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In this work, the grain size distribution from the experimental dataset was not available and therefore not taken into account. Consequently, another way to represent similarities between the grain size distribution from experimental data and digital rocks (geological models) was chosen, see details in figure 5.1.
Figure 5.1. Hypothetical curves in blue and red color. Geological model grain size distribution vs experiments. Porosity vs permeability plot (upper right corner)
It is important to notice that even if the experimental data values observed in a porosity vs permeability plot are coincide with the geological models data point created, does not mean a unique solution for their grain size distribution. In other words, even matching the points from figure 5.1 (porosity vs permeability plot), there are still some uncertainties regarding how accurate and well represented the grain size distributions from the experimental datasets could match the geological models created. In figure 5.1, hypothetical curves show differences in their distribution. This is only one hypothetical scenario but the differences in reality could be worse. Differences in the grain size distribution will also affect the pore size distribution.
5.2. Comparison with Pore-scale Displacement Mechanism.
The pore-scale events by which fluids invade the network are snap-off, piston-type and pore filling. These events and their respective behavior through the pore networks were explained in the theory section.
Hughes and Blunt  found different flow patterns when snap-off and piston-type displacement occurred. Based on their findings, Nguyen et al.  stated that the order in which piston-type and snap-off displacements occur determines the pattern of the displacement and therefore the shape of the relative permeability curve and the value of the residual oil saturation. In that way, if the displacement is dominated by snap-off, relative permeabilities are low while residual oil saturation is high. In case the displacement is dominated by piston-type events, the trapping would be low (Sor low) and relative permeability high.
In order to illustrate the effect of the displacement mechanisms, the figures 5.2 and 5.3 are presented. They describe the relationship between the water end point relative permeability Krw(SOR) and the residual oil saturation Sor. Figure 5.2 shows the oil wet pore fraction in colors red and blue with values of 1 and 0 respectively and figure 5.3 three wettability conditions.
Figure 5.3 shows that during wettability condition 3 the data points are distributed in a curve shape form where the highest oil-wet pore fraction occurrence is in the upper left part. In contrast, in figure 5.2b in wettability condition 2 the data points have a more triangular shape where the highest oil-wet pore fraction occurrence is in the bottom part of the triangle. Figure 5.2a shows wettability condition 1 where the data points have a shape in between the two conditions previously mentioned. Notice that when the oil-wet pore fraction is relatively close to 0 the majority of the data points for the three wettability conditions fall in the same area.
Figure 5.4 shows that wettability condition 3 is the only one which presents highest values of the water end point relative permeability while residual oil saturation is keep at the lowest values. This behavior suggests that the displacements are dominated by piston-type where there exists low oil trapping . In wettability condition 2 the data points are the only ones that present a trend with low water end point relative permeability while the residual oil saturation is getting higher if oil-wet pore fraction increases. This behavior suggests that the displacement is dominated by snap-off mechanism , however, there is a not clear behavior when at low water end point relative permeability the residual oil saturation remains low. Wettability condition 1 shows a mixed behavior, no dominant displacement could be established from the figures.
Figure 5.2. Water end point relative permeability Krw,(SOR) vs Sor . a) Network flow models having wettability condition 1.b) Network flow models having wettability condition 2
Figure 5.3. Water end point relative permeability Krw,(SOR) vs Sor . Network flow models having wettability conditions 3.
Figure 5.4. Water end point relative permeability Krw,(SOR) vs Sor . All wettability conditions for all the netweork flow models are presented.
5.3. Wettability Effects
In chapter 4, a different set of variables was used to identify trends or evident effects from the pore network flow models data set when plotting them in a IAH vs IUSBM graph but no major correlations were found with most of the variables. In among the variables used, contact angles, oil-wet pore fraction, and distributing oil-wet elements based on pore size: uncorrelated with pore size (random), preferable large-pores and preferable small-pores are closely analyzed in this section.
Figure 5.5. Amott water Iw and oil indices Io versus oil-wet pore fraction for all wettability conditions (including all pore network flow models dataset).
