(see SPT survey)

When flow rate is being intentionally varied in harmonic cycles with sandface amplitude  and cycling frequency :

q(t) = q_0 \, \sin ( \omega  \, t )

then bottom-hole pressure will follow the same variation pattern:

p_{wf}(t) = p_0 \, \sin ( \omega \, [ t - t_{\Delta} ] )

with a bottom-hole pressure amplitude  and the time delay . 

It takes some time (3-5 cycles ) for pressure to develop a stabilized response to rate variations.

The pressure delay   and associated dimensionless phase shift  represent the inertia effects from the adjoined reservoir and characterized by formation pressure diffusivity:

\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  is related to formation transmissibility:

\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  to formation properties :

X =r_w \, \sqrt{ \frac{\omega}{\chi} }

\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 } } 

\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  and dimensionless radius  are found from  –  and then  is substituted to  to calculate transmissibility .

In case of a low frequency pulsations:

\omega \ll 0.00225 \, \frac{ \chi }{ r_w^2} \quad \Longleftrightarrow \quad X \ll 0.15

the equations  –  can be explicitly resolved in terms of transmissibility and diffusivity:

\chi = 0.25 \, \omega \, \gamma^2 \, r_w^2 \, \exp \frac{\pi}{2 \, {\rm tg} \Delta }

\sigma = \frac{q_0}{8 \, p_0 \, \sin \Delta}

where .

The above analytical approach (either  –  or   – ) 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:

q(t) = q_0 + \sum_{n=0}^\infty q_n \, \sin ( \omega_n  \, t )

and the pressure response becomes complex as well: 

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 best fit to a complex system of non-linear algebraic equations:

X_n =r_w \, \sqrt{ \frac{\omega_n}{\chi} }

\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 } } 

\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  and  play important academic role and provide fast track estimations in SPT engineering.