Role of relative permeability hysteresis in modified salinity water flooding

Many laboratory and field scale trials have shown that modified salinity water flooding increases the mobility of oil and improves oil recovery. However, field scale simulation of the process guided by core flooding data leads to inaccurate prediction of oil recovery if the effect of the saturation history of the reservoir on relative permeability is ignored. Here, we address this problem by proposing three models illustrating the interplay among wettability alteration, hysteresis effect, and fluid flow transport in porous media. The models provide the variation of imbibition and secondary drainage saturation curves versus salinity caused by wettability alteration. Our simulations show that including hysteresis together with the wettability alteration process can better quantify the re-trapping of the oil that is mobilized due to wettability alteration of the reservoir. We further show that in the capillary transition zone, where water saturation varies from top to bottom of the reservoir, oil recovery continues for a long time, although the water cut starts early, in comparison with a reservoir with an initial uniform water saturation. Despite many experiments conducted for advanced waterflooding, our observations show that measuring the secondary drainage and imbibition saturation curves is essential and serves as a valuable input for the field scale simulation of modified salinity waterflooding of water flooded reservoirs with highly non-uniform saturation history.


Introduction
Modified salinity water (MSW) flooding is widely applied to enhance hydrocarbon production due to its availability, affordability, and environmental-friendliness, and its potential in reducing the residual oil [19,26,32].The chemical interaction of brines of different composition alters the mobility of oil and water in the reservoir.The accurate fieldscale simulation of MSW flooding requires an understanding of the mobility of different phases and the hysteresis effects.The impact of the different formulations of these phenomena on the field-scale numerical simulations of MSW flooding is the main focus of this work.
The laboratory and field experiments demonstrate that lowering or modifying brine salinity leads to additional oil production [21,23,24,35].These observations triggered the analysis of salinity's impact on the oil recovery in both carbonate and sandstone reservoirs.Surfactant flooding also can enhance the water injection sweep efficiency by reducing the interfacial tension between oil and water [20].Hot water injection induces a wettability change in the reservoir, hence, may lead to good recoveries [1].To link these observations to the recovery in the field scale, the experiments are used to tune models of phase mobility implemented in reservoir simulators.
The empirical relative permeability is commonly employed to relate the saturation of fluids phases with their movement in the porous media.Fluid movement for each phase depends not only on the saturation of each phase but also on the saturation history (so-called hysteresis).Hysteresis is an essential factor for modeling of water injection into the reservoirs and plays a crucial role in displacing oil towards the production wells.For example, significant unrealistic results may be obtained by reservoir modeling, particularly in gas reservoirs, when the impacts of hysteresis are ignored [7].Erroneous recovery factors up to twice as high as the field data is observed without considering the effect of hysteresis and by only using the drainage data in a gas reservoir with a strong water drive [17].Also, hysteresis is vital for modeling the capillary transition zone, where water saturation varies from top to bottom of the reservoir in the presence of substantial oil.Neglecting the hysteresis effect in such a model leads to a significant underestimation of oil recovery [13].
Experimental results [33][34] from capillary hysteresis measurements and the history-matching of unsteady-state coreflooding experiments show that the injection of MSW leads to more substantial capillary and relative permeability hysteresis compared with the injection of high salinity water.This implies that different hysteresis parameters are required for high and low salinity water.Fig. 1 illustrates why hysteresis becomes essential during the injection of MSW under the wettability alteration process.For example, in the case of injection of MSW in tertiary mode (i.e. after conventional seawater (SW) flooding), a new oil bank is formed due to the mobilization of new oil by MSW and therefore increasing oil saturation (Fig. 1) downstream.Fig. 1b shows that the second "oil bank" formed by MSW injection moves in a mixing zone, which may have a different secondary drainage curve than the initially formed oil bank by SW injection.From a modelling perspective, one needs to decide whether new imbibition and secondary-drainage saturation curves should be used for MSW injection and how they are influenced during the dynamic alteration of the mobility of phases often attributed to the compositional wettability change.Furthermore, most reservoirs, particularly carbonates, show oil-wet and mixed-wet wettability with high remaining oil saturation, making them good candidates for MSW flooding.Oil-wet and mixed-wet systems, as well as capillary transition zone and three-phase systems, regularly display even more hysteresis between imbibition and secondary-drainage curves that result in different flow behavior without including relative permeability hysteresis [14].On this account, another question to consider is whether the laboratory results (spontaneous or forced imbibition without considering hysteresis) can represent advanced water flooding processes for a full-field scale where the hysteresis effect may play a critical role because of nonuniform distribution of water saturation.
This article investigates the impact of hysteresis in relative permeability during dynamic fluid phase mobility alteration due to MSW flooding and its impact on the ultimate oil recovery.This article follows with an introduction of hysteresis in relative permeability, change of mobility of phases due to wettability alteration, and MSW flooding.Then several numerical test cases in different scales are introduced to illustrate the role of relative permeability hysteresis.Finally, we discuss the simulation results under the various conditions of initial water saturation and relative permeability hysteresis.

