Computers and Geotechnics 73 (2016) 72–82

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Computers and Geotechnics

journal homepage: www.elsevier .com/ locate/compgeo

Research Paper

Evaluating slope stability uncertainty using coupled Markov chain

http://dx.doi.org/10.1016/j.compgeo.2015.11.0210266-352X/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +86 27 6877 2496; fax: +86 27 6877 4295.E-mail address: qixiaohui@whu.edu.cn (X.-H. Qi).

Dian-Qing Li a, Xiao-Hui Qi a,⇑, Zi-Jun Cao a, Xiao-Song Tang a, Kok-Kwang Phoon b, Chuang-Bing Zhou c

a State Key Laboratory of Water Resources and Hydropower Engineering Science, Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering (Ministry ofEducation), Wuhan University, 8 Donghu South Road, Wuhan 430072, PR ChinabDepartment of Civil and Environmental Engineering, National University of Singapore, Blk E1A, #07-03, 1 Engineering Drive 2, Singapore 117576, Singaporec School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, PR China

a r t i c l e i n f o

Article history:Received 5 March 2015Received in revised form 8 October 2015Accepted 20 November 2015

Keywords:Slope stabilityGeological uncertaintyCoupled Markov chainFinite element stress-based method

a b s t r a c t

Geological uncertainty appears in the form of one soil layer embedded in another or the inclusion ofpockets of different soil type within a more uniform soil mass. Uncertainty in factor of safety (FS) andprobability of failure (Pf) of slope induced by the geological uncertainty is not well studied in the past.This paper presents one approach to evaluate the uncertainty in FS and Pf of slope in the presence of geo-logical uncertainty using borehole data. The geological uncertainty is simulated by an efficient coupledMarkov chain (CMC) model. Slope stability analysis is then conducted based on the simulated heteroge-neous soils. Effect of borehole layout schemes on uncertainty evaluation of FS and Pf is investigated. Theresults show that borehole within influence zone of the slope is essential for a precise evaluation of FSstatistics and Pf. The mean of FS will converge to the correct answer as the borehole number increases.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

It is well known that soil heterogeneity or spatial variability ofsoil properties plays a significant role in theperformance of geotech-nical systems such as a slope (e.g. [6,7,10,12,17,19,25,26,28]). Atpresent, studies on spatial variability in geotechnical engineeringmainly focused on inherent variability within one nominally homo-geneous layer [11,14,16,20,22]. The inherent variability is the varia-tion of soil property parameters from one point to another due todifferentdeposition conditions and loadinghistories [4,21]. It occursin a soil mass that belongs to the same material type. However,another form of soil heterogeneity, namely geological uncertainty,also exists in reality (e.g. [5,9]). It appears in the formof one soil layerembedded in another or inclusion of pockets of different soil typewithin a more uniform soil mass [4]. Some attention has been paidto this kind of uncertainty. For example, Tang et al. [27] introduceda renewal process to describe the probabilistic nature of a soil stra-tum consisting of two distinct material types. Halim [8] evaluatedthe reliability of geotechnical systems considering the uncertaintyof geological anomaly. Herein the geological anomaly refers to thecase of pockets of different soil type includedwithin amore uniformsoil mass. The occurrence of geological anomalies in space is mod-eled by a Poisson process in this study. Kohno et al. [15] studiedthe system reliability of a tunnel running through two rock types.

Similarly, the occurrence of the less dominant rock in the two wasalso modeled by a Poisson process. The limitations of these studyare quite obvious. Both the renewal process and the Poisson processcan only model the two simplest forms of geological uncertainty.Only two types of soil are involved in these forms. Themore generalform, namely layers with more than twomaterial types embeddingeach other, cannot be handled by these two processes.

In reality, geological uncertainty typically involves more thantwo soil types embedding each other in a layered profile (e.g.[5,9]). There are very limited studies on how this form of uncer-tainty affects the factor of safety (FS) and probability of failure(Pf) of a slope. This paper aims to evaluate the uncertainty in FSand Pf of a slope in the presence of geological uncertainty usingborehole data. Coupled Markov chain (CMC) is an effective modelto simulated geological uncertainty. To facilitate the applicationof this model in geotechnical practice, Qi et al. [23] proposed apractical method to estimate one key input of the CMC model,i.e. horizontal transition probability matrix (HTPM). Based on thismethod, this paper applies the CMC model to a slope problem.The borehole database for the Perth Central Business District, Wes-tern Australia, is adopted to simulate the geological uncertaintyconditional on known stratigraphy given in the boreholes. Threetypes of soils (clay, silt and sand) are present in this database.Inherent variability is not considered within each layer. Based onthe simulated heterogeneous soils, the FS of slope can be calculatedusing the finite element-strength reduction method. Monte Carlosimulation of slope stability analysis is conducted using different

D.-Q. Li et al. / Computers and Geotechnics 73 (2016) 72–82 73

borehole layout schemes. The effect of borehole layout schemes onthe FS statistics (including mean and standard deviation) and Pf isanalyzed. To overcome the limitation of too less borehole number,some virtual boreholes are created to further investigate the role ofborehole number and borehole location in evaluating the uncer-tainty in FS of slope.

