FINITE ELEMENT METHODS FOR CONVECTION DOMINATED TRANSPORT PROBLEMS
POTEMPA, THOMAS CHARLES
Doctor of Philosophy
Physical processes obey one or more conservation laws. Engineers and scientists have derived by manipulation of these conservation laws differential equations which govern many physical systems. These differential equations are either elliptic, parabolic, or hyperbolic in form. State-of-the-art numerical methods accurately approximate the solutions of elliptic and parabolic differential equations, but difficulties arise when approximating the solutions of many hyperbolic differential systems which describe certain convection dominated physical phenomena. The finite element method for the solution of partial differential equations is given a physical interpretation for potential flow problems, heat flow problems, and mass conservation problems. The differential equations describing these systems represent elliptic, parabolic, and hyperbolic differential systems. The finite element model of the mass conservation law is modified using physical interpretations to represent an analog of the upstream weighted finite difference method. From these physical interpretations, a method for modeling single phase multi-component areal flow in a porous media is developed for both incompressible and compressible flow. This method models only the convective mass transfer, since the magnitude of the numerical diffusion added by the method is expected to be far greater than the magnitude of the physical diffusion present in the actual physical situation. This method represents a novel idea in taking a mass balance over a computational molecule which differs from the computational molecule that is employed in the finite difference method with upstream weighting. This finite element computational molecule is a region in R('3) defined by a basis element for the bilinear splines in R('2). This method has several nice properties. The method exhibits a maximum principle, and is easily implemented in state-of-the-art numerical models of fluid flow in a porous media. The method is computationally fast. The grid orientation which is present in the finite difference method does not appear in this model. The compositional profiles generated by this method are similar to the compositional profiles generated by a finite element method which converges to the true solution of the unit mobility ratio incompressible miscible displacement problem.