Network Fluid Modeling Results Analyzed Biology Essay

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In this section, the results obtained from the network fluid modeling are analyzed and compared to laboratory SCAL data correlations [1, 2]. The results are presented in three separate sections, wettability characterization, relative permeability and capillary pressure. All of them are considered to have a distinct importance for fluid flow through porous media. In the first section, the three wettability conditions previously established in section 2 are analyzed using different variables in order to characterized and compare them with experimental results. The following section, describes the network fluid modeling results compared to the work of Smits and Jing [2]. Finally, the section three shows the capillary pressure results from primary drainage and primary imbibition of the networks which are compared to Cense's work [1].

The data obtained from the simulator were fitted using the Corey model for relative permeability, Brooks & Corey approach for drainage [16] and the Skjaeveland model [17] for imbibition capillary pressure. For fitting procedure refer to appendix B.

4.1. Correlation Approach

Using the same correlation approach described in previous experimental works [1, 2], most of the SCAL parameters show a correlation with the pore geometrical factor. The reason is for a given rock-type the main characteristics are the pore geometry and pore size distributions that in turn control permeability. This factor which is originated from the Kozeny-Carman equation reflects the flow and trapping characteristics of the pore networks [2]. Another effect studied in the correlations, is the wettability effect. In this work, contact angle, oil-wet pore fraction, distribution of oil wet pores and their effects in wettability index are used to characterize wettability.

4.2 Wettability Characterization Analysis

Reservoir sandstone rocks usually fall into a large group of intermediate wettability. In the industry there is no such direct test for wettability indication, however, the empirical indicator of wettability measurements most widely used include Amott [21] and USBM [22] as well as the combination of the Amott/USBM methods.

There are some discrepancies between both indexes. The wettability index IAH is defined to be between -1 and 1 whereas this range does not apply to IUSBM, therefore numerical differences for strongly-wetted conditions are expected. Another difference could be seen once they are compared over a wide range of weakly-wetted and intermediate conditions, since IAH reflects spontaneous imbibition whereas IUSBM is derived from the drainage capillary pressure curves; however the relationship between these two is not clear yet. This issue was addressed by Dixit et al. [51] in the late 90's, finding an analytical relationship between the Amott-Harvey index and the USBM index for various wetting scenarios based on oil-wet pore fraction, called in this report as α1 , and their distribution within the pore space. In their work, they assumed a uniform pore size distribution, single contact angles (0° in the water-wet and 180° 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) case, where oil-wet pores are uncorrelated with size.

Mixed-wet case with large pores being oil-wet (MWL).

Mixed-wet case where small pore are oil-wet (MWS).

It was not until 2006, that the experimental conformation of different wettability classes was introduced by Skauge et al. [52], where experimental confirmation of the existence of wettability classes was found using a set of cores from sandstone reservoirs in the North Sea and subsequently acquiring environmental SEM results from USBM and Amott-Harvey wettability indices. Their theoretical explanation of the existence of this wettability classification in real reservoir is described in three possible events. Firstly, the presence of little affinity of the water phase due to mineralogy of reservoir surface. Secondly, the adsorption mechanism found in polar components that may destabilized or precipitate larger molecules. Thirdly possibility described by Skauge as the discontinuity of the water film on the rock surface [52].

Based on Dixit analytical relationship [51] and Skauge [52] experimental findings, wettability classification of the available pore network models are analyzed. In order to establish the relationship between IAH and IUSBM , first a table with variables tested for their impact are summarized within table 4.1.

Table 4.1. Variables varied in the network simulator.

Here, a total of 3210 network flow models are presented to characterize wettability using both index, IAH and IUSBM. The e-Core software allows selecting the distribution of oil-wet pore to be preferentially large pores, preferentially small pores or random. In this way, a third from the total of network flow models generated, correspond to a different distribution for the oil invaded pores which are oil-wet. From figure 4.1 to figure 4.3, a total of 102 points are presented (selected from the 3210 network flow models) in each IUSBM vs IAH plot and representing 34 different geological models (Fontainenbleau, Bentheim, Berea and North Sea reservoir).

