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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
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}
mathblock
where
anchor
 LaTeX Math Inline
phinalignmentleft\phi_n = 1 + c_\phi \, (p-p_i)  + 0.5 \, c^2_\phi \, (p-
bodyp_i
)^2 
...
t
initial formation pressure: 
 LaTeX Math Inline
bodyp_i = p(
0)
formation pressure at  time moment 
 LaTeX Math Inline
bodyt
 
 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
body
p_i = p(0)
V_\phi = V \cdot \phi_i
initial open pore volume of the main pay (excluding the aquifer and gas cap)
initial formation pressure
 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
V_e
\phi_i = \phi(p_i)
initial porosityinitial drainage volume
 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\downarrow%7D_W(t)
Cumulative water injection 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_G(t)
Cumulative 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_%7BWAQ%7D(t)
Cumulative water influx from 
 aquifer
 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
initial oil saturation
 LaTeX Math Inline
body--uriencoded--Q%5e%7B\downarrow%7D_%7BGCAP%7Dt)
Cumulative gas influx from 
 gas cap
 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
bodyR_s(p)
Solution GOR as functions of reservoir pressure 
 LaTeX Math Inline
bodyp
 LaTeX Math Inline
bodyB_g(p)
Gas formation volume factor as functions of reservoir pressure 
 LaTeX Math Inline
body
B_w
p
 LaTeX Math Inline
bodyR_v(p)
Vaporized Oil Ratio as functions of reservoir pressure 
 LaTeX Math Inline
bodyp
Water formation volume factor
 LaTeX Math Inline
bodyR_{sn}, \; R_{vn}B_w(p)
normalized cross-phase exchange ratios as functions of 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):
...
which closes equation 
 LaTeX Math Block Reference
anchorMatBal
 for the pressure 
 LaTeX Math Inline
bodyp(t)
.
...
Approximations
...
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 = c_r + \frac{1}{p} 
g = \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) ]