Role of uncertainity in soil hydraulic properties in rainfall induced landslides

ROLE OF UNCERTAINTY IN SOIL HYDRAULIC PROPERTIES IN RAINFALL-INDUCED LANDSLIDES ANASTASIA MARIA SANTOSO NATIONAL UNIVERSITY OF SINGAPORE 2011 ROLE OF UNCERTAINTY IN SOIL HYDRAULIC PROPERTIES IN RAINFALL-INDUCED LANDSLIDES ANASTASIA MARIA SANTOSO (B. Eng, Institute of Technology Bandung) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2011 Who carves a channel for the downpour, and hacks a way for the rolling thunder, so that rain may fall on lands …? Whose skill details every cloud and tilts the flasks of heaven until the soil cakes into a solid mass and clods of earth cohere together? The LORD’s reply to Job, Job 38: 25-26, 37-38 (The Jerusalem Bible) DEDICATION To Er. Timotius Santoso and Ellen D. Widjaja, M.D. My dear Po and Mo, When I remember the guidance and kindness I have received during my PhD study, I feel almost ungrateful to my professors in offering this thesis not to them, but to you. But it cannot be otherwise1. For I consider this thesis, humble though it is, not a mere product of four years of study, but a part of my life’s work. Hence it seems right that it shall be dedicated to those to whom I owe so much in life. For all the good things that I have enjoyed, which of those does not come through you? A high regard for education and learning, the will to our best, the faith that has ever sustained me, the maxim that we should make our own path instead of following others’: all these things I have learnt from you. And what parents could be more generous than you? Few parents let their daughter go off to pursue her dreams, and an academic pursuit must have seemed strange to entrepreneurs like you. Yet you have stood by me through the dark days of Qualifying Examination preparation, shared my joy when my papers were published (though you did not fail to utter your astonishment upon knowing that publications bring no direct financial gain!), and even managed to remain proud of me and my choices. So I humbly hope that the close of my PhD journey may bring you satisfaction and joy. It surely brings a great joy to me, but still greater is the joy that comes from the privilege to remain, dearest Father and Mother, Your little daughter, Anastasia This style of dedication was used by C.S. Lewis in his Preface to Paradise Lost. I employ it here as it suits my purpose. i ACKNOWLEDGEMENT I would like to express my gratitude to my advisors, Professor Quek Ser Tong and Professor Phoon Kok Kwang, for their guidance and encouragement throughout my PhD study. Working with them has been rewarding and enjoyable, though certainly not easy. Through many pleasant conversations with them, I have also learnt many things beyond academic matters. I would like to thank the examiners of my thesis, Associate Professor Harry Tan Siew Ann (National University of Singapore), Dr. Michael Beer (National University of Singapore), and Professor Craig H. Benson (University of Wisconsin-Madison), for spending their valuable time reading my thesis and making insightful suggestions. Special thanks are also due to Dr. Beer and Dr. Goh Siang Huat (National University of Singapore) for their kind encouragements throughout my graduate study. I would like to thank Dr. Cheng Yonggang, not only for generously sharing with me his knowledge on unsaturated seepage, but also for his friendship. I have also learned much about rainfall-induced landslides from Dr. Muthusamy Karthikeyan, and I thank him for this. The research scholarship and the President Graduate Fellowship from the National University of Singapore are gratefully acknowledged. My gratitude also goes to those from my former university (Institute of Technology Bandung) who first inspired me to pursue a doctoral study. Chief among them is Associate Professor Sindur P. Mangkoesoebroto, who first showed me that an academic life can be rewarding. Professor Bambang Budiono has also encouraged me to continue my study, and Associate Professor Indra Djati Sidi further encouraged my interest in reliability. I cherish the warm friendship of fellow research students in the structural and geotechnical group. I choose not to mention names, lest I forget some dear friends. Yet one could not be left unmentioned: Ms. D.D. Thanuja Krishanthi Kulathunga. I am deeply grateful to my sister, Mady Naomi, M.D., and her sweet little family. Our chats have always reminded me that there is more to life than research and study. Lastly, my warm gratitude goes to my dear Chris (Mr. Christian S. Sanjaya), who has been a true friend who overlooks my failures and rejoices in my successes. ii TABLE OF CONTENTS TITLE PAGE DEDICATION i ACKNOWLEDGMENTS ii TABLE OF CONTENTS iii viii SUMMARY LIST OF TABLES x LIST OF FIGURES xi LIST OF SYMBOLS xvi Chapter Introduction 1.1. Rainfall-induced Slope Failures 1.2. Reliability Analysis of Unsaturated Slope 1.3. Objectives of Study 1.4. Organization of Thesis 12 Chapter Literature Review 15 2.1. Uncertainty in Analysis of