The procedure adopted in this part in order to compare the different wettability scenarios to the lab experiments starts with the variation of α1, oil-wet pore fraction (0< α1<1) for each wettability condition. A distribution is presented in each figure representing the frequency of how the oil-wet pore fraction was assigned to each model.

The α1 variation in each figure is shown as a color scale where completely red is assumed to be 1 and completely blue is assumed to be 0. At the same time, different geometric figures represent the different geological models used to generate the pore networks. Figures such as: square shape is used for Bentheim, a rhomb shape for the Fontainebleau models, circle for Berea and triangles for the North Sea reservoir models.

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 respect to the oil-wet pore distribution. The upper line represents the mixed wet large, the middle one the fractional wet and the lower one the mixed wet small. In these analytical relationships, the values of Rmin and Rmax assigned for these lines correspond to 0.3 and 40 µm respectively and assuming a volume exponent of 2.

The figure 4.1(a) shows a distribution where the major oil-wet pore fraction occurrence corresponds when α1 is greater than 0.6. The effect is shown in figure 4.1 (b) where more points with red color appeared. In contrast, in figure 4.1 (c) shows a different distribution where the major oil-wet pore fraction occurrence corresponds to α1 lower than 0.6. As can be seen from figures 4.1 (d), more points with blue color appeared compared to figure 4.1 (b).

d)

Figure 4.1. Oil-wet pore fraction frequency distribution for the three wettability conditions. Higher frequencies values for α1 > 0.6 (a) and α1 < 0.6 (c) and their IAH and IUSBM relationship (b) and (d) respectively.

From figure 4.2 (a), an oil-pore fraction distribution only made with α1 lower or equal than 0.6 is shown and their respective IAH vs IUSBM relationship generated in figure 4.2 (b). The result mostly shows data points in quadrant I (reader refer to figure 4.4 (a)). In contrast, figure 4.2(c) shows an oil-pore fraction distribution where α1 is always greater or equal than 0.6 showing data points in quadrants (I,II,III and IV).

Figure 4.2. Equal oil-wet pore fraction frequency distribution for the three wettability conditions, showing oil-wet pores distribution when (a) α1 ≤ 0.6 and (c) α1 ≥ 0.6. IAH and IUSBM relationship generated from each oil-wet pore fraction distribution (b) and (d).

Figure 4.3 (a) shows a random distribution of the oil-wet pore fraction α1 for all the wettability conditions. The result observed in figures 4.3 (b,c,d and e) were generated using the α1 random distribution of figure 4.3 (a).

d) e)

Figure 4.3. Oil-wet pore fraction with a random distribution used to generate different IUSBM vs IAH plots in order to analize different effects. (b) Oil wet pore fraction from 0 to 1. (c) Wettability conditions effect. (d) Coordination number effect. (e) Oil-wet pore size preferentially large, small and uncorrelated with size effect.

One important effect was shown when the three wettability conditions were presented (figure 4.3c). Wettability condition 3 presented the most oil-wet tendency when data points fall into quadrant II and III. In contrast, wettability condition 2 showed the most water-wet system tendency in which most of its data points fall into quadrant I. Wettability condition 1 showed a mix-wet system with data points in quadrants I,II,III mostly. Taking into account figure 4.3c, the oil-wet pore fraction α1 in figure 4.3a as well have influence in the wettability on the network showing more oil-wet condition when the α1 is close to 1 and water-wet condition when α1 is relatively close to 0.

Figure 4.3 (d) shows the coordination number (z) effect of each pore network model where the blue color represent when z > 3.8 and the black color to z ≤ 3.8. A greater number of data points for z ≤ 3.8 fall into the black continues lines. For the wettability types (random, small, large) in figure 4.3 (e), the data points are scattered over the diagram and they do not follow the wettability classes trend lines proposed by Dixit.

4.2.1. Comparison of the network flow results with the experimental data

Using the same oil-wet pore fraction random distribution as figure 4.3 (a), the network flow data points are shown in figure 4.4 (b). Also, the experimental data from Cense's work and Smits & Jings.

