Probabilistic Based on Conditional Random Field

Linxue Zhao Master of Engineering (Architecture & ) Bachelor of Engineering (Engineering Mechanics)

https://orcid.org/0000-0003-2595-7987

A thesis submitted for the degree of Master of Philosophy at The University of Queensland in 2020 School of Civil Engineering Abstract

Slope stability is always among the most important and attended geotechnical research topics. The heterogeneity and uncertainty of the properties of geotechnical materials and the complexity of actual slope engineering highlight the significance of studying slope stability at the probabilistic level. Yet most existing evaluation systems are based on deterministic conceptions. Superior to a definite safety factor ( FS ) provided by the deterministic analysis, statistical evaluation indexes obtained from the probabilistic analysis usually offer a more scientific, reliable, and realistic estimation on the slope safety. The random field theory is the most widely used methodology for characterizing the spatial variability and uncertainty of , and the application of conditional random field (CRF) is an improvement on it, which is adopted herein.

In this study, the random fields of properties are generated using the Covariance matrix decomposition (CMD) method, and according to related theory, the heterogeneous and uncertain characteristics are defined by several statistical parameters. Further, with the application of CRF, some known soil property values at certain conditioning locations are taken into account when generating random fields. Since CRF can make better use of known information of the geotechnical object, it is supposed to provide a more realistic and convincing result. The generation of CRF is realized through the Kriging technique. Then for each single simulation of the soil property’s field, a corresponding safety factor and other failure consequences are calculated by adopting a numerical limit analysis (NLA) OptumG2. At last, the probabilistic analysis is carried out through the direct Monte Carlo simulation (MCS), leading to some statistical results describing the slope safety, such as the distribution of FS , the failure probability ( Pf ), and the reliability index (  ).

There are several research emphases involved in this study, which are all based on the probabilistic analysis framework. At first, comparative analysis between the traditional unconditional random field method and the CRF method verifies the superiority of CRF, and the comparison between various sampling schemes reveals the effect of sampling points’ layout on the conditioning performance. Then the main part is parametric analyses concerning three different slope models. They are conducted to investigate the influence of statistical parameters of soil properties on the slope reliability, the form of random fields, and the overall state of slope stability. Involved slope models and soil properties include the undrained ( Su ) for an undrained slope, the cross-correlated ( c ) and internal frictional angle (  ) for a cohesion-frictional slope, and the ( K s ) for a dam slope. Furthermore, there is a discussion about if the additional consideration of failure consequence can improve the slope reliability definition which

i only incorporates the failure probability. In addition, with the CRF of K s , the parametric sensitivity of the seepage situation in the slope is also studied.

Through comparative analysis, the superiority of the CRF method lies in its more consistent estimations on the field form and the slope stability. Yet the effect on the estimation on the slope stability’s mean level is uncertain and much depends on the soil property level at the conditioning locations. Since Pf (characterizing the slope reliability) is an outcome of the distribution of FS (describing the variability and the mean level of slope stability estimation), the influence brought by applying CRF on the slope reliability estimation is also uncertain. Next, the evaluation of sampling schemes also adopts the standards just mentioned. It is found that increasing the number of sampling points and setting them at more influential locations (the locations where the failure is more likely to occur) lead to a better performance of the scheme.

The statistical parameters covered in this study don’t significantly influence the mean level of slope stability, except that a higher coefficient of variation (COV) of Su brings a negative effect. The sensitivity of the variability of slope stability to the statistical parameters largely conforms with that of the variability of field form. Specifically speaking, the correlation length or the ratio between horizontal and vertical correlation lengths of Su has a two-sided effect on the variability of slope stability, so that as the two parameters increase, the variability first rises then turns gentle or even declines. Meanwhile, with increasing COV of Su , the variability keeps going up. Then for another slope model, with the cross-correlation between c and  becoming less negative or more positive, the variability of slope stability increases. Next, the effects of the correlation length and

COV of K s qualitatively follow the laws concerning Su . Finally, the conclusions concerning slope reliability result from the above influence laws about the overall state of slope stability.

For defining the slope reliability, taking the failure consequence into account in addition to the failure probability doesn’t bring significant improvement, so it is not recommended. Some other notable conclusions are about the seepage in the dam slope model. The correlation length and COV of K s both negatively affect the overall flux level in the slope. Meanwhile, with the correlation length going up, the variability of seepage situation first increases then decreases, and with a rising COV, it constantly ascends. The above-mentioned conclusions are of great significance at both theoretical and practical levels.

ii Declaration by author

This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis.

I have clearly stated the contribution of others to my thesis as a whole, including statistical assistance, survey design, data analysis, significant technical procedures, professional editorial advice, financial support and any other original research work used or reported in my thesis. The content of my thesis is the result of work I have carried out since the commencement of my higher degree by research candidature and does not include a substantial part of work that has been submitted to qualify for the award of any other degree or diploma in any university or other tertiary institution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award.

I acknowledge that an electronic copy of my thesis must be lodged with the University Library and, subject to the policy and procedures of The University of Queensland, the thesis be made available for research and study in accordance with the Copyright Act 1968 unless a period of embargo has been approved by the Dean of the Graduate School.

I acknowledge that copyright of all material contained in my thesis resides with the copyright holder(s) of that material. Where appropriate I have obtained copyright permission from the copyright holder to reproduce material in this thesis and have sought permission from co-authors for any jointly authored works included in the thesis.

iii Publications included in this thesis

No publications included.

Submitted manuscripts included in this thesis

No manuscripts submitted for publication.

Other publications during candidature

Rao, P.P., Zhao, L.X., Liu, Y. and Li, L. (2018). Extended 3D stability analysis of a slope reinforced with piles using upper-bound limit analysis method. Advanced Engineering Sciences, 50(6), 184–192. (in Chinese) Rao, P.P., Zhao, L.X., Chen, Q.S. and Nimbalkar, S. (2019). Three-dimensional slope stability analysis incorporating coupled effects of pile reinforcement and reservoir drawdown. International Journal of Geomechanics, 19(4), 06019002. Rao, P.P., Zhao, L.X., Chen, Q.S. and Nimbalkar, S. (2019). Three-dimensional limit analysis of slopes reinforced with piles in soils exhibiting heterogeneity and anisotropy in cohesion. Soil Dynamics and Engineering, 121, 194-199.

Contributions by others to the thesis

Dr. Dorival Pedroso and Dr. Alexander Scheuermann advised on some aspects of the thesis, including the research direction and the methodology.

Statement of parts of the thesis submitted to qualify for the award of another degree

No works submitted towards another degree have been included in this thesis.

Research involving human or animal subjects

No animal or human subjects were involved in this research.

iv Acknowledgments

My deepest gratitude goes to my principal supervisor, Dr. Dorival Pedroso, and my associate supervisor, Dr. Alexander Scheuermann, for their support, guidance, and suggestions on my research work. Their patience, understanding, and tolerance as I faced struggles is valuable for me.

I am grateful to Dr. Dilum Fernando for his help during the early stage of my research and Mr. Sebastian Quintero for his laboratory assistance. Also, I would like to thank Dr. David Lange, Dr. Matthew Mason, Dr. Mehdi Serati, and Professor Mark Hickman for their advice on my research direction. In addition, I appreciate the help from the staff and technicians in the School of Civil Engineering at The University of Queensland.

I would also like to thank my colleagues, friends, and roommates for making my stay in Australia an enjoyable experience.

Finally, I can’t be more grateful to my parents for their encouragement and financial support. I wish to dedicate this thesis to them.

v Financial support

No financial support was provided to fund this research.

Keywords slope stability, reliability analysis, conditional random field, Monte Carlo simulation, numerical limit analysis, probability of failure, sampling point layout, risk assessment, cross-correlation, seepage analysis

Australian and New Zealand Standard Research Classifications (ANZSRC)

ANZSRC code: 090501, Civil , 100%

Fields of Research (FoR) Classification

FoR code: 0905, Civil Engineering, 100%

vi Table of Contents

Abstract...... i Declaration by author...... iii Publications included in this thesis...... iv Submitted manuscripts included in this thesis...... iv Other publications during candidature...... iv Contributions by others to the thesis...... iv Statement of parts of the thesis submitted to qualify for the award of another degree...... iv Research involving human or animal subjects...... iv Acknowledgments...... v Financial support...... vi Keywords...... vi Australian and New Zealand Standard Research Classifications (ANZSRC)...... vi Fields of Research (FoR) Classification...... vi Table of Contents...... vii List of Figures...... xi List of Tables...... xv List of Abbreviations...... xvi Chapter 1 Introduction...... 1 1.1 Background and significance...... 1 1.2 Literature review...... 3 1.2.1 Analytical probabilistic analysis of slope stability...... 4 1.2.2 Probabilistic analysis based on the random field theory...... 6 1.2.3 Probabilistic analysis based on CRF...... 9 1.2.4 Development of the numerical limit analysis (NLA)...... 11 1.3 Introduction to the theory...... 13 1.3.1 Random field theory...... 13 1.3.2 Conditional random field...... 13 1.3.3 Monte Carlo simulation...... 14 1.3.4 Numerical limit analysis and OptumG2 software...... 14 1.4 Methodology, research scope, and research objectives...... 20

vii Chapter 2 Sampling scheme selection of conditional random field...... 22 2.1 Introduction...... 22 2.2 Generation of random fields...... 22 2.2.1 Generation of unconditional random fields...... 22 2.2.1.1 Covariance matrix decomposition (CMD)...... 23 2.2.1.2 Local average subdivision (LAS)...... 24 2.2.2 Generation of conditional random fields...... 25 2.2.3 Mathematica programs for random field generation by CMD...... 27 2.3 Problem definition...... 29 2.3.1 Basic model and sampling scheme definition...... 29 2.3.2 Deterministic analysis based on the mean...... 30 2.4 Field form analysis...... 31 2.4.1 Typical unconditional random fields...... 31 2.4.2 Real field...... 32 2.4.3 Typical conditional random fields for different sampling schemes...... 34 2.5 Sampling efficiency and slope reliability...... 36 2.6 Conclusions...... 40 Chapter 3 Conditional random field analysis on an undrained slope...... 42 3.1 Introduction...... 42 3.2 Problem definition...... 43 3.2.1 Parametric analysis definition...... 43 3.2.2 Deterministic analysis based on various mean values...... 43 3.3 Real fields...... 44 3.3.1 Real fields with changing  ...... 45 Su

3.3.2 Real fields with changing l ...... 45 Su

3.3.3 Real fields with changing  ...... 45 Su

3.3.4 Real fields with changing COV ...... 46 Su

3.4 Field form analysis...... 51 3.4.1 Typical conditional random fields with changing l ...... 51 Su

3.4.2 Typical conditional random fields with changing  ...... 52 Su

viii 3.4.3 Typical conditional random fields with changing COV ...... 52 Su

3.5 Parametric analysis aimed at the slope reliability...... 56 3.5.1 Parametric analysis on l ...... 56 Su

3.5.2 Parametric analysis on  ...... 60 Su

3.5.3 Parametric analysis on COV ...... 63 Su

3.6 Parametric analysis aimed at the “risk”...... 66 3.7 Conclusions...... 71 Chapter 4 Conditional random field analysis on a cohesion-frictional slope...... 72 4.1 Introduction...... 72 4.2 Generation of two cross-correlated random fields...... 72 4.3 Problem definition...... 74 4.3.1 Basic model and parametric analysis definition...... 74 4.3.2 Deterministic analysis based on the mean...... 75 4.4 Verification...... 76 4.5 Field form analysis...... 77 4.5.1 Paired real fields...... 78 4.5.2 Typical random field pairs...... 80 4.5.3 Typical failure modes...... 87 4.6 Parametric analysis aimed at the slope reliability...... 88 4.7 Conclusions...... 91 Chapter 5 Conditional random field analysis on a dam slope with stochastic hydraulic conductivity...... 92 5.1 Introduction...... 92 5.2 Problem definition...... 93 5.2.1 Basic model and parametric analysis definition...... 93 5.2.2 Deterministic analysis based on the mean...... 94 5.3 Field form analysis...... 95 5.3.1 Real fields...... 101 5.3.2 Flux fields...... 102 5.3.3 Typical conditional random fields...... 102 5.4 Parametric analysis aimed at the overall state of slope stability and the seepage situation103

ix 5.4.1 Parametric sensitivity of the overall state of slope stability...... 103 5.4.2 Parametric sensitivity of the seepage situation...... 105 5.5 Conclusions...... 108 Chapter 6 Conclusions and recommendations...... 109 6.1 Conclusions...... 109 6.2 Recommendations...... 111 References...... 113

x List of Figures

Fig. 1.1 Lower-bound element and upper-bound element...... 16

Fig. 1.2 Surface and body forces acting on soil mass...... 16

Fig. 2.1 Local average subdivision in 2D...... 23

Fig. 2.2 Slope model for the random field of Su ...... 27

Fig. 2.3 Failure mode for deterministic analysis of the undrained slope...... 28

Fig. 2.4 Typical unconditional random fields of Su ...... 29

Fig. 2.5 Typical failures of the unconditional case...... 29

Fig. 2.6 Real field of Su ...... 30

Fig. 2.7 Information extracted from the real field of Su ...... 30

Fig. 2.8 Failure mode for the real field of Su ...... 30

Fig. 2.9 Typical conditional random fields of Su for Scheme1...... 31

Fig. 2.10 Typical failures of Scheme1...... 32

Fig. 2.11 Typical conditional random fields of Su for Scheme4...... 32

Fig. 2.12 Typical failures of Scheme4...... 32

Fig. 2.13 Typical conditional random fields of Su for Scheme7...... 33

Fig. 2.14 Typical failures of Scheme7...... 33

Fig. 2.15 Probability density functions of FS for different schemes...... 36

Fig. 3.1 Failure modes for deterministic analyses of the undrained slope...... 42

Fig. 3.2 Real fields of S with different  : (a)   45kPa , (b)   50kPa , (c) u Su Su Su   55kPa ...... 45 Su

Fig. 3.3 Real fields of S with different l : (a) l  3m , (b) l  7m , (c) l 15m ...... 46 u Su Su Su Su

Fig. 3.4 Real fields of S with different  : (a)   3 , (b)   7 , (c)   15 ...... 47 u Su Su Su Su

xi Fig. 3.5 Real fields of S with different COV : (a) COV  0.3, (b) COV  0.5 , (c) u Su Su Su COV  0.7 ...... 48 Su

Fig. 3.6 Typical conditional random fields of S with different l : (a) l  3m , (b) l  7m , (c) u Su Su Su l 15m ...... 51 Su

Fig. 3.7 Typical conditional random fields of S with different  : (a)   3 , (b)   7 , (c) u Su Su Su   15 ...... 52 Su

Fig. 3.8 Typical conditional random fields of S with different COV : (a) COV  0.3, (b) u Su Su COV  0.5 , (c) COV  0.7 ...... 53 Su Su

Fig. 3.9 Influence of l on COV ...... 55 Su FS

Fig. 3.10 Influence of l on the 80% median interval of FS ...... 57 Su

Fig. 3.11 Influence of l on P ...... 57 Su f

Fig. 3.12 Influence of  on COV ...... 59 Su FS

Fig. 3.13 Influence of  on the 80% median interval of FS ...... 60 Su

Fig. 3.14 Influence of  on P ...... 60 Su f

Fig. 3.15 Influence of COV on COV ...... 62 Su FS

Fig. 3.16 Influence of COV on the 80% median interval of FS ...... 63 Su

Fig. 3.17 Influence of COV on P ...... 63 Su f

Fig. 3.18 Influence of l on Risk...... 67 Su

Fig. 3.19 Influence of  on Risk...... 67 Su

Fig. 3.20 Influence of COV on Risk...... 68 Su

Fig. 3.21 Typical failures corresponding to various failure masses...... 68

Fig. 4.1 Slope model for the random fields of cross-correlated c and  ...... 73

Fig. 4.2 Failure mode for the deterministic analysis of the cohesion-frictional slope...... 74

xii Fig. 4.3 Verification concerning the influence of c, on  FS ...... 74

Fig. 4.4 Verification concerning the influence of c, on Pf ...... 75

Fig. 4.5 Paired real fields of c and  with different c, : (a) c,  0 , (b) c,  -0.5 , (c)

c,  0.5 ...... 77

Fig. 4.6 Typical unconditional random field pairs of c and  with different c, : (a) c,  0 ,

(b) c,  -0.5 , (c) c,  0.5 ...... 81

Fig. 4.7 Typical conditional random field pairs of c and  with different c, : (a) c,  0 , (b)

c,  -0.5 , (c) c,  0.5 ...... 84

Fig. 4.8 Typical failures for the conditional case with c,  0.5 ...... 85

Fig. 4.9 Influence of c, on COVFS ...... 86

Fig. 4.10 Influence of c, on the 80% median interval of FS ...... 88

Fig. 4.11 Influence of c, on Pf ...... 88

Fig. 5.1 Slope model for the random field of K s ...... 92

Fig. 5.2 Failure mode for the deterministic analysis of the dam slope...... 93

Fig. 5.3 Degree of saturation for the deterministic analysis of the dam slope...... 93

Fig. 5.4 Real fields of K with different l : (a) l  0.5m , (b) l  3m , (c) l  7m ...... 94 s Ks v K s v K s v K s v

Fig. 5.5 Real fields of K with different COV : (a) COV  0.3, (b) COV  0.5 , (c) s K s K s K s COV  0.7 ...... 95 K s

Fig. 5.6 Flux fields with different l : (a) l  0.5m , (b) l  3m , (c) l  7m ...... 96 Ks v K s v K s v K s v

Fig. 5.7 Flux fields with different COV : (a) COV  0.3, (b) COV  0.5 , (c) COV  0.7 K s K s K s K s ...... 97

Fig. 5.8 Typical conditional random fields of K with different l : (a) l  0.5m , (b) s Ks v K s v l  3m , (c) l  7m ...... 98 K s v K s v

xiii Fig. 5.9 Typical conditional random fields of K with different COV : (a) COV  0.3, (b) s K s K s COV  0.5 , (c) COV  0.7 ...... 99 K s K s

Fig. 5.10 Influence of COV and l on COV ...... 102 K s Ks v FS

Fig. 5.11 Influence of COV and l on the value range of FS ...... 102 K s Ks v

Fig. 5.12 Spatial distributions of phreatic surfaces for different l : (a) l  0.5m , (b) l  3m , Ks v K s v K s v (c) l  7m ...... 104 K s v

Fig. 5.13 Spatial distributions of phreatic surfaces for different COV : (a) COV  0.3, (b) K s K s COV  0.5 , (c) COV  0.7 ...... 105 K s K s

xiv List of Tables

Table 2.1 Resulting indexes of various schemes...... 35

Table 3.1 Influence of l on  ...... 55 Su FS

Table 3.2 Table 3.2 Influence of l on  ...... 55 Su FS

Table 3.3 Influence of l on  ...... 58 Su

Table 3.4 Influence of  on  ...... 59 Su FS

Table 3.5 Influence of  on  ...... 59 Su FS

Table 3.6 Influence of  on  ...... 61 Su

Table 3.7 Influence of COV on  ...... 61 Su FS

Table 3.8 Influence of COV on  ...... 62 Su FS

Table 3.9 Influence of COV on  ...... 64 Su

Table 3.10 Influence of l on the average consequence (m2)...... 66 Su

Table 3.11 Influence of  on the average consequence (m2)...... 66 Su

Table 3.12 Influence of COV on the average consequence (m2)...... 66 Su

Table 4.1 The known data of the five sampling points for different c, ...... 78

Table 4.2 Influence of c, on FS ...... 86

Table 4.3 Influence of c, on  FS ...... 86

Table 4.4 Influence of c, on  ...... 89

Table 5.1 Influence of COV and l on  ...... 101 K s Ks v FS

Table 5.2 Influence of COV and l on  (10-3)...... 101 K s Ks v FS

xv List of Abbreviations

BLUE: best linear unbiased estimate

CMD: Covariance matrix decomposition

COV: coefficient of variation

CPT:

CRF: conditional random field

DEM: discrete element method

FDM: finite differential method

FELA: finite-element limit analysis

FEM: finite element method

FORM: first-order reliability method

FOSM: first-order second-moment method

LA: limit analysis

LAS: Local average subdivision

LEM: limit equilibrium method

MCS: Monte Carlo simulation

NLA: numerical limit analysis

PDF: probability density function

RFEM: random finite element method

RSM: response surface method

SS: subset simulation

TGU: total global uncertainty

VST: vane shear test

xvi Chapter 1 Introduction

1.1 Background and significance

(1) Probabilistic analysis of slope stability

Natural geotechnical materials usually exhibit spatial variability and uncertainty on their inherent properties. Such characteristics are affected by various factors during their formation processes, such as properties of their parent materials, weathering and erosion processes, transportation agents, and conditions of sedimentation (Vanmarcke 1977; Phoon and Kulhawy 1999a; Mitchell and Soga 2005). In addition to inherent spatial variability of soils, various uncertainties that arise during geotechnical site characterization also affect the estimation of soil properties (Christian et al. 1994; Phoon and Kulhawy 1999a), including measurement errors arising from imperfect test equipment or procedural-operator errors, statistical uncertainty resulted from an insufficient number of tests, and transformation uncertainty associated with the transformation models used to interpret test results. The stochastic characters mentioned above subsequently influence the analysis of geotechnical structures. For all this, the variability and uncertainty in soil properties can be rationally incorporated using probabilistic theory and statistical properties.

