q_O = q_{Oo} + q_{Og}



q_G = q_{Gg} + q_{Go}



q_W = q_{Ww}



Following the definition of Solution GOR (Rs) and Vaporized Oil Ratio (Rv) :


R_s = q_{Go}/q_{Oo}



R_v = q_{Og}/q_{Gg}


so that:


q_O = q_{Oo} + R_v \, q_{Gg}



q_G = q_{Gg} + R_s \, q_{Oo}



q_W = q_{Ww}



Following the definition of Oil formation volume factor (Bo) , Gas formation volume factor (Bg) and Water formation volume factor (Bw):


q_{Oo}= \frac{q_o}{B_o}



q_{Gg} = \frac{q_g}{B_g}



q_{Ww} = \frac{q_w}{B_w}


so that:


q_O = \frac{q_o}{B_o} + R_v \,\frac{q_g}{B_g}



q_G = \frac{q_g}{B_g} + R_s \, \frac{q_o}{B_o}



q_W = \frac{q_w}{B_w}


and solving the above system of equations leads to:


q_o = \frac{B_o \cdot (q_O - R_v \, q_G)}{1- R_v \, R_s}



q_п = \frac{B_п \cdot (q_G - R_s \, q_O)}{1- R_v \, R_s}



q_w = B_w \cdot q_w



The oil phase  includes oil component  and gas component  so that the oil phase mass flux is:

m_o = m_{Oo} + m_{Go}

The gas phase  includes gas component  and oil component  so that the gas phase mass flux is:

m_g = m_{Gg} + m_{Og}

The water phase  includes water component  only so that the water phase mass flux is:

m_w = m_{Ww}

→


m_o = \rho_O \cdot q_{Oo} + \rho_G \cdot q_{Go}



m_g = \rho_G \cdot q_{Gg} + \rho_O \cdot q_{Og}



m_w = \rho_W \cdot q_{Ww}


→


m_o = \rho_O \cdot q_{Oo} + \rho_G \cdot R_s \, q_{Oo}



m_g = \rho_G \cdot q_{Gg} + \rho_O \cdot R_v \, q_{Gg}



m_w = \rho_W \cdot q_{Ww}


→


m_o = (\rho_O + \rho_G \cdot R_s) \cdot q_{Oo}



m_g = (\rho_G + \rho_O \cdot R_v) \cdot q_{Gg}



m_w = \rho_W \cdot q_{Ww}


→


m_o = (\rho_O + \rho_G \cdot R_s) \cdot \frac{q_o}{B_o}



m_g = (\rho_G + \rho_O \cdot R_v) \cdot \frac{q_g}{B_g}



m_w = \rho_W \cdot \frac{q_w}{B_w}


→


\rho_o = \frac{\rho_O + \rho_G \cdot R_s}{B_o}



m_g = \frac{\rho_G + \rho_O \cdot R_v}{B_g}



m_w = \frac{\rho_W}{B_w}




The total mass flow of all phases:

\dot m = \dot m_o + \dot m_g + \dot m_w =  (\rho_O + \rho_G \cdot R_s) \cdot \frac{q_o}{B_o} + (\rho_G + \rho_O \cdot R_v) \cdot \frac{q_g}{B_g} + \rho_W \cdot \frac{q_w}{B_w}

→

\dot m = (\rho_O + \rho_G \cdot R_s) \cdot \frac{q_O - R_v \, q_G}{1-R_v \, R_s} + (\rho_G + \rho_O \cdot R_v) \cdot \frac{q_G - R_s \, q_O}{1- R_v \, R_s} + \rho_W \cdot q_W

→

\dot m = \frac{ (\rho_O + \rho_G \cdot R_s)\cdot (q_O - R_v \, q_G) +  (\rho_G + \rho_O \cdot R_v) \cdot (q_G - R_s \, q_O) }{1-R_v \, R_s} + \rho_W \cdot \frac{q_w}{B_w}

→

\dot m = \frac{ \rho_O \, q_O \, (1- R_v \, R_s) +  \rho_G \, q_G  \, (1- R_v \, R_s) }{1-R_v \, R_s} + \rho_W \cdot q_W

→

\dot m = \rho_O \cdot q_O +  \rho_G \cdot q_G  + \rho_W \cdot q_W = \dot m_O  + \dot m_G + \dot m_W

→

\dot m = \dot m_o + \dot m_g + \dot m_w = \dot m_O + \dot m_G + \dot m_W

which means that total mass flux of all fluid phases is equal to the total mass flux of all fluid components.

As volatile oil model does not assume water-component exchange between phases the equality  can be broken down into two equalities:


\dot m_{HC} = \dot m_o + \dot m_g = \dot m_O + \dot m_G



 \dot m_w = \dot m_W