**2. Experiments**

#### *2.1. Experimental Design*

Bohai oilfield is a typical fluvial facies reservoir with high permeability and porosity. According to the property variation range of the main reservoir, the law of high waterflooding PVs' relative permeability of different properties (244~9380 mD) was studied. A total of 9 groups of waterflooding experiments were carried out using typical reservoir cores. The simulated oil viscosity is 28 mPa·s. Standard brine used in the experiments has a salinity of 10,000 mg/L. The test fluid was designed according to the actual reservoir oil and water properties of the Q32-6 oilfield. The waterflooding rate was 1.0 mL/min. The maximum waterflooding PVs is 2000 PVs.

#### *2.2. Sample and Devices*

In this study, natural cores with different permeability (244–4814.8 mD) were selected to carry out high waterflooding PVs displacement experiments. We describe the cores used in the experiment in Figure 1. All of these cores are moderately wet. Table 1 shows a summary of cores conditions.


**Table 1.** Summary of the experiment samples.

**Figure 1.** Cores used in the experiments.

The experimental process consists of five parts: pressure control system, temperature control system, core holder, metering, and data acquisition system (Figure 2).

**Figure 2.** Schematic representation of the experimental apparatus of the relative permeability measurement.

The relative permeability of different cores at high waterflooding PVs was measured.

#### *2.3. Experimental Procedure*

In this experiment, the core is vacuumed and then saturated with standard saltwater, followed by oil flooding to create irreducible water conditions, and finally, continuous water flooding at a specified temperature. The variation of oil and water at the outlet end with time and the displacement pressure difference is recorded. The specific steps are as follows:

A. The simulated formation water is prepared and stands for 1 day, then filtered and loaded into an intermediate container.

B. The cores treated by washing oil and salt are dried, weighed, vacuumed, and saturated with simulated formation water.

C. The core is loaded into the core holder, the irreducible water is established by simulated oil flooding, and the oil phase permeability under the condition of irreducible water is measured.

D. Simulated formation water is used to displace the core with constant speed. The data of breakthrough time, cumulative oil production, cumulative liquid production, displacement velocity, and the displacement pressure difference between the two ends of the core are accurately recorded.

E. At the beginning, encryp<sup>t</sup> the record and gradually lengthen the time interval of the record with the continuous decrease in oil production. The experiment was finished after a water injection of 2000 PVs.

#### **3. Experimental Result**

#### *3.1. High Waterflooding PVs Relative Permeability*

Nine groups of high waterflooding PVs' relative permeability curves show similar characteristics. Here, we take the relative permeability curve of Core #4 as an example (Figure 3, other high waterflooding PVs relative permeability curves see Appendix A). It can be seen that with the increase in water saturation, the relative permeability of the oil phase decreases slowly, while that of the water phase increases sharply, and the increase in the water phase is much larger than the decrease in the oil phase. As waterflooding PVs increase from 50 PVs to 2000 PVs, water saturation increases from 60.8 to 75.2. If water waterflooding PVs stop at 50 PVs, oil-phase relative permeability is considered 0, and the water saturation is 60.8. Water-phase relative permeability is 13.4. If the waterflooding PVs increase to 2000 PVs, the relative permeability of the water phase increases from 13.4 to 66.4, and the relative permeability of the oil phase still decreases slowly with the water saturation increases from 60.8 to 66.4. During the increase in waterflooding PVs, the mobility of the oil phase is very low, accompanied by a continuous decrease, and the mobility of the water phase increases sharply.

**Figure 3.** Relative permeability curve of #4 (K = 1793.1 mD).

The waterflooding characteristic curve of no. 4 natural core at high waterflooding PVs (2000 PV) was obtained (Figure 4). The waterflooding characteristic curve upturns after the water cut reaches 98%. Before the "inflection point", the waterflooding PVs is 67.26 PV, the water consumption is 3.36%, the cumulative time is 941.44 min, and the oil displacement efficiency is 55.74%. From the "inflection point" to the end of the experiment, another 1933 PV water was injected, which accounted for 96.64% of the total water consumption. The cumulative time was 27,052 min, and the oil displacement efficiency increased by 13%. It is suggested that after the waterflooding characteristic curve is upturned, the water consumption increases sharply, but the oil displacement efficiency increases slowly, and the development effect gradually deteriorates.

