Accurate Resolution of Physical Dispersion in the Multidimensional Numerical Modeling of Miscible and Chemical Displacement

Abstract

Summary This paper presents a finite-difference scheme for numerical modeling of the convection-dominated flows common in miscible or chemical flooding in porous media. The proposed difference operator possesses desirable phase and amplitude properties capable of resolving sharp fronts with negligible over- and undershoot. The scheme is generalized to multidimensions. Difference analogs have been derived for variable grid sizes for large-scale compositional simulations. Application to 3D miscible displacement problems and comparison with other standard industry methods have demonstrated the scheme's effectiveness in controlling numerical diffusion and grid-orientation sensitivity without any significant additional computational effort. Introduction Diffusion and convection play important roles in multicomponent, multiphase flow through porous media. Convection is often the dominant process, however, making the solution of the basic governing equations numerically challenging. In particular, in the modeling of EOR processes, which are generally characterized by relatively low levels of physical dispersion, application of standard central-difference methods to the first-derivative convection operator results in oscillatory solutions. To remedy this situation, several difference schemes have been suggested for the convection term, one of the most popular being to use one-sided upstream differencing. The truncation error introduced by this method, however, causes severely degraded solutions. Application of this upwind scheme to problems that entail propagating a sharp or unstable front (such as miscible displacement or chemical transport in porous media) produces unacceptably smeared results. This dissipation of fronts, called numerical diffusion, has received considerable attention in the reservoir simulation literature. The two basic approaches that have been explored are higher-order methods and variations of the method of characteristics. In many cases, higher-order methods are complex, are significantly more expensive than the first-order approximations usually applied, and exhibit numerical instability, particularly in cases with little or no capillarity or dispersion (problems with steep gradients and sharp fronts). The method of characteristics typically requires adaptive grids or other special front-tracking procedures. Another aspect of truncation error in multidimensional flow problems is grid-orientation sensitivity. Standard upstream difference techniques applied to the simulation of adverse-mobility-ratio miscible displacements yield fronts that are spatially distorted and that exhibit a strong dependence on the orientation of the finite-difference grid. Several schemes have been proposed to minimize grid-orientation effects. These principally involve nine-point finite differencing and improved calculations of interblock fluid mobilities. Some of these schemes greatly alleviate grid-orientation sensitivity but contribute little to reducing numerical diffusion. In this paper, we present a simple, accurate, and computationally efficient scheme suitable for large-scale reservoir simulations that significantly reduces numerical diffusion and grid-orientation sensitivity.