Figure 4.4. Quadrants distribution for IAH and IUSBM relationship (a). IAH and IUSBM data points from Cense [6], Smits & Jing [32] and selected network fluid models using oil-wet pore fraction of figure 5.3 (a) and random pore size distribution. Dashed lines indicate theoretical boundaries for mixed wet large mixed, fractionally wet and mixed wet small (ν=2, rmin=0.3 µm , rmax =40 µm).

As a summary of figure 4.4, some few points could be drawn, such as:

A considerable quantity of data points fall in quadrant I for network flow models in comparison with the experimental results, suggesting more water wet influence due to the inclusion of oil-wet pore fraction of less than 0.5.

A few data points from network flow models fall in quadrant III suggesting an oil wet influence in the group of data.

Quadrant III and IV are the most populated for experimental results, a possible reason might be associated with the exposure to oil-based drilling muds that could cause the smaller pores become oil-wet; this wettability alteration in the experiments was reported by Yan et al. [53].

4.2.2. Section Summary

To conclude this section, differences between the experimental wettability results and the pore networks fluid flow models were seen and discussed when plotting them in a wettability index IUSBM vs IAH graph. The strongest effects observed in wettability of the pore network flow models occurred when the oil-wet pore fraction α1 and contact angles or wettability conditions (1,2 and 3) were varied. When α1 is equal or lower than 0.6 or only wettability condition (2) is present, the pore network flow models are considered to be water-wet. In contrast, when α1 is greater than 0.6 or wettability (1 or 3) are present, the wettability of the networks flow models are considered to be intermediate to oil-wet.

Comparing to theoretical and experimental findings, the pore network flow models fail to show neither the wettability classes [51, 52] when distributing oil-wet elements based on pore size using the software e-Core nor the effect of the coordination number in the networks [51]. In addition, the experimental dataset have intermediate wettability characteristics (figure 4.4b) whereas none of the wettability conditions have a stronger only intermediate wettability.

4.3. Relative Permeability

Simulated relative permeability curves were fitted with the Corey relative permeability model, using a Levenberg-Marquardt fitting algorithm [54].

From lab experiments results, there are some considerations that have to be kept in mind regarding the correlations made by Smits & Jing [2], as can be taken from the following:

should not exceed 250 and only be used with realistic values of φ and k ; the wettability range of the dataset is approximately ranging from 0.3 to 0.7 and their dependency has not been tested outside the range, extrapolation to W=0 and W=1 gives values as expected for completely water-wet or completely oil-wet; all the relative permeability curves have been normalized with respect to Kro_cw such that Kro_cw= 1 (by definition); relative permeability correlations were established for primary imbibition.

Note that wettability has been defined between 0 and 1, to be 0 if the rock is completely water-wet, 1 if the rock is completely oil-wet and between 0 and 1 for intermediate / mixed-wet rock. Hence: W =0.5*(1-w) [2]

Also, all the correlations obtained from lab experiments of all six Corey parameters against wettability, has been made with 4 data points, either WUSBM or WAH are available from those four data points (represented as black color in figures 4.9, 4.11, and 4.13) whereas 10 data points (represented as black circles) from the same correlations are plotted in figures 4.8, 4.10 and 4.12 but using a constant wettability value of 0.40 for sandstone rocks. According to Smits & Jing [2], wettability data could not be obtained from the majority of the fields therefore a constant value for that reason was used.

Table 4.2 Results of regressions from Smits & Jing and Simulations from the three different wettability conditions. All values are averages from all samples and models.

Table 4.2, shows the SCAL parameters obtained from the network flow models and from experimental data. The figures 4.5, 4.6 and 4.7 show the corresponding relative permeability functions in linear scale (a) and in logarithmic scale (b). The latter is plotted to examine the small values for the relative permeability that become indistinguishable from zero on a linear scale. The pink color curves represent the experimental kro and krw . From figures 4.5-4.7 it can be seen that the match of the model curves (wettability 1, 2 and 3) to the experimental curves are fairly good over a broad saturation range within experimental error, however, the end points show slight differences.

b)

Figure 4.5. Relative permeability curves kro and krw comparison from wettability condition 1 and Smits & Jing results. Both axis in linear scale (a) and same figure but relative permeability axis in logarithmic scale (b)

In general, the irreducible water saturation for all three wettability conditions is lower than the experiments results and residual oil saturation is somewhat higher.

