Abstract

The differential equation for flow of gases in a porous medium is non-linear and cannot be solved by strictly analytical methods. Previous studies in the literature have obtained analytical solutions to this equation by linearization (i.e. treating viscosity and compressibility as constant).

In this study, the solution for the non-linear gas flow equation is obtained Using the semi-analytical technique developed by Kale and Mattar which solves the non-linear equation by the method of perturbation.

Results obtained, for prediction of pressure drawdown in gas reservoirs, indicate that the solution of the linearized form of the equation is valid for both low and high permeability reservoirs.

Introduction

The flow of gas through porous media is represented by the well known diffusivity equation. In the past (Aronofsky and Jenkins, 1953) assumptions of ideal gas behaviour led to the use of "pressure squared" as the variable for analysis of gas flow rate. The standard approach was to use the same fluid flow equations for gas or oil flow but in terms of p for oil flow and in terms of p2 for gas flow. Later it was shown that the use of p instead of p2 was as appropriate under certain conditions for gas flow relationships (Mathews and Russel, 1967). AI-Hussainy and Ramey (1966) introduced yet a third variable, Ψ, known as the pseudo-pressure or real gas potential which allowed for variations in µ and z, the viscosity and compressibility factor of natural gas. The relationship of p and p2 to Ψ is explained in Aziz et. al. (1976).

Since the diffusivity equation for gas flow in terms of the pseudo-pressure involves the least number of assumptions, it is considered to be the most rigorous of the three treatments (p, p2, Ψ). However, even though Ψ accounts for variations in µz (Viscosity x compressibility factor) the resulting differential equation. Equation 1, (ERCB 1975) still contains a nonlinear term, µc, (viscosity x compressibility). Equation (1) (Available in full paper)

Where ΔPD is a dimensionless pressure drop, and is defined in the nomenclature along with the other symbols.

Because µc is a function of Ψ, this equation is nonlinear and cannot be solved analytically. Al-Hussainy and Ramey (1966) assumed µc to be constant at initial conditions, µici, and solved the resulting linear partial differential equation. The solution is the familiar Exponential Integral, Ei, solution.

The dependence of µc on p, p2 or Ψ was investigated in a paper by Mattar (1979) and it was found that under certain circumstances, the change in µc can be substantial therefore, assumption of a constant µc to obtain a solvable differential equation may be questionable.

It is the intent of this publication to solve the non-linear Equation 1 by the method of perturbation without assuming µc to be a constant. Since Ψ is the variable that introduces the least number of assumptions, it will be used in preference to p or p2.

Extent of non-linearity – Variation of µc with Ψ

The variation of µc with Ψ for natural gas is discussed in detail by Mattar (1979).

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