Combining the radial basis function Eulerian and Lagrangian schemes with geostatistics for modeling of radionuclide migration through the geosphere

To assess the long-term safety of a radioactive waste disposal system, mathematical models are used to describe groundwater flow, chemistry, and potential radionuclide migration through geological formations. A number of processes need to be considered, when predicting the movement of radionuclides through the geosphere. The most important input data are obtained from field measurements, which are not available for all regions of interest. For example, the hydraulic conductivity as an input parameter varies from place to place. In such cases, geostatistical science offers a variety of spatial estimation procedures. Methods for solving the solute transport equation can also be classified as Eulerian, Lagrangian and mixed. The numerical solution of partial differential equations (PDE) has usually been obtained by finite-difference methods (FDM), finite-element methods (FEM), or finite-volume methods (FVM). Kansa introduced the concept of solving partial differential equations using radial basis functions (RBF) for hyperbolic, parabolic, and elliptic PDEs. The aim of this study was to present a relatively new approach to the modeling of radionuclide migration through the geosphere using radial basis function methods in Eulerian and Lagrangian coordinates. In this study, we determine the average and standard deviation of radionuclide concentration with regard to variable hydraulic conductivity, which was modelled by a geostatistical approach. Radionuclide concentrations will also be calculated in heterogeneous and partly heterogeneous 2D porous media. (C) 2004 Elsevier Ltd. All rights reserved

Multiscale Methods for Stochastic Collocation of Mixed Finite Elements for Flow in Porous Media

This thesis contains methods for uncertainty quantification of flow in porous media through stochastic modeling. New parallel algorithms are described for both deterministic and stochastic model problems, and are shown to be computationally more efficient than existing approaches in many cases.First, we present a method that combines a mixed finite element spatial discretization with collocation in stochastic dimensions on a tensor product grid. The governing equations are based on Darcy's Law with stochastic permeability. A known covariance function is used to approximate the log permeability as a truncated Karhunen-Loeve expansion. A priori error analysis is performed and numerically verified.Second, we present a new implementation of a multiscale mortar mixed finite element method. The original algorithm uses non-overlapping domain decomposition to reformulate a fine scale problem as a coarse scale mortar interface problem. This system is then solved in parallel with an iterative method, requiring the solution to local subdomain problems on every interface iteration. Our modified implementation instead forms a Multiscale Flux Basis consisting of mortar functions that represent individual flux responses for each mortar degree of freedom, on each subdomain independently. We show this approach yields the same solution as the original method, and compare the computational workload with a balancing preconditioner.Third, we extend and combine the previous works as follows. Multiple rock types are modeled as nonstationary media with a sum of Karhunen-Loeve expansions. Very heterogeneous noise is handled via collocation on a sparse grid in high dimensions. Uncertainty quantification is parallelized by coupling a multiscale mortar mixed finite element discretization with stochastic collocation. We give three new algorithms to solve the resulting system. They use the original implementation, a deterministic Multiscale Flux Basis, and a stochastic Multiscale Flux Basis. Multiscale a priori error analysis is performed and numerically verified for single-phase flow. Fourth, we present a concurrent approach that uses the Multiscale Flux Basis as an interface preconditioner. We show the preconditioner significantly reduces the number of interface iterations, and describe how it can be used for stochastic collocation as well as two-phase flow simulations in both fully-implicit and IMPES models

The use of the mesh free methods (radial basis functions) in the modeling of radionuclide migration and moving boundary value problems

Recently, the mesh free methods (radial basis functions-RBFs) have emerged as a novel computing method in the scientific and engineering computing community. The numerical solution of partial differential equations (PDEs) has been usually obtained by finite difference methods (FDM), finite element methods (FEM) and boundary elements methods (BEM). These conventional numerical methods still have some drawbacks. For example, the construction of the mesh in two or more dimensions is a nontrivial problem. Solving PDEs using radial basis function (RBF) collocations is an attractive alternative to these traditional methods because no tedious mesh generation is required. We compare the mesh free method, which uses radial basis functions, with the traditional finite difference scheme and analytical solutions. We will present some examples of using RBFs in geostatistical analysis of radionuclide migration modeling. The advection-dispersion equation will be used in the Eulerian and Lagrangian forms. Stefan's or moving boundary value problems will also be presented. The position of the moving boundary will be simulated by the moving data centers method and level set method

A probabilistic model of a porous heat exchanger

This paper presents a probabilistic one-dimensional finite element model for heat transfer processes in porous heat exchangers. The Galerkin approach is used to develop the finite element matrices. Some of the submatrices are asymmetric due to the presence of the flow term. The Neumann expansion is used to write the temperature distribution as a series of random variables, and the expectation operator is applied to obtain the mean and deviation statistics. To demonstrate the feasibility of the formulation, a one-dimensional model of heat transfer phenomenon in superfluid flow through a porous media is considered. Results of this formulation agree well with the Monte-Carlo simulations and the analytical solutions. Although the numerical experiments are confined to parametric random variables, a formulation is presented to account for the random spatial variations