Pore-scale Network Modeling of Three-Phase Flow Based on Thermodynamically Consistent Threshold Capillary Pressures. II. Results

Transcription

1 Transp Porous Med (2017) 116: DOI /s Pore-scale Network Modeling of Three-Phase Flow Based on Thermodynamically Consistent Threshold Capillary Pressures. II. Results Arsalan Zolfaghari 1 Mohammad Piri 1 Received: 8 February 2016 / Accepted: 20 December 2016 / Published online: 21 January 2017 Springer Science+Business Media Dordrecht 2017 Abstract We use the model described in Zolfaghari and Piri (Transp Porous Media, 2016) to predict two- and three-phase relative permeabilities and residual saturations for different saturation histories. The results are rigorously validated against their experimentally measured counterparts available in the literature. We show the relevance of thermodynamically consistent threshold capillary pressures and presence of oil cusps for significantly improving the predictive capabilities of the model at low oil saturations. We study systems with wetting and spreading oil layers and cusps. Three independent experimental data sets representing different rock samples and fluid systems are investigated in this work. Different disordered networks are used to represent the pore spaces in which different sets of experiments were performed, i.e., Berea, Bentheimer, and reservoir sandstones. All three-phase equilibrium interfacial tensions used for the simulation of three-phase experimental data are measured and used in the model s validation. We use a fixed set of parameters, i.e., the input network (to represent the pore space) and contact angles (to represent the wettability state), for all experiments belonging to a data set. Incorporation of the MSP method for capillary pressure calculations and cusp analysis significantly improves the agreement between the model s predictions of relative permeabilities and residual oil saturations with experimental data. Keywords Three-phase flow in porous media Pore-scale network modeling Cusp formation Threshold capillary pressure Wettability Abbreviations EB G/O Energy balance Gas/oil B Mohammad Piri mpiri@uwyo.edu Arsalan Zolfaghari azolfagh@uwyo.edu; arsalan.zolfaghari@gmail.com 1 Dept. 3295, 1000 E. University Avenue, Laramie, WY 82071, USA

2 1140 A. Zolfaghari, M. Piri LC Layer collapse LF Layer formation O/W Oil/water SGI Secondary gas injection W/ With W/o Without 1 Introduction Network models are developed to predict two- and three-phase relative permeabilities for multi-phase fluid flow in porous media. Over the last two decades, authors have presented several models incorporating various levels of pore-scale physics. A detailed review of the pore-scale network models and their features is provided elsewhere (Piri and Blunt 2005a; Zolfaghari and Piri 2016). Here, we briefly list studies in which network models were developed as a tool to predict experimentally measured three-phase relative permeabilities and residual saturations. Fenwick and Blunt (1998a, b) used regular pore networks to study three-phase flow in water-wet porous media. In particular, they investigated the mechanisms relevant to high oil recovery during various gravity drainage experiments. They used angular pores and allowed layer drainage in their model to mimic quadratic characteristic form for the oil relative permeability at low saturations observed experimentally. The simulated three-phase relative permeabilities and saturation paths were compared qualitatively with those from experimental measurements. Authors found that three-phase flow properties are strong functions of saturation histories, displacement mechanisms, and spreading coefficients. Lerdahl et al. (2000) stochastically modeled the main processes of sandstone formation and transformed the results into a pore network. The authors compared the simulated relative permeabilities successfully against experimental data by Oak (1990). Piri and Blunt (2005b) compared their predictions of oil, water, and gas three-phase relative permeabilities and the corresponding saturation histories against the experimental counterparts in Berea sandstone reported by Oak (1990). Using a Berea sandstone network constructed by Øren et al. (1998), their overall predictions for different phases relative permeabilities were favorable. They, however, systematically overpredicted oil relative permeabilities at low oil saturations where the flow is dominated by spreading oil layers. As a result, the residual oil saturation at the end of gas injection was underestimated. The comparison of the model presented by Al-Dhahli et al. (2013) against the same experimental data continued to overpredict oil relative permeabilities at low oil saturations. Consequently, the experimental residual oil saturation at the end of gas injection was not predicted accurately. Three-phase gas relative permeabilities were not reported for the same experiment. The authors, however, showed improved predictions of oil relative permeability compared to those presented by Piri and Blunt (2005b). By matching the measured two-phase relative permeabilities and capillary pressures for a relatively simple pore network, Svirsky et al. (2007) predicted three-phase relative permeabilities for the reported experimental measurements in water-wet Berea sandstone (Oak 1990). Neglecting the conductance of spreading oil layers in the model, the authors underpredicted three-phase oil relative permeabilities during gas injections. Their respective saturation paths did not agree with the experimentally measured values. In this study, we use three independent sets of experimental data from literature to rigorously validate the prediction of our model, particularly at low oil saturation where the

