Mathematical model of Capacitance Resistance Model (CRM)
The model equation is:
q^{\uparrow}(t) + \tau \cdot \frac{ d q^{\uparrow}}{ dt } = f \cdot q^{\downarrow}(t) - \gamma \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 | |
| Reservoir Storage |
The and
constants are related to some primary well and reservoir properties:
\gamma = c_t \, V_\phi |
\tau = \frac{\gamma}{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 equation can be integrated explicitly:
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] |
and written in equivalent form:
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] |
The objective function is:
E[\tau, \gamma, f] = \sum_k \big[ q^{\uparrow}(t_k) - \tilde q^{\uparrow}(t_k) \big]^2 \rightarrow \min |
The basic constraints are:
\tau \geq 0 , \quad \gamma \geq 0, \quad f \geq 0 |
The additional constraints may be imposed as:
f \leq 1 |
which means that a part of injection () is going away from the reservoir drained by producer.
The model equation is:
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} |
This equation can be integrated explicitly:
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] |
The objective function is:
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 |
The constraints are:
\tau_n \geq 0 , \quad \gamma_n \geq 0, \quad f_{nm} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{nm} \leq 1 |
The model equation is:
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] |
The objective function is:
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 |
The constraints are:
\tau_j \geq 0 , \quad \gamma_n \geq 0, \quad f_{ij} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{ij} \leq 1 |
The model equation is:
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) |
The objective function is:
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 |
The constraints are:
\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 |
where
p_{nr}(0) = p_n(0) + (\tau_n / \gamma_n) \cdot q^{\uparrow}_n(0) |
is the initial formation pressure.
The equation can be re-written with explicit form of initial formation pressure:
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) |
where could be both producer
or injector
.
If is known then it can be fixed during the search loop which normally improves the quality of future production forecasts.
Some extensions to conventional CRM model can be found in XCRM – Liquid-Control Cross-well Capacitance Resistance Model @model.
Some extensions to conventional CRM model can be found in Explicit Linear Production Model
Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Capacitance Resistance Model (CRM)
Production – Injection Pairing @ model
[ Slightly compressible Material Balance Pressure @model ]