# Wettability of Carbonate Reservoir Rocks:
## 2 Wettability Measurement Methods
[39] assumed an equal viscosity for oil and water (viscosity ratio equal to 1) to eliminate the influence of the viscosity ratio on water fractional flow (\(f_{w}\)) and proposed changing Equation (6) into  
\[f_{w}^{\mu_{w}=\mu_{o}}=\frac{1}{1+\frac{k_{ro}}{k_{rw}}} \tag{7}\]  
In the plot of normalized water fractional flow (based on Equation (7)) against normalized water saturation \(S_{wn}\) Equation (8), a sample with a curve below the 1:1 line is assumed to be water-wet, otherwise oil-wet. The normalized water saturation is given by  
\[S_{wn}=\frac{S_{w}-S_{w}}{1-S_{ar}-S_{mc}} \tag{8}\]  
in which \(S_{w}\) is the water saturation, and \(S_{wc}\) is the interstitial water saturation. [PERSON] [15] investigated the validity of [PERSON]'s rules of thumb and showed that while the third rule is generally unreliable, the first rule is suitable. Moreover, he showed that the second rule needed a modification. [PERSON] [15] pointed out that using 50% water saturation as a reference value in the [PERSON]'s second rule is unrealistic and defined a reference crossover saturation (RCS) as follows:  
\[RCS=\frac{1}{2}+\frac{S_{wc}-S_{ar}}{2} \tag{9}\]  
where \(S_{wc}\) and \(S_{ar}\) are both expressed as fractions. According to the modified [PERSON]'s second rule established by [PERSON] [15], the crossover point of relative permeability curves lies to the right of RCS in water-wet rocks, while the crossover point is expected to be located at the left of the RCS in oil-wet systems. Following the above modification, [PERSON] [15] proposed a new wettability index by combining [PERSON]'s first rule and the modified [PERSON]'s second rule as follows:  
\[I_{L}=\alpha\left(\frac{0.3-k_{rw@ROS}}{0.3}\right)+\beta\left(\frac{0.5-k_{rw@ ROS}}{0.5}\right)+\frac{CS-RCS}{1-S_{ar}-S_{mc}} \tag{10}\]  
where \(I_{L}\) is the Lak index and \(k_{rw@ROS}\) is the water relative permeability at residual oil saturation. Recall that relative permeability is defined as the effective permeability divided by oil permeability at the interstitial water saturation. In Equation (10), \(CS\) is the crossover saturation in fraction, and \(\alpha\) and \(\beta\) are constant coefficients determined based on \(k_{rw@ROS}\) as follows:  
\[\text{If }\left\{ \begin{array}{l}k_{rw@ROS}<0.3,\text{ }then\text{ }\alpha=0.5,\text{ }\beta=0\\ 0.3\leq k_{rw@ROS}\leq 0.5,\text{ }then\text{ }\alpha=\beta=0\\ k_{rw@ROS}>0.5,\text{ }then\text{ }\alpha=0,\text{ }\beta=0.5\end{array} \right. \tag{11}\]  
Lak index varies from \(-1\) (strongly oil-wet systems) to 1 (strongly water-wet systems). In another study, [PERSON] et al. [13] showed that the areas below water and oil relative permeability curves could be used to estimate the average wettability as:  
\[I_{ML}=\frac{A_{o}-A_{w}}{A_{o}+A_{w}} \tag{12}\]where \(A_{o}\) and \(A_{w}\) are respectively the areas below oil and water relative permeability curves and \(I_{ML}\) is the modified Lak wettability index which varies from \(-1\) in strongly oil-wet rocks to \(1\) in strongly water-wet samples. The near zero values of \(I_{ML}\) will indicate intermediate wettability.