Consider a well-reservoir system consisting of:
- producing well W1 draining the reservoir volume V_{\phi, 1}
- water injecting well W2 supporting pressure in reservoir volume V_{\phi, 2} which includes the drainage volume V_{\phi, 1} of producer W1 and potentially other producers.
The drainage volume difference \delta V_{\phi} = V_{\phi, 2} - V_{\phi, 1} >0 may be related to the fact that water injection W2 is shared between V_{\phi, 1} and another reservoir or with another producer.
Case #1 – Constant flowrate production q_1 = \rm const >0
The pressure response \delta p_1 in producer W1 to the variation \delta q_2 of injection volume in injector W2:
| (1) | \delta p_1 = - p_{u,\rm 21}(t) \cdot \delta q_2 |
Case #2 – Constant BHP p_1 = \rm const
The flowrate response
\delta q_1 in producer W1 to the variation
\delta q_2 of injection volume in injector W2:
| (2) | \delta q_1 = - \frac{p_{u,\rm 11}(t)}{p_{u,\rm 21}(t)} \cdot \delta q_2 |
where t is time since the water injection rate has changed by the \delta q_2 value.
Pseudo-steady state flow
| (3) | \delta q_1 = -\frac{c_{t,I} V_{\phi, I}}{c_t V_{\phi}} \cdot \delta q_2 |
Steady state flow
| (4) | \delta q_1 = -\frac{c_{t,I} V_{\phi, I}}{c_t V_{\phi}} \cdot \delta q_2 |