Content

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.

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

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:

(1)	$\begin{array}{l}\displaystyle q^{\uparrow}(t) = f \, q^{\downarrow}(t) - \tau \cdot \frac{ d q^{\uparrow}}{ dt } - \beta \cdot \frac{d p_{wf}}{dt}\end{array}$

where

$\begin{array}{l}q^{\uparrow}(t)\end{array}$	total surface production
$\begin{array}{l}q^{\downarrow}(t)\end{array}$	total surface injection
$\begin{array}{l}p_{wf}(t)\end{array}$	average bottomhole pressure in producers
$\begin{array}{l}f\end{array}$	unitless constant, showing the share of injection which actually contributes to production
$\begin{array}{l}\tau\end{array}$	time-measure constant, related to well productivity [ s/Pa ]
$\begin{array}{l}\beta\end{array}$	storage-measure constant, related to reservoir volume and compressibility [ m³/Pa ]

The $\begin{array}{l}\tau\end{array}$ and $\begin{array}{l}\beta\end{array}$ constants are related to some primary well and reservoir characteristics:

(2)	$\begin{array}{l}\displaystyle \beta = c_t \, V_\phi\end{array}$

(3)	$\begin{array}{l}\displaystyle \tau = \frac{\beta}{J} = \frac{c_t V_\phi}{J}\end{array}$

where

$\begin{array}{l}c_t\end{array}$	total formation-fluid compressibility
$\begin{array}{l}V_\phi = \phi \, V_R\end{array}$	drainable reservoir volume
$\begin{array}{l}V_R\end{array}$	total rock volume within the drainage area
$\begin{array}{l}\phi\end{array}$	average effective reservoir porosity
$\begin{array}{l}J\end{array}$	total fluid productivity index

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

(4)	$\begin{array}{l}\displaystyle c_t = c_r + s_w c_w + s_o c_w + s_g c_g\end{array}$

where

$\begin{array}{l}c_r\end{array}$	rock compressibility
$\begin{array}{l}c_w, \, c_o, \, c_g\end{array}$	water, oil, gas compressibilities
$\begin{array}{l}s_w, \, s_o, \, s_g\end{array}$	water, oil, gas formation saturations

$\begin{array}{l}V^\circ_{\phi}\end{array}$

Derivation

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

(5)	$\begin{array}{l}\displaystyle J = \frac{q_{\uparrow}(t)}{p_r(t) - p_{wf}(t)} = \rm const\end{array}$

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

(6)	$\begin{array}{l}\displaystyle p_r(t) = p_{wf}(t) + J^{-1} q_{\uparrow}(t)\end{array}$

The second assumption is that drainage volume of producers-injectors system is finite and constant in time:

(7)	$\begin{array}{l}\displaystyle V_\phi = V_{rocks} \phi = \rm const\end{array}$

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

(8)	$\begin{array}{l}\displaystyle c_t \equiv \frac{1}{V_{\phi}} \cdot \frac{dV_{\phi}}{dp} = \rm const\end{array}$

which can be easily integrated:

(9)	$\begin{array}{l}\displaystyle V_{\phi}(t) =V^\circ_{\phi} \cdot \exp \big[ - c_t \cdot [p_i - p_r(t)] \big]\end{array}$

where $\begin{array}{l}p_i\end{array}$ is field-average initial formation pressure, $\begin{array}{l}V^\circ_{\phi}\end{array}$ is initial drainage volume,

$\begin{array}{l}p_r(t)\end{array}$ – field-average formation pressure at time moment $\begin{array}{l}t\end{array}$ ,

$\begin{array}{l}V_{\phi}(t)\end{array}$ is drainage volume at time moment $\begin{array}{l}t\end{array}$ .

(10)	$\begin{array}{l}\displaystyle \frac{dV_{\phi}}{dp} = c_t \, V_{\phi} \ \cdot\end{array}$

The change in drainage volume $\begin{array}{l}dV_{\phi}\end{array}$ is leading to formation pressure variation

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

The last equation can be rewritten as:

(12)	$\begin{array}{l}\displaystyle \int_0^t q_{\uparrow}(\tau) d\tau - f \int_0^t q_{\downarrow}(\tau) d\tau = c_t \, V_\phi \, [p_i - p_r(t)]\end{array}$

and differentiated

(13)	$\begin{array}{l}\displaystyle q_{\uparrow}(\tau) = f q_{\downarrow}(\tau) - c_t \, V_\phi \, \frac{d p_r(t)}{d t}\end{array}$

and substituting $\begin{array}{l}p_r(t)\end{array}$ from productivity equation (6):

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

which leads to (1).

The target function is:

(15)	$\begin{array}{l}\displaystyle E[\tau, \beta, f] = \sum_k \big[ q^{\uparrow}(t_k) - \tilde q^{\uparrow}(t_k) \big]^2 \rightarrow \min\end{array}$

The constraints are:

(16)	$\begin{array}{l}\displaystyle \tau \geq 0 , \quad \beta \geq 0, \quad 0 \leq f \leq 1\end{array}$

CRMP – Multi-tank Producer-based Capacitance Resistance Model

(17)	$\begin{array}{l}\displaystyle q^{\uparrow}_j (t) = \sum_i^{n_i} f_{ij} q^{\downarrow}_i(t) - \tau_j \cdot \frac{ d q^{\uparrow}_j}{ dt } - \beta_j \cdot \frac{d p_j}{dt}\end{array}$

The target function is:

(18)	$\begin{array}{l}\displaystyle E[\tau, \beta, f] = \sum_k \sum_j \big[ q^{\uparrow}_j(t_k) - \tilde q^{\uparrow}_j(t_k) \big]^2 \rightarrow \min\end{array}$

The constraints are:

(19)	$\begin{array}{l}\displaystyle \tau_j \geq 0 , \quad \beta_j \geq 0, \quad f_{ij} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{ij} \leq 1\end{array}$

ICRM – Multi-Tank Integrated Capacitance Resistance Model

(20)	$\begin{array}{l}\displaystyle Q^{\uparrow}_j (t) = \sum_i^{n_i} f_{ij} Q^{\downarrow}_i(t) - \tau_j \cdot \big[ q^{\uparrow}_j(t) - q^{\uparrow}_j(0) \big] - \beta_j \cdot \big[ p_j(t) - p_j(0) \big]\end{array}$

The target function is:

(21)	$\begin{array}{l}\displaystyle E[\tau, \beta, f] = \sum_k \sum_j \big[ Q^{\uparrow}_j(t_k) - \tilde Q^{\uparrow}_j(t_k) \big]^2 \rightarrow \min\end{array}$

The constraints are:

(22)	$\begin{array}{l}\displaystyle \tau_j \geq 0 , \quad \beta_j \geq 0, \quad f_{ij} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{ij} \leq 1\end{array}$

References

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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CRM – Capacitance Resistance Model

Definition

Application

Advantages

Limitations

Technology

CRM – Single-Tank Capacitance Resistance Model

CRMP – Multi-tank Producer-based Capacitance Resistance Model

ICRM – Multi-Tank Integrated Capacitance Resistance Model

References