Multidimensional Numerical Dispersion

Numerical dispersion can cause a smearing of otherwise sharp saturation fronts. The usual methods of estimating the magnitude of the smearing effect in one dimension (1D) are shown to apply in two and three dimensions (2 and 3D) as well. Besides the smearing effect, numerical dispersion affects the finite-difference solution of a multidimensional flow problem by rotating the principal flow axes. A method for determining the importance of the rotation effect is discussed. Numerical illustrations are included. Introduction Most reservoir simulation models available today obtain solutions to fluid flow equations-usually nonlinear partial differential equations-by replacing derivatives with finite-difference approximations. The use of these approximations, derived by manipulating Taylor's series, introduces an error known as truncation error. For many problems, the error is small and the approximate solutions of the subsequent finite-difference equations are sufficiently accurate for engineering purposes. However, truncation error can cause significant solution inaccuracies for certain types of problems. Examples of these problems include miscible floods and immiscible floods in which viscous forces are much larger than capillary forces. The most common example of the latter is the Buckley-Leverett problem with capillary pressure set to zero. A relatively simple equation that exhibits the effect of truncation error is the 1D convection-dispersion equation, (1) where the constant coefficients phi, K, and v are porosity, the dispersion coefficient, and velocity, respectively. The solution, S, of Eq. 1 may be saturation or concentration. The finite-difference solution of Eq. 1 introduces truncation error that can smear an otherwise sharp saturation front as if additional physical dispersion were present. This smearing, caused by truncating Taylor's series, often is called numerical dispersion or numerical diffusion. Truncation error studies often begin with a 1D convection-dispersion equation, such as Eq. 1, after the space and time coordinates (x and t) are redefined to remove two of the three constant coefficients (phi, K, and v). It appears that the effects of numerical dispersion in more than 1D have not been studied analytically, though attempts to minimize numerical dispersion in 2D -- particularly the grid orientation effect -- are numerous. All the attempts in more than 1D are empirical in the sense that the degree of success of the proposed numerical dispersion reduction method is determined relative to a case that does not include the proposed method. It is the purpose of this paper to present an analysis of the effects of numerical dispersion on the finite-difference numerical solution of the multidimensional convection-dispersion equation. The analytic results will provide a better understanding of the role that numerical dispersion plays in 2- or 3D; they can be used for estimating numerical dispersion effects, and they can provide a standard analytic basis for evaluating the degree of success of proposed numerical dispersion reduction methods. Analytical expressions for multidimensional numerical dispersion (MND) coefficients, derived by performing a truncation error analysis on the 3D convection-dispersion equation, are presented. This analysis is analogous to that used by Lantz in 1D. The significance of the results then is examined. SPEJ P. 143^