In this work, the grain size distribution from the experimental dataset was non available and hence non taken into history. Consequently, another manner to stand for similarities between the grain size distribution from experimental informations and digital stones ( geological theoretical accounts ) was chosen, see inside informations in figure 5.1.
Figure 5.1. Conjectural curves in bluish and ruddy colour. Geological theoretical account grain size distribution V experiments. Porosity vs permeableness secret plan ( upper right corner )
It is of import to detect that even if the experimental information values observed in a porousness V permeableness secret plan are coincide with the geological theoretical accounts informations point created, does non intend a alone solution for their grain size distribution. In other words, even fiting the points from figure 5.1 ( porousness vs permeableness secret plan ) , there are still some uncertainnesss sing how accurate and good represented the grain size distributions from the experimental datasets could fit the geological theoretical accounts created. In figure 5.1, conjectural curves show differences in their distribution. This is merely one conjectural scenario but the differences in world could be worse. Differences in the grain size distribution will besides impact the pore size distribution.
The pore-scale events by which fluids invade the web are snap-off, piston-type and pore filling. These events and their several behaviour through the pore webs were explained in the theory subdivision.
Hughes and Blunt [ 55 ] found different flow forms when snap-off and piston-type supplanting occurred. Based on their findings, Nguyen et Al. [ 58 ] explained that the form of the comparative permeableness curve and the value of the residuary oil impregnation are related to the order in which Piston type and snap-off supplantings occurs. These supplantings mechanisms find the different flow forms. In that manner, if the supplanting is dominated by snap-off, comparative permeablenesss exhibit low values whereas the residuary oil impregnation show marks of the antonym. In instance the supplanting is dominated by piston-type events, the caparison would be low ( Sor depression ) and comparative permeableness high.
In order to exemplify the consequence of the supplanting mechanisms, the figures 5.2 and 5.3 are presented. They describe the relationship between the H2O terminal point comparative permeableness Krw ( SOR ) and the residuary oil impregnation Sor. Figure 5.2 shows the oil moisture pore fraction in colourss ruddy and bluish with values of 1 and 0 severally and calculate 5.3 three wettability conditions.
Figure 5.3 shows that during wettability status 3 the information points are distributed in a curve form signifier where the highest oil-wet pore fraction happening is in the upper left portion. In contrast, in figure 5.2b in wettability status 2 the information points have a more triangular form where the highest oil-wet pore fraction happening is in the bottom portion of the trigon. Figure 5.2a shows wettability status 1 where the information points have a form in between the two conditions antecedently mentioned. Notice that when the oil-wet pore fraction is comparatively close to 0 the bulk of the information points for the three wettability conditions fall in the same country.
Figure 5.4 shows that wettability status 3 is the lone one which presents highest values of the H2O terminal point comparative permeableness while residuary oil impregnation is keep at the lowest values. This behavior suggests that the supplantings are dominated by piston-type where there exists low oil pin downing [ 55 ] . In wettability status 2 the information points are the lone 1s that present a tendency with low H2O terminal point comparative permeableness while the residuary oil impregnation is acquiring higher if oil-wet pore fraction additions. This behavior suggests that the supplanting is dominated by snap-off mechanism [ 55 ] , nevertheless, there is a non clear behaviour when at low H2O terminal point comparative permeableness the residuary oil impregnation remains low. Wettability status 1 shows a assorted behaviour, no dominant supplanting could be established from the figures.
Figure 5.2. Water terminal point comparative permeableness Krw, ( SOR ) V Sor. a ) Network flow theoretical accounts holding wettability status 1.b ) Network flow theoretical accounts holding wettability status 2
Figure 5.3. Water terminal point comparative permeableness Krw, ( SOR ) V Sor. Network flow theoretical accounts holding wettability conditions 3.
Figure 5.4. Water terminal point comparative permeableness Krw, ( SOR ) V Sor. All wettability conditions for all the netweork flow theoretical accounts are presented.
5.3. Wettability Effectss
In chapter 4, a different set of variables was used to place tendencies or apparent effects from the pore web flow theoretical accounts informations set when plotting them in a IAH V IUSBM graph but no major correlativities were found with most of the variables. In among the variables used, contact angles, oil-wet pore fraction, and administering oil-wet elements based on pore size: uncorrelated with pore size ( random ) , preferred large-pores and preferred small-pores are closely analyzed in this subdivision.
Figure 5.5. Amott H2O Iw and oil indices Io versus oil-wet pore fraction for all wettability conditions ( including all pore web flow theoretical accounts dataset ) .
