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 flowrate variation \delta q_2 in injector W2:
| (1) | \delta p_1 = - p_{u,\rm 21}(t) \cdot \delta q_2 |
where
t | time since the water injection rate has changed by the \delta q_2 value. |
p_{u,\rm 21}(t) | cross-well pressure transient response in producer W1 to the unit-rate production in injector W2 |
|---|
Case #2 – Constant BHP p_1 = \rm const
The flowrate response
\delta q_1 in producer W1 to the flowrate variation
\delta q_2 in injector W2:
| (2) | \delta q_1 = - \frac{p_{u,\rm 21}(t)}{p_{u,\rm 11}(t)} \cdot \delta q_2 |
where
t | time since the water injection rate has changed by the \delta q_2 value. |
p_{u,\rm 21}(t) | cross-well pressure transient response (CTR) in producer W1 to the unit-rate production in injector W2 |
|---|---|
p_{u,\rm 11}(t) | drawdown pressure transient response (DTR) in producer W1 to the unit-rate production in the same well |
For the finite-volume drain V_{\phi} < \infty the flowrate response factor \delta q_1 / \delta q_2 is constant:
| (3) | \delta q_1 / \delta q_2 = f_{21} = \frac{c_{t,2} V_{\phi, 2}}{c_{t,1} V_{\phi, 1}} = \rm const |