**Figure 4.** Type A waterflood characteristic curve of #4.

The experimental results of relative permeability at high waterflooding PVs show that increasing waterflooding PVs can improve oil displacement efficiency. However, the mobility of the water phase increases sharply at high waterflooding PVs, especially after the "inflection point" of a waterflooding characteristic curve appears; the same production of oil needs to consume more injection water. Therefore, before the "inflection point" appears in the waterflooding characteristic curve, it is the key to increasing waterflooding recovery efficiency economically and effectively. After the "inflection point", appropriate measures can be taken to control the profile and plug water in order to improve the oil displacement contribution of injected water. Injection–production correlation analysis and flow diagnosis [5–9] can effectively help to identify high water consumption brands and provide a basis for better waterflooding development of oilfields.

#### *3.2. Relationship between Residual Oil Saturation and Waterflooding PVs*

The amplitude of oil displacement efficiency improvement in different waterflooding PVs was counted (Figure 5). When the waterflooding PVs were increased from 50 PV to 500 PV, the average oil displacement efficiency was increased by 10.5%; when the displacement was further increased from 500 PV to 1000 PV, the oil displacement efficiency was increased by 1.5%; when the displacement was further increased from 1000 PV to 2000 PV, the oil displacement efficiency was only increased by 0.9%. The ultimate oil displacement efficiency is closely related to permeability. With the increase in permeability, the final displacement efficiency and the improved range of displacement efficiency increase. In the actual oilfield development process, the oil displacement efficiency under 2000 PV can be regarded as the ultimate displacement efficiency under the waterflooding state.

**Figure 5.** The range of oil displacement efficiency improvement in different water waterflooding PVs.

#### *3.3. Mechanism Analysis*

The reason for high waterflooding PVs' relative permeability of high water cut stage shows these characteristics: (1) With the increase in water saturation in the process of waterflooding, the oil phase is continuously segmented, and the degree of discontinuity is continuously intensified, which is characterized by large quantity, small volume, and strong dispersion. (2) The increase in discontinuity is due to the increase in discontinuous flow patterns, such as cluster flow and porous flow, and the oil flow becomes more and more dispersed. (3) With the increase in water saturation, the proportion of discontinuous flow morphology (droplet flow, membrane flow, columnar flow, and porous flow) continues to increase, while the proportion of continuous flow morphology (cluster flow) gradually decreases. (4) In the continuous phase, the cluster flow has strong mobility and relatively high relative permeability, while the remaining oil of the other four forms has weak mobility and low relative permeability in the discontinuous phase, while the inflection point of the relative permeability curve occurs when the discontinuous phase with low relative permeability begins to develop. In other words, inhibiting the transformation and development of the continuous phase and discontinuous phase and maintaining the proportion of the continuous phase at a certain level can delay the occurrence of the inflection point of the phase permeability curve to some extent, which is helpful in improving the recovery factor [36–39].

#### **4. Numerical Simulation**

According to the experimental results, there is a significant difference between the relative permeability of high waterflooding PVs and the traditional relative permeability, especially in the high water cut stage. Therefore, based on MRST, we introduce the high waterflooding PVs' relative permeability into the traditional numerical simulator (the new numerical simulator), update the relative permeability curve by automatically converting the injected water multiple, and compare the results with the traditional numerical simulator (a constant phase permeability curve is used in the whole simulation process). We set up a numerical simulation model: a heterogeneous model with high permeability bands along the diagonal distribution. The numerical simulation, calculating results obtained from the modified simulator, is compared with that from a traditional numerical simulator. The basic parameters of the numerical simulation model are shown in Table 2, and the relative permeability curve used in the numerical simulation model is shown in Figure 3.


**Table 2.** Simulation model parameters.

#### *4.1. High Water Waterflooding PVs Relative Permeability Characterization*

MRST (MATLAB Reservoir Simulation Toolbox) is free, open-source software for reservoir modeling and simulation, developed primarily by the Computational Geosciences group in the Department of Mathematics and Cybernetics at SINTEF Digital. It provides a friendly environment for the realization of applications of new understanding or mechanisms on the influence of two-phase flow in the reservoir. It was used to write the empirical formula of residual oil saturation, respectively, into the simulator, which calculates the dynamic residual oil saturation of each grid at each time step and ensures that the value is less than the limit value of residual oil saturation.

