Corresponding author: O. Gharbi, Department of Earth Science and Engineering, Imperial College London, Prince Consort Rd., London SW7 2BP, UK. (email@example.com)
 We use pore network modeling to study the impact of wettability and connectivity on waterflood relative permeability for a set of six carbonate samples. Four quarry samples are studied, Indiana, Portland, Guiting, and Mount Gambier, along with two subsurface samples obtained from a deep saline Middle Eastern aquifer. The pore space is imaged in three dimensions using X-ray microtomography at a resolution of a few microns. The images are segmented into pore and solid, and a topologically representative network of pores and throats is extracted from these images. We then simulate quasi-static displacement in the networks. We represent mixed-wet behavior by varying the oil-wet fraction of the pore space. The relative permeability is strongly dependent on both the wettability and the average coordination number of the network. We show that traditional measures of wettability based on the point where the relative permeability curves cross are not reliable. Good agreement is found between our calculations and measurements of relative permeability on carbonates in the literature. This work helps establish a library of benchmark samples for multiphase flow and transport computations. The implications of the results for field-scale displacement mechanisms are discussed, and the efficiency of waterflooding as an oil recovery process in carbonate reservoirs is assessed depending on the wettability and pore space connectivity.
 Relative permeability curves describe the averaged flow behavior of immiscible fluids and are universally used in large-scale flow predictions for applications in improved oil recovery, carbon dioxide storage and contaminant transport. Several excellent reviews that have investigated the physics of relative permeability are available [Raza et al., 1968; Anderson, 1987; Morrow, 1990]. In these papers it is demonstrated that relative permeability depends on wettability, pore structure and connectivity.
 The direct measurement of relative permeability is often expensive and, for a single field, generally limited to a single displacement sequence on a limited number of samples at one wettability condition. Digital core analysis is a complement to this activity allowing predictions of multiphase properties to be made for different samples, displacement sequences and wettability to be made easily, once verified against good quality measurements on benchmark samples. The recent development of high-resolution imaging capabilities has enabled different rock types to be studied that serve as the basis for predictions of flow and transport properties [Arns et al., 2004; Knackstedt et al., 2006; Al-dhahli et al., 2011; Blunt et al., 2012].
 It is estimated that more than half of the world's remaining recoverable reserves of conventional oil are contained in carbonate reservoirs [Ahlbrandt et al., 2005]. Several experimental investigations and field results have shown that a large number of carbonate reservoirs present a heterogeneous wettability where a fraction of the pore space is water-wet and the remaining fraction is oil-wet [Treiber et al., 1971]. Brown and Fatt  proposed the term fractional wettability to define a heterogeneous wettability where the wetting preference of the surface is randomly distributed throughout the pore space. Salathiel introduced the term mixed wettability where larger pores are oil-wet whereas the smaller pores remain water-wet. We will use the term mixed wettability to characterize a heterogeneous wettability where the wetting preference of the porous medium is randomly distributed.
 Pore network modeling can successfully predict multiphase flow properties for sandstones, including the effects of wettability [Dixit et al., 1999; van Dijke and Sorbie, 2002; Al-Futaisi and Patzek, 2003; Øren and Bakke, 2003; Valvatne and Blunt, 2004]. Several recent studies have discussed improving traditional network modeling techniques to study carbonates. Sok et al. presented a multiscale imaging methodology that better accounts of pore structure and connectivity in carbonates. Pore space registration was used to combine 3-D and 2-D images using microcomputed tomography, focused ion beam, and backscattered scanning electron microscopy.Ioannidis and Chatzis  presented a dual network model that accounts of interconnected channels in vuggy carbonates by superimposing vugs on matrix blocks. They compared the model predictions to measurements of capillary pressures. Dong et al. compared four different network extraction and reconstruction methods (medial axis, velocity based, grain recognition and maximal ball algorithms) on 3-D images of sandstones and carbonates. Waterflood relative permeability predictions for water-wet cases showed similar behavior for carbonates. They concluded that generally, maximal ball extraction and velocity based algorithms give similar predictions of multiphase properties.
