A Critical Comparison of Finite-difference Interblock Mobility Approximations In Numerical Reservoir Simulation
Authors: Ko S.C.M. , Buchanan W.L. , Vinsome P.K.W. 
Published: March 2010

Abstract
A systematic comparison of eighteen different interblock mobility methods, four previously known and fourteen newly developed in this work, is mode using standard five-point finite-difference techniques to study the influence of inter-block mobility approximations on numerical dispersion and grid orientation in the numerical solution. Specially designed numerical experiments as well as conventional five-spot pattern simulations were performed to provide a common basis for the evaluation of these methods.
An improved interblock mobility method which uses centralized upstream fractional flow and harmonic total mobility (CUF & HTM) is proposed for use. Application of this new method can reduce both numerical dispersion of the flood fronts and sensitivity of predicted areal displacement performance to grid orientation as compared to the commonly used two-point upstream method.
In addition, guidelines on the limiting shock mobility ratio (Ms) for safe use of the previously known and the proposed CUF & HTM methods were developed. In general, use of all methods investigated in this study together with the standard five-point finite-difference techniques should be applied with caution for multidimensional simulation of reservoir performance with Ms ≥ 2.

Introduction
The use of finite-difference techniques in numerical reservoir simulation has been greatly advanced since the early development some twenty years ago. Despite the extensive history and wide acceptance of the methods in numerical reservoir modelling, two important and fundamental problems remain practically unresolved. They are the long-recognized difficulties associated with numerical dispersion and grid orientation.
Many previous attempts(1–7) to solve the problem associated with numerical dispersion or diffusion in the finite-difference solution of hyperbolic problems have resulted in only partial success. The problem becomes particularly severe in multidimensional simulations, because of practical limitations on the number of grid blocks.
For another class of special problems involving adverse mobility ratio piston-like displacements, the divergence of the numerical solutions depending on the orientation of the grid system has yet to be dealt with satisfactorily. Much effort(7·12) has been spent previously in an attempt to resolve this problem in finite-differencing, but a simple, effective and economical method which covers the whole range of adverse mobility ratios has yet to be developed for the commonly used five-point schemes. Furthermore, the relationships between numerical dispersion and grid orientation effects have not been qualitatively or quantitatively explained to date.
In this work, we attempted to address these two problems by studying systematically all available and some new interblock mobility methods in multidimensional reservoir simulation using the standard five-point finite-difference technique. A total of 18 interblock mobility methods were investigated. It was also our objective to establish guidelines for safe use of all commonly used and some new interblock mobility methods in areal pattern flood simulations.

Theory and Methods

General
To simplify the mathematical analysis, let us consider two immiscible and incompressible fluids (oil and water) flowing through a porous medium. For incompressible rock, and negligible capillary and gravity effects, the continuity equations for the water and oil phases are given by
(Equation in full paper)