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 bottom-hole 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 |
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Case #2 – Constant BHP: p_1 = \rm const
Assume that the flowrate \delta q_1(t) in producer W1 is being adjusted to compensate the bottom-hole pressure variation \delta p_1(t) in response to the flowrate variation \delta q_2 in injector W2 so that bottom-hole pressure in producer W1 stays constant at all times \delta p_1(t) = \delta p_1 = \rm const.
In this case, flowrate response \delta q_1 in producer W1 to the flowrate variation \delta q_2 in injector W2 is going to be:
(4) | \delta q_1(t) = - \frac{\dot p_{u,\rm 21}(t)}{\dot p_{u,\rm 11}(t)} \cdot \delta q_2 |
where
t | time since injector's W2 rate has changed by \delta q_2. |
\dot p_{u,\rm 21}(t) | time derivative of cross-well pressure transient response (CTR) in producer W1 to the unit-rate production in injector W2 |
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\dot p_{u,\rm 11}(t) | time derivative of drawdown pressure transient response (DTR) in producer W1 to the unit-rate production in the same well |
For the finite-volume drain V_{\phi,1} \leq V_{\phi,2} < \infty the flowrate response factor \delta q_1 / \delta q_2 is getting stabilised over time as:
(11) | \delta q_1 / \delta q_2 = f_{21} = \frac{c_{t,2} V_{\phi, 2}}{c_{t,1} V_{\phi, 1}} = \rm const |
If pressure in producer W1 is supported by several injectors then:
(14) | \delta q_1 =\sum_k f_{k1} \delta q_k |
which makes one of the key assumptions in Capacitance Resistance Model (CRM).
See also
[ DTR ] [ CTR ] [ Capacitance Resistance Model (CRM) ]