Consider a well-reservoir system (Fig. 1) 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 W0 with total sandface flowrate q_0(t) <0, supporting pressure in reservoir volume V_{\phi, 0}
The injection drainage volume
V_{\phi, 0} includes the drainage volume
V_{\phi, 1} of producer W1 and may be equal to it
V_{\phi, 0} = V_{\phi, 1} or may be bigger
V_{\phi, 0} > V_{\phi, 1} in case injector W0 supports other producers {W1 .. WN}:
V_{\phi, 0} = \sum_{k=1}^N V_{\phi, k}.
Fig. 1. Location map of injector-producer pairing with 4 producers {W1, W2, W3, W4} and one injector W0. |
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_0 in injector W0:
(1) | \delta p_1 = - p_{u,\rm 21}(t) \cdot \delta q_0 |
where
t | time since the water injection rate has changed by the \delta q_0 value. |
p_{u,\rm 01}(t) | cross-well pressure transient response in producer W1 to the unit-rate production in injector W0 |
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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 W0 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_0 in injector W0 is going to be:
(7) | \delta q_1(t) = - \frac{\dot p_{u,\rm 01}(t)}{\dot p_{u,\rm 11}(t)} \cdot \delta q_0 |
where
t | time since injector's W0 rate has changed by \delta q_0. |
\dot p_{u,\rm 01}(t) | time derivative of cross-well pressure transient response (CTR) in producer W1 to the unit-rate production in injector W0 |
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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,0} < \infty the flowrate response factor \delta q_1 / \delta q_0 is getting stabilised over time as:
(14) | \delta q_1 / \delta q_0 = - f_{01} = - \frac{V_{\phi, 1}}{ V_{\phi, 0}} = \rm const |
The response delay in time still exists but in usual time-scales of production analysis it becomes negligible and one can consider
(14) as constant in time.
In case injector W0 supports only one producer W1, then both wells drain the same reservoir volume
V_{\phi, 0} = V_{\phi, 1} so that
(14) leads to:
(17) | \delta q_1 = -\delta q_0 |
which means that producer W1 with constant BHP and finite-reservoir volume will eventually vary its rate at the same volume as injector W0.
In case injector W0 supports many producers {W1 .. WN} then all injection shares towards producers are going to sum up to a unit value:
(18) | \sum_{k=1}^N f_{0k} = 1 |
with constant coefficients f_{0k} \geq 0, \ {k=\{i..N \} }, unless there is a thief injection outside the drainage area of all producers and in this case:
(19) | \sum_{k=1}^N f_{0k} < 1 |
If pressure around producer W1 is supported by several injectors
N_{\rm inj} > 1 then production response in producer W1 is going to be:
(20) | \delta q_1 =-\sum_i f_{i1} \delta q_i |
with constant coefficients f_{i1} \geq 0, \ {i=\{0..N_{\rm inj} \} }.
The equations (18), (19) and (20) make one of the key assumptions in Capacitance Resistance Model (CRM).
It is important to note that CRM assumption that injector W0 may drain bigger volume than producer W1 V_{\phi, 0}> V_{\phi, 1} is a misnomer in most practical cases.
When wells (producers and injectors) are placed into the same connected reservoir volume they drain the same total volume V_\phi all together and all UTRs will have the same LTR asymptotic:
(21) | p_{u,\rm ik}(t \rightarrow \infty ) \rightarrow \frac{t}{\rm RS}, \quad \forall i \in N_{\rm inj}, k \in N. |
where \rm RS = \int_V c_t \, \phi \, dV is total reservoir storage connecting all the wells.
Moreover, if each well is placed in different reservoir volumes which are only connected through wellbores then again they will all drain the same volume which is the sum of all connected volumes through the wellbores and all UTRs will again trend to the same LTR asymptotic.
In order to relate true UTRs (from numerical grid simulations or from deconvolution) to the CRM injection share constants f_{ik} one needs to implement a certain workflow:
- Start with true UTRs \displaystyle p_{u, ik}(t) with the same LTR asymptotic \displaystyle p_{u, ik}(t) \rightarrow \frac{t}{RS}.
- Select injector W0
- Select producer W1
- Perform two convolution tests to account for the impact from {W2 .. WN} production on to DTR_11
p_{u, 11}(t) and CTR_01
p_{u, 01}(t):
- Test #1 – DTR_11
- Calculate historically-averaged rate for each producer: \displaystyle q^*_k = \frac{1}{N_k} \sum_{m=1}^{N_k} q_k(t_m)
- Calculate interfering DTR_11: \displaystyle p^*_{u, 11}(t) = p_{u, 11}(t) + \sum_{k \neq 1} p_{u, k1}(t) \cdot q^*_k, meaning that injector W0 is shut-down and all producers are working with constant rates q^*_k, except producer W1 which is working with unit-rate
- Test #2 – CTR_01
- Calculate historically-averaged rate for each producer: \displaystyle q^*_k = \frac{1}{N_k} \sum_{m=1}^{N_k} q_k(t_m)
- Calculate interfering CTR_01: \displaystyle p^*_{u, 01}(t) = p_{u, 01}(t) + \sum_{k} p_{u, k1}(t) \cdot q^*_k , meaning that injector W0 is working with unit-rate and all producers are working with constant rates q^*_k
- Test #1 – DTR_11
- Calculate injection share constant: \displaystyle f_{01} = \frac{\dot p^*_{01}(t)}{\dot p^*_{11}(t)} \Bigg|_{t \rightarrow \infty} as LLS over equation: \displaystyle \dot p^*_{01}(t) = f_{01} \cdot \dot p^*_{11}(t)
- Perform two convolution tests to account for the impact from {W2 .. WN} production on to DTR_11
p_{u, 11}(t) and CTR_01
p_{u, 01}(t):
- Repeat the same for other producers (starting from point 2a onwards)
- Select producer W1
- Repeat the same for other injectors (starting from point 2 onwards)
Again it is important to note a difference between
- CRM assumptions (constant PI, constant drainage volumes with no flow boundaries and constant total compressibility) – which may or may not take place and hence may or may not make CRM applicable in a specific case
and
- CRM concept of mismatching drainage volumes between producers and injectors which is just a terminology and does not exert restrictions on well-reservoir system
See also
[UTR] [ DTR ] [ CTR ] [ Capacitance Resistance Model (CRM) ]