An Improved Numerical Technique for Solving Multidimensional Miscible Displacement Equations
Authors: Chaudhari Narayan M. 
Published: December 2007

Abstract
Difference analog equations of high-order accuracy describing stable miscible displacement are presented. The high-order-difference scheme eliminates almost all the numerical smearing, which is the result of the normally used approximations, and leaves only the effect of the physical dispersion in the solution. In a one-dimensional system, the technique involves an addition of a negative dispersion term to the continuity equation. The negative dispersion term, which is called the numerical dispersion coefficient, depends upon the flow velocity, the time-step size and the block size. The procedure is extended to multidimensional systems. As a check, comparisons of the computed results with analytical solutions in one- and two-dimensional systems are made. The error-function solution in a one-dimensional miscible system and a five-spot fractional flow curve computed from the potentiometric model are considered as the analytical potentiometric model are considered as the analytical solutions in the two cases. Based on the computed results, it has been concluded that the effect of the term with the negative numerical dispersion coefficient accounts for almost all the truncation error within the practical limits of oilfield problems.

Introduction
Miscible flooding is one of the most efficient hydrocarbon recovery processes known at the present time. Its application in the field, however, present time. Its application in the field, however, remains limited because of economic requirements. Recently, the development of vertical miscible floods has made the process economically attractive. The result is that many oil companies have proposed and started such floods in suitable reservoirs. The amount of solvent required to flood an entire reservoir miscibly can be roughly calculated by analytical methods. Since these methods are derived for a one-dimensional system, some important questions remain unanswered. Some of these questions are:How will the solvent bank develop under the proposed injection scheme?What will be the condition of the solvent bank at breakthrough?
These questions can only be answered by two- or three-dimensional studies. Known numerical techniques simulating miscible displacement give considerable smearing of concentration in the liquid-liquid transition zone when applied to a Held case. The effect of numerical smearing in this case is significantly larger than the effect of physical dispersion. Therefore, in the past, immiscible simulators with controlled numerical past, immiscible simulators with controlled numerical smearing were used to simulate miscible floods. For example, in one case the two-phase saturation profile was displaced as near piston-like as possible, profile was displaced as near piston-like as possible, neglecting mixing due to molecular diffusion and convective dispersion. In another case, capillary pressure and relative permeability functions were pressure and relative permeability functions were modified to simulate the dispersion effect. It was stated that the second method is applicable to a system where numerical smearing is considerably less than the effect of physical dispersion. This limits the use of the second method to a system such as gas-gas displacement. In this article a high-order difference scheme for miscible equations is given. This scheme eliminates most of the numerical smearing, leaving only the effect of physical dispersion in the transition zone. The procedure is developed for one- and two-dimensional systems. Calculated results are compared with analytical solutions. It is shown that the procedure can be extended to a three-dimensional system.

FORMULATION OF THE PROBLEM
The differential equations governing a miscible displacement process for an incompressible system are the continuity equations and the flow equations. Numerical solutions of this set of equations have been described by Peaceman and Rachford and Warren et al. Their methods are adequate to simulate miscible displacement in laboratory systems. If one applies them to a field-size system, significant truncation errors are introduced in the results. Usually these errors are large compared with the effect of physical dispersion on the results.
SPEJ
P. 277