Iso-Permeability Point Trail Method to Determine the Relative Permeability Curve for a New Amphiphilic Polymer Flooding

: Amphiphilic-polymer ﬂooding, which can increase water viscosity, decrease oil viscosity, and improve oil displacement efﬁciency, is a promising oil exploitation method for heavy oil. Due to oil–water emulsiﬁcation, shear-thinning, and changes in oil viscosity when determining the relative permeability data of new amphiphilic polymers, the conventional J.B.N. method is not accurate. This paper presents a new method called the iso-permeability point trial method to determine the relative permeability curve by combining the J.B.N. method, the Corey model


Introduction
A new amphiphilic multifunctional oil displacement agent, EHA, has been developed to effectively displace mobile heavy oils with viscosities of 150-500 mPa•s or even thousands of mPa•s, as in the Bohai oil reservoir [1,2].A variety of active groups were introduced into the side chain of the acrylamide skeleton, which can both improve the viscosity of the water phase and displacement efficiency [1,[3][4][5].Meanwhile, it can effectively disassemble the accumulation of the asphaltene molecular layer of heavy oil, thus reducing the viscosity of heavy oil.A number of laboratory simulation experiments show that EHA flooding can significantly enhance the recovery of mobile heavy oil, and it is a potential ideal oil displacement agent for heavy oil field development [3][4][5][6][7][8][9][10][11].However, it is very important to understand the influence mechanism of EHA on the relative permeability curve when applying it in numerical simulation and pilot trials.
Generally, the relative permeability curve can be acquired from core flooding, the capillary curve method, the empirical equation method, the field data inversion method, etc. [12], and core flooding is the commonly used method.There are two kinds of measurement methods, including steady-state and unsteady-state methods, which have different theoretical bases.Based on Darcy's formula, the steady-state method can measure the relative permeability in a relatively wide saturation range.The principle of this method is simple, and the results obtained are highly reliable.However, the test process is timeconsuming, so it is seldom used and it is only used in early research [13].For example, Schneider and Owens (Schneider and Owens, 1980) used the steady-state displacement mode to determine the polymer relative permeability curves of 18 outcrops and formation cores with different permeabilities [14].Tang et al. used Hirasaki's model (Hirasaki, 1975) to determine a set of polymer displacement relative permeability experiment results by the steady-state mode [15].The unsteady-state mode is based on the proximal propulsion theory of Buckley-Leverett's one-dimensional two-phase water displacement.Only flow data under constant pressure or constant flow velocity are recorded in the experimental process.Compared with the steady-state method, much measurement time can be saved, so it is the most commonly used experimental method to acquire a relative permeability curve.Salman et al. (1979) rederived the polymer solution fractional flow equation, based on which he rederived the modified J.B.N. [16].However, the effective viscosity model of the polymer solution is based on the Blake-Kozney equation; therefore, it is a constant in the process of displacement, so it is relatively simple.Lei (1994) calculated the oil-phase relative permeability based on the traditional J.B.N. method and then determined the water phase that contained polymer permeability using the oil-phase relative permeability data correction formula, which is mainly based on the capillary number model considering the rheological property of the polymer solution and the correction coefficient [17].However, this method fails to account for the plugging effect of the polymer solution to the core face and the initial pressure selection in the experimental test; thus, the stability of the experimental results is questionable.Shi (2001) further modified the polymer solution's effective viscosity by multiplying water viscosity and the ratio of resistance coefficient to residual resistance coefficient [18].This method needs to determine the coefficient of resistance and residual resistance of polymer solution in porous media, which not only enhances the complexity and difficulty of the experiment but also brings more possibility of experimental error due to the application of parallel core samples.
The literature review shows that the most concerning problem in measuring the relative permeability curve by the unsteady-state technique is how to solve the relative permeability curve according to experimental results and improve the accuracy of differential data processing in the calculation [19][20][21][22][23][24][25][26][27].The J.B.N. method is the most commonly applied method for calculating the relative permeability under the unsteady-state mode, and it is also the recommended method of the Chinese national standard (GB/T 28912-2012 [28]).However, a differential calculation is involved in the J.B.N. method [29], which is difficult in practical operation, especially in the displacement process of amphiphilicpolymer flooding [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44].There are two main reasons: (1) the produced liquid usually has an emulsification effect, and measurement errors for oil and water occur easily.At the same time, because of the high viscosity and dark color of crude oil, the wall-hanging phenomenon can be usually observed in the liquid receiving container, which affects the measurement accuracy of oil and water.Therefore, to obtain a clear oil-water interface and improve measurement accuracy, it is necessary to conduct high-speed centrifugal separation for the produced liquid which is difficult to measure.(2) To acquire the relative permeability data of the novel amphiphilic polymer, EHA, the displacing phase viscosity is used when conducting the classical J.B.N. method.However, the EHA solution system is a non-Newtonian fluid and the actual flow viscosity is directly affected by the shear rate, so there is a huge distinction between apparent viscosity in the beaker and actual viscosity in porous media.The commonly used viscosity processing methods include apparent viscosity, Darcy viscosity, produce fluid viscosity, and other methods.Some of these methods do not consider the displacement fluid flow condition in the core, and some do not consider the existence of oil in fluid flow; therefore, the obtained data are not accurate enough [45,46].
Considering that it is difficult to accurately obtain amphiphilic-polymer solution viscosity in the process of measuring the relative permeability curve, a mathematical method is adopted and the fitting derivation is carried out based on relative permeability curve characteristics.The viscosity value of the EHA solution is avoided, that is, the target curve is acquired in a mathematical way.

