5.4.1.1: Stabilized Flow of Gas to a Vertical Production Well in Terms of Pressure

To develop the inflow performance relationships for gas, we will need to use the definition of the gas formation volume factor, $B_g$, Equation 3.68:

or,

Since Equation 5.08 is written in terms of bbl/day, we need use the form of $B_g$ in bbl/SCF. Substituting Equation 5.09 into Equation 5.08 results in:

Now, if we assume that the term, $\frac{p}{{\mu }_{g}Z}$ , is constant with pressure, then we can remove it from the integral (in essence, this is what we did for the oil equation). While at first glance, this may seem like a bad assumption because we are removing p and two pressure dependent functions, ${\mu }_g$ and $Z$ , from an integration with respect to pressure; within certain pressure ranges it is not a bad assumption. This is because we are not interested in how the individual components in the group, $\frac{p}{{\mu }_{g}Z}$ , behave with respect to pressure but are concerned with how the group behaves in its entirety. I will discuss this in more detail later in this lesson. If we remove this group from the integral in Equation 5.10, then we obtain:

Performing the two integrations in Equation 5.11:

or,

Rearranging Equation 5.13 results in:

or,

Now, if we use the normal U.S. definitions of Standard Pressure $\left(p_{SC}=14.7psi\right)$ and Standard Temperature $\left(T_{SC}=520°R\right)$, then we have:

Equation 5.14c is the steady-state inflow performance relationship for single-phase gas production. All that remains is to discuss how to evaluate the term, $\left(\overline{\frac{p}{{\mu }_{g}Z}}\right)$. Typically, we do this by evaluating the average pressure as the arithmetic mean pressure:

and use that average pressure to evaluate ${\mu }_g$ and $Z$. Because all of the pressure terms in this equation are written directly as $“p”$, we refer to this formulation as the Inflow Performance Relationship for Gas in Terms of Pressure. As with the oil inflow performance relationship, we can add a skin factor to account for damage or stimulation and write similar equations in terms of $\overline{p}$ (rather than $p_e$ ) for the pseudo steady-state flow regime.