0 10 20 30 40 50 60 70

0

20

40

60

80

100

D23

D20'D78'D75' D23'

D20

D77'D76'

D78

D75

D77

boreholes projective boreholes

y (m

)

x (m)

D76

(a) Relative location of the boreholes

2. Coupled Markov chain model

Coupled Markov chain (CMC) is a random process model, whichcan simulate the geological uncertainty involving more than twotypes of soils embedding each other [3]. The model is theoreticallysimple, explicit and computationally efficient. It is a coupled pro-duct of two one-dimensional Markov chains. One describes thesequence of soil states in horizontal direction, and the other in ver-tical direction. Herein the soil state refers to soil type, such as sand,clay and silt. For each one-dimensional Markov chain, the probabil-ity of transitions between different soil states are denoted by onetransition probability matrix, i.e. horizontal transition probabilitymatrix (HTPM, Ph) for the horizontal Markov chain, and verticaltransition probability matrix (VTPM, Pv) for the vertical Markovchain. Both matrices have a size of m ⁄m, where m (m P 2) isthe total number of soil state involved. For example, the elementin ith row, jth column of VTPM, namely pvij , denotes the probabilityof transition from soil state i (Si) to soil state j (Sj) in verticaldirection.

The basic idea of CMC is as follows. As shown in Fig. 1, thedomain to be modeled is discretized into a number of cells withthe same size. The state of cell (i, j) (i > 1, i = column number;j > 1, j = row number) depends on the states of the cells on thetop [cell (i, j � 1)], left [cell (i � 1, j)] and rightmost [cell (Nx, j),Nx = the sum of cell columns] of the current cell. Soil states onthe leftmost column [i.e. the cells (1, j), j = 1,. . ., Nz, Nz = the sumof cell rows] (considered as left boundary of the simulationdomain), rightmost column [i.e. the cells (Nx, j), j = 1,. . ., Nz] (consid-ered as right boundary of the simulation domain) and top row [i.e.the cells (i,1), i = 1,. . ., Nx] are fixed. The former two are revealed bytwo boreholes while the latter is directly observable from theground surface. They can be used as conditional information tosimulate the states of the other cells inside the domain. The depen-dence of the cell states is described in terms of transition probabil-ities as

plr;kjq ¼phlkp

hðNx�iÞkq pvrk

Pmf¼1p

hlf p

hðNx�iÞfq pvrf

ð1Þ

where plr,k|q is the probability that cell (i, j) is in state Sk, given thatcell (i � 1, j), (i, j � 1) and (Nx, j) is in state Sl, Sr and Sq; ph

lk and pvrk arethe corresponding elements of the horizontal and vertical transition

probability matrices, Ph and Pv; phðNx�iÞkq [phðNx�iÞ

fq ] is the probability of

1, 1 Nx, 1

1, Nz Nx,Nz

i, j-1

i-1, j i, j Nx, j

Fig. 1. Numbering system in a two-dimensional domain for the coupled Markovchain.

transition from Sk (Sf) to Sq in (Nx � i) steps in the horizontaldirection. It is the corresponding elements of (Ph)(Nx�i), i.e. thematrix obtained by multiplying HTPM by itself (Nx � i) times.

3. Borehole data

Some borehole data from Central Business District, Perth, Wes-tern Australia are collected for geological uncertainty simulation inthis paper. The relative location and stratigraphy of the boreholesare plotted in Fig. 2. As shown by Fig. 2(a), the boreholes are scat-tered distributed within a 70 m � 100 m area. To construct a two-dimensional model of slope, all the boreholes need to be projectedto a line that is parallel to the sliding direction of the slope. Sincethere are no real slopes nearby the borehole area, a projection lineparallel to x axis is assumed to be the sliding direction of a slope.The locations of the projected borehole are obtained as shown inFig. 2(a). For brevity, the boreholes from left to right (i.e. boreholesD760, D770, D750, D780, D230, D200) are re-labeled as boreholes 1, 2,3, 4, 5, 6 from hereon. Fig. 2(b) illustrates the stratigraphy revealedby the boreholes. As shown by Fig. 2(b), three types of soil (i.e. clay,silt and sand) are involved. The material in the top layer is sand inall boreholes. Borehole lengths vary from 22.2 m to 29.2 m. Themaximum interval distance between the boreholes in x directionis 70 m. The minimum thickness of the geological unit revealedby boreholes is 0.3 m [see borehole 4 in Fig. 2(b)].

