Consider a well-reservoir system consisting of:
- producing well W1 with total sandface flowrate q_1(t)>0 and bhp p_1(t)>0, draining the reservoir volume V_{\phi, 1}
- water injecting well W2 with total sandface flowrate q_2(t) <0, 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 in producer W1 is being automatically adjusted by \delta q_1(t) to compensate the bottom-hole pressure variation \delta p_1(t) in response to the total sandface 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 petroleum practice this happens when the formation is capable to deliver more fluid than the current lift settings in producer so that the bottom-hole pressure in producer is constantly kept at minimum value defined by the lift design..
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{V_{\phi, 2}}{ V_{\phi, 1}} = \rm const |
In case when injector W2 supports only one producer W1 then
V_{\phi, 2} = V_{\phi, 1} and
\delta q_1 = \delta q_2, which means that producer W1 with constant BHP and finite-reservoir volume will eventually vary its rate at the same volume as injector W2 (the response delay in time becomes irrelevant at long terms of the conventional production analysis).
If pressure in producer W1 is supported by several injectors
N_{\rm inj} > 1 then over a long period of time one can assume:
(14) | \delta q_1 =\sum_k f_{k1} \delta q_k |
with constant coefficients f_{1k}, \ {k=\{1..N_{\rm inj} \} }, which makes one of the key assumptions in Capacitance Resistance Model (CRM).
See also
[ DTR ] [ CTR ] [ Capacitance Resistance Model (CRM) ]