Derivation of Material Balance Pressure @model

(1)	$\begin{array}{l}\displaystyle \frac{\partial (\rho_A \, \phi) }{\partial t} + \nabla (\rho_A \, {\bf u}_A) = 0, \quad A = \{ O, G, W \}\end{array}$

(2)	$\begin{array}{l}\displaystyle \int_V \, \frac{\partial (\rho_A \, \phi) }{\partial t} \, dV = - \int_V \, \nabla (\rho_A \, {\bf u}_A) \, dV = - \int_{\partial V} \, \rho \, {\bf u}_A \, d {\bf A}\end{array}$

(3)	$\begin{array}{l}\displaystyle V \cdot \delta (\rho_A \, \phi) = \delta \, m_A\end{array}$

(4)	$\begin{array}{l}\displaystyle V \cdot \delta \left( \phi \, \sum_\alpha \rho_{A,\alpha} \, s_\alpha \right) = \mathring{\rho}_A \cdot \delta \, q_A, \quad \alpha = \{ o, g, w \}\end{array}$

(5)	$\begin{array}{l}\displaystyle \delta \left( \phi \, \sum_\alpha \frac{\rho_{A,\alpha}}{\mathring{\rho}_A} \, s_\alpha \right) = V^{-1} \cdot \delta \, q_A\end{array}$

(6)	$\begin{array}{l}\displaystyle \delta \left( \phi \, \sum_\alpha \frac{\mathring{V}_{A,\alpha}}{V_\alpha} \, s_\alpha \right) = V^{-1} \cdot \delta \, q_A\end{array}$

(7)	$\begin{array}{l}\displaystyle \delta \left( \phi \, \sum_\alpha \xi_{A,\alpha}\, s_\alpha \right) = V^{-1} \cdot \delta \, q_A\end{array}$

(8)	$\begin{array}{l}\displaystyle \xi_{A,\alpha} = \frac{\mathring{V}_{A,\alpha}}{V_\alpha}\end{array}$

(9)	$\begin{array}{l}\displaystyle \xi_{A, \alpha} = \frac{\mathring{V}_{A,\alpha}}{V_\alpha}\end{array}$

(10)	$\begin{array}{l}\displaystyle \xi_{O, o} = \frac{\mathring{V}_{O, o}}{V_o} = \frac{1}{B_o}\end{array}$

(11)	$\begin{array}{l}\displaystyle \xi_{O, g} = \frac{\mathring{V}_{O, g}}{V_g} = \frac{\mathring{V}_{O, g}}{\mathring{V}_{G, g}} \cdot \frac{\mathring{V}_{G, g}}{V_g} = \frac{R_s }{B_g}\end{array}$

(12)	$\begin{array}{l}\displaystyle \xi_{G, o} = \frac{\mathring{V}_{G, o}}{V_o} = \frac{\mathring{V}_{G, o}}{\mathring{V}_{O, o}} \cdot \frac{\mathring{V}_{O, o}}{V_o} =\frac{R_s }{B_o}\end{array}$

(13)	$\begin{array}{l}\displaystyle \xi_{G, g} = \frac{\mathring{V}_{G, g}}{V_g} = \frac{1}{B_g}\end{array}$

(14)	$\begin{array}{l}\displaystyle \xi_{W, w} = \frac{\mathring{V}_{W, w}}{V_w} = \frac{1}{B_w}\end{array}$

Next step is to write the equations explicitly for MBO fluid.

(15)	$\begin{array}{l}\displaystyle \delta \left[ \phi \cdot \left( \xi_{O,o} \, s_o + \xi_{O,g} \, s_g \right) \right] = V^{-1} \, \delta q_O\end{array}$

(16)	$\begin{array}{l}\displaystyle \delta \left[ \phi \cdot \left( \xi_{G,o} \, s_o + \xi_{G,g} \, s_g \right) \right] = V^{-1} \, \delta q_G\end{array}$

(17)	$\begin{array}{l}\displaystyle \delta \left[ \phi \cdot \xi_{W,w} \, s_w \right] = V^{-1} \, \delta q_W\end{array}$

(18)	$\begin{array}{l}\displaystyle \phi \cdot \left[ \frac{1}{B_o} \, s_o + \frac{R_v}{B_g} \, s_g \right] = V^{-1} \, \delta q_O + \phi_i \cdot \left[ \frac{1}{B_{oi}} \, s_o + \frac{R_{vi}}{B_{gi}} \, s_{gi} \right]\end{array}$

