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The simulation is based on the following equation:
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| anchor | 1CRMST |
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| alignment | left |
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q^{\uparrow}(t) = f \, q^{\downarrow}(t) - \tau \cdot \frac{ d q^{\uparrow}}{ dt } - \beta \cdot \frac{d p_{wf}}{dt} |
where
| total surface production |
| total surface injection |
| average bottomhole pressure in producers |
| unitless constant, showing the share of injection which actually contributes to production |
| time-measure constant, related to well productivity [ s/Pa ] |
| storage-measure constant, related to reservoir volume and compressibility [ m3/Pa ] |
The
and constants are related to some primary well and reservoir characteristics:| LaTeX Math Block |
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\beta = c_t \, V_\phi |
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\tau = \frac{\beta}{J} = \frac{c_t V_\phi}{J} |
where
| total formation-fluid compressibility |
| drainable reservoir volume |
| total rock volume within the drainage area |
| average effective reservoir porosity |
| total fluid productivity index |
Total formation compressibility is a linear sum of reservoir/fluid components:
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c_t = c_r + s_w c_w + s_o c_w + s_g c_g |
where
| rock compressibility |
| water, oil, gas compressibilities |
| water, oil, gas formation saturations |
p_{wf}(t) | average bottomhole pressure in producers | | | LaTeX Math Inline |
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body | \beta
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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 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_{rocks} \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} = \frac{1}{V_{\phi}} \cdot \frac{1}{p_i - p_r(t) } \cdot \Bigg[ \int_0^t q_{\uparrow}(\tau) d\tau - f \int_0^t q_{\downarrow}(\tau) d\tau \Bigg] = \rm const |
where – field-average initial formation pressure, – field-average formation pressure at time moment ,
The last equation can be rewritten as: | 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) d\tau - f q_{\downarrow}(\tau) d\tau = - c_t \, V_\phi \, \frac{d p_r(t)}{d t} |
and using substituting from productivity equation | LaTeX Math Block Reference |
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:| LaTeX Math Block |
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| q_{\uparrow}(\tau) d\tau - f q_{\downarrow}(\tau) d\tau = - c_t \, V_\phi \, \bigg[ \frac{d p_{wf}(t)}{d t} + J^{-1} \frac{d q_{\uparrow}}{d t} \bigg] |
which leads to | LaTeX Math Block Reference |
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The target function is:
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E[\tau, \beta, f] = \sum_k \big[ q^{\uparrow}(t_k) - \tilde q^{\uparrow}(t_k) \big]^2 \rightarrow \min |
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