In order to give an penetration into how the three different wettability conditions are impacting the Amott oil and H2O wettability index while altering the oil-wet pore fraction, figure 5.5 is presented. The Iw showed a reasonably tendency in between the wettability conditions when was varied. A threshold can be observed from
either in the Amott oil or Amott H2O index. In the Amott oil, three clearly increasing tendencies are ascertained whereas in the Amott H2O all of them are diminishing in a similar mode.
The wettability status 2 shows a level tendency until has reached the highest values, whereas status 3 addition after and status 1 after with a less steep. The differences seen in the oil Amott index are a verification of the effects in wettability conditions. The oil-wet pore fraction, every bit good, has influence in the wettability of the theoretical accounts.
Figure 5.6. Wettability status 3 in a ) Io V Iw secret plan and B ) in a 3-D secret plan adding
Figure 5.6 shows the consequence of the oil-wet elements distribution based on pore size. It can be seen that oil-wet pores that are preferentially little have the lowest value of Io but at the same clip the highest value of Iw per each increase of oil-wet pore fraction. Whereas oil-wet pores preferentially in big 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 regard to the other two. These effects have been observed before by Man & A ; Jing [ 59 ] proposing that for the same oil-wet preferentially distributed in big pores is more oil-wet than the others in footings of comparative surface countries that become oil-wet. On the other manus, the USBM index did non demo any clear grounds of correlativity with pore sizes that might be the ground why wettability tendencies did non corresponds with theoretical findings ( the graph is non showed ) .
The caparison mechanism has been described by several writers [ 8, 9, 29 ] . In this work the wettability theoretical account proposed by AE?ren [ 14 ] is used and explained in item in the theory subdivision. It is of import to observe that this theoretical account depends on Amott indices Io, Iw, pore size distribution, Swi, Sor, , contact angles which have been analyzed during the present work. In add-on, some restrictions were seen from this theoretical account in comparing with the web flow theoretical accounts, such as:
I ) Traping parametric quantity a2 ( equation 3.16 ) has been derived for an interval of.
two ) Accessibility map uses contact angle distribution of: water-wet pores and oil-wet pores.
In footings of comparative permeablenesss, the oil Corey advocate or curve form factor was undervaluing the experimental dataset demoing lower values. The ground might be related to:
– Poor connectivity of the oil stage or more snap-off supplanting in the pores ( largely seen for wettability 2 ) .
– Dissimilarities associated with intrinsic heterogeneousnesss ; this job was stated at the beginning of this subdivision.
. In contrast, parametric quantities H2O Corey advocate Nw and the terminal point to H2O comparative permeableness krw ( Sor ) showed a dissension with the pore web flow theoretical accounts.
5.4. Capillary Pressure Curve Shape Factor – Primary Drain
It was shown in figure 4.16, that discrepancies between the experimental tendency and the pore web flow theoretical accounts were important when Ten was below 55. These disagreements might be related to the following possible grounds:
I ) Differences in pore size distribution or pore-throat size: The span of the experimental dataset shown in figure 4.16 is really big in comparing with the pore web flow theoretical account informations points. Some points fell inside the experimental mistake bounds but others outdoors largely for values of X lower than 55. To do a just comparing, figure 5.7 shows a subset of few experimental informations points that reasonably coincided with the pore web informations points in the porousness V permeableness secret plan ( left side ) and their several informations points in the curve form factor a V X ( right side ) .
Figure 5.7. Subset of experimental and pore web flow theoretical accounts data points. a ) Porosity vs permeableness secret plan. B ) secret plan a V X. Blue square represents the pore web and the pink the experimental datasets.
Therefore, the grain size distribution and how the grains are sorted are of import cardinal characteristics during drainage because they are responsible keys for the pore size distribution. At the beginning of the procedure, the biggest pores are invaded with the non-wetting fluid. Subsequently, the in-between size pores are filled during the tableland of the curve and eventually some of the smallest pores are reached by the non-wetting fluid at the terminal of the procedure.
In such a manner, when a porous media exhibit a narrow pore-throat size, intending good sorted, the pore size distribution index will demo big values ( steep incline toward lower impregnations ) whereas when the porous media showed broad pore-throat size, intending ill sorted, the pore size distribution index tend to hold little values ( smoother slope towards lower impregnations ) [ 16 ] . This consequence is showed in figure 5.8. Differences in pore size distributions between the pore webs and experimental datasets will finally impact the form of the drainage curve.
