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| The first assumption of CRM is that productivity index of producers stays constant in time: | LaTeX Math Block |
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| J = \frac{q_{\uparrow}(t)}{p_r(t) - p_{wf}(t)} = \rm const |
which can be re-written as explicit formula for formation pressure: | LaTeX Math Block |
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| p_r(t) = p_{wf}(t) + J^{-1} q_{\uparrow}(t) |
The second assumption is that drainage volume of producers-injectors system is finite and constant in time: | LaTeX Math Block |
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| V_\phi = V_r \phi = \rm const |
The third assumption is that total formation-fluid compressibility stays constant in time: | LaTeX Math Block |
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| c_t \equiv \frac{1}{V_{\phi}} \cdot \frac{dV_{\phi}}{dp} = \rm const |
which can be easily integrated: | LaTeX Math Block |
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| V_{\phi}(t) =V^\circ_{\phi} \cdot \exp \big[ - c_t \cdot [p_i - p_r(t)] \big] |
where is field-average initial formation pressure, is initial drainage volume,
– field-average formation pressure at time moment , is drainage volume at time moment .
Equation | LaTeX Math Block Reference |
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can be rewritten as:| LaTeX Math Block |
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| dV_{\phi} = c_t \, V_{\phi} \, dp |
The dynamic variations in drainage volume
are due to production/injection:| LaTeX Math Block |
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| dV_{\phi}= \int_0^t q_{\uparrow}(\tau) d\tau - f \int_0^t q_{\downarrow}(\tau) d\tau |
and leading to corresponding formation pressure variation: | LaTeX Math Block |
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| dp = p_i - p_r(t) |
thus making | LaTeX Math Block Reference |
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become:| LaTeX Math Block |
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| \int_0^t q_{\uparrow}(\tau) d\tau - f \int_0^t q_{\downarrow}(\tau) d\tau = c_t \, V_\phi \, [p_i - p_r(t)] |
and differentiated | LaTeX Math Block |
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| q_{\uparrow}(\tau) = f q_{\downarrow}(\tau) - c_t \, V_\phi \, \frac{d p_r(t)}{d t} |
and substituting from productivity equation | LaTeX Math Block Reference |
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:| LaTeX Math Block |
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| q_{\uparrow}(\tau) = f q_{\downarrow}(\tau) - c_t \, V_\phi \, \left[ \frac{d p_{wf}(t)}{d t} + J^{-1} \frac{d q_{\uparrow}}{d t} \right] |
which leads to | LaTeX Math Block Reference |
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This equation The equation
| LaTeX Math Block Reference |
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can be integrated explicitly:
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