(see SPT survey)
When flow rate is being intentionally varied in harmonic cycles with sandface amplitude q_0 and cycling frequency \omega = \frac{2 \, \pi}{T}:
| (1) | q(t) = q_0 \, \sin ( \omega \, t ) |
then bottom-hole pressure will follow the same variation pattern:
| (2) | p_{wf}(t) = p_0 \, \sin ( \omega \, [ t - t_{\Delta} ] ) |
with a bottom-hole pressure amplitude p_0 and the time delay t_{\Delta}.
It takes some time (3-5 cycles t \geq 3 \, T) for pressure to develop a stabilized response to rate variations.
The pressure delay t_{\Delta} and associated dimensionless phase shift \Delta = \omega \, t_{\Delta} represent the inertia effects from the adjoined reservoir and characterized by formation pressure diffusivity:
| (3) | \chi = \Big < \frac{k}{\mu} \Big > \frac{1}{\phi \, c_t} |
The diffusion nature of pressure dictates that amplitude of pressure variation is proportional to amplitude of sandface flowerate variation and the ratio \frac{p_0}{q_0} is related to formation transmissibility:
| (4) | \sigma = \Big < \frac{k}{\mu} \Big > \, h |
The exact solution of differential diffusion equation for vertical well with negligible well storage and infinite boundary homogeneous reservoir can be represented by a system of non-linear algebraic equations, relating field-measured parameters \big\{ \frac{q_0}{p_0}, \, t_{\Delta} \big\} to formation properties \{ \sigma, \, \chi \}:
| (5) | X =r_w \, \sqrt{ \frac{\omega}{\chi} } |
| (6) | \Delta = \omega \, t_{\Delta} = \frac{\pi}{4} - arctg{ \frac{Ker_1 X \cdot Kei \, X - Ker_1 X \cdot Kei \, X }{Ker_1 X \cdot Kei \, X +Ker_1 X \cdot Kei \, X } } |
| (7) | \sigma =\frac{1}{2 \pi} \, \frac{q_0}{p_0} \, \sqrt{ \frac{Ker^2 X + Kei^2 X}{Ker_1^2 X + Kei_1^2 X} } |
The above equations assume that diffusivity \chi and dimensionless radius X are found from (5) – (6) and then X is substituted to (7) to calculate transmissibility \sigma.
In case of a low frequency pulsations:
| (8) | \omega \ll 0.00225 \, \frac{ \chi }{ r_w^2} \quad \Longleftrightarrow \quad X \ll 0.15 |
the equations (5) – (7) can be explicitly resolved in terms of transmissibility and diffusivity:
| (9) | \chi = 0.25 \, \omega \, \gamma^2 \, r_w^2 \, \exp \frac{\pi}{2 \, {\rm tg} \Delta } |
| (10) | \sigma = \frac{q_0}{8 \, p_0 \, \sin \Delta} |
where \Delta = \omega \, t_{\Delta}.
The above analytical approach (either (5) – (7) or (9) – (10)) is rarely helpful in practise.
The field operations are very finnicky and difficult to follow the pre-desgined sequence of clean harmonic pulsations.
As result, the flowrate variation becomes a complex sum of harmonics:
| (11) | q(t) = q_0 + \sum_{n=0}^\infty q_n \, \sin ( \omega_n \, t ) |
and the pressure response becomes complex as well:
| (12) | p_{wf}(t) = p_0 + \sum_{n=0}^\infty p_n \, \sin ( \omega_n \, [ t - t_{\Delta_n} ] ) |
The use of analytical formulas requires fourier transformation to identify the key harmonics from the raw data with a manual control from analyst and a certain amount of subjectivism on which harmonics to pick up for calculating the transmissibility and diffusivity.
Once the harmonics are identified one need to search for the \{ \sigma, \, \chi \} best fit to a complex system of non-linear algebraic equations:
| (13) | X_n =r_w \, \sqrt{ \frac{\omega_n}{\chi} } |
| (14) | \Delta_n = \omega_n \, t_{\Delta} = \frac{\pi}{4} - arctg{ \frac{Ker_1 X_n \cdot Kei \, X_n - Ker_1 X_n \cdot Kei \, X_n }{Ker_1 X_n \cdot Kei \, X_n +Ker_1 X_n \cdot Kei \, X_n } } |
| (15) | \sigma =\frac{1}{2 \pi} \, \frac{q_n}{p_n} \, \sqrt{ \frac{Ker^2 X_n + Kei^2 X_n}{Ker_1^2 X_n + Kei_1^2 X_n} } |
In practice, the most efficient methodology to interpret the SPT data is via fitting numerical model to the raw pressure-rate data.
Still, formulas (9) and (10) play important academic role and provide fast track estimations in SPT engineering.