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Finally, we will investigate the use of the real gas pseudo-pressure in the diffusivity equation. For this, we differentiate the definition of the pseudo-pressure integral, **Equation 5.19**, using the fundamental theorem of calculus:

$$\frac{\partial m\left(p\right)}{\partial p}=2\frac{p}{{\mu}_{g}Z}$$

or,

$$\partial p=\frac{{\mu}_{g}Z}{2p}\partial m\left(p\right)$$

Substituting **Equation 5.42b** into **Equation 5.39** results in:

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

Again, we evaluate the ${c}_{g}{\mu}_{g}$ product at either the initial pressure, ${p}_{i}$ , or the average pressure, $\overline{p}$. In this development, we have:

- $\text{0.006328}$ is an equation constant (5.615 x 0.001127)
- $r$ is the radial coordinate in a radial-cylindrical coordinate system, ft
- $p$ is the pressure, psi
- $m\left(p\right)$ is the real gas pseudo-pressure, psi
^{2}/cp - $\varphi $ is the porosity of the reservoir, fraction
- ${c}_{g}$ is the gas compressibility, 1/psi
- ${\mu}_{g}$ is the gas viscosity, cp
- ${k}_{g}$ is the effective permeability to gas, md
- $t$ is time, days