PNG 301
Introduction to Petroleum and Natural Gas Engineering 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:

B g =Z T r   p sc T sc   p in f t 3 /SCF
Equation 5.09a


B g =Z T r   p sc 5.615  T sc   p in bbl/SCF
Equation 5.09b

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:

r w r e 1 r  dr= 0.001127( 2π )( 5.615 ) T SC   k g   h q g   T r   p SC Pwf Pe p μ g   Z  dp
Equation 5.10

Now, if we assume that the term, p μ 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, μ 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, p μ 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:

r w r e 1 r dr= 0.001127( 2π )( 5.615 ) T SC   k g   h q g   T r   p SC ( p μ g  Z ¯ ) p wf p e dp
Equation 5.11

Performing the two integrations in Equation 5.11:

[ lo g e ( r ) ] r w r e = 0.001127( 2π )( 5.615 ) T SC   k g  h q g   T r   p SC ( p μ g  Z ¯ ) [ p ] p wf p e
Equation 5.12


lo g e ( r e r w )= 0.001127( 2π )( 5.615 ) T SC   k g  h q g   T r   p SC ( p μ g  Z ¯ )( p e p wf )
Equation 5.13

Rearranging Equation 5.13 results in:

q g = 0.001127( 2π )( 5.615 ) T SC   k g   h T r   p SC ( p μ g   Z ¯ ) ( p e p wf ) log e ( r e r w )
Equation 5.14a


q g = 0.03976   T SC   k g   h T r   p SC ( p μ g   Z ¯ ) ( p e p wf ) lo g e ( r e r w )
Equation 5.14b

Now, if we use the normal U.S. definitions of Standard Pressure ( p SC =14.7 psi ) and Standard Temperature ( T SC =520°R ), then we have:

q g = 1.4065   k g   h T r ( p μ g   Z ¯ ) ( p e p wf ) lo g e ( r e r w )
Equation 5.14c

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, ( p μ g  Z ¯ ) . Typically, we do this by evaluating the average pressure as the arithmetic mean pressure:

p ¯ = ( p e + p wf ) 2
Equation 5.15

and use that average pressure to evaluate μ 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 p ¯ (rather than p e ) for the pseudo steady-state flow regime.