In case of harmonic periodic pulsations and sufficiently long pressure-rate delay and a simple diffusion model (single-bed homogeneous reservoir without boundary) the pressure pulse response can be approximated by analytical model:
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q= \sum_k q_1k \cdot \cos \left(\frac{2 \pi \, k \, t}{T} \right) |
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p = \sum_k p_1k \cdot \cos \left(\frac{2 \pi \, k \, t}{T} + \delta_k \right) |
where
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distance between the pint of flow variation and point of pressure response, being:
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| k-th 1st harmonic amplitude of flowrate variation |
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| p_1k = \frac{q_0k}{\sigma} ... |
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1st k-th harmonic amplitude of pressure response to the flowrate variation |
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| \delta_k = \frac{\pi}{8} + \frac{L}{\sqrt{\chi \, T}} |
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phase shift caused by pressure response delay to the flowrate variation
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| \sigma = \left< \frac{k}{\mu} \right> h |
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formation transmissbility |
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| \chi = \left< \frac{k}{\mu} \right> \frac{1}{c_t \, \phi} |
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formation pressure diffusivity |
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