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.
CRM does not pretend to predict reserves distribution as dynamic model does.
It only provides hints for misperforming wells and sectors which need a further focus.
CRM can only be tuned for injectors with a rich history of rates variations.
CRM only works 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 | |
share of injection which actually contributes to production | |
average bottomhole pressure in producers | |
The first assumption of CRM is that productivity index of producers stays constant in time:
which can re-written as explicit formula for formation pressure:
The second assumption is that drainage volume of producers-injectors system is finite and constant in time:
The third assumption is that total formation-fluid compressibility stays constant in time:
where – field-average initial formation pressure, – field-average formation pressure at time moment , The last equation can be rewritten as:
and differentiated
and using from productivity equation :
|
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 |
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