Slope stability analysis has always been one of the most attended geotechnical problems. Further, in the field of probabilistic geotechnical analysis, slope stability analysis seems to also have attracted more attention in existing studies than any other geotechnical application. In the context of probabilistic analysis, the stability of some certain simulations of the objective slope is assessed in different ways of deterministic analysis, and the slope reliability is frequently measured by the reliability index (  ) and the probability of failure ( Pf ) which is defined as the probability of the performance requirements not being satisfied. The comparative study between deterministic and probabilistic analyses reveals the necessity of the latter. A slope with a deterministic safety factor ( FS ) considerably larger than 1 could possess an unacceptably high probability of failure. In another case, the average safety factor derived from the probabilistic analysis is smaller than the safety factor resulted from the deterministic analysis adopting the mean level of soil parameters (Hicks and Samy 2002). Namely the deterministic analysis tends to give a non-conservative result in the estimation of slope stability. The evaluation modes offered by probabilistic analysis provide a more scientific and believable reference for both academic and practical applications.

(2) Random field theory and conditional random field

With the spatial variability and uncertainty of geotechnical materials drawing more and more attention, information quantifying such uncertainty and variability must be incorporated to allow the

1 establishment of a probabilistic evaluation system associated with various geotechnical contexts. To address this issue, Vanmarcke (1977, 1983) developed the random field theory to characterize such kind of spatial variability and uncertainty. To be more precise, the random field theory is for simulating heterogeneous continuous fields of stochastic property parameters exhibiting certain random distribution characteristics and spatial correlation. Based on the random field theory, not only the characteristics of uncertainty and variability can be defined comprehensively by several statistical parameters, but also some probabilistic evaluation indexes concerning slope reliability are available by estimating plenty of random field realizations governed by the given statistical parameter setting.

Site investigation data are generally available in practical engineering, even though the amount of data may not be abundant. These data may come from field vane shear test (VST) or cone penetration test (CPT), and reflect the reference true values of soil properties at certain locations, which should keep invariant in every random field simulation. The traditional unconditional random field simulation, which neglects these known information, may overestimate the uncertainty of the concerned random fields during the simulation. Additionally, such a waste of site investigation effort will also affect the responses, such as the distribution of FS and the probability of failure, of the whole slope system. Hence, it is of practical significance to take the known data into consideration in reliability-based slope analysis.

To address this issue, the application of the conditional random field (CRF) makes an adjustment by taking full advantage of known information of the geotechnical object. That is, both the actual measurements and their statistics are incorporated in conditional random fields. Such treatment leads to models closer to reality and subsequently reduces the level of uncertainty in stability analysis. So it is expectable that the upgrade to CRF will lead to a more realistic and scientific slope reliability estimation. Nevertheless, there are quite limited studies using conditional simulation in slope reliability estimation.

(3) Monte Carlo simulation and numerical limit analysis

In the last few decades, several probabilistic analysis methods have been developed to deal with various uncertainties in slope engineering rationally, such as first-order second-moment method (FOSM) (Tang et al. 1976; Christian et al. 1994; Hassan and Wolff 1999), first-order reliability method (FORM) (Low and Tang 1997; Low et al. 1998), and Monte Carlo simulation (MCS) method (El-Ramly et al. 2002, 2005; Griffiths and Fenton 2004). Compared with the analytical methods (FORM, FOSM), MCS gains supreme popularity in the slope reliability analysis due to its robustness and conceptual simplicity.

2 For the deterministic analysis stage, such as each single realization of the random field analysis and the calculation based on the mean level of parameters, limit equilibrium method (LEM), finite element method (FEM), or even some analytical formulations for slopes with simple failure modes, can be applied to get a single FS or the relation between FS and random variables. In the existing probabilistic research on slope stability, the limit analysis (LA) method is seldom used in the deterministic stage, let alone its numerical analysis form. The numerical limit analysis (NLA) is a reliable method for analyzing slope stability based on the theorem of limit analysis and can provide strict upper-bound and lower-bound solutions. Since the theoretical behind this method is distinguished from FEM and LEM, it’s reasonable to investigate its application on probabilistic slope stability analysis.

Based on the above research background and status, as as from the perspective of the advancement of methodology and the innovation of the research scheme, the core concept concerning methodology in this study is the application of NLA and MCS to the probabilistic analysis of slope stability, while the characterization of spatial variability and uncertainty is through the CRF method. For different geotechnical contexts, parametric sensitivity analyses will be carried out with different emphases. Relevant conclusions are of considerable practical reference value and academic significance. The background and significance of each research emphasis will be specified in the following chapters.

1.2 Literature review

In most existing studies on geotechnical engineering, it is assumed that the properties of geotechnical materials are deterministic without random spatial variability. However, it’s logical to be convinced of their spatial variability and randomness due to various factors during the formation process and property estimation process of geotechnical materials, which is also proved by plenty of in-situ measurements and laboratory tests (Chiasson et al. 1995; Lumb 1966; Phoon and Kulhawy 1999a). To address the influence of such probabilistic characters, fundamental early works appeared in the 1970s (Matsuo and Kuroda 1974; Alonso 1976; Tang et al. 1976) and have developed continuously. Vanmarcke (1977, 1983) proposed and developed the random field theory to characterize the spatial variability of geotechnical materials, by which the correlation of a soil property at different locations (i.e. autocorrelation) was incorporated rationally.

3 To carry out the reliability-based assessment using  and Pf , several methods are prevailing over others, such as FOSM, FORM, MCS, the subset simulation (SS), the response surface method (RSM). The basic theories and developments of these methods are introduced concisely herein.

1.2.1 Analytical probabilistic analysis of slope stability

(1) First-order second-moment method (FOSM)

FOSM is a relatively simple method for performing probabilistic slope stability analysis. It requires a deterministic analytical model (performance function) for slope stability calculation, which is differentiable for all uncertain variables involved in the model. Assume that g(x) denotes the performance function of slope stability used to calculate FS , in which x  (x1, x2 ,..., xn ) is a set of random variables representing uncertain model parameters in the performance function, with mean values i ,i 1,2,..., n . Consider, for example, that FS is normally distributed. By FOSM, the mean value FS and the standard deviation  FS of FS are calculated as (Cao et al. 2017)

FS  g(1, 2 ,..., n ) (1.1)

2 n  g  n n g g    2       FS  i    ij i j (1.2) i1  xi  i1 ji xi x j

in which  i ,i 1,2,..., n are standard deviations of random variables xi ,i  1,2,..., n , ij is the correlation coefficient between two different uncertain parameters xi and x j .

The reliability index  is defined as

 1   FS (1.3)  FS

Subsequently, the slope failure probability Pf is calculated as

Pf  1 ( ) (1.4) in which Φ is the cumulative distribution function of a standard Gaussian random variable (Baecher and Christian 2003). In addition, FOSM usually uses a predefined critical slip surface of slope failure and does not account for uncertainties in slope failure mode.

After the early exploration of the calculation formulation, the application of FOSM to slope reliability analysis was investigated continuously. Hassan and Wolff (2000) discussed the application of three different kinds of slice methods along with mean-value FOSM and advanced

4 FOSM. Gui et al. (2000) incorporated the effect of stochastic hydraulic conductivity and discussed different boundary conditions using FOSM. Farah et al. (2011) employed FOSM in the comparison analysis between RFEM (random finite element method) and LEM (Bishops simplified method) and performed an optimization strategy to look for the critical slip surface.

(2) First-order reliability method (FORM)

By FORM, the reliability index  is interpreted as a measure of the distance between the peak of the multivariate distribution of the uncertain variables (the joint distribution of x  (x1, x2 ,..., xn ) ) and the critical point on the failure boundary in a dimensionless space. It is calculated as (Hasofer and Lind 1974; Ang and Tang 1984; Low and Tang 1997; Baecher and Christian 2003)

T  x    1  x      min      (1.5) xiF      

in which F represents the failure domain;   (1, 2 ,..., n ) is a mean vector of uncertain variables;   (1, 2 ,..., n ) is a standard deviation vector of uncertain variables; and  is the correlation matrix of uncertain variables. Note that the equivalent mean values and standard deviations of random variables should be used when uncertain variables are not normally distributed (Ang and Tang 1984).

Eq. (1.5) has been implemented in a spreadsheet environment with the aid of the built-in optimization tool “Solver” for probabilistic slope stability analysis by Low and Tang (1997), Low et al. (1998). Center coordinates and radius of circular slip surfaces were also considered as additional optimization variables by Low (2003) so that variation of potential critical slip surfaces was implicitly factored in the analysis. However, because of the limitation of the optimization tool used,

FORM tends to overestimate  and underestimate Pf (Wang et al. 2009).

Then there were some attempts to improve FORM, such as combining two types of FORM (Low et al. 2011), employing different optimization methods (Tang 2012; Tun 2016, 2018), considering spatial variability by using “equivalent parameters” (Li X.Y. et al. 2017), simplifying FORM so that neither transformation of correlated random variables nor optimization tools were required (Ji et al. 2018). Salloum et al. (2011) considered seismic effect using upper-bound limit analysis. Metya and Bhattacharya (2014, 2016a, 2016b) used Sequential Quadratic Programming (SQP) to search the critical slip surface by traditional slices methods. System reliability analysis considering multiple failure modes was performed (Low et al. 2011; Xiao et al. 2017).

5 1.2.2 Probabilistic analysis based on the random field theory

(1) Monte Carlo simulation (MCS) method

MCS is a numerical process of repeatedly calculating a mathematical or empirical operator, in which the variables within the operator are random or contain uncertainties with prescribed probability distributions (Ang and Tang 2007; Wang 2011; Wang et al. 2011). The result from each repetition of the numerical process is treated as a sample of the operator’s true solution, analogous to a sample observed in a physical experiment. For every realization of the random sample based on the prescribed probability distributions, the deterministic mathematical operator g(x) is used to calculate one possible value of FS . Then enough plenty of simulations are carried out to obtain numerous possible values of FS or other objective output indexes so that some related statistical analysis can be performed to estimate probabilistic results like Pf and  .

Some comparative analysis of MCS with FORM (Griffiths et al. 2008; Ji et al. 2012) and FOSM (El-Ramly et al. 2002; Jha 2015) demonstrated its superiority. It can be performed together with various types of deterministic methods, such as limit equilibrium methods (LEM), finite element method (FEM), and direct analytical methods for simple failure modes, etc.

Griffiths and Fenton (2000) presented a nonlinear finite element program for slope reliability analysis. Then Griffiths and Fenton (2004), Griffiths et al. (2007, 2009a), Huang et al. (2010, 2013a) further developed the random finite element method (RFEM) and rationalized its superiority and accuracy by comparison analysis with FORM. Unlike FORM, RFEM can take into account spatial correlation rigorously and allow slope failure mode to develop naturally along the least resistance path. Also, RFEM tends to provide a more conservative result on slope reliability estimation compared with FORM. Hicks and Samy (2002) preliminarily researched the influence of random field’s statistics on slope reliability. Le et al. (2012, 2019) and Tang et al. (2018) incorporated the random variation of seepage parameters and of the rainfall pattern respectively. Luo and Bathurst (2017) used on slope crest as an index to assess slope reliability and they (2016, 2018) also investigated the reliability of geosynthetic-reinforced slopes. As for the application of 3D models, Griffiths et al. (2009b) extended the RFEM code to 3D condition and compared 2D and 3D RFEM; Hicks and Spencer (2010), Hicks et al. (2014), Li Y.J. et al. (2015) discussed three types of failure modes with different horizontal correlation lengths and compared the results with earlier literature; Liu Y. et al. (2018) investigated the most pessimistic cross-section among the 3D slope to ensure a conservative result.

Cho (2007) tried to look for the critical failure surface and then carried out probabilistic slope stability analysis by Spencer’s LEM. Later, more research was implemented about the search of

6 critical failure surface and selection of representative slip surfaces (Zhang et al. 2011a; Li L. et al. 2013, 2014; Chu et al. 2015). Dayal et al. (2011) and Chen and Chang (2011) investigated practical engineering cases by software SLOPE/W considering specific multiple practical conditions. Still along with LEM, a different assessment procedure (Salgado and Kim 2014) and some advancements on MCS (Javankhoshdel and Bathurst 2014; Li D.Q. et al. 2015a; Mojtahedi et al. 2018) were proposed concerning the robustness and efficiency. Meanwhile, the reliability of both FEM and LEM was verified by the comparison between them (deWolfe et al. 2011; Javankhoshdel et al. 2017; Amoushahi et al. 2018).

When it comes to some particular types of slope models, such as rock slope (Gravanis et al. 2014; Pantelidis et al. 2015; Zhao et al. 2016) and infinite slope (Griffiths et al. 2011; Li D.Q. et al. 2014; Prakash et al. 2019), the failure mode is assumed to be planar, so that FS is available through some direct analytical calculation. Then corresponding probabilistic analysis can be carried out accordingly. Furthermore, Srivastava and Sivakumar Babu (2008) and Pandit et al. (2018) applied the finite differential method (FDM) to some numerical parametric analysis and a project case respectively.

(2) Subset Simulation (SS)

However, MCS to some extent suffers from a lack of resolution and efficiency, especially at small probability levels. To address this, an advanced MCS variant called “subset simulation” (SS) has been used for probabilistic slope stability analysis to improve MCS (Au et al. 2010). Subset simulation takes advantage of conditional probability and Markov Chain Monte Carlo Simulation (MCMCS) method to compute small tail probability efficiently (Au and Beck 2001, 2003). In other words, it amplifies the sample section around the target failure region to improve efficiency and resolution. It expresses a small probability event as a sequence of intermediate events with larger conditional probability and employs specially designed Markov Chains to generate conditional samples of these intermediate events until the final target failure region is achieved.

The application of SS was well developed in recent years. Li D.Q. et al. (2016a, 2017) optimized traditional combination of RFEM and MCS by SS with small failure probability and generalized it by selecting representative failure events. Furthermore, Li D.Q. et al. (2016b) and Xiao et al. (2016) carried out SS on a two-layered slope example combining FEM and LEM and extended it to 3D conditions. Huang et al. (2017) presented a SS method based on RFEM that is free of the calculation of every single FS , resulting in higher efficiency and lower accuracy. Jiang et al. (2017) systematically analyzed correlated planar failure modes of rock slopes. Yuan et al. (2019) took into account random rainfall processes and the saturated hydraulic conductivity when studying an infinite slope.

7 (3) Response surface method (RSM)

The idea of response surface method (RSM) is originally proposed by Box and Wilson (1951) as a statistical technique to determine the optimum conditions for chemical research. Later on, as an independent step supplementing and improving the aforementioned reliability methods, RSM has been applied to slope reliability problems with implicit performance functions. By conducting a sequence of designed experiments, the RSM uses a computationally efficient model to approximate the original analysis model obtained from methods like LEM or FEM. To explicitly construct the relationship between the response and the input variables, a simple high order polynomial function (Wong 1985; Xu and Low 2006; Li L. and Chu 2015a) or a Hermite polynomial chaos expansion (Li D.Q. et al. 2011a; Jiang et al. 2014) may be used.

The advantage of RSM is that an approximate explicit performance function can be used repeatedly in a structural response analysis in a timely fashion, effectively avoiding the computationally expensive process of implementing a detailed analysis by, for example, the nonlinear FEM. However, this also means that the accuracy of this method depends on how well the original performance function is approximated (Li Y.J. 2017).

Much attention was paid to multiple RSM considering various slip surfaces in multi-layered slopes. For example, RFEM (Li D.Q. et al. 2015b), LEM (Jiang et al. 2015), FDM (Li D.Q. et al. 2015c) were applied to address this issue respectively; Jiang and Huang (2016) additionally performed SS for small probability condition; Liu et al. (2018a, 2018b) tried to simplify RSM by a variance reduction technique; Li L. and Chu (2015a, 2015b) compared single RSM and multiple RSM with an in-house LEM code. Meanwhile, the explicit performance function was constructed in different, more efficient ways, such as Kriging-based RSM (Zhang et al. 2013; Ma et al. 2017) and multivariate adaptive regression spline (MARS) based RSM (Liu and Cheng 2016; Liu et al. 2018c; Metya et al. 2017). Li D.Q. et al. (2016c) summarized four types of RSM and compared their accuracy and efficiency with direct MCS and FORM.

(4) Other research hotspots of reliability-based slope analysis and the potential of NLA

Except for the above prevailing methods, some other probabilistic analysis methods were proposed for estimate slope reliability, such as point estimate method (PEM) (Hassan and Wolff 2000; Ahmadabadi and Poisel 2015, 2016), Copula-based sampling method (CBSM) (Wu 2013; Xu and Zhou 2018), perturbation-spectral stochastic method (Mousavi Nezhad et al. 2011), jointly distributed random variable (JDRV) method (Johari et al. 2013; Johari and Khodaparast 2015; Johari and Mousavi 2018).

8 In the aforementioned review in accordance with research methods, some special engineering or natural conditions are involved. Besides that, there are some other studies on various influential factors, such as system analysis aiming at multi-layered slopes (Li D.Q. et al. 2011b), seismic effect (Yeh and Rahman 1998; Johari and Khodaparast 2015; Huang H.W. et al. 2018), random rainfall condition (Zhang et al. 2014; Dou et al. 2014; Zhang L. et al. 2018), spatial variety of parameters’ mean value (Li D.Q. et al. 2014; Zhu et al. 2017), the influence of piles (Zhang et al. 2017), etc.

The latest research tendency of reliability-based slope analysis concentrates on several hotspots. Firstly, researchers paid more attention to the failure mode for more systematic assessment of slope failure risk (Zhu et al. 2015; Chu et al. 2015; Li D.Q. et al. 2016a; Xiao et al. 2016; Qi and Li 2018), especially in 3D contexts (Hicks and Spencer 2010; Hicks et al. 2014). Secondly, due to the importance of characterizing soil properties, growing research focused on back analysis which is based on Bayesian Theorem (El-Ramly et al. 2005; Wang et al. 2010; Zhang et al. 2010, 2011b; Yang et al. 2018; Jiang et al. 2018a). The back analysis intends to find a set of model parameters that would result in the observed performance of geostructures. Thirdly, the application of conditional random field was proved to provide a more authentic slope model. The research status of CRF is introduced in the next Section in more detail.

Despite the numerous existing research achievements about probabilistic analysis of slope stability, the application of limit analysis method to this issue is pretty sparse, either at the analytical or numerical level. Pan et al. (2017) adopted analytical limit analysis to build the response surface based on sparse polynomial chaos expansions, in order to analyze 3D Hoek-Brown rock slopes. With established NLA programs, Huang et al. (2013b) assessed the risk (expected sliding mass) of a slope vulnerable to multiple failure modes; Kasama and Whittle (2016) discussed the failure modes systematically; Gomes et al. (2017) took into account the uncertainties in depth and soil hydraulic parameters for estimating a 3D slope; Brahmi et al. (2018) considered the load inclination when probabilistically estimating the bearing capacity of strip footings on slope crest. Due to its superiority over other methods, NLA deserves more consideration at the deterministic stage of probabilistic analysis. Later on, there will be a review on the development history of NLA, yet it mainly remains within the scope of deterministic analysis.

