Mathematical model of dynamic wellbore storage effects is based on the idea that if surface rate changes 

< \delta q_s \, B > = \delta q_t + C_S \, \frac{dp_{wf}}{dt}

where  

In stationary conditions the surface fluid volumes 

< \delta q_s \, B > = \delta q_t

or

< \delta q_s \, B > = B_w \, \delta  q_W + (B_o - R_s \, B_g) \, \delta q_O + (B_g - R_v \, B_o) \, \delta q_G = \delta q_t

for multi-phase fluid production

For constant wellbore storage the early time pressure response (ETR) build up is charcaterizsed by linera dependence on time:

p_{wf}(t) = p_{wf}(0) - \frac{q_s}{C_S}  \, t

The formula 

The actual form of the function   

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: 

C_S = c \, V_{wb}

where 

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 

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

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

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

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 

C_S = \frac{A}{\rho \, g}

where

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

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

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

Varying wellbore storage 

See Also

Petroleum Industry / Upstream / Subsurface E&P Disciplines / Well Testing / Pressure Testing / Wellbore Storage