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Modelling facility for field-average formation pressure 

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bodyp(t)
 at any time moment 
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bodyt
 as response to production flowrates history, which in case of MBO fluid takes form:

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anchorMatBal
alignmentleft
\phi_n(p) = \frac{B_o - R_s \, B_g}{1- R_s \, R_v} \cdot F_O 
+\frac{ B_g - R_v \, B_o}{1- R_s \, R_v} \cdot F_G 
+B_w  \, F_W
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anchorphin
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\phi_n = \exp \left[ c_\phi \, (p-p_i)  \right] \approx 1 + c_\phi \, (p-p_i)  + 0.5 \, c^2_\phi \, (p-p_i)^2
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anchorGO
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F_O = V_
e^
\phi^{-1} \, \delta \, Q_O +
\left[
F_{Oi}
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anchorGO
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F_{Oi} = \frac{s_{oi}}{B_{oi}}  + \frac{R_{vi}\, s_{gi}}{B_{gi}}
\right] 
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anchordQO
alignmentleft
\delta \, Q_O = - Q^{\uparrow}_O
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anchorGG
alignmentleft
F_G = V_
e^
\phi^{-1} \, \delta \, Q_G + 
\left[
F_{Gi}
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anchorGO
alignmentleft
F_{Gi} = \frac{R_{si}\, s_{oi}}{B_{oi}}  + \frac{ s_{gi}}{B_{gi}}
\right]
 
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anchordGG
alignmentleft
\delta \, Q_G = Q^{\downarrow}_G - Q^{\uparrow}_G + Q^{\downarrow}_{GCAP}
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anchorGW
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F_W = V_
e^
\phi^{-1} \, \delta \, Q_W + F_{Wi} 
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anchorGO
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F_{Wi} = \frac{ s_{wi}}{B_{wi}} 
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anchordGW
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\delta \, Q_W = Q^{\downarrow}_W - Q^{\uparrow}_W + Q^{\downarrow}_{WAQ}
where
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bodyp_i
initial formation pressure: 
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bodyp_i = p(0)
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body--uriencoded--Q%5e%7B\uparrow%7D_O(t)
Cumulative oil production by the time moment 
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bodyt
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bodyV_e \phi = V \cdot \phi_i
initial drainage volume of Main Pay open pore volume of the main pay (excluding the aquifer and gas cap)
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body--uriencoded--Q%5e%7B\uparrow%7D_G(t)
Cumulative gas production by the time moment 
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bodyt
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body\phi_i = \phi(p_i)
initial porosity
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body--uriencoded--Q%5e%7B\uparrow%7D_W(t)
Cumulative water production by the time moment 
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bodyt
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bodyc_\phi
pore compressibility 
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body--uriencoded--Q%5e%7B\downarrow%7D_W(t)
Cumulative water injection by the time moment 
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bodyt
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body--uriencoded--s_%7Bwi%7D
initial water saturation
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body--uriencoded--Q%5e%7B\downarrow%7D_G(t)
Cumulative gas injection by the time moment 
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bodyt
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body--uriencoded--s_%7Bgi%7D
initial  gas saturation
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body--uriencoded--Q%5e%7B\downarrow%7D_%7BWAQ%7D(t)
Cumulative water influx from Aquifer Expansion by the time moment 
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bodyt
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body--uriencoded--s_%7Boi%7D
initial oil saturation: 
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body--uriencoded--s_%7Boi%7D = 1 - s_%7Bwi%7D - s_%7Bgi%7D
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body--uriencoded--Q%5e%7B\downarrow%7D_%7BGCAP%7Dt)
Cumulative gas influx from Gas Cap expansion by the time moment 
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bodyt




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bodyB_o(p)
Oil formation volume factor as functions of reservoir pressure 
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bodyp
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bodyR_s(p)
Solution GOR as functions of reservoir pressure 
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bodyp
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bodyB_g(p)
Gas formation volume factor as functions of reservoir pressure 
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bodyp
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bodyR_v(p)
Vaporized Oil Ratio as functions of reservoir pressure 
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bodyp
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bodyB_w(p)
Water formation volume factor as functions of reservoir pressure 
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bodyp
 


...
In some specific cases equation 
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anchorMatBal
 can be explicitly integrated with the accuracy sufficient for practical applications:
Slightly compressibility flow
Low pressure dry gas
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body
\{ 
--uriencoded--c_t = c_\phi + c_
e = {
%7B\rm 
 const}, \ c_t = {\rm const} \}
fluid%7D = %7B\rm const%7D
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body--uriencoded--c_
t = c_r + \frac{1}{p} 
g = \sim \
frac{1}{p}
frac%7B1%7D%7Bp%7D
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anchorQ6XP7
alignmentleft
p(t)  = p_i + \frac{\Delta Q(t)}{V_
e
\phi \cdot c_t}


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anchor3J3AD
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p(t)  = p_i \exp \left[ \frac{\Delta Q(t)}{V_
e
\phi} \right]
where 
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body\Delta Q
 is Cumulative Voidage Replacement Balance (CVRB):
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anchorDQ
alignmentleft
\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)

The above approximations sometime allow using simple graphical methods for rough estimation of drainage volume 
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bodyV_e
 and associated Hydrocarbon Reserves.
...
[ Derivation of Material Balance Pressure @model ]
[ Modified Black Oil fluid @model (MBO) ]