Methods and Theories
On the basis of the result of the literature research and the aforementioned analysis, in order to determine the relative permeability curve for the new type of amphiphilic-polymer oil displacement agent, the following improvements have been carried out: first, adopting the transient method to determine the phase flow rate and pressure data acquisition.One of the most important steps in the data acquisition stage is to conduct high-speed centrifugal separation for the produced fluid, so as to ensure the accuracy of the final measured data [47][48][49].Secondly, in the calculation process, it is necessary to adapt mathematical methods.Relying on the characteristic of the relative permeability curve, fitting derivation is conducted.To avoid using the constantly changing viscosity, an iso-permeability point trail method for relative permeability calculation was proposed, and the target curve was automatically calculated using Python programming.The list of relevant methods is shown in Table 1.
Table 1.An improved method for measuring the relative permeability curve of the new amphiphilicpolymer EHA solution by the unsteady-state method.

Existing Problems Solutions Operation Method
Oil-water measurement Centrifugation and weighing

Calculation of relative permeability curve
Iso-permeability point trial method Python programming In the process of solving relative permeability by unsteady-state mode, oil or waterphase viscosity and other data are not needed to compute the oil-phase relative permeability (Equation ( 1)), but the viscosity of each phase is needed in calculating the water-phase relative permeability (Equation (2)).Therefore, in the process of calculating the relative permeability of EHA displacement, the oil-phase relative permeability and the corresponding water saturation can be directly computed from experimental data according to Equations (1) and (3).
where K ro (S we ) is the oil-phase relative permeability at the outlet saturation; K rw (S we ) is the water-phase relative permeability at the outlet saturation; V(t) is the dimensionless cumulative volume of the water injection.V(t) = V t /V p , where V t is the cumulative volume of the water injection.V p is the total pore volume of the core sample.V o (t) is the dimensionless cumulative oil production.V o (t) = V o /V p , where V o is the cumulative oil production; f o (S we ) is the oil cut at the outlet (the volume percentage of oil production in total liquid output); f w (S we ) is the water cut at the outlet (the volume percentage of water output in total liquid output); S we is the water saturation at exit, decimal; S wi is irreducible water saturation.µ o is the dynamic viscosity of oil, mPa•s.µ w is the dynamic viscosity of water, mPa•s.I is the ratio of flow capacity at any moment to the initial moment.K is the absolute core permeability.M is the cross-section area of the experimental core, cm 2 .L is the length of the experimental core, cm.Q(t) is liquid output at time t, cm 3 /s.∆p(t) is the differential pressure of the experimental core at time t, 10 −1 MPa.v is the flow rate, cm/s.With the oil-phase relative permeability curve being solved, K rowc , S or , and S wc can be obtained according to Equation ( 5) in the Corey model and experimental data of the unsteady method.In the same way, K rwor , S or , and S wc can also be determined using Equation (6) to calculate K rw in the oil and water relative permeability Corey model and experimental data of the unsteady method.Now, the only unknown terms in the K rw equation are K rw , S w , and n w .
where K rowc is the oil phase relative permeability under S wc .K rwor is the water-phase relative permeability under S or .S wc is the irreducible water saturation.S or is the residual oil saturation.n o and n w are exponential terms in the Corey model.
