In this subdivision, the consequences obtained from the web fluid mold are analyzed and compared to laboratory SCAL informations correlativities [ 1, 2 ] . The consequences are presented in three separate subdivisions, wettability word picture, comparative permeableness and capillary force per unit area. All of them are considered to hold a distinguishable importance for fluid flow through porous media. In the first subdivision, the three wettability conditions antecedently established in subdivision 2 are analyzed utilizing different variables in order to qualify and compare them with experimental consequences. The undermentioned subdivision, describes the web fluid patterning consequences compared to the work of Smits and Jing [ 2 ] . Finally, the subdivision three shows the capillary force per unit area consequences from primary drainage and primary imbibition of the webs which are compared to Cense ‘s work [ 1 ] .
The informations obtained from the simulator were fitted utilizing the Corey theoretical account for comparative permeableness, Brooks & A ; Corey attack for drainage [ 16 ] and the Skjaeveland theoretical account [ 17 ] for imbibition capillary force per unit area. For fitting process refer to appendix B.
4.1. Correlation Approach
Using the same correlativity attack described in old experimental plants [ 1, 2 ] , most of the SCAL parametric quantities show a correlativity with the pore geometrical factor. The ground is for a given rock-type the chief features are the pore geometry and pore size distributions that in bend control permeableness. This factor which is originated from the Kozeny-Carman equation reflects the flow and pin downing features of the pore webs [ 2 ] . Another consequence studied in the correlativities, is the wettability consequence. In this work, contact angle, oil-wet pore fraction, distribution of oil moisture pores and their effects in wettability index are used to qualify wettability.
4.2 Wettability Characterization Analysis
Reservoir sandstone stones normally fall into a big group of intermediate wettability. In the industry there is no such direct trial for wettability indicant, nevertheless, the empirical index of wettability measurings most widely used include Amott [ 21 ] and USBM [ 22 ] every bit good as the combination of the Amott/USBM methods.
There are some disagreements between both indexes. The wettability index IAH is defined to be between -1 and 1 whereas this scope does non use to IUSBM, hence numerical differences for strongly-wetted conditions are expected. Another difference could be seen once they are compared over a broad scope of weakly-wetted and intermediate conditions, since IAH reflects self-generated imbibition whereas IUSBM is derived from the drainage capillary force per unit area curves ; nevertheless the relationship between these two is non clear yet. This issue was addressed by Dixit et Al. [ 51 ] in the late 90 ‘s, happening an analytical relationship between the Amott-Harvey index and the USBM index for assorted wetting scenarios based on oil-wet pore fraction, called in this study as I±1, and their distribution within the pore infinite. In their work, they assumed a unvarying pore size distribution, individual contact angles ( 0A° in the water-wet and 180A° in the oil-wet pores ) and ignored the effects of pore connectivity. The distinguish wettability types from Dixit et Al. work are as follows:
Fractionally-wet ( FW ) instance, where oil-wet pores are uncorrelated with size.
Mixed-wet instance with big pores being oil-wet ( MWL ) .
Mixed-wet instance where little pore are oil-wet ( MWS ) .
It was non until 2006, that the experimental conformation of different wettability categories was introduced by Skauge et Al. [ 52 ] , where experimental verification of the being of wettability categories was found utilizing a set of nucleuss from sandstone reservoirs in the North Sea and later geting environmental SEM consequences from USBM and Amott-Harvey wettability indices. Their theoretical account of the being of this wettability categorization in existent reservoir is described in three possible events. First, the presence of small affinity of the H2O stage due to mineralogy of reservoir surface. Second, the surface assimilation mechanism found in polar constituents that may destabilise or precipitate larger molecules. Third possibility described by Skauge as the discontinuity of the H2O movie on the stone surface [ 52 ] .
Based on Dixit analytical relationship [ 51 ] and Skauge [ 52 ] experimental findings, wettability categorization of the available pore web theoretical accounts are analyzed. In order to set up the relationship between IAH and IUSBM, foremost a tabular array with variables tested for their impact are summarized within table 4.1.
Table 4.1. Variables varied in the web simulator.
Here, a sum of 3210 web flow theoretical accounts are presented to qualify wettability utilizing both index, IAH and IUSBM. The e-Core package allows choosing the distribution of oil-wet pore to be preferentially big pores, preferentially little pores or random. In this manner, a 3rd from the sum of web flow theoretical accounts generated, correspond to a different distribution for the oil invaded pores which are oil-wet. From figure 4.1 to calculate 4.3, a sum of 102 points are presented ( selected from the 3210 web flow theoretical accounts ) in each IUSBM V IAH secret plan and stand foring 34 different geological theoretical accounts ( Fontainenbleau, Bentheim, Berea and North Sea reservoir ) .