Rainfall-induced Landslides 15 2.2. Probability Model of Uncertain Soil Properties 21 2.2.1.Soil-Water Characteristic Curve 23 2.2.2. Soil Saturated Hydraulic Conductivity 24 2.2.3. Other Soil Properties 25 2.3. Simulation of Uncertain Soil Properties 27 2.4. Reliability Assessment 29 iii 2.4.1. Reliability Estimation Techniques 32 2.4.2. Subset Simulation 37 2.5. Concluding Remarks 38 Chapter Unsaturated Seepage and Slope Stability Analysis 39 3.1. Governing Equation of Seepage Through Unsaturated Soil 41 3.1.1. Constitutive Equations of Unsaturated Soil 3.2. Transient Analysis of Unsaturated Seepage 43 46 3.2.1. Finite Element Formulation of Richards Equation 46 3.2.2. Program THFELA 49 3.3. Steady State Analysis of Unsaturated Seepage 52 3.4. Stability of Unsaturated Infinite Slope 54 3.5. Validation of Numerical Models 58 3.5.1.Validation of Finite Element Seepage Analysis 58 3.5.3. Validation of Infinite Slope Model 62 3.6. Performance Function in Probabilistic Analysis of Slope 3.6.1. Illustration of Multiple Failure Modes 65 66 3.7. Concluding Remarks 69 Chapter Probabilistic Characterization of Soil Properties 70 4.1. Available Probability Models of SWCC 73 4.2. SWC Curve-Fitting 77 4.2.1. Measurement Data 77 4.2.2. SWCC Equations 78 4.2.3. Determination of Curve-Fitting Parameters 80 iv 4.2.4. Statistics of SWCC Parameters 4.3. Lognormal Joint Probability Model of SWCC 82 87 4.3.1. Lognormal Random Variable 88 4.3.2. Lognormal Random Vector 90 4.4. Validation of Lognormal Random Vector Model 91 4.4.1. Normalized SWCC 91 4.4.2. Non-normalized SWCC 98 4.5. Alternate Probability Models of SWCC 4.6.Probability Model of Saturated Hydraulic Conductivity 99 100 4.6.1. Marginal Distribution of ks 100 4.6.2. Spatial Correlation of ks 104 4.7. Concluding Remarks 109 Chapter Modified Metropolis-Hastings Algorithm for Efficient 111 Subset Simulation 5.1. Subset Simulation 114 5.2. Original Metropolis-Hastings Algorithm 117 5.3. Modified Metropolis-Hastings Algorithm 124 5.4. Verification of Modified Metropolis-Hastings Algorithm 127 5.4.1. Transition Probability 128 5.4.2. Reversibility condition 131 5.4.3. Chain-correlation 132 5.4.4. Error of estimators 134 5.5. Infinite Slope Examples 139 5.5.1. Example 1: Undrained Analysis v 139 5.5.2. Example 2: Transient seepage analysis 143 5.6. Concluding Remarks 145 Chapter Effects of Soil Spatial Variability on Seepage and Slope 147 Stability 6.1. Steady State Seepage – Saturated Soils 151 6.1.1. Flux Boundary Problem 152 6.1.2. Head Boundary Problem 156 6.2. Steady State Seepage – Unsaturated Soils 158 6.2.1. Clayey Soil 159 6.2.2. Sandy Soil 165 6.3. Transient Seepage – Unsaturated Soils 165 6.3.1. Clayey Soil 166 6.3.2. Sandy Soil 169 6.4. Factor of Safety and Probability of Slope Failure 171 6.4.1. Factor of Safety 171 6.4.2. Probability of Slope Failure 174 6.5. Uncertain SWCC 177 6.5.1. Uncertain SWCC and Spatially Variable ks 178 6.5.2. Uncertain SWCC and Deterministic ks 183 6.6. Concluding Remarks 184 Chapter Conclusions 187 7.1. Conclusions 187 7.2. Recommendations 190 vi References 192 Appendices 206 Appendix A. SWCC Parameters of Clayey Soils 206 Appendix B. Counter Example for Modified Metropolis-Hastings 207 Algorithm Appendix C. Pressure Head for Infiltration Flux Exceeding Saturated 209 Hydraulic Conductivity Appendix D. List of Publications 212 vii International Journal for Numerical Methods in Engineering, 52: 10291043. 2001. Hurtado, J.E. Structural Reliability: Statistical Learning Perspectives. Springer, Berlin. 2004. Johnson, G., Crumbley, W.S., and Boutwell, G.P. Field verification of clay liner hydraulic conductivity, In: Waste Containment Systems: Construction, Regulation, and Performance, ASCE, 226-245. Johnson, N. L., Kotz, S. & Balakrishnan, N. Continuous Univariate Distributions, Volume 1, John Wiley & Sons, New York. 1994. Kool, J.B. & Parker, J.C. Development and evaluation of closed-form expressions for hysteretic soil hydraulic properties, Water Resources Research, 23(1): 105-114. 1987. KSTAT Standards Committee [H.A. Loaiciga (chair)], Standard Guideline for Fitting Saturated Hydraulic Conductivity using Probability Density Functions, ASCE / EWRI Standard 50-2008, ASCE, Reston. 2008a. KSTAT Standards Committee [H.A. Loaiciga (chair)], Standard Guideline for Calculating the Effective Saturated Hydraulic Conductivity, ASCE / EWRI Standard 51-2008. 2008b. Lambe, T.W. and Whitman, R.V. Soil Mechanics, New York: John Wiley & Sons. 1969. Leij, F.J., Alves, W.J., van Genuchten, M.Th., and Williams, J. R. The UNSODA Unsaturated Soil Hydraulic Database User’s Manual, version 1.0. EPA-600-R-96-095, U.S. EPA, Cincinnati, Ohio. 1996. Leong, E.C. and Rahardjo, H. Review of soil water characteristic curve equations, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 123(12):1106-1117. 1997a. Leong, E.C. and Rahardjo, H. Permeability functions for unsaturated soils, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 123(12):1118-1126. 