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q^{\uparrow} (t) =\exp(-t/\tau) \cdot \left[ \ q^{\uparrow} (0) +
\tau^{-1} \gamma \cdot \big( p(0) - p(t) \cdot \exp(t/\tau) \big)
+\tau^{-1} \cdot \int_0^t \exp(s/\tau) \left[ f \cdot q^{\downarrow}(s) + \gamma \cdot p(s) \right] ds \ \right]= |
The objective function is:
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q^{\uparrow}_n (t) =\exp(-t/\tau_n) \cdot \left[ \ q^{\uparrow}_n (0) + \tau_n^{-1} \cdot \int_0^t \exp(s/\tau_n) \left[ \sum_m f_{nm} \cdot q^{\downarrow}_m(s) - \gamma_n \frac{dp_n}{ds} \right] ds \ \right] |
The objective function is:
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The model equation is:
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| anchor | J57KHQCRM |
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| alignment | left |
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p_n(t) = p_n(0) - \tau_n / \gamma_n \cdot \big[ q^{\uparrow}_n(t) - q^{\uparrow}_n(0) \big] - \gamma_n^{-1} \cdot Q^{\uparrow}_n (t) + \gamma_n^{-1} \cdot \sum_m f_{nm} \ Q^{\downarrow}_m(t) |
The objective function is:
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\tau_n \geq 0 , \quad \gamma_n \geq 0, \quad f_{nm} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{ij} \leq 1 |
See Also
where
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p_{nr}(0) = p_n(0) + (\tau_n / \gamma_n) \cdot q^{\uparrow}_n(0) |
is the initial formation pressure.
The equation
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can be re-written with explicit form of initial formation pressure:| LaTeX Math Block |
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p_n(t) = p_{nr}(0) + (\tau_n / \gamma_n) \cdot q^{\uparrow}_n(t) + \gamma_n^{-1} \cdot \sum_m f_{nm} \ Q_m^{\downarrow}(t) |
where could be both producer | LaTeX Math Inline |
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| body | --uriencoded--Q_m%5e%7B\uparrow%7D |
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or injector | LaTeX Math Inline |
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| body | --uriencoded--Q_m%5e%7B\downarrow%7D |
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.
If
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| body | --uriencoded--p_%7Bnr%7D(0) |
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is known then it can be fixed during the search loop which normally improves the quality of future production forecasts.
XCRM – Liquid-Control Cross-well Capacitance Resistance Model
Some extensions to conventional CRM model can be found in XCRM – Liquid-Control Cross-well Capacitance Resistance Model @model.
ELPM – Explicit Linear Production Model
Some extensions to conventional CRM model can be found in Explicit Linear Production Model
See Also
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Petroleum Industry / Upstream / Production / Subsurface Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Capacitance Resistance Model (CRM)
Production – Injection Pairing @ model
[ Slightly compressible Material Balance Pressure @model ]
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