According to the literature, under an intermediate range of water saturation, the water saturation and oil-water phase relative permeability ratio satisfy the relationship in Equation ( 7) [13].
In constant rate displacement, Equation ( 9) can be derived according to water-phase fractional flow Equation (8).
Combining Equations ( 7) and ( 9), we can get Equation (10) can be transformed to be The values measured by the unsteady-state method were substituted into Equation (11) and plotted ln as a function of S w .It can be seen that, within the range of medium water saturation, there is a linear relationship between ln f o 1− f o and S w .B is the slope of the line and lnA − ln µ o µ w is the intercept of the line.Both slopes and intercepts can be calculated according to the data in the form of coordinate maps.Now, assume a saturation value S w at an iso-permeability point of relative permeability, then substitute it into Equation ( 5) to obtain the value of K ro (S w ).At the iso-permeability point, K rw (S w ) = K ro S w ; therefore, the K rw (S w ) value can be obtained.At this point, the value of n w can be obtained by substituting S w into Equation (6).If the assumed iso-permeability point saturation is the actual iso-permeability point saturation, the relationship between water-phase relative permeability and water saturation can be obtained by substituting the n w value into Equation (6).Therefore, the key point here is how to determine the value of iso-permeability point saturation, S w .
The method of determining the saturation of the iso-permeability point is given below: Energies 2023, 16, 7362 5 of 14 (1) Assume an initial iso-permeability point, S wx and a step size of the iso-permeability point variation, ∆S w (0.01 ≤ ∆S w ≤ 0.05).( 2) According to Equation ( 5), calculate the corresponding values of oil relative permeability, K ro (S wx ), K ro (S wx + ∆S w ), and K ro (S wx − ∆S w ) under iso-permeability point S wx , and S wx + ∆S w and S wx − ∆S w .(3) Calculate the n w value with K ro (S wx ) = K rw (S wx ) at the iso-permeability point and Equation ( 6).The expression of the water-phase relative permeability in the Corey model can also be acquired, which can be used to solve 7), coefficients A and B can be obtained simultaneously.
The obtained value is denoted as B , and it is compared with the slope from the regression graph.
002, a minimal positive real number), then the assumed iso-permeability point is the actual iso-permeability point.
002, a minimal positive real number), then the assumed iso-permeability point is not appropriate, and it is necessary to re-guess the iso-permeability point.Then go back to Step (1), and recalculate until it meets B = B or ||B | − |B|| < ε, then the assumed iso-permeability point at this time is the actual iso-permeability point.( 5) Then, according to Equation ( 5), the value of K ro (S wx ) can be obtained.Since at the iso-permeability point, K rw (S wx ) = K ro (S wx ), the value of K rw (S wx ) can be calculated.At this point, the value of the water relative permeability index in the Corey model can be obtained by plugging S wx into Equation ( 6).Finally, the relationship between the water-phase relative permeability and water saturation can be acquired by plugging n w into Equation ( 6).Now the complete expression of water relative permeability in the Corey model can be obtained.
In order to realize the above algorithm, a cyclic calculation program is programmed through Python 3.

Experimental Steps
Combined with the conventional approaches for determining relative permeability by the unsteady-state technique and the improved technique mentioned previously in this article, the steps for determining the relative permeability curve of the new amphiphilicpolymer displacement are as follows: (1) Experimental Condition

