UNCERTAINTY ANALYSIS OF ROCK FAILURE BEHAVIOUR USING AN INTEGRATION OF THE PROBABILISTIC COLLOCATION METHOD AND ELASTO-PLASTIC CELLULAR AUTOMATON

The Karhunen-Loeve （KL） expansion and probabilistic collocation method （PCM） are combined and applied to an uncertainty analysis of rock failure behavior by integrating a self- developed numerical method （i.e., the elastic-plastic cellular automaton （EPCA））. The results from the method developed are compared using the Monte Carlo Simulation （MCS） method. It is concluded that the method developed requires fewer collocations than MCS method to obtain very high accuracy and greatly reduces the computational cost. Based on the method, the elasto- plastic and elasto-brittle-plastic analyses of rocks under mechanical loadings are conducted to study the uncertainty in heterogeneous rock failure behaviour.

Conditional Simulation of Flow in Heterogeneous Porous Media with the Probabilistic Collocation Method

A stochastic approach to conditional simulation of flow in randomly heterogeneous media is proposed with the combination of the Karhunen-Loeve expansion and the probabilistic collocation method(PCM).The conditional log hydraulic conductivity field is represented with the Karhunen-Loeve expansion,in terms of some deterministic functions and a set of independent Gaussian random variables.The propagation of uncertainty in the flow simulations is carried out through the PCM,which relies on the efficient polynomial chaos expansion used to represent the flow responses such as the hydraulic head.With the PCM,existing flow simulators can be employed for uncertainty quantification of flow in heterogeneous porous media when direct measurements of hydraulic conductivity are taken into consideration.With illustration of several numerical examples of groundwater flow,this study reveals that the proposed approach is able to accurately quantify uncertainty of the flow responses conditioning on hydraulic conductivity data,while the computational efforts are significantly reduced in comparison to the Monte Carlo simulations.

A Comparative Study of Stochastic Collocation Methods for Flow in Spatially Correlated Random Fields

Stochastic collocation methods as a promising approach for solving stochastic partial differential equations have been developed rapidly in recent years.Similar to Monte Carlo methods,the stochastic collocation methods are non-intrusive in that they can be implemented via repetitive execution of an existing deterministic solver without modifying it.The choice of collocation points leads to a variety of stochastic collocation methods including tensor product method,Smolyak method,Stroud 2 or 3 cubature method,and adaptive Stroud method.Another type of collocation method,the probabilistic collocation method(PCM),has also been proposed and applied to flow in porous media.In this paper,we discuss these methods in terms of their accuracy,efficiency,and applicable range for flow in spatially correlated random fields.These methods are compared in details under different conditions of spatial variability and correlation length.This study reveals that the Smolyak method and the PCM outperform other stochastic collocation methods in terms of accuracy and efficiency.The random dimensionality in approximating input random fields plays a crucial role in the performance of the stochastic collocation methods.Our numerical experiments indicate that the required random dimensionality increases slightly with the decrease of correlation scale and moderately from one to multiple physical dimensions.