(b) Soil layer in the boreholes

Fig. 2. The relative location and stratigraphy of the boreholes in Perth city,Australia.

Table 2Vertical transition probability matrices for various borehole layout schemes.

State 1 (clay) 2 (sand) 3 (silt)

(a) Scheme 3 [boreholes (1,4,6)]1 (clay) 0.940 0.060 0.0002 (sand) 0.060 0.915 0.0263 (silt) 0.000 0.033 0.967

(b) Scheme 4A [boreholes (1,3,4,6)]1 (clay) 0.942 0.058 0.0002 (sand) 0.057 0.926 0.0173 (silt) 0.000 0.033 0.967

(c) Scheme 4B [boreholes (1,2,5,6)]1 (clay) 0.871 0.129 0.0002 (sand) 0.093 0.872 0.0353 (silt) 0.000 0.071 0.929

(d) Scheme 6 [boreholes (1,2,3,4,5,6)]1 (clay) 0.943 0.058 0.0002 (sand) 0.053 0.931 0.0153 (silt) 0.000 0.026 0.974

Table 3Horizontal transition probability matrices for various borehole layout schemes.

State 1 (clay) 2 (sand) 3 (silt)

(a) Scheme 3 [boreholes (1,4,6)]1 (clay) 0.975 0.025 0.0002 (sand) 0.025 0.964 0.0113 (silt) 0.000 0.014 0.986

(b) Scheme 4A [boreholes (1,3,4,6)]1 (clay) 0.976 0.024 0.0002 (sand) 0.024 0.969 0.0073 (silt) 0.000 0.014 0.986

(c) Scheme 4B [boreholes (1,2,5,6)]1 (clay) 0.971 0.029 0.0002 (sand) 0.021 0.972 0.0083 (silt) 0.000 0.015 0.985

(d) Scheme 6 [boreholes (1,2,3,4,5,6)]1 (clay) 0.970 0.030 0.0002 (sand) 0.028 0.964 0.0083 (silt) 0.000 0.013 0.987

74 D.-Q. Li et al. / Computers and Geotechnics 73 (2016) 72–82

4. Realizations of coupled Markov chain

The collected Australian boreholes are adopted to illustrate theeffect of borehole layout scheme on soil transition simulation. Fourborehole layout schemes are designed based on the boreholes. Theborehole data in each scheme are used to estimate the VTPM,HTPM as well as to provide the conditional information. In otherwords, four sets of VTPMs and HTPMs are respectively estimatedto simulate CMC for the four borehole layout schemes. Note thatthe first-order Markovian property is the prerequisite of the adop-tion of CMC model for geological uncertainty simulation. Hence, ahypothesis test is conducted first to investigate the Markovianorder of the soil transitions. The result shows that the soil transi-tions in the underlying boreholes can be described by first-orderMarkov chain. The test procedure is omitted here. Details of thetest procedure can be found in Qi et al. [23].

4.1. Borehole layout schemes

Table 1 summarizes the four borehole layout schemes. Theseschemes are specially designed to reflect the effect of both bore-hole number and location. As shown in Table 1, each scheme is acombination of several boreholes. To implement the CMC simula-tion method, the design of borehole layout scheme needs to obeyone rule. As illustrated by Eq. (1) in part 2, the stratigraphy ofthe boreholes at both leftmost and rightmost cell columns of theCMC model domain is indispensable in the conditional simulation.Hence, the borehole layout scheme must include the boreholes atthe two outmost columns. Besides, it must be noted that theCMC simulation is conducted in the intervals between any twoadjacent boreholes. In other words, the conditional information isprovided by all the boreholes in the borehole layout scheme.

4.2. Transition probability matrix estimation

Four vertical transition probability matrices and horizontaltransition probability matrices are estimated using the methodproposed by Qi et al. [23]. Following Elfeki and Dekking [3], theheight of the CMC cell is set to be the minimum thickness of geo-logical unit revealed by boreholes, i.e. 0.3 m. The cell length is setto 0.9 m. This value is larger than the vertical sampling intervaldue to the relative large scale of soil transition variability in hori-zontal direction than that in vertical direction [23]. The estimatedVTPMs and HTPMs are listed in Tables 2 and 3, respectively. Asshown in Table 2 (Table 3), the differences in the VTPMs (HTPMs)for different borehole layout schemes are tiny.

