5.4.2.2: The Diffusivity Equation for a Gas in Radial-Cylindrical Coordinates in Terms of Pressure

As we have already seen, in the pressure range of $p>3,000psi$ , the group $\frac{p}{{\mu }_{g}Z}$ can be considered to be approximately constant. With this simplification, we can remove the group from the spatial derivative on the left-hand side of Equation 5.37 to obtain:

This is the diffusivity equation for real gases in terms pressure, p. While this may at first glance appear to be linearized, it is not. Both the gas compressibility term, $c_g=\frac{1}{p}-\frac{1}{Z}\frac{dZ}{dp}$ , and the gas viscosity term, ${\mu }_g$ , are pressure dependent. When using the pressure formulation, we typically evaluate these terms at either the initial pressure condition, $p_i$ , or the average condition, $\overline{p}$. (Note: other more rigorous methods are available for the evaluation of the $c_g{\mu }_g$ product; however, they are beyond the scope of this course.)