Bayesian updating for progressive excavation of high rock slopes using multi-type monitoring data
Introduction
A great number of large hydropower stations have been built in China in recent years, such as those at Dagangshan, Jinping, Laxiwa and Xiluodu. Due to the typically deep valleys and the 100–300 m high dams, large-scale rock slopes are often required to be excavated. Detailed stability analysis of these slopes is required in the design of the excavation (Li et al., 2016b; Chen et al., 2017; Jiang and Zhou, 2017) which, in turn, needs reasonable estimates of the relevant geomechanical parameters. These parameters can be estimated based on field and laboratory testing data, but the latter may have significant uncertainties due to sample disturbance and measurement errors. In addition, due to budget and time constraints, a limited amount of testing is often performed which explores only a small fraction of the construction site. If the uncertain geomechanical parameters are used directly in the numerical analysis of the excavation, the calculation result may be unrealistic (Tang et al., 1999; Zhang et al., 2014a, Zhang et al., 2014b).
Due to the severe consequences associated with slope failure in large hydropower projects, monitoring systems for the displacement and stress states of the excavated slopes are usually implemented, and a large amount of data are collected. These monitoring data can be used through back analysis to predict the stability of future excavations. One advantage of the back analysis method is that some of the uncertainty associated with using laboratory testing data can be reduced (Tang et al., 1999), and back analysis based on field measurements has already been widely applied in geotechnical engineering practice (Lv et al., 2017). In traditional deterministic back analyses, the derived mechanical parameters have fixed values (Vardakos et al., 2012; Jiang et al., 2018a, Jiang et al., 2018b; Sun et al., 2018). However, due to the heterogeneous and discontinuous nature of many geomaterials, it is more physically more meaningful to define the mechanical properties as random variables with means and standard deviations. A further limitation of deterministic back analysis that it is usually implemented by minimizing some measure of the difference between the field measurement and the calculation result (Sun et al., 2018), thereby ignoring the influence of the measurement error. In contrast, the Bayesian method accounts for the uncertainty in the geotechnical parameters, while the measurement error is also considered by using a likelihood function (Juang et al., 2012; Li et al., 2016a, Li et al., 2016b). Moreover, it also combines the prior knowledge based on geotechnical experience. The Bayesian updating method has been widely used in various geotechnical engineering problems including slope stability (Li et al., 2016a, Li et al., 2016b; Ering and Babu, 2016; Liu et al., 2018; Yang et al., 2018), failure reliability updating (Schweckendiek et al., 2014), pile capacity (Huang et al., 2016), braced excavations (Juang et al., 2012) and site characterization (Wang et al., 2016a, Wang et al., 2016b, Wang et al., 2016c; Yang et al., 2017).
Several case studies of the Bayesian updating method for simple soil slopes have been described, but the literature on the Bayesian method applied to rock slope engineering is very limited (Zhang et al., 2014a, Zhang et al., 2014b; Li et al., 2016a, Li et al., 2016b; Zhou et al., 2017; Jiang et al., 2018a, Jiang et al., 2018b). Li et al., 2016a, Li et al., 2016b identified the geomechanical parameters of the high rock slope located at Longtan Hydropower Station in China by the Bayesian method, where only the monitoring data for the horizontal displacement were used to perform the update. In contrast, Bayesian updating was performed using multiple observations in some studies, but these observations were derived from hypothetical numerical models rather than actual monitored data (Peng et al., 2014; Zhang et al., 2018). However, for this excavation, various monitoring data was obtained on site, including the slope surface deformation monitored by total stations, the rock mass deformation measured by multi-point extensometers, and the anchorage forces recorded by anchored cable dynamometers. Not surprisingly, parameters updated using only one type of measurement may be misleading. For example, if the same parameters are updated using the monitoring data for the displacements or anchorage forces, two quite different results can be obtained. It is thus desirable, where possible, to combine multiple sources of data when performing back analyses.
Because of the complex geological conditions of many high rock slopes, a variety of different rock mass and structural plane materials may be included in a numerical model. As a result, the number of material parameters and degrees of freedom may both be large. If all the material parameters are considered within a complicated numerical model, Bayesian updating may become very complex, time consuming, and impractical. Therefore, parametric sensitivity analysis is often needed before the Bayesian updating process to reduce the number of random variables. In addition, the response surface method (Li et al., 2015), support vector machines (Li et al., 2016a, Li et al., 2016b) and artificial neural networks (Mollon et al., 2009) are often used in back analysis to approximate the non-linear relationship between the deduced parameters (such as deformation modulus and cohesion) and the geotechnical response (such as the factor of safety and displacement) in order to reduce the computational burden.
The Markov chain Monte Carlo simulation (MCMC) is widely used to sample the posterior target distribution, while the Gibbs and the Metropolis-Hasting algorithms are two traditional methods that are employed to establish Markov chains (Gill, 2002). Although research in geotechnical engineering has applied these two traditional algorithms to Bayesian inference, mathematicians have proposed several mew approaches to improve the efficiency and accuracy of MCMC simulation. For example, the DiffeRential Evolution Adaptive Metropolis (DREAM) algorithm has been shown to be more effective in solving posterior sampling problems (Vrugt, 2016).
