Published on *PNG 520: Phase Behavior of Natural Gas and Condensate Fluids* (https://www.e-education.psu.edu/png520)

The isothermal compressibility of a fluid is defined as follows:

$${c}_{f}=-\frac{1}{V}{\left(\frac{\partial V}{\partial \rho}\right)}_{T}$$

This expression can be also given in term of fluid density, as follows:

$${c}_{f}=-\frac{1}{\rho}{\left(\frac{\partial \rho}{\partial P}\right)}_{T}$$

For **liquids**, the value of isothermal compressibility is very small because a unitary change in pressure causes a very small change in volume for a liquid. In fact, for slightly compressible liquid, the value of compressibility (c_{o}) is usually assumed independent of pressure. Therefore, for small ranges of pressure across which c_{o} is nearly constant, Equation (18.16) can be integrated to get:

$${c}_{o}\left(p-{p}_{b}\right)=\mathrm{ln}\left(\frac{{\rho}_{o}}{{\rho}_{ob}}\right)$$

In such a case, the following expression can be derived to relate two different liquid densities $({\rho}_{o}\text{,}\rho \text{ob)}$
at two different pressures (p, p_{b}):

$${\rho}_{o}={\rho}_{ob}\left[1+{c}_{o}(p-{p}_{b})\right]$$

The Vasquez-Beggs correlation is the most commonly used relationship for c_{o}.

For **natural gases,** isothermal compressibility varies significantly with pressure. By introducing the real gas law into Equation (18.16), it is easy to prove that, for gases:

$${c}_{g}=\frac{1}{P}-\frac{1}{Z}{\left(\frac{\partial Z}{\partial P}\right)}_{r}$$

Note that for an ideal gas, c_{g} is just the reciprocal of the pressure. “c_{g}” can be readily calculated by graphical means (chart of Z versus P) or by introducing an equation of state into Equation (18.19).