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LaTeX Math Block
anchorMatBal
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(B_o - R_s \, B_g) \, G_O +(B_g - R_V \, B_o) \, G_G + (B_w  \, G_W - \phi_n )\, (1- R_s \, R_v)  = 0
LaTeX Math Block
anchorGO
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G_O = V_e^{-1} \, \delta \, Q_O + \left[ \frac{s_{oi}}{B_{oi}}  + \frac{R_{vi}\, s_{gi}}{B_{gi}}\right] 
LaTeX Math Block
anchordQO
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\delta \, Q_O = - Q^{\uparrow}_O
LaTeX Math Block
anchorGG
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G_G = V_e^{-1} \, \delta \, Q_G + \left[ \frac{R_{si}\, s_{oi}}{B_{oi}}  + \frac{ s_{gi}}{B_{gi}}\right] 
LaTeX Math Block
anchordGG
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\delta \, Q_G = Q^{\downarrow}_G - Q^{\uparrow}_G + Q^{\downarrow}_{GCAP}
LaTeX Math Block
anchorGW
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G_W = V_e^{-1} \, \delta \, Q_W +  \frac{ s_{wi}}{B_{wi}} 
LaTeX Math Block
anchordGW
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\delta \, Q_W = Q^{\downarrow}_W - Q^{\uparrow}_W + Q^{\downarrow}_{WAQ}
LaTeX Math Block
anchorphin
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\phi_n = 1 + c_\phi \, (p-p_i)  + 0.5 \, c^2_\phi \, (p-p_i)^2 

where

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

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bodyt

LaTeX Math Inline
bodyV_e

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
bodyc_\phi

pore compressibility 

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
body--uriencoded--s_%7Bwi%7D

initial water saturation

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_%7Bgi%7D

initial gas 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_%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_%7BWAQ%7D(t)

Cumulative water influx from Aquifer 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)

Cumulative gas influx from Gas Cap expansion by the time moment

LaTeX Math Inline
bodyt

LaTeX Math Inline
bodyB_g(p)

Gas formation volume factor 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



LaTeX Math Inline
bodyR_s(p), \, R_v(p)

 Solution GOR and Vaporized Oil Ratio as functions of reservoir pressure

LaTeX Math Inline
bodyp










The MatBal equation 

LaTeX Math Block Reference
anchorMatBal
 is often 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):

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