3 Pore-scale Network Modeling of Three-Phase Flow Based 1141 stability of the wetting and spreading oil layers is simply the most important controlling factor. The impact of thermodynamically consistent threshold capillary pressures and cusp analysis (Zolfaghari and Piri 2016) is investigated by comparing our results against experimental data for both two- and three-phase systems. For each data set, we use one set of contact angles to represent the wettability state of the system. No additional parameters are used for the simulations of different experiments in the same data set. A modified saturation path tracking technique is used to follow the experimental saturation path during gas injection. For those experiments where water saturation reached lower values than the irreducible water saturation of the medium, a special evaporation module is developed to follow the exact saturation path of the experiment. This ensures that relative permeability of each phase at a particular saturation uniquely corresponds to an actual saturation state reached during the experiment. We also compare the results against those predicted by other pore network models. The paper is concluded by a set of final remarks. 2 Validation Against Experimental Data The network model described in Zolfaghari and Piri (2016) is used to simulate two- and three-phase relative permeabilities, capillary pressures, and saturation paths for three groups of experiments performed in different sandstone samples with different wettabilities. Table 1 lists different groups of experimental data used for model validation in this work. We first examine the effect of wetting layer formation (LF) and collapse (LC) on two-phase relative permeabilities and residual saturations. This is done by comparing the results against experimental data reported by Valvatne and Blunt (2004) for waterflooding in an oil-wet reservoir sandstone. We then validate the model against three-phase data, where spreading LF and LC as well as cusp analysis impact the characteristics of relative permeability curves and residual oil saturations. We compare the modeling results against two sets of experimental data generated with weakly water-wet Berea and Bentheimer sandstone core samples. Following the approach proposed by Piri and Blunt (2005b), for each experimental data set, we first match two-phase steady-state measurements of relative permeabilities while honoring Bartell and Osterhof (1927) constraint among contact angles and interfacial tensions. After fixing the system s wettability state, we simulate various three-phase flow processes and compare the simulated three-phase relative permeabilities, residual oil saturations, and saturation paths against their experimental counterparts. 2.1 Two-Phase Flow in an Oil-Wet Reservoir Sandstone As mentioned earlier, an experimental data set representing an oil-wet reservoir sandstone (Valvatne and Blunt 2004) is selected to investigate the effect of thermodynamically consistent threshold capillary pressures and stability of wetting oil layers on two-phase flow properties. The steady-state relative permeabilities were reported for two core samples with fairly similar permeabilities (approximately 300 md). We did not have access to the original rock samples used in the experiments to generate high-resolution images of the pore space and construct equivalent pore networks. Therefore, we used the network modification process proposed by Valvatne and Blunt (2004) to modify a Berea sandstone network used by various authors (see, for instance, Refs. Valvatne and Blunt 2004; Piri and Blunt 2005a; Valvatne et al. 2005; Piri and Blunt 2005b; Suicmez et al. 2007, 2008) to reproduce primary drainage capillary pressure data. We assumed that the main topological features of the Berea pore network and the sandstone samples used in the experiments are similar.

4 1142 A. Zolfaghari, M. Piri Table 1 Three different groups of experimental data used for model validation No. Rock type Wettability Data set Type of LF Type of LC References 1. Reservoir sandstone Strongly oil wet Two-phase Wetting oil layers MSP & geometric Valvatne and Blunt (2004) 2. Berea sandstone Weakly water wet Two- and three-phase Spreading oil layers MSP, geometric, & cusp Oak (1990) 3. Bentheimer sandstone Weakly water wet Two- and three-phase Spreading oil layers MSP, geometric, & cusp Alizadeh and Piri (2014)

5 Pore-scale Network Modeling of Three-Phase Flow Based 1143 Fig. 1 Network modification for the simulation of experimental data. a Pore and throat size distributions for the original Berea network are modified to simulate, b the experimental capillary pressure data for oil primary drainage Experimentally measured primary drainage capillary pressure data were used to modify inscribed radius of the individual elements. The modified network was then used to simulate two-phase relative permeability curves from waterflooding. Experimental values of interfacial tension and contact angles were not reported. Following the methodology described by Valvatne and Blunt (2004), throat size distribution was tuned by a factor representing the percentage of change. The factor could be fixed or linearly regressed between the maximum and minimum values determined by the user. This approach ensures that the ranking order of the throat sizes remains the same after modification. Each pore radius is then modified to maintain the same aspect ratio as those of the pores in the original network. We performed a careful analysis of the parameters and selected the best values to match the capillary pressure data. Figure 1a illustrates the pore and throat size distributions for both initial and modified pore networks. The simulated oil/water capillary pressure generated using the modified pore network is shown in Fig. 1b. The oil/water interfacial tension and PD contact angles used in this simulation were 48 mn/m and 0, respectively. After wettability alteration, core exhibits strongly oil-wet characteristics with low residual oil saturation and Amott water index of 0. We established an initial water saturation similar to the experimental value of 3%. We use MSP-Geometrical-a as the layer collapse rule (see Zolfaghari and Piri 2016 for details). The advancing oil/water contact angles were uniformly distributed between 93 and 172. According to the two-phase thermodynamically consistent threshold capillary pressure results presented in Zolfaghari and Piri (2016), stability of wetting oil layers strongly depends on the wettability. MSP method restricts the span of oil/water contact angles and capillary pressures over which wetting oil layers can exist. Specifically, the new model predicts that the wetting layers are stable on more oil-wet surfaces and at higher oil/water capillary pressures (Zolfaghari and Piri 2016) and, hence, the contact angles used in this work are greater than those reported in Valvatne and Blunt (2004). This is also consistent with the range of advancing contact angles used by Ryazanov et al. (2010) for the simulation of similar data utilizing a network of pores and throats with regular cross sections. Figure 2 shows the simulated oil and water relative permeabilities during waterflooding. As it is shown in Fig. 2b, implementation of thermodynamically consistent threshold capillary pressures for LF and LC not only captures relative permeability trends but also accurately simulates residual oil saturation at the end of waterflood. The simulated Amott water and oil indices after wettability alteration are 0 and 0.145, respectively. These values are in good agreement with their reported exper-

