OIL

(1)	$\begin{array}{l}\displaystyle k_{row}(s_o) = k_{rowc} \cdot \bigg[ \frac{ s_o - s_{orw} }{ 1- s_{wi} - s_{orw} } \bigg]^{n_{ow}}\end{array}$

(2)	$\begin{array}{l}\displaystyle k_{rwo}(s_w) = k_{rwoc} \cdot \bigg[ \frac{ s_w - s_{wco} }{ 1- s_{wco} - s_{orw} } \bigg]^{n_{wo}}\end{array}$

$\begin{array}{l}s_o\end{array}$

current oil saturation, $\begin{array}{l}s_o = 1 - s_w\end{array}$

$\begin{array}{l}s_w\end{array}$

current water saturation, $\begin{array}{l}s_w = 1 - s_o\end{array}$

$\begin{array}{l}s_{orw}\end{array}$

residual oil saturation to water displacement, below which oil is immobile

$\begin{array}{l}s_{wсo}\end{array}$

critical water saturation to oil displacement, below which water is immobile

$\begin{array}{l}k_{rowc}\end{array}$

maximum oil relative permeability to water displacement

$\begin{array}{l}k_{rwoc}\end{array}$

maximum water relative permeability to oil displacement

$\begin{array}{l}n_{ow}\end{array}$

oil relative permeability curvature to water displacement

$\begin{array}{l}n_{wo}\end{array}$

water relative permeability curvature to oil displacement

where $\begin{array}{l}s_{wi}\end{array}$ is initial water saturation which maybe equal to critical $\begin{array}{l}s_{wсo}\end{array}$ or less than critical $\begin{array}{l}s_{wi} < s_{wco}\end{array}$ like for example in petroleogenetic rocks.

The model assumes no gas presence in pores.

Fig. 1. Typical Oil+Water RPM Corey @model

The alternative form of the Oil+Water RPM Corey @model can be presented as a function of normalized water saturation $\begin{array}{l}s\end{array}$ :

(3)	$\begin{array}{l}\displaystyle s = \frac{s_w - s_{wi}}{1-s_{wi}-s_{orw}}\end{array}$

which changes between $\begin{array}{l}s = 0\end{array}$ for initial water saturation $\begin{array}{l}s_w = s_{wi}\end{array}$ and $\begin{array}{l}s = 1\end{array}$ for maximum water saturation $\begin{array}{l}s_w = 1- s_{orw}\end{array}$ .

In this case equations (1) and (2) take form:

OIL

WATER

(4)	$\begin{array}{l}\displaystyle k_{row}(s_o) = k_{rowc} \cdot (1-s)^{n_{ow}}\end{array}$

(5)	$\begin{array}{l}\displaystyle k_{rwo}(s_w) = k^_{rwoc} \cdot (s - s^)^{n_{wo}}\end{array}$

(6)	$\begin{array}{l}\displaystyle s^* = \frac{s_{wco}-s_{wi}}{1-s_{wi}-s_{orw}}\end{array}$

(7)	$\begin{array}{l}\displaystyle k^*_{rwoc} = k_{rwoc} \cdot \left( \frac{1-s_{wi}-s_{orw}}{1-s_{wco}-s_{orw}} \right)^{n_{wo}}\end{array}$

and fractional flow function is going to be:

(8)	$\begin{array}{l}\displaystyle f_w = \frac{M_{rwo}}{M_{rwo} + M_{row}} = \frac{(s-s^)^{n_{wo}}}{(s-s^)^{n_{wo}} + g \cdot (1-s)^{n_{ow}}}\end{array}$

(9)	$\begin{array}{l}\displaystyle \dot f_w = \frac{d f_w}{ds} = g \cdot (s-s^)^{n_{wo}-1} \cdot \frac{ n_{wo} (1-s)^{n_{ow}} + n_{ow} (s-s^) (1-s)^{n_{ow}-1}} {\left[ (s-s^*)^{n_{wo}} + g \cdot (1-s)^{n_{ow}} \right]^2}\end{array}$

where

(10)	$\begin{array}{l}\displaystyle g = \frac{M_{rowc}}{M_{rwoc}} \cdot \left( \frac{1-s_{wco}-s_{orw}}{1-s_{wi}-s_{orw}} \right)^{n_{wo}}\end{array}$

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See also