1.2.3 Probabilistic analysis based on CRF

As a highly effective improvement to the random field theory, the conditional random field (CRF) has been widely used in various geotechnical contexts including the slope reliability estimation. As mentioned earlier, when adopting the CRF method, both some measured data at certain sampling locations and the statistical characteristics of the soil property studied need to be taken into account. So the issue of data acquisition and processing is essential for the application of CRF. Namely, the

9 methods to determine statistical parameters based on the original data are worth exploring. Phoon and Kulhawy (1999a, 1999b) comprehensively specified the procedure of determining the statistical parameters from the measured data, with the three primary sources of geotechnical uncertainties all taken into consideration, namely inherent variability, measurement error, and transformation uncertainty. Fenton (1999) tried to model the random field based on CPT data. Uzielli et al. (2005) attempted to characterize normalized CPT parameters’ statistics using a finite-scale weakly stationary random field model. Lloret-Cabot et al. (2014) compared two methods of estimating the correlation length, the second one of which involved a new strategy combining information from CRFs with the first one. Gong et al. (2017) presented a framework for optimization of site investigation program, within which the robustness of the statistical characterization of geotechnical properties and the investigation effort are optimized. Cai et al. (2017) studied the vertical spatial variability of the soil behavior type index based on the database collected from testing sites of four different geologic formations. Jiang et al. (2018a) explored the application of the Bayesian Updating approach to learn the distribution of spatially varying soil properties and to update the slope reliability with multiple sources of test data. The technique for generating CRF is another key issue involved. The monograph of “Risk Assessment in Geotechnical Engineering” systematically introduced relative methodologies (Fenton and Griffiths 2008). Jiang et al. (2018b) proposed a simplified approach for generating CRFs of spatially varying soil properties in slope reliability analysis. In Namikawa (2016), a conditional multivariate normal distribution was used to express CRF generation.

There is also extensive comparative research between unconditional and conditional random field methods, verifying the superiority of CRF. Wu et al. (2009) discussed to what extent the CRF method improves the unconditional method, under different property variability conditions. The comparison in Kim and Sitar (2013) was between three approaches, namely the unconditional approach considering only the spatial variability, not the measurement-related uncertainties, the unconditional approach considering both, and the conditional approach. Liu et al. (2017) carried out a parametric analysis on soil properties’ statistics not only with a comparison between conditional and unconditional cases but also with the study on the effects of different layouts of sampling points on CRF simulation. Indeed, after confirming the superiority of CRF, the next related question is the efficiency of different sampling point layouts, namely the optimization of sampling scheme. To be specific, related studies tried to assess how the number, location, and interval of CPT affect the sampling efficiency (van den Eijnden 2010; Lloret-Cabot et al. 2012; Li Y.J. et al. 2016; Yang et al. 2017; Jiang et al. 2018b).

10 Among the above-mentioned literature, most CRF simulations were implemented on soil strength parameters, yet in some other works, the random field theory was used to simulate other variable parameters. These parameters include the permeability (Griffiths and Fenton 1997), the ore reserve (Frimpong and Achireko 1998), an original state parameter (Hicks and Onisiphorou 2005), three indexes used as indicators for decision-making in arriving at the preliminary founding depth of piles in Hong Kong (Dasaka and Zhang 2012), etc. Furthermore, some additional factors were taken into consideration combining with the inherent variability of soil properties, such as Haldar and Babu (2008) carrying out reliability estimations for pile foundations, Popescu et al. (2005b) investigating the effects of soil heterogeneity on the liquefaction potential subjected to seismic loading, and the geological uncertainty being simulated by a coupled Markov chain model in Deng et al. (2017) and Li D.Q. et al. (2016c).

Different soil property parameters usually exhibit some certain cross-correlation between them and such cross-correlation is sometimes taken into account in the literature. For example, Popescu et al. (2005b) considered the negative cross-correlation between two index soil properties, namely the overburden stress-normalized cone tip resistance and the index; Lo and Leung (2017), Liu et al. (2017) and Gong et al. (2018) all established the cross-correlated random fields of the cohesion and the frictional angle; Johari and Gholampour (2018) gave the regression correlation coefficients between some property parameters of unsaturated soil. Next, besides FS defining the slope stability, there are some other ways to describe the slope state, such as the settlement of slope face under certain load (Hicks and Onisiphorou 2005) and the total global uncertainty of the field (Lloret-Cabot et al. 2012), etc. In some of the above studies, the slope failure consequence or the spatial distribution of failure modes was investigated along with Pf and FS ’s distribution, such as Liu et al. (2017) and Johari and Gholampour (2018). Some other studies concerning CRF also exist, focusing on various research emphases. For instance, Li X.Y. et al. (2016) characterized the variability of geologic profiles by CRF, within which the nonstationary field was transformed into stationary one by removing a lower-order polynomial trend; Liu et al. (2017) and Lo and Leung (2017) respectively employed SS and RSM combined with the conditional simulation of random field, etc.

1.2.4 Development of the numerical limit analysis (NLA)

The limit analysis (LA) is always extensively employed in the research of multiple geotechnical fields including slope stability due to its calculative efficiency and conceptual soundness. Its realization at the numerical level, the numerical limit analysis (NLA), further exploits its application range, namely problems with heterogeneous soil profiles, anisotropic strength

11 characteristics, fissured soils, discontinuities, complicated boundary conditions, and complex loading (Sloan 2013).

Among the earliest proposals about NLA theory were Anderheggen and Knöpfel (1972) and Bottero et al. (1980). They proposed coarse linear programming to solve the simple plate-bending problem and soil mechanical problems respectively. Later on, Sloan et al. made a significant contribution to the development of NLA. Sloan (1989) established a whole theoretic framework for upper-bound NLA and presented plastic multiplier rates to describe the velocity field along with velocity vectors. Then this approach was improved by treating velocity discontinuities at all edges between elements as unknowns rather than setting shear directions of limit number of velocity discontinuities as prior (Sloan and Kleeman 1995). In later research, the former linear approximation of yield criterion was abandoned and non-linear programmings for both upper-bound (Lyamin and Sloan 2002a) and lower-bound (Lyamin and Sloan 2002b) NLA were established, among which a comprehensive set of constraints arose. Furthermore, this upper-bound method was formulated in terms of stresses based on duality theory and this modification brought better efficiency and applicability (Krabbenhoft et al. 2005; Yuan and Du 2018). When it came to the optimization procedure, second-order cone programming (SOCP) was proved suitable to this context rather than earlier active set algorithm and quasi-Newton method (Krabbenhoft et al. 2007). Sloan (2013) then made a summary of NLA theory and introduced its application to several common geotechnical problems like foundation, slope, , etc.

After the method of NLA was established, it was fully employed on deterministic slope stability analysis. Meanwhile, researchers tended to incorporate other environmental conditions and geotechnical contexts, such as (Chen et al. 2004), linearly varied strength parameter (Li A.J. 2009a), seismic effect (Li A.J. 2009b; Yang and Chi 2014; Belghali et al. 2017), two-layered slopes (Lim et al. 2015), blasting damage zones (Qian et al. 2017). For rock slope, a different element type (Durand et al. 2006) and the Hoek-Brown criterion (Li A.J. et al. 2008; Lim et al. 2017; Belghali et al. 2017) were adopted to seek a modification. The superiority of NLA was also indicated by some comparison with FEM (Tschuchnigg et al. 2015a, 2015b; Carrión et al. 2017) and analytical LA (Belghali et al. 2017). Nevertheless, as mentioned in Section 1.2.2, NLA was seldom used in the probabilistic analysis of slope stability.

12 1.3 Introduction to the theory

1.3.1 Random field theory

Generally speaking, there are two considerations involved in the formulation of the random field theory, namely “variability at one point” and “spatial dependence”. On one hand, at every single specific position, the parameter value is a variable governed by a certain probability density function (PDF) with relative independence, such as a Gaussian distribution. On the other hand, the values of two positions are not totally independent and in fact exhibit quite a correlation depending on their relative positions. Further, the joint PDF is independent of spatial position, that is, it depends just on relative positions of the points. This assumption, known as weak stationarity, implies that the marginal, or point, PDF is constant in space.

The first concern in the use of random field theory is to select a proper probability distribution for the field. Despite many available probability distributions exist, the lognormal distribution possesses some unique advantages and is widely employed in modelling the inherent properties of soils and rocks. An outstanding advantage of the lognormal distribution is its simplicity, because it is derived from a simple nonlinear transformation of the classic Gaussian distribution. Furthermore, the lognormal distribution guarantees that the random variable is always positive. Then for a specific case, we need the mean and the standard deviation to determine the distribution.

Secondly, it is the covariance function that governs the correlation pattern among the whole field, telling us how rapidly the field varies in space. There are several alternatives (Fenton and Griffiths 2008) with the covariance function usually being chosen. Besides the mean and the standard deviation, the covariance function also involves the correlation length (also sometimes known as “scale of fluctuation”). It is defined as a separation distance, within which the soil property shows a relatively strong correlation from point to point.

1.3.2 Conditional random field

The establishment of CRF is a process of simulating unmeasured spatial locations conditioned on the known values at measured locations. Namely, for generating CRF, both the actual measurements and their statistics are incorporated, instead of using only the statistics (e.g., mean, standard deviation, and correlation length) in unconditional simulations.

Since there are few applications of conditional simulation in slope reliability analysis, the comprehensive investigation of CRF is proposed as a main innovative point herein. Specifically, the thesis tries to demonstrate CRF’s superiority and necessity over unconditional analysis, to carry out parametric analyses on random field characteristics for different slope models, and to investigate

13 multiple evaluation indexes. In general, it’s aimed at proposing a more realistic and scientific slope reliability evaluation system.

For investigating parametric sensitivity in different geotechnical contexts at the theoretical level, the slope models under study are hypothetical with reference to the literature and related standards. As for the known data at measured locations, they are actually extracted from an unconditional simulation which is implemented based on the predefined statistics. That is to say, this unconditional random field is treated as the “real” field for reference.

1.3.3 Monte Carlo simulation

One primary goal of probabilistic analysis is to estimate means, variances, and probabilities associated with the response of complex systems to random inputs. While it is traditionally preferable to evaluate these probabilistic response by analytical methods (FORM, FOSM), where possible, systems defying analytical solutions often receive our attention. Relevant simulation techniques are ideal since they are conceptually simple and lead to direct results.

Consider, for example, probabilistic slope stability analysis in which the mathematical operator g(x) involves the calculation of FS and the judgment of failure occurrence ( FS 1). The direct MCS starts with the characterization of statistics of uncertainty properties involved, as well as the slope geometry and other necessary information, followed by generating nMC sets of random samples according to the prescribed conditions. Based on the deterministic model g(x) and the nMC samples, nMC possible values of FS can be obtained. Then, the statistical analysis is performed to estimate Pf and  . Also, MCS can be used to estimate the moments, mainly the mean value and the variance of the response.

MCS, however, not only lacks efficiency and resolution at small probability levels, as mentioned earlier, but also does not offer insights into the relative contributions of uncertainties of input parameters to the output response (Baecher and Christian 2003).

1.3.4 Numerical limit analysis and OptumG2 software

In this work, the deterministic stability evaluation for every single realization of random field is carried out by OptumG2 which is developed based on NLA theory. NLA is rooted in the concept of a discrete formulation just like the finite element method, so it is also called finite-element limit analysis (FELA). Based on the discretized velocity field or stress field, the formulations of the objective function, the unknowns to be optimized, and various constraints, are established according to the theorem of upper-bound and lower-bound limit analysis. Unlike the analytical formulation of limit analysis adopting relatively limited velocity fields or stress fields, NLA can search the optimal

14 solution and generate the critical field more freely. At last, the problem is abstracted to solving an optimization procedure.

(1) General limit analysis theory

The limit analysis is based on the plastic bounding theorems, which were developed by Drucker et al. (1951, 1952), and assumes a perfectly plastic material, small deformations, and an associated flow rule. The last assumption, which is usually termed the normality rule, implies that the plastic

p strain rates ij are normal to the yield surface f ( ij ) .

The lower-bound theorem requires the computation of a statically admissible stress field. Such a stress field means that it satisfies equilibrium equations, stress boundary conditions, and the yield criterion. For a perfectly plastic material model with an associated flow rule, it can be concluded that the load supported by a statically admissible stress field is a lower bound on the true limit load. By contrast, the upper-bound theorem is based on the principle aimed at a kinematically admissible velocity field satisfying the velocity boundary conditions and the plastic flow rule. For such a velocity field, an upper bound on the limit load is available by equating the power done by the external loads to the power dissipated internally by plastic deformation. Then the maximum lower bound and minimum upper bound solution can be determined by corresponding optimization procedures.

The key feature of the upper and lower bound theorems is that they facilitate the computation of rigorous bounds. Furthermore, some other principles, which reproduce the governing equations, are alternative. Rather than benefit the computation of rigorous bounds, these principles show a different advantage, that is they suggest various “compromise” solutions, which isn’t rigorously bounded but tend to be closer to the true solution than either of the rigorous upper and lower bound solutions. These principles are generally referred to as mixed ones and typically incorporate both the stresses and the velocities as primary variables.

(2) Discretization technology

The solid elements available in OptumG2 include not only elements leading to rigorous upper and lower bounds on the exact solution but also “mixed” elements that are often more accurate but do not result in rigorously bounded solutions.

With the satisfaction of all the aforementioned constraints, the obvious candidate for a lower-bound element is a triangle with a linear variation of the stresses between the corner nodes and a constant displacement throughout the element, as shown in Fig. 1.1a (see Lyamin and Sloan 2002a). As for

15 an upper-bound element, a triangle with quadratic interpolation of displacements and linear interpolation of stresses (Fig. 1.1b) is adoptable (see Lyamin and Sloan 2002b).

(a) (b)

Fig. 1.1 Lower-bound element and upper-bound element

For the numerical analysis in this work, a kind of mixed element named 15-node Gauss is adopted. It’s with quartic interpolation of velocities and cubic interpolation of stresses. Although not rigorous, the results produced by this element tend to converge from above.

(3) Finite element lower-bound formulation

A valid formulation of the lower-bound method is briefly presented below. Based on the formulation of Lyamin and Sloan (2002a), some important modifications are introduced to address stress discontinuities and highlight the fundamental differences between FELA and displacement finite-element analysis.

Fig. 1.2 Surface and body forces acting on soil mass

Fig. 1.2 exhibits a soil mass, with volume V and boundary area A , subject to fixed surface stresses (tractions) t acting on the boundary At , as well as an unknown set of tractions q acting

16 on the boundary Aq . In practice t and q might correspond to a predefined surcharge and an unknown bearing capacity, respectively. A set of fixed body forces g is also shown in Fig. 1.2, and unknown body forces h act over the volume V . The former is typically a prescribed unit weight, while the latter, which refers to an unknown body force capacity, is pretty useful in estimating the stability of slopes, and excavations.

Linear elements are employed to discretize the domain. Based on the corresponding mesh, a lower-bound calculation aims to find a statically admissible stress field σ that satisfies equilibrium throughout V , balances the prescribed tractions t on At , nowhere violates the yield criterion f so that f (σ)  0 , and maximizes the collapse load. In summary, the finite element lower-bound formulation can be abstracted to an optimization problem as below.

The objective function to be maximized is

T T c1 σq  c2 h (1.6)

where σq is the global vector of unknown nodal stresses along the boundary Aq corresponding to tractions q , c1 is a vector of constants transforming σq to the local vector of unknown tractions, c2 is a vector of element areas and h is the global vector of unknown element body loads.

The constraints are in accordance with conditions the stress field must satisfy.

T T T e e e Continuum equilibrium [B1 B2 B3 ]σ  (h  g ) (1.7a)

Discontinuity equilibrium Sσ a  Sσ b (1.7b)

Stress boundary conditions Sσt  t (1.7c)

Yield condition f (σ i )  0 (1.7d)

where the terms Bi are the standard strain-displacement (compatibility) matrices, the superscript e means each single element, S is a matrix transforming global vector of nodal stress to local one, σ a ,σ b are nodal stresses of pairs of adjacent nodes of neighbouring elements, the stresses for boundary nodes σt must match the prescribed surface tractions t , the yield condition is satisfied at every point in the domain if the inequality constraint f (σ i )  0 is imposed at each node i .

The unknowns to be optimized are the nodal stresses σ and element body loads h.

17 (4) Finite element upper-bound formulation

An efficient formulation of the upper-bound finite-element method is briefly described below. This follows the formulation developed by Lyamin and Sloan (2002b), with essential modifications for handling velocity discontinuities as introduced by Krabbenhoft et al. (2005).

Referring to Fig. 1.2 again, an upper-bound calculation seeks to find a velocity field u that satisfies compatibility, the flow rule and the velocity boundary conditions w on the surface area

Aw , and minimizes the internal power dissipation (due to plastic shearing) minus the power done by the fixed external loads. Then an upper bound on the limit load can be obtained by equating the optimized value of this difference to the power expended by the external loads. In summary, the finite element upper-bound formulation can be abstracted to an optimization problem as below.

The objective function to be minimized is

σ T Bu  cTu (1.8) where σ is a global vector of unknown element stresses, u is a global vector of unknown nodal

e T e T e velocities, B is the element compatibility matrix defined by Βi  A Bi , B   B is a global compatibility matrix, σ T Bu is the power dissipated by plastic shearing in the continuum and discontinuities, c T u is the rate of work done by fixed tractions and body forces.

The constraints are in accordance with conditions the velocity field must satisfy.

Flow rule conditions B eue   ef (σ e ) (1.9a)

Plastic multiplier times element area  e  0 (1.9b)

Consistency condition  e f (σ e )  0 (1.9c)

Velocity boundary conditions Tuw  w (1.9d)

Yield condition f (σ e )  0 (1.9e) where  e is the plastic multiplier  times the area for element e , f (σ e ) is the yield function for element e , T is a matrix transforming global vector of nodal velocity to local one, the velocities for boundary nodes uw must match the prescribed surface velocities w .

The unknowns to be optimized are nodal velocities u, element stresses σ e and plastic multipliers  .

18 (5) Strength reduction

The strength reduction analysis in OptumG2 is realized by calculating a strength reduction factor by which the material parameters need to be reduced in order to attain a state of incipient collapse. A factor greater than 1 thus implies a stable system while a factor less than 1 implies that additional strength is required to prevent collapse.

Depending on the model under investigation, one or more parameters are reduced in some particular way that leads to the value of the parameters actually causing collapse. For instance, for Mohr-Coulomb model, c and tan are reduced by the same amount. ( c and  refer to the cohesion and the internal frictional angle respectively.) This reduction strategy is consistent with the commonly used definition of the safety factor FS :

c tan FS   (1.10) cred tanred

In other words, the strength reduction factor is viewed as the safety factor. In this work, the safety factor FS is used to characterize slope stability on deterministic level.

The other reduction strategy related to our research is about Tresca model, which is carried out on the shear strength Su :

S FS  u (1.11) Su red

(6) Comparison between OptumG2 and other software

Prevailing existent calculation tools for slope stability analysis are based on LEM and FEM, such as SLOPE/W (LEM), (FEM), and various self-developed codes by researchers. However, as a practical, efficient, and accurate method, NLA is highly competitive compared with anyone of them. Also, OptumG2, which is developed based on NLA theory, possesses at least the following several advantages. Firstly, OptumG2 offers a more convenient and computationally efficient approach since it directly investigates the critical state of a given model without resorting to process-oriented or incremental analysis. Unlike the displacement-based finite-element method, it doesn’t require the simulation and interpretation of the complete nonlinear load-deformation response up to collapse (Popescu et al. 2005a). Secondly, OptumG2 can provide rigorous upper-bound (UB) and lower-bound (LB) solutions for various types of geotechnical problems, which bracket the true solution. In addition, this numerical tool also provides the option of 15-node triangular mixed element from Gauss family for FELA, which is adopted in this work. Thirdly, compared with LEM and analytical limit analysis, OptumG2 is of better applicability, so that it can

19 handle complex problems with irregular geometries, varying soil properties, loadings, and boundary conditions (Sloan 2013; Raj et al. 2018). Moreover, it has a technique of automatic mesh adaptivity which helps to obtain a more precise answer. In summary, the NLA software combines the power of finite element discretization for handling complicated soil conditions and the bounding capability of lower and upper bound plastic limit theorems.

