Synonyms: Compressibility factor = Z-factor

Disclaimer: Not to be confused with Compressibility $\begin{array}{l}c\end{array}$ .

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

(1)	$\begin{array}{l}\displaystyle Z = \frac{p \, V_m}{R \, T} = \frac{p}{\rho} \cdot \frac{M}{R \, T}\end{array}$

where

$\begin{array}{l}p\end{array}$	fluid pressure	$\begin{array}{l}V_m = V/\nu\end{array}$	fluid molar volume
$\begin{array}{l}T\end{array}$	fluid temperature	$\begin{array}{l}V\end{array}$	fluid volume
$\begin{array}{l}\rho\end{array}$	fluid density	$\begin{array}{l}\nu\end{array}$	amount of substance
$\begin{array}{l}R\end{array}$	gas constant	$\begin{array}{l}M\end{array}$	molar mass of a fluid

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

(2)	$\begin{array}{l}\displaystyle Z(T, p) = Z^{\circ} \cdot \frac{\rho^{\circ} \, T^{\circ}}{p^{\circ}} \cdot \frac{p}{\rho(T, p) \, T}\end{array}$

where $\begin{array}{l}()^{\circ}\end{array}$ means reference conditions, usually Standard Conditions (STP).

Z-factor can be used to calculate fluid density $\begin{array}{l}\rho\end{array}$ and Formation Volume Factor (FVF) $\begin{array}{l}B\end{array}$ as:

(3)	$\begin{array}{l}\displaystyle \rho(T, p) = \rho^{\circ} \cdot \frac{Z^{\circ} \, T^{\circ}}{p^{\circ}} \cdot \frac{p}{Z(T, p) \, T}\end{array}$

(4)	$\begin{array}{l}\displaystyle B(T, p) = \frac{\rho^{\circ}}{\rho(T, p)} = \frac{p^{\circ} }{Z^{\circ} \, T^{\circ}} \cdot \frac{Z(T, p) \, T}{p}\end{array}$

Z-factor is related to fluid compressibility $\begin{array}{l}c\end{array}$ as:

(5)	$\begin{array}{l}\displaystyle c(p) = \frac{1}{p} - \frac{1}{Z} \frac{dZ}{dp}\end{array}$

(6)	$\begin{array}{l}\displaystyle Z(p) = Z_0 \cdot \frac{p}{p_0} \cdot \exp \left[ - \int_{p_0}^p c(p) dp \right]\end{array}$

Derivation

(7)

$\begin{array}{l}\displaystyle c = \frac{1}{\rho} \frac{d\rho}{dp} = \frac{d \ln \rho}{dp} = \frac{d }{dp} \left( \ln \left(\frac{p}{Z} \right) \right) = \frac{Z}{p} \cdot \frac{d }{dp} \left(\frac{p}{Z} \right) = \frac{Z}{p} \cdot \left( \frac{1}{Z} + p \cdot \frac{d }{dp} \left( \frac{1}{Z} \right) \right) = \frac{1}{p} - \frac{1}{Z} \frac{dZ}{dp}\end{array}$

Rewriting (5):

(8)	$\begin{array}{l}\displaystyle \frac{d \ln Z}{dp} = \frac{1}{p} - c(p) \rightarrow \ln \frac{Z}{Z_0} = \ln \frac{p}{p_0} - \int_{p_0}^p c(p) \, dp\end{array}$

which arrives to (6).

The Z-factor value for Ideal Gas is strictly unit: $\begin{array}{l}Z(T, p) = 1\end{array}$ .

For many real gases (particularly for the most compositions of natural gases) the Z-factor is trending towards unit value ( $\begin{array}{l}Z \rightarrow 1\end{array}$ ) while approaching the STP.

For incompressible fluids the Z-factor is trending to linear pressure dependence ( $\begin{array}{l}Z \rightarrow a \cdot p\end{array}$ ) with pressure growth.

Modelling Z-factor $\begin{array}{l}Z(T,p)\end{array}$ as a function of fluid pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$ is based on Equation of State.

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

Page tree

References