In order to give an insight into how the three different wettability conditions are affecting the Amott oil and water wettability index while changing the oil-wet pore fraction, figure 5.5 is presented. The Iw showed a fairly trend in between the wettability conditions when was varied. A threshold can be observed from
either in the Amott oil or Amott water index. In the Amott oil, three clearly increasing trends are observed whereas in the Amott water all of them are decreasing in a similar manner.
The wettability condition 2 shows a flat trend until has reached the highest values, whereas condition 3 increase after and condition 1 after with a less steep. The differences seen in the oil Amott index are a confirmation of the effects in wettability conditions. The oil-wet pore fraction, as well, has influence in the wettability of the models.
Figure 5.6. Wettability condition 3 in a) Io vs Iw plot and b) in a 3-D plot adding
Figure 5.6 shows the effect of the oil-wet elements distribution based on pore size. It can be seen that oil-wet pores that are preferentially small have the lowest value of Io but at the same time the highest value of Iw per each increment of oil-wet pore fraction. Whereas oil-wet pores preferentially in large pores represents the highest value of Io and the lowest of Iw. The oil-wet pores uncorrelated with size have middle values of Io and Iw with respect to the other two. These effects have been observed before by Man & Jing  suggesting that for the same oil-wet preferentially distributed in large pores is more oil-wet than the others in terms of relative surface areas that become oil-wet. On the other hand, the USBM index did not show any clear evidence of correlation with pore sizes that might be the reason why wettability trends did not corresponds with theoretical findings (the graph is not showed).
The trapping mechanism has been described by several authors [8, 9, 29]. In this work the wettability model proposed by ÃËœren  is used and explained in detail in the theory section. He extended Dixitââ‚¬â„¢s  analytical relationships between Amott wettability index and for a group of capillary tubes by including accessibility effects, phase trapping and contact angle. It is important to note that this model depends on Amott indices Io, Iw, pore size distribution, Swi, Sor, , contact angles which have been analyzed during the present work. In addition, some limitations were seen from this model in comparison with the network flow models, such as:
i) Trapping parameter a2 (equation 3.16) has been derived for an interval of .
ii) Accessibility function uses contact angle distribution of: water-wet pores and oil-wet pores .
In terms of relative permeabilities, the oil Corey exponent or curve shape factor was underestimating the experimental dataset showing lower values. The reason might be related to:
- Poor connectivity of the oil phase or more snap-off displacement in the pores(mostly seen for wettability 2).
- Dissimilarities associated with intrinsic heterogeneities; this problem was stated at the beginning of this section.
. In contrast, parameters water Corey exponent Nw and the end point to water relative permeability krw(Sor) showed a disagreement with the pore network flow models.
5.4. Capillary Pressure Curve Shape Factor - Primary Drainage
It was shown in figure 4.16, that discrepancies between the experimental trend and the pore network flow models were significant when X was below 55. These discrepancies might be related to the following possible reasons:
i) Differences in pore size distribution or pore-throat size: The span of the experimental dataset shown in figure 4.16 is very large in comparison with the pore network flow model data points. Some points fell inside the experimental error bounds but others outside mostly for values of X lower than 55. To make a fair comparison, figure 5.7 shows a subset of few experimental data points that fairly coincided with the pore network data points in the porosity vs permeability plot (left side) and their respective data points in the curve shape factor a vs X (right side).
Figure 5.7. Subset of experimental and pore network flow models data points. a) Porosity vs permeability plot. b) plot a vs X. Blue square represents the pore network and the pink the experimental datasets.
Therefore, the grain size distribution and how the grains are sorted are important key features during drainage because they are responsible keys for the pore size distribution. At the beginning of the process, the biggest pores are invaded with the non-wetting fluid. Subsequently, the middle size pores are filled during the plateau of the curve and finally some of the smallest pores are reached by the non-wetting fluid at the end of the process.
In such a way, according to Brooks & Corey , when a porous media have narrow pore-throat size, meaning well sorted, the pore size distribution index tend to exhibit large values (steep slope toward lower saturations) whereas when the porous media showed wide pore-throat size, meaning poorly sorted, the pore size distribution index tend to have small values (smoother slope towards lower saturations).This effect is showed in figure 5.8. Differences in pore size distributions between the pore networks and experimental datasets will eventually affects the shape of the drainage curve.
