Natural Depletion

Oil Depletion

The EUR during the natural oil depletion can be assessed with the following formula:

(1)	$\begin{array}{l}\displaystyle EUR_{ND} = \frac{Q_o}{V_o} = \frac{ (p_i - p_{wf}) \, c_t}{(1-s_{wi})\, B_o} = \frac{ (p_i - p_{wf}) }{(1-s_{wi})\, B_o} \, \big( c_r + s_{wi} c_w + (1-s_{wi})c_o \big)\end{array}$

where $\begin{array}{l}p_{wf}\end{array}$ is flowing bottom-hole pressure, $\begin{array}{l}p_i\end{array}$ – initial formation pressure, $\begin{array}{l}B_o\end{array}$ – formation volume factor for oil, $\begin{array}{l}Q_o\end{array}$ – cumulative oil production, $\begin{array}{l}V_o\end{array}$ – STOIIP, $\begin{array}{l}s_{wi}\end{array}$ – initial water saturation in oil pay.

EUR Deduction

The total compressibility of oil saturated formation

(2)	$\begin{array}{l}\displaystyle c_t = \frac{1}{V_{\phi}} \frac{\partial V_{\phi}}{\partial p} = c_r + s_{wi} c_w + (1-s_{wi})c_o\end{array}$

and can be split into rock, water, oil components:

(3)	$\begin{array}{l}\displaystyle c_t = c_r + s_{wi} c_w + (1-s_{wi})c_o\end{array}$

For low compressible oil compressibility can be assumed constant $\begin{array}{l}c_t = \rm const\end{array}$ and the volume reduction can be related to pressure decline as:

(4)	$\begin{array}{l}\displaystyle \frac{\delta V_\phi}{V_\phi} = c_t \, \delta p = c_t \, (p_i - p_{wf \, min})\end{array}$

(5)	$\begin{array}{l}\displaystyle \delta V_\phi = Q_o \, B_o\end{array}$

and

(6)	$\begin{array}{l}\displaystyle V_o = s_o \, V_\phi = (1-s_{wi}) \, V_\phi\end{array}$

hence

(7)	$\begin{array}{l}\displaystyle \frac{Q_o \, B_o \, (1-s_{wi})}{V_o} = c_t \, (p_i - p_{wf \, min})\end{array}$

and

(8)	$\begin{array}{l}\displaystyle EUR = \frac{Q_o}{V_o} = \frac{ (p_i - p_{wf \, min}) \, c_t}{(1-s_{wi})\, B_o}\end{array}$

For the naturally flowing wells the production bottom hole pressure can be assessed as:

(9)	$\begin{array}{l}\displaystyle p_{wf} = p_s + \rho_g \, g\, h + \bigg( 1- \frac{\rho_g}{\rho_o} \bigg) \, p_b\end{array}$

where $\begin{array}{l}p_s\end{array}$ – tubing-head pressure defind by the production athering system, $\begin{array}{l}h\end{array}$ – is the true vertical deoth at formation top, $\begin{array}{l}\{ \rho_o, \, \rho_g \}\end{array}$ – oil and gas densities, $\begin{array}{l}p_b\end{array}$ – bubble-point pressure.

Gas Depletion

The Expected Ultimate Recovery during the natural gas depletion can be assessed with the following formula:

(10)	$\begin{array}{l}\displaystyle EUR_{GD} = \frac{Q_g}{V_g} = 1- \frac{p_{wf}}{p_i}\end{array}$

Water flooding

Motivation = maintain formation pressure at sweep interface

The Expected Ultimate Recovery during the waterflood sweep can be assessed with the following formula:

(11)	$\begin{array}{l}\displaystyle EUR_{WF} = E_S \, E_D + (1-E_S) EUR_{ND}\end{array}$

where $\begin{array}{l}E_D\end{array}$ – displacement efficiency, $\begin{array}{l}E_S\end{array}$ – sweep efficiency.

Sweep efficiency

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Water-Oil displacement efficiency

Displacement efficiency defines the fraction of the initial pore-saturating fluid displaced in a the piston sweep by invaded fluid

$\begin{array}{l}\displaystyle E_D = 1- \frac{V_{FLUID, \ INITITAL} - V_{FLUID, \ INVADED}}{V_{FLUID, \ INITITAL}}\end{array}$

Gas flooding

Motivation = maintain formation pressure at sweep interface with gas in case of high water mobility $\begin{array}{l}\frac{k_{rw}}{\mu_w} \gg \frac{k_{ro}}{\mu_o}\end{array}$ which makes watrflood inefficient.

Gas displacement efficiency

(12)	$\begin{array}{l}\displaystyle E_D = \frac{1-s_{wi}-s_{org}}{1-s_{wi}}\end{array}$

where $\begin{array}{l}s_{wi}\end{array}$ – inititial water in oil pay, $\begin{array}{l}s_{org}\end{array}$ – residual oil to gas sweep.

WAG flooding

Motivation = maintain formation pressure at sweep interface with alternating inejction of water and gas in case of high residual oil to water sweep is high $\begin{array}{l}s_{orw}\end{array}$ and gas sweep is less than to water sweep $\begin{array}{l}s_{org} < s_{orw}\end{array}$ .

(13)	$\begin{array}{l}\displaystyle E_D = \frac{1-s_{wi}-s_{org}}{1-s_{wi}}\end{array}$

Chemical EOR

Motivation = maintain formation pressure at sweep interface with chemical injection and reduce residual oil to EOR sweep $\begin{array}{l}s_{or \, eor} < s_{orw}\end{array}$ .

(14)	$\begin{array}{l}\displaystyle E_D = \frac{1-s_{wi}-s_{ori}}{1-s_{wi}}\end{array}$

where $\begin{array}{l}s_{wi}\end{array}$ – inititial water in oil pay, $\begin{array}{l}s_{ori}\end{array}$ – residual oil to injection sweep.

CО2 injection

Reference

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