Definition

The Capacitance-Resistance Model is a class of mathematical models relating the production rate history to the offset injection rate history with or without account fro the producers bottom-hole pressure variation.

In case the bottom-hole pressure data is available via Permanent Downhole Gauge (PDG) records it can be  considered as the PDG interpretation facility in terms of injection → production connectivity.

In case he bottom-hole pressure data is not available  the bottom-hole pressure is considered constant over time.

q_j (t) = \sum_i^{n_i} f_{ij} I_i(t)  - \tau_j \, \frac{ d q_j(t) }{ dt }  - J_j \, \tau_j \, \frac{d p_{wf, j }(t)}{dt}

p_{wf}(t) = p_r(t) - \frac{q(t)}{J}

p_r(t) = p_i + \frac{1}{c_t \, V_{\phi}} \, \Bigg[ \int_0^t q_i(\tau) d\tau - \int_0^t q(\tau) d\tau  \Bigg]