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Modelling facility for field-average formation pressure
at any time moment as response to production flowrates history, which in case of
MBO fluid takes form
: LaTeX Math Block |
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anchor | MatBal |
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alignment | left |
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| \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 |
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| \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 |
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LaTeX Math Block |
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| F_O = V_\phi^{-1} \, \delta \, Q_O + |
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\left[ \frac{s_{oi}}{B_{oi}} + \frac{R_{vi}\, s_{gi}}{B_{gi}}\right] | LaTeX Math Block |
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| F_{Oi} = \ |
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left[ \frac{s_{oi}}{B_{oi}} + \frac{R_{vi}\, s_{gi}}{B_{gi}} |
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\right] | LaTeX Math Block |
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| \delta \, Q_O = - Q^{\uparrow}_O |
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LaTeX Math Block |
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| F_G = V_\phi^{-1} \, \delta \, Q_G + |
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\left[ \frac{R_{si}\, s_{oi}}{B_{oi}} + \frac{ s_{gi}}{B_{gi}}\right] | LaTeX Math Block |
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| F_{Gi} = \ |
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left[ \frac{R_{si}\, s_{oi}}{B_{oi}} + \frac{ s_{gi}}{B_{gi}} |
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\right] | LaTeX Math Block |
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| \delta \, Q_G = Q^{\downarrow}_G - Q^{\uparrow}_G + Q^{\downarrow}_{GCAP} |
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LaTeX Math Block |
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| F_W = V_\phi^{-1} \, \delta \, Q_W + |
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\frac{ swi}}{B_{wi}} | LaTeX Math Block |
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| F_{Wi} = \frac{ s_{wi}}{B_{wi}} |
| LaTeX Math Block |
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| \delta \, Q_W = Q^{\downarrow}_W - Q^{\uparrow}_W + Q^{\downarrow}_{WAQ} |
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where
| | LaTeX Math Inline |
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body | --uriencoded--Q%5e%7B\uparrow%7D_O(t) |
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LaTeX Math Inline |
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body | V_\phi = V \cdot \phi_i |
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| initial open pore volume of the main pay (excluding the aquifer and gas cap) | LaTeX Math Inline |
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body | --uriencoded--Q%5e%7B\uparrow%7D_G(t) |
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| | LaTeX Math Inline |
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body | --uriencoded--Q%5e%7B\uparrow%7D_W(t) |
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| pore compressibility | LaTeX Math Inline |
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body | --uriencoded--Q%5e%7B\downarrow%7D_W(t) |
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LaTeX Math Inline |
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body | --uriencoded--s_%7Bwi%7D |
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| initial water saturation | LaTeX Math Inline |
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body | --uriencoded--Q%5e%7B\downarrow%7D_G(t) |
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LaTeX Math Inline |
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body | --uriencoded--s_%7Bgi%7D |
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| | LaTeX Math Inline |
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body | --uriencoded--Q%5e%7B\downarrow%7D_%7BWAQ%7D(t) |
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| Cumulative water influx from Aquifer Expansion by the time moment |
LaTeX Math Inline |
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body | --uriencoded--s_%7Boi%7D |
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| initial oil saturation: LaTeX Math Inline |
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body | --uriencoded--s_%7Boi%7D = 1 - s_%7Bwi%7D - s_%7Bgi%7D |
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| LaTeX Math Inline |
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body | --uriencoded--Q%5e%7B\downarrow%7D_%7BGCAP%7Dt) |
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| Cumulative gas influx from Gas Cap expansion by the time moment |
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The MatBal equation
LaTeX Math Block Reference |
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can be
re-written in the following popular form:...
LaTeX Math Block |
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anchor | MatBal_formula |
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alignment | left |
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p = p_i + \frac{\delta Q}{c_\phi \, V_\phi} + \delta p_i |
...
LaTeX Math Block |
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anchor | MatBal_formula |
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alignment | left |
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\delta p_i = \frac{ B_{og} \, F_{Oi} + B_{go} \, F{Gi} + B_w \, F_W -1}{c_\phi}
= \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 |
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B_{og} = \frac{B_o - R_s \, B_g}{1- R_s \, R_v} |
...
LaTeX Math Block |
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B_{go} = \frac{ B_g - R_v \, B_o}{1- R_s \, R_v} |
where
...
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The MatBal equation
LaTeX Math Block Reference |
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can be complemented by constant
PI model of Bottom-Hole Pressure ( for
producers and
LaTeX Math Inline |
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body | p^{\downarrow}_{wf}(t) |
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for
injectors):...
| Low pressure dry gas |
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LaTeX Math Inline |
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body | \{ --uriencoded--c_t = c_\phi + c_e = {%7B\rm const}, \ c_t = {\rm const} \}fluid%7D = %7B\rm const%7D |
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| LaTeX Math Inline |
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body | --uriencoded--c_t = c_r + \frac{1}{p} g = \sim \frac{1}{p}frac%7B1%7D%7Bp%7D |
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LaTeX Math Block |
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| p(t) = p_i + \frac{\Delta Q(t)}{V_\phi \cdot c_t} |
| LaTeX Math Block |
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| p(t) = p_i \exp \left[ \frac{\Delta Q(t)}{V_\phi} \right] |
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where
is Cumulative Voidage Replacement Balance (CVRB): LaTeX Math Block |
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| \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) |
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The above approximations sometime allow using simple graphical methods for rough estimation of drainage volume
and associated
Hydrocarbon Reserves.
...