Approximation of Material Balance Pressure @model for slightly compressibility flow:

(1)	$\begin{array}{l}\displaystyle p(t) = p_i + \frac{\Delta Q(t)}{V_\phi \cdot c_t}\end{array}$

(2)

$\begin{array}{l}\displaystyle \Delta Q = - \frac{B_o - R_s \, B_g}{1- R_s \, R_v} \cdot \, Q^{\uparrow}_O + \frac{ B_g - R_v \, B_o}{1- R_s \, R_v} \cdot \, \left( Q^{\downarrow}_G - Q^{\uparrow}_G + Q^{\downarrow}_{GCAP} \ \ \right) + B_w \, \left( Q^{\downarrow}_W - Q^{\uparrow}_W + Q^{\downarrow}_{WAQ} \ \ \right)\end{array}$

where

$\begin{array}{l}\Delta Q(t)\end{array}$	Cumulative Voidage Replacement Balance (CVRB) over time $\begin{array}{l}t\end{array}$
$\begin{array}{l}V_\phi = V \cdot \phi_i\end{array}$	initial drainage volume of the main pay (excluding the aquifer and gas cap)
$\begin{array}{l}\phi_i = \phi(p_i)\end{array}$	initial porosity
$\begin{array}{l}c_t = c_\phi + c_o \, s_{oi} + c_g \, s_{gi} + c_w \, s_{wi}\end{array}$	total compressibility
$\begin{array}{l}c_\phi\end{array}$	pore compressibility
$\begin{array}{l}s_{wi}\end{array}$	initial water saturation
$\begin{array}{l}s_{gi}\end{array}$	initial gas saturation
$\begin{array}{l}s_{oi}\end{array}$	initial oil saturation: $\begin{array}{l}s_{oi} = 1 - s_{wi} - s_{gi}\end{array}$
$\begin{array}{l}c_o, \, c_g, \, c_w\end{array}$	fluid compressibility of water phase, oil phase and gas phase

The equations (1) and (2) are often used in express assessment of thief water production share $\begin{array}{l}\Omega^{\uparrow}_W = Q^{\uparrow}_{W,{\rm true}} \, / \,Q^{\uparrow}_W\end{array}$ and thief water injection share $\begin{array}{l}\Omega^{\downarrow}_W = Q^{\downarrow}_{W,{\rm true}} \, / \,Q^{\downarrow}_W\end{array}$ :

(3)	$\begin{array}{l}\displaystyle p_i - p(t) = \alpha \cdot Q^{\uparrow}_O(t) + \beta \cdot Q^{\uparrow}_W(t) - \gamma \cdot Q^{\downarrow}_W(t)\end{array}$

(4)	$\begin{array}{l}\displaystyle \alpha > 0, \quad \beta > 0, \quad \gamma > 0\end{array}$

(5)	$\begin{array}{l}\displaystyle V_\phi = \frac{ 1 }{ \alpha \cdot c_t} \cdot \frac{B_o - R_s \, B_g}{1- R_s \, R_v}\end{array}$

(6)	$\begin{array}{l}\displaystyle \Omega^{\uparrow}_W = \frac{Q^{\uparrow}_{W,{\rm true}}}{Q^{\uparrow}_W} = \frac{\beta}{\alpha} \cdot \frac{1}{B_w} \cdot \frac{B_o - R_s \, B_g}{1- R_s \, R_v}\end{array}$

(7)	$\begin{array}{l}\displaystyle \Omega^{\downarrow}_W = \frac{Q^{\downarrow}_{W, {\rm true}}}{Q^{\downarrow}_W} =\frac{\gamma}{\alpha} \cdot \frac{1}{B_w} \cdot \frac{B_o - R_s \, B_g}{1- R_s \, R_v}\end{array}$

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Slightly compressible Material Balance Pressure @model

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