In case the whole wellbore volume is filled with fluid at the moment of opening or closing the well at surface the wellbore fluid compressibility is going to be:

c = \frac{1}{V_{wb}} \frac{\delta V_{wb}}{\delta p} 

In the very first moments the surface fluid will only compress (for injectors) or decompress (for producers) the wellbore fluid column without communication with subsurface formation thus leading to the following correlation:

\frac{dp}{dt} = \frac{1}{c \, V_{wb}} \frac{dV}{dt} =\frac{1}{c \, V_{wb}} \frac{dV}{dt}

or

\delta q_s = \delta q_t - c \, V_{wb} \, \frac{dp_{wf}}{dt}

providing that

Comparing

The true vertical pressure difference between two points of a rising fluid column is:

\Delta p = \rho g \Delta h = \frac{\rho g}{A} \Delta V

The pressure build up then:

\frac{dp}{dt} = \frac{\rho g}{A} \frac{dV}{dt} = \frac{\rho g}{A} q

or

\delta q_s = \delta q_t - \frac{A}{\rho g} \, \frac{dp_{wf}}{dt}

providing that

Comparing