Minimization of Grid Orientation Effects Through Use of Higher Order Finite Difference Methods

Abstract

ABSTRACT Grid orientation effects can arise in simulations of recovery processes when the mobility ratio of the displacement is unfavorable. These effects result from the application of numerical solution techniques to equations describing physically unstable displacement processes. In theory, the introduction of stabilizing terms (such as physical dispersion in the case of a miscible displacement) in the governing equations acts to minimize these effects. In practice, however, the numerical dispersion resulting from the first order discretization of the convective terms overwhelms the physically dispersive terms, and the stabilizing effects of these terms are not accurately modeled. In this paper, we apply higher order, shock capturing, finite difference methods to the simulation of miscible and immiscible displacement processes. This will be shown to reduce numerical dispersion to a level where the effects of the stabilizing terms can be accurately resolved numerically, resulting in the minimization of grid orientation effects. Two different families of higher order methods are investigated. The first such method, a so-called second order TVD (total variation diminishing) scheme, reduces numerical dispersion somewhat relative to the usual first order scheme. The second method investigated, a third order ENO (essentially non-oscillatory) scheme, reduces numerical dispersion even further, to an extent such that the stabilizing terms can be accurately resolved. This will be seen to nearly eliminate grid orientation effects in some cases in which the usual first order finite difference approach yields highly grid dependent solutions and the second order approach yields slightly grid dependent solutions.