Solution of a Non-linear Gas Flow Equation By the Perturbation Technique

Abstract

Abstract The flow of gas in porous media is represented by a non-linear differential equation, even when the pseudo-pressure concept is used. Analytical solutions, in the past, involved linearization of the coefficient by assuming µc constant at initial conditions. This paper solves the non-linear equation by the method of Perturbation, and demonstrates that the effect of linearization is, for all practical purposes, equivalent to having a small negative skin. The validity of the pseudo-pressure approach using µici instead of µu is thus confirmed. 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 appropriate under certain conditions for gas flow relationships (Mathews and Russel, 1967). Al-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 Z2 the viscosity and compressibility factor of natural gas. The relationship of p and p2 to ψ is explained in Aziz et al. (1976). Because 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 µ z (viscosity x compressibility factor), the resulting differential equation, Equation 1 (ERCB 1975), still contains a non-linear term, µ c (viscosity x compressibility). (Equation 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 non-linear and cannot be solved analytically. AI-Hussainy and Ramey (1966) assumed µc to be constant at initial conditions, uici, and solved the resulting ice 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, the assumption of a constant µC to obtain a solvable differential equation may be questionable. It is the intent of this paper to solve the non-linear Equation 1 by the method of perturbation without assuming µc to be a constant. Because it 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). However, a typical relationship is shown in Figure 2.