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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| 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-\tau) dq_1(\tau) - \int_0^t p_{u,\rm 21}(t-\tau) dq_2(\tau) = p_i - \int_0^t p_{u,\rm 21}(t-\tau) dq_2(\tau) |
Consider a step-change in injector's W2 flowrate at zero time , which can be written as: | LaTeX Math Inline |
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| body | dq_2(\tau) = \delta q_2 \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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| anchor | Case2_PSS_p11 |
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| alignment | left |
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| \delta p_1(t) = p_1(t)-p_i = - \int_0^t p_{u,\rm 21}(t-\tau) \delta q_2 \cdot \delta(\tau) \, d\tau = - p_{u,\rm 21}(t) \cdot \delta q_2 |
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