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.
Problem #1
Assuming producer is working with constant flowrate q_1 = \rm const >0, quantify the pressure response in producer W1 to the unit variation of injection volume in injector W2.
| (1) | \delta p_1 = - p_{\rm CTR}(t) \cdot \delta q_I |
Problem #2
Assuming producer is working with constant BHP
p_1 = \rm const, quantify the flowrate response in producer W1 to the unit variation of injection volume in injector W2.
| (2) | \delta q = - \frac{p_{\rm DTR}(t)}{p_{\rm CTR}(t)} \cdot \delta q_I |
where t is time since the water injection rate has changed by the \delta q_I value.
Pseudo-steady state flow
| (3) | \delta q = -\frac{c_{t,I} V_{\phi, I}}{c_t V_{\phi}} \cdot \delta q_I |
Steady state flow
| (4) | \delta q = -\frac{c_{t,I} V_{\phi, I}}{c_t V_{\phi}} \cdot \delta q_I |