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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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p_n(t) = p_{nr}(0) + (\tau_n / \gamma_n)  \cdot  q^{\uparrow}_n(t)  - \gamma_n^{-1} \cdot Q^{\uparrow}_n (t) + \gamma_n^{-1} \cdot \sum_m f_{nm}  \ Q_m^{\downarrow}(t)   

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XCRM  – Liquid-Control Cross-well Capacitance Resistance Model

p_n(t) = p_{nr}(0) + (\tau_n / \gamma_n)  \cdot  q_n(t)  - \gamma_n^{-1} \cdot Q_n (t) + \gamma_n^{-1} \cdot  
 \sum_{m \neq n}f_{nm}  \ Q_m(t) 

Q_m(t) =  \int q_m(t) \, dt

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 ModelIn regular case , the initial formation pressure at datum is the same for all wells:  LaTeX Math Inlinebody--uriencoded-- p_%7Bnr%7D(0) = p_i = %7B\rm const%7D, \ \forall n

See Also

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Petroleum Industry / Upstream /  Production / Subsurface Production / Field Study & Modelling / Production Analysis / Capacitance Resistance Model (CRM)

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