Fluid volume consisting of two or more contacting phases

In most practical cases in Petroleum Industry it will be a 3-phase fluid (water, oil and gas).

The total fluid density of multiphase fluid is given by following equation:

(1)	$\begin{array}{l}\displaystyle \rho = \sum_\alpha s_\alpha \, \rho_\alpha\end{array}$

where

$\begin{array}{l}\rho_\alpha\end{array}$

Density of $\begin{array}{l}\alpha\end{array}$ -phase

$\begin{array}{l}s_\alpha\end{array}$

volumetric share of $\begin{array}{l}\alpha\end{array}$ -phase

Derivation

(2)	$\begin{array}{l}\displaystyle \rho = \frac{m}{V} = \frac{1}{V} \, \sum_\alpha m_\alpha = \sum_\alpha \frac{m_\alpha}{V}= \sum_\alpha \frac{V_\alpha}{V} \cdot \frac{m_\alpha}{V_\alpha} = \sum_\alpha s_\alpha \, \rho_\alpha\end{array}$

The total fluid compressibility of multiphase fluid is given by following equation:

(3)	$\begin{array}{l}\displaystyle c = \sum_\alpha s_\alpha \, c_\alpha\end{array}$

where

$\begin{array}{l}\rho_\alpha\end{array}$

fluid compressibility of $\begin{array}{l}\alpha\end{array}$ -phase

$\begin{array}{l}s_\alpha\end{array}$

volumetric share of $\begin{array}{l}\alpha\end{array}$ -phase

Derivation

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

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See also