In stationary conditions the surface fluid volumes q_{sf} and sandface volumes q are related through formation volume factor B :
| q_{sf} = qB |
If well is opened at surface with flow rate q then it will take some time before the pressure disturbance reach the bottom hole and force the inflow from formation with flow rate q_{sf} :
| (1) | q_{sf} = qB - C_S \, \frac{dp_{wf}}{dt} |
where C_S – is called wellbore storage.
Same happens if the well is shut 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 qB = C_S \, \frac{dp_{wf}}{dt} 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 C_S(p_{wf}) 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 c – fluid compressibility, V_{wb} – 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 c(p) = \rm const at all pressures and if well has no integrity issues the wellbore volume V_{wb} will remain constant in time leading to a constant wellbore storage C_S = \rm const.
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:
| C_S = \frac{A}{\rho} |
where \rho – fluid density, A – wellbore cross-sectional area available for flow.