Methodology
In the simulation of MSW flooding, the common practice is to assign a new set of relative permeability curves to the MSW-oil system.Here, the method applied in reservoir simulations for switching between relative permeabilities of MSW-oil and SW-oil or FW-oil, the concept and formulation of hysteresis, and the procedure for simulating MSW flooding along with the hysteresis effect are described.Furthermore, several numerical test cases are presented to evaluate the hysteresis impact on the performance of MSW flooding.

Relative permeability hysteresis
Land [16] links the hysteresis phenomena to the oil trapping in porous media and then comes up with its effect on relative permeability.Since then, all developed models for representing hysteresis have been based on Land model, such as Carlson and Killough methods.Carlson and Killough methods are used as an industry-standard approach to capture the hysteresis effect, enabling different saturation functions for drainage and imbibition processes.In the Carlson method, the scanning curves are produced in parallel with the imbibition curve [6].Therefore, the scanning curves can be obtained by merely shifting the imbibition curve to the point where the maximum non-wetting phase saturation (S hys ni ) cuts the drainage curve (Fig. 2).However, in the Killough method, more input data and calculations are required.To obtain the scanning curve for a specific saturation, the following equation is used for a given where S nr is the residual saturation of the scanning curve and S Dra nr is the residual non-wetting saturation of the drainage curve(Fig.2), and C is calculated by.
where S max n is the maximum non-wetting saturation and S imb nr is the residual non-wetting saturation of the imbibition curve.Knowing the residual saturation of the scanning curve (S nr ), the relative permeability for a particular saturation (S n ) on the scanning curve is obtained by.
where k Imb rn and k Dra rn denote the relative permeability values on the bounding imbibition and drainage curves, respectively, and.

Relative permeability hysteresis and wettability alteration
The change of mobility of the fluid phases are often linked to the wettability alteration of the rock due to the different physicochemical interactions between the rock and the injected and formation water.The wettability of the rock surface controls the location, spatial distribution, and fluid flow in a porous medium, thus impacting relative permeability [4].Therefore, wettability alteration, as one of the mechanisms suggested for MSW flooding, changes the initial relative permeability curves of the fluids in the reservoir.The new shape of the relative permeability curve depends on the ability of MSW to interact with the rock surface, oil, and formation brine, which alters the affinity of the mineral surface from oil-wet towards the water-wet [22].As a rule of thumb, moving from an oil-wet state towards a water-wet state increases Fig. 3. Effect of wettability on relative permeability parameters; the wetting fluid is generally distributed in the small pores or as a thin film on the mineral surface, while non-wetting fluid occupies the center of the large pores [4], which means that the wetting fluid tends to move through the small pore with lower permeability and give more space to the non-wetting fluid in the large pores.The relative permeability of a non-wetting fluid is higher compared with the wetting fluid, e.g. in the water-wet system, oil flows easily in the larger pores, and in most cases, the relative permeability of oil generally approaches the absolute permeability because the water fills only small pores without blocking the oil flow.Irreducible water saturation increases in a water-wet system as more water remains in the small pores and forms a thin film over the mineral surfaces (Raza et al., 1968).