Some comparative analyses in the literature between OptumCE and other methods for different geotechnical contexts validated its capacity and reliability. In Kasama and Whittle (2011), the comparison with FEM showed same qualitative features in bearing capacity estimation but suggested that the FEM calculations tend to overestimate the probability of failure at large correlation lengths. For the context of slope stability analysis, OptumCE (2019) and Yingchaloenkitkhajorn (2019) found that the OptumG2 results are generally in good agreement with the results by LEM programs (Slide, SVSLOPE, AutoSLOPE). Furthermore, there were several studies comparing OptumG2 with analytical limit analysis approaches, which validated both the numerical and analytical methods (Chwała and Puła 2020; Senent et al. 2020; Zhang Z. et al. 2018).

1.4 Methodology, research scope, and research objectives

The research background and significance of probabilistic analysis on slope stability by the CRF method have been stated earlier. The thesis topic is determined based on its significance and the research experience and interest of the author. Then the method framework and the overall research scheme are determined regarding the research status of this subject. As introduced in the Literature review, related research status includes widely used methods, various research priorities with corresponding significance, and limitations of existing studies. Meanwhile, the research scope’s originality and the methodology’s superiority and applicability also need consideration.

The overall methodology is presented below. The conditional random fields are generated by combining Covariance matrix decomposition (CMD) and the Kriging technique. The deterministic stability evaluation for every single realization of random field is carried out by OptumG2 which is developed based on the limit analysis theory. Then the direct Monte Carlo simulation (MCS) is realized by manually repeating the calculation of FS , leading to some target probabilistic indexes describing slope reliability.

The research scope covered by this thesis can be summarized as the following. Firstly, comparison analyses are carried out between an unconditional case and several conditional cases with different sampling schemes, by which we can find out the superiority and significance of CRF and the effect

20 of sampling points’ layout concerning the sampling efficiency and the slope reliability estimation. Secondly, parametric analyses based on different slope models are conducted to investigate the influence of soil properties’ statistical characteristics on the slope reliability, along with the form of CRF and the overall state of slope stability. The author also tries to illuminate the underlying mechanism of the influence law. In addition, there is a discussion about if considering both the failure probability and the failure consequence can amend the slope reliability estimation, compared with considering only the failure probability. At last, the sensitivity of the seepage situation to the stochastic permeability is studied. Implementing these research proposals with the above method framework is innovative and pioneering.

The objective of this thesis can be divided into three main aspects. Firstly, the author aims to propose a novel method framework for the probabilistic analysis of slope stability, including trying to improve the evaluation index of slope reliability. Secondly, the verification of CRF’s superiority and some guidelines for optimizing the sampling point setting can be expected. Thirdly, the main research content concerning the parametric analysis is supposed to provide some related conclusions with regularity. They are about the sensitivity of the slope state to the statistical characteristics of soil properties. In short, it is hopeful that the outcomes of this thesis have a certain guiding significance to the measurement point setting and the slope state estimation, especially the slope reliability estimation, in practice. Meanwhile, relevant influence laws and underlying mechanisms are of great theoretical significance.

21 Chapter 2 Sampling scheme selection of conditional random field

2.1 Introduction

The significance and rationality of the conditional random field (CRF) are already warranted by considerable studies and discussions. Logically speaking, by making full use of the available information, the conditional approach can provide more credible field simulations. Results of calculation and analysis also indicate that it is an effective tool for reducing the uncertainties in slope safety assessment. Here probabilistic analysis will be conducted for both conditional and unconditional approaches to further verify the superiority of CRF. Furthermore, with the same slope model, different conditional sampling schemes will be investigated for optimizing the sampling efficiency.

In the previous literature review, the deficiency of traditional unconditional random field method and the significance of CRF are already revealed (Deng et al. 2017; Liu et al. 2017; Lo and Leung 2017; Johari and Gholampour 2018; Jiang et al. 2018b). As for the layout of measurement points, most existing studies are in the form of CPT boreholes, and what we care about is their number, location, and interval (Li Y.J. et al. 2016; Yang et al. 2017; Lloret-Cabot et al. 2012; Jiang et al. 2018b). In some other studies, the measurement points are in the form of discrete single points and the extracted data are relatively limited (Liu et al. 2017; Gong et al. 2017). The sampling scheme discussion herein is inspired by these studies.

In this chapter, with a slope model in the literature, the distribution of FS , the failure probability

Pf and some other evaluation indexes will be calculated for the unconditional case and seven sampling schemes. By the comparison between condition and unconditional cases and the determination of the optimal scheme, we will find out to what extent the CRF method is superior to the unconditional method and how the layout (number, location, interval) of sampling points affects the sampling efficiency and the slope reliability estimation. Meanwhile, for clarity, the generation procedures for unconditional and conditional random fields will be introduced in detail, which are employed throughout the whole thesis.

2.2 Generation of random fields

2.2.1 Generation of unconditional random fields

Various random-field generator algorithms are available (Fenton and Griffiths 2008): Moving-average (MA) method, Discrete Fourier transform (DFT) method, Covariance matrix decomposition (CMD), Fast Fourier transform (FFT) method, Turning-bands method (TBM), Local

22 average subdivision (LAS) method, and so on. Among these methods, the most widely-used two are CMD and LAS. In my early programming attempts, both methods were programmed by the scientific calculation software of Mathematica, and CMD is finally chosen for this study considering its applicability and calculation efficiency. The procedures for both CMD and LAS are introduced as the following, and the Mathematica programs for CMD are exhibited later in this section.

2.2.1.1 Covariance matrix decomposition (CMD)

The implementation strategy for a random-field generator involves not only the algorithm itself but also some pre simplifying process. For example, the random field Y t defined by the transformation Y t  e X t  , will have a lognormal distribution if X t is normally distributed.

Suppose that X t has stationary mean  X and standard deviation  X at each point t . That is,

X t can be derived from the transformation X t   X  X Zt with Zt being standard-normally distributed. Moreover, its correlation function  X   is closely corresponding

CX   to its covariation function CX   as  X    2 , which characterizes the correlation between  X any two random variables at different points with a distance of  .

Of course the mean, standard deviation and covariance structure of the resulting field Y t is different, yet can be nonlinearly derived by following relational expressions.

1    2 X 2 X Y  e (2.1)

1 2  X   X 2 2  X  Y e e 1 (2.2)

 2   e X X 1     (2.3) Y  2 e X 1

In turn, for a lognormal random field to be generated, with given Y ,  Y , Y   (involving the correlation length lY ), corresponding terms for the normal random field X t are available through the reverse derivations of Eq. (2.1) to Eq. (2.3). In practice, the random-field generator is actually implemented on a standard normal random field Zt with zero mean, unit variance and

Z     X  .

Here CMD comes in as a direct method of generating stationary random fields in a pointwise manner. In the prescribed covariance structure CZ ti  t j   CZ  ij  , ti ,i 1,2,..., n are discrete

23 points in the field and  ij is the lag vector between the points ti and t j . If C is a positive definite covariance matrix with elements Cij  CZ  ij  , then the vector of correlated random variables Zi can be produced according to

Z  LU (2.4) where L is a lower triangular matrix satisfying LLT  C (typically obtained using Cholesky decomposition), and U is a vector of n -independent standard normal random variables. In particular, U is actually the source of uncertainty and diversity of the fields.

At last, the desired lognormal random field can be obtained through the transformation of

    Y t  e X X Z t . Then the repetitive operation of the generator will lead to amounts of random fields, meeting the need of MCS.

2.2.1.2 Local average subdivision (LAS)

The LAS process (Fenton 1990; Fenton and Vanmarcke 1990) proceeds in a top-down recursive manner. Initially, the random field domain is represented by a single cell. In 2D condition, it is then uniformly subdivided into 4 smaller cells of the same size. The child cells of the last stage are parent cells for the next stage. The subdivision process is carried out stage by stage, until the required subdivision level of the random field is attained.

Fig. 2.1 Local average subdivision in 2D

Fig. 2.1 shows the subdivision of a parent cell at an arbitrary location within a larger random field

i i i i domain. The parent cells shown in the figure are denoted as Z  {Z1 , Z2 ,..., Z9} and the child cells

24 i1 i1 i1 i1 i1 are denoted as Z  {Z1 , Z2 , Z3 , Z4 } . Fig. 2.1 only illustrates the subdivision of the parent

i cell Z5 . The algorithm will be presented here using vector notation (Fenton 1994).

i1 i1 i1 i1 i1 The values of Z  {Z1 , Z2 , Z3 , Z4 } are obtained by adding a random component to a weighted mean. The mean term is derived from a best linear unbiased estimate (BLUE) using a 3×3

i i i i neighborhood of the parent cell values, i.e. Z  {Z1 , Z2 ,..., Z9}. Hence,

Z i1  AT Z i  LU (2.5) where U is a random vector with independent standard normal elements.

The covariance matrices are defined as

T R  E[Z i Z i ] (2.6a)

T S  E[Z i Z i1 ] (2.6b)

T B  E[Z i1Z i1 ] (2.6c) where E[.] denotes the expectations which are derived using local average theory (i.e. variance reduction, see Vanmarcke 1977, 1983) over the cell domains.

The matrix A is determined by

A  R1S (2.7) while the lower triangular matrix L satisfies

LLT  B  S T A (2.8)

At the beginning of the LAS process, a random global average is prescribed. Then, the 4 smaller cells subdivided from the global domain should meet the requirement that local averages average to preserve the global or parent average value.

2.2.2 Generation of conditional random fields

As for taking known property values of specific sampling points into consideration while establishing random fields, the Kriging technique comes in. The general procedure of generating CRF in this work is presented as the following (Fenton and Griffiths 2008), which in fact combines CMD and Kriging technique.

When investigating the probabilistic nature of a particular model, we often have experimental information available that should be reflected in each simulation. For example, suppose we are

25 investigating the stability of a slope, and we have some soil property values known at certain spatial locations. The soil properties should only be random between the sample locations, becoming increasingly random with distance from the sample locations. It’s worth noting that all known data are standardized according to the transformation in Section 2.2.1 to match up the generation of standard normally distributed fields. Of course, reverse transformation is required at last to obtain the desired lognormal CRFs.

A random field which takes on certain known values at specific points in the field is a conditional random field, Zc t . It takes on specific values zt  at the sample locations t ,  1,2,..., nk , where nk is the number of sample locations. To accomplish the conditional simulation, the random field will be separated into two parts spatially: t ,  1,2,..., nk , being those points at which the random field takes on deterministic values zt , and t , 1,2,..., n  nk , being those points of the random field at which we wish to simulate realizations of their possible random values. The subscript  denotes known values, while  denotes unknown values to be simulated. n is the total number of points in the field to be simulated.

The conditional random field is simply formed from three components:

Zc t  Zu t[Zk t Z s t] (2.9)

where Zu t is an unconditional simulation, Zk t is the BLUE of field based on known values at t , Z s t is the BLUE of field based on unconditional simulation values at t . At each t ,

Zk t   zt  and Z s t   Zu t .

The unconditional simulation Zu t can be produced using the CMD method introduced in Section

2.2.1.1. The BLUE field Zk t and Z s t, are determined respectively by

nk Zk t     zt    (2.10)  1

nk Z s t      Zu t    (2.11)  1

for t (points at which the random field is still random), where  is the unconditional field mean at t ,  is the unconditional field mean at t ,  is weighting coefficients to be discussed.

The weighting coefficients,  , are determined from

26   C 1b (2.12)

where  is the vector of  , C is the nk  nk matrix of covariances between the unconditional random field values at the known points. The matrix C has components Cij  Cov[Zu ti , Zu t j ] for i, j 1,2,..., nk . Finally, b is a vector of length nk containing the covariances between the unconditional random field values at the known points and the prediction point. It has components b  Cov[Zu t , Zu t ] for  1,2,..., nk .

2.2.3 Mathematica programs for random field generation by CMD

(1) Generating program for unconditional random fields

27 (2) Generating program for the real field

(3) Generating program for conditional random fields (Scheme 7)

28 2.3 Problem definition

2.3.1 Basic model and sampling scheme definition

An undrained slope model with the value lattice is shown in Fig. 2.2, whose geometric and basic soil property parameters are set with reference to Yang et al. (2017). The undrained slope is inclined to the horizontal at angle 1  26.6 (2:1 slope), with height H1 10m , and soil unit weight

3  1  20.0kN / m , which are all held constant. The undrained shear strength Su is assumed to be lognormally distributed with the mean   50kPa and the standard deviation   25kPa . So Su Su the coefficient of variation is COV   /   0.5 . The spatial correlation length l is fixed at Su Su Su Su 10m and assumed to be equal in the horizontal and vertical directions in this chapter, i.e. l  l 10m . Meanwhile, a kind of theoretical autocorrelation function, the 2D Markov Su h Su v correlation function, is utilized for this slope model and defined as

  2 2 y   ,  exp x   (2.13) 1 x y  l l   Su h Su v 

where  x  xi  x j and  y  yi  y j are the distances between two points in the horizontal and vertical directions, respectively.

29 In CMD, the interval between adjacent value points is 0.5m, forming a 120*40 dot matrix. For one set of probabilistic analysis, 500 Monte Carlo runs will be conducted.

2H 2H CPT3

CPT1 H

CPT2 2H

H

1 20 60 90 120

6H

Fig. 2.2 Slope model for the random field of Su

Suppose three cone penetration tests (CPTs) have been conducted at the locations shown in Fig.2.2 and will provide site investigation data for CRF generation. For Su ’s unconditional random fields, with no investigation data taken into consideration, the generation will be performed following the aforementioned CMD procedure. For conditional random fields, the “measurements” of Su by the CPTs are extracted from an unconditional simulation with predefined statistical parameters. Namely, this unconditional random field is treated as the “real” field throughout the analysis process. Then the CRF generator using the Kriging technique can provide CRFs matching the known data at the locations.

To achieve the research objective of this chapter, eight sampling schemes are adopted for random field generation, i.e. Scheme0 (no CPT considered, unconditional), Scheme1 (conditioned on CPT1), Scheme2 (conditioned on CPT2), Scheme3 (conditioned on CPT3), Scheme4 (conditioned on CPT1 and CPT2), Scheme5 (conditioned on CPT1 and CPT3), Scheme6 (conditioned on CPT2 and CPT3), Scheme7 (conditioned on CPT1, CPT2, and CPT3).

2.3.2 Deterministic analysis based on the mean

The deterministic analysis based on the mean Su , 50kPa, is conducted for reference. The FS is calculated to be 1.45, which is close to the result in Yang et al. (2017), 1.47. The corresponding failure mode is shown as Fig. 2.3, which is in fact the shear dissipation distribution among the slope model while failure happens.

30 Fig. 2.3 Failure mode for deterministic analysis of the undrained slope

2.4 Field form analysis

Before the quantitative analysis on sampling efficiency and slope reliability, here some typical random fields and corresponding failure modes are exhibited to visually reflect the form of random fields and the conditioning effect of sampling points. The exhibition includes the unconditional case, the virtual real field, and three representative sampling schemes with different borehole numbers. Following the methods in Section 2.2, the random fields are generated in the form of a series of property values. Then they are imported into OptumG2 by “Map”, and the whole field is filled automatically by the software.

2.4.1 Typical unconditional random fields

Four typical random fields of Su for the unconditional case are shown in Fig. 2.4, and the corresponding failure modes in Fig. 2.5. To keep consistency and visual comparability, all the value ranges for random fields in this Section are set as consistent as possible, as shown by the color tape in Fig. 2.4. It can be seen that the forms of the four random fields are relatively stochastic with no apparent similarity. After all, they are generated with no conditioning and just following the predefined statistics.

The value of Su among the field is around the mean, 50kPa. Yet, the upper limit (about 120kPa) is further away from the mean than the lower limit (about 15kPa), which matches up with characteristics of the lognormal distribution. The color lumps in the random fields are the reflection of the correlation between points. Within the correlation length, the soil property shows a relatively strong correlation from point to point. Furthermore, because the horizontal and vertical correlation lengths are set the same, the shapes of color lumps don’t show apparent orientation, like longer horizontally than vertically, or vice versa.

31 (a) (b)

(c) (d)

Fig. 2.4 Typical unconditional random fields of Su

As for the four typical failures, they can’t reveal all possibilities of failure modes since the number is limited. But they still show various failure modes, such as different failure scales, toe-failure (Fig. 2.5c) and base-failure (Fig. 2.5a), and compound mode (Fig. 2.5d).

(a) (b)

(c) (d) Fig. 2.5 Typical failures of the unconditional case

2.4.2 Real field

The measured information for conditioning is extracted from the virtual real field shown in Fig. 2.6, which is generated through CMD as an ordinary unconditional random field. All the conditional random fields are conditioned on all the three CPTs or part of them. That is to say, only the sampling points in Fig. 2.7 actually affect the generation of CRFs. Of course, for different schemes,

32 acting CPTs are different. As mentioned in the last Section, the author tries to apply the Su value range of the real field figure to all the random field figures in this section.

The FS for real field is 1.11 and its failure mode is illustrated in Fig. 2.8.

Fig. 2.6 Real field of Su

Fig. 2.7 Information extracted from the real field of Su

Fig. 2.8 Failure mode for the real field of Su

33 2.4.3 Typical conditional random fields for different sampling schemes

For investigating the random fields and failure modes of conditional cases, Scheme1 (CPT1), Scheme4 (CPT1, 2), and Scheme7 (CPT 1, 2, 3) are selected as represents to study the effect of sampling points’ number. It can be seen from Fig. 2.9, Fig. 2.11, and Fig. 2.13 that, no matter for which scheme, the property values remain consistent with the real field at all sampling points. (Slight differences in color are due to the slight discordance of value ranges.) At prediction locations, the property values still show some uncertainty.

By the comparison between random fields in this Section and those in Fig. 2.4, the four random fields for conditional cases share more similarities than the unconditional case, especially at the locations around the sampling points. Furthermore, as the number of boreholes rises from 1 to 3, the conditioning effect of known data gets stronger, the field forms become closer to each other and more consistent with the real field. This demonstrates that with the adoption of conditional simulation and with more conditioning information considered, the uncertainty of soil property’s value field is reduced and the random fields become closer to reality. Such improvement is supposed to benefit the slope reliability analysis.

The failure modes for all cases (Fig. 2.10, Fig. 2.12, Fig. 2.14) exhibit considerable diversity. Not only the failure scales are changing, but also the failure modes contain different varieties, such as some compound failures with more than one failure surfaces and some local shear failures around the slope toe.

(a) (b)

(c) (d)

Fig. 2.9 Typical conditional random fields of Su for Scheme1

34 (a) (b)

(c) (d) Fig. 2.10 Typical failures of Scheme1

(a) (b)

(c) (d)

Fig. 2.11 Typical conditional random fields of Su for Scheme4

(a) (b)

(c) (d) Fig. 2.12 Typical failures of Scheme4

35 (a) (b)

(c) (d)

Fig. 2.13 Typical conditional random fields of Su for Scheme7

(a) (b)

(c) (d) Fig. 2.14 Typical failures of Scheme7

2.5 Sampling efficiency and slope reliability

In this section, the influences of considering known data and employing different sampling schemes on the evaluation of sampling efficiency and slope reliability will be under discussion. For each scheme, the distribution of FS is available through MCS, as well as FS ’s mean value ( FS ), coefficient of variation ( COVFS ) and the failure probability ( Pf ). The reliability index  has a relation with Pf as Pf 1   , where  symbolizes the cumulative distribution function of the standard normal distribution.