The process adopted in this portion in order to compare the different wettability scenarios to the lab experiments starts with the fluctuation of I±1, oil-wet pore fraction ( 0 & lt ; I±1 & lt ; 1 ) for each wettability status. A distribution is presented in each figure stand foring the frequence of how the oil-wet pore fraction was assigned to each theoretical account.
The I±1 fluctuation in each figure is shown as a colour graduated table where wholly ruddy is assumed to be 1 and wholly bluish is assumed to be 0. At the same clip, different geometric figures represent the different geological theoretical accounts used to bring forth the pore webs. Figures such as: square form is used for Bentheim, a diamond form for the Fontainebleau theoretical accounts, circle for Berea and trigons for the North Sea reservoir theoretical accounts.
The continues black lines show in each figure ( right side ) , represents the analytical relationships derived by Dixit at Al. [ 51 ] between IUSBM and IAH for each of the wettability type with regard to the oil-wet pore distribution. The upper line represents the assorted moisture big, the in-between one the fractional moisture and the lower one the assorted moisture little. In these analytical relationships, the values of Rmin and Rmax assigned for these lines correspond to 0.3 and 40 Aµm severally and presuming a volume advocate of 2.
The figure 4.1 ( a ) shows a distribution where the major oil-wet pore fraction happening corresponds when I±1 is greater than 0.6. The consequence is shown in figure 4.1 ( B ) where more points with ruddy colour appeared. In contrast, in figure 4.1 ( degree Celsius ) shows a different distribution where the major oil-wet pore fraction happening corresponds to I±1 lower than 0.6. As can be seen from figures 4.1 ( vitamin D ) , more points with bluish colour appeared compared to calculate 4.1 ( B ) .
vitamin D )
Figure 4.1. Oil-wet pore fraction frequence distribution for the three wettability conditions. Higher frequences values for I±1 & gt ; 0.6 ( a ) and I±1 & lt ; 0.6 ( degree Celsius ) and their IAH and IUSBM relationship ( B ) and ( vitamin D ) severally.
From figure 4.2 ( a ) , an oil-pore fraction distribution merely made with I±1 lower or equal than 0.6 is shown and their several IAH V IUSBM relationship generated in figure 4.2 ( B ) . The consequence largely shows information points in quadrant I ( reader refer to calculate 4.4 ( a ) ) . In contrast, figure 4.2 ( degree Celsius ) shows an oil-pore fraction distribution where I±1 is ever greater or equal than 0.6 demoing informations points in quarter-circles ( I, II, III and IV ) .
Figure 4.2. Equal oil-wet pore fraction frequence distribution for the three wettability conditions, demoing oil-wet pores distribution when ( a ) I±1 a‰¤ 0.6 and ( degree Celsius ) I±1 a‰? 0.6. IAH and IUSBM relationship generated from each oil-wet pore fraction distribution ( B ) and ( vitamin D ) .
Figure 4.3 ( a ) shows a random distribution of the oil-wet pore fraction I±1 for all the wettability conditions. The consequence observed in figures 4.3 ( B, degree Celsius, vitamin D and vitamin E ) were generated utilizing the I±1 random distribution of figure 4.3 ( a ) .
vitamin D ) vitamin E )
Figure 4.3. Oil-wet pore fraction with a random distribution used to bring forth different IUSBM V IAH secret plans in order to analize different effects. ( B ) Oil moisture pore fraction from 0 to 1. ( degree Celsius ) Wettability conditions consequence. ( vitamin D ) Coordination figure consequence. ( vitamin E ) Oil-wet pore size preferentially big, little and uncorrelated with size consequence.
One of import consequence was shown when the three wettability conditions were presented ( figure 4.3c ) . Wettability status 3 presented the most oil-wet inclination when information points fall into quadrant II and III. In contrast, wettability status 2 showed the most water-wet system inclination in which most of its informations points fall into quadrant I. Wettability status 1 showed a mix-wet system with informations points in quarter-circles I, II, III largely. Taking into history figure 4.3c, the oil-wet pore fraction I±1 in figure 4.3a every bit good have influence in the wettability on the web demoing more oil-wet status when the I±1 is near to 1 and water-wet status when I±1 is comparatively close to 0.
Figure 4.3 ( vitamin D ) shows the coordination figure ( omega ) consequence of each pore web theoretical account where the blue colour represent when omega & gt ; 3.8 and the black colour to z a‰¤ 3.8. A greater figure of informations points for omega a‰¤ 3.8 autumn into the black continues lines. For the wettability types ( random, little, big ) in figure 4.3 ( vitamin E ) , the information points are scattered over the diagram and they do non follow the wettability classes tendency lines proposed by Dixit.
4.2.1. Comparison of the web flow consequences with the experimental informations
Using the same oil-wet pore fraction random distribution as figure 4.3 ( a ) , the web flow informations points are shown in figure 4.4 ( B ) . Besides, the experimental information from Cense ‘s work and Smits & A ; Jings.
