Effects of Gas Property Variations On the Flow of Gases In Linear Systems
Authors: Ambastha A.K. , Rahman N.M. Anisur 
Published: March 2010

Abstract
This paper presents a detailed analysis of pressure profiles as modified by gas property variations across a rectilinear system nder large pressure-drop situations. Required numerical solutions for this study have been obtained by including the non-linear pressure gradient-squared term in the partial differential equation for linear flow of gases under steady-state and transient conditions.
This study shows that neither a pressure-averaged nor a volume-averaged compressibility-like term can adequately describe the effects of gas property variations on actual pressure profiles for all situations. Numerically-simulated gas flow data under large pressure-drop conditions have been analysed using traditional approaches to quantify the effects that resemble a "velocity coefficient" (β) as a result of gas property variations. A brief investigation of transient pressure profiles is also presented to suggest a time criterion for the attainment of steady-state conditions for gas flow in linear systems.

Background
Often, for gas flow under high-velocity conditions, observed pressure drop is larger than that predicted by Darcy's law. It is necessary to include a non-linear, quadratic term in Darcy's law to account for the additional pressure drop. Thus, the extra pressure drop for high-velocity gas flow situations can be modelled using a differential form of the Forchheimer(l) equation as derived by Green and Duwez(2):
Equation (1) Available In Full Paper.
An integrated form of Equation (1) for linear flow of gases in a core of length L, assuming constant gas viscosity and deviation factor, is(3):
Equation (2) Available In Full Paper.
Equation (2) suggests that a Cartesian graph of (p12 - p22)/qsc
vs. qscwill be a straight line whose slope and intercept are related to β) and k, respectively. Equation (2) has been widely-accepted for evaluating the velocity coefficient β) (also known as inertial resistance coefficient, turbulence factor, etc. in the petroleum literature) using experimental data under high-velocity gas flow conditions. Velocity coefficient (β)) is regarded as a rock property.
Also, β) is considered to be independent of the flowing fluid properties and the geometrical dimensions of the porous medium. Some of the key studies related to high-velocity gas flow appear n References (4) through (7).
Temeng and Home(8) developed some analytical solutions for the flow of gases in a rectilinear porous medium including the non-linear term in the pressure form of the diffusivity equation. The diffusivity equation was obtained by using Darcy's law as the appropriate rate equation and universal gas law as the equation of state. The non-linear coefficient multiplying the pressure-gradient squared term in the diffusivity equation used by Temeng and Home(8) contains the effects of gas property variations across a linear system. They used a pressure-averaged compressibility-like term to obtain a constant coefficient for the pressure-gradient squared term in the partial differential equation. They solved analytically the following two steady-state problems:Constant rate inlet condition as per Darcy's law and constant pressure outlet condition, andConstant pressure at both the inlet and the outlet.