Experimental Steps
Combined with the conventional approaches for determining relative permeability by the unsteady-state technique and the improved technique mentioned previously in this article, the steps for determining the relative permeability curve of the new amphiphilicpolymer displacement are as follows: (1) Experimental Condition  Preparation of oil displacement system solution: firstly, 5000 mg/L mother solution of EHA was prepared, and aged for 12-24 h in the same condition.Then dilute the solution to the target concentration, use Wuyin agitator at grade 1 and shear of 20 s to simulate the shear effect of the solution at the near well zone.It can be used after shear defoaming.Prior to testing and utilization, filter it using a sand core funnel to prevent plugging during injection.5 Experimental core model: homogeneous core model with a size of 4.5 cm × 4.5 cm × 30 cm. 6 Experimental basic instruments: displacement micro pump, pressure sensor, produced liquid collection device, pressure acquisition system, desktop highspeed centrifuge, as shown in Figure 2. (2) Gas permeability: the horizontal valve is added to both ends of the core.The gas permeability test device is adopted to log the permeability of the experimental core.The gas flow velocity is calculated through the gas meter, so as to determine the gas permeability.
(3) Vacuum pumping and saturate in water, then calculate core porosity, Φ. (4) Water permeability: first, connect the core gripper to the displacement system, at the same time open the thermostat, set the temperature at 65 • C, and start the experiment after the temperature and confining pressure are stabilized.Note: the pressure system has to be air-free to ensure the stability of the pressure measurement.(5) Saturated oil: after water permeability is measured, close the water injection pipeline, open the oil injection pipeline, keep the thermostatic chamber at 65 • C, and saturate the core with oil under simulated reservoir conditions.(6) Oil displacement experiment: 1 Open the pump and inject the oil displacement system at a flow velocity of 0.5 mL/min, make sure the solution fills the pipeline and drains to the outlet of the pipeline.The pressure displayed on the experimental system by the voltage sensor connected to the six-way valve is recorded as the initial pressure when the active water pipeline is not connected to the core inlet end; 2 Connect the displacement pipeline to the core inlet, adjust the flow velocity of the pump, then start the displacement experiment and continuously monitor the displacement pressure; 3 Inject the displacement solution into the experimental core at a velocity of 0.5 mL/min.At the beginning of the displacement experiment, the liquid time needs to be recorded once every 5 min at a flow velocity of 0.5 mL/min.Meanwhile, watch the water breakthrough time of the outlet.At the beginning of the water coming out, the liquid metering frequency needs to increase with a time Energies 2023, 16, 7362 7 of 14 interval of 2 min.In order to measure conveniently, 2 mL water can be added to the 10 mL test tube in advance; 4 When the measured water cut exceeds 90%, change the liquid metering frequency to 10 min until the pressure is stable and no oil is produced.The pressure in this process must be stabled in at least 1 PV.(2) Gas permeability: the horizontal valve is added to both ends of the core.The gas permeability test device is adopted to log the permeability of the experimental core.
The gas flow velocity is calculated through the gas meter, so as to determine the gas permeability.
(3) Vacuum pumping and saturate in water, then calculate core porosity, Φ. (4) Water permeability: first, connect the core gripper to the displacement system, at the same time open the thermostat, set the temperature at 65 °C, and start the experiment after the temperature and confining pressure are stabilized.Note: the pressure system has to be air-free to ensure the stability of the pressure measurement.① Open the pump and inject the oil displacement system at a flow velocity of 0.5 mL/min, make sure the solution fills the pipeline and drains to the outlet of the pipeline.The pressure displayed on the experimental system by the voltage sensor connected to the six-way valve is recorded as the initial pressure when the active water pipeline is not connected to the core inlet end; ② Connect the displacement pipeline to the core inlet, adjust the flow velocity of the pump, then start the displacement experiment and continuously monitor the displacement pressure; ③ Inject the displacement solution into the experimental core at a velocity of 0.5 mL/min.At the beginning of the displacement experiment, the liquid time needs to be recorded once every 5 min at a flow velocity of 0.5 mL/min.Meanwhile, watch the water breakthrough time of the outlet.At the beginning of the water coming out, the liquid metering frequency needs to increase with a time interval of 2 min.In order to measure conveniently, 2 mL water can be added to the 10 mL test tube in advance; ④ When the measured water cut exceeds 90%, change the liquid metering frequency to 10 min until the pressure is stable and no oil is produced.The pressure in this process must be stabled in at least 1 PV.