6 1144 A. Zolfaghari, M. Piri Fig. 2 Comparison of the measured (Valvatne and Blunt 2004) and simulated oil and water relative permeabilities during waterflooding in an oil-wet reservoir sandstone: a a linear scale and b a semilog scale to highlight the prediction of residual non-wetting phase saturation imental counterparts (i.e., I w = 0andI o = 0.14 reported by Valvatne and Blunt 2004). One should note that in these experiments, the sample used with the centrifuge method for capillary pressure measurements was different than those used for the relative permeability tests. That explains the difference between connate water saturations in Figs. 1band2. The prediction of residual oil saturation with our model was possible through MSP analysis of the formation and collapse of wetting oil layers. A network model without MSP analysis could still simulate relative permeabilities trend but would underestimate residual oil saturation, see, for instance, simulation of the same experiments by Valvatne and Blunt (2004). The residual oil saturation was underpredicted. In that work, the oil layers were formed and collapsed based on geometric criteria. In other words, the same set of contact angles that captures relative permeabilities trend do not lead to prediction of residual oil saturation unless MSP analysis is implemented for the displacements. 2.2 Three-Phase Flow in Berea Sandstone In order to validate our model for three-phase flow properties, we compare its predictions against three-phase steady-state relative permeabilities, saturation paths, and residual oil saturations reported by Oak (1990)andAlizadeh and Piri (2014). Both data sets use steadystate measurement techniques in which simultaneous injections of two or three phases at different ratios are used to create various experimental saturation paths. In this section, we focus on the Oak s experimental data set (Oak 1990). This experimental data set includes twoand three-phase flow data in Berea sandstone core samples and has been previously studied by Øren et al. (1998), Piri and Blunt (2005b), and Al-Dhahli et al. (2013). Piri and Blunt (2005b) matched two-phase oil/water and gas/oil imbibition relative permeability curves by adjusting relevant advancing contact angles. Having advancing oil/water and gas/oil contact angles, they used Bartell and Osterhof constraint to obtain a distribution for the gas/water contact angles. They successfully matched two-phase gas/water relative permeability with the calculated contact angle distribution. In this work, we use oil/water and gas/oil contact angle distributions similar to those used by Piri and Blunt (2005b). We, however, measured experimentally three-phase equilibrium interfacial tensions for the fluids used by Oak at the experimental pressure and temperature (Table 2). We then used the Bartell and Osterhof

7 Pore-scale Network Modeling of Three-Phase Flow Based 1145 Fig. 3 Comparison of the measured (Oak 1990) and simulated two-phase relative permeabilities in Berea sandstone for the a oil/water and b gas/oil imbibition processes Table 2 Three-phase equilibrium interfacial tensions and spreading coefficient (mn/m) measured for fluids used by Oak (1990) in experiments on high-permeability Berea sandstone samples Fluids σ ow σ go σ gw C s Dodecane with 10 vol% iodooctane brine nitrogen Table 3 Contact angles in degree used for the simulation of various two- and three-phase experiments in Berea sandstone θ PD ow θ r ow θ a ow θ r go θ a go θ r gw θ a gw constraint to obtain gas/water contact angle distribution, which is slightly different than that used by Piri and Blunt (2005b). Table 3 lists the contact angles used in this section. To represent the pore space in Berea sandstone, we use a similar network as the one in Piri and Blunt (2005a, b). To construct this network, a process-based technique founded on stochastic reconstruction of the main sandstone forming processes (Bakke and Øren 1997; Øren et al. 1998; Øren and Bakke 2002) was used. With these contact angle distributions, we were able to match the experimentally measured two-phase relative permeabilities (see Figs. 3a, b and 4). For all fluid pairs, initial saturations of the relevant drainage experiments have been matched. As reported in different experimental studies (Oak 1990; Alizadeh and Piri 2014), gas/oil imbibition relative permeabilities are different than those of experiments in which water is present, i.e., oil/water and gas/water imbibitions. The gas/oil imbibition results were obtained with 80% clay volume of the original network (see Fig. 3b). We assume a smaller fraction of the clay volume contributes to the saturations. The clays are reported to have extremely small pore sizes (i.e., in order of several nanometers). The layer spacing of clay minerals can vary with the clay structure and its water content. As an example, the layer spacings for Wyoming montmorillonites are reported to be between 1 and 1.5nm for different water content values (Boek et al. 1995). Such small pore sizes enhance water retention in clay minerals even under very small controlled relative

