Non-Darcy Transient Radial Gas Flow Through Porous Media
Authors: Dranchuk P.M. , Flores J. 
Published: December 2007

Abstract
A computer program was developed for solving the nonlinear partial differential equations describing transient radial gas flow through porous media. Specifically, the program can handle the four sets of boundary conditions encountered in a finite radial system. The program has been employed to obtain pressure distributions and mass fluxes as functions of time and position for the constant terminal rate case and the constant terminal pressure case, in porous media having different values of permeability, inertial resistance coefficient, slip coefficient, and rock compressibility; the program takes into account changes of gas properties with pressure. Results indicate that each of these rock properties can, under certain conditions, appreciably affect the transient response. For instance, it was observed that slip, rather than being a laboratory curiosity, can affect the transient response in a typical tight reservoir.

Introduction
The flow of gases through porous media has been a subject of investigation for many years. The method of attack has consisted of developing simplified models describing steady flow and combining them with the continuity equation in order to describe transient behavior. The simplified steady-flow models were gradually made more sophisticated and were in turn used to develop models which were better able to describe transient behavior in real systems. Because of its relative simplicity, linear flow has usually been studied first, with the results obtained being then used to solve the radial case. Thus, the nonlinear partial differential equations describing the transient flow of an ideal gas through a linear porous medium were first solved by means of finite-difference methods. This was followed by a more detailed treatment of the linear case and a treatment of the radial system. Others proposed solutions for the radial case. The manner in which molecular streaming or slip altered the transient in linear systems was examined theoretically by Collins and Crawford and by Aronofsky and verified experimentally by Wallick and Aronofsky. The influence of nonideal gas behavior on the transient response for the linear case was examined by Aronofsky and Ferris. The effect of such behavior on the radial case was initially investigated by Aronofsky and Porter and later by Al-Hussainy et al. Non-Darcy How and its effect on the transient response was examined by Swift and Kiel and Tek et al. To this point the relaxations regarding the initial simplified model had consisted of adding one effect; viz, slip, real gas behavior, or non-Darcy flow. The introduction of two complicating effects was first applied to the transient response by Eilerts et al, who studied condensate reservoirs by allowing non-Darcy flow and real gas behavior. The manner in which inertial and slip effects together influenced steady linear gas flow was studied by Dranchuk and Sadiq and by Dranchuk and Kolada. These same two effects were incorporated in the transient model for the linear case by Dranchuk and Chwyl. Recently, a model was developed describing steady radial gas flow in which slip, inertial effects, and real gas behavior are permitted simultaneously; the model has been successfully applied to laboratory data. On the basis of this model and the realization that the interactions between the various effects in a nonlinear partial differential equation can be unpredictable, this study was undertaken. The object was to develop a transient radial gas flow model in which slip, inertial effects, real gas behavior, and rock compressibility are permitted to operate simultaneously; to solve the resulting model for various boundary conditions; and to examine the influence on the resulting transient response of these complicating effects.

MATHEMATICAL MODEL
The purpose of this study was accomplished by observing the behavior of some hypothetical gas reservoirs.
SPEJ
P. 129