Compressibility of multiphase fluid in thermodynamic equilibrium at a given pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$ is a linear sum of its single-phase components:

(1)	$\begin{array}{l}\displaystyle c_f(p, T) = \sum_{\alpha} s_\alpha \cdot c_\alpha(p,T)\end{array}$

where

$\begin{array}{l}s_\alpha\end{array}$	$\begin{array}{l}\alpha\end{array}$ -phase volume share, subjected to $\begin{array}{l}\sum_{\alpha} s_\alpha = 1\end{array}$
$\begin{array}{l}c_\alpha(p, T)\end{array}$	$\begin{array}{l}\alpha\end{array}$ -phase compressibility as function of pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$

Derivation

The total multiphase volume:

(2)	$\begin{array}{l}\displaystyle V = \sum V_\alpha\end{array}$

where $\begin{array}{l}V_\alpha\end{array}$ are volumes, occupied by individual phases.

The volume fraction of individual phase is defined as:

(3)	$\begin{array}{l}\displaystyle s_\alpha = \frac{V_\alpha}{V}\end{array}$

This leads to:

(4)	$\begin{array}{l}\displaystyle c_f = \frac{1}{V} \, \frac{\partial V}{\partial p} = \frac{1}{V} \sum_\alpha \frac{\partial V_\alpha}{\partial p} = \sum_\alpha \frac{V_\alpha}{V} \, \frac{1}{V_\alpha} \frac{\partial V_\alpha}{\partial p} = \sum_\alpha s_\alpha \, c_\alpha\end{array}$

In most popular practical case of a 3-phase fluid model this will be:

(5)	$\begin{array}{l}\displaystyle c_f = s_w \, c_w + s_o \, c_o + s_g \, c_g\end{array}$

where $\begin{array}{l}\{ w, \, o, \, g \}\end{array}$ mean water phase, oil phase and gas phase.

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See also