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Due to the heterogeneity of natural groundwater systems, any quantitative description of aquifer hydraulic properties is subject to uncertainty. Consequently, prediction of groundwater contaminant transport is also subject to uncertainty. Stochastic approaches to transport simulation quantify this uncertainty in terms of random variables and processes. An important practical consideration in the application of such methods is their large computational cost. In recent years the first-order reliability method (FORM) has been introduced as a possible technique for obtaining stochastic results with low computational expense. Specifically, the implementation of FORM known as advanced FORM (AFORM) has been shown to produce reasonably accurate results when applied to simple problems. However, recently published results indicate that the computational burden of AFORM can equal, or even exceed, that of Monte Carlo simulation when applied to groundwater contamination problems with a large number of variables. If FORM is to be a viable alternative, the computational costs of the method must be lowered. In this work we propose two alternative implementations of FORM that have a higher computational efficiency. The primary numerical difficulty that arises in AFORM is locating the linearization point, a procedure that requires the solution of a non-linearly constrained optimization problem. We minimize this difficulty by zoning spatially variable aquifer parameters and by defining a new linearization point that can be found more easily. The new approaches are shown to produce results that are comparable to those obtained with AFORM when applied to a one-dimensional transport problem. Future work will be aimed at generalizing the procedures described herein.