8 1146 A. Zolfaghari, M. Piri Fig. 4 Comparison of the measured (Oak 1990)and simulated two-phase gas/water imbibition s relative permeabilities in Berea sandstone humidities (Delage et al. 1998; Cui et al. 2002). In other words, the nanopore space within the clay structure may still be saturated with water even during gas/oil experiments. This is consistent with the experimental observations of higher wetting phase relative permeabilities for gas/oil imbibitions in comparison with those of oil/water and gas/water experiments. We study numerous three-phase experiments reported by Oak (1990).The same set of contact angles, obtained based on the two-phase results (Table 3), is used. We first present our results for one of the experiments that has been studied by other authors to compare the predictions. This would allow us to illustrate the improvement in model s predictive capabilities on three-phase oil relative permeability at low oil saturations and residual oil saturation. We then proceed to validating our results against other experiments in the same data set. But first, we present equilibrium interfacial tensions measured for the fluid systems used by Oak (1990)andAlizadeh and Piri (2014) Measurements of Equilibrium Interfacial Tensions Accurate measurement of interfacial tensions and consequently spreading coefficient affect all cusp analysis and MSP calculations for three-phase flow modeling. Therefore, in this work, we have measured the three-phase equilibrium interfacial tensions for the fluids used by Oak (1990) at experimental pressure and temperature. The values for the fluid system used by Alizadeh and Piri (2014) are reported from that reference. Interfacial tensions were measured by pendant drop/rising bubble technique for dodecane (with 10 vol% iodooctane), brine (with 5 wt% sodium chloride, 0.5 wt% calcium chloride, and 10 wt% cesium chloride), and nitrogen. All three phases were pre-equilibrated for 12 h prior to the measurements. The density values were 63.57, , and kg/m 3 for the equilibrated gas, oil, and brine phases, respectively. Axisymmetric drop shape analysis technique was used to analyze the images of the bubbles/drops. More details about the experimental apparatus can be found elsewhere (see Saraji et al. 2013). The measurement cell was pressurized with one fluid, while the other fluid was slowly injected into the cell through a needle. Different needle sizes were used for each pair of fluids to maintain a bond number of approximately 1. At least 20 independent measurements were performed for each pair of fluids. The average equilibrium interfacial tension values for dodecane/brine, nitrogen/dodecane, and nitrogen/brine are listed in Table 2.

9 Pore-scale Network Modeling of Three-Phase Flow Based 1147 Fig. 5 Comparison of the measured (Oak 1990) and simulated three-phase saturation path of the secondary gas injection for experiment 10 in sample 14 Fig. 6 Comparison of the measured (Oak 1990) and simulated three-phase oil relative permeabilities of the secondary gas injection for experiment 10 in sample Three-Phase Relative Permeabilities and Residual Saturations The new implemented capabilities to handle formation and collapse of layers and cusps make the model more sensitive to the magnitude of initial oil/water capillary pressures for gas injection processes. Oil/water capillary pressure can be kept unchanged during gas injection, if we do not use saturation path tracking steps.

10 1148 A. Zolfaghari, M. Piri Fig. 7 Impact of different models/parameters on the simulated three-phase oil relative permeabilities for experiment 10 in sample 14 reported by Oak (1990). In this figure, w/, w/o, EB, and Cs denote with, without, energy balance, and spreading coefficient, respectively We start with comparing our results with those of experiment 10 in sample 14 of Oak s data set. The saturation path of this experiment indicates that it is a gas injection flow test after a primary oil drainage. A closer investigation reveals that the oil volume flow rate had been reduced by a factor of 8 before gas injection was started (i.e., at the end of primary oil drainage). This explains the higher oil saturation reported for this experiment compared to those of the other primary drainage tests listed for the same sample with similar oil and water flow rates. Reducing the oil flow rate at the end of primary drainage reduces oil pressure drop and consequently reduces the capillary pressure (Pini et al. 2012; Pini and Benson 2013; Akbarabadiand Piri 2015; Zolfaghari2014). To compare our simulated relative permeabilities with those of the experiment, we followed three steps: (1) modeled a primary drainage to an initial brine saturation similar to that of the experiment, (2) adjusted the capillary pressure according to oil flow rate changes in the experiment as mentioned earlier, and (3) modeled gas injection. We then compare the simulated three-phase relative permeabilities with those of the experiment. Figure 5 shows the experimental and simulated saturation paths for experiment 10 in sample 14. The comparison of the simulated oil relative permeability with its experimental counterpart for this experiment is illustrated in Fig. 6. In this plot, the results by Piri and Blunt (2005a, b) andal-dhahli et al. (2013) are also shown for comparison. As it is seen, incorporation of thermodynamically consistent spreading oil LF and LC as well as cusp configurations has significantly improved model s predictive capabilities particularly at low oil saturations. In addition to an improved relative permeability prediction, the model produces closer agreement with the experimental residual oil saturation at the end of gas injection process. Similar to two-phase oil relative permeability results (see Sect. 2.1), previous models tend to underestimate the residual oil saturation by allowing the oil layers collapse at higher gas/oil capillary pressures. As mentioned earlier, the agreement shown in Fig. 6 has been achieved through incorporation of LF, LC, and cusp analysis based on energy balance. A set of three-phase equilibrium interfacial tensions were measured in our laboratory for brine, dodecane, and nitrogen, the

