Dimensionless spatial correlation length,
p f , C
, R
pf
pfd
pfs R C
Fig. 17. Variation of failure probabilities, consequence and risk with spatial correlation lengths (Example 2).
-0.4 -0.2 0 0.2-1
-0.8
-0.6
-0.4
-0.2
0
Pore water pressure (kPa)
Dep
th (
m)
= 0.125
Analyticalu
w<0
uw
0
-0.4 -0.2 0 0.2-1
-0.8
-0.6
-0.4
-0.2
0
Pore water pressure (kPa)
= 1.00
-0.4 -0.2 0 0.2-1
-0.8
-0.6
-0.4
-0.2
0
Pore water pressure (kPa)
= 8.00
-0.4 -0.2 0 0.2-1
-0.8
-0.6
-0.4
-0.2
0
Pore water pressure (kPa)
= 100.00
Fig. 18. Scatter plots of porewater pressure at failure for selected spatial correlation lengths (Example 2). (For interpretation of the references to colour in this figure, the reader is referredto the web version of this article.)
113A. Ali et al. / Engineering Geology 179 (2014) 102–116
0.96 0.98 1-1
-0.8
-0.6
-0.4
-0.2
0
Se
Dep
th (
m)
= 0.125
Se
< 1
Se
= 1
0.96 0.98 1-1
-0.8
-0.6
-0.4
-0.2
0
Se
= 1.00
0.96 0.98 1-1
-0.8
-0.6
-0.4
-0.2
0
Se
= 8.00
0.96 0.98 1-1
-0.8
-0.6
-0.4
-0.2
0
Se
= 100.00
Fig. 19. Scatter plots of effective degree of saturation (Se) at failure, for selected spatial correlation lengths (Example 2). (For interpretation of the references to colour in this figure, thereader is referred to the web version of this article.)
-15 -10 -5 0-1
-0.8
-0.6
-0.4
-0.2
0
Pore water pressure (kPa)
Dep
th(m
)
0.5 1 1.5 2 2.5 3-1
-0.8
-0.6
-0.4
-0.2
0
Factor of safety
Dep
th(m
)
Initial conditionsDeep failure, t
f=0.086 days
Shallow failure, tf=0.023 days
Fig. 20. Illustration of shallow and deep failures due to generation of a positive pore water pressure (Example 2).
0 1 2 3 4-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
Ks (m/day)
Dep
th (
m)
Deep failureShallow failure
Fig. 21. Sample random fields for deep and shallow failures due to generation of positivepore water pressures (Θ = 0.125).
114 A. Ali et al. / Engineering Geology 179 (2014) 102–116
Fig. 22 shows the histogram of failure depths Df for different spatialcorrelation lengths Θ. As far as shallow failures are concerned, theyare much deeper in Example 2 compared to Example 1 at all spatialcorrelation lengths. The histogram clearly illustrates the reason for anincrease in consequence compared to Example 1, as the shallow failuresare much deeper.
5. Conclusion
The risk of rainfall induced landslides is studied quantitatively, basedon the logic that the consequence should be assessed individually foreach failure. When the saturated hydraulic conductivity is modelled as arandom field, it was shown that the probability of failure increases asthe spatial correlation length increases. However, when consequence offailure (measured here by the depth of the failure) is accounted for, a crit-ical spatial correlation length exists at which the risk is maximum. Thisconfirms clearly that the consequence should be assessed individuallyfor a rational risk assessment. The triggeringmechanisms for a rainfall in-duced landslide have also been highlighted by the pore water pressuredistributions at failure. When failure occurs due to generation of positivepore water pressure, the risk tends to be higher than when the failureoccurs due to loss of suction.
0 200 400 600-1
-0.8
-0.6
-0.4
-0.2
0D
epth
(m
)
Frequency
= 0.125
0 200 400 600-1
-0.8
-0.6
-0.4
-0.2
0
Frequency
= 1.00
0 200 400 600-1
-0.8
-0.6
-0.4
-0.2
0
Frequency
= 8.00
0 200 400 600-1
-0.8
-0.6
-0.4
-0.2
0
Frequency
= 100.00
Fig. 22. Histogram of failure depths for selected spatial correlation lengths (Example 2).
115A. Ali et al. / Engineering Geology 179 (2014) 102–116
6. Notations
a scaling suctionc′ effective cohesionC consequenceD slope depthDf depth of failureFN normal forceFT tangential forceFS factor of safetyg gaussian equivalent of the log-normal saturated hydraulic
conductivity random fieldI rainfall intensityK hydraulic conductivityKr relative hydraulic conductivityKs saturated hydraulic conductivityKsmin maximum value of Ks which can cause a shallow failureKsnf minimum value of Ks which can cause a shallow failurem van Genuchten model parametern soil porositynf number of simulations resulting in slope failuresnsim number of simulationsnfd number of simulations resulting in deep slope failuresnfs number of simulations resulting in shallow slope failuresN van Genuchten model parameterpf probability of failurepfd probability of deep failurepfs probability of shallow failureq flux infiltrating the slopeR riskS degree of saturationSe effective degree of saturationSr residual degree of saturationt rainfall durationtf first passing time of failureu pore pressure headuw pore water pressureua pore air pressureW weight of failing soil massz slope normal directionZ standard normal variateα slope angleγ unit weight of soilγs unit weight of soil solidsγw unit weight of waterθ soil volumetric water contentθr residual water contentθs saturated water content
θ lnKs spatial correlation length of logarithm of saturated hydraulicconductivity
Θ normalized spatial correlation length of logarithm of saturat-ed hydraulic conductivity
μKsmean saturated hydraulic conductivity
μ lnKsmean of logarithm of saturated hydraulic conductivity
νKs coefficient of variation of saturated hydraulic conductivityρ(z) correlation functionσ total normal stressσ ′ effective normal stressσKs standard deviation of saturated hydraulic conductivityσ lnKs
standard deviation of logarithm of saturated hydraulicconductivity
τ soil shear stressτf soil shear strengthϕ ′ effective friction angleΦ cumulative standard normal distribution functionχ coefficient of effective stress
Acknowledgements
The authors wish to acknowledge the support of the AustralianResearch Council (CE11001009) in funding the Centre of Excellencefor Geotechnical Science and Engineering.
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