36 Two other indexes, the sampling efficiency index ( I se ) and the total global uncertainty (TGU) reduction, can also reflect the efficiency of a sampling scheme. (See Li Y.J. et al. 2016 and

Lloret-Cabot et al. 2012) I se is defined as

 FSu I se  (2.14)  FSc

where  FSu is the standard deviation of FS for the unconditional simulation, and  FSc is for the conditional simulation under consideration.

TGU means the global uncertainty for the domain under investigation. It is defined as the integral of the variance of the soil property under study over the domain. For the example in this chapter, it can be calculated as the sum of Su ’s variances for all value points in the slope domain. Then the TGU reduction here is the reduction relative to the TGU of unconditional case. So it can be expressed as

 TGU c  1 100% (2.15)  TGU u 

where TGU u is the TGU for the unconditional simulation, and TGU c is for the conditional simulation under consideration.

All related results are put together in Table 2.1 for the sake of analysis.

For analyzing the sampling efficiency of different schemes, the most acting indexes should be

COVFS , I se , and TGU reduction. Generally speaking, as the CPT number rises from 0 to 3, both

I se and TGU reduction go up, while COVFS decreases. Such changes mean that with more field information under consideration, the uncertainty of random fields and therewith the variability of FS are reduced, as well the sampling efficiency is practically enhanced. The above law is also in good agreement with the field form analysis in the last section. When comparing schemes with the same borehole number, like Scheme1, 2, 3 or Scheme4, 5, 6, it’s found that there is also some kind of tendency. It can be told from Table 2.1 that the positive roles of the three CPTs in this example decrease in the order of CPT1, CPT2, CPT3. According to COVFS and I se , the sampling efficiency of Scheme4 is even pretty close to Scheme7 and obviously higher than Scheme 5 and 6. That is, if only one CPT is available, CPT1 is better than CPT2 and then CPT3. Such kind of difference mainly depends on to what extent the sampling points are involved in the failure occurring zone. In this slope model, for most failures, CPT1 is more related to the failure mode, and thus its conditioning effect is more influential. In this sense, CPT2 is not as important as CPT1, and CPT3 is even less.

37 Table 2.1 Resulting indexes of various schemes

Scheme Scheme Scheme Scheme Scheme Scheme Scheme Case Unconditional 1 2 3 4 5 6 7

Pf (%) 12.6 27.4 19.8 11.4 20.2 11 6.8 7.2

 1.146 0.601 0.849 1.206 0.834 1.227 1.491 1.461

Mean of FS 1.210 1.092 1.147 1.223 1.109 1.173 1.243 1.164

Standard

deviation of 0.183 0.159 0.162 0.177 0.122 0.144 0.159 0.119 FS

COV of FS 15.14 14.61 14.13 14.50 10.97 12.27 12.78 10.22 (%)

I se 1 1.149 1.131 1.033 1.507 1.274 1.153 1.540

TGU (106 kPa2) 2.272 1.882 2.043 2.060 1.778 1.812 1.957 1.635

TGU reduction 0 17.19 10.11 9.32 21.77 20.26 13.86 28.03 (%)

When it comes to the mean of FS , there seems to be no apparent trend along with changing sampling scheme. Except that FS for unconditional case (1.210) perfectly matches the literature reference (Yang et al. 2017), the results for conditional cases are totally different. This is quite understandable because the whole FS level for a set of CRF analysis largely depends on the real field, whose selection is largely stochastic. So FS for conditional cases is incomparable from work to work. Here comes another question that if the extent to which FS ’s distribution centers on the real field’s FS should be a key criterion on judging the quality of a sampling scheme. The answer given here is no, or not very much. As aforementioned, not all information of the real field is extracted and works in the generation of CRFs. So if the soil property level of some sampling points is biased, it’s possible that the result is even farther from real field’s stability level when considering these points. And the level of sampling points is pretty random, uncontrollable, and thus possible to be highly biased. Although a scheme with more information considered doesn’t necessarily mean that the whole FS distribution centers on the real field’s FS more, the corresponding overall tendency is still undoubted. So it doesn’t contradict the analysis in the last

38 paragraph. Theoretically speaking, such kind of doubt only exists when sampling points take small portion of the field, and if the portion is large enough that the sampling points’ level can represent the real field’s level, there won’t be doubt. However, either in this study or in all related literature, the sampling points can only be small part of the field. Moreover, in practice, the in-situ measurements are limited due to the cost and practicability, and the real field for practical engineering is in fact unknowable. So it’s more rational to expect the CRFs’ level to center around the sampling points’ level than the real field’s level. Like in this example, the FS rises from Scheme1 to Scheme2 then Scheme3, as well as from Scheme 4 to Scheme5 then Scheme6. Such differences are because of the different Su levels of CPT1, CPT2 and CPT3, which are apparent in

Fig. 2.7. As for Pf , it is essentially the outcome of FS and COVFS . It’s easy to understand that if the majority of FS ’s distribution is greater than 1, Pf is positively correlated to COVFS and negatively correlated to FS . As a result, among the seven conditional cases, Pf decreases with rising borehole number roughly, while when comparing results within the same borehole number, the effect of FS on Pf is greater than COVFS .

Fig. 2.15 Probability density functions of FS for different schemes

The whole FS distributions for representative schemes (the unconditional case and Scheme1, 4, 7) are illustrated in Fig. 2.15, from which we can directly observe the differences on the statistics of

39 FS and on Pf . It’s obvious that with more CPTs taken into account, the shape of FS ’s PDF turns slimmer and taller, which indicates that the uncertainty in the estimation is significantly reduced. With the unstable mean value of FS in addition, Pf changes accordingly. What’s more, the PDF of FS stretches further towards the positive direction, which is similar to the characteristic of the lognormal distribution.

In summary, the application of CRF does make sense for the simulation of slope model. Meanwhile, in order to enhance the sampling efficiency of the sampling scheme and reduce the uncertainty in the estimation of slope stability, it’s better to increase the number of measurement points and set them at the locations where the failure tends to occur, such as beneath the slope face, around the slope toe or the slope crest. In addition, within the highly influential zone, there should be some distance between the measurement points to capture more geotechnical information. In the calculation example of this chapter, Scheme7 is the optimal one and will be adopted in the parametric analysis of the next chapter.

2.6 Conclusions

In the chapter, the generation procedures for unconditional and conditional random fields are introduced at first. Then based on an undrained slope model, the superiority of CRF and the performances of various sampling schemes are investigated by observing typical random fields, comparing the sampling efficiency, and discussing the slope reliability estimation. Some key points can be concluded.

(1) The statistical characteristics of soil property can be reflected by some features of typical random fields. With increasing boreholes taken into account, the uncertainty in the random field of soil property is reduced, which can be proved by field forms becoming closer and the TGU going down. Meanwhile, the failure modes for one set of random fields show high diversity.

(2) The sampling efficiency goes up when incorporating more sampling points. As for reducing the uncertainty in the estimation of slope stability, considering more conditioning information also plays a strongly positive role. Furthermore, analysis within the same borehole number indicates that setting the sampling points at more influential locations, i.e. the locations where the failure is more likely to occur, will benefit the slope reliability estimation.

(3) The superiority of the CRF method over the unconditional method is confirmed.

(4) The global stability level of one set of conditional random fields is not necessarily around the level of their real field. So the mean of FS is not stable when the sampling points under

40 consideration change. The failure probability is an outcome of the distribution of FS , which is influenced by both the mean and COV of FS .

41 Chapter 3 Conditional random field analysis on an undrained slope

3.1 Introduction

With the verification of CRF’s rationality and a relatively superior sampling scheme offered by the previous chapter, a comprehensive parametric analysis using the CRF method will be carried out in the chapter, based on the same undrained slope model. When researching some soil property at the probabilistic level by random field theory, its value distribution over the whole field is characterized by some statistical parameters. The parametric analyses involving these parameters can reveal the sensitivity of slope reliability to the random field’s statistical characteristics, which speaks volumes for the sake of both theoretical understanding and practical reference.

For the undrained soil, the shear strength ( Su ) is the most essential strength parameter, and there have been many random field analyses conducted on it, such as Hicks and Samy (2002), Wu et al. (2009), Kim and Sitar (2013), Yang et al. (2017), Jiang et al. (2018b). As introduced in Section 1.2.3, a big part of them involves the application of CRF. However, there hasn’t been a comprehensive parametric analysis to study the influence of the statistical parameters of Su ’s CRF on the slope reliability. Furthermore, the limit analysis method is seldom used to calculate single FS in existing related research.

In this chapter, parametric sensitivity analysis is carried out on the correlation length, the ratio between horizontal and vertical correlation lengths, and the coefficient of variation of the undrained shear strength ( Su ). The overall state of slope stability is reflected by the distribution of FS , including the average level shown by the mean of FS and the variability shown by COV of FS.

Then the FS distribution will result in the failure probability ( Pf ) to define the slope reliability. What’s more, the forms of typical random fields under different parameter settings will be comparatively analyzed to help understand the mechanism behind the slope reliability’s sensitivity to statistical parameters. At last, another evaluation criterion of slope reliability named “risk” will be discussed, by which the failure consequence can be taken into account in addition to the failure probability. Since the extent of damage is different from failure to failure, maybe Pf can’t fully reflect the slope reliability by itself. Such extended study can explore the possibility of improving the slope reliability evaluation criterion.

42 3.2 Problem definition

3.2.1 Parametric analysis definition

The undrained slope model in the previous chapter is also employed here with the optimal sampling scheme (Scheme7). So in this chapter, all analysis is based on the CRF method. The slope geometry, soil unit weight, and the value lattice are set the same as Fig. 2.2. Su is still assumed to be lognormally distributed, governing by the 2D Markov correlation function. Meanwhile, the statistical parameters for Su will serve as the ones under investigation in the parametric analysis, including its mean value  , coefficient of variation COV   /  , correlation length l , Su Su Su Su Su ratio between horizontal and vertical correlation lengths   l / l . Su Su h Su v

As for the resulting indexes, the distribution of FS , as well as its mean value and COV, can still be used to reflect the overall state of slope stability. Then the slope reliability is characterized by the failure probability Pf and the reliability index  . After that, to consider both the failure probability and the failure consequence when estimating the slope reliability, another index, the

“risk” ( Ri ), will be taken into discussion. The detailed definitions of the failure consequence and the “risk” are presented later.

Firstly to analyze the effect of the correlation length, with setting COV  0.5 and  1 , the Su Su value of l will take 0.3H , 0.5H , 0.7H , 1.0H , 1.5H respectively on the premise of Su v 1 1 1 1 1  taking 40, 45, 50, 55 (kPa). The other two sets of parametric analysis are based on the same Su values of  . One is about changing  (3, 5, 7, 10, 15) with COV  0.5 and l  0.1H , Su Su Su Su v 1 and the other changing COV (0.3, 0.4, 0.5, 0.6, 0.7) with  1 and l  0.3H . Su Su Su v 1

3.2.2 Deterministic analysis based on various mean values

The deterministic analyses based on various mean values of Su are conducted for reference. The safety factors for Su taking 40, 45, 50, 55 (kPa) are 1.16, 1.31, 1.45, 1.60, respectively. The corresponding failure modes are shown in Fig. 3.1.

43 (a) Su = 40kPa (b) Su = 45kPa

(c) Su = 50kPa (d) Su = 55kPa Fig. 3.1 Failure modes for deterministic analyses of the undrained slope

3.3 Real fields

For different parameter settings, the real field of course can’t remain unchanged. In this section, the forms of real fields for various parameter settings will be exhibited and comparatively discussed.

Here an interesting dialectical relationship about the treatment of real fields that shouldn’t be overlooked is real fields' difference and consistency. On one hand, for a set of soil property statistics, the corresponding real field is essentially one unconditional random field governed by the given statistics. Naturally, different statistical parameter settings would yield different real fields, thus resulting in different sets of known data. Nevertheless, on the other hand, all real fields involved in the parametric sensitivity discussion should maintain some consistent pattern to avoid the influence brought by the inherent uncertainty of real field generation. Since for one parameter setting, all CRFs are conditioned on some information extracted from the real field, the uncertainty in the form of real field will affect the probabilistic results about slope reliability to some extent. With such kind of influence, the rationality and effectiveness of the parametric analysis can’t be guaranteed. To achieve matching up all real fields with the same pattern, the independent standard-normally distributed vector U (see Section 2.2.1.1) for all real field generation programs should remain invariant.

Since few references concerning parametric sensitivity analyses based on CRF exist, the above treatment on real fields is largely original. Similar treatment can be found in Liu et al. (2017), yet wasn’t illuminated clearly.

44 In the parametric analysis, the first set of parameters is defined as the baseline case, which will offer a baseline real field and a corresponding vector Ub . In the following process of parametric analysis, the baseline vector Ub will be used to generate the real fields for other parameter settings. The baseline case for this analysis is   50kPa , COV  0.5 , l  l 10m . Su Su Su h Su v

Later in this section, four sets of real fields will be exhibited, with changing  , l ,  , and Su Su Su COV , respectively. We can observe the consistency in the pattern of the fields and the differences Su brought by the changing parameters.

 3.3.1 Real fields with changing Su

With fixed COV  0.5 ,  1, l  l  7m , the value of  is set as 45kPa, 50kPa, and Su Su Su h Su v Su

55kPa respectively for the three figures in Fig. 3.2. Naturally, the overall level of Su among the whole field increases with  rising up. (The observation of a field need to take the Su corresponding value range into consideration, which is different from field to field.) In the meantime, it’s also obvious that the whole patterns of the three fields are pretty similar, with close distributions of high-value zones and low-value zones.

l 3.3.2 Real fields with changing Su

With fixed   50kPa , COV  0.5 ,  1 , l is set as 3m, 7m, and 15m respectively for Su Su Su Su the three figures in Fig. 3.3. The overall patterns of the three real fields still maintain some certain similarities, yet the color lumps tend to become bigger from (a) to (c). An apparent example is the high-value zone at the top-left corner. That is because with a longer correlation length, the property value will exhibit a higher extent of spatial correlation, i.e. it will be correlated within a larger range in the field. The typical feature of a field with a longer correlation length is that the correlation zones tend to be farther extended.

 3.3.3 Real fields with changing Su

With fixed   50kPa , COV  0.5 , l  1m ,  is set as 3, 7, and 15 respectively for the Su Su Su v Su three figures in Fig. 3.4. It can be seen that because the fields are not anymore equally correlated in the horizontal and vertical directions, the similarity between these real fields turns less clear. Nevertheless, some features alike can still be captured, such as the high-value zones near the slope crest, near the bottom of CPT3, and at the bottom-right corner.

45 With fixed l and increasing l , the vertical dimension of correlation zones doesn’t change Su v Su h greatly, while the horizontal dimension gets larger so that the correlation zones become horizontally longer. In Fig. 3.4a, the color lumps are relatively small and the horizontal extensions are not that apparent. In Fig. 3.4b, the horizontal extensions of correlation zones indicate the unequal correlations in the horizontal and vertical directions. In Fig. 3.4c, such kind of development is further promoted.

COV 3.3.4 Real fields with changing Su

With fixed   50kPa ,  1, l  l  3m , COV is set as 0.3, 0.5, and 0.7 respectively Su Su Su h Su v Su for the three figures in Fig. 3.5. We can see that as expected the three real fields show considerable similarity in form. An evident difference between the three figures is about their value ranges, which proves their different degrees of internal deviation. With almost the same distribution of correlation zones, the greater COV is, the more apart the S values spread from the mean level, Su u i.e. the higher the overall deviation is.

The above analysis sheds some light on how the field form embodies the statistical characteristics, and further, how the statistical parameters affect the field form. Also, the efforts to keep the consistency of the pattern of real fields are verified. Next, the typical conditional random fields corresponding to these real fields will help to further elucidate the field form.

46 (a)   45kPa Su

(b)   50kPa Su

(c)   55kPa Su

Fig. 3.2 Real fields of S with different  u Su

47 (a) l  3m Su

(b) l  7m Su

(c) l 15m Su

Fig. 3.3 Real fields of S with different l u Su

48 (a)   3 Su

(b)   7 Su

(c)   15 Su

Fig. 3.4 Real fields of S with different  u Su

49 (a) COV  0.3 Su

(b) COV  0.5 Su

(c) COV  0.7 Su

Fig. 3.5 Real fields of S with different COV u Su

50 3.4 Field form analysis

The influence of  is intuitive and easy to understand, so related analysis won’t be carried out in Su this section. Except that, for each parameter setting in the last section, four typical conditional random fields are listed to verify the effects of statistical parameters on the form of CRFs at a more general level. More important is the parametric analysis aimed at the relationship between multiple CRFs of one parameter setting. The four CRFs will adopt the same value range as the corresponding real field does, and of course, the property values at sampling points are conditioned by the corresponding real field.

l 3.4.1 Typical conditional random fields with changing Su

The parameter setting is corresponding to Section 3.3.2. The forms of individual CRFs also follow the phenomenon presented in Section 3.3.2. That is, the correlation zones are larger with greater l . Su

What matters more is the relationship between the four fields. In Fig. 3.6a, the color lumps in figures are relatively small and evenly distributed. So although the field forms are pretty different from field to field, the influencing ranges of single correlation zones are limited, and thus the differences in form are less considerable from an overall perspective. In Fig. 3.6b, the field form still shows a high level of variability. Meanwhile, the differently distributed high-value and low-value correlation zones are of larger scales. The randomness in field form is strengthened by the greater influencing ranges of correlation zones. Indeed, visually speaking, the four figures in Fig. 3.6b are pretty diverse from each other. When it comes to Fig. 3.6c, with even larger correlation zones, the field forms don’t become more diverse. Instead, the four fields in Fig. 3.6c are pretty similar. The similarity can be told by the large high-value zone on the left and the low-value zone on the right. The opposite tendency is because of the conditioning effect of sampling points. With a pretty great correlation length, the known information can influence a broader area, so that the randomness of field form is weakened. In fact, some randomly deviated correlation zones still exist away from the CPTs, such as the bottom-left area in the third figure.

In summary, the effect of varying correlation length can be divided into two opposite parts. On one hand, a greater correlation length means larger randomly deviated correlation zones, thus strengthening the randomness in field form. Under such kind of effect, the variability of field form will be enhanced. On the other hand, the conditioning effect of the known data at sampling points could be strengthened by a greater correlation length. The field form of a broader area will converge so that there will be an opposite effect on the variability. The later kind of effect can’t be considered when using the unconditional random field method.

51 Conclusively, the overall effect of correlation length is not monotonic. The two kinds of effects are both reflected in Fig. 3.6 and will be further discussed later with reference to the slope reliability analysis.

 3.4.2 Typical conditional random fields with changing Su

The parameter setting is corresponding to Section 3.3.3. The forms of individual CRFs also follow the phenomenon presented in Section 3.3.3. That is, the correlation zones are horizontally longer with greater  . Su

The law about the relationship between CRFs is analogical to that of l . In essence, changing  Su Su with fixed l means changing l . So from Fig. 3.7a to Fig. 3.7c, with rising  , the two kinds Su v Su h Su of effects are both reflected to some extent. The consistency of field form brought by the conditioning effect of CPT data can be observed in the four figures of Fig. 3.7b around CPT2. Meanwhile, the four distinctive figures in Fig. 3.7c fully demonstrate the variability of field form caused by the randomly deviated and distributed correlation zones.

COV 3.4.3 Typical conditional random fields with changing Su

The parameter setting is corresponding to Section 3.3.4. The forms of individual CRFs also follow the phenomenon presented in Section 3.3.4. That is, the overall deviation of Su is higher with greater COV . Su

Through the comparison between the three sets of figures in Fig. 3.8, we can tell that the forms of CRFs turn more different with COV going up. The four figures in Fig. 3.8a show some Su unexpected similarity in form, which is indicated by the similar distributions of high-value and low-value zones. In contrast, the field form in Fig. 3.8b and Fig. 3.8c exhibits a high degree of diversity. While a few local similarities can still be found between some two fields in Fig. 3.8b, the four fields in Fig. 3.8c are totally different in form.

