\frac{\partial (\rho_A \, \phi) }{\partial t} + \nabla (\rho_A \, {\bf u}_A) = 0, \quad A = \{ O, G, W \}

Mass conservation for each fluid component individually

\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}

Integrate over the entire pay volume

V \cdot \delta (\rho_A \, \phi) = \delta \, m_A

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 \}

\delta \left( \phi \, \sum_\alpha \frac{\rho_{A,\alpha}}{\mathring{\rho}_A} \, 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

 \delta \left( \phi \, \sum_\alpha \xi_{A,\alpha}\, s_\alpha \right) =  V^{-1} \cdot \delta \, Q_A

  \xi_{A,\alpha} =  \frac{\mathring{V}_{A,\alpha}}{V_\alpha}

For the MBO fluid:

\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}

where

\phi_n(p) = \phi(p) / \phi_i

With new definitions:

\frac{1}{B_o} \, s_o + \frac{R_v}{B_g} \, s_g  = F_O/\phi_n

F_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  = F_G/\phi_n

F_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  = F_W/\phi_n

F_W  = V_e^{-1} \, \delta Q_W + \frac{1}{B_{wi}} \, s_{wi}

The equations can be finally explicitly expressed in terms of reservoir saturations:

s_o = \frac{B_o \, (F_O - R_v \, F_G)}{\phi_n \, (1- R_s \, R_v)}

s_g = \frac{B_g \, (F_G - R_s \, F_O)}{\phi_n \, (1- R_s \, R_v)}

s_w = \frac{B_w \, F_W}{\phi_n}

Now summing up and taking into account that one arrives to a single equation:

\frac{B_o \, (F_O - R_v \, F_G)}{\phi_n \, (1- R_s \, R_v)} +  \frac{B_g \, (F_G - R_s \, F_O)}{\phi_n \, (1- R_s \, R_v)}  +  \frac{B_w \, F_W}{\phi_n}  =1

B_o \, (F_O - R_v \, F_G) + B_g \, (F_G - R_s \, F_O)  + B_w \, F_W \, (1- R_s \, R_v)  = \phi_n \, (1- R_s \, R_v)

(B_o - R_s \, B_g) \, F_O +(B_g - R_V \, B_o) \, F_G + (B_w  \, F_W - \phi_n )\, (1- R_s \, R_v)  = 0

See Also