1997b. Li, K.S. and Lumb, P. Probabilistic design of slopes, Canadian Geotechnical Journal, 24:520-535. 1987. Lu, N. and Godt, J. Infinite-slope stability under steady unsaturated seepage conditions, Water Resources Research, 44, W11404. 2008. Lu, N. and Griffiths, D.V. Profiles of steady-state suction stress in unsaturated soils, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 130(10): 1063-1076. 2004. Lu, N. and Likos, W.J. Unsaturated Soil Mechanics, New York: Wiley. 2004. 198 Luckner, L, van Genuchten, M.Th. & Nielsen, D.R. A consistent set of parametric models for the two-phase flow of immiscible fluids in the subsurface, Water Resources Research 25(10): 2187-2193. 1989. Ma, H.-F. and Ang, A.H.-S., Reliability analysis of redundant ductile structural systems. Structural Research Series No. 494, University of Illinois at Urbana-Champaign. 1981. Melchers, R.E. Importance sampling in structural systems, Structural Safety, 6:3-10. 1987. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., and Teller, A.H. Equation of state calculations by fast computing machines, The Journal of Chemical Physics, 21(6):1087-1092. 1953. Miller, C.J., Yesiller, N., Yaldo, K., and Merayyan, S. Impact of soil type and compaction conditions on SWCC, Journal of Geotechnical and Geoenvironmental Engineering, 128(9), 733-742. 2002. Montgomery, D.C. and Runger, G.C. Applied Statistics and Probability for Engineers, John Wiley & Sons, Singapore. 2003. Morgenstern, N.R. and Price, V.E. The analysis of the stability of general slip surfaces. Geotechnique, 15(1): 79-93. 1965. Mostyn, G.R. and Li, K.S. Probabilistic slope analysis – state-of-play, In Probabilistic Methods in Geotechnical Engineering, Eds. K.S. Li and S.C.R. Lo. Balkema, Rotterdam. 1993. Mualem, Y. A new model for predicting the hydraulic conductivity of unsaturated porous media, Water Resources Research, 12:593-622. 1976. Okimura, T., Torii, N., Osaki, Y., Nanbu, M. and Haraguchi, K. Improvement of prediction accuracy of system of real-time type hazard map of slope failure disasters caused by heavy rainfalls. In Proceedings of the 17th Southeast Asian Geotechnical Conference, Li.J.C.C and Lin, M.L (Eds), Volume 2, pp. 189-192. 2010. Olson, R.E. and Daniel, D.E. Measurement of the hydraulic conductivity of fine-grained soils, Permeability and Groundwater Contaminant Transport, ASTM Special Tech. Publ. 746, Eds. T.F. Zimmie and C.O. Riggs. 1981. Pan, L. and Wierenga, P.J. A transformed pressure head-based approach to solve Richards’ equation for variably saturated soils, Water Resources Research 31(4):925-931. 1995. Paniconi, C. and Putti, M. A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems, Water Resources Research, 30(12):3357-3374. 1994. 199 Papadimitriou, C., Beck, J.L. and Katafygiotis, L.S. Asymptotic expansions for reliabilities and moments of uncertain dynamic systems, Journal of Engineering Mechanics, ASCE, 123(12): 1219-1229. 1997. Parent, S.E., Cabral, A., & Zornberg, J.G. Water retention curve and hydraulic conductivity function of highly compressible materials, Canadian Geotechnical Journal, 44, 1200-1214. 2007. Phoon, K-K., Kulhawy, F. H., and Grigoriu, M. D. Reliability based design of foundations for transmission line structures. Electric Power Research Institute, Palo Alto, Report TR-105000. 1995. Phoon, K.K. and Kulhawy, F.H. Characterization of geotechnical variability, Canadian Geotechnical Journal, 36(4): 612-624. 1999a. Phoon, K. K. and Kulhawy, F.H. Evaluation of geotechnical property variability, Canadian Geotechical Journal, 36(4): 625-639. 1999b. Phoon, K.K., Huang, S.P. and Quek, S.T. Simulation of second-order processes using Karhunen-Loeve expansion, Computers and Structures, 80(12): 1049-1060. 2002a. Phoon, K. K., Huang S. P. & Quek, S. T. Implementation of Karhunen-Loeve expansion for simulation using a wavelet-Galerkin scheme, Probabilistic Engineering Mechanics, 17(3): 293-303. 2002b. Phoon, K.K., Quek, S.T. and An, P. Identification of statistically homogeneous soil layers using modified Bartlett statistics, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 129(7): 649659. 2003. Phoon, K. K. General Non-Gaussian Probability Models for First-Order Reliability Method (FORM): A State-of-the-Art Report, ICG Report 20042-4 (NGI Report 20031091-4), International Centre for Geohazards, Oslo. 2004. Phoon K. K., Huang, H. W. and Quek, S. T. Simulation of strongly nonGaussian processes using Karhunen-Loeve expansion, Probabilistic Engineering Mechanics, 20(2), 188-198. 2005. Phoon K. K. Bootstrap estimation of sample autocorrelation functions. In Proc. GeoCongress 2006: Geotechnical Engineering in the Information Technology Age, Eds. D.J. DeGroot, J.T. DeJong, J.D. Frost & L.G. Braise, ASCE, Reston, CDROM. 2006a. Phoon, K.K. Modeling and simulation of stochastic data. In Proc. GeoCongress 2006: Geotechnical Engineering in the Information Technology Age, Eds. D.J. DeGroot, J.T. DeJong, J.D. Frost & L.G. Braise, ASCE, Reston, CDROM. 