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q^{\uparrow} (t) =\exp(-t/\tau)  \cdot \left[ \ q^{\uparrow} (0) + \tau^{-1} \cdot  \int_0^t \exp(s/\tau) \left[ f \cdot q^{\downarrow}(s) - \gamma \frac{dp}{ds} \right] ds   \ \right]

The objective function isand written in equivalent form:

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E[\tau, \gamma, f] = \sum_k \big[q^{\uparrow} (t) =\exp(-t/\tau)  \cdot \left[ \ q^{\uparrow} (t_k0) -+ 
\tilde q^{\uparrow}(t_k) \big]^2   \rightarrow \min 

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\tau \geq  0 , \quad \gamma \geq 0,  \quad  f \geq 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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E[\tau, \gamma, f] = \sum_k \big[ q^{\uparrow}(t_k) - \tilde q^{\uparrow}(t_k) \big]^2   \rightarrow \min 


The basic constraints are:

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\tau \geq  0 , \quad \gamma \geq 0,  \quad  f \geq 0


The additional constraints may be imposed as:

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f \leq 1

which means that a part of injection (

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) is going away from the reservoir drained by producer.

CRMP – Multi-Injector Capacitance Resistance Model


The model equation is:

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q^{\uparrow}_n (t) +  \tau_n \cdot  \frac{ d q^{\uparrow}_n}{ dt }=
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f \leq 1

which means that a part of injection (

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body1 - f
) is going away from the reservoir drained by producer.

CRMP – Multi-Injector Capacitance Resistance Model

The model equation is:

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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)  - \gamma_n  \cdot  \frac{d p_n}{dt}

This equation can be integrated explicitly:

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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(st)  - \gamma_n  \cdot  \frac{dpd p_n}{ds} \right] ds \ \right]dt}


This equation can be integrated explicitlyThe objective function is:

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E[\tauq^{\uparrow}_n, (t) \gamma=\exp(-t/\tau_n, f_{nm}] = \sum_k \sum_n \big[) \cdot \left[ \  q^{\uparrow}_n (t_k0) + \tau_n^{-1}  \tildecdot q^{\uparrow}_n(t_kint_0^t \exp(s/\tau_n) \big]^2   \rightarrow \min left[ \sum_m  f_{nm} \cdot  q^{\downarrow}_m(s) - \gamma_n \frac{dp_n}{ds} \right] ds  \right]


The objective function isThe constraints are:

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E[\tau_n \geq  0 ,  \quad \gamma_n \geq 0,  \quad f_{nm}] \geq  0 , \quad= \sum_k \sum_i^{N^n \big[ q^{\uparrow}} f_{nm} \leq 1

ICRM  – Integrated Multi-Injector Capacitance Resistance Model

_n(t_k) - \tilde q^{\uparrow}_n(t_k) \big]^2   \rightarrow \min 


The constraints areThe model equation is:

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Q^{\uparrow}tau_n (t) = \sum_n \geq  0 ,  \quad \gamma_n \geq 0,  \quad f_{nm} Q^{\downarrow}_n(t)  - \tau_n \cdot \big[ q^{\uparrow}_n(t) - q^{\uparrow}_n(0) \big]  - \gamma_n \cdot \big[ p_n(t) - p_n(0) \big]\geq  0 , \quad \sum_i^{N^{\uparrow}} f_{nm} \leq 1

ICRM  – Integrated Multi-Injector Capacitance Resistance Model


The model equation isThe objective function is:

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E[\tau_n, \gamma_n,Q^{\uparrow}_n (t) = \sum_n f_{nm}] =  \sum_k \sum_n Q^{\downarrow}_n(t)  - \tau_n \cdot \big[ Q^q^{\uparrow}_n(t_k) - \tilde Q^q^{\uparrow}_n(t_k0) \big]^2  - \rightarrowgamma_n \min 

The constraints are:

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\tau_j \geq  0 ,  \quad \gamma_n \geq 0,  \quad f_{ij} \geq  0 , \quad \sum_i^{N^{\uparrow}} f_{ij} \leq 1

QCRM  – Liquid-Control Multi-Injector  Capacitance Resistance Model

cdot \big[ p_n(t) - p_n(0) \big]


The objective function is:

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E[\tau_n, \gamma_n, f_{nm}] =  \sum_k \sum_n \big[ Q^{\uparrow}_n(t_k) - \tilde Q^{\uparrow}_n(t_k) \big]^2   \rightarrow \min 


The constraints areThe model equation is:

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p_n(t) = p_n(0) - \tau_n / \tau_j \geq  0 ,  \quad \gamma_n  \cdot\geq 0,  \big[quad q^f_{\uparrow}_n(t) - q^{\uparrow}_n(0) \big]  - \gamma_n^{-1} \cdot Q^ij} \geq  0 , \quad \sum_i^{N^{\uparrow}_n (t) + \gamma_n^{-1} f_{ij} \cdot \sum_m f_{nm} Q^{\downarrow}_m(t)  leq 1


QCRM  – Liquid-Control Multi-Injector  Capacitance Resistance Model


The model equation isThe objective function is:

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E[\taup_n, \gamma_n, f_{nm}] =  \sum_k \sum_n(t) = p_n(0) - \tau_n / \gamma_n  \cdot \big[ pq^{\uparrow}_n(t_k) - \tilde pq^{\uparrow}_n(t_k0) \big]^2  - \rightarrowgamma_n^{-1} \min 

The constraints are:

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\tau_n \geq  0 ,  \quad \gamma_n \geq 0,  \quadcdot Q^{\uparrow}_n (t) + \gamma_n^{-1} \cdot \sum_m f_{nm} \geq  0 , \quad \sum_i^{N^{\uparrow}} f_{ij} \leq 1 Q^{\downarrow}_m(t)  


The objective function is:

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E[\tau_n, \gamma_n, f_{nm}] =  \sum_k \sum_n \big[ p_n(t_k) - \tilde p_n(t_k) \big]^2   \rightarrow \min 


The constraints are:

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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,  \quad p_{nr}(0) > 0


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:

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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

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 could be both producer
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or injector
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.


If 

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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 Production / Field Study & Modelling / Production Analysis / Capacitance Resistance Model (CRM)

Production – Injection Pairing @ model

[ Slightly compressible Material Balance Pressure @model ]

Show If
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CRM as MDCV @model

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