Fig. 4.
Relative permeability of oil and water (High and modified salinity) for drainage and imbibition used in the current study a) model-1, only imbibition curve is salinity dependent b) model-2, the salinity causes an increase of maximum oil relative permeability for both imbibition and drainage curves c) model-3, in addition to the change of maximum oil relative permeability for both imbibition and drainage curves, S wc is also a function of salinity.(HSW: high salinity water such as formation water and seawater, MSW: modified salinity water, Dra: drainage, Imb: imbibition).
Several experimentalists have investigated the effect of wettability alteration on the fluid phases mobility during EOR process.The experimental results of Tang and Morrow [30] and the study of Jerauld et al. [14] show the same water relative permeability for both modified and high salinity water, while the residual oil saturation is lower in modified salinity injection compared with high salinity.Besides, the coreflood experiments conducted by Geffen et al. [11] and Braun and Holland [5] for water-wet systems and Fatemi and Sohrabi [10] for mixed-wet systems showed that the hysteresis effect is much larger for the non-wetting phase (oil) compared to wetting phase (water), particularly for low IFT oil types.
Recent works of history matching of coreflooding experiments [18,25,28,33] and the change of relative permeability parameters due to the wettability alteration process (Fig. 3) have led us to consider three sets of relative permeability where the maximum imbibition and drainage relative permeability of oil can be independent of (Fig. 4a) or dependent on (Fig. 4b and c) salinity.All models can represent changing from oil-wet/mixed-wet state to water-wet state by reducing 4% of residual oil saturation.In model-2, the salinity causes an increase of maximum oil relative permeability (both k 0 ro and k 0 rw ).However, in Model-3, in addition to the change in k 0 ro and k 0 rw , we also change the corresponding S wc as a function of salinity.Among these three models, Fig. 4c shows a high potential of modified salinity for changing the wettability from oil-wet to strongly water-wet condition.Also, we assume that water relative permeability does not show hysteresis.However, since we assume wettability alteration as the primary mechanism in MSW flooding, water relative permeability of modified salinity is shifted to the right.This implies that water relative permeability is the same at high salinity and modified salinity residual oil saturation.
Table 1 shows Corey parameters for the relative permeability illustrated in Fig. 4.

Salinity dependence of relative permeability
A common method in MSW flooding simulation is to change the relative permeability and capillary pressure curves as a function of salinity.Jerauld et al. [14] suggested interpolating between low salinity and high salinity curves to obtain the oil and water permeability and capillary pressure based on the salinity in each grid block (Fig. 5a).Therefore, the final relative permeability and capillary pressure values at a specific saturation are calculated through the weighted average of low salinity and high salinity curves (Fig. 5b).A detailed review on the topic of interpolation between high salinity and low salinity can be found in Al-Ibadi et al. [2] and Hosseinzadeh et al. [12].Note that tentimes diluted SW (10TDSW) was chosen based on the experiment conducted by Mokhtari et al. [23].

Table 1
Corey relative permeability parameters for model-1, model-2, and model-3 (HSW: high salinity water such as formation water and seawater, MSW: modified salinity water) (two significant digits are given for better reproducibility of our numerical results).

Configuration of numerical experiments
We implement a new method in MATLAB, enabling Schlumberger ECLIPSE 100 to simulate the salinity effect in the presence of the hysteresis effect.First, by activating the hysteresis option, Eclipse is run to solve multiphase flow and salt transport in each time step.Then, in the MATLAB interface, the new salt concentration in each grid block is used to assign new saturation functions for imbibition and secondary drainage curve to each grid block.It is done by updating the ECLIPSE data file in each time step.This procedure captures both the effect of wettability alteration and the hysteresis.The conceptual model used in this study (Fig. 7) bears a close resemblance to the one proposed by Taheriotaghsara [29].For the sake of simplicity, the same relative permeability is assigned for formation water (FW) and seawater as high salinity water.We only consider relative permeability hysteresis due to its essential impact on mobility ratio toward the oil recovery without including the capillarity effect.It is also assumed that water and oil viscosity is the same to eliminate/reduce the numerical instabilities due to the combined effect of viscosity ratios and relative permeability change [32].Table 2 shows the rock and fluid properties that are used for the simulation of MSW flooding.