4.3. Effect of borehole layout scheme on soil transition simulation

To illustrate the effect of borehole layout schemes on soil tran-sition simulation, two typical realizations of CMC for the boreholelayout schemes 4A and 4B are plotted in Fig. 3. Note that the CMCrealizations for each scheme are simulated using the correspond-ing HTPM and VTPM, and conditional on the corresponding bore-

Table 1Different borehole layout schemes considered in the study.

holes. For example, the VTPM in Table 2(b) and the HTPM inTable 3(b) are adopted for the CMC simulation of scheme 4A, andthe conditional information is provided by boreholes 1, 3, 4, 6. Inthese realizations, the model domain is a rectangular area with ax, z range of [0 m,70.2 m] and [0 m,26.7 m], respectively. TheCMC cells are 0.9 m in length and 0.3 m in height. The soil stateat the top cell row of the CMC is assumed to be sand because thetop soil revealed by all the boreholes is sand. The three soil statesare represented by three different colors (see the legends). Asshown in Fig. 3, the difference between the realizations for thesame borehole layout scheme mainly lies in the soil state withinthe zone where boreholes are absent. For example, the distribu-tions of clay layer between boreholes 4 and 6 are different forthe two realizations of scheme 4A [see Fig. 3(a1,a2)]. For realiza-tion 1, the clay layers above z = 12 m and below z = 12 m is

clay sand siltLegend: clay sand siltLegend:

Borehole 4 Borehole 6Borehole 3Borehole 1

x (m)

z (m

)

26.7

18

6

0

12

24

70.2271890 36 45 54 63

(a1) Realization 1 for scheme 4A [BH (1, 3, 4, 6)] Borehole 4 Borehole 6Borehole 3Borehole 1

x (m)

z (m

)

26.7

18

6

0

12

24

70.2271890 36 45 54 63

(a2) Realization 2 for scheme 4A [BH (1, 3, 4, 6)]

6eloheroB5eloheroB2eloheroBBorehole 1

x (m)

z (m

)

26.7

18

6

0

12

24

70.2271890 36 45 54 63

(b1) Realization 1 for scheme 4B [BH (1, 2, 5, 6)] 6eloheroB5eloheroB2eloheroBBorehole 1

x (m)

z (m

)

26.7

18

6

0

12

24

70.2271890 36 45 54 63

(b2) Realization 2 for scheme 4B [BH (1, 2, 5, 6)]

Fig. 3. Realizations of coupled Markov chain for different borehole layout schemes.

D.-Q. Li et al. / Computers and Geotechnics 73 (2016) 72–82 75

x (m)

z (m

)

26.7

18

6

0

12

24

0 9 18 27 70.236 45 54 63

6eloheroB5eloheroB4eloheroB2eloheroB Borehole 3Borehole 1

Fig. 4. Slope model and borehole locations.

Table 4Parameters for various types of soil for slope stability analysis.

Soiltypes

Unit weight,c (kN/m3)

Cohesion,c (kPa)

Frictionangle, / (�)

Elasticmodulus, E(MPa)

Poisson’sratio, m

Clay 20 18 25 30 0.30Sand 20 2 33 50 0.30Silt 20 6 28 30 0.30

3 4 5 60.90

0.95

1.00

1.05

1.10

Scheme 6[BH (1 2 3 4 5 6)]

Scheme 3[BH (1 4 6)]

Scheme 4B[BH (1 2 5 6)]

Scheme 4A[BH (1 3 4 6)]

μFS

number of boreholes (BH)

(a) μ FS

0.05

0.06

76 D.-Q. Li et al. / Computers and Geotechnics 73 (2016) 72–82

separated by sand layer. However, the clay layer is connected inthe interval z = [2 m, 24 m] for realization 2. In addition, therealizations for different borehole layout scheme maybe quitedifferent. For example, the clay (sand) layer of the realizationsfor scheme 4B is much thinner (thicker) than that for scheme 4A[compare Fig. 3(b1,b2) with Fig. 3(a1,a2)]. This mainly resultsfrom the soil state-fixing effect of the boreholes. As shown by theFig. 2(b), the clay layer around z = 20 m revealed by borehole 5 isthinner than that revealed by other boreholes.

