Methods for Increased Accuracy in Numerical Reservoir Simulators
Authors: Todd M.R. , O'Dell P.M. , Hirasaki G.J. 
Published: December 2007

Abstract
This paper proposed the use of two-point upstream weighting of fluid mobility as an alternative to the generally employed single-point approximation Use of the two-point formula results in the reduction of both numerical dispersion of flood fronts and the sensitivity of predicted areal displacement performance to grid orientation. Stability analysis performance to grid orientation. Stability analysis provides the time-step limitation for control of provides the time-step limitation for control of solution oscillations. This together with limitations for control of overshoot and truncation error provides a practical basis for the automatic selection of time steps.

Introduction
As an indication of the growing concern for controlling the total cost of large-scale reservoir simulations, the emphasis of a number of recent publications has been directed toward increasing publications has been directed toward increasing computing efficiency. In this paper, two methods to increase the computing efficiency of reservoir simulators are described. The use of two-point upstream weighting of fluid mobility is described and compared with the commonly used single-point upstream approximation. The two-point approximation generally requires fewer grid blocks to obtain a given accuracy than does the single-point approximation. In addition, the calculated performance of areal models is less sensitive to grid performance of areal models is less sensitive to grid orientation when using the two-point approximation.
Computing efficiency is also increased with the use of an automatic time-step selector. Time-step limitations are described in this paper for controlling stability, overshoot (negative saturations), and truncation error. In general, these limitations change each time step as conditions change. If any of the limitations is exceeded, the results of the simulation may be meaningless. An automatic time-step selector detects and avoids running difficulties by using the proper time-step size. Using these methods, simulation proper time-step size. Using these methods, simulation results are obtained with less expenditure of engineering and computer time.

Two-Point Approximations For Fluid Mobility
The majority of general-purpose reservoir simulators reportedly in use today are based on the solution of finite-difference analogs to the conservation equations describing multiphase flow in porous media. Thus, the continuous domain of a reservoir is divided into a number of discrete blocks, and solutions for pressure and saturations are obtained at the grid block centers (or grid points). Central-difference approximations are normally used for the spatial derivatives in the discrete formulation of the conservation equations. As described below, this scheme necessitates the evaluation of flow coefficients (kk,/muB) at the planes separating adjacent grid blocks. As fluid and planes separating adjacent grid blocks. As fluid and reservoir properties are only defined at grid points, some method must be devised for approximating interblock flow coefficients based on values at the grid points.
Of the terms that make up the flow coefficients, only the saturation-dependent relative permeability changes rapidly enough from grid block to grid block to cause significant difficulty. Although several weighting schemes have been employed in the past for evaluating the relative permeability at a block face, only single-point upstream weighting appears to be in general use. Unfortunately, use of this weighting scheme is well known to cause excessive numerical dispersion of flood fronts. In addition, areal displacement performance is found to be quite sensitive to the grid orientation for grid meshes of practical extent for large-scale reservoir simulations. This has been demonstrated qualitatively by Garrett and will be described both qualitatively and quantitatively later in this paper.
As an alternative to single-point weighting of relative permeability, a two-point weighting of relative permeability, a two-point scheme is now described which results in both reduced numerical dispersion of flood fronts and decreased sensitivity of predicted areal performance to grid orientation.