We have already discussed, the stabilized production of gas is similar to the flow of oil; however, due to the compressible nature of gas, we must consider the pressure dependencies of the gas properties more rigorously than we did for oil. I will again start with the steady-state inflow performance relationship.

For steady-state analysis, we need to make the following assumptions:

- Flow is steady-state and Darcy’s Law applies and all of the assumptions inherent Darcy’s Law are valid in the system
- The drainage volume is radial-cylindrical (see
**Figure 4.01**) - The drainage volume is bounded in the interior with a cylindrical well (radius equal to ${r}_{w}$ ) and kept at flowing well pressure of ${p}_{wf}$
- The drainage volume is bounded on the exterior with a cylindrical boundary (radius equal to ${r}_{e}$ ) which has an external pressure of ${p}_{e}\text{}\left({p}_{wf}{p}_{e}\right)$
- The drainage volume is bounded on the top and bottom (constant height) no-flow boundaries
- Within the drainage volume, the rock properties are homogeneous (uniform with location)
- Within the drainage volume, the rock properties are isotropic (uniform in all directions)
- Flow is horizontal

With these assumptions, we can use the single-phase version of Darcy’s Law:

In **Equation 5.03**, we are using the * effective* permeability to gas in the presence of a water phase to allow for an irreducible, immobile water phase in the reservoir. Now, for radial flow, we have:

Substituting into Darcy’s Law, we have:

Separating variables and integrating results in:

Now, using the assumption of a homogeneous permeability, we have:

To this point, the derivation of the stabilized inflow performance relationship for gas is identical to the derivation for oil. In the derivation of the inflow performance relationship for oil, we assumed that ${\mu}_{o}\text{}{B}_{o}$ were constant, and we removed them from the integral. As stated earlier, the properties, ${\mu}_{g}$ and ${B}_{g}$ , are pressure dependent properties due to the compressible nature of gas and may need to be treated differently from the treatment of the oil properties. These differences manifest themselves in how we treat the integral in **Equation 5.08**.

There are three methods used in the industry to evaluate the integral in **Equation 5.08**. These methods lead to three different formulations for the inflow performance relationship for gas. These are:

- Stabilized Production of Gas in Terms of Pressure
- Stabilized Production of Gas in Terms of Pressure-Squared
- Stabilized Production of Gas in Terms of Pseudo-Pressure

In addition to these three analytical inflow performance relationships, we will discuss one common empirical inflow performance relationship, the Rawlins and Schellhardt ** Backpressure** or

**Deliverability Equation**^{[1]}.

[1] Rawlins, E.L. and Schellhardt, M.A. 1935. Backpressure Data on Natural Gas Wells and Their Application to Production Practices, 7. Monograph Series, U.S. Bureau of Mines.