Mathematical model of Capacitance Resistance Model (CRM)
CRM – Single-Tank Capacitance Resistance Model
The model equation is:
(1) | q^{\uparrow}(t) = f \, q^{\downarrow}(t) - \tau \cdot \frac{ d q^{\uparrow}}{ dt } - \beta \cdot \frac{d p_{wf}}{dt} |
where
q^{\uparrow}(t) | average surface production per well |
q^{\downarrow}(t) | average surface injection per well |
p_{wf}(t) | average bottomhole pressure in producers |
f | unitless constant, showing the share of injection which actually contributes to production |
\tau | time-measure constant, related to well productivity |
\beta | storage-measure constant, related to dynamic drainage volume and total compressibility |
The \tau and \beta constants are related to some primary well and reservoir properties:
(2) | \beta = c_t \, V_\phi |
(3) | \tau = \frac{\beta}{J} = \frac{c_t V_\phi}{J} |
where
c_t | total formation-fluid compressibility |
V_\phi = \phi \, V_R | drainable reservoir volume |
V_R | total rock volume within the drainage area |
\phi | average effective reservoir porosity |
J | total fluid productivity index |
Total formation compressibility is a linear sum of reservoir/fluid components:
(4) | c_t = c_r + s_w c_w + s_o c_w + s_g c_g |
where
c_r | rock compressibility |
c_w, \, c_o, \, c_g | water, oil, gas compressibilities |
s_w, \, s_o, \, s_g | water, oil, gas formation saturations |
The objective function is:
(16) | E[\tau, \beta, f] = \sum_k \big[ q^{\uparrow}(t_k) - \tilde q^{\uparrow}(t_k) \big]^2 \rightarrow \min |
The constraints are:
(17) | \tau \geq 0 , \quad \beta \geq 0, \quad 0 \leq f \leq 1 |
CRMP – Multi-tank Producer-based Capacitance Resistance Model
The model equation is:
(18) | q^{\uparrow}_n (t) + \tau_n \cdot \frac{ d q^{\uparrow}_n}{ dt }= \sum_m f_{nm} \cdot q^{\downarrow}_m(t) - \beta_n \cdot \frac{d p_n}{dt} |
This equation can be integrated explicitly:
(19) | q^{\uparrow}_n (t) =\tau_n^{-1} \exp(-t/\tau_n) \cdot \int_0^t \exp(s/\tau_n) \left[ \sum_m f_{nm} q^{\uparrow}_m(s) - \beta_n \frac{dp_n}{ds} \right] ds |
The objective function is:
(20) | E[\tau, \beta, f] = \sum_k \sum_n \big[ q^{\uparrow}_n(t_k) - \tilde q^{\uparrow}_n(t_k) \big]^2 \rightarrow \min |
The constraints are:
(21) | \tau_n \geq 0 , \quad \beta_n \geq 0, \quad f_{nm} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{nm} \leq 1 |
ICRM – Multi-Tank Integrated Capacitance Resistance Model
The model equation is:
(22) | 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 objective function is:
(23) | 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:
(24) | \tau_j \geq 0 , \quad \beta_j \geq 0, \quad f_{ij} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{ij} \leq 1 |
PCRM – Liquid-Control Capacitance Resistance Model
The model equation is:
(25) | p_n(t) = p_n(0) - \tau_n / \beta_n \cdot \big[ q^{\uparrow}_n(t) - q^{\uparrow}_n(0) \big] - \beta_n^{-1} \cdot Q^{\uparrow}_n (t) + \beta_n^{-1} \cdot \sum_m f_{nm} Q^{\downarrow}_m(t) |
(26) | 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 objective function is:
(27) | 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:
(28) | \tau_j \geq 0 , \quad \beta_j \geq 0, \quad f_{ij} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{ij} \leq 1 |
See Also
Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Capacitance Resistance Model (CRM)
Production – Injection Pairing @ model