A possible explanation for this phenomenon is that, when COV gets greater, the deviation of S Su u in one field is higher and the range of all possibilities of field forms widens. Thus under the same level of conditioning effect, there tend to be more differences from field to field.

The above analysis on typical CRFs needs to combine with the later slope reliability analysis. The comprehensive consideration of them will help to understand the mechanism of the parametric sensitivities involved in this chapter.

52 (a) l  3m Su

(b) l  7m Su

(c) l 15m Su

Fig. 3.6 Typical conditional random fields of S with different l u Su

53 (a)   3 Su

(b)   7 Su

(c)   15 Su

Fig. 3.7 Typical conditional random fields of S with different  u Su

54 (a) COV  0.3 Su

(b) COV  0.5 Su

(c) COV  0.7 Su

Fig. 3.8 Typical conditional random fields of S with different COV u Su

55 3.5 Parametric analysis aimed at the slope reliability

After the field form analysis in the previous two sections, the core parametric analysis aimed at the slope reliability will be conducted in this section. The parameters under consideration and their value ranges are already presented in detail in Section 3.2. The influence of the mean of Su is explicit and predictable according to the common sense. So here taking different values of  as Su premises is for a more comprehensive and reliable investigation into the sensitivity to other statistical parameters.

To describe the result, several indexes are adopted. The coefficient of variation (COV) and the standard deviation ( FS ) of FS have a similar function, which is to define the variability of FS .

Here the result of COVFS is presented in charts for clarity. Meanwhile, the results of  FS and

FS are recorded in tables for reference. The failure probability Pf and the reliability index  both characterize the slope reliability, and there is an explicit relation between them (see Eq. (1.4)).

The difference is that Pf negatively characterizes the slope reliability, while  positively. Here

Pf is presented in charts and  in tables.

The entire presentation of the distributions of FS is not convenient. In this chapter, the 80% median interval of FS will be used to partly characterize the distribution. With dropping the highest 10% and lowest 10% values of FS , the left interval can characterize the variability of FS by its width and the overall level of FS by its position. The 80% median intervals can be shown in a form of strip chart, visually reflecting the FS distribution characteristics with varying parameters.

l 3.5.1 Parametric analysis on Su

According to the results in Table 3.1, the mean level of FS doesn’t change significantly with varying l , while it naturally gets higher with greater  . This demonstrates that the mean level Su Su of slope stability is not sensitive to the correlation length of Su . In this calculation example, the mean level of FS is below 1 for  taking 40kPa, and above 1 for the other three sets. Su

As shown in Fig. 3.9 and Table 3.2, from the perspective of COVFS , the variability of FS looks at the same level with different premises of  , while from the perspective of  , the variability Su FS of FS goes up with rising  . This is on the ground that COV takes the mean into account when Su defining variability. In the following analysis, COV will be taken as the primary index defining variability.

56 Table 3.1 Influence of l on  Su FS

l (m) Su 3 5 7 10 15  (kPa) Su 40 0.953 0.935 0.941 0.934 0.955

45 1.051 1.050 1.037 1.056 1.058

50 1.193 1.182 1.166 1.184 1.161

55 1.297 1.287 1.275 1.293 1.302

Table 3.2 Influence of l on  Su FS

l (m) Su 3 5 7 10 15  (kPa) Su 40 0.055 0.080 0.091 0.100 0.125

45 0.066 0.090 0.104 0.126 0.119

50 0.081 0.093 0.112 0.131 0.157

55 0.082 0.096 0.127 0.163 0.163

Fig. 3.9 Influence of l on COV Su FS

57 What we care about more is the effect of l , which is shown by the broken lines in Fig. 3.9. On Su the whole, COV tends to increase with rising l , except that for the sets of  taking 45kPa FS Su Su and 55kPa, COV changes in a contrary way with relatively great l (from 10m to 14m). This FS Su indicates that to some extent, the variability of slope stability becomes higher when the correlation length gets greater. Meanwhile, when the correlation length is great enough, its growth may result in lower slope stability variability. The underlying mechanism is the same with that mentioned in the field form analysis. That is, when l gets longer, the deviated values of S among the field Su u will affect the overall strength of slope to a greater extent, because the influencing ranges of randomly deviated correlation zones will be larger. Then both the field form and the slope stability will turn less consistent. Nevertheless, when l is long enough for the conditioning effect of Su sampling points to make a difference, a greater l will lead to a larger range of field partly Su determined by the known information, thus reducing the uncertainty of field form and slope stability. Both the analyses on the typical CRFs and the variability of slope stability reveal the two-sidedness of the effect of l . It’s notable that, in the parametric analysis based on Su unconditional random field method in the literature, the conditioning effect of known property values doesn’t exist, so the general conclusion is the variability of slope stability is simply positively related to the correlation length.

The effect of l on the overall state of slope stability can be comprehensively and visually Su reflected by the 80% median interval charts, as shown in Fig. 3.10. To avoid the unclarity brought by the graphics overlapping, the four strips in the chart are plotted separately. With the line of FS 1 plotted for reference, the enhancement of the overall level of FS with greater  can Su still be observed easily. From the observation of every single strip, we can tell that the overall heights of the strips don’t change a lot with rising l , while the widths increase at first then flatten Su out or fall slightly. Such phenomenons indicate the same law with the above analysis concerning the mean level and the variability of slope stability.

58 Fig. 3.10 Influence of l on the 80% median interval of FS Su

Fig. 3.11 Influence of l on P Su f Finally, the failure probability is the consequence of FS distribution and can be partly reflected by the 80% median interval charts. The general and inevitable rule is that, with the majority of FS

59 distribution below 1, Pf is negatively related to FS and COVFS , and with the majority of FS distribution above 1, Pf is negatively related to FS and positively related to COVFS . In the calculation example of this chapter, with reference to Fig. 3.11, the sensitivity of P to l is f Su developed accordingly. Specifically, for the set of   40kPa , with  1 , P decreases Su FS f when l gets longer, which is on the ground that  doesn’t change significantly and COV Su FS FS goes up. For the other three sets, also under the combined influence of FS and COVFS , with an increasing l , P goes up at first and then probably goes down or tends to be gentle. In addition, Su f the overall levels of P are separately distinguished according to the different values of  . f Su

Table 3.3 Influence of l on  Su

l (m) Su 3 5 7 10 15  (kPa) Su 40 -0.900 -0.793 -0.681 -0.589 -0.375

45 0.739 0.565 0.369 0.337 0.502

50 2.257 1.977 1.645 1.522 1.019

55 ∞ ∞ 2.257 1.852 1.881

Pf characterizes the slope reliability negatively, while  in Table 3.3 positively. The parametric sensitivity law won’t be repeated from the positive perspective of slope reliability here.

 3.5.2 Parametric analysis on Su

The influence mechanism of  is similar to that of l to a great extent. From Table 3.4, we can Su Su tell that with increasing  ,  has an inconspicuous falling trend, yet basically it remains Su FS stable. So the mean level of slope stability is not sensitive to  , either. Su

The influence of  on COV is also two-sided. When  is relatively great, its negative Su FS Su correlation with COV is significantly reflected in Fig 3.12. Generally speaking, with  FS Su increasing, COVFS firstly increases, and then decreases or becomes steady, which shows the changing tendency of the variability of slope stability. The reason is that with rising  and Su unchanged l , the factor of deviating from mean level and the conditioning effect of known Su v

60 information will be simultaneously promoted in the horizontal direction. Hence the influence of  appears non-monotonic. Su Table 3.4 Influence of  on  Su FS

 Su 3 5 7 10 15  (kPa) Su 40 0.953 0.948 0.941 0.934 0.934

45 1.077 1.054 1.052 1.043 1.051

50 1.192 1.175 1.161 1.159 1.159

55 1.313 1.299 1.287 1.284 1.261

Table 3.5 Influence of  on  Su FS

 Su 3 5 7 10 15  (kPa) Su 40 0.048 0.054 0.057 0.056 0.057

45 0.044 0.055 0.061 0.063 0.060

50 0.052 0.062 0.065 0.068 0.061

55 0.059 0.069 0.073 0.072 0.071

Fig. 3.12 Influence of  on COV Su FS

61 The 80% median interval charts in Fig. 3.13 verify the above statement. The overall heights of the strips remain steady and the widths firstly increase then become stable or reduce slightly.

Fig. 3.13 Influence of  on the 80% median interval of FS Su

Fig. 3.14 Influence of  on P Su f

62 Table 3.6 Influence of  on  Su

 Su 3 5 7 10 15  (kPa) Su 40 -0.986 -0.938 -0.871 -1.175 -1.054

45 1.538 0.986 0.962 0.687 0.863

50 ∞ ∞ ∞ 2.409 ∞

55 ∞ ∞ ∞ ∞ ∞

Since P for  taking 50kPa and 55kPa is almost 0, the information offered by Fig. 3.14 is f Su relatively limited. Nevertheless, the P changing tendencies for  taking 50kPa and 55kPa f Su basically conform to the aforementioned principles. For the set of   40kPa , with  1, P Su FS f decreases initially then increases as  gets higher. For the set of   45kPa , with  1, the Su Su FS changing tendency is the opposite.

COV 3.5.3 Parametric analysis on Su

It is already found in the field form analysis that with rising COV , not only the S values in Su u individual fields become more deviated, but also the difference between fields gets more significant. Such a growing difference subsequently results in the enhancement of the variability of slope stability, which is demonstrated by the rising tendency of COVFS in Fig. 3.15. As shown in Fig. 3.15, for all four sets of analysis, COV goes up monotonically with increasing COV . FS Su

Another important finding is from Table 3.7. That is, as COV increases, the mean of FS goes Su down clearly and steadily. This demonstrates that the mean level of slope stability will drop down when Su is with more deviation.

Table 3.7 Influence of COV on  Su FS

COV Su 0.3 0.4 0.5 0.6 0.7  (kPa) Su 40 1.057 1.013 0.946 0.887 0.853

45 1.177 1.130 1.067 1.001 0.958

50 1.324 1.257 1.193 1.122 1.050

55 1.453 1.370 1.303 1.229 1.154

63 Table 3.8 Influence of COV on  Su FS

COV Su 0.3 0.4 0.5 0.6 0.7  (kPa) Su 40 0.050 0.050 0.070 0.067 0.076

45 0.053 0.051 0.067 0.068 0.072

50 0.054 0.061 0.081 0.083 0.090

55 0.058 0.070 0.095 0.102 0.098

Fig. 3.15 Influence of COV on COV Su FS The change of the overall state of slope stability is clearly presented in Fig. 3.16. The decreasing heights of strips indicate the decline of the mean level of slope stability. In the meantime, the strips widen steadily, meaning the monotonic negative effect of COV on the consistency of slope Su stability.

As presented by Fig. 3.17, no matter how high the overall level of Pf is, the sensitivity of slope reliability to COV is pretty consistent. That is, with increasing COV , the slope reliability will Su Su significantly decrease. This can be explained as the following. For the three sets of  taking Su

45kPa, 50kPa and 55kPa, the majority of FS is above 1, so the decreasing FS and increasing COV will lead to a higher P synergistically. For the set of  taking 40kPa, with the FS f Su

64 majority of FS close to or below 1, obviously because the effect of FS is greater than that of COV , P is still positively correlated to COV . FS f Su

Fig. 3.16 Influence of COV on the 80% median interval of FS Su

Fig. 3.17 Influence of COV on P Su f

65 Table 3.9 Influence of COV on  Su

COV Su 0.3 0.4 0.5 0.6 0.7  (kPa) Su 40 0.938 0.202 -0.713 -1.626 -1.943

45 ∞ ∞ 1.195 0.015 -0.668

50 ∞ ∞ 2.257 1.476 0.681

55 ∞ ∞ ∞ 2.014 1.665

The analysis on the overall state of slope stability and the slope reliability in this section is closely related to the previous analysis on the field form. They supplement each other for understanding the mechanism of parametric sensitivity.

3.6 Parametric analysis aimed at the “risk”

Most probabilistic analyses on slope stability are aimed at determining the failure probability Pf , with the full probability distribution of FS also determined. The concept of “risk” takes this type of analysis one step further by not only considering the failure probability but also quantifying the failure consequences. It’s possible that, in some cases, Pf is relatively small, yet once failure happens, it does great damage. Then the slope safety tends to be overestimated if only considering the small Pf .

The evaluation index “risk” ( Ri ) of slope failure is conventionally calculated as (Li D.Q. et al. 2016a)

N f

Ri   PiCi  Pf C (3.1) i1

where Pi 1/ NT and Ci are the probability and consequence of the failure mode corresponding to the i -th failure sample during MCS, NT and N f refer to the total number of MCS

N f realizations and the number of failure samples. C  Ci / N f means the average consequence of i1 different failure modes. As pointed out by Huang et al. (2013b), the consequence of slope failure depends on the sliding mass volume, which can, therefore, be taken as an “equivalent” quantity to

66 quantify the consequence of slope failure in slope risk assessment. In OptumG2, the volume of sliding soil mass in each Monte Carlo run is recorded for the purpose of risk evaluation.

The parametric analysis here follows the value ranges mentioned in Section 3.2. Then the comparison between results about Ri and preceding indexes (like Pf ) will reveal if it makes sense to consider failure consequence coupled with failure probability when talking about slope reliability.

The two factors involved are the failure probability and the average consequence. The former one is presented in the last section, and the later one can be obtained by averaging the sliding masses of failure cases. In this study, all analysis is carried out at the two-dimensional level, so the unit for the average consequence and the risk is m2. The average consequences with varying l ,  and Su Su COV on the premises of different  are recorded in Table 3.10, Table 3.11 and Table 3.12. It Su Su can be seen that when  is higher, the average consequence of failure gets smaller. That is to say, Su the average damage caused by failure is less serious. This tendency collaborates with the change of

Pf on affecting the slope safety. Namely a smaller C and a lower Pf both lead to a safer slope.

However, with changing l ,  and COV , C doesn’t change significantly. This indicates Su Su Su that the average expectation of slope failure’s damage is not sensitive to the three statistical parameters.

The changes of Ri , resulted from Pf and C , are illustrated in Fig. 3.18, Fig. 3.19 and Fig. 3.20. Compared with Fig. 3.11, Fig. 3.14 and Fig. 3.17, we can tell that the changing law of Ri with varying statistical parameters is almost identical to that of Pf . So the descriptions of slope reliability offered by these two indexes are much the same. This is understandable since from the perspective of  , the change of matches with that of P , while from the perspective of other Su C f parameters, C remains stable itself thus making no difference.

Finally, it is worth noting that the similarity between the Ri and Pf results demonstrate the consistency of calculations.

67 Table 3.10 Influence of l on the average consequence (m2) Su

l (m) Su 3 5 7 10 15  (kPa) Su 40 586.0 536.6 511.9 546.6 513.7

45 529.0 485.6 502.6 499.5 495.6

50 451.8 468.1 470.6 430.1 407.5

55 − − 355.2 293.5 332.3

Table 3.11 Influence of  on the average consequence (m2) Su  Su 3 5 7 10 15  (kPa) Su 40 596.0 553.0 521.6 528.0 521.7

45 495.9 504.9 477.9 460.8 465.7

50 − − − 431.5 −

55 − − − − −

Table 3.12 Influence of COV on the average consequence (m2) Su

COV Su 0.3 0.4 0.5 0.6 0.7  (kPa) Su 40 622.6 601.3 567.7 548.0 545.4

45 − − 509.6 530.2 529.7

50 − − 451.8 484.8 496.2

55 − − − 465.8 466.5

68 Fig. 3.18 Influence of l on Risk Su

Fig. 3.19 Influence of  on Risk Su

69 Fig. 3.20 Influence of COV on Risk Su In addition, some typical failure modes corresponding to various failure masses and their safety factors are listed for reference. The failures shown in Fig. 3.21 are under the parameter setting of   45kPa , COV  0.6 ,  1, l  0.3H . Su Su Su Su v 1

(a) Failure mass=330.85m2, FS =0.903 (b) Failure mass=437.95m2, FS =0.992

(c) Failure mass=481.45m2, FS =0.981 (d) Failure mass=497.37m2, FS =0.94

(e) Failure mass=556.95m2, FS =0.956 (f) Failure mass=704.35m2, FS =0.93 Fig. 3.21 Typical failures corresponding to various failure masses

70 3.7 Conclusions

The core content of this chapter is the parametric analysis on the statistical parameters of Su ’s random field. To be specific, the author investigate the influence of statistical parameters on the field form, the slope reliability, and the slope failure “risk”. Some conclusions and inspirations can be summarized as the following.

(1) With a greater correlation length, the correlation zones in random fields tend to be farther extended, leading to a two-sided effect on the variability of the form of CRF. When the correlation length is relatively small, its growth will enhance the variability of field form. Yet when it reaches a stage that is large enough to reflect the conditioning effect of the known data at sampling points, its growth will bring the opposite effect. The later kind of effect can’t be taken into account by using the unconditional random field method, so when the correlation length is relatively great, the superiority of CRF over the unconditional method is more highlighted.

(2) Associated with the effect on field form, the effect of increasing correlation length on the variability of slope stability is also not monotonic. That is, the variability of slope stability first rises, then turns gentle or even drops. Meanwhile, the mean level of slope stability is not sensitive to the correlation length. As a result, the changing tendency of the slope reliability can be expressed as follows: when the mean level of FS is above 1, with the correlation length increasing, the slope reliability goes down at first and then probably goes up or tends to be gentle; when the mean level of FS is below 1,the changing tendency is the opposite.

(3) The effect of the ratio between horizontal and vertical correlation lengths is analogical to that of the correlation length since increasing ratio with fixed vertical correlation length means increasing horizontal correlation length. With a greater correlation length ratio, the correlation zones become horizontally longer. The sensitivities of the variability of field form, the overall state of slope stability, and the slope reliability to the correlation length ratio are highly similar to those to the correlation length.

(4) With a greater COV of Su : the Su values throughout the field spread apart more from the mean level; the form of CRF turns less consistent; the mean level of slope stability drops down and the variability rises up; finally, the slope reliability significantly decreases.

(5) For the calculation example investigated in this chapter, taking the failure consequence into account coupled with the failure probability when defining the slope reliability doesn’t provide a significant improvement compared with only considering the failure probability. It’s found that the average failure consequence, i.e. the average expectation of slope failure’s damage, is not sensitive to the statistical parameters. Thus the changing law of “risk” with varying statistical parameters is almost identical to that of the failure probability.

71 Chapter 4 Conditional random field analysis on a cohesion-frictional slope

4.1 Introduction

Geotechnical properties are often found to correlate with each other. A representative example of such kind of correlation, which is widely observed and draws much attention from researchers, is discussed in this chapter. That is, for the cohesion-frictional soil, the two classical strength parameters, the cohesion c and the internal frictional angle  , are usually considered cross-correlated. The extent of correlation between c and  and whether the correlation is positive or negative can be characterized by the cross-correlation coefficient c, . So the parametric analysis on c, will reveal the sensitivity of the slope reliability to the degree of correlation. The parametric analysis here is based on the conditional simulation of cross-correlated random fields of c and  .

Among existing probabilistic analyses on c  soil, most of them took the cross-correlation between the two parameters into account (Wu 2013; Gong et al. 2018; Liu et al. 2017; Deng et al. 2017; Johari and Gholampour 2018), while several others didn’t (Zhang et al. 2014). Other kinds of cross-correlations, like those between undrained soil strength and unit weight (Javankhoshdel et al. 2017), soil strength parameters and Young’s modulus (Lo and Leung 2017), soil strength parameters and pore water pressure ratio (Zhang et al. 2010), five van Genuchten-Mualem soil hydraulic parameters (Gomes et al. 2017), were also under investigation in the literature. Nevertheless, there hasn’t been a study on the sensitivity of the cohesion-frictional slope reliability to c, under the framework of the CRF method and NLA.

The core task of this chapter is to investigate the influence of the cross-correlation between c and  on the overall state of slope stability and the slope reliability. To achieve it, the procedure of generating two cross-correlated conditional random fields with a given cross-correlation coefficient and their respective statistical parameters is presented at first. Then the typical forms of paired random fields are exhibited and analyzed to help to understand how c, exerts an influence. What’s more, the parametric analysis in this chapter is also conducted based on the unconditional random field method as a contrast, to further verify the significance of the conditional random field.