2006b. Phoon, K.K., Chen, J. R. & Kulhawy, F. H. Characterization of model uncertainties for augered cast-in-place (ACIP) piles under axial 200 compression. In Foundation Analysis & Design: Innovative Methods, Geotechnical Special Publication 153, Eds. R. L. Parsons, L. M. Zhang, W. D. Guo, K. K. Phoon & M. Yang, ASCE, Reston, 82-89. 2006. Phoon, K. K., Chen, J. R. & Kulhawy, F. H. Probabilistic hyperbolic models for foundation uplift movement. In Probabilistic Applications in Geotechnical Engineering, Geotechnical Special Publication 170, Eds. K. K. Phoon, G. A. Fenton, E. F. Glynn, C. H. Juang, D. V. Griffiths, T. F. Wolff & L. M. Zhang, ASCE, Reston, CDROM. 2007. Phoon, K.K.(Ed). Reliability-based Design in Geotechnical Engineering: Computations and Applications. Taylor and Francis. 2008. Phoon, K.K., Santoso, A. M and Cheng, Y.G. Probabilistic Analysis of Soil Water Characteristic Curves from Sandy Clay Loam. In Proceedings of GeoCongress 2008: The Challenge of Sustainability in the Geoenvironment, Geotechnical Special Publication 178, ASCE, Reston, VA. 2008. Phoon, K.K., Toll, D.G., and Karthikeyan, M. Study on the Effects and Impacts of Climate Change on Singapore, Volume V: Slope Stability. Tropical Marine Science Instistute, National University of Singapore. 2009. Phoon, K.K., Santoso, A.M. and Quek, S.T. Probabilistic analysis of soil water characteristic curves. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 136 (3): 445-455. 2010. Popescu, R., Deodatis, G. and Prevost, J.H. Simulation of homogeneous nonGaussian stochastic vector fields, Probabilistic Engineering Mechanics, 13(1): 1-13. 1998. Puig, B., Poirion, F. and Soize, C. Non-Gaussian simulation using Hermite polynomial expansion: convergences and algorithms, Probabilistic Engineering Mechanics, 17(3): 253-264. 2002. Quek, S.T. and Ang, A.H.-S., Reliability analysis of structural systems by stable configuration, Journal of Structural Engineering, ASCE, 116(10):2656-2670. 1990. Rackwitz, R. Reviewing probabilistic soils modelling, Computers and Geotechnics, 26: 199-223. 2000. Rahardjo, H., Lim, T.T., Chang, M.F. and Fredlund, D.G. Shear strength characteristics of a residual soil, Canadian Geotechnical Journal, 32: 6077. 1995. Rahardjo, H., Li, X.W., Toll, D.G., and Leong, E.C. The effect of antecedent rainfall on slope stability, Geotechnical and Geological Engineering, 19:371-399. 2001. 201 Rahardjo,H., Ong,T.H., Rezaur, R.B., and Leong, E.C. Factors controlling instability of homogeneous soil slopes under rainfall, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 133(12): 15321543. 2007. Rezaur, R.B., Rahardjo, H., Leong, E.C. and Lee T.T. Hydrologic behavior of residual soil slopes, Journal of Hydrologic Engineering, ASCE, 8(3): 133144. 2003. Roberts, G. O. and Rosenthal. J.S. Examples of adaptive MCMC, Journal of Computational and Graphical Statistics, 18(2): 349-367. 2009. Rubinstein, R.Y. Simulation and the Monte-Carlo Method. New York: John Wiley and Sons, Inc. 1981. Santoso, A.M., Phoon, K.K. and Quek, S.T. Flow of water through spatially heterogeneous soil, In Proceedings of the 17th South East Asian Geotechnical Conference: Geo-engineering for Natural Hazard Mitigation and Sustainable Development, Eds. J.C.C. Li and M-L. Lin, Taiwan Geotechnical Society, Taiwan, 2010, Volume II, pp. 249 – 253. 2010 a. Santoso, A.M., Phoon, K.K. and Quek, S.T. System reliability estimation using subset simulation with modified Metropolis-Hastings algorithm. In Proceedings of the 5th ASRANet International Conference, Edinburgh, CDROM. 2010 b. Santoso, A.M., Phoon, K.K. and Quek, S.T. Modified Metropolis-Hastings algorithm with reduced chain-correlation for efficient subset simulation, Probabilistic Engineering Mechanics, doi: 10.1016/j.probengmech.2010.08.007. 2010 c. Santoso, A.M., Phoon K.K. and Quek, S.T., Probability models for SWCC and hydraulic conductivity, 14th Asian Regional Conference on Soil Mechanics and Geotechnical Engineering, May 23 – 27, 2011, Hong Kong, China. 2011 a. Santoso, A.M., Phoon K.K. and Quek, S.T., Effects of spatial variability on rainfall-induced slope failure, Sixth MIT Conference on Computational Fluid and Solid Mechanics, June 15-17, 2011, Massachusetts, USA (to be published). 2011 b. Sakamoto, S. and Ghanem, R. Polynomial chaos decomposition for the simulation of non-Gaussian nonstationary stochastic processes, Journal of Engineering Mechanics, ASCE, 128(2): 190-200. 2002. Schueller, G.I. and Stix, R. A critical appraisal of methods to determine failure probabilities, Structural Safety, 4:293-309. 1987. Schueller, G.I., Bucher, C.G., Bourgund, U. And Ouypornprasert, W. On efficient computational schemes to calculate structural failure probabilities, Probabilistic Engineering Mechanics, 4(1): 10-18. 1989. 202 Schueller, G.I., Pradlwarter, H.J., and Koutsourelakis, P.S. A critical appraisal of reliability estimation procedures for high dimensions, Probabilistic Enginering Mechanics, 19:463-474. 2004. Schueller, G.I. and Pradlwarter, H.J. Benchmark study on reliability estimation in higher dimensions of structural systems – an overview, Structural Safety, 29:167-182. 