Initial water saturation distribution
We simulate the two-phase flow under both uniform and nonuniform initial water saturation to obtain results that are easy to analyse and realistic.To that aim, we define three states (Fig. 8) with the same average initial water saturation of 0.3 as follows: • Case 1: constant distribution of initial water saturation between injector and producer • Case 2: initial water saturation randomly distributed between 0.1 and 0.5 • Case 3: initial water saturation varies from 0.5 to 0.1, starting from the injector to the producer • Case 4: initial water saturation varies from 0.1 to 0.5, starting from the injector to the producer (opposite to case 3) • Case 5: vertical capillary transition zone between oil and water zone varies from 0.1 to 0.626 Vertical capillary transition zones can be present in oil reservoirs (e. g., the North Sea chalk reservoirs) [3].The transition zone is usually saturated with oil and connate water at the top and fully saturated with water at the base.The thickness of the transition zone varies from a few meters to several hundred meters that usually contains a substantial amount of oil [13].The transition zone between oil and water zones with a height of 80 m (case 5) was inspired by Halfdan reservoir located approximately 250 km west of the Danish west coast in the North Sea (Fig. 8b).

Results
Here, we first highlight the importance of the hysteresis effect during the injection of MSW flooding.Then the effect of initial distribution of water saturation, changing the relative permeability due to wettability alteration towards water-wet condition, and the performance of MSW flooding under secondary or tertiary model are discussed.Note that all simulations are performed in 1D unless stated otherwise.

Constant initial water saturation (model-1 and model-2)
Three different constant uniform initial water saturation (S wi = 0.1, 0.3, and 0.5) are chosen in order to investigate their impact on the shape of scanning curves.Fig. 10 presents the simulated scanning curves of Kilough method between the bounding curves in the middle and last grid blocks of the 1D model under seawater injection.When S wi = 0.1, the results show that the calculated scanning curve continues following the bounding imbibition curve until it reaches the residual oil saturation.However, for the other two cases (S wi = 0.3 and 0.5), the calculated scanning curves start at the bounding drainage curve and scans down to  the bounding imbibition curve but with different residual oil saturation.Note that the same water relative permeability is used for drainage and imbibition.
Obviously, the residual oil saturation is reached more rapidly in the middle grid block (Fig. 10a and b). and takes longer for the last grid block (Fig. 10c and d).Remarkably, oil relative permeability has a high value as the initial water saturation decreases ) at the same injected PV.Comparing two initial wettability conditions (model-1 and model-2 in Fig. 4), regarding the use of for secondary and tertiary MSW flooding, it is found that the oil moves slightly quicker in model-2 than model-1 because oil relative permeability for model-2 is higher than model-1 at the same injected PV (Fig. 11 and Fig. 12).In the secondary flooding (Fig. 11), the calculated scanning curve leaves the bounding drainage curve at S wi = 0.3, moving to the bounding imbibition curve until the MSW front reaches the middle of the reservoir.Next, due to the impact of MSW, a new bounding imbibition curve (model-1 and model-2) and drainage curve (model-2) are assigned to those grid blocks that are affected by the change of salinity.MSW triggers the scanning curve a bit up and then falls slightly towards the newly allocated bounding imbibition curve until it reaches the residual oil saturation, which is smaller than residual oil to seawater.In tertiary flooding (Fig. 12), due to the formation of a new oil bank by the MSW, the calculated scanning curve for MSW follows the same path of SW up to a certain point and then declines gradually to new residual oil saturation.