4.2 Generation of two cross-correlated random fields

When considering two cross-correlated properties simultaneously in slope reliability investigation, two corresponding sets of random fields need to be generated. For every point among the slope

72 model, the values of the two properties should follow some predefined correlation. Meanwhile, the random field for either property need to obey its own statistics. Fenton and Griffiths (2003) outlined the simulation of cross-correlated random fields, making use of the lower triangular matrix, Lρ , from the Cholesky decomposition of the correlation matrix.

Following the simplifying transformation in Section 2.2, the generation of two cross-correlated lognormal random fields Y1t and Y2 t is equivalent to dealing with two corresponding standard-normally distributed fields Z1t and Z2 t . (The correlation functions and the correlation lengths of Y1t and Y2 t are assumed to be the same.) Since the mean and the standard deviation of normal random fields don’t really make a difference to the cross-correlation coefficient, the relation between the coefficients is expressed as follows:

 2 2  X1  X 2  ln Y ,Y e 1e 11  1 2      (4.1) Z1 ,Z2 X1 ,X 2   X1 X 2

where the notations concerning X1t and X 2 t also follow the meanings in Section 2.2.

The cross-correlation coefficient (1  1) is used to characterize the correlation between the two random fields. Three extreme cases  = −1, 0, and 1, are corresponding to completely negatively correlated, uncorrelated, and completely positively correlated, respectively.

The correlation matrix between Z1t and Z2 t, assumed to be stationary, is formed as

 1   Z1 ,Z2 ρΖ ,Ζ    (4.2) 1 2  1  Z1 ,Z2 

Then compute the Cholesky decomposition of ρ , i.e. find a lower triangular matrix L Ζ1 ,Ζ 2 ρΖ1 ,Ζ2

T such that Lρ Lρ  ρΖ ,Ζ . The form of L can be easily derived as Ζ1 ,Ζ2 Ζ1 ,Ζ2 1 2 ρΖ1 ,Ζ2

 1 0  Lρ   2  (4.3) Ζ1 ,Ζ2 ρ 1 ρ  Ζ1,Ζ2 Ζ1,Ζ2 

The next step is to generate two independent standard-normally distributed random fields, G1t and G2 t . Here the correlation function G   is the same as Z   . ( Z     X   , and

 X   is derived from Eq. (2.3).) Then the underlying point-wise correlated random fields are formed as

73 Z1t G1t    L   (4.4) ρΖ1 ,Ζ2 Z2 t G2 t

In fact, here the random field generators mentioned in Section 2.2 are directly implemented on the generation of G1t and G2 t , subsequently leading to Z1t and Z2 t , then transforming to the desired Y1t and Y2 t . This also implies that in the implementation of Kriging technique, the known information extracted from the real fields of Y1t and Y2 t need to be transformed into

 X  X Z t  the framework of G1t and G2 t through the inverse operation of Y t  e and Eq. (4.4).

4.3 Problem definition

4.3.1 Basic model and parametric analysis definition

Referring to the slope model in Liu et al. (2017), the parametric analysis will focus on the influence of the cross-correlation coefficient c, . As seen from Fig. 4.1, the cohesion-frictional slope has a height of H 2 10m and a slope angle of  2  45 , and the slope consists of a single soil layer

3 with a unit weight of  2  20.0kN / m . Following Cho (2010) and Li D.Q. et al. (2015b), the cohesion c and the frictional angle  are modeled as cross-correlated lognormal random fields. The mean values of c and  are 9kPa and 27°, respectively, and their COV are 0.3 and 0.2, respectively. The horizontal and vertical correlation lengths are chosen as lh  20m and lv  2m for both random fields of c and  . A 2D single exponential autocorrelation function is applied to this slope model and defined as

    y   ,  exp x   (4.5) 2 x y    lh lv 

where  x  xi  x j and  y  yi  y j are the distances between two points in the horizontal and vertical directions, respectively.

The point interval of CMD is 0.5m, forming a 60*30 dot matrix. For one set of probabilistic analysis, 500 Monte Carlo runs will be conducted.

74 H H

A 3 m B H 3 1.5H m C 3 m D 0.5H 3 m E 5m 5m 5m 5m

3H

Fig. 4.1 Slope model for the random fields of cross-correlated c and 

As always emphasized, a conditional random field simulation requires known data of soil properties at some particular locations. However, with the same situation of the undrained model in Chapter 3, there are no real measured samples available because the slope in question is hypothetical. Likewise, virtual “real” fields are used to offer known data at sampling points. Fig. 4.1 also shows the layout of the five virtual sampling points, whose horizontal and vertical intervals are 5m and 3m. The site investigation data for CRF generation are extracted from these five points.

In this chapter, the only changing parameter is c, , with a value range of (-0.7, -0.5, -0.3, 0, 0.3, 0.5). For comparison, the analysis will be carried out based on both the unconditional random field and CRF. The employment of resulting indexes will follow the preceding setting in Chapter 3. Namely, the 80% median interval of FS , along with its mean value and COV, will describe the overall state of slope stability. Then the failure probability Pf and the reliability index  will define the slope reliability.

4.3.2 Deterministic analysis based on the mean

The deterministic analysis based on the mean, i.e. c  9kPa and   27 , is conducted for reference. The FS is calculated to be 1.07. The corresponding failure mode is shown as Fig. 4.2.

75 Fig. 4.2 Failure mode for the deterministic analysis of the cohesion-frictional slope

4.4 Verification

In order to verify the methodology employed in this study, a comparison analysis is conducted with reference to some results in Liu et al. 2017. For this section, the slope model and all parameter values follow the setting in the literature. The basic model in Liu et al. (2017) is with the mean values of c and  as 10kPa and 30° and other parameters are identical to the slope model presented in the last section. To ensure the comparability of the results of CRF analysis, the known data of the five sampling points for various c, also follow the setting in the literature (see Table 4 in Liu et al. 2017).

Fig. 4.3 Verification concerning the influence of c, on  FS

76 Fig. 4.4 Verification concerning the influence of c, on Pf

The comparison is carried out on the change in  FS and Pf with respect to c, . As seen in Fig. 4.3 and Fig. 4.4, the results obtained by the method of this study match well with the reference.

Generally speaking, both  FS and Pf become lower when using the CRF method instead of the unconditional method. Moreover, they both increase with rising c, .

In the literature, with the parameter setting of c 10kPa and   30 , the failure probabilities for different cases tend to be low. The subset simulation (SS) technique was performed to calculate very low failure probabilities. Without SS, traditional MCS can only get a result of zero or very close to zero when dealing with such low Pf cases, so that no tendency can be reflected. For the conditional case in Fig. 4.4, for example, the results in the literature reach a super low magnitude (10-3 to 10-10) and the results obtained by this study are just zero. The emphasis of this study is to investigate the parametric sensitivity of Pf to c, , so the mean values of the strength parameters are slightly adjusted to make the Pf results more observable.

4.5 Field form analysis

The parameter under investigation in this chapter is the cross-correlation coefficient c, , so the analysis on the field form should emphasize the cross-correlation between the paired fields of c

77 and  , including if it is positive or negative and its level. The field forms with c, taking 0, -0.5, 0.5 are exhibited here, corresponding to no correlation, some negative correlation, some positive correlation, respectively. In following figures, the first one of every field pair is the random field for c and the second one for  . For the sake of comparison, the author tries to keep the value ranges uniform for all figures in this section (see Fig. 4.5).

In this section, the real fields are shown firstly to reveal how c, affects the field form with some consistent pattern. Then for both unconditional and conditional cases, four typical random field pairs for each c, value will be listed to further show such kind of effect, to verify the conditioning effect in CRFs, and most essentially, to find out how c, affects the variability of field form.

4.5.1 Paired real fields

To meet the requirement of parametric analysis, the paired real fields of c and  for different

c, need to be determined for generating corresponding sets of CRFs. The paired real fields should maintain some consistent pattern, and related reasons are already specified in Section 3.3. What needs to be locked here are the independent standard-normally distributed vectors U for

G1t and G2 t (see Section 2.2.1.1 and Section 4.2). The baseline case for this study is

c,  0 .

With a deeper insight into the generator in Section 4.2, especially Eq. (4.3) and Eq. (4.4), it’s not hard to understand that the first lognormal field Y1t (i.e. the field of c ) is entirely derived from

G1t , while the second one Y2 t (i.e. the field of  ) is partially derived from G2 t and partially affected by G1t according to the value of c, . This is how the cross-correlation is reflected in the generator.

This mechanism is also reflected in the forms of real fields. Since the pattern (i.e. the vectors U for G1t and G2 t ) is consistent for all real field pairs, the influence brought by different c, values on the field form is clearly shown by Fig. 4.5. In Fig. 4.5a, c,  0 and there is no correlation between the c field and the  field. For the convenience of explanation, four feature areas are circled in the figures. In Fig. 4.5b and Fig. 4.5c, as mentioned just above, the c fields are the same with Fig. 4.5a. Meanwhile, in Fig. 4.5b ( c,  0.5 ), the  field is negatively correlated to the c field, and there are indeed some features to prove it. For example, in area 1, under the negatively correlated influence of the level of c value, the level of  value turns lower

78 compared with Fig. 4.5a; likewise, in area 2 and area3, the level of  value gets higher and even lower on a low basis, respectively; in area 4, the level of c value is moderate thus making no great impact. In Fig. 4.5c ( c,  0.5 ), the mechanism is similar to Fig. 4.5b, and the difference is the influence of the level of c value is no longer negatively correlated. For example, in area 3, the level of  value becomes less low.

(a) c,  0

(b) c,  -0.5

(c) c,  0.5

Fig. 4.5 Paired real fields of c and  with different c,

79 Extracted from the real fields, the known data of the five sampling points for various c, are listed in Table 4.1 for reference. Similar treatment can be found in the literature, yet without detailed explanation.

Table 4.1 The known data of the five sampling points for different c,

c, c (kPa)  (°)

A B C D E A B C D E

-0.7 8.76 9.41 8.92 12.05 8.87 27.24 24.39 25.52 23.37 26.50

-0.5 8.76 9.41 8.92 12.05 8.87 27.57 24.45 25.51 24.80 26.72

-0.3 8.76 9.41 8.92 12.05 8.87 27.77 24.62 25.56 26.17 26.89

0 8.76 9.41 8.92 12.05 8.87 27.94 25.01 25.71 28.17 27.08

0.3 8.76 9.41 8.92 12.05 8.87 27.96 25.54 25.93 30.10 27.21

0.5 8.76 9.41 8.92 12.05 8.87 27.88 25.98 26.11 31.33 27.26

4.5.2 Typical random field pairs

Both for the unconditional and conditional cases (Fig. 4.6 and Fig. 4.7), when conducting the comparison between (a), (b), and (c), the influence of c, mentioned just above can still be observed. In (a) figures, there is no cross-correlation between the paired c field and  field, or just some coincidental and surface correlation. Meanwhile, in (b) and (c) figures, some certain negative or positive correlation features are pretty obvious in most individual field pairs. Some exceptional pairs exist, such as Fig. 4.6c4 and Fig. 4.7b3, because c, taking -0.5 and 0.5 only means being correlated to some extent and there is inherent independence between G1t and

G2 t.

Considering the conditioning effect of sampling points, it is expected that the forms of the four field pairs in Fig. 4.7a are closer than the four in Fig. 4.6a, as well as the comparisons concerning (b) and (c). However, such kind of effect is not apparent in the resulting figures, probably because the number of the sampling points is relatively small and thus the known information is limited. For the three sets in Fig. 4.7, although there are some features showing the consistency of field form, exceptions do exist. At least, the similarity around the five sampling points can be guaranteed.

80 (a1)

(a2)

(a3)

(a4)

(a) c,  0

81 (b1)

(b2)

(b3)

(b4)

(b) c,  -0.5

82 (c1)

(c2)

(c3)

(c4)

(c) c,  0.5

Fig. 4.6 Typical unconditional random field pairs of c and  with different c,

83 (a1)

(a2)

(a3)

(a4)

(a) c,  0

84 (b1)

(b2)

(b3)

(b4)

(b) c,  -0.5

85 (c1)

(c2)

(c3)

(c4)

(c) c,  0.5

Fig. 4.7 Typical conditional random field pairs of c and  with different c,

86 What we concern the most is how c, affects the variability of field form. That is, we need to observe to what extent the forms of the four field pairs are close, and more importantly, how the situation changes among (a), (b), and (c). It can be seen that, in either Fig. 4.6 or Fig. 4.7, there is no obvious tendency for this issue. In other words, for each one of the three sets, the four field pairs are just different with a degree showing no tendency among (a), (b), and (c). This is understandable since the cross-correlation between two parameters is not supposed to affect the variability in the field form of either one.

It's worth noting that the analysis of this section remains on the level of field form and doesn’t consider the field on the level of physical meaning. No tendency for the variability of field form is not equal to no tendency for the variability of slope stability. This question will be further discussed in the following analysis concerning the overall state of slope stability.

4.5.3 Typical failure modes

Four different typical failure modes for the case of conditional random field and c,  0.5 are shown in Fig. 4.8. Most failures follow the modes in Fig. 4.8c and Fig. 4.8d, in which the failure surface passes the slope toe or somewhere near the slope toe. A minority of failures are like the modes in Fig. 4.8a and Fig. 4.8b, in which the failure surface passes the slope face.

(a) (b)

(c) (d)

Fig. 4.8 Typical failures for the conditional case with c,  0.5

87 4.6 Parametric analysis aimed at the slope reliability

This section will investigate the sensitivity of the overall state of slope stability and the slope reliability to the cross-correlation coefficient c, . To start with, as shown in Table 4.2, the mean of FS maintains a stable level with changing c, , for both the unconditional and conditional cases. So the cross-correlation between c and  doesn’t have a significant influence on the mean level of slope stability.

Table 4.2 Influence of c, on FS

c, -0.7 -0.5 -0.3 0 0.3 0.5

Unconditional 1.048 1.054 1.052 1.047 1.047 1.049

Conditional 1.051 1.049 1.058 1.055 1.053 1.052

Table 4.3 Influence of c, on  FS

c, -0.7 -0.5 -0.3 0 0.3 0.5

Unconditional 0.066 0.092 0.097 0.106 0.134 0.144

Conditional 0.033 0.038 0.049 0.054 0.062 0.076

Fig. 4.9 Influence of c, on COVFS

88 The variability of slope stability is characterized by the COV of FS in Fig. 4.9 and the standard deviation of FS in Table 4.3. Here the analysis is primarily based on the result of COVFS . There are mainly two apparent rules about the variability of slope stability. Firstly, it is significantly smaller for CRF than for the unconditional random field; secondly, it keeps increasing as c, rises from -0.7 to 0.5. The reason for the former law is already presented in Chapter 2. That is, under the conditioning effect, the conditional random fields for a parameter setting tend to be closer, thus leading to more consistent states of the slope. Nevertheless, the mechanism for the later law is different. The change of c, from -0.7 to 0.5 can be treated as a coherent process, as well as two separated ones. In other words, when the correlation is negative ( c,  0 ), the lower the correlation is (the smaller the c, ’s absolute value is), the greater the variability of slope stability is; when the correlation is positive, the contrary is the case. The relative mechanism is specified as the following.

For understanding the underlying mechanism, the case of c,  0 is treated as a benchmark. In this case, the random fields of c and  affect the slope stability independently. As known to all, the two parameters c and  are both positively related to slope safety. Then in the cases of

c,  0 , as pointed out in Section 4.5, the field forms of c and  have some negative cross-correlation. Namely, at some zones or points where the c value is relatively high, the  value tends to be relatively low and vice versa (see Fig. 4.6b and Fig. 4.7b). This will cause the result that, at these zones or points, the strength property’s deviation caused by the strength parameters’ deviation is neutralized. So the strength property will remain relatively stable, and thus the slope stability will be more consistent compared with the no-correlation case. The greater the negative correlation is, the stronger such kind of effect is and the more consistent the slope stability is. As for the cases of c,  0 , as shown in Fig. 4.6c and Fig. 4.7c, the field forms of c and  are positively cross-correlated. Here the coupled effect of the strength parameters’ deviation changes from neutralization to synergy. So the random deviation of strength property among the field is strengthened and promoted, and thus the slope stability will be less consistent.

The overall state of slope stability is visually reflected by the 80% median interval charts in Fig. 4.10. We can see that the strip of the conditional case is much narrower than that of the unconditional case. Meanwhile, the two strips both turn wider with increasing c, .

The slope reliability is negatively characterized by Pf in Fig. 4.11. Again, Pf is the consequence of the distribution of FS , i.e. the overall state of slope stability. The relationship between them is specified in Section 3.5. In summary, the final law emerging here is that the slope reliability

89 obtained through the CRF method is higher than the unconditional method, and the slope reliability decreases with increasing c, .

Fig. 4.10 Influence of c, on the 80% median interval of FS

Fig. 4.11 Influence of c, on Pf

90 Table 4.4 Influence of c, on 

c, -0.7 -0.5 -0.3 0 0.3 0.5

Unconditional 0.700 0.625 0.553 0.485 0.300 0.243

Conditional 1.728 1.329 1.195 0.970 0.915 0.700

The analysis in this section extends and deepens the preceding field form analysis. When giving the features of field form some physical meaning, we find out that although there isn’t any tendency for the variability of field form, the variability of slope stability does have some sensitivity to c, .

4.7 Conclusions

The generation procedure for two cross-correlated random fields is firstly introduced, and then the methodology of this chapter is verified by a comparison with the results in the literature. The parametric analysis is conducted on the cross-correlation coefficient between cohesion and internal frictional angle. The parametric sensitivities of the form of paired random fields and the slope reliability are analyzed based on both conditional and unconditional cases.

(1) For the case of no cross-correlation between the paired c field and  field ( c,  0 ), there are no apparent features showing relevance in form. For the case of negative cross-correlation

( c,  0.5 ), there are some certain negative correlation features. Namely, at some zones or points where c is relatively high,  tends to be relatively low and vice versa. For the case of positive cross-correlation ( c,  0.5 ), the situation is just the opposite.

(2) For the calculation example in this chapter, the form of paired random fields doesn’t exhibit apparent smaller variability when the CRF method is applied, probably because relatively few sampling points provide limited known information. What’s more, the variability of the form of paired random fields is not obviously influenced by the cross-correlation between c and  .

(3) The cross-correlation between c and  doesn’t have a significant influence on the mean level of slope stability. The variability of slope stability is much smaller for CRF than for the unconditional random field. Meanwhile, the variability of slope stability increases with c, rising from -0.7 to 0.5, i.e. with the cross-correlation first becoming less negative then turning more positive. The slope reliability obtained by the CRF method is higher than the unconditional method, and most important of all, the slope reliability decreases with increasing c, .

91 Chapter 5 Conditional random field analysis on a dam slope with stochastic hydraulic conductivity

5.1 Introduction

The permeability is a key factor affecting the slope stability, and the hydraulic conductivity K s is a key parameter characterizing the permeability property. Also, the hydraulic conductivity is used to calculate all of the hydraulic parameters associated with slope performance, including the seepage rate and amount, the degree of saturation, the pressure head, the pore-water pressure, and therefore, the state of . So incorporating the effect of the variability and uncertainty of K s into slope stability analysis will result in more realistic estimations. Here within the framework of the

CRF method again, the author will investigate the influence of the statistical parameters of K s on estimating the overall state of slope stability and the seepage situation in the slope.