2007. Schweizer, B. Thirty years of copulas. In Advances in Probability Distributions with Given Marginals, Kluwer, Dordrecht, Netherlands, pp. 13-50. 1991. Sheng, D., Gens, A., Fredlund, D.G. and Sloan, S.W. Unsaturated soils: from constitutive modelling to numerical algorithms, Computers and Geotechnics, 35: 810-824. 2008. Shinozuka, M. and Deodatis, G. Simulation of stochastic processes by spectral representation, Applied Mechanics Reviews, 44(4): 191-203. 1991. Sillers W.S. and Fredlund D.G. Statistical assessment of soil-water characteristic curve models for geotechnical engineering, Canadian Geotechnical Journal, 38:1297-1313. 2001. Soralump, S. Rainfall-triggered landslide: from research to mitigation practice in Thailand. In Proceedings of the 17th Southeast Asian Geotechnical Conference, Li.J.C.C and Lin, M.L (Eds), Volume 2, pp. 153-158. 2010. Srinilta, S., Nielsen, D.R. and Kirkham, D. Steady flow of water through a two-layer soil, Water Resources Research, 5: 1053-1063. 1969. Srivastava, R. and Yeh, J. T.-C. Analytical solutions for one dimensional transient infiltration toward the water table in homogeneous and layered soils, Water Resources Research, 27(5): 753-762. 1991. Sudicky, E.A. A natural gradient experiment on solute transport in a sand aquifer: spatial variability of hydraulic conductivity and its role in the dispersion process, Water Resources Research, 22(13): 2069-2083. 1986. Swartzendruber, D. Water flow through a soil profile as affected by the least permeable layer, Journal of Geophysical Research, 65(12): 4037-4042. 1960. Takagi, S. Analysis of the vertical downward flow of water through a twolayered soil, Soil Sciences, 90: 98-103. 1960. Tan, T.S., Phoon, K.K. and Chong, P.C. Numerical study of finite element method based solutions for propagation of wetting fronts in unsaturated soil, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 130(3):254-263. 2004. Tang, W.H, Yucemen, M.S., and Ang, A.H.-S. Probability-based short term design of soil slopes, Canadian Geotechnical Journal, 13:201-215. 1976. 203 Tang, W.H., Zhang, J. and Zhang, L.M. Characterizing geotechnical model uncertainty. In Proc. 10th International Conference on Structural Safety and Reliability, Osaka, Japan. H. Furuta, D.M. Frangopol and M. Shinozuka, eds. Taylor and Francis, 2010. Thomas, H.R. and Zhou, Z. Minimum time-step size for diffusion problem in FEM analysis, International Journal of Numerical Methods in Engineering, 40:3865-3880. 1997. Tierney, L., Markov chains for exploring posterior distributions, The Annals of Statistics 22(4): 1701-1728. 1994. Tierney, L. and Mira, A. Some adaptive Monte Carlo methods for Bayesian inference, Statistics in Medicine 18: 2507-2515. 1999. Toll, D.G. Rainfall-induced landslides in Singapore, Proc. Institution of Civil Engineers: Geotechnical Engineering, 149(4):211-216. 2001. Tsaparas, I., Rahardjo, H., Toll, D.G. and Leong, E.C. Controlling parameters for rainfall-induced landslides, Computers and Geotechnics, 29: 1-27. 2002. Tsaparas, I., Rahardjo, H., Toll, D.G. and Leong, E.C. Infiltration characteristics of two instrumented residual soil slopes, Canadian Geotechnical Journal, 40(5): 1012-1032. 2003. Uzielli, M. & Phoon K. K. Some observations on assessment of Gaussianity for correlated profiles. In Proceedings of GeoCongress 2006: Geotechnical Engineering in the Information Technology Age, Eds. D.J. DeGroot, J.T. DeJong, J.D. Frost & L.G. Braise, ASCE, Reston, CDROM. 2006. Van Genuchten, M.T. A close-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Science Society of America Journal, 44(5):892-898. 1980. Vanapalli, S.K. and Fredlund, D.G. Comparison of different procedure to predict unsaturated soil shear strength, Proc. Advances in Unsaturated Geotechnics, GSP No. 99, C.D. Shackelford, S.L Houston and N.Y. Chang, Eds., ASCE, Reston, Va., pp. 195-209. 2000. Vanmarcke, E.H. Probabilistic modeling of soil profiles, Journal of the Geotechnical Engineering Division, 103:1227-1246. 1977a. Vanmarcke, E.H. Reliability of earth slopes, Journal of the Geotechnical Engineering Division, ASCE, 103:1247-1265. 1977b. Vanmarcke, E.H. Random Fields: Analysis and Synthesis. MIT Press, Cambridge, Massachusetts. 1983. Wang, Y., Cao, Z., and Au, S.K. Efficient Monte Carlo Simulation of parameter sensitivity in probabilistic slope stability analysis. Computers and Geotechnics, 37(7-8): 1015-1022. 2010. 204 Warrick, A.W. and Yeh, T.-C. J. One-dimensional, steady vertical flow in a layered soil profile, Advances in Water Resources, 13(4): 207-210. 1990. Yang, J., Zhang, R., Wu, J. and Allen, M.B. Stochastic analysis of adsorbing solute transport in two-dimensional unsaturated soils, Water Resources Research, 32(9): 2747-2756. 1996. Yang, Tsung-yuan. Random Vibration of Structures. New York, Wiley .1986. Yeh, T.-C. J., Gelhar, W.L. and Gutjahr, A.L. Stochastic analysis of unsaturated flow in heterogeneous soils: 1. Statistically isotropic media, Water Resources Research, 21(4): 447-456. 