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q_O = q_{Oo} + q_{Og}



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q_G = q_{Gg} + q_{Go}



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q_W = q_{Ww}



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


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R_s = q_{Go}/q_{Oo}



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R_v = q_{Og}/q_{Gg}


so that:


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



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



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q_W = q_{Ww}



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


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



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



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


so that:


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



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



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q_W = \frac{q_w}{B_w}


and solving the above system of equations leads to:


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



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



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q_w = B_w \cdot q_w



The oil phase 

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

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m_o = m_{Oo} + m_{Go}

The gas phase 

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includes gas component 
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body--uriencoded--()_%7BGg%7D
and oil component 
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 so that the gas phase mass flux is:

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m_g = m_{Gg} + m_{Og}

The water phase 

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

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m_w = m_{Ww}


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



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



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m_w = \rho_W \cdot q_{Ww}



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



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



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m_w = \rho_W \cdot q_{Ww}



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



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



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m_w = \rho_W \cdot q_{Ww}



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



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



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



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



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



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m_w = \frac{\rho_W}{B_w}




The total mass flow of all phases:

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\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}

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\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

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\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}

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