Figure 4.4. Quadrants distribution for IAH and IUSBM relationship ( a ) . IAH and IUSBM informations points from Cense [ 6 ] , Smits & A ; Jing [ 32 ] and selected web fluid theoretical accounts utilizing oil-wet pore fraction of figure 5.3 ( a ) and random pore size distribution. Dashed lines indicate theoretical boundaries for assorted wet big assorted, fractionally wet and assorted moisture little ( I?=2, rmin=0.3 Aµm, rmax =40 Aµm ) .
As a sum-up of figure 4.4, some few points could be drawn, such as:
A considerable measure of informations points autumn in quadrant I for web flow theoretical accounts in comparing with the experimental consequences, proposing more H2O wet influence due to the inclusion of oil-wet pore fraction of less than 0.5.
A few informations points from web flow theoretical accounts fall in quadrant III proposing an oil moisture influence in the group of informations.
Quadrant III and IV are the most populated for experimental consequences, a possible ground might be associated with the exposure to oil-based boring claies that could do the smaller pores become oil-wet ; this wettability change in the experiments was reported by Yan et Al. [ 53 ] .
4.2.2. Section Summary
To reason this subdivision, differences between the experimental wettability consequences and the pore webs fluid flow theoretical accounts were seen and discussed when plotting them in a wettability index IUSBM V IAH graph. The strongest effects observed in wettability of the pore web flow theoretical accounts occurred when the oil-wet pore fraction I±1 and contact angles or wettability conditions ( 1,2 and 3 ) were varied. When I±1 is equal or lower than 0.6 or merely wettability status ( 2 ) is present, the pore web flow theoretical accounts are considered to be water-wet. In contrast, when I±1 is greater than 0.6 or wettability ( 1 or 3 ) are present, the wettability of the webs flow theoretical accounts are considered to be intermediate to oil-wet.
Comparing to theoretical and experimental findings, the pore web flow theoretical accounts fail to demo neither the wettability classes [ 51, 52 ] when administering oil-wet elements based on pore size utilizing the package e-Core nor the consequence of the coordination figure in the webs [ 51 ] . In add-on, the experimental dataset have intermediate wettability features ( figure 4.4b ) whereas none of the wettability conditions have a stronger lone intermediate wettability.
4.3. Relative Permeability
Fake comparative permeableness curves were fitted with the Corey comparative permeableness theoretical account, utilizing a Levenberg-Marquardt adjustment algorithm [ 54 ] .
From lab experiments consequences, there are some considerations that have to be kept in head sing the correlativities made by Smits & A ; Jing [ 2 ] , as can be taken from the followers:
should non transcend 250 and merely be used with realistic values of I† and K ; the wettability scope of the dataset is about runing from 0.3 to 0.7 and their dependence has non been tested outside the scope, extrapolation to W=0 and W=1 gives values as expected for wholly water-wet or wholly oil-wet ; all the comparative permeableness curves have been normalized with regard to Kro_cw such that Kro_cw= 1 ( by definition ) ; comparative permeableness correlativities were established for primary imbibition.
Note that wettability has been defined between 0 and 1, to be 0 if the stone is wholly water-wet, 1 if the stone is wholly oil-wet and between 0 and 1 for intermediate / mixed-wet stone. Hence: W =0.5* ( 1-w ) [ 2 ]
Besides, all the correlativities obtained from lab experiments of all six Corey parametric quantities against wettability, has been made with 4 informations points, either WUSBM or WAH are available from those four informations points ( represented as black colour in figures 4.9, 4.11, and 4.13 ) whereas 10 informations points ( represented as black circles ) from the same correlativities are plotted in figures 4.8, 4.10 and 4.12 but utilizing a changeless wettability value of 0.40 for sandstone stones. Harmonizing to Smits & A ; Jing [ 2 ] , wettability informations could non be obtained from the bulk of the Fieldss therefore a changeless value for that ground was used.
Table 4.2 Consequences of arrested developments from Smits & A ; Jing and Simulations from the three different wettability conditions. All values are norms from all samples and theoretical accounts.
Table 4.2, shows the SCAL parametric quantities obtained from the web flow theoretical accounts and from experimental informations. The figures 4.5, 4.6 and 4.7 show the corresponding comparative permeableness maps in additive graduated table ( a ) and in logarithmic graduated table ( B ) . The latter is plotted to analyze the little values for the comparative permeableness that become identical from nothing on a additive graduated table. The pink colour curves represent the experimental kro and krw. From figures 4.5-4.7 it can be seen that the lucifer of the theoretical account curves ( wettability 1, 2 and 3 ) to the experimental curves are reasonably good over a wide impregnation scope within experimental mistake, nevertheless, the terminal points demo little differences.
