ABSTRACT
A numerical method is developed for solving the convection-diffusion equation arising from the conservation equations used to represent miscible displacement processes. This method is superior to the conventional finite difference methods in that it introduces only a negligible amount of numerical dispersion, in contrast to the significant numerical smearing introduced by most finite difference techniques.
The construction of the method is based on the method of operator splitting; the first fractional step solves the convection equation with zero diffusion, whereas the second step accounts for the diffusive part of the convection-diffusion equation. Unlike conventional finite difference methods, which approximate the convection term by some form of upwind differencing scheme, the present approach calculates the convective flow by the random choice method in the first fractional step. The advantage of this approach is that it propagates the solution correctly without smearing out the displacement front, regardless of the grid size. This desirable feature is absent in most finite difference techniques, and as a result they are unable to model the physical dispersion accurately. In the second fractional step of the method, the parabolic character of the diffusion term is appropriately approximated by the conventional central differencing scheme.
The combined scheme is unconditionally stable. Extensive numerical tests show that the method is able to approximate the convection-diffusion equation accurately without introducing any significant numerical dispersion. No spurious oscillation is observed even at small values of the diffusion coefficient. In the limit when the diffusion coefficient approaches zero, a perfectly sharp displacement front is calculated. Extension of the method to multidimensional problems is possible by the use of operator splitting, which reduces the problem to fractional steps of locally one dimensional problem.
