Modelling facility for field-average average formation pressure
at any time moment as response to production flowrates history, which in case of MBO fluid takes form:Swfrac{ds_w}{dt}1}{A_eh_e\phi_e(p)left[ q^{\downarrow}_w(t) - q^{\uparrow}_w(t) + q^{\downarrow}_{WAQ}(t) \right] - 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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r s_ww s_w \right] \frac{dp}{dt}
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| F_O = V_\phi^{-1} \, \delta \, Q_O + F_{Oi} |
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So\frac{ds_w}{dt1A_e \, h_e \, \phi_e(p)} \left[ q^{\downarrow}_o(t) - q^{\uparrow}_o(t) \right] - \left[ c_r(p) s_o +c_o(p) s_o \right] \frac{dp}{dt}
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\frac{ds_w}{dt} =\frac{1}{A_e \, h_e \, \phi_e(p)} \left[ q^{\downarrow}_g(t) - q^{\uparrow}_g(t) + q^{\downarrow}_{GC}(t) \right] - \left[ c_r(p) s_w +c_g(p) s_g \right] \frac{dp}{dt}
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s_w + s_o + s_g = 1 |
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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
| \Delta Q --uriencoded--Q%5e%7B\uparrow%7D_O(t) |
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full-field cumulative reservoir fluid balance A_e | drainage areaQ^{\uparrow}_t--uriencoded--Q%5e%7B\uparrow%7D_G(t) |
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full-field cumulative offtakesh_e | effective formation thickness averaged over drainage areaQ^{\downarrow}_t--uriencoded--Q%5e%7B\uparrow%7D_W(t) |
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full-field cumulative intakes_e(p)effective porosity as function of formation pressure p(t) | Q^{\downarrow}_{GC}--uriencoded--Q%5e%7B\downarrow%7D_G(t) |
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cumulative volumetric inflow from Gas Cap Expansionc_(p)total compressibility as function of formation pressure p(t)Q^{\downarrow}_{WAQ}--uriencoded--Q%5e%7B\downarrow%7D_%7BWAQ%7D(t) |
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cumulative volumetric inflow The direct consequence of the above equations:
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...
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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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The MatBal equation
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is often can be complemented by constant
PI model of Bottom-Hole Pressure ( 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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| p^{\uparrow}_{wf, k}(t) = p(t) - {J^{\uparrow}_k}^{-1} \cdot \frac{dQ^{\uparrow}_k}{dt} |
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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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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
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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 Drive and Gas Cap Drive drive models models which are normally based on the relationsrelated to pressure dynamics
:
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
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In some specific cases equation
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can be explicitly integrated with the accuracy sufficient for practical applications:\{ --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 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
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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
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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) ]