Results
This part provides some comparison and analysis of the Corey model method and the J.B.N. method under the iso-permeability point trail method using a 4.5 cm × 4.5 cm × 30 cm homogeneous core for different heavy-oil viscosities, core permeabilities, and amphiphilicpolymer concentrations.The J.B.N. method and iso-permeability point trial method are used to measure and analyze the difference in relative permeability curves between amphiphilicpolymer displacement and water displacement.According to the research parameter range, the selected experimental conditions and results of the relative permeability curve are represented in Table 3 and Figure 3, respectively.The Corey model is also used to match the experimental data.The solid lines in Figure 3 represent the results calculated by the iso-permeability point trial calculation method based on the Corey model.The circle points represent the treatment results of the iso-permeability point trial calculation based on the J.B.N. method.From Figure 3a-i, we can find that the water relative permeability can be well obtained by using the iso-permeability point trail method.The water relative permeability obtained by the iso-permeability point trail method is almost the same on the basis of the J.B.N. method and the Corey model method, while there are some differences in the oil relative permeability based on these two methods.The difference is caused by the characteristics of the J.B.N. method and the Corey model.In terms of water relative permeability curve calculation, the displacement phase viscosity is needed when using the classical J.B.N. method.As a non-Newtonian fluid, the viscosity of the EHA system is related to the shear rate in the actual flow process of the core, and there is a great difference between apparent viscosity and the actual viscosity in porous media.The classical J.B.N. method fails to consider the fluid flow condition in the core, as well as the change of oil caused by EHA in the flow, so the data obtained may not be accurate.However, when combining the isopermeability point trial method with the J.B.N. method, the changing displacement fluid viscosity is no longer needed.Instead, the characteristics of the relative permeability curve are used to fit the curve and the final curve are more smooth and reasonable.The timeconsuming and truncation phenomenon caused by the non-smooth relative permeability curve in the simulation can be prevented.

The Effect of EHA Concentration
The core permeability was fixed at 2000 mD and the crude oil viscosity was fixed at 250 mPa•s.The relative permeability curves (including water flooding) of the new amphiphilic-polymer solution under different concentrations were obtained, and they are plotted in Figure 4.The experimental results are shown in Table 4. From Figure 3a-i, we can find that the water relative permeability can be well obtained by using the iso-permeability point trail method.The water relative permeability obtained by the iso-permeability point trail method is almost the same on the basis of the J.B.N. method and the Corey model method, while there are some differences in the oil relative permeability based on these two methods.The difference is caused by the characteristics of the J.B.N. method and the Corey model.In terms of water relative permeability curve calculation, the displacement phase viscosity is needed when using the classical J.B.N. method.As a non-Newtonian fluid, the viscosity of the EHA system is related to the shear rate in the actual flow process of the core, and there is a great difference between apparent viscosity and the actual viscosity in porous media.The classical J.B.N. method fails to consider the fluid flow condition in the core, as well as the change of oil caused by EHA in the flow, so the data obtained may not be accurate.However, when combining the isopermeability point trial method with the J.B.N. method, the changing displacement fluid viscosity is no longer needed.Instead, the characteristics of the relative permeability curve are used to fit the curve and the final curve are more smooth and reasonable.The timeconsuming and truncation phenomenon caused by the non-smooth relative permeability curve in the simulation can be prevented.

The Effect of EHA Concentration
The core permeability was fixed at 2000 mD and the crude oil viscosity was fixed at 250 mPa•s.The relative permeability curves (including water flooding) of the new amphiphilic-polymer solution under different concentrations were obtained, and they are plotted in Figure 4.The experimental results are shown in Table 4.    Comparing the relative permeability curve of EHA flooding with water flooding, it is found that the water relative permeability for EHA flooding is smaller than that of water flooding at the same water saturation because the viscosity-increasing ability of EHA reduces the flowability of the water phase.The interfacial activity of EHA enables crude oil to be peeled off and migrate easily.The flowability of oil is enhanced.Under the same water saturation, the oil's relative permeability during EHA displacement is larger than that of Energies 2023, 16, 7362 10 of 14 water flooding.EHA makes the water phase have the same flowability as the oil phase at higher water saturation, so the iso-permeability point moves to the right.EHA flooding can effectively enhance heavy oil recovery with lower residual oil saturation compared to water flooding.
By comparing relative permeability curves of the EHA flooding at different concentrations, the following rules are found: (1) The bigger the concentration of EHA is, the smaller the residual oil saturation and the wider the two-phase seepage zone are.As the EHA concentration rises, the solution viscosity increases, oil-water interfacial tension lowers, and the corresponding oilincreasing capacity is stronger, so the saturation of residual oil reduces.Since the oil used in the experiment to saturate cores with identical permeability is the same, the irreducible water saturation is similar.So the higher the concentration of the EHA solution, the lower the saturation of the residual oil, and the corresponding span of the two-phase region expands.(2) The higher the concentration of EHA is, the lower the water relative permeability is, and the higher the oil relative permeability becomes under the same water saturation.
The bigger the concentration of EHA is, the larger the aggregate size is, and the stronger the flow resistance in the porous medium becomes, resulting in the decrease of water relative permeability and an increase of oil relative permeability.
(3) The iso-permeability point shifted to the right with the increase in EHA concentration.
With the rise of EHA concentration, the water-phase relative permeability decreases, and that of the oil phase increases.Meanwhile, the residual oil saturation point shifts to the right, which makes the iso-permeability point shift to the right as well.