Figure 4.6. Relative permeability curves kro and krw comparison from wettability condition 2 and Smits & Jing results. Both axis in linear scale (a) and same figure but relative permeability axis in logarithmic scale (b)

Figure 4.7. Relative permeability curves kro and krw comparison from wettability condition 3 and the results of Smits & Jing results. Both axis in linear scale (a) and same figure but relative permeability axis in logarithmic scale (b)

Therefore, even though the values for all of the six Corey parameters from simulation results are different from the lab experiments (table 4.2), is still possible to get a fairly good approximation.

Table 4.3. Wettability Index from Amot-Harvey and USBM for the three wettability conditions and the wettability index range obtained from lab experiments.

Note that in the experiments used in the work of Smits & Jing [2] wettability was often not measured. In the cases it was measured, it sometimes was derived from the USBM method, in others by Amott or NMR. For that reason, both wettability indexes are used and shown during the analysis with the six Corey parameters.

4.3.1 Corey exponents: water exponent Nw and oil exponent No

The Corey model approach has proposed that the non-wetting phase and wetting phase relative permeability can be described by the power law model [16]. The Corey exponents Nw and No describe the curvature between the end-points, i.e. the shape of the curves.

Figure 4.8 shows how the water Corey exponent changes with respect to the pore geometrical factor under the influence of α1 variation (0≤ α1 ≤ 1). The blue color in the pore network data points represents α1 closer to 0 and the red color α1 closer to 1. All the results from the network flow models are included in this figure. The black data points represent the experimental data from Smits & Jing [2] work and the black line their trend. From the figure, it can be seen that the line representing the experimental trend falls into an area of water-wet to mixed-wet wettability trend of the network flow models data points. In general, figure 4.8 suggests that the water Corey parameter tend to the maximum values when the models are oil-wet.

Figure 4.8. Water Corey exponent as a function of the pore geometrical factor.

The previous figure 4.8 represented a general view of all the pore network flow models whereas the following figures (4.9, 4.11 and 4.13) represent what is happening in each wettability condition using the oil-wet pore fraction with a random distribution (figure 4.3a) among the 34 models created.

In Figure 4.9, the relation between the exponent Nw and wettability index WUSBM is shown for all wettability conditions. The black line represents the correlation made by lab experiments whereas the rest denotes the network flow models data correlations. The x axis is an experimental relationship between wettability index and the pore geometrical factor found by Shell Exploration & Production B.V.

Figure 4.9. Corey exponent Nw vs Jing's [2] correlation using WUSBM (a) and WIAH (b). All the three conditions, w. condition 1, w. condition 2 and w.condition3 are plotted in color blue, cyan and purple respectively.

As figure 4.9 shows, in one hand, wettability conditions 1 and 3 are not in agreement with laboratory experiment correlations. On the other hand, wettability condition 2 showed a relative a fair trend with respect to the latter. Also, there are few data points available to obtain the correlation from lab experiments comparing to the scatter cloud of data points for each condition. According to Jing [2], Nw should decrease when wettability and geometrical factor increase. One of the reason could be assigned to the lower Swc or Sir average values show in table 4.2 for the three wettability conditions comparing to the experimental data. The initial water saturation tends to affect the shape of the relative permeability curves [55].

Figure 4.10, shows how the oil Corey exponent changes with respect to the pore geometrical factor under the influence of α1 variation (0≤ α1 ≤ 1). The blue color in the pore network data points represents α1 closer to 0 and the red color α1 closer to 1. All the results from the network flow models are presented in this figure. The black data points represent the experimental data from Smits & Jing [2] work and the black line their experimental trend.

Figure 4.10. Oil Corey exponent No vs pore geometrical factor.

The experimental data points fall in the area which the pore network model data points suggest that the core samples should be considered mixed to oil-wet.

In figure 4.11, all the network flow model data points resulting from all wettability conditions using WUSBM and WAH are in fair agreement with the trend of experiment correlations, however, the average values of the pore network models data points tend to underestimate the No results from the experimental dataset. One reason might be associated with Sir differences shown in table 5.2 between wettability conditions and experiments. It was reported by Øren at al. [56] that matching the Sir of the pore network with the experiments produces a better approximation of the oil relative permeability curves.

