Simulation of Stratified Waterflooding by Pseudo Relative Permeability Curves
Authors: Hearn C.L. 
Published: December 2007

In two-dimensional areal numerical simulation of injection projects, the use of relative permeability curves that do not properly describe vertical effects in the reservoir call lead to incorrect results. Here is a way to calculate carves that reflect reservoir stratification and that can be used in two-dimensional reservoir simulators to approximate the three-dimensional solution.

Introduction
Numerical reservoir simulation has seen increasing use as a reservoir engineering tool. An important consideration in using numerical models, particularly in studies involving pressure maintenance, is the vertical definition of the reservoir. An accurate description of vertical sweep efficiency is important if the model is to simulate field performance. The only rigorous way of modeling vertical effects is to use a three-dimensional reservoir simulator, but, this technique is not always practical. To avoid excessive data preparation time and computing expense it is preferable to use a two-dimensional areal model. preferable to use a two-dimensional areal model. Unfortunately, a two-dimensional areal calculation implies uniform reservoir properties and fluid saturations throughout the reservoir thickness - an assumption that clearly is incorrect for most reservoirs. This means the input data to the two-dimensional model must be adjusted so that it will approximate vertical effects. Previous authors have shown that, where vertical effects are dominated by gravity segregation, two-dimensional areal reservoir simulators can be used to approximate the three-dimensional solution through the use of appropriate pseudo relative permeability and capillary pressure functions. permeability and capillary pressure functions. However, vertical sweep may be dominated, not by gravity forces, but by viscous flow forces due to vertical permeability variation. This paper presents a method for developing pseudo relative permeability curves for two-dimensional simulation of fluid displacement projects where vertical sweep is primarily affected by permeability variation. primarily affected by permeability variation. A common method for approximating the effect of vertical permeability variation in displacement projects such as waterflooding is to assume that the projects such as waterflooding is to assume that the reservoir is stratified. To obtain a general model, the stratified model, which describes the vertical efficiency, is usually combined with a method for approximating areal sweep. Experience has shown that such a model can successfully simulate and predict the performance of pattern waterfloods. However, such a model may not be applicable when the well patterns are not uniform or when there is significant areal variation in reservoir properties. In such cases, a two-dimensional numerical model is convenient for simulating reservoir performance, using the pseudo relative permeability performance, using the pseudo relative permeability concept presented here. The pseudo relative permeability functions are based on a mathematical model for calculating vertical efficiency using a stratified reservoir concept. The stratified model is basically similar to those described by Hiatt and by Warren and Cosgrove. Viscous rather than gravity forces are assumed to dominate the vertical sweep. The model is thus more applicable to water-drive than to gas-injection projects where density differences may be large. The theoretical basis for using the pseudo relative permeability curves in numerical simulators can be permeability curves in numerical simulators can be seen by analysis of the stratified model.
JPT
P. 805