3 4 5 60.01

0.02

0.03

0.04

Scheme 6[BH (1 2 3 4 5 6)]

Scheme 4B[BH (1 2 5 6)]

Scheme 3[BH (1 4 6)]

σ FS

number of boreholes (BH)

Scheme 4A[BH (1 3 4 6)]

(b) σ FS

3 4 5 61E-3

0.01

0.1

1

Scheme 6[BH (1 2 3 4 5 6)]

Scheme 4B[BH (1 2 5 6)]

Scheme 3[BH (1 4 6)]

Pf

number of boreholes (BH)

Scheme 4A[BH (1 3 4 6)]

(c) Pf

Fig. 5. FS statistics and Pf associated with various borehole layout schemes.

5. Illustrative example

Site investigation is crucial to the design practice of geotechni-cal engineering. For example, Ching and Phoon [1] studied thevalue of geotechnical site investigation in reduction of constructioncost using reliability-based design method. Factor of safety (FS)and probability of failure (Pf) are important indicators for a slope.They reflect the safety level of the slope. Hence, this paper aims tostudy the effect of site investigation effort on reducing the uncer-tainty in FS and Pf of slope as geological uncertainty is considered.The site investigation effort refers to the number of boreholes inthis study. Since various borehole layout schemes can be designed,it is natural to question whether various borehole layout schemesare equally efficient in reducing the uncertainty in FS and Pf. Thisissue is addressed through Monte Carlo simulation of slopestability.

It is worth noting that the geological uncertainty must be effec-tively considered in the slope stability analysis. This is achieved byfirstly mapping the soil states of the simulated CMC to a slopemodel. As shown in Fig. 4, the slope model is divided into manyregions with the same size and arrangement as the CMC cells(see Figs. 3 and 4). Herein the region is an area within which thesoil is homogeneous. The soil parameters of the soil states for theCMC cell are assigned to the corresponding regions. Afterward,the slope stability analysis is performed using a commercial soft-ware, namely Abaqus. A finite element-strength reduction methodembedded in the software is adopted for slope stability analysis.Based on this method, Monte Carlo simulation of slope stabilityin the presence of geological uncertainty is conducted using a

FS=1.017

x (m)

z (m

)

26.7

18

6

0

12

24

2.07728190 36 45 54 63

Borehole 4 Borehole 6Borehole 1

(a) Scheme 3 [boreholes (1 4 6)]

x (m)

z (m

)

26.7

18

6

0

12

24

2.07728190 36 45 54 63

Borehole 4 Borehole 6Borehole 3Borehole 1

claysandsilt

x (m)

z (m

)

26.7

18

6

0

12

24

70.2271890 36 45 54 63

Borehole 4 Borehole 6Borehole 3Borehole 1

FS=1.086

(b) Scheme 4A [boreholes (1 3 4 6)]

Fig. 6. Typical realizations of coupled Markov chain and the corresponding contour of plastic strain for various borehole layout schemes.

D.-Q. Li et al. / Computers and Geotechnics 73 (2016) 72–82 77

x (m)

z (m

)

26.7

18

6

0

12

24

70.2271890 36 45 54 63

6eloheroB5eloheroB2eloheroBBorehole 1

claysandsilt

x (m)

z (m

)

26.7

18

6

0

12

24

70.2271890 36 45 54 63

6eloheroB5eloheroB2eloheroBBorehole 1

FS=0.941

(c) Scheme 4B [boreholes (1 2 5 6)]

x (m)

z (m

)

26.7

18

6

0

12

24

70.2271890 36 45 54 63

6eloheroB5eloheroB4eloheroB2eloheroB Borehole 3Borehole 1

claysandsilt

FS=1.004

x (m)

z (m

)

26.7

18

6

0

12

24

70.2271890 36 45 54 63

6eloheroB5eloheroB4eloheroB2eloheroB Borehole 3Borehole 1

(d) Scheme 6 [boreholes (1 2 3 4 5 6)]

Fig. 6 (continued)

78 D.-Q. Li et al. / Computers and Geotechnics 73 (2016) 72–82

Table 5HTPM and VTPM used for creating one realization of CMC.

State 1 (clay) 2 (sand) 3 (silt)

(a) HTPM1 (clay) 0.939 0.031 0.0312 (sand) 0.039 0.939 0.0223 (silt) 0.039 0.022 0.939

(b) VTPM1 (clay) 0.860 0.070 0.0702 (sand) 0.090 0.860 0.0503 (silt) 0.090 0.050 0.860

x (m)

z (m

)

26.7

18

6

0

12

24

70.2271890 36 45 54 63

Fig. 7. One arbitrary CMC realization used for creating virtual boreholes.

