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Modelling facility for field-average formation pressure 

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

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




 LaTeX Math Inline
bodyB_o(p)
Oil formation volume factor as functions of reservoir pressure 
 LaTeX Math Inline
bodyp
 LaTeX Math Inline
body--uriencoded--Q%5e%7B\downarrow%7D_%7BGCAP%7Dt)R_s(p)
Solution GOR as functions of Cumulative gas influx from Gas Cap expansion by the time moment reservoir pressure 
 LaTeX Math Inline
body
t
p
 LaTeX Math Inline
bodyB_g(p)
Gas formation volume factor as functions of reservoir pressure 
 LaTeX Math Inline
bodyp
 LaTeX Math Inline
body
R_
v(p)
Water formation volume factor as Vaporized Oil Ratio as functions of reservoir pressure 
 LaTeX Math Inline
bodyp
 LaTeX Math Inline
bodyR_s(p), \, R_vB_w(p)
 Solution GOR and Vaporized Oil Ratio as Water formation volume factor as functions of reservoir pressure 
 LaTeX Math Inline
bodyp
 




The MatBal equation 
 LaTeX Math Block Reference
anchorMatBal
 is often can be complemented by constant PI model of Bottom-Hole Pressure (
 LaTeX Math Inline
bodyp^{\uparrow}_{wf}(t)
 for producers and 
 LaTeX Math Inline
bodyp^{\downarrow}_{wf}(t)
 for injectors):
...
In some specific cases equation 
 LaTeX Math Block Reference
anchorMatBal
 can be explicitly integrated with the accuracy sufficient for practical applications:
Slightly compressibility flow
Low pressure dry gas
 LaTeX Math Inline
body
\{ 
--uriencoded--c_t = c_\phi + c_
e = {
%7B\rm 
 const}, \ c_t = {\rm const} \}
fluid%7D = %7B\rm const%7D
 LaTeX Math Inline
body--uriencoded--c_
t 
g = 
 c_r + 
 \
frac{1}{p} \
sim \
frac{1}{p}
frac%7B1%7D%7Bp%7D
 LaTeX Math Block
anchorQ6XP7
alignmentleft
p(t)  = p_i + \frac{\Delta Q(t)}{V_
e
\phi \cdot c_t}


 LaTeX Math Block
anchor3J3AD
alignmentleft
p(t)  = p_i \exp \left[ \frac{\Delta Q(t)}{V_
e
\phi} \right]
where 
 LaTeX Math Inline
body
...
\Delta Q
 is Cumulative Voidage Replacement Balance (CVRB):
 LaTeX Math Block
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 This allows using simple graphical methods for estimating drainage volume 
 LaTeX Math Inline
bodyV_e
 and associated Hydrocarbon Reserves.
See Also
...
Petroleum Industry / Upstream /  Production / Subsurface Production / Field Study & Modelling / Production Analysis / Material Balance Analysis (MatBal)
...
[ Derivation of Material Balance Pressure @model ]
[ Modified Black Oil fluid @model (MBO) ]