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:
Does not pretend to predict reserves distribution as dynamic model does |
Only provides hints for misperforming wells and sectors which need a further focus |
Only provides hints for misperforming wells and sectors which need a further focus |
Can only be tuned for injector-producer pairing with a rich history of injection rates variations |
Can only work at long times and only in areas with limited drainage volume |
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.
The CRM model is trying
The simulation is based on the following equation:
q^{\uparrow}(t) = f \, q^{\downarrow}(t) - \tau \cdot \frac{ d q^{\uparrow}}{ dt } - \beta \cdot \frac{d p_{wf}}{dt} |
where
total surface production | |
total surface injection | |
average bottomhole pressure in producers | |
unitless constant, showing the share of injection which actually contributes to production | |
time-measure constant, related to well productivity [ s/Pa ] | |
storage-measure constant, related to reservoir volume and compressibility [ m3/Pa ] |
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 target 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 |
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 target 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 |
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 target 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 |