Figure 5.8. Example of primary drainage capillary force per unit area V impregnation
two ) Dynamic effects:
Lenormand [ 60 ] showed in a 2-D pore web made of canals ( form of pores and pharynxs ) that the wetting stage at the terminal of the primary drainage will be trapped in the web. However, he confirmed that a leak mechanism is observed when the wetting fluid is seeking to get away via the corners and its measure is related to the rapid rate of drainage and unstable viscousness.
Harmonizing to Barenblatt [ 61 ] , a dynamic consequence exists in a fluid/fluid interface. This consequence needs a relaxation clip in order to carry through the equilibrium at certain force per unit area conditions. The redistribution of fluids in the pore-space is non instantaneous during impregnation fluctuation and hence capillary force per unit area and comparative permeableness procedures are non merely dependant on unstable impregnation.
Hassanizadeh and Gray [ 62, 63 ] proposed a generalisation of the capillary force per unit area vs impregnation relationship including dynamic effects as:
where is the capillary force per unit area at equilibrium conditions, is the dynamic capillary force per unit area, is the capillary or capillary damping coefficient and is the clip derived function of the impregnation.
Other writers such as Das et Al. [ 64 ] , plotted the differences between dynamic capillary force per unit area and the capillary force per unit area in equilibrium. The pertinence of the equation 5.1 is higher when the mean impregnation and lessening. In other words, from medium to take down H2O impregnation degrees the consequence of the clip derivative is stronger to the capillary force per unit area curves. Therefore, the dynamic coefficient is a nonlinear map and increases as impregnation decreases. This consequence was shown [ 64 ] in a homogenous porous medium with all right and harsh sand where a higher dynamic coefficient was observed in the all right sand compared with the coarse sand at same H2O impregnation.
5.5. Capillary force per unit area parametric quantities – Imbibition
Some disagreements were seen from old subdivision when compared pore web flow theoretical accounts capillary force per unit area parametric quantities under imbibition with the experimental information.
Here, a closer expression is taken at the relationship between the parametric quantities that integrate the Skjaeveland theoretical account for imbibition capillary force per unit area curve.
In figure 5.9, the H2O curve form factor aw showed values greater than 1.5 and indiscriminately distributed up to values closer to six when ( ruddy colour ) whereas the H2O entry force per unit area cw presented values closer to zero. This consequence suggests that the H2O subdivision of the Skjaeveland theoretical account ( first term of the equation that includes cw and aw ) is non lending to the general equation when the wettability of the pore web flow theoretical accounts is more oil-wet. In contrast, values of aw closer to one hold a higher values of cw and proposing a drum sander 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 instance might be related with the narrow pore pharynxs that was found every bit good for the curve form factor a in drainage.
Figure 5.9. Variation of in a H2O curve form factor aw versus the H2O entry force per unit area cw secret plan.
Comparing the experimental informations consequences on aw, the value was considered changeless aw=0.2 due to their consequences were non really dependable from an experimental point of position.
Analyzing the oil curve form factor ao, the same secret plan was used with its several entry force per unit area carbon monoxide, as shon in figure 5.10. The oil curve form factor ao showed values greater than 0 and indiscriminately distributed up to values closer to 8 when ( bluish to light violet colour ) whereas the oil entry force per unit area carbon monoxide presented values closer to zero. That consequence suggests that the oil subdivision of the Skjaeveland theoretical account is non lending to the general equation when the wettability of the pore web flows theoretical accounts are more water-wet. In contrast, values of ao between 0 and 1.5 were seen when whereas higher values of cw suggested a drum sander curve and an intermediate to oil-wet conditions.
Figure 5.10. Variation of in an oil curve form factor ao versus the oil entry force per unit area carbon monoxide secret plan
Figure 5.11, shows a closer expression at the relationship of ao V for values of. It can be observed that the highest values of ( ruddy colour ) are reached at the lowest values of ao & lt ; 0.4 proposing a more oil-wet status and from values of ao between 0.4 to 1.5 the intermediate 1s. In add-on, the three wettability conditions are shown in figure 5.13. Wettability status 3 showed the most oil-wet status of all whereas the wettability status 2 suggests a more intermediate-wet status. Compared to the experimental dataset, the nucleus sample are suggested to be intermediate-wet with a value of ao=0.837.
Figure 5.11. Variation of in an oil curve form factor ao versus the oil entry force per unit area carbon monoxide secret plan.