 However, only a few studies have investigated the relative permeability of carbonates. Al-Kharusi and Blunt presented a predictive workflow for carbonate samples based on 2-D scanning electron microscopy images, statistical reconstruction, network extraction and modeling that was applied to a reservoir sample.Zhao et al. used pore network modeling to assess the impact of wettability for networks extracted from different types of rock: a sand pack, a Middle Eastern sandstone, Berea and a granular carbonate. Their results showed that for mixed-wet samples, optimal recovery occurs when a small fraction of the medium is water-wet.Bauer et al.  developed a dual method to account, in an averaged sense, for microporosity in carbonates and presented good predictions of relative permeability.
 The main objective of this study is to analyze relative permeability for a set of carbonate samples with different connectivity and pore structure. The impact of wettability and connectivity on multiphase properties is then highlighted and the generic behavior of relative permeability in carbonates is discussed. First, we briefly present our methodology and present the results obtained from the network models for different wettability. This is followed by a comparison of the computational results with measurements in the literature. Finally, we discuss the implications for field-scale recovery and we assess waterflooding efficiency in carbonate reservoirs.
2. Materials and Methods
 Six carbonate limestone samples are studied, four of which are quarry samples that were obtained from different locations: Portland limestone, an oolitic limestone of Jurassic age, containing peloids with interparticular porosity, which is a standard building material; Indiana limestone, a Mississippian age grainstone, that contains bivalve shell and peloids; Guiting carbonate, a Jurassic limestone, composed of 80% calcite and 17% quartz, where the pore space shows evidence of dissolution; and Mount Gambier limestone from the Oligocene age from Australia, composed of fragments of coral with some calcite. In addition two subsurface samples obtained from a high-salinity aquifer in the Middle East are investigated: Middle Eastern carbonate 1 (ME1) and Middle Eastern carbonate 2 (ME2).
 Small cylindrical cores (5 mm diameter and 25 mm in length) were drilled from larger cores at the native state. Upon the wet drilling, the samples were dried at an oven for approximately 1 h at 30°C to ensure that no liquid was trapped inside the pore space. Dry scans of selected carbonate samples were made at the SYRMEP (Synchrotron Radiation for Medical Physics) beam line at the Elettra synchrotron in Trieste, Italy. The images were obtained using energies between 27–30 keV [Kaiser et al., 2010]. Two detectors were employed, with different effective pixel sizes 4.5 μm (for Portland and Mount Gambier) and 3.85 μm for the rest of the samples. Images were recorded with a CCD camera located at a distance of 50 cm from the sample. The CCD camera binned the results, giving a final resolution of 9 μm (for Portland and Mount Gambier) and 7.7 μm for the rest of the samples. For each topographic scan 1200 projections of the sample were acquired for equally spaced rotation angles over a total rotation of 1800; the scan lasted approximately 4 h. In-house software at the SYRMEP beam line was used to reconstruct the 3-D images and to eliminate noise and ring artifacts. The reconstructed images were composed of around 6003 voxels. Figure 1shows 2-D sections through the 3-D images obtained.
 The 3-D images were first cropped into 3-D cubic images of around 3503 voxels. The exact image dimensions are summarized Table 1. The 3-D images were subsequently segmented into binary images based on a histogram analysis using Otsu's thresholding algorithm in ImageJ software [Sahoo et al., 1988]. Based on the binary images, networks of pores and throats were extracted using the maximal ball algorithm [Dong and Blunt, 2009]. The 3-D visualizations of the extracted networks are shown inFigure 2.
The average coordination number is the average number of throats connected to each pore. The average aspect ratio is the average of the ratio of the pore radius to the mean radius of the throats connected to it. The permeability is computed from a flow simulation through the network.
Voxel resolution (μm)
Number of voxels
Physical volume (mm3)
Number of pores
Number of throats
Total number of elements
Average coordination number
Maximum pore radius (μm)
Average pore radius (μm)
Average aspect ratio
3.23 × 10−14
1.37 × 10−13
5.69 × 10−13
9.40 × 10−13
3.72 × 10−13
2.20 × 10−11
 A detailed description of the extracted networks is provided in Table 1. The samples cover a wide range of average coordination numbers: ME1 and Portland are poorly connected with coordination numbers of approximately 2.5, whereas Guiting and Mount Gambier are highly connected with average coordination numbers of 5.1 and 7.4, respectively. As we show later, the average coordination number (average number of throats connected to a single pore) is a key determinant of relative permeability and residual saturation. It is derived from the network extraction analysis and is an indicator of the connectivity of the void space. The pore and throat distributions of the networks are presented in Figures 3 and 4.