1985 a. Yeh, T.-C. J., Gelhar, W.L. and Gutjahr, A.L. Stochastic analysis of unsaturated flow in heterogeneous soils: 2. Statistically anisotropic media with variable α, Water Resources Research, 21(4): 457-464. 1985 b. Yeh, T.-C. J., Gelhar, W.L. and Gutjahr, A.L. Stochastic analysis of unsaturated flow in heterogeneous soils: 3. Oservations and applications, Water Resources Research, 21(4): 465-471. 1985 c. Yeh, T.-C.J. One-dimensional steady infiltration in heterogeneous soils, Water Resources Research, 25(10): 2149-2158. 1989. Zaslavsky, D. Theory of unsaturated flow into a non-uniform soil profile, Soil Sciences, 97: 400-410. 1964. Zhang, L.L., Zhang, L. M., and Tang, W.H. Importance of considering correlations among parameters of soil-water characteristic curve. In Applications of Statistics and Probability in Civil Engineering, Eds. Der Kiureghian, Madanat, and Pestana, Millpress, Rotterdam. 2003. Zhang, L.L., Fredlund, D.G., Zhang, L.M., and Tang, W.H. Numerical study of soil conditions under which matric suction can be maintained, Canadian Geotechnical Journal, 41: 569-582. 2004. Zhang, L.L., Zhang, J., Zhang, L.M., and Tang, W.H. Back analysis of slope failure with Markov chain Monte Carlo simulation, Computers and Geotechnics, 37(7-8): 905-912. 2010. 205 Appendix A. SWCC Parameters of Clayey Soils It is stated in Section 4.3 that the SWCC parameters “a” and “n” of clayey soils can be modeled by a lognormal and shifted lognormal distribution, respectively. The histogram and the lognormal distribution fit for clayey soils are shown in Fig. A.1. The dotted curves represent the lognormal distribution. 0.8 Mean = S.D. = COV = Skewness = Kurtosis = N= p-value= 0.6 0.4 0.2 0.314 kPa-1 0.267 kPa-1 0.85 0.705 2.473 17 0.032 Relative Frequency Relative Frequency 0.8 0.0 0.4 0.2 a parameter (kPa-1) 0.4 0.2 Mean = S.D. = COV = Skewness = Kurtosis = N= p-value> 1.093 0.035 0.03 1.048 4.414 17 0.25 0.0 0.8 Relative Frequency Relative Frequency 0.6 0.6 0.471 kPa-1 0.359 kPa-1 0.76 0.744 2.642 24 0.03 0.0 0.8 Mean = S.D. = COV = Skewness = Kurtosis = N= p-value= 0.6 0.4 0.2 a parameter (kPa-1) Mean = S.D. = COV = Skewness = Kurtosis = N= p-value> 1.123 0.046 0.04 0.448 2.561 24 0.25 0.0 1.0 1.1 1.2 1.3 1.4 1.5 1.6 n parameter 1.0 1.1 1.2 1.3 1.4 1.5 1.6 n parameter (a) (b) Fig. A.1. Empirical distributions of SWCC parameters: (a) Clay (b) Silty clay. 206 Appendix B. Counter Example for the Modified Metropolis-Hastings Algorithm In Chapter 5, it is stated that the Metropolis-Hastings cannot be modified in an arbitrary way. Any modified algorithm must have a transition probability which satisfies the reversibility condition and the simulated samples must follow the target distribution. It has been shown that the modified MetropolisHastings algorithm proposed in this study satisfies both requirements. A counter example is presented here to illustrate that a transition probability that does not satisfy the reversibility condition will produce simulated samples which deviate from the target distribution. For the target distribution of p(X | X < ci), the algorithm of the counter example is as follows: 1. Start from an initial sample X1 = x < ci. 2. Generate a pre-candidate sample y following the proposal PDF. 3. Accept y as the candidate sample with probability α = min{1, φ(y)IFi(y)/φ(x)}. This can be done in steps: a. Generate u~uniform[0,1]. If u < min{1,φ(y)/φ(x)}, take y as the candidate sample, X ' = y. Otherwise return to Step 2. b. Check the location of X ' (requires performance function calculation). If y < ci, accept it as the next sample along the chain, X2 = y. Otherwise return to Step 2. 4. Repeat steps – to get the next sample X3,…, XN / Nc along the chain. N / Nc is the chain length. 207 The reversibility condition is measured by the infinity norm of matrix (T –T '). For this counter example, the infinity norm of matrix (T – T ') obtained from numerical simulation with a sample size of 10000 is 0.68. This is far from the norm of matrix T for the original and the modified algorithm, which is around 0.07. symmetric. This high value of norm indicates that matrix T is not Thus, the counter example does not satisfy the reversibility condition. To verify the distribution of the simulated conditional samples, a single realization of the Markov chain with length of 1000 is simulated. The target distribution is a truncated standard normal, X | X < -1.28. Figure B.1. shows the empirical CDF of Markov chain samples simulated using the original, modified, and the counter algorithm. The analytical CDF of a truncated normal distribution is also shown. As apparent from the figure, the original and the modified algorithm follow the target distribution while the counterexample does not. 1.0 Analytical Original Modified Counter example F(X) 0.8 0.6 0.4 0.2 0.0 -4 -3 -2 -1 X Fig. B.1. Comparison of the empirical cumulative distribution of samples simulated using original, modified, and counter Metropolis-Hastings algorithm. 