Previous studies have proposed the first-order reliability method (FORM) as an approach to quantitative stochastic analysis of subsurface transport. Most of these considered only simple analytical models of transport in homogeneous media. Studies that looked at more-complex, heterogeneous systems found FORM to be computationally demanding and were inconclusive as to the accuracy of the method. Here we show that FORM is poorly suited for computing point concentration cumulative distribution functions (cdfs) except in the case of a constant or monotonically increasing solute source. FORM is better equipped to predict transport in terms of the cumulative mass flux across a control surface. As a demonstration, we use FORM to estimate the cumulative mass flux cdf in two- dimensional, random porous media. Adjoint sensitivity theory is employed to minimize the computational burden. In addition, properties of the conductivity covariance and distribution are exploited to improve efficiency. FORM required eight times less CPU time than Monte Carlo simulation to generate the results presented. The accuracy of FORM is found to be minimally affected by the size of the initial solute body and the solute travel distance. However, the accuracy is significantly influenced by the degree of heterogeneity, providing an accurate estimate of the cdf when there is mild heterogeneity (σInK = 0.5) but a less accurate estimate when there is stronger heterogeneity (σInK = 1.0).

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/97WR00660

The first-order reliability method of predicting cumulative mass flux in heterogeneous porous formations

Stochastic modelling and risk analysis of groundwater pollution using FORM coupled with automatic differentiation

Groundwater flow and contaminant transport modelling is carried out to predict the concentrations of radionuclides in groundwater beneath the tailings pond. This is used as a tool for radiological impact assessment studies. The geo-hydrological parameters involved in the modelling exhibit inherent uncertainty associated with their values. This uncertainty may get propagated to the model output, i.e. the dose to the public. The propagation of the uncertainty in the input parameters to the output is modelled using suitable methods of probabilistic analysis. Response surface method coupled with first-order reliability method is used to develop a methodology for estimating the probability that the dose rate value through drinking water pathway at a location around the tailings pond exceeds the WHO guidelines for drinking water, termed as the probability of exceedance of acceptable levels. This method also gives the estimate of sensitivity of the probability of exceedance to the different input parameters. It is observed that the probability of failure decreases as the distance from tailings pond centre is increased and beyond a distance of around 0.5 km from tailings pond centre, the probability reduces to zero. The work also brings out the importance of quantifying the uncertainty in case of actual field problems where there is wide variation in the values of the various parameters within the domain under study.

Response Surface Method Coupled with First-Order Reliability Method Based Methodology for Groundwater Flow and Contaminant Transport Model for the Uranium Tailings Pond Site

A general probabilistic fatigue crack growth prediction methodology for accurate and efficient damage prognosis is proposed in this paper. This methodology consists two major parts. First, the realistic random loading is transformed to an equivalent constant amplitude loading process based on a recently developed mechanism model. This transformation avoids the cycle-by-cycle calculation of fatigue crack growth under variable amplitude loading. Following this, an inverse first-order reliability method (IFORM) is used to evaluate the fatigue crack growth at an arbitrary reliability level. Inverse FORM method does not require a large number of function evaluations compared to the direct Monte Carlo simulation. Computational cost is significantly reduced and the proposed method is very suitable for real-time damage prognosis. Numerical examples are used to demonstrate the proposed method. Various experimental data under variable amplitude loading are collected and model predictions are compared with experimental data for model validation.

/pdf/efficient-probabilistic-methods-for-real-time-fatigue-damage-3mazilx5zf.pdf

Efficient probabilistic methods for real-time fatigue damage prognosis

A new methodology for concurrent dynamic analysis and structural fatigue prognosis is proposed in this paper. The proposed methodology is based on a novel small timescale formulation of material fatigue crack growth that calculates the incremental crack growth at any arbitrary time within a loading cycle. It defines the fatigue crack kinetics based on the geometric relationship between the crack-tip opening displacement and the instantaneous crack growth rate. The proposed crack growth model can be expressed as a set of first-order differential equations. The structural dynamics analysis and fatigue crack growth model can be expressed as a coupled hierarchical state-space model. The dynamic response (structural level) and the fatigue crack growth (material level) can be solved simultaneously. Several numerical problems with single-degree-of-freedom and multiple-degree-of-freedom cases are used to show the proposed methodology. Model predictions are validated by using coupon testing data from open literatu...

Inverse First-Order Reliability Method for Probabilistic Fatigue Life Prediction of Composite Laminates under Multiaxial Loading

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.

Assessing uncertainty in subsurface solute transport: Efficient first-order reliability methods.

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Assessing uncertainty in subsurface solute transport: Efficient first-order reliability methods.