In the current work, a Bayesian framework for updating geomechanical parameters with multiple types of field measurements is presented and applied to high rock slope excavations. The parameters that proved to have an important influence on the displacement and anchorage force in a sensitivity analysis are defined as random variables. The FLAC3D software (Itasca Consulting Group, 2000) is adopted to simulate progressive excavation of rock slopes. A back propagation neural network is then used to approximate the time-consuming numerical simulations. The multi-step excavation case of the left bank slope at the Lianghekou Hydropower Station in China is used to illustrate the proposed Bayesian framework. The framework is discussed for three cases: using only displacement measurements, using only anchorage force measurements, and using both types of measurements simultaneously. The posterior distributions of the random variables in Bayesian inference were searched using a multi-chain MCMC simulation algorithm (DREAM). The variables are updated step-by-step, based on the excavation sequence, until the 20th excavation step. The updated variables are then used to predict the geotechnical response for the last excavation step (the 21st excavation step). Section 2 describes the basic theory of Bayesian inference, while Section 3 presents the proposed Bayesian updating framework procedure for the excavated rock slope. Section 4 introduces the geological setting and the numerical model used for the case study. The calculation results for the three monitoring conditions are given in Section 5, while the discussion and conclusions are presented in Section 6.
Section snippets
Framework for Bayesian updating
Let θ = [θ1, θ2, … , θd] denote the vector of the key geomechanical parameters, where all of them are defined as random variables. Further, let Y = [y1, y2, … , ym] denote the vector of field observations at the timeststep = [1, 2, … , m]. The relationship between the model functions H(θ) = [h1(θ), h2(θ), … , hm(θ)] and field observations Y = [y1, y2, … , ym] iswhere ε = [ε1, ε2, … , εm] is an m-vector of measurement errors and hi(θ) is a simulator of yi.
By adopting Bayesian inference, the prior
Parametric sensitivity analysis
In order to reduce the number of random variables and the computational time before the Bayesian updating, a sensitivity analysis was performed to identify the parameters that have an important influence on the displacement and the anchorage force. The selected parameters are defined as random variables and will be updated in the Bayesian framework. Shen and Abbas (2013) proposed a sensitivity analysis approach as followswhere λk
Project background
The Lianghekou hydropower station is situated on the Yalong River in Yajiang County of Ganzi Prefecture, Sichuan Province, China. The project is located where the Yalong River joins the Qingda River and the Xianshui River. (Fig. 2). It is the tallest earth-rockfill dam in China with a maximum height of 295 m. The reservoir capacity is 10.154 × 109 m3 at the normal water level of 2865 m. The project is currently under construction and will be completed in 2023.
Results
For the cohesion and deformation modulus parameters of the class III1, III2, IV1, IV2, V1, V2 rock masses and the structural planes f1, f4, f8, f9, f10, f11, f12, gb01–04 in Table 2, Table 3, a total of 28 sensitivity analysis were performed based on Eqs. (7), (8). E(III2) has the highest relative sensitivity and is thus the most sensitive parameter, followed by c(III1). The critical value is conservatively determined to be 5% in this study. Parameters with relative sensitivities greater than
Discussion and conclusions
The uncertainties in geomechanical parameters have a significant impact on the design, construction and risk assessment of rock slopes. However, traditional deterministic back analysis methods cannot take into account these uncertainties and ignore the effects of measurement errors. In this study, a probabilistic back analysis framework is presented that integrates parametric sensitivity analysis, a BP neural network and the Bayesian method. The application of the proposed methodology is
Acknowledgments
The work reported in this paper has received financial support from the National Natural Science Foundation of China (Grant No. U1765207) and the Natural Science Foundation of Hubei Province (Grant No. 2016CFA083). The first author would like to thank the China Scholarship Council (CSC) for its financial support for his joint Ph.D. at the ARC Centre of Excellence for Geotechnical Science and Engineering at the University of Newcastle, Australia.
References (35)
- et al.
Automatic extraction of blocks from 3D point clouds of fractured rock
Comput. Geosci.
(2017) - et al.
Probabilistic back analysis of rainfall induced landslide a case study of Malin landslide, India
Eng. Geol.
(2016) - et al.
Updating reliability of single piles and pile groups by load tests
Comput. Geotech.
(2016) - et al.
A rigorous solution for the stability of polyhedral rock blocks
Comput. Geotech.
(2017) - et al.
Bayesian updating of slope reliability in spatially variable soils with in-situ measurements
Eng. Geol.
(2018) - et al.
Efficient 3-D reliability analysis of the 530 m high abutment slope at Jinping I Hydropower Station during construction
Eng. Geol.
(2015) - et al.
Probabilistic back analysis based on Bayesian and multi-output support vector machine for a high cut rock slope
Eng. Geol.
(2016) - et al.
Updating performance of high rock slopes by combining incremental time-series monitoring data and three-dimensional numerical analysis
Int. J. Rock Mech. Min. Sci.
(2016) - et al.
Sequential reduction of slope stability uncertainty based on temporal hydraulic measurements via the ensemble Kalman filter
Comput. Geotech.
(2018) - et al.
Stability analysis of earthquake-induced rock slope based on back analysis of shear strength parameters of rock mass
Eng. Geol.
(2017)
Slope safety evaluation by integrating multi-source monitoring information
Struct. Saf.
Updating piping reliability with field performance observations
Struct. Saf.
Rock slope reliability analysis based on distinct element method and random set theory
Int. J. Rock Mech. Min. Sci.
A back-analysis method using an intelligent multi-objective optimization for predicting slope deformation induced by excavation
Eng. Geol.
Reliability in back analysis of slope failures
Soils Found.
Parameter identification in numerical modeling of tunneling using the Differential Evolution Genetic Algorithm (DEGA)
Tunn. Undergr. Space Technol.
Markov chain Monte Carlo simulation using the DREAM software package: theory, concepts, and MATLAB implementation
Environ. Model Softw.
Cited by (43)
A Bayesian multi-model inference methodology for imprecise moment-independent global sensitivity analysis of rock structures
2024, Journal of Rock Mechanics and Geotechnical EngineeringAn efficient physics-guided bayesian framework for predicting ground settlement profile during excavations in clay
2024, Journal of Rock Mechanics and Geotechnical Engineering