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| LaTeX Math Block |
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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) |
where
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} |
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| 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 |
| | | |
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
.
Variations
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In some specific cases equation
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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} |
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| LaTeX Math Block |
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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
See Also
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Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis
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