Definition

The Capacitance-Resistance Model (CRM) is a class of mathematical models relating the production rate history to the offset injection rate history with ability to account for the producers bottom-hole pressure variation.

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

Application

• 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

Limitations

CRM does not pretend to predict pressure and reserves distribution as 3D dynamic model does.

It only provides hints for misperforming wells and sectors which need a further focus.

Technology

CRM is built around production data 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. 

The CRM analysis is

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

CRM – Single-Tank Capacitance Resistance Model

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}{dt}

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_j \}, \{ \beta_j \}, \{ f_{ij} \}] = \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_j \}, \{ \beta_j \}, \{ f_{ij} \}] = \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

p_{wf}(t) = p_r(t) - \frac{q(t)}{J}

p_r(t) = p_i + \frac{1}{c_t \, V_{\phi}} \, \Bigg[ \int_0^t q_i(\tau) d\tau - \int_0^t q(\tau) d\tau  \Bigg]

References

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