Specific 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 bottomhole pressure data

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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

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Limitations

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

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Motivation

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Production rate in producing well depends on its productivity index 

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. 

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CRM – Single-Tank Capacitance Resistance Model

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 }  - J \cdot \tau \cdot \frac{d p}{dt}p_{wf}(t) \right)

and as such depends on completion/lift settings (defining 

It keeps declining due to the offtakesThe target function is:

\sum_k \big[ q^p_e(t) = p_e[q_1^{\uparrow}(t_k) - \tilde q^, q_2^{\uparrow}(t), q_3^{\uparrow}(t_k), \big]^2   \rightarrow \min 

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dots]

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

q^{\uparrow}_j p_e(t) = \sum_i^{n_i} f_{ij} q^p_e[q_1^{\downarrow}_i(t)  - \tau_j \cdot  \frac{ d q^{\uparrow}_j}{ dt }  - J_j \cdot  \tau_j \cdot  \frac{d p_j}{dt}

The target function is:

\sum_k \sum_j \big[ q^{\uparrow}_j(t_k) - \tilde q^{\uparrow}_j(t_k) \big]^2   \rightarrow \min 

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 - q^{\uparrow}_j(0) \big]  - J_j \cdot  \tau_j \cdot  \big[ p_j - p_{wf, \, j }(0) \big]

The target function is:

\sum_k \sum_j \big[ Q^{\uparrow}_j(t_k) - \tilde Q^{\uparrow}_j(t_k) \big]^2   \rightarrow \min 

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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2

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

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