Numerical Modeling With A Moving Coordinate System; Application To Flow Through Porous Media

Abstract

A new numerical technique for solving convective diffusive equations is developed. Convective diffusive equations arise from mass conservation relations in flow through porous media. When convection dominates diffusion the equations are very difficult to solve numerically, because inaccurate oscillating solutions can result. Simulation of enhanced oil recovery processes requires solution of several coupled convective diffusive equations and they can not be adequately solved because of these numerical difficulties.

An analysis and comparison of numerical methods are performed. Criteria are given for the spatial mesh size ([delta]x) to eliminate the oscillations for weighted residual methods. The new solution technique minimizes the oscillations.

The differential equations are transformed into a moving coordinate system with a time-dependent velocity, which eliminates the influence of the convective term, but makes the boundary location change in time. Both linear and nonlinear, one- and two-dimensional equations are solved with physically realistic parameters. Methods for determining the moving coordinate system velocity and changing the location of the boundary conditions are described. Orthogonal collocation on finite elements, finite difference and the Galerkin finite element method are applied. Several integration schemes including variable timestep schemes are used. Comparison of the accuracy and the solution cost to exact solutions and schemes with a fixed coordinate system are made. The feasibility of solving coupled nonlinear elliptic-parabolic problems is discussed. A new solution technique for solving elliptic equations in both a fixed and a moving coordinate system is developed.

The moving coordinate system is applicable to both one- and twodimensional problems at a cost savings of a factor 20-50 for physical realistic parameters.