Mathematical model of Capacitance Resistance Model (CRM)

Content

CRM – Single-Injector Capacitance Resistance Model

The model equation is:

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

where

$\begin{array}{l}q^{\uparrow}(t)\end{array}$	average surface production per well
$\begin{array}{l}q^{\downarrow}(t)\end{array}$	average surface injection per well
$\begin{array}{l}p_{wf}(t)\end{array}$	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
$\begin{array}{l}\gamma\end{array}$	Reservoir Storage

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

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

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

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

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 be 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)	$\begin{array}{l}\displaystyle V_\phi = V_r \phi = \rm const\end{array}$

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 $\begin{array}{l}t\end{array}$ .

Equation (8) can be rewritten as:

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

The dynamic variations in drainage volume $\begin{array}{l}dV_{\phi}\end{array}$ are due to production/injection:

(11)	$\begin{array}{l}\displaystyle dV_{\phi}= \int_0^t q_{\uparrow}(\tau) d\tau - f \int_0^t q_{\downarrow}(\tau) d\tau\end{array}$

and leading to corresponding formation pressure variation:

(12)	$\begin{array}{l}\displaystyle dp = p_i - p_r(t)\end{array}$

thus making (10) become:

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

and differentiated

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

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

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

which leads to (1).

The equation (1) can be integrated explicitly:

(16)	$\begin{array}{l}\displaystyle q^{\uparrow} (t) =\exp(-t/\tau) \cdot \left[ \ q^{\uparrow} (0) + \tau^{-1} \cdot \int_0^t \exp(s/\tau) \left[ f \cdot q^{\downarrow}(s) - \gamma \frac{dp}{ds} \right] ds \ \right]\end{array}$

and written in equivalent form:

(17)	$\begin{array}{l}\displaystyle q^{\uparrow} (t) =\exp(-t/\tau) \cdot \left[ \ q^{\uparrow} (0) + \tau^{-1} \gamma \cdot \big( p(0) - p(t) \cdot \exp(t/\tau) \big) +\tau^{-1} \cdot \int_0^t \exp(s/\tau) \left[ f \cdot q^{\downarrow}(s) + \gamma \cdot p(s) \right] ds \ \right]\end{array}$

The objective function is:

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

The basic constraints are:

(19)	$\begin{array}{l}\displaystyle \tau \geq 0 , \quad \gamma \geq 0, \quad f \geq 0\end{array}$

The additional constraints may be imposed as:

(20)	$\begin{array}{l}\displaystyle f \leq 1\end{array}$

which means that a part of injection ( $\begin{array}{l}1 - f\end{array}$ ) is going away from the reservoir drained by producer.

CRMP – Multi-Injector Capacitance Resistance Model

The model equation is:

(21)	$\begin{array}{l}\displaystyle q^{\uparrow}_n (t) + \tau_n \cdot \frac{ d q^{\uparrow}_n}{ dt }= \sum_m f_{nm} \cdot q^{\downarrow}_m(t) - \gamma_n \cdot \frac{d p_n}{dt}\end{array}$

This equation can be integrated explicitly:

(22)	$\begin{array}{l}\displaystyle q^{\uparrow}_n (t) =\exp(-t/\tau_n) \cdot \left[ \ q^{\uparrow}_n (0) + \tau_n^{-1} \cdot \int_0^t \exp(s/\tau_n) \left[ \sum_m f_{nm} \cdot q^{\downarrow}_m(s) - \gamma_n \frac{dp_n}{ds} \right] ds \right]\end{array}$

The objective function is:

(23)	$\begin{array}{l}\displaystyle E[\tau_n, \gamma_n, f_{nm}] = \sum_k \sum_n \big[ q^{\uparrow}_n(t_k) - \tilde q^{\uparrow}_n(t_k) \big]^2 \rightarrow \min\end{array}$

The constraints are:

(24)	$\begin{array}{l}\displaystyle \tau_n \geq 0 , \quad \gamma_n \geq 0, \quad f_{nm} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{nm} \leq 1\end{array}$

ICRM – Integrated Multi-Injector Capacitance Resistance Model

The model equation is:

(25)	$\begin{array}{l}\displaystyle Q^{\uparrow}_n (t) = \sum_n f_{nm} Q^{\downarrow}_n(t) - \tau_n \cdot \big[ q^{\uparrow}_n(t) - q^{\uparrow}_n(0) \big] - \gamma_n \cdot \big[ p_n(t) - p_n(0) \big]\end{array}$

The objective function is:

(26)	$\begin{array}{l}\displaystyle E[\tau_n, \gamma_n, f_{nm}] = \sum_k \sum_n \big[ Q^{\uparrow}_n(t_k) - \tilde Q^{\uparrow}_n(t_k) \big]^2 \rightarrow \min\end{array}$

The constraints are:

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

QCRM – Liquid-Control Multi-Injector Capacitance Resistance Model

The model equation is:

(28)	$\begin{array}{l}\displaystyle p_n(t) = p_n(0) - \tau_n / \gamma_n \cdot \big[ q^{\uparrow}_n(t) - q^{\uparrow}_n(0) \big] - \gamma_n^{-1} \cdot Q^{\uparrow}_n (t) + \gamma_n^{-1} \cdot \sum_m f_{nm} \ Q^{\downarrow}_m(t)\end{array}$

The objective function is:

(29)	$\begin{array}{l}\displaystyle E[\tau_n, \gamma_n, f_{nm}] = \sum_k \sum_n \big[ p_n(t_k) - \tilde p_n(t_k) \big]^2 \rightarrow \min\end{array}$

The constraints are:

(30)	$\begin{array}{l}\displaystyle \tau_n \geq 0 , \quad \gamma_n \geq 0, \quad f_{nm} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{ij} \leq 1, \quad p_{nr}(0) > 0\end{array}$

where

(31)	$\begin{array}{l}\displaystyle p_{nr}(0) = p_n(0) + (\tau_n / \gamma_n) \cdot q^{\uparrow}_n(0)\end{array}$

is the initial formation pressure.

The equation (28) can be re-written with explicit form of initial formation pressure:

(32)	$\begin{array}{l}\displaystyle p_n(t) = p_{nr}(0) + (\tau_n / \gamma_n) \cdot q^{\uparrow}_n(t) + \gamma_n^{-1} \cdot \sum_m f_{nm} \ Q_m^{\downarrow}(t)\end{array}$

where $\begin{array}{l}Q_m\end{array}$ could be both producer $\begin{array}{l}Q_m^{\uparrow}\end{array}$ or injector $\begin{array}{l}Q_m^{\downarrow}\end{array}$ .

If $\begin{array}{l}p_{nr}(0)\end{array}$ is known then it can be fixed during the search loop which normally improves the quality of future production forecasts.

XCRM – Liquid-Control Cross-well Capacitance Resistance Model

Some extensions to conventional CRM model can be found in XCRM – Liquid-Control Cross-well Capacitance Resistance Model @model.

ELPM – Explicit Linear Production Model

Some extensions to conventional CRM model can be found in Explicit Linear Production Model

References

Sayarpour, M., Zuluaga, E., Kabir, C.S., and Lake, L.W. (2007). The Use of Capacitance-Resistive Models for Rapid Estimation of Waterflood Performance and Optimization. Paper SPE 110081 presented at the SPE Annual Technical Conference and Exhibition, Anaheim, California, 11-14 November, doi.org/10.2118/110081-MS

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.org/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.org/10.2118/177106-MS

Page tree

CRM – Single-Injector Capacitance Resistance Model

CRMP – Multi-Injector Capacitance Resistance Model

ICRM – Integrated Multi-Injector Capacitance Resistance Model

QCRM – Liquid-Control Multi-Injector Capacitance Resistance Model

XCRM – Liquid-Control Cross-well Capacitance Resistance Model

ELPM – Explicit Linear Production Model

See Also

References

Page tree

Capacitance-Resistivity Model (CRM) @model

CRM – Single-Injector Capacitance Resistance Model

CRMP – Multi-Injector Capacitance Resistance Model

ICRM – Integrated Multi-Injector Capacitance Resistance Model

QCRM – Liquid-Control Multi-Injector Capacitance Resistance Model

XCRM – Liquid-Control Cross-well Capacitance Resistance Model

ELPM – Explicit Linear Production Model

See Also

References