Figure 4.5. Relative permeableness curves kro and krw comparing from wettability status 1 and Smits & A ; Jing consequences. Both axis in additive graduated table ( a ) and same figure but comparative permeableness axis in logarithmic graduated table ( B )
In general, the irreducible H2O impregnation for all three wettability conditions is lower than the experiments consequences and residuary oil impregnation is slightly higher.
Figure 4.6. Relative permeableness curves kro and krw comparing from wettability status 2 and Smits & A ; Jing consequences. Both axis in additive graduated table ( a ) and same figure but comparative permeableness axis in logarithmic graduated table ( B )
Figure 4.7. Relative permeableness curves kro and krw comparing from wettability status 3 and the consequences of Smits & A ; Jing consequences. Both axis in additive graduated table ( a ) and same figure but comparative permeableness axis in logarithmic graduated table ( B )
Therefore, even though the values for all of the six Corey parametric quantities from simulation consequences are different from the lab experiments ( table 4.2 ) , is still possible to acquire a reasonably good estimate.
Table 4.3. Wettability Index from Amot-Harvey and USBM for the three wettability conditions and the wettability index scope obtained from lab experiments.
Note that in the experiments used in the work of Smits & A ; Jing [ 2 ] wettability was frequently non measured. In the instances it was measured, it sometimes was derived from the USBM method, in others by Amott or NMR. For that ground, both wettability indexes are used and shown during the analysis with the six Corey parametric quantities.
4.3.1 Corey advocates: H2O advocate Nw and oil advocate No
The Corey theoretical account attack has proposed that the non-wetting stage and wetting stage comparative permeableness can be described by the power jurisprudence theoretical account [ 16 ] . The Corey advocates Nw and No depict the curvature between the end-points, i.e. the form of the curves.
Figure 4.8 shows how the H2O Corey advocate alterations with regard to the pore geometrical factor under the influence of I±1 fluctuation ( 0a‰¤ I±1 a‰¤ 1 ) . The bluish colour in the pore web informations points represents I±1 closer to 0 and the ruddy colour I±1 closer to 1. All the consequences from the web flow theoretical accounts are included in this figure. The black information points represent the experimental information from Smits & A ; Jing [ 2 ] work and the black line their tendency. From the figure, it can be seen that the line stand foring the experimental tendency falls into an country of water-wet to mixed-wet wettability tendency of the web flow theoretical accounts data points. In general, figure 4.8 suggests that the H2O Corey parametric quantity tend to the maximal values when the theoretical accounts are oil-wet.
Figure 4.8. Water Corey advocate as a map of the pore geometrical factor.
The old figure 4.8 represented a general position of all the pore web flow theoretical accounts whereas the undermentioned figures ( 4.9, 4.11 and 4.13 ) represent what is go oning in each wettability status utilizing the oil-wet pore fraction with a random distribution ( figure 4.3a ) among the 34 theoretical accounts created.
In Figure 4.9, the relation between the advocate Nw and wettability index WUSBM is shown for all wettability conditions. The black line represents the correlativity made by lab experiments whereas the remainder denotes the web flow theoretical accounts data correlativities. The ten axis is an experimental relationship between wettability index and the pore geometrical factor found by Shell Exploration & A ; Production B.V.
Figure 4.9. Corey exponent Nw V Jing ‘s [ 2 ] correlativity utilizing WUSBM ( a ) and WIAH ( B ) . All the three conditions, w. status 1, w. status 2 and w.condition3 are plotted in colour blue, cyan and purple severally.
As figure 4.9 shows, in one manus, wettability conditions 1 and 3 are non in understanding with laboratory experiment correlativities. On the other manus, wettability status 2 showed a relation a just tendency with regard to the latter. Besides, there are few informations points available to obtain the correlativity from lab experiments comparing to the spread cloud of informations points for each status. Harmonizing to Jing [ 2 ] , Nw should diminish when wettability and geometrical factor addition. One of the ground could be assigned to the lower Swc or Sir mean values show in table 4.2 for the three wettability conditions comparing to the experimental information. The initial H2O impregnation tends to impact the form of the comparative permeableness curves [ 55 ] .
Figure 4.10, shows how the oil Corey advocate alterations with regard to the pore geometrical factor under the influence of I±1 fluctuation ( 0a‰¤ I±1 a‰¤ 1 ) . The bluish colour in the pore web informations points represents I±1 closer to 0 and the ruddy colour I±1 closer to 1. All the consequences from the web flow theoretical accounts are presented in this figure. The black information points represent the experimental information from Smits & A ; Jing [ 2 ] work and the black line their experimental tendency.
Figure 4.10. Oil Corey advocate No V pore geometrical factor.
The experimental information points autumn in the country which the pore web theoretical account informations points suggest that the nucleus samples should be considered assorted to oil-wet.