D.-Q. Li et al. / Computers and Geotechnics 73 (2016) 72–82 79

non-intrusive stochastic procedure. Steps of this procedure are notpresented here. Details are given elsewhere [13,18].

5.1. Number of Monte Carlo realizations for calculating the mean andstandard deviation of FS

The realization number, Nsim, is an important factor in MonteCarlo simulation. A large Nsim will produce too much computationtime and effort. Hence, the realization number of Monte Carlo sim-ulation is studied herein. The Australian boreholes (1,4,6)described in part 3 are adopted to conditionally simulate the soilstates used for slope stability analysis. Similar simulation strategyas part 4 is adopted. In other words, the VTPM in Table 2(a) andHTPM in Table 3(a) are set as input of CMC simulation. The CMCcell size is 0.9 m ⁄ 0.3 m and soil state in top cell row is still sand.The simulated soil states are mapped to a slope model to representvarious soil materials. A slope model with a size of 70.2 m ⁄ 26.7 mis adopted for slope stability analysis (see Fig. 4). The slope in themodel is 13.5 m in length and 13.5 m in height. A set of soil param-

Table 6Different borehole layout schemes consisting of virtual b

borehole layout scheme

Inf

6

8A

8B

12A

12B

eters including unit weight (c), cohesion (c), friction angle (u), elas-tic modulus (E) and Poisson’s ratio (m) for clay, sand and silt soilsare summarized in Table 4. Note that Drucker–Prager (DP) modelis adopted as constitute model of the soils. This model is more suit-able to simulate granular materials than Mohr–Coulomb (MC) con-stitute model, because the MC model may bring converge problemin the finite element analysis. The parameters of DP model can beconverted from the parameters of MC model. For plain strain prob-lem and non-associated flow, the relations between DP and MCparameters are:

tanb ¼ffiffiffi3

psin/ ð2Þ

dc¼

ffiffiffi3

pcos/ ð3Þ

where b is the slope of the linear yield surface for DP model. It iscommonly referred to as the friction angle of the material; d isthe cohesion of DP model; u and c are the friction angle and cohe-sion of MC model. Details of the parameter conversion can be foundin Abaqus Analysis User’s Manual (Abaqus 6.12, Section 23.3.1).

One method to determine a proper realization number is con-ducting parameter study (e.g. [2]). Generally, the statistics of FSfluctuate with the Monte Carlo realization number. The fluctuationdecreases with increasing realization number. A proper Nsim can beobtained when the statistics of FS achieve a steady level. Hence, aparameter analysis is carried out to determine a proper Nsim.10,000 and 500 realizations of Monte Carlo simulation are respec-tively conducted conditional on boreholes (1,4,6). Mean and stan-dard deviation of FS are simply calculated based on the resulting10,000 or 500 FSs. The obtained mean and standard deviation ofFS, namely (lFS, rFS), associated with 10,000 and 500 simulationsare (1.057,0.042) and (1.057,0.041), respectively. The differences

oreholes.

luence zone

6 7 8 9 10 11 121.105

1.110

1.115

1.120

1.125

1.130

1.135

FS from Monte Carlo simulation "correct" FS

RE: relative error

Scheme 12ARE=0.54%

Scheme 12BRE=0.18%

Scheme 6RE=1.44%

Scheme 8ARE=1.26%

Scheme 8BRE=0.51%

μFS

number of boreholes (BH)

"correct" FS=1.130

(a) μ FS

6 7 8 9 10 11 120.000

0.005

0.010

0.015

0.020

Scheme 12B

Scheme 12A

Scheme 8B

Scheme 6

σFS

number of boreholes (BH)

Scheme 8A

(b) σ FS

Fig. 8. FS statistics associated with various virtual borehole layout schemes.

80 D.-Q. Li et al. / Computers and Geotechnics 73 (2016) 72–82

between these two sets of results are minor. Therefore, 500 realiza-tions of Monte Carlo simulation are enough for the underlyingslope stability problem. In addition, as pointed out by Elfeki [2],the number of Monte Carlo realizations for problem involving geo-logical uncertainty may be much less than that involving Gaussianrandom field. Actually, he analyzed a problem of groundwater flowand transport in geological heterogeneous soils. It is concludedthat 50 realizations are sufficient for convergence. Hence,Nsim = 500 realizations of Monte Carlo simulation are adopted tocalculate the mean and standard deviation of FS for the slope stud-ied herein.