Wettability alteration towards strongly water-wet state (model-3)
In this section, we model a water flooding process that changes the rock wettability towards the strongly water-wet condition under the effect of hysteresis.In the strongly water-wet condition, the maximum oil relative permeability in k r -S w plot moves to the right and up because the water is located in the small pores and therefore does not block the oil flow through the larger pores.Thus, the oil relative permeability is high and frequently reaches the rock's absolute permeability [4].Fig. 13 illustrates that when the dynamic wettability alteration occurs towards a strongly water-wet condition (model-3), less oil is produced compared with model-2 and model-1.When the MSW is injected into a reservoir with S wi = 0.3, the scanning curve leaves the primary drainage curve towards the imbibition curve (Fig. 14).The MSW imbibition and drainage curves are assigned to the grid block influenced by the change of salinity of MSW.Therefore, the scanning curve touches MSW imbibition curve earlier due to the fact that the MSW imbibition curve is shifted to the right.As a result, the scanning curve follows the MSW imbibition up to its residual oil saturation, which means that no further remaining oil saturation is mobilized.It implies that some of the remaining oil that is released by MSW is trapped again by snap-off or Fig. 14.Relative permeability hysteresis of modified salinity injection when wettability alteration changes towards the strongly water-wet condition (model-3) during secondary MSW flooding at grid block i = 250 when the initial water saturation is 0.3 a) relative permeability hysteresis b) fractional flow curve (water-cut).The color bar on the left side illustrates the timing of simulated data in the number of injected pore volumes.bypass trapping phenomena, which is in consistent with the experiment results [33].

Non-uniform initial water saturation (case 2-4)
Given the initial water saturation distribution, some grid blocks initially experience an imbibition process, then drainage or vice versa.From Fig. 15, it can be seen that residual oil saturation for Case 3 is smaller than Case 4 due to the starting point of initial water saturation.For Case 4, initial water saturation around the injection well is similar to the irreducible water saturation, which causes the scanning curve to start from the imbibition curve.However, for Case 3, the scanning curve leaves the drainage curve towards the imbibition curve, leading to new residual oil saturation (Eq.( 1)).Consequently, the ultimate recovery factor for Case 3 is higher compared with Case 2 and Case 4 (Fig. 16).The most striking result shown in Fig. 15 is that the shock front for Case 3 is ahead of the other two cases since the initial water saturation around the injection well is higher, equivalent to the high relative permeability of water.

MSW performance in the transition zone (case 1 and 5)
In this section, the transition zone depicted in Fig. 8b with average water saturation of 30% is used to investigate the impact of hysteresis during the MSW flooding process in a 2D model.Fig. 17a demonstrates that most injected seawater preferentially moves to the bottom of the transition zone, where the reservoir is close to residual oil saturation, and subsequently, the relative permeability of water is high in this region.Compared with seawater injection, MSW injection shows rotated omega-shaped behavior with an adequately more stable shock front than that formed by formation water ahead of MSW shock front (Fig. 17b).This leads to a uniform sweep of remaining oil towards the oil production well in a transition zone where water saturation is distributed unevenly.The stabilized omega-shaped behavior of the MSW shock front can be explained as follows: when MSW is injected as secondary flooding, formation water is mobilized and travels ahead of the MSW front, creating a large water saturation difference between the top and bottom of the transition zone.In contrast, the MSW bank moves behind the    formation water bank, where most oil is swept with formation water, making water saturation distribution more uniform.Note that higher viscosity contrast between oil and water worsens the sweep efficiency; therefore, we assume the viscosity of both water and oil are the same to study only the relative permeability impact.
The comparison of oil recovery between the transition zone (case 5) and constant uniform water saturation (case 1) with the same initial water saturation of 30% shows that MSW injection in the transition zone produces slightly more oil in the long term (Fig. 18).Although water is produced at the beginning of the injection of MSW into the transition zone, it takes longer to reach 100% water production compared to the case with initial constant uniform water saturation.

