To develop the inflow performance relationships for gas, we will need to use the definition of the gas formation volume factor, , Equation 3.68:
Since Equation 5.08 is written in terms of bbl/day, we need use the form of in bbl/SCF. Substituting Equation 5.09 into Equation 5.08 results in:
Now, if we assume that the term, , 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, and , 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, , 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:
Rearranging Equation 5.13 results in:
Now, if we use the normal U.S. definitions of Standard Pressure and Standard Temperature , 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, . Typically, we do this by evaluating the average pressure as the arithmetic mean pressure:
and use that average pressure to evaluate and . Because all of the pressure terms in this equation are written directly as , 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 (rather than ) for the pseudo steady-state flow regime.