5.2. Effect of borehole layout scheme on evaluation of the FS andreliability of slope

This part conducts some slope stability analyses using variousborehole layout schemes. Firstly, the effect of different boreholelayout schemes in Table 1 on FS and probability of failure (Pf) ofslope is investigated. These borehole layout schemes consist of realboreholes described in part 3. Note that the number of the realboreholes is limited and the spacings between the boreholes arefixed. Secondly, to overcome these limitations, some virtual bore-hole layout schemes with more number of boreholes are createdfrom one arbitrary realization of CMC. These virtual layoutschemes are adopted to further study the role of borehole numberand location on evaluating the FS of slope. The result can be used todesign an effective borehole layout scheme in geotechnicalpractice.

5.2.1. Effect of borehole layout schemes on FS and reliability of slope –analysis using real boreholes

In this part, Monte Carlo simulations are carried out based onthe borehole layout schemes (see Table 1) consisting of real bore-holes described in part 3. The mean of FS (lFS) associated with thefour schemes are plotted in Fig. 5(a). As shown in Fig. 5(a), theincreasing in borehole number does not ensure a monotonoustrend of lFS. For example, the lFS for four boreholes [boreholes(1,3,4,6)] is higher than both three and six boreholes [i.e. bore-holes (1,4,6) and (1,2,3,4,5,6)]. This mainly results from the dif-ferent soil distributions associated with various borehole layoutschemes, which can be seen in Fig. 6. Fig. 6 plots several typicalrealizations of coupled Markov chain and the corresponding con-tour of plastic strain for various borehole layout schemes. The con-tour of plastic strain is customarily used to determine the positionof critical slip surface. As shown in Fig. 6, the CMC realizationsassociated with boreholes (1,4,6) and (1,2,3,4,5,6) have moresands than the realization associated with borehole (1,3,4,6) [seethe rectangles in Fig. 6(a,d)]. Since the sand has a relatively lowshear strength compared with clay (see Table 4), the FS for the real-izations of boreholes (1,4,6) and (1,2,3,4,5,6) is lower than therealization for boreholes (1,3,4,6).

In addition, the lFS for the borehole layout schemes with thesame number of borehole but different borehole locations are alsodifferent. This indicates that the borehole location plays a signifi-cant role in evaluating the FS of slope. As shown in Fig. 5, the lFS

for boreholes (1,3,4,6) is 1.073 while the lFS for boreholes(1,2,5,6) is 0.947. The 1.073 is closer to the more accuracy result,i.e. the lFS = 1.016 for boreholes (1,2,3,4,5,6) than 0.947. This isprobably because the boreholes (1,3,4,6) have more boreholes ininfluence zone of slope than boreholes (1,2,5,6). Herein the influ-ence zone refers to the area which the performance of geotechnicalstructures highly depend on [24]. Obviously, the influence zone canbe viewed as the volume around the critical slip surface for a slopestability problem. As shown by the four realizations in the Fig. 6,the critical slip surface is approximated located in the interval ofx = [27 m,50 m]. Apparently, boreholes 3 and 4 is nearer to theinfluence zone than boreholes 2 and 5. The uncertainty in the soildistribution in influence zone can be more effectively reduced byboreholes 3 and 4 than boreholes 2 and 5. This phenomenon canbe clearly observed in Fig. 6. As shown by Fig. 6(c), the sand layer(marked with two rectangles) in the realization associated withboreholes (1,2,5,6) is much thicker than that in other realizations.Consequently, the resulting lFS for boreholes (1,2,5,6) is less accu-racy than boreholes (1,3,4,6). Hence, the boreholes around theinfluence zone should be drilled as a priority in order to effectivelyevaluate the safety of a slope.

Fig. 5(b) and (c) plot the standard deviation of FS (rFS) and Pfassociated with the four schemes. Note that the Pf is calculatedusing more Monte Carlo realizations (i.e. 10,000 realizations) thanlFS and rFS. This number of realizations is sufficient for calculatinga Pf with enough accuracy for the underlying problem, because theminimum Pf for the four schemes is as high as 7.01%. As seen fromFig. 5(b) and (c), similar observations to the lFS are found. Theincreasing in borehole number does not ensure a decreasing trendof rFS and Pf. Possible reason may lie in that the added boreholesintroduce more uncertainty in soil distributions, because the soillayers revealed by different boreholes are quite different [seeFig. 2(b)]. To sum up, there is no monotonous trend of FS statisticsand Pf with borehole number. Since no obvious trend for the FSstatistics is observed in Fig. 5, it is natural to question whetherthe FS statistics converge as the borehole number furtherincreases. Note that the number of the collected borehole is limitedand the spacings between the boreholes are fixed. Hence, some vir-tual boreholes are created in the next part. In this way, the conver-gence of FS statistics with borehole number can be studied.

clay sand siltLegend: clay sand siltLegend:

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(a) The “real” soil distribution

virtual boreholes

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(b) One realization of CMC using six virtual boreholes (scheme 6)

x (m)

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virtual boreholes

(c) One realization of CMC using twelve virtual boreholes (scheme 12B)

Fig. 9. The ‘‘real” soil distribution and CMC realizations using virtual boreholes.

