Consider a well-reservoir system consisting of:
- producing well W1 draining the reservoir volume
- water injecting well W2 supporting pressure in reservoir volume which includes the drainage volume of producer W1 and potentially other producers.
The drainage volume difference
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| body | \delta V_{\phi} = V_{\phi, 2} - V_{\phi, 1} >0 |
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may be related to the fact that water injection
W2 is shared between
and another reservoir or with another producer.
Case #1 – Constant flowrate production
The pressure response
in producer
W1 to the flowrate variation
in injector
W2:
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\delta p_1 = - p_{u,\rm 21}(t) \cdot \delta q_2 |
where
Case #2 – Constant BHP
The flowrate response
in producer
W1 to the flowrate variation
in injector
W2:
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\delta q_1 = - \frac{p_{u,\rm 21}(t)}{p_{u,\rm 11}(t)} \cdot \delta q_2 |
where
| Expand |
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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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| anchor | Case2_PSS_p11 |
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| alignment | left |
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| p_1(t) = p_i - \int_0^t p_{u,\rm 11}(t) dq_1 - \int_0^t p_{u,\rm 21}(t) dq_2 = \rm const |
The time derivative is going to be | LaTeX Math Block |
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| anchor | Case2_PSS_p11 |
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| alignment | left |
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| \dot p_1(t) = - \left( \int_0^t p_{u,\rm 11}(t) dq_1 \right)^{\cdot} - \left( \int_0^t p_{u,\rm 21}(t) dq_2 \right)^{\cdot} = 0 |
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For the finite-volume drain
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| body | V_{\phi,1} \leq V_{\phi,2} < \infty |
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the flowrate response factor
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| body | \delta q_1 / \delta q_2 |
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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_2 = f_{21} = \frac{c_{t,2} V_{\phi, 2}}{c_{t,1} V_{\phi, 1}} = \rm const |
which makes one of the key assumptions in Capacitance Resistance Model (CRM).
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For the finite-volume reservoir | LaTeX Math Inline |
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| body | V_{\phi,1} \leq V_{\phi,2} < \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,1} V_{\phi, 1}} |
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| anchor | Case2_PSS_p21 |
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| alignment | left |
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| p_{u,\rm 21}(t \rigtharrow \infty) \rightarrow \frac{t}{c_{t,2} 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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one arrives to | LaTeX Math Block Reference |
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