11 Pore-scale Network Modeling of Three-Phase Flow Based 1149 fluid system that had been used by Oak in the flow experiments. The calculated spreading coefficient was negative based on the measured interfacial tension (see Sect ). Here, we investigate the impact of each factor to illustrate the resulting variations in the simulated relative permeabilities for experiment 10 in sample 14. This allows us to demonstrate that only after all the pertinent features are incorporated, one can properly simulate oil relative permeabilities at low oil saturations. Figure 7 presents simulation results for four cases designed to demonstrate the sensitivity of the relative permeabilities to presence/absence of different parameters/capabilities. For comparison, we also include the predictions made by Piri and Blunt (2005a, b)andal-dhahli et al. (2013) for the same experiment. In this figure, EB stands for energy balance analysis for all displacements including LF and LC. W/ and w/o denote with and without, respectively. Cs represents spreading coefficient for the set of interfacial tensions used. As it is seen, any simulation results without energy balance or cusp analysis fail to quantitatively produce the three-phase oil relative permeability and residual oil saturation. We first turn off both LF and LC based on energy balance and cusp analysis, and at the same time use the measured interfacial tensions in the simulation ( W/o EB & Cusp, Cs = measured or case 1). This means that the layers will form and collapse based on geometric criteria. The goal is to find out whether the measured interfacial tensions and the resulting negative spreading coefficient would improve the agreement with the data compared to the results presented by Piri and Blunt (2005b). In this case, new LF/LC and cusp capabilities were not used. Next, we use the interfacial tensions and spreading coefficient used in Piri and Blunt (2005b) (i.e., σ ow = 48, σ go = 19, and σ gw = 67 mn/m) as well as new LF/LC capability (based on energy balance) but without cusp formation ( W/ EB & w/o Cusp, Cs = 0 or case 2). In case 3, we repeat case 2 with this difference that we use measured interfacial tension and spreading coefficient ( W/ EB, w/o Cusp, Cs=measured ). And finally, we simulate a gas injection with all new capabilities included but we use a less negative spreading coefficient ( W/ EB & Cusp, Cs = 0.2 or case 4). In this case, we run the model with both energy balance and cusp analysis for a non-spreading system with a slightly negative spreading coefficient, i.e., 0.2 mn/m. A slightly negative spreading coefficient is used to make the system as close as possible to completely spreading while using cusp analysis. It shows that cusp analysis has the most prominent impact on oil relative permeability at low oil saturations and the residual oil saturations at the end of gas injection. According to thermodynamically consistent threshold capillary pressures (Zolfaghari and Piri 2016), cusp forms and collapses at much lower gas/oil capillary pressures. As it is concluded from Fig. 7, all new developments including energy balance, cusp analysis, and representative values of three-phase equilibrium interfacial tensions affect the model s prediction of three-phase oil relative permeability as well as residual oil saturation. The best agreement in Fig. 7 was obtained only after incorporation of all the relevant factors mentioned earlier. Also, one should note that the simplified representations of pore geometries used in the model could possibly impact conductance of the layers and, hence, their connectivities. Layer conductances in real pore bodies could be reduced due to the variable cross sections, surface roughnesses, and corner geometries. Figure 8 shows the comparison of simulated and measured three-phase gas relative permeability for the same experiment (experiment 10 in sample 14). Impact of the new methodology for handling oil layers/cusps is minimal on the relative permeability of the most non-wetting phase, particularly if gas to water displacements do not compete with gas to oil displacements during gas injection process, which is the case in this experiment as initial water saturation was low prior to the injection of gas. As it is seen, all simulated results show good agreement with those of the experiment.

12 1150 A. Zolfaghari, M. Piri Fig. 8 Comparison of the measured (Oak 1990) and simulated three-phase gas relative permeabilities of the secondary gas injection for experiment 10 in sample 14 Next, we study experiments 25 in sample 13, 13 in sample 14, and 7 in sample 14 from Oak s data set (Oak 1990). The last two experiments relate to gas injections after primary oil drainage, waterflood, and secondary oil drainage, but with a rather important difference with the experiment we have studied earlier. In these experiments, water saturation reaches below the irreducible values reported for the primary drainage experiments in the same core samples. This reduction often occurs after the first introduction of gas into the core sample, i.e., beginning of the gas injection process. After an initial drop, water saturation remains fairly constant throughout the gas injection experiment. We think this indicates water evaporation due to the injection of not fully equilibrated gas at the beginning of experiments. Oak had used fresh gas for the experiments as opposed to recirculation of fully equilibrated gas. In order to account for this reduction in water saturation prior to gas injection, we have developed a procedure in which the saturated clay volume is slightly tuned to match experimental water saturation during gas injection. With this approach, we compare our results against those of three different gas injection experiments. First, we simulate a secondary gas injection process with initial oil saturation of 71.9%, i.e., experiment 25 in sample 13. Since there was no change in oil flow rate before the start of gas injection, no adjustment is made to the oil/water capillary pressure at the end of simulated primary drainage. As mentioned earlier, we use a technique to match water saturation during gas injection process. This allows us to follow the same saturation path as the experiment (see Fig. 9a). The minimum observed water saturation for experiment 25 in sample 13 is 22.5%, which is slightly below irreducible water saturation at the end of primary drainage for the same sample, i.e., 24.6%. Figures 9b and 9c illustrate the simulated oil and gas relative permeabilities for this experiment. As it is seen, we obtain a good agreement with the experimental data, particularly for oil relative permeability at low oil saturations. Both MSP and cusp analysis have been used in the simulation of all experiments presented in this section. Experiment 13 in sample 14 is selected to test the model for simulation of a tertiary gas injection process with initial oil saturation of 73.1%. We think this experiment is a gas injection process after waterflooding followed by a secondary oil drainage, since the closest

