## 3 Results  
A total of 2000 data points were used to calculate the effective parameters of eight types of 2-D rock samples: Bandera gray, Bentheimer, Berea, Berea upper gray, Berea sister gray, Castlegate, Leopard, and Buff Berea. Each rock sample had 250 2-D rock images. The absolute porosity distribution in Figure 6 shows the range of absolute porosity values and indicates the homogeneity of the rocks. The quartiles were \(Q1=0.23\), \(Q2=0.26\), and \(Q3=0.29\) within the porosity range of sandstone. The effective porosity distribution is shown in Figure 7.  
The effective porosity quartiles of \(Q1=0.12\), \(Q2=0.15\), and \(Q3=0.18\) were smaller compared with the quartile values for absolute porosity. The absolute porosity (\(y\)-axis) was plotted against the effective porosity (\(x\)-axis) in a scatter plot in Figure 8, showing that the relationship between these two porosities is directly proportional with a positive gradient, indicating that higher absolute porosity generally corresponds to higher effective porosity. However, the data points are below the blue line that represents the \(45^{\circ}\) line, the equilibrium line where absolute porosity equals effective porosity; therefore, the effective porosity in this case, is always smaller than the absolute porosity. The tortuosity results are shown in Figure 9 for each rock sample. The \(x\)-axis represents tortuosity values, and the \(y\)-axis represents the frequency or number of distributed data points. The red color represents the average value, the blue color represents the lowest value, and the green color represents the highest value of tortuosity. The Bandera gray rock shows the lowest tortuosity range, with a log-normal distribution evident from the rightward elongation indicating an asymmetrical distribution. The skewness value of this distribution is also positive (right-skewed), meaning the data is more distributed in the lower range. All distributions for the Bentheimer rock show a log-normal shape and are asymmetrical with positive skewness (right-skewed). All distributions are  
Figure 6: Distribution of absolute porosity. Figure 7: Distribution of effective porosity. asymmetrical with positive skewness (right-skewed) for the Berea rock, indicating the data are distributed more in the lower range. The Berea sister gray rock shows the lowest tortuosity range, with each skewness value being positively greater than 1 (right-skewed). The Berea upper gray rock's lowest tortuosity exhibits a gamma distribution with positive skewness. The tortuosity results for the Buff Berea rock are positively skewed with an asymmetrical distribution. Likewise, the Leopard rock values are positively skewed (right-skewed). The Castlegate rock exhibits the lowest range distribution, approaching normal distribution, indicated by a skewness value of 0.09. The plots in Figure 10 illustrate the relationships between effective porosity, permeability, and tortuosity. In Figure 10a, a strong inverse correlation (\(R^{2}=0.81\)) is observed between effective porosity and tortuosity, indicating that, as effective porosity increases, the pathways for fluid flow become less complex, reducing tortuosity. Figure 10b shows a positive correlation (\(R^{2}=0.7\)) between effective porosity and permeability, suggesting that higher effective porosity results in better fluid transmission through the rock. In Figure 10c, a moderate inverse correlation (\(R^{2}=0.50\)) is found between tortuosity and permeability, implying that more convoluted pathways (higher tortuosity) hinder fluid flow, resulting in lower permeability. Overall, these plots highlight the interrelated nature of these parameters, where higher effective porosity generally leads to lower tortuosity and higher permeability, facilitating easier fluid movement through the rock. The plots in Figure 11 indicate weak correlations between average throat size and the effective parameters of porosity, permeability, and tortuosity.