Modelling facility for field-average formation pressure
at any time moment as response to production flowrates history:
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| anchor | MatBal |
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| alignment | left |
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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) + Q^{\downarrow}_{GC}(t) + Q^{\downarrow}_{AQ}(t) |
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| c_t(s,p) = c_r + c_w s_w + c_o s_o + c_g s_g + s_o [ R_{sp} + (c_r + c_o) R_{sn} ] + s_g [ R_{vp} + R_{vn}(c_r + c_g) ] |
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| s_w(t) = s_{wi} + (1-s_{wi}) \cdot \rm RFO(t)/E_S |
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| s_g(t) = s_{wi} + (1-s_{wi}) \cdot \rm RFG(t)/E_S |
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where
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| | | full-field cumulative offtakes by the time moment |
| | | full-field cumulative intakes by the time moment |
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| body | Q^{\downarrow}_{GC}(t) |
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| cumulative gas influx from Gas Cap Expansion |
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| body | Q^{\downarrow}_{AQ}(t) |
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| cumulative water influx from Aquifer Expansion |
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| body | --uriencoded--s_%7Bwi%7D |
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| Initial water saturation | | Oil and Gas Recovery Factor |
The MatBal equation
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is often 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):
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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} |
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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 |
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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
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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 models which are normally related to pressure dynamics
:
which closes equation
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for the pressure
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Variations
In some specific cases equation
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can be explicitly integrated: | Low pressure dry gas |
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| body | \{ \phi_e = {\rm const}, \ c_t = {\rm const} \} |
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| body | c_t = c_r + \frac{1}{p} \sim \frac{1}{p} |
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| p(t) = p_i + \frac{\Delta Q(t)}{V_e \cdot c_t} |
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| p(t) = p_i \exp \left[ \frac{\Delta Q(t)}{V_e \cdot c_t} \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 ]
