\frac{\partial (\rho_A \, \phi) }{\partial t} + \nabla (\rho_A \, {\bf u}_A) = 0, \quad A = \{ O, G, W \}
\int_V \, \frac{\partial (\rho_A \, \phi) }{\partial t} \, dV = - \int_V \, \nabla (\rho_A \, {\bf u}_A) \, dV = - \int_{\partial V} \, \rho \, {\bf u}_A \, d {\bf A}
V \cdot \delta (\rho_A \, \phi) = \delta \, m_A \Rightarrow V \cdot \delta \left( \phi \, \sum_\alpha \rho_{A,\alpha} \, s_\alpha \right) = \mathring{\rho}_A \cdot \delta \, q_A, \quad \alpha = \{ o, g, w \}
\Rightarrow \delta \left( \phi \, \sum_\alpha \frac{\rho_{A,\alpha}}{\mathring{\rho}_A} \, s_\alpha \right) = V^{-1} \cdot \delta \, q_A \Rightarrow \delta \left( \phi \, \sum_\alpha \frac{\mathring{V}_{A,\alpha}}{V_\alpha} \, s_\alpha \right) = V^{-1} \cdot \delta \, q_A
\delta \left( \phi \, \sum_\alpha \frac{\mathring{V}_{A,\alpha}}{V_\alpha} \, s_\alpha \right) = V^{-1} \cdot \delta \, q_A
For the MBO fluid:
\xi_{A, \alpha} = \frac{\mathring{V}_{A,\alpha}}{V_\alpha}
\xi_{O, o} = \frac{\mathring{V}_{O, o}}{V_o} = \frac{1}{B_o}
\xi_{O, g} = \frac{\mathring{V}_{O, g}}{V_g} = \frac{\mathring{V}_{O, g}}{\mathring{V}_{G, g}} \cdot \frac{\mathring{V}_{G, g}}{V_g} = \frac{R_s }{B_g}
\xi_{G, o} = \frac{\mathring{V}_{G, o}}{V_o} = \frac{\mathring{V}_{G, o}}{\mathring{V}_{O, o}} \cdot \frac{\mathring{V}_{O, o}}{V_o} =\frac{R_s }{B_o}
\xi_{G, g} = \frac{\mathring{V}_{G, g}}{V_g} = \frac{1}{B_g}
\xi_{W, w} = \frac{\mathring{V}_{W, w}}{V_w} = \frac{1}{B_w}
Next step is to write the equations explicitly for MBO fluid.
\delta \left[ \phi \cdot \left( \xi_{O,o} \, s_o + \xi_{O,g} \, s_g \right) \right] = V^{-1} \, \delta q_O
\delta \left[ \phi \cdot \left( \xi_{G,o} \, s_o + \xi_{G,g} \, s_g \right) \right] = V^{-1} \, \delta q_G
\delta \left[ \phi \cdot \xi_{W,w} \, s_w \right] = V^{-1} \, \delta q_W
\phi \cdot \left[ \frac{1}{B_o} \, s_o + \frac{R_v}{B_g} \, s_g \right] = V^{-1} \, \delta q_O + \phi_i \cdot \left[ \frac{1}{B_{oi}} \, s_o + \frac{R_{vi}}{B_{gi}} \, s_{gi} \right]
\phi \cdot \left[ \frac{R_s}{B_o} \, s_o + \frac{1}{B_g} \, s_g \right] = V^{-1} \, \delta q_G + \phi_i \cdot \left[ \frac{R_{si}}{B_{oi}} \, s_o + \frac{1}{B_{gi}} \, s_{gi} \right]
\phi \cdot \frac{1}{B_w} \, s_w = V^{-1} \, \delta q_W + \phi_i \cdot \frac{1}{B_{wi}} \, s_{wi}
\phi_n \cdot \left[ \frac{1}{B_o} \, s_o + \frac{R_v}{B_g} \, s_g \right] = V_e^{-1} \, \delta q_O + \left[ \frac{1}{B_{oi}} \, s_o + \frac{R_{vi}}{B_{gi}} \, s_{gi} \right]
\phi_n \cdot \left[ \frac{R_s}{B_o} \, s_o + \frac{1}{B_g} \, s_g \right] = V_e^{-1} \, \delta q_G + \left[ \frac{R_{si}}{B_{oi}} \, s_o + \frac{1}{B_{gi}} \, s_{gi} \right]
\phi_n \cdot \frac{1}{B_w} \, s_w = V_e^{-1} \, \delta q_W + \frac{1}{B_{wi}} \, s_{wi}
\phi_n(p) = \phi(p) / \phi_i
With new definitions:
\frac{1}{B_o} \, s_o + \frac{R_v}{B_g} \, s_g = G_O/\phi_n
G_O = V_e^{-1} \, \delta q_O + \left[ \frac{1}{B_{oi}} \, s_o + \frac{R_{vi}}{B_{gi}} \, s_{gi} \right]
\frac{R_s}{B_o} \, s_o + \frac{1}{B_g} \, s_g = G_G/\phi_n
G_G = V_e^{-1} \, \delta q_G + \left[ \frac{R_{si}}{B_{oi}} \, s_o + \frac{1}{B_{gi}} \, s_{gi} \right]
\frac{1}{B_w} \, s_w = G_W/\phi_n
G_W = V_e^{-1} \, \delta q_W + \frac{1}{B_{wi}} \, s_{wi}
The equations can be finally explicitly express sturations:
s_o = \frac{B_o \, (G_o - R_v \, G_G)}{\phi_n \, (1- R_s \, R_v)}
s_g = \frac{B_g \, (G_G - R_s \, G_O)}{\phi_n \, (1- R_s \, R_v)}
s_w = \frac{B_w \, G_W}{\phi_n}
Now summing up and taking into account that one arrives to a single equation:
\frac{B_o \, (G_o - R_v \, G_G)}{\phi_n \, (1- R_s \, R_v)} + \frac{B_g \, (G_G - R_s \, G_O)}{\phi_n \, (1- R_s \, R_v)} + \frac{B_w \, G_W}{\phi_n} =1
B_o \, (G_o - R_v \, G_G) + B_g \, (G_G - R_s \, G_O) + B_w \, G_W \, (1- R_s \, R_v) = \phi_n \, (1- R_s \, R_v)
(B_o - R_s \, B_g) \, G_O +(B_g - R_V \, B_o) \, G_G + (B_w \, G_W - \phi_n )\, (1- R_s \, R_v) = 0
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normalized cross-phase exchange derivatives as functions of reservoir pressure and temperature