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Discontinuous Galerkin Methods for Convection-Diffusion Equations and Applications in Petroleum Engineering
This dissertation contains research in discontinuous Galerkin (DG) methods applying to convection-diffusion equations. It contains both theoretical analysis and applications. Initially, we develop a conservative local discontinuous Galerkin (LDG) method for the coupled system of compressible miscible displacement problem in two space dimensions. The main difficulty is how to deal with the discontinuity of approximations of velocity, u, in the convection term across the cell interfaces. To overcome the problems, we apply the idea of LDG with IMEX time marching using the diffusion term to control the convection term. Optimal error estimates in L∞(0, T; L2) norm for the solution and the auxiliary variables will be derived. Then, high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes will be developed. There are three main difficulties to make the concentration of each component between 0 and 1. Firstly, the concentration of each component did not satisfy a maximum-principle. Secondly, the first-order numerical flux was difficult to construct. Thirdly, the classical slope limiter could not be applied to the concentration of each component. To conquer these three obstacles, we first construct special techniques to preserve two bounds without using the maximum-principle-preserving technique. The time derivative of the pressure was treated as a source of the concentration equation. Next, we apply the flux limiter to obtain high-order accuracy using the second-order flux as the lower order one instead of using the first-order flux. Finally, L2-projection of the porosity and constructed special limiters that are suitable for multi-component fluid mixtures were used. Lastly, a new LDG method for convection-diffusion equations on overlapping mesh introduced in  showed that the convergence rates cannot be improved if the dual mesh is constructed by using the midpoint of the primitive mesh. They provided several ways to gain optimal convergence rates but the reason for accuracy degeneration is still unclear. We will use Fourier analysis to analyze the scheme for linear parabolic equations with periodic boundary conditions in one space dimension. To investigate the reason for the accuracy degeneration, we explicitly write out the error between the numerical and exact solutions. Moreover, some superconvergence points that may depend on the perturbation constant in the construction of the dual mesh were also found out.
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Removal of Asphaltene from Crude Oil by the Electro-deposition Method
The deposition of asphaltene on interior pipe surfaces is considered a challenging flow assurance issue. By exploiting the net charge of the asphaltene molecule, it can be effectively removed from the flow stream by means of electro–deposition before transportation to prevent later deposition. To evaluate this concept, electro–deposition of asphaltene from a synthetic oil using varying electric field strength and solvent type was studied. Both native charge (positive) and induced charge (negative) of asphaltene were observed in this study; the inducement of charge was realized with higher current density. Asphaltene solubility in the synthetic oil was increased by increasing the polar/non-polar solvent ratio revealing a higher conductivity of the synthetic oil. In addition to electric field strength and synthetic oil composition, the asphaltene concentration and synthetic oil viscosity also affected the electro-deposition.