Periodic Pressure Pulsations

In case of 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:

(1)	$\begin{array}{l}\displaystyle q= \sum_k q_k \cdot \cos \left(\frac{2 \pi \, k \, t}{T} \right)\end{array}$

(2)	$\begin{array}{l}\displaystyle p = \sum_k p_k \cdot \cos \left(\frac{2 \pi \, k \, t}{T} + \delta_k \right)\end{array}$

where

$\begin{array}{l}L\end{array}$

distance between the pint of flow variation and point of pressure response, being:

well radius $\begin{array}{l}L=r_w\end{array}$ for Self-Pulse Test
distance between generating and receiving well $\begin{array}{l}L= \sqrt{ \left({\bf r}_{\rm Generator} - {\bf r}_{\rm Receiver} \right ) ^2}\end{array}$ for Pressure Pulse Interference Test

$\begin{array}{l}q_k\end{array}$

k-th harmonic amplitude of flowrate variation

(3)	$\begin{array}{l}\displaystyle p_k = \frac{q_k}{\sigma} ...\end{array}$

k-th harmonic amplitude of pressure response to the flowrate variation

(4)	$\begin{array}{l}\displaystyle \delta_k = \frac{\pi}{8} + \frac{L}{\sqrt{\chi \, T}}\end{array}$

phase shift caused by pressure response delay to the flowrate variation

(5)	$\begin{array}{l}\displaystyle \sigma = \left< \frac{k}{\mu} \right> h\end{array}$

formation transmissbility

(6)	$\begin{array}{l}\displaystyle \chi = \left< \frac{k}{\mu} \right> \frac{1}{c_t \, \phi}\end{array}$

formation pressure diffusivity

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Periodic Pressure Pulsations