 Thin section analysis and mercury injection porosimetry of some of the samples indicate that the samples contain microporosity. There are small pores below the resolution of the scans (7.7 μm and 9 μm) and the grains are, in most cases, themselves porous. In our methodology we neglect the role of microporosity in the computation of multiphase properties: we assume that microporosity is poorly connected and does not impact our relative permeability predictions. In reality, most of the microporosity will remain water filled throughout the displacement sequence. This will result in an apparent connate or irreducible water saturation which we do not capture and may lead to a larger water relative permeability than calculated here [Øren et al., 1998; Blunt et al., 2002]. This is discussed later in the paper.
2.2. Methodology and Overview of the Network Simulations
 Capillary controlled displacement is simulated using the pore network model developed by Valvatne and Blunt . Initially the medium is assumed to be filled with the wetting phase (brine) and oil is then injected. After oil invasion, we alter the wettability of the pore spaces in direct contact with oil to represent mixed-wet conditions. Waterflooding is then simulated and relative permeability curves are generated.
 We study the impact of wettability in mixed-wet media where some fraction, f, of the pore space occupied by oil is made oil-wet and a fraction 1 − f remains water-wet. We vary the oil-wet fraction from zero (a strongly water-wet case) to 1 (strongly oil-wet rock). In addition to modeling mixed-wet media, this methodology reproduces wettability alteration which is due to asphaltene deposition/precipitation in carbonates. This alteration, governed by oil composition, brine salinity and rock mineralogy is difficult to predict a priori. Where oil has been in contact with the carbonate surface (pores and throats), random contact angles with no spatial correlation are assigned with different distributions, given inTable 2, for the water-wet and oil-wet pores and throats.
Table 2. Input Parameters for Relative Permeability Computationsa
 The 3-D networks are composed of individual elements (pores and throats) with circular, triangular or square cross-sectional shapes. Using square or triangular-shaped networks elements allows for the explicit modeling of wetting layers where nonwetting phase occupies the center of the element and wetting phase remains in the corners. The pore space in carbonates is highly irregular with water remaining in the grooves and crevices after primary oil flooding due to capillary forces. The wetting layers might not be more than a few microns in thickness, with little effect on the overall saturation or flow. Their contribution to wetting phase connectivity is, however, of vital importance, ensuring low residual wetting phase saturation by preventing trapping. Wetting layers of water are always present in the corners, while layers of oil sandwiched between water in the corners and water in the center can be observed in oil-wet regions. Layer drainage is when oil flows in these layers, allowing, slowly, very low saturations to be reached.
 Five wettability distributions are studied: f = 0, f = 0.25, f = 0.5, f = 0.75 and f = 1. Figure 5shows a completely water-wet case (f = 0) as a reference. As expected, water remains in the smallest portions of the pore space, giving very low water relative permeability and significant trapping of oil in the larger pores at the end of waterflooding, mainly caused by snap-off. In the case of poorly connected carbonates (ME1, Portland and Indiana limestones), up to 75% of the pore space can be trapped. However, for the better connected networks, namely ME2, Guiting and Mount Gambier, the water relative permeability is higher and there is less trapping (there are more pathways for the oil to escape), although the residual saturation is around 40% or higher in all cases.
Figure 6shows a mixed-wet case with f = 0.25. A small fraction of oil-wet pores tends to increase the amount of oil trapping, particularly in the less connected networks where now there is little or no range of saturation when two phases flow simultaneously, except very slow flow in wetting layers. The water phase connectivity is reduced and the water relative permeability is in general lower than the strongly water-wet case. The water-wet regions fill first in a capillary-controlled displacement at the pore scale: these are the small pores and poorly connected; however, they surround most of the oil-wet pores that are then trapped. These pores cannot then be displaced during forced water injection, which explains the increase in residual oil saturation. Here again, for the highly connected networks, the water relative permeability is higher since the water has more possible pathways through the system and there is both spontaneous and forced displacement by water.