D.-Q. Li et al. / Computers and Geotechnics 73 (2016) 72–82 81

5.2.2. Effect of borehole layout schemes on FS of slope – analysis usingvirtual boreholes

In this part, the effect of borehole layout scheme on FS of slopeis further studied using virtual boreholes. Firstly, one arbitraryrealization of CMC is simulated. This arbitrary realization can beviewed as the ‘‘real” soil distribution. Slope stability is conductedbased on this realization. The obtained FS is considered to be the‘‘correct” FS of the slope. Secondly, virtual boreholes are createdby placing several vertical lines on the simulated realization.New set of HTPM and VTPM is estimated using the virtual bore-holes. On this basis, 500 new realizations of CMC are simulatedconditional on the virtual boreholes. Thirdly, Monte Carlo

simulation of slope stability is conducted with the 500 realizationsof CMC. The obtained lFS is compared with the ‘‘correct”answer. The trend of FS statistics with virtual layout scheme isinvestigated.

One set of HTPM and VTPM shown in Table 5 is used to simulategeological uncertainty. One arbitrary CMC realization is plotted inFig. 7. This realization is used to create virtual boreholes. The slopemodel in Fig. 4 and the soil parameters listed in Table 4 areadopted for slope stability. The resulting FS for this realization(‘‘real” soil distribution) is 1.130. It is viewed as the ‘‘correct” FSof the slope. Five virtual borehole layout schemes are designed toinvestigate the effect of both borehole number and borehole

82 D.-Q. Li et al. / Computers and Geotechnics 73 (2016) 72–82

location. The locations of virtual boreholes for each scheme areplotted in Table 6. As seen in Table 6, the boreholes in schemes6, 8A, 12A are evenly distributed in the x directions. For scheme8B, 12B, more virtual boreholes are placed in the influence zone.The FS statistics associated with the five virtual borehole layoutschemes are plotted in Fig. 8. As shown in Fig. 8, as the boreholenumber increases, the lFS gradually converges to the ‘‘correct”answer. For example, the relative error of FS is 1.44% for six virtualboreholes (scheme 6), while the error reduces to 0.18% for twelvevirtual boreholes (scheme 12B). Herein the relative error is definedas |‘‘correct” FS- estimated lFS|/‘‘correct” FS. Meanwhile, rFS gener-ally decreases with increasing borehole number. These phenomenaactually can be expected. As more boreholes are used, the simu-lated CMC are more similar to the ‘‘real” soil distribution. Thiscan be well observed in Fig. 9, which plots the ‘‘real” soil distribu-tion and two arbitrary realizations of CMC using different numberof virtual boreholes. In addition, the FS statistics associated withthe virtual borehole layout schemes with the same borehole num-ber but different borehole locations are still different. Obviously,the virtual borehole layout scheme with more virtual boreholesin the influence zone has more accurate FS statistics (see Fig. 8).This further highlights the importance of selecting proper boreholelocation. In other words, the boreholes in influence zone are moreimportant than the boreholes outside the influence zone.

6. Conclusions

This paper investigates the effect of borehole layout scheme onevaluating uncertainty in the factor of safety and the probability offailure of slope caused by geological uncertainty. Coupled Markovchain conditioning on the borehole data of Perth city, Australian isadopted to model the geological uncertainty. Monte Carlo simula-tion of slope stability analysis is carried out using the finite element–strength reduction method. The effectiveness of various boreholelayout schemes inevaluating theuncertainty in the FS andPf of slopeis investigated. The following conclusions are tentatively drawn.

(1) The evaluation of FS statistics and Pf of a slope depends onthe design of borehole layout schemes. An increasing bore-hole number does not ensure a monotonic lFS or a decreas-ing rFS and Pf. Boreholes within influence zone of slope aremore important than the boreholes outside influence zonefor a precise evaluation of FS statistics and Pf.

(2) As the borehole number keeps increasing, themean of FS con-verges to the ‘‘correct” answer.Meanwhile, the standarddevi-ation generally decreases with increasing borehole number.

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