Figure 4.11. Oil Corey exponent No vs Jing [32] correlation using WUSBM (a) and WIAH (b).

The x axis is an experimental relationship between wettability index and the pore geometrical factor found by Shell Exploration & Production B.V.

4.3.2. Relative Permeability End points: Kro_Swc and Krw_Sor

The end-point for water relative permeability is known to be between 0.1 and 0.4 for a water-wet rock and close to 1.0 for completely oil-wet rock [2]. The end-point oil relative permeability is assumed to be equal to 1.0 by definition (industry standard) and was used in such a way in this work.

Figure 4.12, shows how the water relative permeability end point changes with respect to the pore geometrical factor under the influence of α1 variation (0≤ α1 ≤ 1). The blue color in the pore network data points represents α1 closer to 0 and the red color α1 closer to 1. All the results from the network flow models are presented in this figure. The black data points represent the experimental data from Smits & Jing [2] work and the black line their experimental trend.

The experimental data points fall into the area where the pore network data points suggest that the core samples should be mixed-wet corresponding to the values observed by Smits & Jings [2]. However, a discrepancy is observed at lower values of Krw,(Sor) (<0.1) where the pore network flow models suggest that is a oil-wet area.

Figure 4.13 (a) and (b), shows all the curves from the pore network models have a similar trend as the experimental results using either WAH or WUSBM. However, wettability condition 2 is not showing correspondence with the others. The water relative permeability end point of wettability condition 2 suggest that is underestimating the experimental value (see figure 5.6 for reference) even if wettability index and pore geometrical factor are increasing (figure 4.13 a).

Figure 4.12. Water end point relative permeability Krw,(Sor) vs pore geometrical factor.

Therefore, it could be inferred that the effect on wettability changes showed in figure 4.12 , where oil-wet is suggested by the pore network flow models data points when Krw,(Sor) < 0.1, is caused mostly by wettability condition 2 behavior when underestimate the values of the experimental Krw,(Sor). The possible reason of that effect might be associated of the short range selected for the advancing oil contact angle θa for wettability condition 2. Hence, if θa increases, the oil film in the corners of oil-wet pores is stable over a large range of capillary pressure [14] and Krw,(Sor) will increase due to oil film connectivity across the network.

Figure 4.13 Correlations from simulation and lab experiment against WUSBM and X (a). Same correlations against WAH and X (b).

The x axis as well as the previous figures is an experimental relationship between wettability index and the pore geometrical factor found by Shell Exploration & Production B.V

4.3.3. Summary

Theory [19, 20] and experimental data [1, 2] refers as:

Water-wet sandstone rock, when the oil Corey exponent has a lower value No ≈ 2 , water Corey exponent Nw≈4 and Krw <0.15.

Oil-wet sandstone rock, when the oil Corey exponent are higher No ≈ 4 , water Corey exponent Nw < 4 and Krw > 0.5.

Table 4.4. Oil Corey exponent No, water Corey exponent Nw, End-point relative permeability Krw and water connate Swc trends when wettability increased (an increase of W reflects an increase in oil-wetness)

The table 4.4 indicates what is the effect (trend) on each of the parameters (No, Nw , Krw, Swc) when wettability increase. The table enclosed experimental data as well as the pore network wettability conditions 1,2 and 3.

Comparing the theory with the results shown in table 4.4 and the figures 4.8-4.12 some conclusions could be drawn:

Contradictory tendency of the water Corey exponent Nw between the theory and the pore network flow models data points. The contradiction suggests that a poor connectivity of the water in the pores exists when the wettability of the network is oil-wet. In addition, the wettability pattern is not very clear for the pore network flow models data points with regards to Krw,(Sor) parameter compare to theory. Combining the parameter Nw and Krw,(Sor) results from the pore network flow models, the water relative permeability curve does not correspond to the theoretical wettability state of the pore networks flow models.

Good agreement with the oil Corey exponent No between theory, experiments and the pore network flow models data points.

