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

(1)	$\begin{array}{l}\displaystyle < \delta q_s \, B > = \delta q_t + C_S \, \frac{dp_{wf}}{dt}\end{array}$

where

$\begin{array}{l}q_s\end{array}$	surface flow rate
$\begin{array}{l}< \delta q_s \, B >\end{array}$	allocation of surface rate to the sandface conditions $\begin{array}{l}< \delta q_s \, B > = B_w \, \delta q_W + (B_o - R_s \, B_g) \, \delta q_O + (B_g - R_v \, B_o) \, \delta q_G\end{array}$
$\begin{array}{l}q_t\end{array}$	total water, oil, gas sandface flowrate
$\begin{array}{l}p_{wf}\end{array}$	bottom-hole pressure
$\begin{array}{l}C_S\end{array}$	constant value called wellbore storage (WBS)

In stationary conditions the surface fluid volumes $\begin{array}{l}q_s\end{array}$ and sandface volumes $\begin{array}{l}q_t\end{array}$ are related through formation volume factor $\begin{array}{l}B\end{array}$ for Simple PVT case:

(2)	$\begin{array}{l}\displaystyle < \delta q_s \, B > = \delta q_t\end{array}$

or

(3)	$\begin{array}{l}\displaystyle < \delta q_s \, B > = B_w \, \delta q_W + (B_o - R_s \, B_g) \, \delta q_O + (B_g - R_v \, B_o) \, \delta q_G = \delta q_t\end{array}$

for multi-phase fluid production (Non-linear multi-phase pressure diffusion @model:3).

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

(4)	$\begin{array}{l}\displaystyle p_{wf}(t) = p_{wf}(0) - \frac{q_s}{C_S} \, t\end{array}$

The formula (1) 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 $\begin{array}{l}C_S(p_{wf})\end{array}$ 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:

(5)	$\begin{array}{l}\displaystyle C_S = c \, V_{wb}\end{array}$

where $\begin{array}{l}c\end{array}$ – fluid compressibility, $\begin{array}{l}V_{wb}\end{array}$ – 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 $\begin{array}{l}c(p) = \rm const\end{array}$ at all pressures and if well has no integrity issues the wellbore volume $\begin{array}{l}V_{wb}\end{array}$ will remain constant in time leading to a constant wellbore storage $\begin{array}{l}C_S = \rm const\end{array}$ .

Derivation

In case the whole wellbore volume is filled with fluid at the moment of opening or closing the well at surface the wellbore fluid compressibility is going to be:

(6)	$\begin{array}{l}\displaystyle c = \frac{1}{V_{wb}} \frac{\delta V_{wb}}{\delta p}\end{array}$

In the very first moments the surface fluid will only compress (for injectors) or decompress (for producers) the wellbore fluid column without communication with subsurface formation thus leading to the following correlation:

(7)	$\begin{array}{l}\displaystyle \frac{dp}{dt} = \frac{1}{c \, V_{wb}} \frac{dV}{dt} =\frac{1}{c \, V_{wb}} \frac{dV}{dt}\end{array}$

or

(8)	$\begin{array}{l}\displaystyle \delta q_s = \delta q_t - c \, V_{wb} \, \frac{dp_{wf}}{dt}\end{array}$

providing that $\begin{array}{l}q_s = 0\end{array}$ until fluid column has reached a surface the surface .

Comparing (8) to (1) one arrives to (5)

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 $\begin{array}{l}q\end{array}$ with the following wellbore storage:

(9)	$\begin{array}{l}\displaystyle C_S = \frac{A}{\rho \, g}\end{array}$

where $\begin{array}{l}\rho\end{array}$ – fluid density, $\begin{array}{l}A\end{array}$ – wellbore cross-sectional area available for flow, $\begin{array}{l}g\end{array}$ – standard gravity.

Derivation

The true vertical pressure difference between two points of a rising fluid column is:

(10)	$\begin{array}{l}\displaystyle \Delta p = \rho g \Delta h = \frac{\rho g}{A} \Delta V\end{array}$

The pressure build up then:

(11)	$\begin{array}{l}\displaystyle \frac{dp}{dt} = \frac{\rho g}{A} \frac{dV}{dt} = \frac{\rho g}{A} q\end{array}$

or

(12)	$\begin{array}{l}\displaystyle \delta q_s = \delta q_t - \frac{A}{\rho g} \, \frac{dp_{wf}}{dt}\end{array}$

providing that $\begin{array}{l}q_s = 0\end{array}$ until fluid column has reached a surface the surface .

Comparing (12) to (1) one arrives to (9)

Page tree

Wellbore storage from fluid compressibility

Wellbore storage from varying fluid level

Varying wellbore storage

See Also

Page tree

Pressure Diffusion Wellbore Storage @model

Wellbore storage from fluid compressibility

Wellbore storage from varying fluid level

Varying wellbore storage

See Also