To develop the inflow performance relationship in terms of pressure, we assumed that the group, $\frac{p}{{\mu}_{g}\text{}Z}$, 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 ${\mu}_{g}\text{}Z$ is relatively constant with pressure and remove it from **Equation 5.10**, leaving:

$${\int}_{{r}_{w}}^{{r}_{e}}\frac{1}{r}}dr=\frac{0.001127\left(2\pi \right)\left(5.615\right){T}_{SC}\text{}{k}_{g}\text{}h}{{q}_{g}\text{}{T}_{r}\text{}{p}_{SC}\text{}{\mu}_{g}\text{}Z}{\displaystyle {\int}_{{p}_{wf}}^{{p}_{e}}p\text{}dp$$

Equation 5.16

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

$${\left[lo{g}_{e}\left(r\right)\right]}_{{r}_{w}}^{{r}_{e}}=\frac{0.001127\left(2\pi \right)\left(5.615\right){T}_{SC}\text{}{k}_{g}\text{}h}{{q}_{g}\text{}{T}_{r}\text{}{p}_{SC}\text{}{\mu}_{g}\text{}Z}{\left[\frac{{p}^{2}}{2}\right]}_{{p}_{wf}}^{{p}_{e}}$$

Equation 5.17

Rearranging **Equation 5.17** results in:

$${q}_{g}=\frac{0.001127\left(2\pi \right)\left(5.615\right){T}_{SC}\text{}{k}_{g}\text{}h}{2\text{}{T}_{r}\text{}{p}_{SC}\text{}{\mu}_{g}\text{}Z}\frac{\left({p}_{e}^{2}-{p}_{wf}^{2}\right)}{{\mathrm{log}}_{e}\left(\frac{{r}_{e}}{{r}_{w}}\right)}$$

Equation 5.18a

or,

$${q}_{g}=\frac{0.01988\text{}{T}_{SC}\text{}{k}_{g}\text{}h}{{T}_{r}\text{}{p}_{SC}\text{}{\mu}_{g}\text{}Z}\frac{\left({p}_{e}^{2}-{p}_{wf}^{2}\right)}{{\mathrm{log}}_{e}\left(\frac{{r}_{e}}{{r}_{w}}\right)}$$

Equation 5.18b

or, after substituting the normal U.S. definitions of ${p}_{SC}$ and ${T}_{SC}$ :

$${q}_{g}=\frac{0.70325\text{}{k}_{g}\text{}h}{{T}_{r}\text{}{\mu}_{g}\text{}Z}\frac{\left({p}_{e}^{2}-{p}_{wf}^{2}\right)}{{\mathrm{log}}_{e}\left(\frac{{r}_{e}}{{r}_{w}}\right)}$$

Equation 5.18c

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*.