4.4. Capillary Pressure

Knowledge of the functional relationship between capillary pressure and saturation is necessary in order to study and solve the equations that govern fluid flow through porous media containing two or more immiscible fluids. In this work, gravitational forces, together with capillary forces of a porous medium, control the distribution as well as the flow of the immiscible phases. The capillary pressure is originated from the interfacial tension or interfacial free energy that exists between two immiscible fluids. It is dependent on the interfacial tension, pore size and contact angle. This section will be subdivided in primary drainage and first imbibition. The equations that govern each trend line, in all the figures, have been removed due to confidentiality protection.

4.4.1. Primary Drainage

The primary drainage Pc curves were fitted with the Brooks-Corey function [16], which is one of the model curve most frequently used. Hence, eq. 1.1 is used for each model. During the fit procedure, two out of three parameters from equation 1.1 are approximated. Therefore, Pe and a are estimated and Sir is assumed to be equal to Swc .A distinction has to be made when Sir and Swc are considered the same. Theoretically they are not because Swc depends on the height of the oil column in the reservoir whereas the Sir does not. In reality Swc is always equal or greater than Sir. Generally, this assumption is commonly made.

Figure 4.14. Correlations between and curve fitting parameter Pe normalized with σ.cos θ. Best fit and error bounds from experiments [6] are shown in blue. The thinner lines represent the error. The pink circles represent the pore network flow models data points.

Note that during this section (primary drainage), the black circles represent the experiments data and their respective best fit is represented by a blue line (equation is shown in each figure). Error bounds are shown with a thinner blue line. The pink circles represent the pore network flow models data points. For primary drainage parameters, such as Pe, Sir and a, the Levenberg-Marquardt fitting algorithm was implemented [52].

Figure 4.15 shows a fair agreement of the pore network flow models data points coinciding with the experimental trend line. Spite of pore network points represents a small area of the complete dataset; the points are located where the majority of the experiment dataset points are.

For Pe and Sir a power law is shown that represents the experiments data and for parameter a a linear trend fit the data, see figures 4.14, 4.15 and 4.16 respectively. Due to confidentiality protection, the equations of all the curves and their respective errors are not shown from this point onwards.

Figure 4.15 Correlation between and Sir. Data points from experiments in black and data points from pore network flow model in pink.

As a comparison, a fair agreement is shown in figure 4.15 between the experiments trend line or best fit and the pore network flow model data points.

Figure 4.16. Correlations between and curve fitting parameter a for first drainage.

The correlation for curve shape parameter a was not in good agreement with the experimental work. For parameter a larger than 0.5, the pore size distribution can be expected to be narrow, giving a larger plateau in the Pc curve. The pore network flow models data points show the curve shape factor between 0.2-0.4 suggesting that there is more uniformity within the grain size in comparison to real rocks, however, this is not completely consistent with the north sea reservoir rock models that have a small grain size distribution compared with the Berea, Fontainebleau and Bentheim models.

4.4.2. Primary Imbibition

The physics 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 pressure coefficients cw and co would be similar to Pe [1].

The comparison between experiments and pore network flow models is shown in figure 4.17. Even though the pore network data points did not fall through the whole span of experiments, it is believe that the pore network data is representative from the experiments. Some limitations were found when trying to create geological models with high porosity because absolute permeability values found were somewhat high.

Figure 4.17. Permeability versus porosity comparing experiments with pore network flow models data points.

It was found that when creating geological models which was desired to have porosities more than 30%, the permeability values observed were really high compare to the permeability range obtained in Eq. 3.2.

It was established experimentally by Dodds [57] that for a random packing of equal spheres the limiting porosity was found to be 36%.

b)

Figure 4.18. Wettability indices USBM versus Amott-Harvey. a) Mixed-wet large, fractionally wet and mixed-wet small pore network results. b) Experiments data versus pore network flow models data points.

Figure 4.18 (a) shows no correlation or trend with the theoretical trend line proposed by Dixit et

al. [51], in theory the fractionally wet data points should fall over or near the middle thick red curve. In the same way, mixed-wet large rock should fall above the upper thin red line and mixed-wet small below the lower thin red line (this was showed previously in the wettability characterization analysis part).

