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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 \, |
| qQ_A, \quad \alpha = \{ o, g, w \} |
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| \Rightarrow \delta \left( \phi \, \sum_\alpha \frac{\rho_{A,\alpha}}{\mathring{\rho}_A} \, s_\alpha \right) = V^{-1} \cdot \delta \, qQ_A \Rightarrow |
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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_{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}q_A, \quad \alpha = \{ o, g, 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 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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| where |
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...
...
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With new definitions:
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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 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 = F_G/\phi_n |
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| F_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 = F_W/\phi_n |
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| F_W = V_e^{-1} \, \delta Q_W + \frac{1}{B_{wi}} \, s_{wi} |
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The equations can be finally explicitly expressed in terms of reservoir saturations:
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| s_o = \frac{B_o \, (F_O - R_v \, F_G)}{\phi_n \, (1- R_s \, R_v)} |
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| s_g = \frac{B_g \, (F_G - R_s \, F_O)}{\phi_n \, (1- R_s \, R_v)} |
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| s_w = \frac{B_w \, F_W}{\phi_n} |
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Now summing up and taking into account that
one arrives to a single equation:| LaTeX Math Block |
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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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| LaTeX Math Block |
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| anchor | MatBal |
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| alignment | left |
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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 |
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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
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| LaTeX Math Inline |
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| body | Q^{\downarrow}_{GC}(t) |
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cumulative gas influx from Gas Cap Expansion
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| LaTeX Math Inline |
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| body | --uriencoded--\phi_%7Bei%7D = \phi_e(p_i) |
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| LaTeX Math Inline |
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| body | Q^{\downarrow}_{AQ}(t) |
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cumulative water influx from Aquifer Expansion
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