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q^{\uparrow}(t) = f \, q^{\downarrow}(t) - \tau \cdot \frac{ d q^{\uparrow}}{ dt } - \betagamma \cdot \frac{d p_{wf}}{dt} |
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\betagamma = c_t \, V_\phi |
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\tau = \frac{\betagamma}{J} = \frac{c_t V_\phi}{J} |
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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 \, \biggleft[ \frac{d p_{wf}(t)}{d t} + J^{-1} \frac{d q_{\uparrow}}{d t} \biggright] |
which leads to | LaTeX Math Block Reference |
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E[\tau, \betagamma, f] = \sum_k \big[ q^{\uparrow}(t_k) - \tilde q^{\uparrow}(t_k) \big]^2 \rightarrow \min |
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\tau \geq 0 , \quad \betagamma \geq 0, \quad 0 \leq f \leq 1 |
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q^{\uparrow}_n (t) + \tau_n \cdot \frac{ d q^{\uparrow}_n}{ dt }= \sum_m f_{nm} \cdot q^{\downarrow}_m(t) - \betagamma_n \cdot \frac{d p_n}{dt} |
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q^{\uparrow}_n (t) =\tau_n^{-1} \exp(-t/\tau_n) \cdot \int_0^t \exp(s/\tau_n) \left[ \sum_m f_{nm} q^{\uparrow}_m(s) - \betagamma_n \frac{dp_n}{ds} \right] ds |
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E[\tau_n, \betagamma_n, f_{nm}] = \sum_k \sum_n \big[ q^{\uparrow}_n(t_k) - \tilde q^{\uparrow}_n(t_k) \big]^2 \rightarrow \min |
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\tau_n \geq 0 , \quad \betagamma_n \geq 0, \quad f_{nm} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{nm} \leq 1 |
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Q^{\uparrow}_jn (t) = \sum_i^{n_i} f_{ijnm} Q^{\downarrow}_in(t) - \tau_jn \cdot \big[ q^{\uparrow}_jn(t) - q^{\uparrow}_jn(0) \big] - \betagamma_jn \cdot \big[ p_jn(t) - p_jn(0) \big] |
The objective function is:
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E[\tau_n, \betagamma_n, f_{nm}] = \sum_k \sum_jn \big[ Q^{\uparrow}_jn(t_k) - \tilde Q^{\uparrow}_jn(t_k) \big]^2 \rightarrow \min |
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\tau_j \geq 0 , \quad \betagamma_jn \geq 0, \quad f_{ij} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{ij} \leq 1 |
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p_n(t) = p_n(0) - \tau_n / \betagamma_n \cdot \big[ q^{\uparrow}_n(t) - q^{\uparrow}_n(0) \big] - \betagamma_n^{-1} \cdot Q^{\uparrow}_n (t) + \betagamma_n^{-1} \cdot \sum_m f_{nm} Q^{\downarrow}_m(t) |
The objective function is:
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Q^{\uparrow}_j (t) = \sum_i^{n_i} f_{ij} Q^{\downarrow}_i(t) - \tau_j \cdot \big[ q^{\uparrow}_j(t) - q^{\uparrow}_j(0) \big] - \beta_j \cdot \big[ p_j(t) - p_j(0) \big] |
The objective function is:
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E[\tau, \beta, fE[\tau_n, \gamma_n, f_{nm}] = \sum_k \sum_jn \big[ Q^{\uparrow}_jn(t_k) - \tilde Q^{\uparrow}_jn(t_k) \big]^2 \rightarrow \min |
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\tau_jn \geq 0 , \quad \betagamma_jn \geq 0, \quad f_{ijnm} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{ij} \leq 1 |
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