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LaTeX Math Block
anchorp
alignmentleft
p(t)  = p_i + \frac{\Delta Q(t)}{V_\phi \cdot c_t}
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)


where

LaTeX Math Inline
body\Delta Q(t)

Cumulative Voidage Replacement Balance (CVRB) over time 

LaTeX Math Inline
bodyt

LaTeX Math Inline
bodyV_\phi = V \cdot \phi_i

initial drainage volume of the main pay (excluding the aquifer and gas cap)

LaTeX Math Inline
body\phi_i = \phi(p_i)

initial porosity

LaTeX Math Inline
body--uriencoded--c_t = c_\phi + c_o \, s_%7Boi%7D + c_g \, s_%7Bgi%7D + c_w \, s_%7Bwi%7D

total compressibility

LaTeX Math Inline
bodyc_\phi

pore compressibility 

LaTeX Math Inline
body--uriencoded--s_%7Bwi%7D

initial water saturation

LaTeX Math Inline
body--uriencoded--s_%7Bgi%7D

initial gas saturation

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
bodyc_o, \, c_g, \, c_w

fluid compressibility of water phase, oil phase and gas phase

...

The equations 

LaTeX Math Block Reference
anchorp
 and
LaTeX Math Block Reference
anchorDQ
 are often used in express assessment of thief water production and water injectionshare
LaTeX Math Inline
body--uriencoded--\Omega%5e%7B\uparrow%7D_W = Q%5e%7B\uparrow%7D_%7BW,%7B\rm true%7D%7D \, / \,Q%5e%7B\uparrow%7D_W
 and thief water injection share
LaTeX Math Inline
body--uriencoded--\Omega%5e%7B\downarrow%7D_W = Q%5e%7B\downarrow%7D_%7BW,%7B\rm true%7D%7D \, / \,Q%5e%7B\downarrow%7D_W
:

LaTeX Math Block
anchorp
alignmentleft
p
(t)
_i - p
_i
(t) = \alpha \cdot Q^{\uparrow}_O(t) + \beta \cdot Q^{\uparrow}_W
+ +
(t) - \gamma \cdot Q^{\downarrow}_W(t)
LaTeX Math Block
anchor
conditions
alignmentleft
\alpha > 0, \quad \beta > 0, \quad \gamma > 0
LaTeX Math Block
anchorPhi
p
alignmentleft
V_\phi =
- 
\frac{ 1 }{ \alpha \cdot c_t}  \cdot \frac{B_o - R_s \, B_g}{1- R_s \, R_v}
LaTeX Math Block
anchor
p
omega_up
alignmentleft
\
omega^
Omega^{\uparrow}_W = \frac{
\beta}
Q^{\
alpha} \cdot \frac{1}{B_w} \cdot \frac{B_o - R_s \, B_g}{1- R_s \, R_v} LaTeX Math Block
anchorp
alignmentleft
\omega^{\downarrow}_W
uparrow}_{W,{\rm true}}}{Q^{\uparrow}_W} = \frac{\
gamma
beta}{\alpha}  \cdot \frac{1}{B_w} \cdot \frac{B_o - R_s \, B_g}{1- R_s \, R_v}
LaTeX Math Block
anchor
DQ
omega_down
alignmentleft
\
Delta Q
Omega^{\downarrow}_W =
-
 
\frac{
B_o - R_s \, B_g}{1- R_s \, R_v} \cdot \, Q^{\uparrow}_O + \frac{ B_g
Q^{\downarrow}_{W, {\rm true}}}{Q^{\downarrow}_W} =\frac{\gamma}{\alpha}  \cdot \frac{1}{B_w} \cdot \frac{B_o - R_
v
s \, B_
o
g}{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)
 Show If
grouparax
 Panel
bgColorpapayawhip
titleARAX
The MatBal equation 
 LaTeX Math Block Reference
anchorMatBal
pageMaterial Balance Pressure @model
 can be re-written as following:
 LaTeX Math Block
anchorMatBal_formula
alignmentleft
p = p_i + \frac{\delta Q}{c_\phi \, V_\phi} + \delta p_i
 LaTeX Math Block
anchorMatBal_formula
alignmentleft
\delta p_i =  \frac{ B_{og} \, F_{Oi} + B_{go} \, F_{Gi} + B_w \, F_W -1}{c_\phi}
 LaTeX Math Block
anchor1
alignmentleft
B_{og} = \frac{B_o - R_s \, B_g}{1- R_s \, R_v}
 LaTeX Math Block
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) 
...
[ Derivation of Slightly compressible Material Balance Pressure @model ]
[ Capacitance-Resistivity Model (CRM) @model ]