In figure 4.11, all the web flow theoretical account informations points ensuing from all wettability conditions utilizing WUSBM and WAH are in just understanding with the tendency of experiment correlativities, nevertheless, the mean values of the pore web theoretical accounts data points be given to undervalue the No consequences from the experimental dataset. One ground might be associated with Sir differences shown in table 5.2 between wettability conditions and experiments. It was reported by Oren at Al. [ 56 ] that fiting the Sir of the pore web with the experiments produces a better estimate of the oil comparative permeableness curves.
Figure 4.11. Oil Corey advocate No V Jing [ 32 ] correlativity utilizing WUSBM ( a ) and WIAH ( B ) .
The x axis is an experimental relationship between wettability index and the pore geometrical factor found by Shell Exploration & A ; Production B.V.
4.3.2. Relative Permeability End points: Kro_Swc and Krw_Sor
The end-point for H2O comparative permeableness is known to be between 0.1 and 0.4 for a water-wet stone and shut to 1.0 for wholly oil-wet stone [ 2 ] . The end-point oil comparative permeableness is assumed to be equal to 1.0 by definition ( industry criterion ) and was used in such a manner in this work.
Figure 4.12, shows how the H2O comparative permeableness terminal point alterations with regard to the pore geometrical factor under the influence of I±1 fluctuation ( 0a‰¤ I±1 a‰¤ 1 ) . The bluish colour in the pore web informations points represents I±1 closer to 0 and the ruddy colour I±1 closer to 1. All the consequences from the web flow theoretical accounts are presented in this figure. The black information points represent the experimental information from Smits & A ; Jing [ 2 ] work and the black line their experimental tendency.
The experimental information points fall into the country where the pore web informations points suggest that the nucleus samples should be mixed-wet matching to the values observed by Smits & A ; Jings [ 2 ] . However, a disagreement is observed at lower values of Krw, ( Sor ) ( & lt ; 0.1 ) where the pore web flow theoretical accounts suggest that is a oil-wet country.
Figure 4.13 ( a ) and ( B ) , shows all the curves from the pore web theoretical accounts have a similar tendency as the experimental consequences utilizing either WAH or WUSBM. However, wettability status 2 is non demoing correspondence with the others. The H2O comparative permeableness terminal point of wettability status 2 suggest that is undervaluing the experimental value ( see figure 5.6 for mention ) even if wettability index and pore geometrical factor are increasing ( figure 4.13 a ) .
Figure 4.12. Water terminal point comparative permeableness Krw, ( Sor ) V pore geometrical factor.
Therefore, it could be inferred that the consequence on wettability alterations showed in figure 4.12, where oil-wet is suggested by the pore web flow theoretical accounts data points when Krw, ( Sor ) & lt ; 0.1, is caused largely by wettability status 2 behaviour when underestimation the values of the experimental Krw, ( Sor ) . The possible ground of that consequence might be associated of the short scope selected for the forward oil contact angle I?a for wettability status 2. Hence, if I?a additions, the oil movie in the corners of oil-wet pores is stable over a big scope of capillary force per unit area [ 14 ] and Krw, ( Sor ) will increase due to oil movie connectivity across the web.
Figure 4.13 Correlations from simulation and lab experiment against WUSBM and X ( a ) . Same correlativities against WAH and X ( B ) .
The x axis every bit good as the old figures is an experimental relationship between wettability index and the pore geometrical factor found by Shell Exploration & A ; Production B.V
Theory [ 19, 20 ] and experimental informations [ 1, 2 ] refers as:
Water-wet sandstone stone, when the oil Corey advocate has a lower value No a‰? 2, H2O Corey advocate Nwa‰?4 and Krw & lt ; 0.15.
Oil-wet sandstone stone, when the oil Corey advocate are higher No a‰? 4, H2O Corey advocate Nw & lt ; 4 and Krw & gt ; 0.5.
Table 4.4. Oil Corey advocate No, H2O Corey advocate Nw, End-point comparative permeableness Krw and H2O connate Swc trends when wettability increased ( an addition of W reflects an addition in oil-wetness )
The tabular array 4.4 indicates what is the consequence ( tendency ) on each of the parametric quantities ( No, Nw, Krw, Swc ) when wettability addition. The tabular array enclosed experimental informations every bit good as the pore web wettability conditions 1,2 and 3.
Comparing the theory with the consequences shown in table 4.4 and the figures 4.8-4.12 some decisions could be drawn:
Contradictory inclination of the H2O Corey advocate Nw between the theory and the pore web flow theoretical accounts data points. The contradiction suggests that a hapless connectivity of the H2O in the pores exists when the wettability of the web is oil-wet. In add-on, the wettability form is non really clear for the pore web flow theoretical accounts informations points with respects to Krw, ( Sor ) parametric quantity comparison to theory. Uniting the parametric quantity Nw and Krw, ( Sor ) consequences from the pore web flow theoretical accounts, the H2O comparative permeableness curve does non match to the theoretical wettability province of the pore webs flow theoretical accounts.