There are already considerable probabilistic analyses involving the permeability property of soils. Griffiths and Fenton (1997) tried to model steady seepage beneath a single sheet pile wall through a three-dimensional soil domain in which the permeability is randomly distributed in space. Gui et al. (2000) quantified the stochastic influence of hydraulic conductivity on the variation of the safety factor for a downstream dam slope, studying its effect on the reliability index. Le et al. (2012) assessed the effect of randomly varied , which leads to a corresponding random variation of both permeability and water retention properties, on the groundwater flow through a flood defense . Le et al. (2019) investigated the effect of rainfall intensity and permeability on the uncertainty of the safety factor and failure size in unsaturated slopes with randomly heterogeneous porosity. Some other works also took the effect of rainfall into account within the contexts of probabilistic slope stability analysis, with spatially variable hydraulic conductivity (Zhang et al. 2014, 2018; Yuan et al. 2019). Gomes et al. (2017) investigated the uncertainty in bedrock depth and soil hydraulic parameters when studying the stability of a real variably-saturated slope in Brazil. Johari and Gholampour (2018) carried out a reliability analysis of unsaturated slopes via CRF simulations of several measured textural and mechanical soil properties, with considering the suction. Yang et al. (2018) proposed a probabilistic back estimation method for characterizing spatial variability of soil hydraulic properties, which is posed within a Bayesian framework.

To be specific, the objective of this chapter is to study the parametric sensitivities of the FS distribution and the spatial variability of phreatic surfaces to the coefficient of variation COV K s and the correlation length l . The relevant field form analysis is also carried out along with the Ks

92 parametric analysis. The research is based on the downstream slope model of a hypothetical earth dam, and the generation of K s ’s random fields follows the methodology presented in Section 2.2.

5.2 Problem definition

5.2.1 Basic model and parametric analysis definition

The embankment dam is a widely used water-retaining structure around the world and the prediction of dam slope stability is of great importance (Fenton and Griffiths 1996). The model of the embankment dam is actually a two-side slope with one side adjacent to a certain water level. To investigate the effect of stochastic hydraulic conductivity on the seepage situation and the probabilistic slope stability, the hypothetical dam model in Fig. 5.1 is established by slightly adjusting the model in Gui et al. (2000). This case is representative of embankment dams built with a single soil on an impervious base with no tailwater. All constant geometric and soil parameters are

1 valued as: dam height H 3  20m , left slope angle 3  33.7 (3:2 slope), right slope angle

2 3 3  21.8 (5:2 slope), water level height H w 12m , soil unit weight  3  20.0kN / m , effective cohesion c  9kPa , effective frictional angle   27 , K ’s mean value   3104 cm / s , s K s correlation length ratio   5 , a material parameter in linear hydraulic model h*  0.5m . A 2D K s autocorrelation function is applied to this dam slope model and defined as

 2 2              x    y  3  x , y  exp      (5.1)   lh  lv     

where  x  xi  x j and  y  yi  y j are the distances between two points in the horizontal and vertical directions, respectively.

The point interval of CMD is 1.0m, forming a 140*30 dot matrix. For one set of probabilistic analysis, 200 Monte Carlo runs will be conducted.

The layout of the nine virtual sampling points is demonstrated in Fig. 5.1. The value ranges of changing vertical correlation length and COV of K are set as l  0.5m,1m,3m,5m,7m and s K s v COV  0.3,0.4,0.5,0.6,0.7 . K s

93 30m 30m 10m 50m 20m m m 5 E 5 m m 5 5 D F m 0 2 m m m 5 G 5

2 C 1 m m 5 B H 5 m m 5 5

m A I m 0 0 1 1

25m 10m 10m 10m 10m 15m 15m 15m 15m 140m

Fig. 5.1 Slope model for the random field of K s

After establishing the dam slope model in OptumG2, with the predefined difference between the water tables on the left-hand and right-hand sides of the dam, as well as “Time Scope” set as “Long Term”, the seepage calculation will provide the steady saturation distribution after a long enough seepage time, as well as the safety factor FS . For the long term analysis, the mean level of K s won’t make a big difference to the results just mentioned. So in this chapter  is set unchanged. K s Another clarification is that the soil shear strength and the hydraulic conductivity are actually correlated, because seepage patterns, which are controlled in part by hydraulic conductivity, determine pore-water pressures and degree of saturation, both of which strongly influence soil shear strength. However, in this chapter, it’s assumed that they are independent and the shear strength is constant.

The fluctuation of FS caused by varying K s is not significant, and thus the failure probabilities for all cases in this chapter are zero. So only the mean and COV of FS will be presented to describe the overall state of slope stability. The distribution of the degree of saturation ( S ) and the location of the phreatic surface can be illustrated like Fig. 5.3 by OptumG2. The phreatic surface indicates the location where the pore water pressure is under atmospheric conditions (i.e. the pressure head is zero). This surface normally separates the saturated (red corresponds to S 1) and dry (blue corresponds to S  0 ) parts of the model, reflecting the seepage pattern through the dam slope. For some parameter settings, contours of many typical phreatic surfaces will be illustrated in one figure to show the spatial variability of the phreatic surface.

5.2.2 Deterministic analysis based on the mean

The deterministic analysis based on the mean value of K s is conducted for reference. (In fact, changing the K s value won’t affect the result.) The corresponding safety factor is 1.207. The failure mode is shown in Fig. 5.2 and the degree of saturation distribution in Fig. 5.3.

94 Fig. 5.2 Failure mode for the deterministic analysis of the dam slope

Fig. 5.3 Degree of saturation for the deterministic analysis of the dam slope

5.3 Field form analysis

In this section, for analyzing the influence of l , the parameter settings are taken as Ks l  0.5m,3m,7m with the premise of COV  0.5 , while for COV , the parameter settings K s v K s K s are taken as COV  0.3,0.5,0.7 with l  3m . Firstly, the comparison between the forms of K s K s v the real fields mainly concerns how the features in field form embody different statistical parameter settings. Secondly, the flux distributions corresponding to the real fields are exhibited to reflect the influence of statistical parameters on the overall flux level. (The flux is commonly known as the seepage velocity, with the unit of m/day.) Thirdly, for each parameter setting, four typical conditional random fields are shown, and the concern is the variability of field form.

For the first and third points, the analyzing perspective and the influence law largely follow the analysis in Section 3.3 and Section 3.4, so related statements are brief here.

95 (a) l  0.5m K s v

(b) l  3m K s v

(c) l  7m K s v

Fig. 5.4 Real fields of K with different l s Ks v

96 (a) COV  0.3 K s

(b) COV  0.5 K s

(c) COV  0.7 K s

Fig. 5.5 Real fields of K with different COV s K s

97 (a) l  0.5m K s v

(b) l  3m K s v

(c) l  7m K s v

Fig. 5.6 Flux fields with different l Ks v

98 (a) COV  0.3 K s

(b) COV  0.5 K s

(c) COV  0.7 K s

Fig. 5.7 Flux fields with different COV K s

99 (a) l  0.5m K s v

(b) l  3m K s v

(c) l  7m K s v

Fig. 5.8 Typical conditional random fields of K with different l s Ks v

100 (a) COV  0.3 K s

(b) COV  0.5 K s

(c) COV  0.7 K s

Fig. 5.9 Typical conditional random fields of K with different COV s K s

5.3.1 Real fields

As stated in Section 3.3, the real fields of K s for different parameter settings are supposed to maintain some consistency in the pattern. Not only such kind of consistency but also the influence of the statistical characteristics on the field form are reflected in Fig. 5.4 and Fig. 5.5. The phenomenon here is analogical to that of the shear strength Su and also conforms to the mathematical meaning of the parameters. That is, it can be seen from Fig. 5.4 that with l Ks v

101 increasing, the correlation zones extend further, and the value of K s is strongly correlated within a longer distance; while Fig. 5.5 shows that as COV goes up, the K value deviates from the K s s mean level to a greater extent.

5.3.2 Flux fields

For the sake of visual comparison, the value ranges for flux distributions are set uniform. Fig. 5.6 and Fig. 5.7 respectively reveal that, other things being equal, the overall flux level in the dam slope model is negatively related to l , as well as COV . The following is an interpretation of the Ks K s underlying mechanism. According to the seepage theory, the flux level in slope largely depends on the permeability of slope soil, which is defined by the hydraulic conductivity K s . When we investigate the distribution of K s within the framework of random field theory, the overall infiltrating property is determined by the relatively low permeability in the model to a great extent. The reason is similar to the Cannikin Law. When l goes up, the far deviated values among the Ks field will affect a greater range and the low-value correlation zones will be larger. When COV K s goes up, the far deviated values will deviate farther. Both will lead to the result that in some zones of the model, the permeability reaches an even lower level. These zones will serve as the “shortest stave” affecting the overall infiltrating property. This is why we have the preceding law about the overall flux level.

5.3.3 Typical conditional random fields

The typical CRFs for different l and COV are shown in Fig. 5.8 and Fig. 5.9. The laws Ks v K s about the variability of field form presented in Section 3.4 hold true again. Specifically, when l Ks is short, the correlation zones are small and evenly distributed, so the differences in form won’t be reflected significantly. Meanwhile, as l gets longer, high-value and low-value correlation zones Ks become larger, and the spatial randomness of K s ’s distribution is amplified, thus enhancing the variability of field form. Then with l getting large enough, the conditioning effect of sampling Ks points makes a big difference, so there is a trend of field forms getting closer. These laws are more or less reflected in Fig. 5.8. As for COV , the variability of field form rises with its increase. As K s seen from Fig. 5.9, the four CRFs in Fig. 5.9c share few similar features in form, compared with those in Fig. 5.9a and Fig. 5.9b.

102 5.4 Parametric analysis aimed at the overall state of slope stability and the seepage situation

5.4.1 Parametric sensitivity of the overall state of slope stability

The results of Monte Carlo simulation show that the FS distributions for this context tend to be concentrated, so the FS distribution is characterized by the value range of FS here instead of the 80% median interval, as illustrated in Fig. 5.11.

Table 5.1 Influence of COV and l on  K s Ks v FS

l (m) Ks v 0.5 1 3 5 7 COV K s

0.3 1.2065 1.2061 1.1991 1.2017 1.2007

0.4 1.2060 1.2065 1.2007 1.1997 1.2013

0.5 1.2070 1.2054 1.1983 1.1977 1.1989

0.6 1.2071 1.2028 1.1979 1.1968 1.1986

0.7 1.2055 1.2054 1.1968 1.1965 1.1969

Table 5.2 Influence of COV and l on  (10-3) K s Ks v FS

l (m) Ks v 0.5 1 3 5 7 COV K s

0.3 3.64 4.61 5.70 6.11 5.83

0.4 4.66 5.93 8.54 7.41 6.35

0.5 6.43 8.14 9.96 9.35 7.57

0.6 6.90 9.51 11.31 10.06 8.67

0.7 8.05 10.61 12.34 11.16 9.45

103 Fig. 5.10 Influence of COV and l on COV K s Ks v FS

Fig. 5.11 Influence of COV and l on the value range of FS K s Ks v

104 We can tell from Table 5.1 that the mean level of slope stability is almost impervious to the statistical characteristics of K s . Then the discussion focuses on the parametric sensitivity of the variability of slope stability. On one hand, the variability of slope stability increases with rising COV , which is demonstrated by the different overall heights of the five broken lines in Fig. 5.10 K s and the different overall widths of the five strips in Fig. 5.11. The reason is that a greater COV K s leads to greater variability of the permeability field, thus resulting in greater variability of slope stability. On the other hand, the variability of slope stability first rises then descends with increasing l , which is reflected by the fluctuation of individual broken lines in Fig. 5.10 and the width Ks v change of individual strips in Fig. 5.11. Its mechanism can be summarized as the two-sided effect of the increasing l . That is, the influence of highly deviated and randomly distributed K values is Ks s promoted and strengthened, while the known data information offered by the sampling points can condition the K value in a larger range. As a result, with rising l , the variability of slope s Ks stability presents a non-monotonic change according to the varying dominant factor.

5.4.2 Parametric sensitivity of the seepage situation

As mentioned earlier, here the phreatic surface serves as a simplified presentation of the overall seepage situation in the dam slope. For a given parameter setting, the phreatic surfaces for 20 randomly selected conditional random fields will be extracted and then presented together to form a spatial distribution of phreatic surfaces. The spatial variability of phreatic surface can characterize the variability of seepage situation.

As shown in Fig. 5.12, in order of (a), (b), and (c), the curve bunch first concentrates, then scatters relatively, and then concentrates again. This demonstrates that with l going up, the variability of Ks seepage situation first increases then decreases. Next, in Fig. 5.13, with COV increasing, the K s curve bunch turns more and more scattered, meaning that the variability of seepage situation keeps increasing. The above laws conform to the laws about the variability of field form and the variability of slope stability. They are all governed by the same mechanism, and it won’t be repeated here.

105 (a) l  0.5m K s v

(b) l  3m K s v

(c) l  7m K s v

Fig. 5.12 Spatial distributions of phreatic surfaces for different l Ks v

106 (a) COV  0.3 K s

(b) COV  0.5 K s

(c) COV  0.7 K s

Fig. 5.13 Spatial distributions of phreatic surfaces for different COV K s

107 5.5 Conclusions

In this chapter, the objective parameter for which we establish conditional random fields is the hydraulic conductivity K s . The parametric analysis is carried out to investigate the effect of statistical characteristics of permeability on the field form, the overall state of slope stability, and the seepage situation in the dam slope. Several main conclusions are summarized here.

(1) The effect of K s ’s correlation length and COV on the field formal characteristics follows the laws concerning Su mentioned in Chapter 3, and it conforms to the mathematical meaning of the parameters. As well, their effect on the variability of field form is analogical to Su . That is, the increase of correlation length brings a two-sided effect, while the increase of COV leads to greater variability.

(2) Other things being equal, the overall flux level in the dam slope is negatively correlated to both the correlation length and COV of K s .

(3) The mean level of slope stability is almost impervious to the statistical characteristics of K s .

The variability of slope stability increases with rising COV of K s , while it first rises then descends with increasing correlation length.

(4) Here the phreatic surface serves as a simplified presentation of the overall seepage situation, so the spatial variability of phreatic surface can characterize the variability of seepage situation. With the correlation length of K s going up, the variability of seepage situation first increases then decreases. With rising COV of K s , the variability of seepage situation constantly increases.

108 Chapter 6 Conclusions and recommendations

6.1 Conclusions

The work involved in this thesis attempts to illuminate two main issues. Firstly, the superiority of the conditional random field method over the unconditional method is verified by the comparisons on the uncertainty of field form, the sampling efficiency, the uncertainty in slope stability assessment, as well as some comparisons in the context of parametric analysis on some soil property parameters. Then, with similar evaluation standards, the sampling scheme is optimized to provide some guidance for sampling point setting. Secondly, parametric analyses are carried out in three different geotechnical contexts, namely an undrained slope with CRF of Su , a cohesion-frictional slope with cross-correlated CRFs of c and  , and a dam slope with CRF of

K s . The core objective is to investigate the influence of statistical characteristics of CRF on the overall state of slope stability (characterized by the distribution of FS ) and the slope reliability

(characterized by the failure probability Pf ). Meanwhile, the parametric sensitivity of the form of CRF is also studied for better understanding the underlying mechanism. Involved statistical parameters include the correlation length ( l ), the ratio between horizontal and vertical correlation lengths (  ), the coefficient of variance (COV), and the cross-correlation coefficient ( c, ). In addition, there are some supplementary studies, such as considering the failure consequence coupled with the failure probability when defining slope reliability and studying how the statistics of K s affect the seepage situation.

In the following, the author summarizes some important conclusions extracted from the whole thesis, hoping to obtain some relevant theoretical laws and some practical references.

(1) The core index characterizing slope reliability is the failure probability Pf , which is actually an outcome of the distribution of the safety factor FS . The general relationship between them is intuitive and easy to understand, and stating it here again is for better presenting relative conclusions. That is, with the majority of FS distribution below 1, Pf is negatively related to

FS and COVFS ; and with the majority of FS distribution above 1, Pf is negatively related to

FS and positively related to COVFS .

(2) The main implication of the superiority of CRF refers to that the CRF method can provide more consistent estimation on the field form and the slope stability thus affecting the estimation of slope reliability, compared with the unconditional method. There are several evidences proving it: less different fields and a lower total global uncertainty (TGU) demonstrating a lower variability of field

109 form, a greater sampling efficiency index ( I se ) demonstrating a higher sampling efficiency, and a smaller COV of FS demonstrating a lower variability of slope stability. The influence on the slope reliability estimation results from the distribution of FS . For example, in the analysis concerning the cross-correlation between c and  , the conditional cases have a lower COVFS and a close FS to the unconditional cases ( FS is mainly above 1), leading to a significantly higher level of slope reliability. In addition, the two-sided effect of the varying correlation length can be comprehensively considered by using the CRF method, especially when the correlation length is relatively large. Such a consideration indeed leads to a modification to the related influence law.

(3) The evaluation of a sampling scheme also adopts the standards just mentioned, i.e. the variability of field form, the sampling efficiency, and the variability of slope stability. It’s found that taking more sampling points into account (i.e. considering more boreholes in this study) and setting the sampling points at more influential locations (i.e. the locations where the failure is more likely to occur) lead to a better performance of the sampling scheme. What’s more, the above measures make more key information of the real field reflected in CRFs, so the FS estimations of CRFs will center on FS of the real field to a greater extent. This will benefit the estimation of the mean level of slope stability and the slope reliability.

(4) With varying statistical parameters of the random field, the general field form changes according to the mathematical meaning of the parameters. For example, with a greater correlation length, the correlation zones tend to extend farther. The statistical parameters in this study have no significant influence on the mean level of slope stability, except that a greater COV of Su leads to a lower mean level. The influence laws concerning the variability of field form and the variability of slope stability conform to each other to a great extent. The law about slope reliability is a subsequent outcome. The specific influence laws are presented as the following. Firstly, with increasing l or  , the variability first rises then turns gentle or even drops, and the slope Su Su reliability goes down at first then probably goes up or tends to be steady when the majority of FS is above 1 (the changing tendency is the opposite when the majority of FS is below 1); with increasing COV , the variability keeps rising, and the slope reliability significantly decreases. Su

Secondly, with c, changing from -0.7 to 0.5, i.e. with the cross-correlation becoming less negative or more positive, the variability increases and the slope reliability decreases. Thirdly, the effect of l and COV on the variability follows the laws concerning S . From the above Ks K s u

110 influence laws, we can tell with what kind of statistical characteristics the slope stability and the field form tend to be more consistent and the slope tends to be more reliable.

(5) Considering the failure consequence coupled with the failure probability when defining the slope reliability doesn’t make sense too much. The average failure consequence, i.e. the average expectation of slope failure’s damage, is not sensitive to the statistical parameters. Thus the influence law about the “risk” is almost identical to that about Pf . So Pf is good enough for characterizing the slope reliability and the modification using the “risk” is not recommended.

(6) Other things being equal, the overall flux level in the dam slope is negatively related to both l and COV . With l going up, the variability of seepage situation first rises then descends. Ks K s Ks With rising COV , the variability of seepage situation constantly increases. K s

6.2 Recommendations

With consideration of the limitations of existing research and this study, here are some recommendations for future studies.

(1) The methodology for probabilistic analysis needs improvement. In this study, the direct Monte Carlo simulation is realized by manually repeating the calculation process of FS . The MCS based on the unconditional random field is already available by the built-in function of OptumG2, which is worth extending to the conditional application. What’s more, it also makes sense to explore the application of the subset simulation (SS) to improve the calculation efficiency and resolution, especially for cases with a small failure probability. In addition, the response surface method (RSM) deserves studying to find a balance between pursuing efficiency and ensuring accuracy.

(2) There are various failure modes for probabilistic analysis, and some of them are spatially complicated. The spatial variability of failure surface is not thoroughly investigated in this study. It is worth further study with the application of CRF.

(3) For analysis of the deterministic stage, the calculation efficiency and accuracy of different methods need a comprehensive comparative analysis. Common methods include the finite element method (FEM), the limit equilibrium method (LEM), the discrete element method (DEM), and the numerical limit analysis (NLA) method.

(4) The determination of the type of variable’s distribution and the type of covariance function is another essential issue in the application of random field theory. The lognormal distribution and

111 three different covariance functions are adopted in this study. The rationality of the selection of them needs some deeper discussion.

(5) The slope models in this study are hypothetical and simplified. Further studies can be carried out based on some more complex models, such as a slope with non-stationary property field, a multi-layered slope, and slopes with other environmental or geotechnical conditions. What’s more, the performance of the method framework presented in this study in practical engineering has yet to be tested.

112 References

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