Comparison of hysteresis methods
Several standard methods are used in commercial reservoir simulators to predict accurate relative permeability hysteresis that scales the scanning curves within the bounding curves at the specified S wi [27].Here, we compare two widely used hysteresis models (Carlson and Killough methods) to predict the relative permeability hysteresis for MSW injection.
As illustrated in Fig. 19, the Carlson approach produces a scanning curve parallel to the imbibition curve.So, one can simply shift the imbibition curve horizontally until it cuts the given S wi .Comparison between Carlson and Killough methods (Fig. 10 and Fig. 19) shows that the scanning curves generated by the Killough method are steeper than the Carlson one.In terms of injection of SW or MSW, the Carlson method predicts lower residual oil saturation and subsequently higher oil recovery in comparison with the Killough method.Fig. 20 shows that the recovery factor indicated by the Carlson procedure is approximately 9 percent higher than the one estimated by the Killough procedure.Future studies on these methods are required to validate the proper hysteresis method during MSW injection.Measuring the mobility of phases in primary drainage and imbibition at various initial water saturations in core flooding experiments helps to elucidate this matter.

Discussions
The present paper aims to investigate the impact of relative permeability hysteresis under the wettability alteration process.Therefore, two mechanisms are involved in the revised model of modified salinity water flooding, wettability alteration and hysteresis.On the one hand, wettability alteration explains the flow behavior when salinity changes from high to low, leading to the detachment of oil from the rock surface, which in turn increases the mobility of oil and facilitates its flow towards the production well.This increase of oil mobility can be incorporated in the model by changing various relative permeability parameters but eventually, we can only alter one set of relative permeability curves for a specific displacement sequence, i.e., when modified salinity displaces oil.On the other hand, hysteresis describes the ability to capture the amount of oil that is trapped during any displacement sequence.We have shown that there are several displacement sequences in MSW flooding that must be captured by a hysteresis model.The experiments conducted by Wang & Alvarado [33][34] show that hysteresis is influenced a lot by salinity change from high to low salinity.They concluded that Low-Salinity brine makes the reservoir rock more water wet, leading to snap-off trapping of oil.As stated in the wettability alteration towards strongly water-wet state (model-3) section, the revised model captured well the experimental observation, which implies that some of the remaining oil that is released by MSW is trapped again by snap-off or bypass trapping.These phenomena cannot be easily described without including hysteresis in the relative permeability observed in the experiments.Moreover, this research has raised many questions that require further experiments for measuring the drainage and imbibition relative permeability curves during advanced water flooding (such as surfactant injection, modified salinity water injection and increasing the temperature of injection water), where wettability of the surface rock changes from oil-wet to water-wet.Such experimental studies help us to develop a better model for predicting the performance of advanced water floodings at the field scale, even though the underlying mechanisms of MSW flooding still have not been clearly identified.On the reservoir scale, heterogeneity of rock negatively affects the areal and volumetric sweep efficiency of waterflooding.However, when MSW is injected as a tertiary flooding method, spatial variation of brine composition is also added to the rock heterogeneity that make the system even more complicated.Changing the wettability of rock surface by shifting the relative permeability towards more water-wet also causes the change of mobility ratio between displaced and displacing fluids that may change the flow path of MSW to the zones unswept during seawater injection.
These results also provide further support for the hypothesis that including the hysteresis not only delivers considerable insight into the trapping phenomena but also may change both drainage and imbibition curves, leading to earlier oil breakthrough.The breakthrough speed is impacted by the maximum relative permeability of oil and water in imbibition and secondary drainage and the distribution of water saturation between injection and production wells, as shown in this study.An example of different speeds of breakthrough curves was observed for the exact same average remaining oil saturation between injection and production wells but different distribution, which is due to the impact of hysteresis.

Conclusions
This study has shown that changing the shape of the relative permeability by MSW for both imbibition and drainage has a considerable impact on the predicted oil recovery, as high as the impact of reducing residual oil saturation.The mobilized oil by model-2, where the wettability alteration causes an increase of maximum oil relative permeability without changing the corresponding irreducible water saturation, is slightly more and moves faster to the production well.One of the more significant findings of this study is that changing the wettability from oil-wet to strongly water-wet condition (model-3), caused by the MSW effect, traps some of the mobilized oil.Consequently, less oil is produced when the rock surface becomes strongly water-wet during the wettability alteration.For the non-uniform initial water saturation distribution, many grid blocks experience an imbibition process followed by drainage or vice versa signifying the importance of including hysteresis (scanning) curves in the model.For the same average initial water saturation, additional oil is obtained at the production well if the scanning curve initially leaves the primary drainage curve towards the imbibition curve.Similar trends are also observed in comparing the transition zone and constant uniform initial water saturation when MSW injection is modelled including hysteresis impact.The results show that extra oil and lower water-cut can be obtained from the long time injection of MSW into the transition zone compared to a reservoir with a uniform initial water saturation.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

