Modelling of advection-dominated transport in fluid-saturated porous media

The modelling of contaminant transport in porous media is an important topic to geosciences and geo-environmental engineering. An accurate assessment of the spatial and temporal distribution of a contaminant is an important step in the environmental decision-making process. Contaminant transport in porous media usually involves complex non-linear processes that result from the interaction of the migrating chemical species with the geological medium. The study of practical problems in contaminant transport therefore usually requires the development of computational procedures that can accurately examine the non-linear coupling processes involved. However, the computational modelling of the advection-dominated transport process is particularly sensitive to situations where the concentration profiles can exhibit high gradients and/or discontinuities. This thesis focuses on the development of an accurate computational methodology that can examine the contaminant transport problem in porous media where the advective process dominates.The development of the computational method for the advection-dominated transport problem is based on a Fourier analysis on stabilized semi-discrete Eulerian finite element methods for the advection equation. The Fourier analysis shows that under the Courant number condition of Cr=1, certain stabilized finite element scheme can give an oscillation-free and non-diffusive solution for the advection equation. Based on this observation, a time-adaptive scheme is developed for the accurate solution of the one-dimensional advection-dominated transport problem with the transient flow velocity. The time-adaptive scheme is validated with an experimental modelling of the advection-dominated transport problem involving the migration of a chemical solution in a porous column. A colour visualization-based image processing method is developed in the experimental modelling to quantitatively determinate the chemical concentration on the porous column in a non-invasive way. A mesh-refining adaptive scheme is developed for the optimal solution of the multi-dimensional advective transport problem with a time- and space-dependent flow field. Such mesh-refining adaptive procedure is quantitative in the sense that the size of the refined mesh is determined by the Courant number criterion. Finally, the thesis also presents a brief study of a numerical model that is capable to capture coupling Hydro-Mechanical-Chemical processes during the advection-dominated transport of a contaminant in a porous medium

Primal dual mixed finite element methods for indefinite advection--diffusion equations

We consider primal-dual mixed finite element methods for the advection--diffusion equation. For the primal variable we use standard continuous finite element space and for the flux we use the Raviart-Thomas space. We prove optimal a priori error estimates in the energy- and the $L^2$-norms for the primal variable in the low Peclet regime. In the high Peclet regime we also prove optimal error estimates for the primal variable in the $H(div)$ norm for smooth solutions. Numerically we observe that the method eliminates the spurious oscillations close to interior layers that pollute the solution of the standard Galerkin method when the local Peclet number is high. This method, however, does produce spurious solutions when outflow boundary layer presents. In the last section we propose two simple strategies to remove such numerical artefacts caused by the outflow boundary layer and validate them numerically.Comment: 25 pages, 6 figures, 5 table

Energy-corrected FEM and explicit time-stepping for parabolic problems

The presence of corners in the computational domain, in general, reduces the regularity of solutions of parabolic problems and diminishes the convergence properties of the finite element approximation introducing a so-called "pollution effect". Standard remedies based on mesh refinement around the singular corner result in very restrictive stability requirements on the time-step size when explicit time integration is applied. In this article, we introduce and analyse the energy-corrected finite element method for parabolic problems, which works on quasi-uniform meshes, and, based on it, create fast explicit time discretisation. We illustrate these results with extensive numerical investigations not only confirming the theoretical results but also showing the flexibility of the method, which can be applied in the presence of multiple singular corners and a three-dimensional setting. We also propose a fast explicit time-stepping scheme based on a piecewise cubic energy-corrected discretisation in space completed with mass-lumping techniques and numerically verify its efficiency

Mathematical modeling and numerical simulation of a bioreactor landfill using Feel++

In this paper, we propose a mathematical model to describe the functioning of a bioreactor landfill, that is a waste management facility in which biodegradable waste is used to generate methane. The simulation of a bioreactor landfill is a very complex multiphysics problem in which bacteria catalyze a chemical reaction that starting from organic carbon leads to the production of methane, carbon dioxide and water. The resulting model features a heat equation coupled with a non-linear reaction equation describing the chemical phenomena under analysis and several advection and advection-diffusion equations modeling multiphase flows inside a porous environment representing the biodegradable waste. A framework for the approximation of the model is implemented using Feel++, a C++ open-source library to solve Partial Differential Equations. Some heuristic considerations on the quantitative values of the parameters in the model are discussed and preliminary numerical simulations are presented