Experiment data points are observed to complete the span for the USBM index axis as well as the pore network flow models data points, however, they differ in the Amott-Harvey index, figure 4.19 (b).

4.4.2.1. Residual Oil Saturation Sor

Up to date, different literatures have shown conflicting evidence with regards to how wettability influences remaining oil saturation. Residual oil saturation is affected by several by factors such as topology of the pore space, rock-fluid interactions (which include wettability, absorption and ion exchange), fluid-fluid properties (such as interfacial tension) and flow process force balance between viscous and capillary forces.

Figure 4.19. Residual oil saturation Sor as function of wettability and pore geometrical factor according to Smits and Jing [2].

The parameter Sor is expected to show high values (around 30%) or more, in strongly water-wet system whereas in a mixed or oil wet rock Sor can have 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 fairly well the trend observed in the experiments [1, 2], although seems that wettability condition 2 has a slightly opposite trend in comparison with the others.

Figure 4.20. Residual oil saturation Sor as a function of WUSBM for all three wettability conditions and experimental data points. A black curve represents the experiments correlation.

Correlations for imbibition are based on wettability index USBM from lab experiments [6]. As well, the same WUSBM from pore network flow models are used in figure 4.20

Figure 4.21. Mobile oil saturation 1- Swc- Sor as a function of initial oil saturation 1-Swc under the influence of of α1 variation (0≤ α1 ≤ 1). The blue color in the pore network data points represents α1 closer to 0 and the red color α1 closer to 1. Experiment data points are shown in black with their respective trend line an error bounds.

The pore network flow model data points suggest in figure 4.22 that the experimental core samples in average are mixed-wet.

b)

Figure 4.22. Mobile oil saturation 1- Swc- Sor (a) as a function of initial oil saturation 1-Swc. The ratio (b) of actual mobile oil saturation and predicted mobile oil saturation versus wettability.

In figure 4.22 (a) it can be observed that all wettability conditions (1, 2, 3) represent fairly well the trend of lab experiments. However, once this correlation is plotted as a mobility ratio against wettability, the trend of wettability condition 2 seems to be contrary.

A set of derivations from So,mob are shown below, starting from figure 4.22 (b) in order to obtain a relationship that relates wettability index and pore geometrical factor with So,mob . The procedure is described as follows:

Assuming that So,mob is a multiplicative function of Soi , Z and Y it follows:

Plotting So,mob as a function of Wusbm, figure 4.22 (b), the function that fits the data points is acquired; but only if the effect of f is dominating over g and h. Then, the ratio between So,mob and f is plotted against g(Z).

Thus, the function of h(Y) is obtained by plotting the ratio against Y.

The procedure applied can be observed in figures 4.23 and 4.24

Figure 4.23. The effect of on the ratio divided by previous trend line

Taking into account the procedure explained from equation 4.1 to equation 4.3, thereafter an equation is derived from the relationship between So,mob ,WUSBM and X . This equation is not shown due to confidentially protection.

From figure 4.23, a fairly good agreement can be seen between all wettability conditions and experiments data. Below, figure 4.24 shows the mobile oil prediction (applying the equation derived from So,mob ,WUSBM and X relationship ) versus actual mobile oil.

Figure 4.24. Mobile oil prediction versus mobile oil of experiment dataset and wettability conditions.

The results of the data points on figure 4.24 shows a fair tendency of the wettability conditions 1,2 and 3 where the condition 3 is the one whose data point correspond closer to the experiments data points.

4.4.2.2. Water Entry Pressure cw and Oil Entry presure co

Primary Imbibition parameters cw and co are considered to be the entry pressure of the process. Values of cw for all wettability conditions were close to zero, e.g. wettability condition 3 value cw = 0.0035 bar (figure 4.26 a). It is suggesting that the dominant entry pressure for the imbibition process is co. However, the value of co is constraint as a negative value of the equation (eq. 1.2). Also, the negative entry pressure for imbibition can never exceed the entry pressure for drainage [1], because the contact angle can be 180° (cos 180° = -1) predominantly at oil wet rocks.