208 Appendix C. Pressure Head for Infiltration Flux Exceeding Saturated Hydraulic Conductivity To study the effects of spatial variability, a numerical example illustrating transient seepage through unsaturated soil with spatially variable ks was presented in Section 6.3. The infiltration flux is fixed at half of the mean value of ks, q = -0.5 ks. The finite element code THFELA is used to perform the transient flow analysis. There are some realizations in which the infiltration flux is higher than the conductivity of the top layer, ks,nl < |q|. THFELA handles such case by changing the boundary condition. Instead of using flux boundary condition of q at the surface, a head boundary condition of h(L) = is used. This head boundary simulates a saturated surface. Thus, in this example with q = -0.5 ks, 34% of the realizations are analyzed with head boundary condition and the rest with flux boundary condition. To ensure that the results show the effects of spatial variability, and not obscured by additional effects of different boundary conditions, only realizations with flux boundary condition (realizations with ks,nl > |q|) were studied in Section 6.3. Hence, the quantiles of pressure head shown in Fig. 6.11 are conditional on ks,nl > |q|. To study the impact of this condition, the pressure head profiles obtained from all realizations of ks are shown in Fig. C.1. 209 z(m) z(m) z(m) -5 -4 -3 -2 -1 Q 25% of h (m) -5 -4 -3 -2 -1 Q 50% of h (m) -5 -4 -3 -2 -1 Q 75% of h (m) -4 -3 -2 -1 Q 75% of h (m) z(m) z(m) z(m) t = days -5 -4 -3 -2 -1 Q 25% of h (m) -5 -4 -3 -2 -1 Q 50% of h (m) =0.1 =0.4 =10 r.var z(m) z(m) z(m) t = 12 days -4 -3 -2 -1 Q 25% of h (m) -3 -2 -1 Q 50% of h (m) -3 -2 -1 Q 75% of h (m) t = 20 days Fig. C.1. (Unconditional) quantiles of pressure head obtained from unsaturated transient seepage analysis with clayey soil and q/μks = -0.5. Comparison of Fig. C.1 (all realizations of ks are considered) to Fig. 6.11 (only realizations that satisfy ks,nl > |q| are considered) shows that the effects of spatial variability is less apparent when all realizations are studied. Up to 12 days, pressure head profile obtained from different values of correlation length and even from random variable model are almost the same. Note that realizations with ks,nl > |q| are analyzed as flux boundary problem while the 210 remaining realizations as head boundary problem. The saturated case in Section 6.1 has highlighted that the relation of correlation length to pressure head may show opposite trend for flux boundary and head boundary problems. This may partially explain why the effects of spatial variability are less clear in Fig. C.1. At elapsed time of 20 days, particularly for Q 75%, it can be seen that soil with higher correlation length of ln ks will have deeper wetting front and higher remaining suction. This is the same trend as observed in the quantiles of pressure head in flux boundary problem (Fig. 6.11). 211 Appendix D. List of Publications Journal papers and special publications: Santoso, A.M., Phoon, K.K. and Quek, S.T. Modified Metropolis-Hastings algorithm with reduced chain-correlation for efficient subset simulation, Probabilistic Engineering Mechanics, 26 (2): 331 – 341. 2011. Santoso, A.M., Phoon K.K. and Quek, S.T., Effects of spatial variability on rainfall-induced slope failure, In Proc. of the Sixth MIT Conference on Computational Fluid and Solid Mechanics, Computers and Structures 89 (11 – 12): 893 – 900. 2011. Phoon, K.K., Santoso, A.M. and Quek, S.T. Probabilistic analysis of soil water characteristic curves, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 136 (3): 445-455. 2010. Santoso, A.M., Phoon, K.K. and Quek, S.T. Reliability analysis of infinite slope using subset simulation, ASCE Geotechnical Special Publication No 186: Contemporary Topics in In Situ Testing, Analysis, and Reliability of Foundations, Eds. M. H. Hussein, M. G. Iskander, D. F. Laefer, pp. 278 - 285. 2009. Phoon, K.K., Santoso, A.M. and Cheng, Y.G. Probabilistic analysis of soil water characteristic curves from sandy clay loam, ASCE Geotechnical Special Publication No 179: Characterization, Monitoring and Modeling of GeoSystems, Eds. M. V. Khire, K. R. Reddy, A. N. Alshawabkeh, pp. 917 925. 2008. Conference papers: Santoso, A.M., Phoon K.K. and Quek, S.T., Probability models for SWCC and hydraulic conductivity, In the Proceedings of the 14th Asian Regional Conference on Soil Mechanics and Geotechnical Engineering, May 23 – 27, 2011, Hong Kong, China. 2011. Santoso, A.M., Phoon, K.K. and Quek, S.T. Reliability analysis of unsaturated soil slopes using subset simulation, In the Proceedings of ICOSSAR 2009: Safety, Reliability and Risk of Structures, Infrastructures and Engineering Systems, Eds. H. Furuta, D.M. Frangopol & M. Shinozuka, Taylor & Francis Group, London (CDROM). 2010. Santoso, A.M., Phoon, K.K. and Quek, S.T. Flow of water through spatially heterogeneous soil, In the Proceedings of the 17th South East Asian Geotechnical Conference: Geo-engineering for Natural Hazard Mitigation and Sustainable Development, Eds. J.C.C. Li and M-L. Lin, Taiwan Geotechnical Society, Taiwan, Volume II, pp. 249 – 253. 