Modelling facility for field-average formation pressure
at any time moment as response to production flowrates history:| LaTeX Math Block |
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| anchor | MatBal |
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| alignment | left |
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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 |
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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] |
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| \delta \, Q_O = - Q^{\uparrow}_O |
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| LaTeX Math Block |
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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 |
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| \delta \, Q_G = Q^{\downarrow}_G - Q^{\uparrow}_G + Q^{\downarrow}_{GCAP} |
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| G_W = V_e^{-1} \, \delta \, Q_W + \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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| LaTeX Math Block |
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| \phi_n = 1 + c_\phi \, (p-p_i) + 0.5 \, c^2_\phi \, (p-p_i)^2 |
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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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| initial oil+gas pay drainage volume (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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| pore compressibility | | LaTeX Math Inline |
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| body | --uriencoded--Q%5e%7B\uparrow%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_W(t) |
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| LaTeX Math Inline |
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| body | --uriencoded--s_%7Bgi%7D |
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| initial gas 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_%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_%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--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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is often 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):| LaTeX Math Block |
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| anchor | BHP_PROD |
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| alignment | left |
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| p^{\uparrow}_{wf, k}(t) = p(t) - {J^{\uparrow}_k}^{-1} \cdot \frac{dQ^{\uparrow}_k}{dt} |
| | LaTeX Math Block |
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| anchor | BHP_INJ |
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| alignment | left |
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| p^{\downarrow}_{wf, \, j}(t) = p(t) - {J^{\downarrow}_j}^{-1} \cdot \frac{dQ^{\downarrow}_j}{dt} |
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| where | where |
| LaTeX Math Inline |
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| body | p^{\uparrow}_{wf, \, k}(t) |
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| | | LaTeX Math Inline |
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| body | p^{\downarrow}_{wf, \, j}(t) |
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| cumulative offtakes from -th producer by the time moment | | cumulative intakes to -th injector by the time moment |
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In practice there is no way to measure the external influx
| LaTeX Math Inline |
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| body | Q^{\downarrow}_{GC}(t) |
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and
| LaTeX Math Inline |
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| body | Q^{\downarrow}_{AQ}(t) |
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so that one need to model them and calibrate model parameters to fit available data on
production flowrates history and
formation pressure data records.
There is a list of various analytical Aquifer Drive and Gas Cap Drive models which are normally related to pressure dynamics
:
which closes equation
| LaTeX Math Block Reference |
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for the pressure
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Variations
In some specific cases equation
| LaTeX Math Block Reference |
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can be explicitly integrated: | Low pressure dry gas |
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| LaTeX Math Inline |
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| body | \{ \phi_e = {\rm const}, \ c_t = {\rm const} \} |
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| | LaTeX Math Inline |
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| body | c_t = c_r + \frac{1}{p} \sim \frac{1}{p} |
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| LaTeX Math Block |
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| p(t) = p_i + \frac{\Delta Q(t)}{V_e \cdot c_t} |
| | LaTeX Math Block |
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| p(t) = p_i \exp \left[ \frac{\Delta Q(t)}{V_e} \right] |
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where
This allows using simple graphical methods for estimating drainage volume
.
See Also
Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Material Balance Analysis (MatBal)
[ Material Balance Pressure Plot ][ FMB Pressure @model]
[ Derivation of Material Balance Pressure @model ]
