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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anchor | BHP_PROD |
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p^{\uparrow}_{wf, k}(t) = p(t) - \frac{1}{J^{\uparrow}_k} \cdot \frac{dQ^{\uparrow}_k}{dt} |
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anchor | BHP_INJ |
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p^{\downarrow}_{wf, k}(t) = p(t) - \frac{1}{J^{\downarrow}_i} \cdot \frac{dQ^{\downarrow}_k}{dt} |
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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 |
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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
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body | --uriencoded--Q%5e%7B\uparrow%7D_O(t) |
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initial formation pressure\Delta Q (t) | full-field cumulative reservoir fluid balance --uriencoded--Q%5e%7B\uparrow%7D_G(t) |
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p^{\uparrow}_{wf}(t) | field-average BHP in producersQ^{\uparrow}_t--uriencoded--Q%5e%7B\downarrow%7D_W(t) |
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full-field cumulative offtakesp^{\downarrow}_{wf}(t) | field-average BHP in injectors | initial water saturation | LaTeX Math Inline |
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body | --uriencoded--Q%5e%7B\downarrow%7D_G(t) |
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body | --uriencoded--s_%7Bgi%7D |
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Q^{\downarrow}_t(t) | full-field cumulative intakes--uriencoded--Q%5e%7B\downarrow%7D_%7BWAQ%7D(t) |
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| Cumulative water influx from Aquifer Expansion by the time moment |
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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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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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\phieeffective porosity as function of formation pressure (t)
The MatBal equation
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can be complemented by constant PI model of Bottom-Hole Pressure (...
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for producers and
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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^{\uparrow}_{wf, k}(t) = p(t) - {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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c_t(p) | total compressibilityas function of formation pressure Q^AQfull-field cumulative volumetric inflow from aquifer expansionttiii
In practice there is no way to measure the external influx
...
and ...
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
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for the pressure .Approximations
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In some specific cases equation In some specific cases MatBal equation
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can be explicitly integrated with the accuracy sufficient for practical applications:Low Ideal rocks and fluids\{ --uriencoded--c_t = c_\phi + c_ |
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e = { const}, \ c_t = {\rm const} \}fluid%7D = %7B\rm const%7D |
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t \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) ]