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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 \, R_v} \cdot F_O 
+\frac{ B_g - R_v \, B_o}{1- R_s \, R_v} \cdot F_G 
+B_w  \, F_W
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
LaTeX Math Block
anchorGO
alignmentleft
F_O = V_\phi^{-1} \, \delta \, Q_O +
\left[ \frac{s_{oi}}{B_{oi}} + \frac{R_{vi}\, s_{gi}}{B_{gi}}\right] 
F_{Oi}
LaTeX Math Block
anchorGO
alignmentleft
F_{Oi} =
\left[
 \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
F_G = V_\phi^{-1} \, \delta \, Q_G + 
\left[ \frac{R_{si}\, s_{oi}}{B_{oi}}  + \frac{ s_{gi}}{B_{gi}}\right] 
F_{Gi}
 LaTeX Math Block
anchorGO
alignmentleft
F_{Gi} = \
left[ \
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
alignmentleft
F_W = V_\phi^{-1} \, \delta \, Q_W + 
 \frac{ s
F_{
wi}}{B_{wi}}
Wi} 
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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}
where
 LaTeX Math Block
anchorMatBal_formula
alignmentleft
p = p_i + \frac{\delta Q}{c_\phi \, V_\phi} + \delta p_i
 LaTeX Math BlockanchorMatBal_formulaalignmentleft
 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_\phi = V \cdot \phi_i
initial open pore volume of the main pay (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\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
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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
 LaTeX Math Inline
body--uriencoded--s_%7Boi%7D
initial oil saturation: 
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body--uriencoded--s_%7Boi%7D = 1 - s_%7Bwi%7D - s_%7Bgi%7D
 LaTeX Math Inline
body--uriencoded--Q%5e%7B\downarrow%7D_%7BGCAP%7Dt)
Cumulative gas influx from 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 
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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
bodyp
 LaTeX Math Inline
bodyR_v(p)
Vaporized Oil Ratio as functions of reservoir pressure 
 LaTeX Math Inline
bodyp
 LaTeX Math Inline
bodyB_w(p)
Water formation volume factor as functions of reservoir pressure 
 LaTeX Math Inline
bodyp
 
The MatBal equation 
 LaTeX Math Block Reference
anchorMatBal
 can be re-written in the following popular form:
\delta 
p
_i =  \frac{ B_{og} \, F_{Oi} + B_{go} \, F_{Gi} + B_w \, F_W -1}{c_\phi}
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anchor1
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B_{og} = \frac{B_o - R_s \, B_g}{1- R_s \, R_v}
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anchor1
alignmentleft
B_{go} = \frac{ B_g - R_v \, B_o}{1- R_s \, R_v}
where
 LaTeX Math Inline
body\delta Q
Cumulative Voidage Replacement Balance (CVRB)
 




The MatBal equation 
 LaTeX Math Block Reference
anchorMatBal
 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):
...
Slightly compressibility flow
Low pressure dry gas
 LaTeX Math Inline
body\{ \phi_e = {\rm const}, \ c_t = {\rm const} \}--uriencoded--c_t = c_\phi + c_%7B\rm 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
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anchorQ6XP7
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p(t)  = p_i + \frac{\Delta Q(t)}{V_\phi \cdot c_t}


 LaTeX Math Block
anchor3J3AD
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p(t)  = p_i \exp \left[ \frac{\Delta Q(t)}{V_\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 drainage volume 
 LaTeX Math Inline
bodyV_e
 and associated Hydrocarbon Reserves.
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