(19)	$\begin{array}{l}\displaystyle \phi \cdot \left[ \frac{R_s}{B_o} \, s_o + \frac{1}{B_g} \, s_g \right] = V^{-1} \, \delta q_G + \phi_i \cdot \left[ \frac{R_{si}}{B_{oi}} \, s_o + \frac{1}{B_{gi}} \, s_{gi} \right]\end{array}$

(20)	$\begin{array}{l}\displaystyle \phi \cdot \frac{1}{B_w} \, s_w = V^{-1} \, \delta q_W + \phi_i \cdot \frac{1}{B_{wi}} \, s_{wi}\end{array}$

(21)	$\begin{array}{l}\displaystyle \phi_n \cdot \left[ \frac{1}{B_o} \, s_o + \frac{R_v}{B_g} \, s_g \right] = V_e^{-1} \, \delta q_O + \left[ \frac{1}{B_{oi}} \, s_o + \frac{R_{vi}}{B_{gi}} \, s_{gi} \right]\end{array}$

(22)	$\begin{array}{l}\displaystyle \phi_n \cdot \left[ \frac{R_s}{B_o} \, s_o + \frac{1}{B_g} \, s_g \right] = V_e^{-1} \, \delta q_G + \left[ \frac{R_{si}}{B_{oi}} \, s_o + \frac{1}{B_{gi}} \, s_{gi} \right]\end{array}$

(23)	$\begin{array}{l}\displaystyle \phi_n \cdot \frac{1}{B_w} \, s_w = V_e^{-1} \, \delta q_W + \frac{1}{B_{wi}} \, s_{wi}\end{array}$

where

(24)	$\begin{array}{l}\displaystyle \phi_n(p) = \phi(p) / \phi_i\end{array}$

With new definitions:

(25)	$\begin{array}{l}\displaystyle \frac{1}{B_o} \, s_o + \frac{R_v}{B_g} \, s_g = G_O/\phi_n\end{array}$

(26)	$\begin{array}{l}\displaystyle G_O = V_e^{-1} \, \delta q_O + \left[ \frac{1}{B_{oi}} \, s_o + \frac{R_{vi}}{B_{gi}} \, s_{gi} \right]\end{array}$

(27)	$\begin{array}{l}\displaystyle \frac{R_s}{B_o} \, s_o + \frac{1}{B_g} \, s_g = G_G/\phi_n\end{array}$

(28)	$\begin{array}{l}\displaystyle G_G = V_e^{-1} \, \delta q_G + \left[ \frac{R_{si}}{B_{oi}} \, s_o + \frac{1}{B_{gi}} \, s_{gi} \right]\end{array}$

(29)	$\begin{array}{l}\displaystyle \frac{1}{B_w} \, s_w = G_W/\phi_n\end{array}$

(30)	$\begin{array}{l}\displaystyle G_W = V_e^{-1} \, \delta q_W + \frac{1}{B_{wi}} \, s_{wi}\end{array}$

The equations can be finally explicitly express sturations:

(31)	$\begin{array}{l}\displaystyle s_o = \frac{B_o \, (G_o - R_v \, G_G)}{\phi_n \, (1- R_s \, R_v)}\end{array}$

(32)	$\begin{array}{l}\displaystyle s_g = \frac{B_g \, (G_G - R_s \, G_O)}{\phi_n \, (1- R_s \, R_v)}\end{array}$

(33)	$\begin{array}{l}\displaystyle s_w = \frac{B_w \, G_W}{\phi_n}\end{array}$

Now summing up and taking into account that $\begin{array}{l}s_o + s_g + s_w = 1\end{array}$ one arrives to a single equation:

(34)	$\begin{array}{l}\displaystyle \frac{B_o \, (G_o - R_v \, G_G)}{\phi_n \, (1- R_s \, R_v)} + \frac{B_g \, (G_G - R_s \, G_O)}{\phi_n \, (1- R_s \, R_v)} + \frac{B_w \, G_W}{\phi_n} =1\end{array}$

(35)	$\begin{array}{l}\displaystyle B_o \, (G_o - R_v \, G_G) + B_g \, (G_G - R_s \, G_O) + B_w \, G_W \, (1- R_s \, R_v) = \phi_n \, (1- R_s \, R_v)\end{array}$

(36)	$\begin{array}{l}\displaystyle (B_o - R_s \, B_g) \, G_O +(B_g - R_V \, B_o) \, G_G + (B_w \, G_W - \phi_n )\, (1- R_s \, R_v) = 0\end{array}$

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