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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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| 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}_{WAQAQ}(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):...
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.
There is a list of various analytical analytical Aquifer Drive and Gas Cap Drive drive models models which are normally based on the relationsrelated to pressure dynamics
:
Q^Fq^{\downarrow}_{GC}(p(t)) |
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Q^Fq^{\downarrow}_{AQ}(p(t)) |
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which closes a set of equations
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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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Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Material Balance Analysis (0D or MatBal)
