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Definition

The Capacitance-Resistance Model (CRM) is a set of mathematical models relating production rate to the bottomhole pressure and offset injection rateSpecific type of Production Analysis (PA) workflow based on correlation between injection rates history and production rate history with account of bottomhole pressure history.

In case the bottom-hole bottomhole pressure data  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.

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Application

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Advantages

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Limitations

CRM does not pretend to predict reserves distribution as dynamic model does.

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Motivation

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Production rate in producing well depends on its productivity index 

CRM can only be tuned for injectors with a rich history of rates variations.

CRM only works at long times and only in areas with limited drainage volume.

Technology

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. 

CRM – Single-Tank Capacitance Resistance Model

The CRM model is trying

The simulation is based on the following equation:

q^q_1^{\uparrow}(t) =J  f\cdot \, q^{\downarrow}left( p_e(t)  - \tau \cdot \frac{ d q^{\uparrow}}{ dt }  - \beta \cdot \frac{d p_{wf}}{dt}

where

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}(t)

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 \right)

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and as such depends on completion/lift settings (defining 

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) and how formation pressure is maintained 

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It keeps declining due to the offtakes

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:

cp_e(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:

p_e[q_1^{\uparrow}(t), q_2^{\uparrow}(t)

, q_3^{\uparrow}(t)

, \

dots]

and maintained by either aquifer or Fluid Injection and in the latter case depends on injection rates:

p_

e(t) = p_

e[q_1^{\downarrow}(t)

,q_2^{\downarrow}(t),q_3^{\downarrow}(t)

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

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

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

V^\circ_{\phi} is initial drainage volume, 

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

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

\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)]

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

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]

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

References

1

2

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The combination of 

The ultimate purpose of CRM is to correlate the long-term (few months or longer) injection rates history with production rates history and BHP history (recorded by PDG).

It is essentially based on the fact that production rate responds to changes in BHP and offset injection.

The major assumptions in CRM model are:

• productivity index of producer stays constant in time
  
  
• dynamic drainage volume of producer is finite and constant in time (which is equivalent to PSS flow regime)
  
  
• total compressibility within the drainage volume of a given producer stays constant in time

Assumption 2 means that interference between producers is fairly constant in time despite the rate variations and their impact on the dynamic drainage volumes.

Technology

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

See Capacitance-Resistivity Model @model

p_{1nn}(t) =J_n^{-1} \left( 1 - \frac{t}{\tau_n} \right)

p_{1nm}(t) = 0

p_{1nm}(t) =  \frac{f_{nm}}{J_n \tau_n} \cdot t

\delta p_{1nn}(t) =  \frac{1}{J_n \tau_n} \cdot t

\delta p_{1nm}(t) = 0

\delta p_{1nm}(t) =  \frac{f_{nm}}{J_n \tau_n} \cdot t

p'_{1nn}(t) = \delta p_{1nn}(t) 

p'_{1nm}(t) = 0

p'_{1nm}(t) = \delta p_{1nm}(t)

See Also

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

[ Capacitance-Resistivity Model @model ]

References

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Arthur Aslanyan, Mathematical aspects of Multiwell Deconvolution and its relation to Capacitance Resistance Model, arxiv.org/abs/2203.01319

Nguyen, A. P., Kim, J. S., Lake, L. W., Edgar, T. F., & Haynes, B. (2011, January 1). Integrated Capacitance Resistive Model for Reservoir Characterization in Primary and Secondary Recovery. Society of Petroleum Engineers. doi:10.2118/147344-MS

Holanda, R. W. de, Gildin, E., & Jensen, J. L. (2015, November 18). Improved Waterflood Analysis Using the Capacitance-Resistance Model Within a Control Systems Framework. Society of Petroleum Engineers. doi:10.2118/177106-MS