Consider a well-reservoir system (Fig. 1) consisting of:
The injection drainage volume
includes the drainage volume
of producer
W1 and may be equal to it
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body | V_{\phi, 0} = V_{\phi, 1} |
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or may be bigger
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body | V_{\phi, 0} > V_{\phi, 1} |
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in case injector
W0 supports
other producers {W1 .. WN}: LaTeX Math Inline |
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body | V_{\phi, 0} = \sum_{k=1}^N V_{\phi, k} |
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Fig. 1. Location map of injector-producer pairing with 4 producers {W1, W2, W3, W4} and one injector W0. |
Case #1 – Constant flowrate production:
The bottom-hole pressure response
in producer
W1 to the flowrate variation
in injector
W0:
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\delta p_1 = - p_{u,\rm 21}(t) \cdot \delta q_2 |
where
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| Consider a pressure convolution equation for the BHP in producer W1 with constant flowrate production at producer W1 and varying injection rate at injector W2 : LaTeX Math Block |
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| p_1(t) = p_i - \int_0^t p_{u,\rm 11}(t-\tau) dq_1(\tau) - \int_0^t p_{u,\rm 01}(t-\tau) dq_0(\tau) = p_i - \int_0^t p_{u,\rm 01}(t-\tau) dq_0(\tau) |
Consider a step-change in injector's W0 flowrate at zero time , which can be written as: LaTeX Math Inline |
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body | dq_0 (\tau) = \delta q_0 \cdot \delta(\tau) \, d\tau |
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| .The responding pressure variation in producer W1 will be: LaTeX Math Block |
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| \delta p_1(t) = p_1(t)-p_i = - \int_0^t p_{u,\rm 21}(t-\tau) \delta q_0 \cdot \delta(\tau) \, d\tau = - p_{u,\rm 01}(t) \cdot \delta q_0 |
which leads to LaTeX Math Block Reference |
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| . |
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Case #2 – Constant BHP:
Assume that the flowrate in producer W1 is being automatically adjusted by
to compensate the
bottom-hole pressure variation
in response to the
total sandface flowrate variation
in injector
W0 so that
bottom-hole pressure in producer
W1 stays constant at all times
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body | \delta p_1(t) = \delta p_1 = \rm const |
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. 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
in producer
W1 to the flowrate variation
in injector
W0 is going to be:
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\delta q_1(t) = - \frac{\dot p_{u,\rm 01}(t)}{\dot p_{u,\rm 11}(t)} \cdot \delta q_0 |
where
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| Consider a pressure convolution equation for the above 2-wells system with constant BHP: LaTeX Math Block |
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| p_1(t) = p_i - \int_0^t p_{u,\rm 11}(t-\tau) dq_1(\tau) - \int_0^t p_{u,\rm 01}(t-\tau) dq_0(\tau) = \rm const |
The time derivative is going to be zero as the BHP in producer W1 stays constant at all times: LaTeX Math Block |
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| \dot p_1(t) = - \left( \int_0^t p_{u,\rm 11}(t-\tau) dq_1(\tau) \right)^{\cdot} - \left( \int_0^t p_{u,\rm 01}(t-\tau) dq_0(\tau) \right)^{\cdot} = 0 |
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| p_{u,\rm 11}(0) \cdot q_1(t) + \int_0^t \dot p_{u,\rm 11}(t-\tau) dq_1(\tau) = - p_{u,\rm 01}(0) \cdot q_0(t) - \int_0^t \dot p_{u,\rm 01}(t-\tau) dq_0(\tau) |
The zero-time value of DTR / CTR is zero by definition LaTeX Math Inline |
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body | p_{u,\rm 11}(0) = 0, \, p_{u,\rm 01}(0) = 0 |
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| which leads to: LaTeX Math Block |
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anchor | Case2_PSS_p11_temp |
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| \int_0^t \dot p_{u,\rm 11}(t-\tau) dq_1(\tau) = - \int_0^t \dot p_{u,\rm 21}(t-\tau) dq_2(\tau) |
Consider a step-change in producer's W1 flowrate and injector's W0 flowrate at zero time , which can be written as LaTeX Math Inline |
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body | dq_1(\tau) = \delta q_1 \cdot \delta(\tau) \, d\tau |
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| .Assume that a lift mechanism in producer automatically adjusts the flowrate to maintain the same flowing bottom-hole and LaTeX Math Inline |
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body | dq_0(\tau) = \delta q_0 \cdot \delta(\tau) \, d\tau |
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| . Substituting this to LaTeX Math Block Reference |
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| leads to: LaTeX Math Block |
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| \int_0^t \dot p_{u,\rm 11}(t-\tau) \delta q_1 \cdot \delta(\tau) \, d\tau = - \int_0^t \dot p_{u,\rm 01}(t-\tau) \delta q_0 \cdot \delta(\tau) \, d\tau |
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| \dot p_{u,\rm 11}(t) \delta q_1 = - \dot p_{u,\rm 01}(t) \delta q_0 |
which leads to LaTeX Math Block Reference |
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| . |
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For the finite-volume drain
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body | V_{\phi,1} \leq V_{\phi,0} < \infty |
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the flowrate response factor
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body | \delta q_1 / \delta q_0 |
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is getting stabilised over time as:
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anchor | Case2_PSS |
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alignment | left |
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\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
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as constant in time.
