EXPLICIT FINITE DIFFERENCE SCHEMES BASED ON THE MODIFIED METHOD OF CHARACTERISTICS AND THEIR APPLICATION TO THE MISCIBLE DISPLACEMENT PROBLEM
Doctor of Philosophy
A class of convection-diffusion parabolic boundary value problems is considered in this study. When convection dominates over diffusion, the solutions of such problems exhibit an almost hyperbolic behavior. Initial conditions with steep gradients propagate in space and time in the form of sharp traveling fronts. Efficient algorithms for solving such problems accurately are developed, analyzed and tested. These schemes are based on the modified method of characteristics using finite differences and explicit time integration, and are consequently named Explicit Modified Method of Characteristics or EMMC schemes. For problems in one space dimension, explicit time steps are taken along the characteristics of the hyperbolic part of the differential operator and the spatial grid is simultaneously adapted to track the front. The diffusion term is discretized by finite differences. Error analysis is performed and a mild stability criterion is derived. This criterion is used for automatic adjustment of the time step. For problems in two space dimensions, the explicit time stepping and the grid adaptation take place along directions approximating those of the characteristics. These directions are chosen so that numerical dispersion is minimized and a rectangular spatial grid is retained at all times. The explicit step discretizes part of the convection terms. The remainder and the diffusion terms are discretized by finite differences. Error analysis is again used to derive a stability criterion for automatic step size adjustment. This criterion is qualitatively similar to the Courant-Friedrichs-Lewy one, but substantially milder. The two dimensional EMMC scheme is combined with a Mixed Finite Elements method 4,7 to solve the concentration and pressure equations modeling the Miscible Oil Displacement in a porous medium. The spatial domain used is a "symmetry element" of the repeated five spot pattern. A variety of numerical tests are performed, including cases of adverse mobility ratios and variable reservoir permeability. The computational results show minimal grid orientation effects, reduced artificial dispersion, very low mass balance errors and convergence of the solution with spatial grid refinement. The total absence of any overshoot or undershoot in the computed solutions constitutes an experimental verification of the stability of the method.