The relationship between oil displacement efficiency and residual oil saturation is as follows:

$$E\_d = \frac{1 - S\_{\rm or} - S\_{\rm wirr}}{1 - S\_{\rm wirr}} \times 100\% \tag{1}$$

In the numerical simulation, residual oil saturation is mainly used to calibrate key parameters such as the relative permeability curve. When the waterflooding PVs are less than 50, the results of conventional waterflooding experiments and relative permeability can be used, and when the water waterflooding PVs are greater than 50, it is necessary to dynamically characterize the residual oil saturation. After processing the experimental data of each group, the relationship between residual oil saturation and waterflooding PVs is established, and the power function is used as the time-varying expression of residual oil saturation (Figure 6).

$$S\_{\text{or}} = A \cdot PV^{B} \tag{2}$$

**Figure 6.** The relationship between the waterflooding PVs and the residual oil saturation.

After fitting 9 groups of experimental data, the values of coefficient A and coefficient B under different permeability are obtained. The expressions of parameter A and parameter B with physical properties and mobility parameters are established (Figures 7 and 8).

$$A = 12.027 \ln\left(\frac{k}{\rho}\right) + 0.1634\tag{3}$$

$$B = 0.0122 \times \ln\left(\frac{k}{\mu}\right) + 0.0937\tag{4}$$

**Figure 7.** The relationship between the value of coefficient A and the ratio of *k/ϕ*.

**Figure 8.** The relationship between the value of coefficient B and mobility.

The empirical formula of residual oil saturation in the oil field is obtained by substituting Equations (3) and (4) into (2).

The actual core experiment and field practice show that there is an extreme value that does not change when the PV number is maximum. Therefore, it is necessary to restrict the extreme value of residual oil saturation. The residual oil saturation under 2000 PV is used as the limit residual oil saturation value.

There is a correlation between irreducible water saturation and core permeability (Figure 9). According to the regression relationship of relevant experimental data in the target oil field, the calculation formula of irreducible water saturation is obtained.

$$S\_{\rm virrr} = -3.116 \times l \lg(k) + 46.868 \tag{5}$$

**Figure 9.** The relationship between the irreducible water saturation and permeability.

The expression of ultimate residual oil saturation in the target oil field can be deduced by simultaneous Equations (1), (2), and (5).

$$S\_{\rm oxjx} = \left(1.266 + 7.626 \lg K - 0.829 \ln \frac{K}{\mu} - 0.486 \lg K \ln \frac{K}{\mu}\right) / 100\tag{6}$$

The model was simulated with MRST, which automatically converted the water multiple, corrected the residual oil saturation endpoint, and calculated the phase permeability iteratively at each step. However, the initially written equation often cannot achieve the best fitting state, and the dynamic residual oil saturation formula needs to be modified iteratively according to the historical fitting effect.

#### *4.2. Numerical Simulation with High Waterflooding PVs Relative Permeability*

In this section, we study the effect of high waterflooding PVs' relative permeability on remaining oil saturation. It is indicated that waterflooding is uniform, and the remaining oil is symmetrically distributed along the injection-production wells' diagonal line in the traditional simulator. The calculation results of the numerical simulator considering the relative permeability of high waterflooding PVs show that the overall oil saturation distribution is more uneven, the water saturation of the high-permeability belt is higher, and the remaining oil saturation of the relatively low-permeability region far from the high-permeability belt is higher than the calculation results of the traditional numerical simulator (Figure 10). This means that the injected water does not flood much into the low-permeability zone of the formation but flows along the high-permeability strip to the production well. In heterogeneous reservoirs, with the continuous enhancement of waterflooding, the high permeability area receives more water and is more likely to be washed out, and the water phase mobility is stronger. The low permeability area does not suffer more erosion due to the increase in water injection, which is consistent with the results of parallel flooding experiments and large 3D physical simulations in the laboratory [31,37,40–42]. Thus, the injected water is more inefficient in the process of increasing waterflooding PVs and further enhances reservoir heterogeneity. Because of the above reasons, the actual reservoir sweep area is more complex, and the remaining oil distribution is far more complex than the traditional numerical simulation results.

