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| \frac{\partial (\rho_A \, \phi) }{\partial t} + \nabla (\rho_A \, {\bf u}_A) = 0, \quad A = \{ O, G, W \} |
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Mass conservation for each fluid component individually |
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| \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} |
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Integrate over the entire pay volume |
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| V \cdot \delta (\rho_A \, \phi) = \delta \, m_A |
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| 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 \} |
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| \delta \left( \phi \, \sum_\alpha \frac{\rho_{A,\alpha}}{\mathring{\rho}_A} \, s_\alpha \right) = V^{-1} \cdot \delta \, q_A |
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| \delta \left( \phi \, \sum_\alpha \frac{\mathring{V}_{A,\alpha}}{V_\alpha} \, s_\alpha \right) = V^{-1} \cdot \delta \, q_A |
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| \delta \left( \phi \, \sum_\alpha \xi_{A,\alpha}\, s_\alpha \right) = V^{-1} \cdot \delta \, q_A |
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| \xi_{A,\alpha} = \frac{\mathring{V}_{A,\alpha}}{V_\alpha} |
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For the MBO fluid:
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\xi_{A, \alpha} = \frac{\mathring{V}_{A,\alpha}}{V_\alpha} |
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| \xi_{O, o} = \frac{\mathring{V}_{O, o}}{V_o} = \frac{1}{B_o} |
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| \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} |
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| \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} |
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| \xi_{G, g} = \frac{\mathring{V}_{G, g}}{V_g} = \frac{1}{B_g} |
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| \xi_{W, w} = \frac{\mathring{V}_{W, w}}{V_w} = \frac{1}{B_w} |
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Next step is to write the equations explicitly for MBO fluid.:
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| \delta \left[ \phi \cdot \left( \xi_{O,o} \, s_o + \xi_{O,g} \, s_g \right) \right] = V^{-1} \, \delta qQ_O |
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| \delta \left[ \phi \cdot \left( \xi_{G,o} \, s_o + \xi_{G,g} \, s_g \right) \right] = V^{-1} \, \delta qQ_G |
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| \delta \left[ \phi \cdot \xi_{W,w} \, s_w \right] = V^{-1} \, \delta qQ_W |
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| \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] |
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| \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] |
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| \phi \cdot \frac{1}{B_w} \, s_w = V^{-1} \, \delta q_W + \phi_i \cdot \frac{1}{B_{wi}} \, s_{wi} |
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| \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] |
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| \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] |
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| \phi_n \cdot \frac{1}{B_w} \, s_w = V_e^{-1} \, \delta q_W + \frac{1}{B_{wi}} \, s_{wi} |
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where |
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| \phi_n(p) = \phi(p) / \phi_i |
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The equations can be finally explicitly express sturationsexpressed in terms of reservoir saturations:
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| s_o = \frac{B_o \, (G_o - R_v \, G_G)}{\phi_n \, (1- R_s \, R_v)} |
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| s_g = \frac{B_g \, (G_G - R_s \, G_O)}{\phi_n \, (1- R_s \, R_v)} |
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| s_w = \frac{B_w \, G_W}{\phi_n} |
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