Mathematical model of dynamic wellbore storage effects is based on the idea that if surface rate changes \delta q_s at certain moment then it will take some time before the pressure disturbance reach the bottomhole and induce sandface flow variance \delta q_t :
| (1) | \delta q_s = \delta q_t - C_S \, \frac{dp_{wf}}{dt} |
where
q_s | surface flow rate |
q_t | total water, oil, gas sandface flowrate |
p_{wf} | bottom-hole pressure |
C_S | constant value called wellbore storage (WBS) |
In stationary conditions the surface fluid volumes q_s and sandface volumes q_t are related through formation volume factor B :
| \delta q_t = \delta q_s |
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 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 q with the following wellbore storage:
| (2) | C_S = \frac{A}{\rho \, g} |
where \rho – fluid density, A – wellbore cross-sectional area available for flow, g – standard gravity.