13 Pore-scale Network Modeling of Three-Phase Flow Based 1151 Fig. 9 Comparison of the measured (Oak 1990) and simulated secondary gas injection of experiment 25 in sample 13 for a the saturation path, and the three-phase b oil and c gas relative permeabilities (c) previous scan numbers in the experimental data set show water injection into the core. Oil saturation is then brought to 73.1% by a secondary oil drainage. We follow the same flooding steps. We then use the procedure mentioned earlier to match the saturation path. Figure 10a shows the comparison of the simulated and experimental saturation paths. Figures 10b and 10c show the agreement between simulated oil and gas relative permeabilities and their experimental counterparts for this flow test. The non-monotonic trend of oil relative permeabilities observed at the end of gas injection is caused by slight variations in the solutions of systems of linear equations used to compute oil pressure at each pore containing a connected oil phase location. These equations are written for the connected phase clusters to compute relative permeabilities (Piriand Blunt 2005a). As the gas injection proceeds, formation of oil layers and cusps creates flow bottlenecks causing significantly different oil pressures in the neighboring pore bodies. Oil/water capillary pressure was not modified for this experiment since oil flow rate had not been changed compared to the previous secondary oil injection experiment. One should note that the capillary pressure reached during a secondary oil drainage is less than that of the primary drainage for a similar saturation. This might be attributed to the contact angle hysteresis after a primary drainage. The lower oil/water capillary pressure explains why oil relative permeabilities fall lower for a gas injection process after a secondary oil drainage than that of secondary gas injection.

14 1152 A. Zolfaghari, M. Piri Fig. 10 Comparison of the measured (Oak 1990) and simulated tertiary gas injection of experiment 13 in sample 14 for a the saturation path, and the three-phase b oil and c gas relative permeabilities (c) The last experiment we study in this group is experiment 7 in sample 14. The difference between this and the previous experiment is that the oil/water capillary pressure has been adjusted according to the change in oil flow rate at the beginning of the gas injection experiment. This adjustment procedure is similar to the one used earlier for experiment 10 in sample 14. As mentioned earlier, experiment 7 is considered as a gas injection after secondary oil drainage. For the simulation purposes, a complete primary drainage establishing S wi of 24.2% is followed by a complete waterflood (S or = 25.7%). Subsequently, a secondary oil injection is performed to match the reported remaining water saturation of 24.7% prior to the introduction of gas in the experiment. Gas injection is then simulated with the full MSP and cusp analysis included. Since the first reported experimental data point after gas injection has lower water saturation (i.e., S w = 20.3%) than S wirr of the system (24%), the evaporation module is invoked to adjust saturated clay volume and match the corresponding water saturation. For the rest of the reported saturation trajectory, water saturation remains fairly constant. The simulated and experimental saturation paths are compared in Fig. 11a for this gas injection experiment. Figures 11b and 11c illustrate the comparison of the simulated and measured oil and gas relative permeabilities, respectively. The agreements are encouraging.

15 Pore-scale Network Modeling of Three-Phase Flow Based 1153 Fig. 11 Comparison of the measured (Oak 1990) and simulated gas injection of experiment 7 in sample 14 for a the saturation path, and the three-phase b oil and c gas relative permeabilities (c) Saturation Path Tracking All the three-phase experiments that we have studied so far, have one thing in common; they all have relatively low initial water saturations. That makes gas to oil displacements the dominant displacement group during gas injection. This situation changes if the initial condition (i.e., before gas injection) has significantly higher water saturation to be displaced by gas. This means that gas could displace both oil and water leading to rather complicated saturation paths. As it is explained in Zolfaghari and Piri (2016), we use a saturation path tracking technique to follow the experimental saturation path in these experiments. Three gas injection experiments with different initial oil saturations were selected. Cusp analysis and MSP method were used in conjunction with saturation path tracking technique to gauge the model s predictive capabilities for a broader range of gas injection experiments. The first experiment in this group relates to a gas injection process with initial oil saturation of 59% established by an oil drainage process (experiment 9 in sample 13). We track the saturation history of the experiment by simulating a primary oil drainage to create the initial condition for the gas injection. At this point, a complete gas injection is modeled using our saturation path tracking algorithm. The algorithm adjusts oil/water capillary pressure to