 When the fractional wettability is 0.5, an equal mix of water-wet and oil-wet pores, at low water saturations, a similar behavior is observed regardless of the connectivity of the pore space (Figure 7). At the beginning of the waterflooding, the water is still poorly connected and flows only through the smallest water-filled pores and thin wetting layers of the pore space; therefore the water relative permeability is low. However, in an equal mix of water-wet and oil-wet fractions of the pore space, depending on the connectivity, an important increase in the water relative permeability is noticeable. After spontaneous imbibition, a significant forced displacement of oil occurs as the oil-wet pores and throats connect through the network. The residual oil saturation is generally lower since oil remains connected in the oil-wet region in layers. This effect is noticeable in the shape of the oil relative permeability for the well-connected samples that show a long region where the oil relative permeability is very low but there is still displacement; this behavior is controlled by slow flow in oil layers [Salathiel, 1973]. The poorly connected samples still show a water-wet controlled behavior, where there is a sharp decrease in the oil relative permeability and significant trapping. Here there is little connectivity of the oil-wet regions and as a consequence layer drainage is unable to achieve low residual saturations. In addition, the maximum water relative permeability varies from very low to very high values dependent on the degree of trapping and the connectivity of the water phase. Where the residual saturation is low, water can fill most of the pore space, and the larger pores in the oil-wet regions, and has a high end point value. A wide range of behavior is seen in this case dependent on the pore structure of the medium.
 When the oil-wet fraction is higher, f = 0.75 (Figure 8), the residual saturation is now very low as the oil remains connected in layers throughout the displacement. The water relative permeability can rise to high values in all cases as the water fills the centers of the larger regions of the pore space. This is a sign of a more typical oil-wet behavior with displacement over a wide saturation range and low relative permeabilities of both oil and water at low saturations of their respective phases, controlled by wetting layer flow [Valvatne and Blunt, 2004]. This behavior is generically similar to network modeling calculations for sandstones [Valvatne and Blunt, 2004; Zhao et al., 2010]. The jumps in some of the curves reflect the relatively small size of the networks studied: improvements in imaging should soon allow larger networks to be constructed.
Figure 9shows the case of the fully oil-wet case (f = 1). The behavior is generally quite similar to that observed for the mixed-wet case (f = 0.75): very low residual oil saturation, a prolonged layer drainage regime (low oil relative permeability at low oil saturation) and high end point water relative permeability.
3.1. Analysis of Results
 In order to summarize the previous description, we analyze the impact of wettability and average coordination number on the relative permeability behavior. The evolution of residual oil saturation with the fractional wettability (Figure 10) shows that the residual oil saturation reaches a maximum for the fractionally wet case with f = 0.25, and then decreases sharply to very low saturations as the medium becomes more oil-wet. Waterflooding gives a high local displacement efficiency for the cases f = 0.75 and f = 1, where the behavior is controlled by oil layers.
 The impact of connectivity on the residual oil saturation is shown Figure 11. The residual oil saturation tends to decrease with increasing connectivity, regardless of wettability.
 One indication of waterflood displacement efficiency that is used to characterize the wettability is the water saturation value at which the oil and water relative permeabilities are equal (Sw where krw = kro) [Craig, 1971]. For water saturations higher than the crossover saturation, waterflooding becomes less efficient, since (for equal viscosities) more water flows than oil. The water saturation at the crossover as a function of wettability for the different carbonate samples is shown in Figure 12. In most cases, the water saturation is highest for the mixed-wet case f = 0.75. This confirms that waterflooding is most effective for mixed-wet carbonates that have preference to an oil-wet behavior. The smallest water saturation at the crossover point is reached for the water-wet and weakly mixed-wet cases (f = 0.25): these are least efficient for waterflooding. This contrasts with traditional analyses of relative permeability that suggests that the crossover point is at more than 50% water saturation for water-wet cases and less than 50% water saturation for mixed-wet or oil-wet samples [Craig, 1971]. We only see this trend in the near oil-wet region; this rule does not apply in general because of the low estimated water relative permeability.