Figure 4.25. Water entry presure cw, of imbibition vs pore geometrical factor. All dataset of pore network flow models and experiments are shown. The thicker black line represent the trend line from experiments and the thin black lines the error bounds.

The pore network flow models data points suggest that the core samples are oil-wet , as can be seen in figure 4.25 where the majority of the red circles are located nearby cw = 0. As cw increase, the wettability changes from oil-wet to water wet.

The individual result from each wettability condition can be observed in figure 4.26 (a). All of them seem to have a close results comparing with the experiments, although the experiments values are lower. Same results can be seen in figure 4.26 (b) when plotting the oil entry pressure co normalized, showing lower values for experiments. This can be explained, from the experimental point of view, that imbibition curves are not very reliable at low saturations and it was found that the curve fit smoothly using zero for cw.

The figure 4.27(a) shows slight differences between the trends observed for wettability conditions and the experimental trend, however, in figure 4.27(b) agreements between both is clearly seen but wettability conditions show lower values.

b)

Figure 4.26. Correlations between and curve fitting parameters cw (a) and normalized co for primary imbibition of all wettability conditions and experimental data (b).

b)

Figure 4.27. Correlations between WUSBM and the ratio (a) and Predicted ratio versus actual data for co (b) where experiments data points are shown against the wettability conditions from the pore network flow models.

4.4.2.3. Oil Curve shape factor ao and water cuve shape factor aw

The parameters ao and aw are known as the curve shape factor for primary imbibition, most specifically nearby the residual oil saturation and connate water saturation respectively. Also, both ao and aw are related to the pore size distribution index at low oil saturation and low water saturation correspondingly. The values from ao were found somewhat lower than experiments. When ao is between 0.1-0.3 the pore network flow models data suggest an oil-wet condition and above that line water-wet s reached, see figure 4.28.

Figure 4.28. Water entry pressure cw, for imbibition versus pore geometrical factor. All dataset of pore network flow models and experiments are shown. The thicker black line represents the trend line from experiments and the thin black lines the error bounds.

Figure 4.29. Correlations between and curve fitting parameter ao for primary imbibition. Wettability group and experiments data is shown.

The average value of ao is lower than lab experiments results, suggesting that some of the reconstructed pore network models has a narrower pore size distribution than the experimental samples and that some of the smaller pores are somewhat not included in the pore network models.

During the fitting procedure of experimental data, parameter aw was fixed at 0.2, the reason is that at low water saturations the curve is not very reliable from experimental point of view. The network flow models data points observed in figure 4.30, show a different behavior; aw values are not constant and somewhat higher that the assumption made during the experiments.

Figure 4.30. Water curve shape factor aw, for imbibition vs pore geometrical factor. All dataset of pore network flow models and experiments are shown. The thicker black line represents the trend line from experiments and the thin black lines the error bounds.

4.4.3. Section Summary

In this section, the capillary pressure parameters for drainage and imbibition were compared with experimental laboratory data in different set of plots against wettability and pore geometrical factor.

In Primary Drainage:

-The capillary entry pressure Pe and the irreducible water saturation Sir parameters were in agreement with experimental data, however, the curve shape factor a showed some disagreements with the average trend of the experimental data. The reasons might be associated with the differences seen in figure 4.17, where absolute permeability and porosity values are in some parts slightly dissimilar.

In Imbibition:

-The residual oil saturation Sor shows agreements between the wettability conditions and experimental data except for the wettability condition 2. The effect infers that the more oil-wet the model becomes, the more Sor is obtained. This might suggest that there is not oil film flow between pores leading to isolated pores.

-The mobile oil vs the initial oil figure suggested that the cores are intermediate-wet which are in agreement with the experimental data.

-The parameters co and cw showed similarities and discrepancies between experiments and pore network flow model data points. Small values of cw were observed inferring low contribution of the water branch to eq. 1.2. and suggesting that the dominant part is the oil branch of the equation. Further analysis of these two parameters can be found in chapter 5 of this report.

-The parameters ao and aw were compared showing differences in both of them with respect to the experimental data. Further analysis of these two parameters ao and aw can be found in chapter 5 of this report.

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