Good understanding with the oil Corey advocate No between theory, experiments and the pore web flow theoretical accounts data points.
4.4. Capillary Pressure
Knowledge of the functional relationship between capillary force per unit area and impregnation is necessary in order to analyze and work out the equations that govern fluid flow through porous media incorporating two or more non-miscible fluids. In this work, gravitative forces, together with capillary forces of a porous medium, command the distribution every bit good as the flow of the non-miscible stages. The capillary force per unit area is originated from the interfacial tenseness or interfacial free energy that exists between two non-miscible fluids. It is dependent on the interfacial tenseness, pore size and contact angle. This subdivision will be subdivided in primary drainage and first imbibition. The equations that govern each tendency line, in all the figures, have been removed due to confidentiality protection.
4.4.1. Primary Drain
The primary drainage Personal computer curves were fitted with the Brooks-Corey map [ 16 ] , which is one of the theoretical account curve most often used. Hence, combining weight. 1.1 is used for each theoretical account. During the fit process, two out of three parametric quantities from equation 1.1 are approximated. Therefore, Pe and a are estimated and Sir is assumed to be equal to Swc.A differentiation has to be made when Sir and Swc are considered the same. Theoretically they are non because Swc depends on the tallness of the oil column in the reservoir whereas the Sir does non. In world Swc is ever equal or greater than Sir. By and large, this premise is normally made.
Figure 4.14. Correlations between and swerve fitting parametric quantity Pe normalized with I?.cos I? . Best tantrum and mistake bounds from experiments [ 6 ] are shown in blue. The dilutant lines represent the mistake. The pink circles represent the pore web flow theoretical accounts data points.
Note that during this subdivision ( primary drainage ) , the black circles represent the experiments informations and their several best tantrum is represented by a bluish line ( equation is shown in each figure ) . Error bounds are shown with a dilutant bluish line. The pink circles represent the pore web flow theoretical accounts data points. For primary drainage parametric quantities, such as Pe, Sir and a, the Levenberg-Marquardt adjustment algorithm was implemented [ 52 ] .
Figure 4.15 shows a just understanding of the pore web flow theoretical accounts data points co-occuring with the experimental tendency line. Spite of pore web points represents a little country of the complete dataset ; the points are located where the bulk of the experiment dataset points are.
For Pe and Sir a power jurisprudence is shown that represents the experiments informations and for parametric quantity a a additive tendency fit the information, see figures 4.14, 4.15 and 4.16 severally. Due to confidentiality protection, the equations of all the curves and their several mistakes are non shown from this point onwards.
Figure 4.15 Correlation between and Sir. Data points from experiments in black and informations points from pore web flow theoretical account in pink.
As a comparing, a just understanding is shown in figure 4.15 between the experiments tendency line or best tantrum and the pore web flow theoretical account informations points.
Figure 4.16. Correlations between and swerve fitting parametric quantity a for first drainage.
The correlativity for curve form parametric quantity a was non in good understanding with the experimental work. For parametric quantity a larger than 0.5, the pore size distribution can be expected to be narrow, giving a larger tableland in the Pc curve. The pore web flow theoretical accounts data points show the curve form factor between 0.2-0.4 proposing that there is more uniformity within the grain size in comparing to existent stones, nevertheless, this is non wholly consistent with the north sea reservoir stone theoretical accounts that have a little grain size distribution compared with the Berea, Fontainebleau and Bentheim theoretical accounts.
4.4.2. Primary Imbibition
The natural philosophies behind the Skjaeveland equation is the same as behind the Brooks-Corey equation. In other words, apart from wettability effects it is expected that the entry force per unit area coefficients cw and carbon monoxide would be similar to Pe [ 1 ] .
The comparing between experiments and pore web flow theoretical accounts is shown in figure 4.17. Even though the pore web informations points did non fall through the whole span of experiments, it is believe that the pore web informations is representative from the experiments. Some restrictions were found when seeking to make geological theoretical accounts with high porousness because absolute permeableness values found were slightly high.
Figure 4.17. Permeability versus porousness comparing experiments with pore web flow theoretical accounts data points.
It was found that when making geological theoretical accounts which was desired to hold porousnesss more than 30 % , the permeableness values observed were truly high comparison to the permeableness scope obtained in Eq. 3.2.
It was established by experimentation by Dodds [ 57 ] that for a random wadding of equal domains the restricting porousness was found to be 36 % .
Figure 4.18. Wettability indices USBM versus Amott-Harvey. a ) Mixed-wet big, fractionally wet and mixed-wet little pore web consequences. B ) Experiments data versus pore web flow theoretical accounts data points.