3.2. Comparison With Measured Data
 We compare our computations to measurements found in the literature on reservoir carbonate samples. The approach is not necessarily predictive as scans of the reservoir samples and an independent measurement of wettability are not available: we simply make an assessment if the estimated connectivity and wettability are plausible for the experimental sample studied. Also, the objective of this comparison is not to have a perfect match between the laboratory measurements and the results of the network modeling by fine-tuning the oil-wet fraction or the contact angles; rather, the goal is to determine if our calculated behavior is supported by the available experimental evidence and discuss the impact of wettability and pore structure on field-scale recovery.
 We study three sets of waterflood relative permeabilities measured on Middle Eastern carbonate reservoir samples. A summary of the petrophysical and geological description of the samples is provided in Table 3.
Table 3. A Summary of the Petrophysical and Geological Descriptions of the Reservoir Samples Found in the Literature
Al-Sayari measured steady state waterflood relative permeability on an aged (restored state) reservoir carbonate sample from the Middle East. Through analysis of thin sections, mercury injection capillary pressure and NMR response, the reservoir sample was described as having a well-connected pore structure with a relatively low fraction of microporosity.
 A similar relative permeability to that measured can be observed for the case of f = 0.25 for the well-connected Guiting and Mount Gambier networks (Figure 13). The relatively low residual oil saturation and the shape of the oil relative permeability curve indicate a mixed-wet behavior. For Guiting, the discrepancies in the water relative permeability can be explained by the unresolved micro porosity.
3.2.2. Case 2
Meissner et al. performed detailed measurements on several samples from the Arab-D reservoir of the Dukhan field, onshore Qatar. They reported the results of several of steady state relative permeability tests for oil/brine and gas/oil systems. Results were reported for both native and the restored state cores. The results were reported in terms of normalized saturations and relative permeabilities:
where , the initial water saturation, is determined after primary drainage and is the residual oil saturation determined by extrapolation of the oil relative permeability as it asymptotically approaches zero. This is only slightly different form the true residual oil saturation that is best determined through the waterflood capillary pressure.
 In this case, to introduce an initial water saturation, we set the maximum primary drainage capillary to be equal to 690 KPa (approximately 100 psi). This value is chosen based on the different capillary pressure measurements that showed a sharp increase in the pressure for an average pressure of around 100 psi.
Figure 14shows a comparison between the four measurements reported of water/oil relative permeability on the native state subsurface cores with the relative permeability generated for the strongly oil-wet case f = 1 for ME1. The suggestion here is that the reservoir is strongly oil-wet with a structure similar to that observed in the subsurface sample from which we extracted a network.
3.2.3. Case 3
Okasha et al. reported unsteady state relative permeability measurements on carbonate reservoir samples from the Arab-D reservoir of the Ghawar field in Saudi Arabia. This is the world's largest conventional oilfield. Three data sets were presented for three samples obtained from different areas of the Ghawar field: Utmaniyah, Hawiyah, and Haradh. Here, since the measured values are presented in a nonnormalized form, we simply compare with the data, without changing the initial water saturation.
Figure 15shows a good agreement between the measurements and the relative permeability generated by network modeling for the mixed-wet Mount Gambier network (f = 0.25) for one of the three samples. Note that we suggest that in this field the wettability and pore structure are different from the subsurface Middle Eastern sample in the previous section.Figure 16shows good agreement for the second measured sample with low-connectivity carbonates, i.e., Portland and ME1 for a strongly oil-wet case. The difference of the wettabilities is evidence of local variations of wettability within the reservoir.
 Good agreement was not obtained for the third sample which had high connate water saturation. This limitation in our modeling will be discussed later.
3.3. Implications for Field-Scale Recovery
 In waterflooding, for oil and water of similar viscosity, the saturation at which the relative permeabilities cross, as discussed above, gives a useful and simple indicator of the recovery. For water saturations beyond the crossover point, more water will be produced than oil. Our simulations (Figure 12) indicate that the optimal waterflood efficiency is observed for a mixed-wet system with a large fraction of oil-wet pores, around 0.75. The highest waterflood efficiency is implied for the less well-connected samples, since in these cases the water relative permeability is very low and this holds back the movement of water, allowing oil to be displaced. For better connected samples, there is less sensitivity to wettability and overall a lower crossover saturation, indicating less favorable recoveries.