Figure 5.8. Example of primary drainage capillary pressure vs saturation
ii) Dynamic effects:
Lenormand  showed using a 2-D pore network made of ducts (shape of pores and throats) that the wetting phase, at the end of the primary drainage, is trapped in the network when the invading non-wetting fluid breaks the continuous path toward the exit. However, he confirmed that a ââ‚¬Å“leakââ‚¬Â mechanism is observed when the wetting fluid is trying to escape via the corners and its quantity is related to the rapid rate of drainage and fluid viscosity.
According to Barenblatt , there is a dynamic effect which is associated with the necessity of a finite relaxation time in a fluid/fluid interface to reach equilibrium at certain pressure conditions. Therefore the redistribution of fluids in the pore-space during saturation variation takes certain time and is not instantaneous. Thus, capillary pressure and relative permeabilities are processes that depend on quantities and therefore are not universal functions of instantaneous fluid saturation only.
Hassanizadeh and Gray [62, 63] proposed a generalization of the capillary pressure vs saturation relationship including dynamic effects as:
where is the capillary pressure at equilibrium conditions , is the dynamic capillary pressure, is the capillary or capillary damping coefficient and is the time derivative of the saturation.
Other authors, Das et al. , who plotted the differences between dynamic capillary pressure and the capillary pressure in equilibrium vs the time derivative of saturation in a two-phase flow 3-D porous media, found that the applicability of the equation 5.1 increases when the average saturation and decrease. In other words, from medium to lower water saturation levels the effect of the time derivative is stronger to the capillary pressure curves. Thus, the dynamic coefficient is a nonlinear function and increases as saturation decreases. This effect was shown in a homogeneous porous medium with fine and coarse sand where a higher dynamic coefficient was observed in the fine sand compared with the coarse sand at same water saturation.
5.5. Capillary pressure parameters - Imbibition
Some discrepancies were seen from previous section when compared pore network flow models capillary pressure parameters under imbibition with the experimental data.
Here, a closer look is taken at the relationship between the parameters that integrate the Skjaeveland model for imbibition capillary pressure curve.
In figure 5.9, the water curve shape factor aw showed values greater than 1.5 and randomly distributed up to values closer to six when (red color) whereas the water entry pressure cw presented values closer to zero. This effect suggests that the water branch of the Skjaeveland model (first term of the equation that includes cw and aw ) is not contributing to the general equation when the wettability of the pore network flow models is more oil-wet. In contrast, values of aw closer to one have a higher values of cw and suggesting a smoother curve and a water-wet conditions. Finally, for intermediate-wet conditions, when , cw shows the highest values whereas aw shows the lowest. The low values of aw for this case might be related with the narrow pore throats that was found as well for the curve shape factor a in drainage.
Figure 5.9. Variation of in a water curve shape factor aw versus the water entry pressure cw plot.
Comparing the experimental data results on aw , the value was considered constant aw=0.2 due to their results were not very reliable from an experimental point of view.
Analyzing the oil curve shape factor ao , the same plot was used with its respective entry pressure co, as shon in figure 5.10. The oil curve shape factor ao showed values greater than 0 and randomly distributed up to values closer to 8 when(blue to light purple color) whereas the oil entry pressure co presented values closer to zero. That effect suggests that the oil branch of the Skjaeveland model is not contributing to the general equation when the wettability of the pore network flows models are more water-wet. In contrast, values of ao between 0 and 1.5 were seen when whereas higher values of cw suggested a smoother curve and an intermediate to oil-wet conditions.
Figure 5.10. Variation of in an oil curve shape factor ao versus the oil entry pressure co plot
Figure 5.11, shows a closer look at the relationship of ao vs for values of . It can be observed that the highest values of (red color) are reached at the lowest values of ao < 0.4 suggesting a more oil-wet condition and from values of ao between 0.4 to 1.5 the intermediate ones. In addition, the three wettability conditions are shown in figure 5.13. Wettability condition 3 showed the most oil-wet condition of all whereas the wettability condition 2 suggests a more intermediate-wet condition. Compared to the experimental dataset, the core sample are suggested to be intermediate-wet with a value of ao=0.837.
Figure 5.11. Variation of in an oil curve shape factor ao versus the oil entry pressure co plot.