Figure 4.18 ( a ) shows no correlativity or tendency with the theoretical tendency line proposed by Dixit et
Al. [ 51 ] , in theory the fractionally wet informations points should fall over or near the in-between midst red curve. In the same manner, mixed-wet big stone should fall above the upper thin ruddy line and mixed-wet little below the lower thin ruddy line ( this was showed antecedently in the wettability word picture analysis portion ) .
Experiment information points are observed to finish the span for the USBM index axis every bit good as the pore web flow theoretical accounts data points, nevertheless, they differ in the Amott-Harvey index, figure 4.19 ( B ) .
220.127.116.11. Residual Oil Saturation Sor
Up to day of the month, different literatures have shown conflicting grounds with respects to how wettability influences staying oil impregnation. Residual oil impregnation is affected by several by factors such as topology of the pore infinite, rock-fluid interactions ( which include wettability, soaking up and ion exchange ) , fluid-fluid belongingss ( such as interfacial tenseness ) and flow procedure force balance between syrupy and capillary forces.
Figure 4.19. Residual oil impregnation Sor as map of wettability and pore geometrical factor harmonizing to Smits and Jing [ 2 ] .
The parametric quantity Sor is expected to demo high values ( around 30 % ) or more, in strongly water-wet system whereas in a assorted or oil moisture stone Sor can hold low values as 10 % or even less [ 2 ] .
The x axis is an experimental relationship between wettability index and pore geometrical factor [ 2 ]
In figure 4.19, all wettability conditions represent reasonably good the tendency observed in the experiments [ 1, 2 ] , although seems that wettability status 2 has a somewhat opposite tendency in comparing with the others.
Figure 4.20. Residual oil impregnation Sor as a map of WUSBM for all three wettability conditions and experimental information points. A black curve represents the experiments correlativity.
Correlations for imbibition are based on wettability index USBM from lab experiments [ 6 ] . As good, the same WUSBM from pore web flow theoretical accounts are used in figure 4.20
Figure 4.21. Mobile oil impregnation 1- Swc- Sor as a map of initial oil impregnation 1-Swc under the influence of of I±1 fluctuation ( 0a‰¤ I±1 a‰¤ 1 ) . The bluish colour in the pore web informations points represents I±1 closer to 0 and the ruddy colour I±1 closer to 1. Experiment information points are shown in black with their several tendency line an mistake bounds.
The pore web flow theoretical account informations points suggest in figure 4.22 that the experimental nucleus samples in norm are mixed-wet.
Figure 4.22. Mobile oil impregnation 1- Swc- Sor ( a ) as a map of initial oil impregnation 1-Swc. The ratio ( B ) of existent nomadic oil impregnation and predicted nomadic oil impregnation versus wettability.
In figure 4.22 ( a ) it can be observed that all wettability conditions ( 1, 2, 3 ) represent reasonably good the tendency of lab experiments. However, one time this correlativity is plotted as a mobility ratio against wettability, the tendency of wettability status 2 seems to be contrary.
A set of derivations from So, rabble are shown below, get downing from figure 4.22 ( B ) in order to obtain a relationship that relates wettability index and pore geometrical factor with So, rabble. The process is described as follows:
Assuming that So, rabble is a multiplicative map of Soi, Z and Y it follows:
Ploting So, rabble as a map of Wusbm, figure 4.22 ( B ) , the map that fits the information points is acquired ; but merely if the consequence of degree Fahrenheit is ruling over g and h. Then, the ratio between So, rabble and degree Fahrenheit is plotted against g ( Z ) .
Therefore, the map of H ( Y ) is obtained by plotting the ratio against Y.
The process applied can be observed in figures 4.23 and 4.24
Figure 4.23. The consequence of on the ratio divided by old tendency line
Taking into history the process explained from equation 4.1 to equation 4.3, thenceforth an equation is derived from the relationship between So, rabble, WUSBM and X. This equation is non shown due to confidentially protection.
From figure 4.23, a reasonably good understanding can be seen between all wettability conditions and experiments informations. Below, figure 4.24 shows the nomadic oil anticipation ( using the equation derived from So, rabble, WUSBM and X relationship ) versus existent nomadic oil.
Figure 4.24. Mobile oil anticipation versus nomadic oil of experiment dataset and wettability conditions.
The consequences of the information points on figure 4.24 shows a just inclination of the wettability conditions 1,2 and 3 where the status 3 is the 1 whose informations point correspond closer to the experiments information points.
18.104.22.168. Water Entry Pressure cw and Oil Entry presure carbon monoxide
Primary Imbibition parametric quantities cw and carbon monoxides are considered to be the entry force per unit area of the procedure. Valuess of cw for all wettability conditions were close to zero, e.g. wettability status 3 value cw = 0.0035 saloon ( figure 4.26 a ) . It is proposing that the dominant entry force per unit area for the imbibition procedure is co. However, the value of carbon monoxide is restraint as a negative value of the equation ( combining weight. 1.2 ) . Besides, the negative entry force per unit area for imbibition can ne’er transcend the entry force per unit area for drainage [ 1 ] , because the contact angle can be 180A° ( cos 180A° = -1 ) preponderantly at oil moisture stones.