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| For the finite-volume reservoir LaTeX Math Inline |
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body | V_{\phi,1} \leq V_{\phi,0} < \infty |
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| the DTR and CTR are both going through the PSS flow regime at late transient times:
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anchor | Case2_PSS_p11 |
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alignment | left |
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| p_{u,\rm 11}(t \rightarrow \infty) \rightarrow \frac{t}{c_t V_{\phi, 1}} |
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anchor | Case2_PSS_p21 |
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alignment | left |
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| p_{u,\rm 01}(t \rightarrow \infty) \rightarrow \frac{t}{c_t V_{\phi,2}} |
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where Substituting LaTeX Math Block Reference |
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| and LaTeX Math Block Reference |
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| in LaTeX Math Block Reference |
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In case injector W0 supports only one producer W1, then both wells drain the same reservoir volume
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body | V_{\phi, 0} = V_{\phi, 1} |
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so that
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leads to:
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\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:
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\sum_{k=1}^N f_{0k} = 1 |
with constant coefficients LaTeX Math Inline |
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body | f_{0k} \geq 0, \ {k=\{i..N \} } |
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, unless there is a thief injection outside the drainage area of all producers and in this case:
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anchor | fokless1 |
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alignment | left |
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\sum_{k=1}^N f_{0k} < 1 |
If pressure around producer W1 is supported by several injectors
then one can assume:
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\delta q_1 =-\sum_k f_{i1} \delta q_i
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with constant coefficients LaTeX Math Inline |
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body | f_{i1} \geq 0, \ {k=\{i..N_{\rm inj} \} } |
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.
The equations LaTeX Math Block Reference |
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, LaTeX Math Block Reference |
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and LaTeX Math Block Reference |
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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 LaTeX Math Inline |
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body | V_{\phi, 0}> V_{\phi, 1} |
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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 all together and all UTRs will have the same LTR asymptotic:
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p_{u,\rm ik}(t \rightarrow \infty ) \rightarrow \frac{t}{\rm RS}, \quad \forall i \in N_{\rm inj}, k \in N. |
where LaTeX Math Inline |
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body | \rm RS = \int_V c_t \, \phi \, dV |
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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 the UTR from numerical grid simulations or from deconvolution to the CRM injection share constants one needs to implement a certain workflow.
- Collect true UTRs
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body | \displaystyle p_{u, ik}(t) |
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with the same LTR asymptotic LaTeX Math Inline |
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body | \displaystyle p_{u, ik}(t) \rightarrow \frac{t}{RS} |
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. - Select injector W0
- Select producer W1
- Perform two convolution tests in virtual space:
- Test #1 – DTR_11
- Calculate historically-averaged rate for each producer:
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body | \displaystyle q^*_k = \frac{1}{N_k} \sum_{m=1}^{N_k} q_k(t_m) |
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- Calculate DTR_11:
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body | \displaystyle p^*_{u, 11}(t) = p_{u, 11}(t) + \sum_{k \neq 1} p_{u, k1}(t) \cdot q^*_k |
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(meaning that injector W0 is shut-down and all producers are working with constant rates , except producer W1 which is working with unit-rate)
- Test #2 – CTR_01
- Calculate historically-averaged rate for each producer:
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body | \displaystyle q^*_k = \frac{1}{N_k} \sum_{m=1}^{N_k} q_k(t_m) |
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- Calculate CTR_01:
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body | \displaystyle p^*_{u, 01}(t) = p_{u, 01}(t) + \sum_{k} p_{u, k1}(t) \cdot q^*_k |
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(meaning that injector W0 is working with unit-rate and all producers are working with constant rates )
- Calculate injection share constant:
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body | \displaystyle f_{01} = \frac{\dot p^*_{01}(t)}{\dot p^*_{11}(t)} \Bigg|_{t \rightarrow \infty} |
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as LLS over equation: LaTeX Math Inline |
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body | \displaystyle \dot p^*_{01}(t) = f_{01} \cdot \dot p^*_{11}(t) |
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- Repeat the same for other producers (starting from point 3 onwards)
- 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) ]