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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 qQ_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 qQ_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 qQ_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 qQ_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 qQ_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 qQ_W + \frac{1}{B_{wi}} \, s_{wi} 
where
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\phi_n(p) = \phi(p) / \phi_i 

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\frac{1}{B_o} \, s_o + \frac{R_v}{B_g} \, s_g  = F_O/\phi_n 
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F_O = V_e^{-1} \, \delta qQ_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  = F_G/\phi_n
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F_G = V_e^{-1} \, \delta qQ_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  = F_W/\phi_n
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F_W  = V_e^{-1} \, \delta qQ_W + \frac{1}{B_{wi}} \, s_{wi} 

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\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 
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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)
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(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

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Petroleum Industry / Upstream /  Production / Subsurface Production / Field Study & Modelling / Production Analysis / Material Balance Analysis (MatBal) / Material Balance Pressure @model