Definition

The Capacitance-Resistance Model (CRM) is 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.

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

Advantages

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

Limitations

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

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

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

where

	total surface production
	total surface injection
	average bottomhole pressure in producers
	unitless constant, showing the share of injection which actually contributes to production
	time-measure constant, related to well productivity [ s/Pa ]
	storage-measure constant, related to reservoir volume and compressibility [ m³/Pa ]

The and constants are related to some primary well and reservoir characteristics:

\beta = c_t \, V_\phi

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

where

	total formation-fluid compressibility
	drainable reservoir volume
	total rock volume within the drainage area
	average effective reservoir porosity
	total fluid productivity index

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

	rock compressibility
	water, oil, gas compressibilities
	water, oil, gas formation saturations

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} = \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

where – field-average initial formation pressure, – field-average formation pressure at time moment ,

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

and substituting from productivity equation :

q_{\uparrow}(\tau) d\tau - f q_{\downarrow}(\tau) d\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

References

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