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| \delta \left[ \phi \cdot \left( \xi_{O,o} \, s_o + \xi_{O,g} \, s_g \right) \right] = V^{-1} \, \delta q_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 q_G |
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| \delta \left[ \phi \cdot \xi_{W,w} \, s_w \right] = V^{-1} \, \delta q_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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DefinitionsWith new definitions:
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| \frac{1}{B_o} \, s_o + \frac{R_v}{B_g} \, s_g = G_O/\phi_n |
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| G_O = 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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| \frac{R_s}{B_o} \, s_o + \frac{1}{B_g} \, s_g = G_G/\phi_n |
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| G_G = 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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| \frac{1}{B_w} \, s_w = G_W/\phi_n |
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| G_W = V_e^{-1} \, \delta q_W + \frac{1}{B_{wi}} \, s_{wi} |
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The equations can be finally explicitly express sturations:
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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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Now summing up and taking into account that
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| \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 |
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| 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) |
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body | --uriencoded--\phi_n(p) = \phi_e(p)/\phi_%7Bei%7D |
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| normalized porosity |
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body | Q^{\downarrow}_{GC}(t) |
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| cumulative gas influx from Gas Cap Expansion | LaTeX Math Inline |
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body | --uriencoded--\phi_%7Bei%7D = \phi_e(p_i) |
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| | initial effective porosity |
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body | Q^{\downarrow}_{AQ}(t) |
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| cumulative water influx from Aquifer Expansion | | | | | | Initial water saturation | | Oil and Gas Recovery Factor |
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