Definition

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A set of mathematical models relating production rate to the bottomhole pressure and offset injection rate.

In case the bottom-hole pressure data is not available it is considered constant over time.

The CRM is trained over historical records of production rates, injection rates and bottom-hole pressure variation.

The major assumptions in CRM model are:

• productivity index of producers stays constant in time
  
  
• drainage volume of producers-injectors system is finite and constant in time
  
  
• total formation-fluid compressibility stays constant in time

Application

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• Assess current production performance
  
  
  • current distribution of recovery against expectations
    
    
  • current status and trends of recovery against expectations
    
    
  • current status and trends of reservoir depletion against expectations
     
  • current status and trends of water flood efficiency against expectations
    
    
  • compare performance of different wells or different groups of wells 
    
    
• Identify and prioritize surveillance opportunities
  
  
• Identify and prioritize redevelopment opportunities

Advantages

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• fast-track
  
  
• based on the robust input data
  
  
• does not involve full-field 3D dynamic modelling and associated assumptions

Limitations

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Technology

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CRM trains linear correlation between variation of production rates against variation of injection rates with account of bottom-hole pressure history in producers.

against material balance and require current FDP volumetrics, PVT and SCAL models. 

The CRM has certain specifics for oil producers, water injectors, gas injectors and field/sector analysis. 

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CRM – Single-Tank Capacitance Resistance Model

The CRM model is trying

The simulation is based on the following equation:

q^{\uparrow}(t) =  f \, q^{\downarrow}(t)  - \tau \cdot \frac{ d q^{\uparrow}}{ dt }  - \beta \cdot \frac{d p_{wf}}{dt}

where

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The 

\beta = c_t \, V_\phi

\tau = \frac{\beta}{J} = \frac{c_t  V_\phi}{J}

where

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Total formation compressibility is a linear sum of reservoir/fluid components:

c_t = c_r +  s_w c_w + s_o  c_w + s_g c_g

where

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

J = \frac{q_{\uparrow}(t)}{p_r(t) - p_{wf}(t)} = \rm const

which can re-written as explicit formula for formation pressure:

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:

V_\phi = V_{rocks} \phi = \rm const

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

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

which can be easily integrated:

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

where

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

The change in drainage volume

c_t \equiv \frac{1}{V_{\phi}} \cdot \frac{dV_{\phi}}{dp} = \frac{1}{V_{\phi}} \cdot \frac{1}{p_i - p_r(t) } \cdot \Bigg[ \int_0^t q_{\uparrow}(\tau) d\tau - f \int_0^t q_{\downarrow}(\tau) d\tau  \Bigg] = \rm const

The last equation can be rewritten as:

\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

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

and substituting

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

which leads to

The target function is:

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

The constraints are:

\tau \geq  0 , \quad \beta \geq 0,  \quad  0 \leq f \leq 1

CRMP – Multi-tank Producer-based Capacitance Resistance Model

q^{\uparrow}_j (t) = \sum_i^{n_i} f_{ij} q^{\downarrow}_i(t)  - \tau_j \cdot  \frac{ d q^{\uparrow}_j}{ dt }  - \beta_j  \cdot  \frac{d p_j}{dt}

The target function is:

E[\tau, \beta, f] = \sum_k \sum_j \big[ q^{\uparrow}_j(t_k) - \tilde q^{\uparrow}_j(t_k) \big]^2   \rightarrow \min 

The constraints are:

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

ICRM  – Multi-Tank Integrated Capacitance Resistance Model

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 target function is:

E[\tau, \beta, f] =  \sum_k \sum_j \big[ Q^{\uparrow}_j(t_k) - \tilde Q^{\uparrow}_j(t_k) \big]^2   \rightarrow \min 

The constraints are:

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

See Also

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Petroleum Industry / Upstream /  Production / Subsurface Production / Field Study & Modelling / Production Analysis

References

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Capacitance-Resistivity Model @model

References

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https://doi.org/10.2118/147344-MS

https://doi.org/10.2118/177106-MS

RAFAEL WANDERLEY DE HOLANDA, CAPACITANCE RESISTANCE MODEL IN A CONTROL SYSTEMS FRAMEWORK: A TOOL FOR DESCRIBING AND CONTROLLING WATERFLOODING RESERVOIRS, 2015.pdf

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