2010. Santoso, A.M., Phoon, K.K. and Quek, S.T. System reliability estimation using subset simulation with modified Metropolis-Hastings algorithm, In the 212 Proceedings of the 5th International ASRANet Conference, Edinburgh, UK (CDROM). 2010. 213 [...]... characterizing the uncertainties in soil- water characteristic curve (SWCC) and the saturated hydraulic conductivity (ks), and investigating their impacts on seepage and stability of unsaturated slope These soil hydraulic properties are among the most critical input parameters which have been recognized in the analysis of rainfall- induced landslides In order to incorporate the uncertainties in the seepage and... research This thesis focuses on the role of uncertainties in the soil hydraulic properties on shallow rainfallinduced landslides It is already known that soil hydraulic properties are among the most important governing factors Given the dearth of prior work, the scope is restricted: (1) statistical characterization of soil hydraulic properties based on measured data available in sufficient quantity from the... for studying rainfall- induced slope failures 1.2 RELIABILITY ANALYSIS OF UNSATURATED SLOPE In addition to the physical complexity, the unsaturated slope analysis is further complicated by the presence of fairly significant uncertainties For instance, uncertainties in the soil properties have been well recognized The sources of uncertainty include the inherent variability in the soil, errors in measurement,... the key function in unsaturated soil mechanics, these statistics are useful for any probabilistic studies involving unsaturated soils 3 In the area of rainfall- induced landslides involving unsaturated soils, this study is among the first that assesses uncertainties and their impacts systematically The results of the probabilistic analysis may provide insights into the impact of uncertainties on unsaturated... Illustration of change in soil matric suction (a) initial condition (b) during dry periods (c) during rainy days (after Phoon et al 2009) 3 2.1 Sources of uncertainty in analysis of rainfall- induced landslides 17 3.1 Typical infinite slope with a weathered layer (after Phoon et al 2009) 58 3.2 Comparison of transient solution from THFELA and analytical steady state solution 60 3.3 Comparison of numerical... through multi-layered soil (e.g Freeze 1975, Bakr et al 1978, Yeh et al 1985 4 a,b,c, Yeh 1989, Fenton and Griffiths 1993, 1997) which have highlighted some of the challenges involved In addition to the uncertainties in the soil properties, there are uncertainties in the climatic boundaries such as the temporal variation in rainfall infiltration Even if the soil properties and the rainfall infiltration have... factor of safety of the slope FSmin minimum factor of safety along the depth of the slope, FSmin = minz{FS(z, X)} g(X) performance function of a system G() one-sided spectral density function of a random field h pressure head in the soil H total head in the soil H hydraulic head difference between the base of the slope and the ground surface {H} vector of total head at nodal points xvi {H}t vector of. .. Probability models of SWCC and ks to be used in parametric study 110 5.1 Example and number of samples used in each verification 128 5.2 Final and intermediate thresholds for various values of PF 134 5.3 Results of the 1D example 136 5.4 Parameters of infinite slope in Example 1 139 5.5 Results of Example 1 140 5.6 Results of Example 2 145 6.1 Summary of parameters of infinite slope 171 x LIST OF FIGURES 1.1... stability A more consistent indicator of slope stability is the probability of slope failure In general engineering problems, failure can be defined as the state where the response or the performance of a system is unsatisfactory In the case of slope stability, failure can be defined as the event of having the factor of safety less than one, given the uncertainties in the soil properties, climatic conditions,... rainfall infiltration based on available data and current state of knowledge With this framework, the significance of incorporating the unsaturated aspect of the problem would be evaluated in a reasonably realistic albeit preliminary way As explained in the previous sections, reliability assessment of rainfall- induced landslide is a complex problem encompassing many challenging aspects Most of these . ROLE OF UNCERTAINTY IN SOIL HYDRAULIC PROPERTIES IN RAINFALL- INDUCED LANDSLIDES ANASTASIA MARIA SANTOSO NATIONAL UNIVERSITY OF SINGAPORE. OF SINGAPORE 2011 ROLE OF UNCERTAINTY IN SOIL HYDRAULIC PROPERTIES IN RAINFALL- INDUCED LANDSLIDES ANASTASIA MARIA SANTOSO (B. Eng, Institute of Technology Bandung) . 2.1. Uncertainty in Analysis of Rainfall- induced Landslides 15 2.2. Probability Model of Uncertain Soil Properties 21 2.2.1 .Soil- Water Characteristic Curve 23 2.2.2. Soil Saturated Hydraulic