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title	Derivation

The first assumption of CRM is that productivity index of producers stays constant in time:

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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:

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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:

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V_\phi = V_r \phi = \rm const

The third assumption is that total formation-fluid compressibility stays constant in time:

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c_t \equiv \frac{1}{V_{\phi}} \cdot \frac{dV_{\phi}}{dp} = \rm const

which can be easily integrated:

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V_{\phi}(t) =V^\circ_{\phi} \cdot \exp \big[ - c_t \cdot  [p_i - p_r(t)] \big]

where

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is field-average initial formation pressure,

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body	V^\circ_{\phi}

is initial drainage volume,

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body	p_r(t)

– field-average formation pressure at time moment

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,

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is drainage volume at time moment

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body	t

.

Equation

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can be rewritten as:

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dV_{\phi} = c_t \, V_{\phi} \, dp

The dynamic variations in drainage volume

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are due to production/injection:

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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:

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dp = p_i - p_r(t)

thus making

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become:

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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

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q_{\uparrow}(\tau)  = f q_{\downarrow}(\tau)  - c_t \, V_\phi \, \frac{d p_r(t)}{d t}

and substituting

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from productivity equation

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:

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q_{\uparrow}(\tau)  = f q_{\downarrow}(\tau)  - c_t \, V_\phi \, \left[ \frac{d p_{wf}(t)}{d t} + J^{-1} \frac{d q_{\uparrow}}{d t} \right]

which leads to

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.

This equation The equation

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can be integrated explicitly:

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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]

...

and 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) + 
\tau^{-1} \gamma \tildecdot  q^{\uparrow}(t_k\big( p(0)  - p(t) \big]^2cdot   \rightarrow \min

The constraints are:

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

...

\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] \leq 1

CRMP – Multi-Injector Capacitance Resistance Model

The model equation is:

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q^{\uparrow}_n (t) +  \tau_n \cdot  \frac{ d = \sum_k \big[ q^{\uparrow}(t_k) - \tilde q^{\uparrow}_n}{ dt }= \sum_m f_{nm} \cdot q^{\downarrow}_m(t)  - \gamma_n  \cdot  \frac{d p_n}{dt}

...

(t_k) \big]^2   \rightarrow \min

The basic constraints are:

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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} q^{\downarrow}_m(s) - \gamma_n \frac{dp_n}{ds} \right] ds \ \right]

The objective function is:

\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 }= \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[ \

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E[\tau_n, \gamma_n, f_{nm}] = \sum_k \sum_n \big[ q^{\uparrow}_n (t_k0) + \tau_n^{-1}  \tildecdot q^{\uparrow}_n(t_kint_0^t \exp(s/\tau_n) \big]^2   \rightarrow \min

The constraints are:

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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E[\tau_n, \gamma_n,

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

ICRM – Multi-Injector Integrated Capacitance Resistance Model

The model equation is:

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Q^n \big[ q^{\uparrow}_n(t_k) - \tilde q^{\uparrow}_n (t_k) = \sum_n f_{nm} Q^{\downarrow}_n(t)  - big]^2   \rightarrow \min

The constraints are:

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\tau_n \cdotgeq  0 ,  \big[quad q^{\uparrow}gamma_n(t) - q^{\uparrow}_n(0) \big]  - \gamma_n \cdot \big[ p_n(t) - p_n(0) \big] \geq 0,  \quad f_{nm} \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_nQ^{\downarrow}_n(t)  - \tau_n \cdot \big[ Q^q^{\uparrow}_n(t_k) - \tilde Q^q^{\uparrow}_n(t_k0) \big]^2  - \rightarrow \min gamma_n \cdot \big[ p_n(t) - p_n(0) \big]

The objective function isThe constraints are:

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E[\tau_j \geq  0 ,  \quadn, \gamma_n \geq 0,  \quad f_{ijnm}] \geq  0= , \quadsum_k \sum_i^{N^{n \big[ Q^{\uparrow}} f_{ij} \leq 1

PCRM – Multi-Injector Liquid-Control 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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p_n(t) = p_n(0) - \tau_n /\tau_j \geq  0 ,  \quad \gamma_n \geq 0, \cdot \big[quad q^f_{\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)  ij} \geq  0 , \quad \sum_i^{N^{\uparrow}} f_{ij} \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[ p_q^{\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,  \quad cdot 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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body	--uriencoded--Q_m%5e%7B\uparrow%7D

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

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Versions Compared

Old Version 55

New Version Current

Key

CRMP – Multi-Injector Capacitance Resistance Model

CRMP – Multi-Injector Capacitance Resistance Model

ICRM – Multi-Injector Integrated Capacitance Resistance Model

ICRM – Integrated Multi-Injector Capacitance Resistance Model

PCRM – Multi-Injector Liquid-Control Capacitance Resistance Model

QCRM – Liquid-Control Multi-Injector Capacitance Resistance Model

XCRM – Liquid-Control Cross-well Capacitance Resistance Model

ELPM – Explicit Linear Production Model

See Also

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Page History

Versions Compared

Old Version 55

New Version Current

Key

CRMP – Multi-Injector Capacitance Resistance Model

CRMP – Multi-Injector Capacitance Resistance Model

ICRM – Multi-Injector Integrated Capacitance Resistance Model

ICRM – Integrated Multi-Injector Capacitance Resistance Model

PCRM – Multi-Injector Liquid-Control Capacitance Resistance Model

QCRM – Liquid-Control Multi-Injector Capacitance Resistance Model

XCRM – Liquid-Control Cross-well Capacitance Resistance Model

ELPM – Explicit Linear Production Model

See Also