Mathematical model of Capacitance Resistance Model (CRM)

Content

CRM – Single-Tank Capacitance Resistance Model

The model equation is:

(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

$\begin{array}{l}q^{\uparrow}(t)\end{array}$	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
$\begin{array}{l}\beta\end{array}$	storage-measure constant, related to dynamic drainage volume and total compressibility

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

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 objective 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

The model equation is:

(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 objective 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

The model equation is:

(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 objective 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

https://doi.org/10.2118/147344-MS

https://doi.org/10.2118/177106-MS

Page tree

CRM – Single-Tank Capacitance Resistance Model

CRMP – Multi-tank Producer-based Capacitance Resistance Model

ICRM – Multi-Tank Integrated Capacitance Resistance Model

See Also

References

Page tree

Capacitance-Resistivity Model @model

CRM – Single-Tank Capacitance Resistance Model

CRMP – Multi-tank Producer-based Capacitance Resistance Model

ICRM – Multi-Tank Integrated Capacitance Resistance Model

See Also

References