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

(1)	$\begin{array}{l}\displaystyle q_o = \frac{ B_o \cdot ( q_O - R_v \, q_G) }{1- R_v \, R_s}\end{array}$

(2)	$\begin{array}{l}\displaystyle q_O = \frac{m_O}{\rho_O}\end{array}$

(3)	$\begin{array}{l}\displaystyle q_g = \frac{ B_g \cdot ( q_G - R_s \, q_O)}{1- R_v \, R_s}\end{array}$

(4)	$\begin{array}{l}\displaystyle q_G = \frac{m_G}{\rho_G}\end{array}$

(5)	$\begin{array}{l}\displaystyle q_w = B_w \cdot q_W\end{array}$

(6)	$\begin{array}{l}\displaystyle q_W = \frac{m_W}{\rho_W}\end{array}$

(7)	$\begin{array}{l}\displaystyle q_t = q_o + q_g + q_w\end{array}$

(8)	$\begin{array}{l}\displaystyle q_t = \frac{B_o - B_g \, R_v}{1-R_v \, R_s} \cdot q_O +\frac{B_g - B_o \, R_v}{1-R_v \, R_s} \cdot q_G + B_w \cdot q_W\end{array}$

(9)	$\begin{array}{l}\displaystyle q_t = \frac{B_o - B_g \, R_v}{(1-R_v \, R_s) \rho_O} \cdot m_O +\frac{B_g - B_o \, R_v}{(1-R_v \, R_s) \, \rho_G} \cdot m_G + \frac{B_w}{\rho_W} \cdot m_W\end{array}$

(10)	$\begin{array}{l}\displaystyle \rho_t = (\dot m_O + \dot m_G + \dot m_G)/q_t\end{array}$

Derivation

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See Also