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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 21}(t-\tau) dq_2(\tau) = \rm const |
The time derivative is going to be zero as the bottom-hole pressurethe 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 21}(t-\tau) dq_2(\tau) \right)^{\cdot} = 0 |
LaTeX Math Block |
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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 21}(0) \cdot q_2(t) - \int_0^t \dot p_{u,\rm 21}(t-\tau) dq_2(\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 21}(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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alignment | left |
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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 W2 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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| and LaTeX Math Inline |
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body | dq_2(\tau) = \delta q_2 \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 21}(t-\tau) \delta q_2 \cdot \delta(\tau) \, d\tau |
LaTeX Math Block |
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anchor | Case2_PSS_p11_temp |
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alignment | left |
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| \dot p_{u,\rm 11}(t) \delta q_1 = - \dot p_{u,\rm 21}(t) \delta q_2 |
which leads to LaTeX Math Block Reference |
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| . |
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LaTeX Math Block |
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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 |
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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:
LaTeX Math Block |
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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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LaTeX Math Block |
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anchor | Case2_PSS_p21 |
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alignment | left |
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| p_{u,\rm 21}(t \rightarrow \infty) \rightarrow \frac{t}{c_{t,2} V_{\phi,2}} |
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where | around producer W1 | c | t | 2} which is jointly drained by producer W1 and injector W2 average drain-area total compressibility of formation around injector W2 |
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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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| . |
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In this case, injector W2 supports only one producer W1 then
LaTeX Math Inline |
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body | V_{\phi, 2} = V_{\phi, 1} |
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and LaTeX Math Inline |
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body | \delta q_1 = \delta q_2 |
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, which means that producer W1 with constant BHP and finite-reservoir volume will eventually vary its rate at the same volume as injector W2.
If pressure in producer W1 is supported by several injectors
then over a long period of time one can assume:
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