Figure 4.25. Water entry presure cw, of imbibition V pore geometrical factor. All dataset of pore web flow theoretical accounts and experiments are shown. The thicker black line represent the tendency line from experiments and the thin black lines the mistake bounds.
The pore web flow theoretical accounts data points suggest that the nucleus samples are oil-wet, as can be seen in figure 4.25 where the bulk of the ruddy circles are located nearby cw = 0. As cw addition, the wettability alterations from oil-wet to H2O moisture.
The single consequence from each wettability status can be observed in figure 4.26 ( a ) . All of them seem to hold a close consequences comparing with the experiments, although the experiments values are lower. Same consequences can be seen in figure 4.26 ( B ) when plotting the oil entry force per unit area carbon monoxide normalized, demoing lower values for experiments. This can be explained, from the experimental point of position, that imbibition curves are non really dependable at low impregnations and it was found that the curve tantrum swimmingly utilizing nothing for cw.
The figure 4.27 ( a ) shows little differences between the tendencies observed for wettability conditions and the experimental tendency, nevertheless, in figure 4.27 ( B ) understandings between both is clearly seen but wettability conditions show lower values.
Figure 4.26. Correlations between and swerve suiting parametric quantities cw ( a ) and normalized carbon monoxide for primary imbibition of all wettability conditions and experimental information ( B ) .
Figure 4.27. Correlations between WUSBM and the ratio ( a ) and Predicted ratio versus existent information for carbon monoxide ( B ) where experiments data points are shown against the wettability conditions from the pore web flow theoretical accounts.
22.214.171.124. Oil Curve form factor ao and H2O cuve form factor aw
The parametric quantities ao and aw are known as the curve form factor for primary imbibition, most specifically nearby the residuary oil impregnation and connate H2O impregnation severally. Besides, both ao and aw are related to the pore size distribution index at low oil impregnation and low H2O impregnation correspondingly. The values from ao were found slightly lower than experiments. When ao is between 0.1-0.3 the pore web flow theoretical accounts data suggest an oil-wet status and above that line water-wet s reached, see figure 4.28.
Figure 4.28. Water entry force per unit area cw, for imbibition versus pore geometrical factor. All dataset of pore web flow theoretical accounts and experiments are shown. The thicker black line represents the tendency line from experiments and the thin black lines the mistake bounds.
Figure 4.29. Correlations between and swerve fitting parametric quantity ao for primary imbibition. Wettability group and experiments informations is shown.
The mean value of ao is lower than lab experiments consequences, proposing that some of the reconstructed pore web theoretical accounts has a narrower pore size distribution than the experimental samples and that some of the smaller pores are slightly non included in the pore web theoretical accounts.
During the fitting process of experimental informations, parametric quantity aw was fixed at 0.2, the ground is that at low H2O impregnations the curve is non really dependable from experimental point of position. The web flow theoretical accounts data points observed in figure 4.30, show a different behaviour ; aw values are non changeless and slightly higher that the premise made during the experiments.
Figure 4.30. Water curve form factor aw, for imbibition V pore geometrical factor. All dataset of pore web flow theoretical accounts and experiments are shown. The thicker black line represents the tendency line from experiments and the thin black lines the mistake bounds.
4.4.3. Section Summary
In this subdivision, the capillary force per unit area parametric quantities for drainage and imbibition were compared with experimental research lab informations in different set of secret plans against wettability and pore geometrical factor.
In Primary Drain:
-The capillary entry force per unit area Pe and the irreducible H2O impregnation Sir parametric quantities were in understanding with experimental informations, nevertheless, the curve form factor a showed some dissensions with the mean tendency of the experimental information. The grounds might be associated with the differences seen in figure 4.17, where absolute permeableness and porousness values are in some parts somewhat dissimilar.
-The residuary oil impregnation Sor shows understandings between the wettability conditions and experimental informations except for the wettability status 2. The consequence infers that the more oil-wet the theoretical account becomes, the more Sor is obtained. This might propose that there is non oil movie flow between pores taking to stray pores.
-The nomadic oil vs the initial oil figure suggested that the nucleuss are intermediate-wet which are in understanding with the experimental information.
-The parametric quantities co and cw showed similarities and disagreements between experiments and pore web flow theoretical account informations points. Small values of cw were observed deducing low part of the H2O subdivision to eq. 1.2. and proposing that the dominant portion is the oil subdivision of the equation. Further analysis of these two parametric quantities can be found in chapter 5 of this study.
-The parametric quantities ao and aw were compared demoing differences in both of them with regard to the experimental information. Further analysis of these two parametric quantities ao and aw can be found in chapter 5 of this study.