Mathematical model of Capacitance Resistance Model (CRM)
The model equation is:
q^{\uparrow}(t) = f \, q^{\downarrow}(t) - \tau \cdot \frac{ d q^{\uparrow}}{ dt } - \beta \cdot \frac{d p_{wf}}{dt} |
where
average surface production per well | |
average surface injection per well | |
average bottomhole pressure in producers | |
unitless constant, showing the share of injection which actually contributes to production | |
time-measure constant, related to well productivity | |
storage-measure constant, related to dynamic drainage volume and total compressibility |
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 objective 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 |
The model equation is:
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 objective 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 |
The model equation is:
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:
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 |
Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Capacitance Resistance Model (CRM)
https://doi.org/10.2118/147344-MS
https://doi.org/10.2118/177106-MS