16 1154 A. Zolfaghari, M. Piri match the experimentally measured saturation path of the experiment (Zolfaghari and Piri 2016). Adjustments are made at constant gas/oil capillary pressures to prevent any oil to gas displacements. Figures 12a and12b illustrate the simulated three-phase oil/water and gas/oil capillary pressures, respectively, for experiment 9 in sample 13. For comparison purposes, the corresponding two-phase capillary pressures are also included. Three-phase oil/water capillary pressure closely follows that of the drainage curve in two-phase flow. This represents oil to water drainage as proposed in the double-displacement mechanisms, where gas pushes oil to invade water-filled elements. Three-phase gas/oil capillary pressures (see Fig. 12b), on the other hand, reflect threshold gas pressures for the most favorable displacements of gas to oil or water. A gas to water invasion could only happen as a single displacement where it produces water at the outlet. Two different mechanisms, on the other hand, could trigger gas to oil displacements. They are single and continuous-continuous double-displacement mechanisms. The experimental and simulated three-phase saturation paths are compared and shown in Fig.13afor experiment 9 in sample 13. Figure13b shows simulated three-phase oil relative permeability during the gas injection process. The model captures the sharp reduction in oil relative permeability as well as the residual oil saturation. Oil/water capillary pressure at the beginning of gas injection process was not modified since the experimental oil flow rate had remained unchanged from the preceding oil injection experiment. One, however, should note that oil/water capillary pressure was modified during gas injection process due to the saturation path tracking algorithm. Cusp analysis and MSP method always use the latest value of the oil/water capillary pressure reached in the simulation. In other words, saturation path tracking impacts energy balance calculation throughout the gas injection process due to changes in oil/water capillary pressure. The simulated gas relative permeability for the same experiment is presented in Fig. 13c. As it is seen, the first experimental gas relative permeability appears at a lower gas saturation than that of the simulated results. This is attributed to the finite size effect associated with the use of small pore networks. When it is compared against other sets of similar experimental data, experiment 9 in sample 13 exhibits much earlier first relative permeability point. The closest experiment for the same sample would be experiment 8, with the first data point gas saturation of almost three times as that observed in experiment 9. This might be related to a lower initial ratio of gas to oil flow rates used in experiment 9 compared to other similar tests (for instance, see experiments 6, 7, and 8 in sample 13 and experiment 5 in sample 14 reported by Oak (1990)). Figure 13d shows the simulation results of water relative permeability for this experiment. All the figures show good agreement with the experimental measurements. The quality of agreements presented in Fig. 13a d demonstrates the significance of saturation path tracking, whereas oil relative permeability agreement in Fig. 13b validates the importance of cusp analysis and MSP method. The next flow test we investigate is experiment 8 in sample 13. Similar to the previous experiment, this is a gas injection process after a primary oil injection. However, initial oil saturation before gas injection is slightly lower, i.e., 57%, and more importantly the gas-to-oil flow rate ratio is much higher than that of experiment 9 in sample 13. Figure 14a shows the experimental and simulated saturation path of the experiment obtained using our saturation path tracking algorithm. For simulation, the exact saturation history, as explained for the experiment 9 above, has been followed. The only difference is the initial oil saturation at the end of primary oil injection (57%). Figure 14b d illustrates the comparison of simulated oil, gas, and water relative permeabilities with their experimental counterparts. The agreements for oil and water relative permeabilities are satisfactory. We, however, slightly overpredict gas relative permeability. This is attributed to much higher initial gas flow rate (5.52 ml/min) used in this experiment compared to that in experiment 9 (0.183 ml/min).

17 Pore-scale Network Modeling of Three-Phase Flow Based 1155 Fig. 12 Comparison of the simulated two- and three-phase a oil/water and b gas/oil capillary pressures. Two-phase results are obtained for the drainage and imbibition cycles of the corresponding pair of fluids. Three-phase capillary pressures are simulated using the saturation path tracking algorithm for experiment 9 in sample 13 The last flow test studied in this group is experiment 7 in sample 13. This is similar to the previous experiment, but with lower initial oil saturation prior to gas injection, i.e., 48%. The lower initial oil saturation translates to lower oil/water capillary pressure at the beginning of gas injection. As before, we do not modify the oil/water capillary pressure at the beginning of gas injection process. Figure 15a shows the experimental saturation path and the simulated counterpart for this experiment. Experimental and simulated relative permeabilities for oil, gas, and water are presented in Fig. 15b d. Similar to the previous experiment, we overpredict gas relative permeability because of the gas flow rate at the

18 1156 A. Zolfaghari, M. Piri (c) Fig. 13 Comparison of the measured (Oak 1990) and simulated gas injection of experiment 9 in sample 13 for a the saturation path, and the three-phase b oil, c gas, and d water relative permeabilities (d) beginning of gas injection (compare Q g = 7.79 ml/min at the beginning of experiment 7 with Q g = ml/min in experiment 9). Figure 16a shows the comparison of the experimental three-phase gas relative permeabilities for all three experiments presented in this section. Similar gas flow rate steps in experiments 7 and 8 lead to identical gas relative permeability trends in these experiments, while experiment 9 shows higher gas relative permeability for a given gas saturation as it used much lower initial gas flow rates. By comparing Figs. 13c and 14c, one notices the importance of gas flow rate, particularly at the beginning of gas injection, on the magnitude of gas relative permeability at a given saturation. This is independently confirmed through experimental measurements in Bentheimer sandstone in which secondary gas injection (SGI) experiments were performed with the same initial saturation but different gas flow rates, see Fig. 16b from Alizadeh and Piri (2014). SGI is used to refer to a process in which gas is injected into a sample that has previously been subjected to primary oil drainage in order to establish the initial water saturation. The injection sequences in both experiments presented in Fig. 16b are identical. The only difference is the capillary numbers, which is about two orders of magnitude higher for the viscous-dominated experiment, i.e., about In the experiment with lower capillary number, gas finds the least resistance path from inlet of the core sample to the outlet. Being the most non-wetting phase, gas occupies the largest pores in the rock sample spanning from the inlet to outlet with the minimum gas

