The Use of Vertical Equilibrium in Two-Dimensional Simulation of Three-Dimensional Reservoir Performance
Authors: Coats K.H. , Dempsey J.R. , Henderson J.H. 
Published: December 2007

Abstract
This paper discusses the use of the Vertical Equilibrium (VE) concept in simulating heterogeneous reservoirs. Where VE criteria are met, this technique allows two-dimensional (2-D) simulation of three-dimensional (3-D) problems with equivalent accuracy, and with attendant substantial savings in data preparation and machine time. The paper presents the VE concept itself and a new dimensionless group as a possible criterion for the validity of VE as applied to thick reservoirs or to reservoirs where the capillary transition zone is a small fraction of thickness. A description of the generation of the appropriate pseudo relative permeability and capillary pressure curves is permeability and capillary pressure curves is presented. presented. In addition to the dimensionless group criterion, an actual comparison of the results of an x-z cross-section and a one-dimensional (1-D) areal run with VE illustrates the validity of the VE concept. Numerical results of such a comparison along with the attendant machine-time requirements are presented. More than an order of magnitude difference in machine-time requirements was experienced. Finally, an actual field case example shows the utility of VE as applied to a reservoir containing one or multiple gas pools residing on a common aquifer.

Introduction
Numerical simulation of reservoir performance currently encompasses a wide variety of recovery processes, reservoir types and purposes or questions processes, reservoir types and purposes or questions to which answers are sought. A feature common to virtually all reservoir simulation studies, however, is the choice of simulation in one, two or three dimensions. Most frequently this choice is one between an areal (x-y) study and a 3-D study. While the areal study is considerably cheaper than a 3-D simulation, the validity or accuracy of the former is often questioned in light of its apparent inability to simulate flow and fluid saturation distributions in the vertical direction. Areal studies are frequently performed with little attention to or understanding of the extent to which the x-y calculations do or can be made to account for this vertical flow and fluid distribution. Previous papers describe a VE assumption or concept which leads to the definition of pseudo relative permeability and capillary pressure curves to be used in areal studies to simulate 3-D flow. A dimensionless group proposed as a criterion for the assumptions validity primarily treats the case of a reservoir where the capillary transition zone is an appreciable fraction of reservoir thickness. This paper neats the case of a reservoir where the capillary capillary transition zone is a small fraction of reservoir thickness (e.g., less than 10 percent). We propose to describe the VE concept as percent). We propose to describe the VE concept as applied to thick reservoirs or to reservoirs where capillary transition zone is a small fraction of thickness; to describe the generation of appropriate relative permeability and capillary pressure curves for such reservoirs to represent 3-D performance by 2-D areal calculations; to propose a new dimensionless group as a criterion for the VE assumptions' validity, obtained from an analysis of countercurrent gravity segregation; and finally, to present a cross-sectional vs 1-D (VE) comparison and a 2-D areal field case study.

THE VERTICAL EQUILIBRIUM CONCEPT
Most oil and gas reservoirs extend distances areally which are at least two orders of magnitude greater than reservoir thickness. Viewed in perspective, these reservoirs appear as "blankets" perspective, these reservoirs appear as "blankets" For a variety of reasons, some valid and some invalid, numerical simulations of such reservoirs are performed occasionally in three dimensions as opposed to only two areal (x-y) dimensions.
SPEJ
P. 63