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In the pressure-squared formulation, we assume that the ${\mu}_{g}Z$ product is constant. We have already seen that this approximation is valid for $p<2,000\text{}psi$. With this simplification, **Equation 5.37** becomes:

$$\frac{1}{r}\frac{\partial}{\partial r}\left(rp\frac{\partial p}{\partial r}\right)=\frac{\varphi {c}_{g}{\mu}_{g}p}{0.006328\text{}{k}_{g}}\frac{\partial p}{dt}$$

Now, we can use the identity:

$$\partial {p}^{2}=2p\partial p$$

or,

$$\partial p=\frac{1}{2p}\partial {p}^{2}$$

Substituting **Equation 5.40b** into **Equation 5.39** results in:

$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial {p}^{2}}{\partial r}\right)=\frac{\varphi {c}_{g}{\mu}_{g}}{0.006328\text{}{k}_{g}}\frac{\partial {p}^{2}}{dt}$$

This is the diffusivity equation for real gases in terms pressure-squared, ${p}^{2}$. Again, the ${c}_{g}{\mu}_{g}$ product represents a non-linear term which we evaluate at either the initial pressure, ${p}_{i}$, or the average pressure, $\overline{p}$.