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 |
The response delay in time still exists but in usual time-scales of production analysis it becomes negligible and one can consider
(11) consant.
In case injector W2 supports only one producer W1 both wells drain the same volume and
V_{\phi, 2} = V_{\phi, 1} so that
(11) leads to:
(14) | \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.
In case injector W2 supports many producers {W1 .. WN } then total injection shares towards producers is going to be unit:
(15) | \sum_{k=1}^N f_{2k} = 1 |
unless there is thief injection outside the drain area of all producers.
If pressure in producer W1 is supported by several injectors
N_{\rm inj} > 1 then over a long period of time one can assume:
(16) | \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) ]