Mathematical model of dynamic wellbore storage effects is based on the idea that if surface rate changes  $//<![CDATA[ \begin{array}{l}\delta q\end{array} //]]>$ at certain moment then it will take some time before the pressure disturbance reach the bottomhole and  induce sandface flow variance  $//<![CDATA[ \begin{array}{l}\delta q_t\end{array} //]]>$ :

where  

In stationary conditions the surface fluid volumes  $//<![CDATA[ \begin{array}{l}q\end{array} //]]>$ and sandface volumes  $//<![CDATA[ \begin{array}{l}q_{sf}\end{array} //]]>$  are related through formation volume factor $//<![CDATA[ \begin{array}{l}B\end{array} //]]>$ :

For mulit=phase case the 

Same happens if the well is shut-in at surface.

The formula  (1) has very generic view and simply states that in the moment of well opening the  surface flow will be zero as  $//<![CDATA[ \begin{array}{l}qB = C_S \, \frac{dp_{wf}}{dt}\end{array} //]]>$ while in time the pressure diffusion is getting stabilised and pressure variation vanishes leading to equalization of the surface and sandface flows.

The actual form of the function    $//<![CDATA[ \begin{array}{l}C_S(p_{wf})\end{array} //]]>$  depends on the particular physics of fluid flow inertial effect and few of them are explained below.

Wellbore storage from fluid compressibility

The simplest case is when borehole is filled with fluid at all times which makes calculation of wellbore storage easy: 

where  $//<![CDATA[ \begin{array}{l}c\end{array} //]]>$ –  fluid compressibility,  $//<![CDATA[ \begin{array}{l}V_{wb}\end{array} //]]>$ – wellbore volume available for flow.

This normally happens for water injectors and gas wells (producers or injectors) at high formation pressure.

In case of water injector the fluid compressibility is constant  $//<![CDATA[ \begin{array}{l}c(p) = \rm const\end{array} //]]>$ at all pressures and if well has no integrity issues the wellbore volume  $//<![CDATA[ \begin{array}{l}V_{wb}\end{array} //]]>$ will remain constant in time leading to a constant wellbore storage $//<![CDATA[ \begin{array}{l}C_S = \rm const\end{array} //]]>$.

Wellbore storage from varying fluid level

In case of oil producers the dynamic fluid level is always below surface and shutting the well down will cause after flow from formation and fluid level rise at constant pace with the flowing wellbore storage:

where $//<![CDATA[ \begin{array}{l}\rho\end{array} //]]>$ –  fluid density,  $//<![CDATA[ \begin{array}{l}A\end{array} //]]>$ – wellbore cross-sectional area available for flow.

Varying wellbore storage