A Two-Layer Three-Dimensional Sirnulator For Studying a Gas Reservoir With? Bottom Water Drive
Authors: Woo P.T. , Emanuel A.S. , Kolada L.J. 
Published: March 2010

Abstract
This paper describes a two-layer, three-dimensional, two-phase, unsteady-state simulator which has been used to model gas reservoirs with bottom water drive. The reservoir-aquifer system is modeled as two layers; with full communication between layers. The gas zone together with a small fraction of the aquifer. is located in the top layer. The main body of the aquifer is located in the bottom layer. Pressure is calculated implicitly using the Line-Successive Over-Relaxation technique, and saturations are calculated explicitly. Test problems show that the simulator has good stability and material balance accuracy for a wide range of conditions. Application of the simulator to model gas cycling in the Kaybob South Beayerhill Lake "A" Pool is given. The mathematical technique is extendable to three-phase reservoirs with bottom water drive.

Introduction
THE RECENT DISCOVERY of the Kaybob South Beaverhill Lake "A" Pool created the need for a simulator which could model depletion and gas cycling in a gas reservoir with bottom water drive. To meet this need, a two-layer, three-dimensional, two-phase, unsteady-state simulator was used to calculate the pressure and saturation distributions in the reservoir. The three-dimensional simulator has good material balance accuracy for computer run times comparable to those of two-dimensional areal simulators. Although the application described in this paper is for a reef-type gas reservoir, the method is extendable to other types of gas or oil reservoirs with bottom water drive.

Review of Literature
To model in detail the effect of bottom water drive, generally a three-dimensional simulator, such as those reported by Coats et al.(IJ, and Breitenbach et al.(2), must be used. A two-dimensional areal simulator(5) would not model correctly the flow of bottom water, because there is no vertical flow path. Givens(4) and Field et al.(5) have described a two-dimensional, single phase areal simulator, which treats bottom water influx by using the van Everdingen and Hurst(6) unsteady- state equation in conjunction with a distribution function. This approach has two limitations. Areal flow of water is not calculated, and vertical flow of water is calculated only indirectly through the distribution function. The present paper presents a two-layer three-dimensional simulator which models water flow in more detail than the simulator described? by Givens and Field et al., and yet maintains the simplicity and economy of a two-dimensional simulator.

Mathematical Model

Model Equations
The partial differential equations for three-dimensional unsteady-state flow of gas and water through a differential volume of a gas reservoir are:
(Equation Available In Full Paper)
The solubility of gas in water and the partial pressure of water in the gas phase are neglected. The potential, Φ, is defined as follows:
(Equation Available In Full Paper)
where D is the depth (positive downward) of the differential volume from a reference plane, and p is the average density of the fluid between the differential volume and the reference plane. The water and gas pressures are related by the capillary pressure, P as follows: