...
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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 |
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The oil phase
includes oil component | LaTeX Math Inline |
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| body | --uriencoded--()_%7BOo%7D |
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and gas component | LaTeX Math Inline |
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| body | --uriencoded--()_%7BGo%7D |
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so that the oil phase mass flux is:| LaTeX Math Block |
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m_o = m_{Oo} + m_{Go} |
The gas phase
includes gas component | LaTeX Math Inline |
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| body | --uriencoded--()_%7BGg%7D |
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and oil component | LaTeX Math Inline |
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| body | --uriencoded--()_%7BOg%7D |
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so that the gas phase mass flux is:| LaTeX Math Block |
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m_g = m_{Gg} + m_{Og} |
The water phase
includes water component | LaTeX Math Inline |
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| body | --uriencoded--()_%7BWw%7D |
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only so that the water phase mass flux is:| LaTeX Math Block |
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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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| LaTeX Math Block |
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| m_g = \rho_G \cdot q_{Gg} + \rho_O \cdot q_{Og} |
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| LaTeX Math Block |
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| m_w = \rho_W \cdot q_{Ww} |
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→
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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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| LaTeX Math Block |
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| m_w = \rho_W \cdot q_{Ww} |
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→
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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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| LaTeX Math Block |
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| m_w = \rho_W \cdot q_{Ww} |
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→
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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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→
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| \rho_o = \frac{\rho_O + \rho_G \cdot R_s}{B_o} |
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| LaTeX Math Block |
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| m_g = \frac{\rho_G + \rho_O \cdot R_v}{B_g} |
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| LaTeX Math Block |
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| m_w = \frac{\rho_W}{B_w} |
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Now it's easy to show that total mass of fluid phases is the same as the total mass of fluid components:
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\dot m = \dot m_o + \dot m_g + \dot m_w = \dot m_O + \dot m_G + \dot m_W |
Here is the breakdown:
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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} |