 This is a somewhat surprising conclusion and implies that waterflooding in mixed to oil-wet carbonates of poor pore space connectivity may be an effective process. This behavior stands in contrast to sandstones, where network modeling studies indicate that more neutrally wet conditions provide optimal recovery [Øren et al., 1998; Valvatne and Blunt, 2004]. Moreover, experimental measurements presented by Jadhunandan and Morrow  have shown that oil recovery by waterflooding in sandstones reaches a maximum at close to neutral wettability.
 However, there is one complexity that we have not addressed. Many carbonate reservoirs are extensively fractured. In these cases, waterflooding rapidly invades the fractures, while recovery from the lower-permeability matrix (the rock studied here) is mediated by a balance of viscous, gravitational and capillary forces. In these circumstances, it is better to have a more water-wet system to allow capillary imbibition into matrix [Blunt et al., 2012]. A discussion of this is beyond the scope of this paper, but we note that both geological structure and multiphase flow properties impact displacement efficiency and overall recovery.
3.4. Limitations of the Analysis
 There are three major limitations of the work we have presented. The first is that the images on which the network extraction is based have a resolution of around 8 μm, meaning that significant fractions of the pore space in intragranular porosity or other small features cannot be resolved. This leads to two problems with our analysis. First, we tend to find rather low water relative permeabilities, since we only allow the water to flow through the larger pores that we capture in the imaging: in reality water is likely to fill most of the micro porosity as well, leading to better connectivity, particularly at low water saturation. Second, for most imposed capillary pressures in primary drainage, as mentioned above, the micro porosity will remain largely water filled, leading to an apparent connate water saturation. This is not captured in our models, which allow the pore space to be drained to effectively zero saturation. To image micro porosity adequately would require resolutions of around 0,1 μm or better, limiting scans to small samples: a realistic model of a carbonate needs to employ a multiscale approach that combines micro porosity, possibly in a statistical sense, with an explicit description of the connectivity of the larger pores [Bauer et al., 2012].
 The second limitation is our treatment of oil layers and pore geometry. We allow oil layers to be present in pores and throats of triangular cross section if it is possible to construct such a layer geometrically. In reality, the formation and collapse of layers is controlled by thermodynamic stability constraints that are, in general, more severe. We tend to underestimate the residual oil saturation in mixed-wet and oil-wet systems. Furthermore, the use of more complex pore shapes, such as mutlipointed stars, allows a richer description of corner and layer flow. These features have been implemented into network models [Ryazanov et al., 2009; Sorbie et al., 2011] but not considered here.
 The third limitation concerns our assignment of contact angle. This is done at random from uniform distributions for water-wet and oil-wet regions. There is not direct confirmation that this is indeed a representative wettability condition. The characterization of wettability at the pore scale remains a topic for future work.
 We have studied the impact of wettability and connectivity on relative permeability for a set of carbonate samples with different pore structures using pore network modeling. Both connectivity and the wetting state of the porous media affect the relative permeability. The impact of wettability alteration is less pronounced in highly connected carbonates. We suggest that waterflooding leads to higher recovery for mixed-wet systems with a high fraction of oil-wet pores (around 75%) compared to strongly water-wet or oil-wet cases. We show that waterflooding leads to lower residual oil saturation in the case of highly connected samples, but that recovery is most effective for the poorly connected samples under mixed-wet conditions.
 A comparison of the generated results with measurements obtained from carbonate reservoirs showed similar behavior and confirmed the capabilities of the network modeling to reproduce the relative permeability observed in mixed and oil-wet reservoir carbonate samples.
 We would like to acknowledge funding from the Qatar Carbonates and Carbon Storage Research Centre (QCCSRC), which is supported jointly by Qatar Petroleum, Shell, and Qatar Science & Technology Park. We also thank Giuliana Tromba and Franco Zanini for their help at the SYRMEP beam line at the Elettra Synchrotron and the two anonymous reviewers.