**Figure 10.** The water saturation distribution at the 50th time step (The left is the numerical simulation results considering the relative permeability of high waterflooding PVs, and the right is the traditional numerical simulation results).

The pressure distribution calculated by the two numerical simulation methods shows that the pressure spreads uniformly from the water injection well to the production well. However, the pressure distribution of the numerical simulation considering the high waterflooding PVs' relative permeability is more uniform, and the pressure difference between the injection well and the production well is smaller than that of the traditional numerical simulation result, indicating that the injected water can flow well to the oil production well along the high permeability channel, so the pressure propagation is more smooth. There is an obvious high-pressure area near the injection well, and an obvious low-pressure area can also be seen near the production well in the traditional numerical simulation result, indicating that the pressure propagation speed is slow (Figure 11). This is also the reason for the difference in saturation distribution.

**Figure 11.** The water saturation distribution at the 50th time step (the left is the numerical simulation results considering the relative permeability of high waterflooding PVs, and the right is the traditional numerical simulation results).

As we have seen, in the initial oil saturation condition, whether considering the high waterflooding PVs' relative permeability or traditional relative permeability, the calculated results of both simulators are consistent. There is anhydrous oil recovery period, and the water cut rises rapidly until point A. When breakthrough occurs in the production well, the water erosion degree of the high permeability channel will be higher and higher, and far more than the low permeability zone. Therefore, after point A, due to lower residual oil saturation of high waterflooding PVs' relative permeability, it is true that more injected water flows to production wells along the high permeability belt, and the sweeping speed of the surrounding low permeability area is slow. When considering the relative permeability of high waterflooding PVs, the water cut of the simulation result is lower than that of the traditional numerical simulation (AB section in Figure 12). As the injection multiple continues to increase, the oil displacement efficiency gradually approaches the limit oil displacement efficiency, and the water cut increases gradually, exceeding the water cut calculated by the traditional numerical simulation.

**Figure 12.** The water cut curve of numerical simulation of traditional simulator and simulator that considering high water waterflooding PVs.

#### **5. Simulation Validations**

In order to verify the accuracy of the calculation results of the new numerical simulator, we compared the logging interpretation saturation of the target layer of the newly drilled passing well with that of the new numerical simulator and the traditional simulator. Taking the reservoir model of a certain area of the Q32-6 oilfield as an example, a fine numerical simulation considering high waterflooding PVs' relative permeability was carried out under the condition of small adjustments of local permeability, conductivity, interlayer plugging performance, and other parameters. The coincidence between the oil saturation of the target layer calculated by the new numerical simulator and the logging interpretation of oil saturation is obviously higher than that calculated by the traditional numerical simulator (Figure 13). The water cut of the whole area fit well (Figure 14). Therefore, the modified simulator could provide a more accurate remaining oil saturation prediction for further developing the potential of the reservoir.

**Figure 13.** The comparison between the actual logging interpretation oil saturation curve and the model oil saturation curve of a passing well.

**Figure 14.** Fitting results of water cut in the whole well area.

## **6. Conclusions**

A total of nine groups of typical cores with different permeability were selected to carry out high waterflooding PVs experiments. The results show that oil displacement efficiency increases 12.9% on average after increasing water waterflooding PVs to 2000 PV, and the potential of the high waterflooding PVs stage cannot be ignored.

The expression of high waterflooding PVs' relative permeability and dynamic displacement efficiency is obtained. The new understanding of high waterflooding PVs' relative permeability is introduced into the numerical simulator. It is considered that in heterogeneous reservoirs, the waterflooding PVs in high permeability areas is much larger than that in low permeability areas. As a result, the oil recovery in the high permeability area and the residual saturation in the low permeability area are underestimated.

Taking the saturation of the key section of the passing well as the verification, the saturation of the passing well at the corresponding time point is in good agreemen<sup>t</sup> with the actual logging interpretation results, and the water cut curve fits well in the whole area.

**Author Contributions:** Conceptualization, S.W.; Formal analysis, J.Z.; Investigation, J.Z.; Methodology, H.H. and S.W.; Software, H.H.; Validation, E.S.; Visualization, X.Y.; Writing—original draft, G.Q.; Writing—review & editing, S.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
