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:

q= \sum_k q_k \cdot \cos \left(\frac{2 \pi \, k \, t}{T} \right)


p = \sum_k p_k \cdot \cos \left(\frac{2 \pi \, k \, t}{T}  + \delta_k \right)

where





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

k-th harmonic amplitude of flowrate variation


p_k = \frac{q_k}{\sigma} ...



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


\delta_k = \frac{\pi}{8} + \frac{L}{\sqrt{\chi \, T}}



phase shift caused by pressure response delay to the flowrate variation


\sigma = \left< \frac{k}{\mu} \right> h



formation transmissbility


\chi = \left< \frac{k}{\mu} \right> \frac{1}{c_t \, \phi}



formation pressure diffusivity