Performance Predictions for Gas Reservoirs Considering Two-Dimensional Unsteady-State Flow
Authors: Carter R.D. 
Published: December 2007

Abstract
Methods are presented for calculating the performance of multiwell gas reservoirs. These methods account for two-dimensional, unsteady-sate flow of a non-ideal gas through a heterogeneous formation. The performance can be obtained for gas reservoirs with known physical properties and configuration subjected to specified development and production schedules. These methods may prove helpful in optimizing the exploitation of gas reservoirs. The numerical solution procedures given for this non-linear problem are extensions of known techniques for the linear diffusion equation. Two of these procedures are based on a new difference equation given by Saul'ev. Limitations and advantages of these procedures are discussed. An example solution is presented and is used to compare the calculation methods.
Introduction
In planning the operation of a multiwell gas reservoir, it is desirable to predict its performance for each alternate development and producing schedule under consideration. This paper presents three numerical procedures that can be used to calculate such gas reservoir performance using a digital computer. These methods account for two-dimensional transient flow of a non-ideal gas within the reservoir. Extension of these methods to three-dimensional flow is possible. The paper is divided into three parts. The first describes the problem for which a solution is wanted and three numerical procedures that can be used to obtain solutions. The second discusses the information necessary to set up a problem. This part will be of greatest interest to the field engineer. The third gives data and results for an example problem.
PROBLEM AND SOLUTION METHODS
In the following development, a gas reservoir is regarded as a sealed, porous, permeable, heterogeneous body in which a single, non-ideal gas phase is flowing. Flow is permitted in the x- and y-horizontal directions, but is considered to be negligible in the vertical or z-direction. The reservoir can be tapped by one or more wells. Methods to be described provide a basis for calculating the complete performance of the reservoir, including rates and cumulative production from the wells, and pressure distribution and decline in the reservoir.
If the isothermal flow of a non-ideal gas obeying Darcy's Law occurs in two dimensions in a region R, the governing equation in this region is:
= .....................(1)
In Eq. 1, k, h and are specified functions of x and y. The quantities and z are specified function of pressure. The term Qg(x, y, t) is a withdrawal term which simulates the presence of wells and should be determinable at any time point to which the solution is carried. However, this does not imply that Qg (x, y, t) is necessarily completely specified at the start of the solution. As will be indicated later, the exact form of Qg (x, y, t) may be part of the solution. A problem is defined by Eq. 1 and appropriate boundary and initial conditions if the various functions discussed above are specified and means are provided for determining Qg (x, y, t). The solution to the problem is the function p (x, y, t) and perhaps Qg (x, y, t). Approximate solutions to problems involving Eq. 1 have been obtained using three finite difference formulations.
SPEJ
P. 35ˆ