Conditional reliability analysis of groundwater flow and subsurface contaminant transport

A framework of numerically robust algorithms adopting a reliability approach for probabilistic finite element analysis of contaminant transport is presented. It focuses on evaluating the probability of "failure" which, as an example, is defined as an event that the concentration at a compliance location exceeds a risk-based limit. The uncertain boundary and initial conditions as well as the material constitutive parameters in the model are treated as spatially correlated random fields, each of which is represented by its own mesh of random variables and the direct data. Their values at the Gaussian points are determined using an optimal linear estimator (OLE). The reliability analysis is performed by importance sampling simulation (ISS) centered at the design points obtained by the first-order reliability method (FORM). In addition to the failure probability, ISS generates other near-failure probability distributions such as that of the concentration at a compliance location. In FORM, the sensitivity derivatives involving the transport equation are computed by the adjoint method and those involving the flow equation are calculated by analytical differentiation. The linearity requirement of the adjoint method dictates the use of streamline-upwind Petrov-Galerkin (SUPG) finite elements for avoiding numerical instability in solving the transport equation. High-order Gaussian integration is used to account for the spatial variation of material constituitive parameters within an element. To ease the intensive memory and computation requirements, the preconditioned conjugate gradient (PCG) and general minimization residual (PGMRes) methods are used to solve the systems of equations associated large-scale problems. Finally, indirect data are incorporated using a Bayesian approach.;Numerical experiments are conducted to examine the performance of this framework for reliability problems with single design points. The results suggest that for obtaining a design point accurate enough for ISS, FORM can be performed efficiently using a relatively coarse random variable mesh, because the design point is generally smooth. Using one or two random variables per correlation length suffices in the numerical examples. Approximating a random variable space by an eigen-subspace further improve the efficiency of FORM. To represent the local correlation structure higher resolution in the subsequent ISS, the mesh can be refined, particularly in the sensitive regions. The design point for the refined mesh is converted from the coarse mesh by OLE. A refined random variable spacing comparable to one-fourth of the correlation length gives suitable level of accuracy. In general, ISS is significantly more efficient than the Monte Carlo simulation for the same level of accuracy. It provides data for estimating the statistical distributions near the reliability target. In contrast to the first- and second-order reliability methods, the accuracy of ISS can be improved asymptotically with more realizations. In comparison to using the finite difference method to obtain the sensitivity derivatives, the use of adjoint method significantly reduces the computational effort.