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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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In practice there is no way to measure the external influx
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| body | Q^q^{\downarrow}_{GC}(t) |
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and
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| body | Q^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.
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which closes the set of equations
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–| LaTeX Math Block Reference |
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for the pressure and saturations | LaTeX Math Inline |
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| body | \{ s_w, \, s_o, \, s_g \} |
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.
Variations
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In some specific cases equation
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can be explicitly integrated:
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