Following the definition of Solution GOR (Rs) and Vaporized Oil Ratio (Rv) :
so that:
q_O = q_{Oo} + R_v \, q_{Gg} |
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q_G = q_{Gg} + R_s \, q_{Oo} |
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Following the definition of Oil formation volume factor (Bo) , Gas formation volume factor (Bg) and Water formation volume factor (Bw):
so that:
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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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} |
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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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The oil phase includes oil component and gas component so that the oil phase mass flux is:
The gas phase includes gas component and oil component so that the gas phase mass flux is:
The water phase includes water component only so that the water phase mass flux is:
→
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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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 = \dot m_o + \dot m_g + \dot m_w = \dot m_O + \dot m_G + \dot m_W |