Modelling facility for field-average saturation | LaTeX Math Inline |
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| body | s(t) = \{ s_w(t), \, s_o(t), \, s_g(t) \} |
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and formation pressure
at any time moment as response to production flowrates history:
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| \frac{ds_w}{dt} =\frac{1}{A_e \, h_e \, \phi_e(p)} \left[ q^{\downarrow}_w(t) - q^{\uparrow}_w(t) + q^{\downarrow}_{WAQ}(t) \right] - \left[ c_r(p) s_w +c_w(p) s_w \right] \frac{dp}{dt}
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| \frac{ds_o}{dt} =\frac{1}{A_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_g}{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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where
The direct consequence of the above equations:
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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) |
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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| 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 |
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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 a set of equations
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–
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for the pressure
and saturations
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| body | \{ s_w, \, s_o, \, s_g \} |
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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
See Also
Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis
Material Balance Analysis (0D or MatBal)