4 )Fig. 1 .Fig. 2 .
Fig. 1.Schematic of core flooding experimental observation during tertiary injection of MSW after SW breakthrough a) saturation profile b) location of oil and brine fronts, assuming that wettability alteration occurs with delay due to ionic adsorption c) potential determining ions (PDIs) concentration profile along the core length (the red color also shows the altered wettability areas after being contacted with MSW).(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 5 .
Fig. 5. a) The interpolation between high and low salinity curves based on the value of total salinity b) linear interpolation between high salinity relative permeability curve (θ = 0) and modified salinity relative permeability curve (θ = 1) used for model-3.

Fig. 7 .
Fig. 7. Schematic of the single-layer model.The water is injected from the left side, pushing oil to the production well on the right side (x-direction).The model is discretized by 500 grid blocks in the x-direction (1D and 2D) and 200 grid blocks in the y-direction (2D) (BC: Boudary condition).

Fig. 8 .Fig. 10 .
Fig. 8. Distribution of initial water saturation with an average of 0.3 (S wi,ave = 0.3) a) water saturation profile between injection and production wells for case 1-4 b) Schematic diagram of the oil-water transition zone with a variety of water saturation distribution (case 5).Note that initial water saturation is uniform in y-direction and x-direction for Case 1-4 and Case 5, respectively.

Fig. 11 .
Fig. 11.Relative permeability hysteresis of modified salinity injection during secondary flooding using Killough method at grid block i = 250 (middle of the domain) when the initial water saturation is 0.3 (S wi = 0.3) a) model-1b) model-2.The color bar on the left side illustrates the timing of simulated data in the number of injected pore volumes.Note that only some simulation data are plotted in the k r -PV injection figures (a and c) to clearly show the saturation color map.

Fig.
Fig. Relative permeability hysteresis of modified salinity injection during tertiary flooding using Killough method at grid block i = 250 when the initial water saturation is 0.3 (S wi = 0.3) a) model-1b) model-2.The MSW injection starts after 0.33 PV injection of SW.The color bar on the left side illustrates the timing of simulated data in the number of injected pore volumes.Note that only some simulation data are plotted in the k r -PV injection figures (a and c) to clearly show the saturation color map.

Fig. 13 .
Fig. 13.Comparison of SW and MSW flooding performance during secondary injection under the effect of hysteresis for different conditions with S wi = 0.3 (w/o = without).

Fig. 15 .
Fig. 15.Water saturation (solid line) and salt concentration (dashed line) profiles of non-uniform initial water saturation between injection and production wells during secondary injection of MSW flooding using Killough method and model-2 when PV injected is 0.2.

Fig. 16 .
Fig. 16.Recovery factor (RF) and water cut (WC) for non-uniform initial water saturation during secondary injection of MSW flooding using Killough method and model-2.

Fig. 17 .
Fig. 17.Water saturation profile of transition zone between injection and production wells during secondary injection of MSW flooding using Killough method when PV injected is 0.2 a) seawater injection b) MSW injection using model-2.

Fig. 18 .
Fig.18.Recovery factor (RF) and water cut (WC) for transition zone (TZ) and constant uniform initial water saturation with an average of 30% during secondary injection of MSW flooding using Killough method and model-2.

Fig.Fig. 20 .
Fig. Relative permeability hysteresis of seawater injection for different initial water saturation using Carlson method at grid block a) i = 250b) i = 500.The model is built with 500 grid blocks in total, where i = 1 and i = 500 represent the injector and producer grid blocks, respectively.The color bar on the left side illustrates the timing of simulated data in the number of injected pore volumes.Note that only some simulation data are plotted in the k r -PV injection figures (a and c) to clearly show the saturation color map.

Table 2
Rock and fluid properties used in the model.