Mathematical model of dynamic wellbore storage effects is based on the idea that if surface rate changes  at certain moment then it will take some time before the pressure disturbance reach the bottomhole and  induce sandface flow variance  :

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

where  

In stationary conditions the surface fluid volumes  and sandface volumes   are related through formation volume factor  for Simple PVT case:

< \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 charcaterized by linear dependence on time:

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

The formula  is empirical and has very generic view simply stating that in the moment of well opening there will be a difference between surface and subsurface flow which is proportional to time derivative of pressure and hence will vanish when pressure stabilises.

The actual form of the function     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: 

C_S = c \, V_{wb}

where  –  fluid compressibility,  – 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  at all pressures and if well has no integrity issues the wellbore volume  will remain constant in time leading to a constant wellbore storage .

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  with the following wellbore storage:

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

where –  fluid density,  – wellbore cross-sectional area available for flow,  – standard gravity.

\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

Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model