19 Pore-scale Network Modeling of Three-Phase Flow Based 1157 (c) Fig. 14 Comparison of the measured (Oak 1990) and simulated gas injection of experiment 8 in sample 13 for a the saturation path, and the three-phase b oil, c gas and d water relative permeabilities (d) saturation possible, i.e., 10.8% in Fig. 16b. In the higher capillary number test, however, viscous pressure drop plays a significant role. Consequently, gas does not necessarily occupy the largest elements particularly in areas closer to the inlet of the core. This causes the gas saturation to increase. The increase in gas saturation is not necessarily due to the largest elements of the rock and does not significantly contribute to the connectivity of the main flow path. These two factors lead to a lower gas relative permeability as a function of its saturation for the viscous-dominated flow regime as observed in Fig. 16b. Comparing experiment 8 with 9 in sample 13 of the Oak data set (Oak 1990), a similar observation could be made about the gas flow rate at the beginning of gas injection experiments. The flow rates at the beginning of gas injection in experiment 9 are Q w = 0.02, Q o = 2.56 and Q g = ml/min. In experiment 8, water and oil flow rates are the same as those in experiment 9 but gas flow rate is increased more than 30 times, i.e., Q g = 5.52 ml/min. Gas relative permeability is higher when is plotted against its saturation for the experiment with lower initial gas flow rate, i.e., experiment 9, as explained earlier for the case of experiments presented by Alizadeh and Piri (2014). This explains the model s overprediction of gas relative permeability depicted in Figs. 14c and15c. The model developed in this work assumes capillary-dominated displacements during any fluid injection process. As a result, the model s prediction of gas relative permeability is more representative of experiments with smaller capillary number, i.e., experiment 9, (see Fig. 13c).

20 1158 A. Zolfaghari, M. Piri (c) Fig. 15 Comparison of the measured (Oak 1990) and simulated gas injection of experiment 7 in sample 13 for a the saturation path, and the three-phase b oil, c gas and d water relative permeabilities (d) Fig. 16 Impact of viscous- and capillary-dominated regimes on the measured gas relative permeabilities for a experiments 7 9 in sample 13 reported by Oak (1990)andb SGI reported by Alizadeh and Piri (2014) Interestingly, higher gas flow rates show very little impact on oil relative permeabilities in the experiments studied here. This is supported by the fact that our simulated results agree with their experimental counterparts in both groups.

21 Pore-scale Network Modeling of Three-Phase Flow Based 1159 Table 4 Three-phase equilibrium interfacial tensions and spreading coefficient (mn/m) measured for fluids used by Alizadeh and Piri (2014) in experiments on Bentheimer sandstone samples Fluids σ ow σ go σ gw C s Soltrol 170-brine-nitrogen Three-Phase Flow in Bentheimer Sandstone Two- and three-phase steady-state measurements of relative permeabilities in Bentheimer sandstone have been recently reported by Alizadeh and Piri (2014). All experiments were performed on two core plugs cut from the same block of Bentheimer sandstone. The core plugs had the same dimensions, cm in length and 3.81 cm in diameter. High-resolution micro-ct images of samples cut from the same block were used to construct the Bentheimer sandstone pore network used in this study. The extracted Bentheimer network is 2.7 mm in each direction with 15,664 pores and 28,095 throats. A 67.6% of all pores and throats have triangular cross sections, while 31.6% and 0.8% are rectangular and circular, respectively. The network has 0.82% clay volume, and an average coordination number of Network porosity and permeability are 23.8% and 2.66 D, respectively. For both samples, the average measured X-ray porosity and brine permeability were 24.4% and 2.64 D, respectively. The fluids used in the experiments were Soltrol 170 with 5 vol% iodooctane, brine (2 wt% calcium chloride, 12 wt% sodium iodide, and 0.01 wt% sodium nitrate), and nitrogen. Interfacial tensions were measured by the authors using the pendant drop/rising bubble technique at experimental pressure and temperature conditions. Table 4 lists the measured equilibrium interfacial tensions Two-Phase Results In this section, we present the comparison of our simulated two-phase flow properties with those measured experimentally for primary oil drainage and oil/water, gas/oil, and gas/water imbibition processes. For primary oil drainage, we assume receding oil/water contact angle on the original solid surface is 0. The network is initially fully saturated with water. Oil is then injected to establish the experimentally measured water saturation. Figure 17a shows the comparison of the experimental and simulated relative permeabilities for the primary oil/water drainage process. To simulate imbibition, we use two imbibition experimental data sets (i.e., oil/water and gas/oil) to find the relevant advancing contact angles on the altered wettability surfaces. We adjust the contact angles to match the pertinent relative permeabilities. Similar to the work of Valvatne and Blunt (2004), two different ranges of contact angles are distributed uniformly for each imbibition process. Before simulation, we assign a target volume fraction of pore bodies to mark two distinct groups of elements in the network. The pores are sorted based on their inscribed radii from the largest to the smallest. From the top of the list, the pores are added to the first group (hereinafter called larger group) until the target volume fraction is reached. The rest of the pores are automatically categorized under the second group (called smaller group). Then, the contact angles are distributed uniformly for each group separately based on the assigned ranges. For each throat, if all neighboring pores belong to a group, we assign it to the same group. Throats with neighboring pores belonging to different groups