Synonyms: Compressibility factor = Z-factor

Disclaimer: Not to be confused with Compressibility .

Dimensionless multiplier describing the deviation of a fluid density from ideal gas estimate under the same pressure & temperature conditions:

Z = \frac{p  \, V_m}{R \, T} = \frac{p}{\rho} \cdot \frac{M}{R \, T}

where

	fluid pressure		fluid molar volume
	fluid temperature		fluid volume
	fluid density		amount of substance
	gas constant		molar mass of a fluid

Alternatively Z-factor can be expressed through the dynamic fluid properties at reference conditions as:

Z(T, p) = Z^{\circ} \cdot \frac{\rho^{\circ} \, T^{\circ}}{p^{\circ}} \cdot \frac{p}{\rho(T, p) \, T}

where means reference conditions, usually Standard Conditions (STP).

Z-factor can be used to calculate fluid density and Formation Volume Factor (FVF) as:

\rho(T, p) = \rho^{\circ} \cdot \frac{Z^{\circ} \, T^{\circ}}{p^{\circ}} \cdot \frac{p}{Z(T, p) \, T}

B(T, p) = \frac{\rho^{\circ}}{\rho(T, p)} =  \frac{p^{\circ} }{Z^{\circ} \, T^{\circ}} \cdot \frac{Z(T, p) \, T}{p}

Z-factor is related to fluid compressibility as:

c(p) = \frac{1}{p} - \frac{1}{Z} \frac{dZ}{dp}

Z(p) = Z_0 \cdot \frac{p}{p_0} \cdot \exp \left[ - \int_{p_0}^p c(p) dp  \right]

c = \frac{1}{\rho} \frac{d\rho}{dp}  = \frac{d \ln \rho}{dp} =  \frac{d }{dp} \left(  \ln  \left(\frac{p}{Z} \right)  \right) = \frac{Z}{p} \cdot \frac{d }{dp} \left(\frac{p}{Z} \right) = \frac{Z}{p} \cdot \left( \frac{1}{Z} + p \cdot \frac{d }{dp} \left( \frac{1}{Z} \right)   \right) = \frac{1}{p}  - \frac{1}{Z} \frac{dZ}{dp}

Rewriting :

\frac{d \ln Z}{dp} = \frac{1}{p} - c(p) \rightarrow \ln \frac{Z}{Z_0} = \ln \frac{p}{p_0} - \int_{p_0}^p c(p) \, dp

which arrives to .

The Z-factor value for Ideal Gas is strictly unit: .

For many real gases (particularly for the most compositions of natural gases) the Z-factor is trending towards unit value () while approaching the STP.

For incompressible fluids the Z-factor is trending to linear pressure dependence () with pressure growth.

Modelling Z-factor as a function of fluid pressure and temperature is based on Equation of State.

There is also a good number of explicit Z-factor Correlations @models.

References

Lateef A. Kareem, New explicit correlation for the compressibility factor of natural gas, 2016