Modelling facility for field-average average formation pressure
and Bottom-Hole Pressure ( for producers and | LaTeX Math Inline |
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| body | p^{\downarrow}_{wf}(t) |
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for injectors) 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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A_e \, h_e \int_{p_i}^p \phi_e(p) \, c_t(p) \, dp = \Delta Q (t) = Q^{\downarrow}_t(t) - Q^{\uparrow}_t(t) + V^{\downarrow}_{GC}(t) + V^{\downarrow}_{AQ}(t) |
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p^{\uparrow}_{wf, k}(t) = p(t) - {J^{\uparrow}_k}^{-1} \cdot \frac{dQ^{\uparrow}_k}{dt} |
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p^{\downarrow}_{wf, k}(t) = p(t) - {J^{\downarrow}_i}^{-1} \cdot \frac{dQ^{\downarrow}_k}{dt} |
where
\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 |
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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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| F_O = V_\phi^{-1} \, \delta \, Q_O + F_{Oi} |
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| F_{Oi} = \frac{s_{oi}}{B_{oi}} + \frac{R_{vi}\, s_{gi}}{B_{gi}} |
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| \delta \, Q_O = - Q^{\uparrow}_O |
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| F_G = V_\phi^{-1} \, \delta \, Q_G + F_{Gi} |
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| F_{Gi} = \frac{R_{si}\, s_{oi}}{B_{oi}} + \frac{ s_{gi}}{B_{gi}} |
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| \delta \, Q_G = Q^{\downarrow}_G - Q^{\uparrow}_G + Q^{\downarrow}_{GCAP} |
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| F_W = V_\phi^{-1} \, \delta \, Q_W + F_{Wi} |
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| F_{Wi} = \frac{ s_{wi}}{B_{wi}} |
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| \delta \, Q_W = Q^{\downarrow}_W - Q^{\uparrow}_W + Q^{\downarrow}_{WAQ} |
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where
| drainage area | | effective formation thickness averaged over drainage areainitial formation pressure\Delta Q (t) | full-field cumulative reservoir fluid balance J^{\uparrow}_k | | --uriencoded--Q%5e%7B\uparrow%7D_G(t) |
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productivity index of k | -th producerQ^{\uparrow}_t| --uriencoded--Q%5e%7B\uparrow%7D_W(t) |
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cumulative offtakes from k | -th producer J^{\downarrow}_i | injectivity Index of i | -th injector |
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| body | --uriencoded--s_%7Bgi%7D |
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Q^{\downarrow}_t(t) | cumulative intakes to -th injector | --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 |
The MatBal equation
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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):| LaTeX Math Block |
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| anchor | BHP_PROD |
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| p^ |
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Q^{\uparrow}_{wf, k}(t) = p(t) |
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full-field cumulative offtakes by the time moment - {J^{\uparrow}_k}^{-1} \cdot \frac{dQ^{\uparrow}_k}{dt} |
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| anchor | BHP_INJ |
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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 |
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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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field-average injectorsdownarrowtfull-field cumulative intakesphiepeffective porosity as function of formation pressure p(t) | Q^{\downarrow}_{GC}()full-field cumulative volumetric inflow from gas cap expansion | total compressibilityas function of formation pressure c_t(p) | p(t) | Q^{AQ}(t)full-field cumulative volumetric inflow from aquifer expansion
In practice there is no way to measure the external influx
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| body | Q^{\downarrow}_{GC}(t) |
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and
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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 Aquifer Drive and Gas Cap drive models Drive models which are normally based on the relationsrelated to pressure dynamics
:
Expansion Expansion FQ^{\downarrow}_{GC}(p(t)) |
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| Q^{\downarrow}_{AQ}(t) = |
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FQ^{\downarrow}_{AQ}(p(t)) |
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) ...
which closes equation
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for the pressure .Approximations
...
In some specific cases equation
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can be explicitly integrated with the accuracy sufficient for practical applications:Low Ideal rocks and fluids\{ \phi_e = {\rm const}, \ c_t = {\rm const} \}| --uriencoded--c_t = c_\phi + c_%7B\rm fluid%7D = %7B\rm const%7D |
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t c_r + frac{1}{p} \frac{1}{p} |
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| p(t) = p_i + \frac{\Delta Q(t)}{V_ |
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e | | LaTeX Math Block |
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| p(t) = p_i \exp \left[ \frac{\Delta Q(t)}{V_ |
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e \cdot c_t 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....
See Also
...
Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Material Balance Analysis (0D or MatBal)MatBal)
[ Material Balance Pressure Plot ][ FMB Pressure @model]
[ Derivation of Material Balance Pressure @model ]
[ Modified Black Oil fluid @model (MBO) ]
