Leopold Vrankar

MODELLING OF RADIONUCLIDE MIGRATION THROUGH THE GEOSPHERE WITH RADIAL BASIS FUNCTION METHOD AND GEOSTATISTICS

Abstract The modelling of radionuclide transport through the geosphere is necessary in the safety assessment of repositories for radioactive waste. A number of key geosphere 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 input parameter, varies from place to place. In such cases geostatistical science offers a variety of spatial estimation procedures. To assess the long term safety of a radioactive waste disposal system, mathematical models are used to describe the complicated groundwater flow, chemistry and potential radionuclide migration through geological formations. The numerical solution of partial differential equations (PDEs) has usually been obtained by finite difference methods (FDM), finite element methods (FEM), or finite volume methods (FVM). Kansa introduced the concept of solving PDEs using radial basis functions (RBFs) for hyperbolic, parabolic and elliptic PDEs. The aim of this study was to present a relatively new approach to the modelling of radionuclide migration through the geosphere using radial basis functions methods and to determine the average and sample variance of radionuclide concentration with regard to spatial variability of hydraulic conductivity modelled by a geostatistical approach. We will also explore residual errors and their influence on optimal shape parameters.

A comparison of the effectiveness of using the meshless method and the finite difference method in geostatistical analysis of transport modeling

Disposal of radioactive waste in geological formations is a great concern with regards to nuclear safety. The general reliability and accuracy of transport modeling depends predominantly on input data such as hydraulic conductivity, water velocity, radioactive inventory, and hydrodynamic dispersion. The most important input data are obtained from field measurements, but they are not always available. One way to study the spatial variability of hydraulic conductivity is geostatistics. The numerical solution of partial differential equations (PDEs) has usually been obtained by finite difference methods (FDM), finite element methods (FEM), or finite volume methods (FVM). These methods require a mesh to support the localized approximations. The multiquadric (MQ) radial basis function method is a recent meshless collocation method with global basis functions. 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 meshless method, which uses radial basis functions, with the traditional finite difference scheme. In our case we determine the average and standard deviation of radionuclide concentration with regard to spatial variability of hydraulic conductivity that was modeled by a geostatistical approach.

The Influence of Geostatistical Data on the Reliability of the Meshless Method in Transport Modeling

The disposal of radioactive waste in geological formation is of great importance for nuclear safety. A number of key geosphere processes need to be considered when predicting the movement of radionuclides through the geosphere. The main goal of this research is to investigate the influence of geostatistical data on reliability and accuracy of computational modelling. We chose the Kansa meshless method that uses radial basis functions as the mathematical solution technique. The aim of this study is to determine the average and sample variance of radionuclide concentration with regard to spatial variability of hydraulic conductivity modelled by geostatistical approach.

Solving One-Dimensional Moving-Boundary Problems with Meshless Method

Many physical processes involve heat conduction and materials undergoing a change of phase. Examples include the safety studies of nuclear reactors, casting of metals, semiconductor manufacturing, geophysics and industrial applications involving metals, oil, and plastics. Due to their wide range of applications the phase change problems have drawn considerable attention of mathematicians, engineers and scientists. These problems are often called Stefan’s or moving boundary value problems. One common feature of phase change problems is that the location of the solid-liquid or solid-solid interface is not known a priori and must be determined during the course of analysis. Mathematically, the interface motion is expressed implicitly in an equation for the conservation of thermal energy at the interface (Stefan’s conditions). This introduces a non-linear character to the system which treats each problem somewhat uniquely. The exact solution of phase change problems is limited exclusively to the cases in which e.g. the heat transfer regions are infinite or semi-infinite one dimensional-spaces. Therefore, solution is obtained either by approximate analytical solution or by numerical methods. Finite-difference methods and finite-element techniques have been used extensively for numerical solutions of moving boundary problems. Recenty, the numerical methods have focused on the idea of using a meshless methodology for the numerical solution of Preprint submitted to Elsevier Science 28 January 2008 partial differential equations based on radial basis functions. One of the common characteristics of all meshless methods is their ability to construct functional approximation or interpolation entirely based on the information given at a set of scattered nodes. In our case we will study solid state phase transformation problem in binary metallic alloys. The numerical solutions will be compared with analytical solutions. Actually, in our work we will examine usefulness of radial basis functions for one-dimensional Stefan’s problems. The position of the moving boundary will be simulated by moving grid method.

Radial Basis Functions Methods for Solving Radionuclide Migration, Phase Change and Wood Charring Problems

Modelling the flow through porous media has a great importance for solving the problems of disposal of radioactive waste. When modelling the flow of contaminated material through the geosphere, it is important to consider all internal processes (e.g. advection, dispersion, retardation) within the geosphere, and external processes associated with the near-field and the biosphere. The general reliability and accuracy of transport modelling depend predominantly on input data such as hydraulic conductivity, water velocity on the boundary, radioactive inventory, hydrodynamic dispersion. The output data are concentration, pressure, etc. The most important input data are obtained from field measurement, which are not available for all regions of interest. In such cases, geostatistical science offers a variety of spatial estimation procedures. A vast variety of important physical processes involving heat conduction and materials undergoing a change of phase may be approached as Stefan problems. One of these processes is the heat transfer involving phase changes caused by solidification or melting, which are important in many industrial applications such as the drilling of high ice-content soil, the storage of thermal energy and the safety studies of nuclear reactors. Due to their wide range of applications, the phase change problems have drawn considerable attention of specialists in different fields of science and engineering. Another problem which seems to be completely different but is in mathematical terms very similar to solidification or melting is charring of wood. After wood is exposed to fire it undergoes thermal degradation. The pyrolysis gases undergo flaming combustion as they leave the charred wood surface. The pyrolysis, charring, and combustion of wood have been presented by (Fredlund, 1993) who performed experiments and numerical analyses. For all physical processesmentioned above, themotion of fluids, phase changes, and pyrolysis processes are governed by a set of partial differential equations (PDES). These governing equations are based upon the fundamental conservation laws. The mass, momentum and energy are conserved in any fluid motions. In most cases, the governing equations are too Radial Basis Functions Methods for Solving Radionuclide Migration, Phase Change and Wood Charring Problems