5.4.1.2: Stabilized Flow of Gas to a Vertical Production Well in Terms of Pressure-Squared

To develop the inflow performance relationship in terms of pressure, we assumed that the group, $\frac{p}{{\mu }_{g}Z}$, was relatively constant in the pressure range of interest, and we removed the entire group from the pressure integral in Equation 5.10. In the pressure-squared formulation, we assume that the product ${\mu }_{g}Z$ is relatively constant with pressure and remove it from Equation 5.10, leaving:

Again, we will see that the ${\mu }_{g}Z$ product can be safely assumed to be relatively constant over a particular pressure range. Performing both integrations in Equation 5.16 results in:

Rearranging Equation 5.17 results in:

or,

or, after substituting the normal U.S. definitions of $p_{SC}$ and $T_{SC}$ :

In this equation, we evaluate ${\mu }_g$ and $Z$ at the arithmetic mean average pressure, Equation 5.15. Again, we can add a skin factor to account for well damage or stimulation and write similar equations in terms of average pressure ${\overline{p}}^2$ for the pseudo-steady state flow regime. Equation 5.18c is the Inflow Performance Relationship for Gas in Terms of Pressure-Squared.