Influence of Core Permeability on Relative Permeability Curve
The new amphiphilic-polymer solution concentration was fixed at 1600 mg/L and the crude oil viscosity was fixed at 250 mPa•s.The relative permeability curves under different core permeability conditions were obtained and they can be seen in Figure 5.The experimental results are shown in Table 5.
flooding at the same water saturation because the viscosity-increasing ability of EHA reduces the flowability of the water phase.The interfacial activity of EHA enables crude oil to be peeled off and migrate easily.The flowability of oil is enhanced.Under the same water saturation, the oil's relative permeability during EHA displacement is larger than that of water flooding.EHA makes the water phase have the same flowability as the oil phase at higher water saturation, so the iso-permeability point moves to the right.EHA flooding can effectively enhance heavy oil recovery with lower residual oil saturation compared to water flooding.
By comparing relative permeability curves of the EHA flooding at different concentrations, the following rules are found: (1) The bigger the concentration of EHA is, the smaller the residual oil saturation and the wider the two-phase seepage zone are.As the EHA concentration rises, the solution viscosity increases, oil-water interfacial tension lowers, and the corresponding oil-increasing capacity is stronger, so the saturation of residual oil reduces.Since the oil used in the experiment to saturate cores with identical permeability is the same, the irreducible water saturation is similar.So the higher the concentration of the EHA solution, the lower the saturation of the residual oil, and the corresponding span of the two-phase region expands.(2) The higher the concentration of EHA is, the lower the water relative permeability is, and the higher the oil relative permeability becomes under the same water saturation.
The bigger the concentration of EHA is, the larger the aggregate size is, and the stronger the flow resistance in the porous medium becomes, resulting in the decrease of water relative permeability and an increase of oil relative permeability.
(3) The iso-permeability point shifted to the right with the increase in EHA concentration.With the rise of EHA concentration, the water-phase relative permeability decreases, and that of the oil phase increases.Meanwhile, the residual oil saturation point shifts to the right, which makes the iso-permeability point shift to the right as well.

Influence of Core Permeability on Relative Permeability Curve
The new amphiphilic-polymer solution concentration was fixed at 1600 mg/L and the crude oil viscosity was fixed at 250 mPa•s.The relative permeability curves under different core permeability conditions were obtained and they can be seen in Figure 5.The experimental results are shown in Table 5.By comparing the EHA flooding relative permeability curves under different core permeability conditions, the following regularities are found: (1) The bigger the core permeability is, the smaller the irreducible water saturation is, the smaller the residual oil saturation becomes, and the wider the two-phase seepage zone.With the rise of core permeability, the average size of the pore throat increases and it is easier for crude oil to enter the pore throat, which makes the irreducible water saturation decrease.With the rise of pore throat size, the shear effect of EHA solution in the core reduces and the aggregate structure is more complete, so the enhanced oil displacement efficiency is improved and the residual oil saturation is reduced.Since the saturation of irreducible water and residual oil decreases with the core permeability, the span interval of the two phases becomes wider.(2) The bigger the permeability is, the higher the water's relative permeability is and the lower the oil's relative permeability becomes.With the rise of permeability, the average pore throat size of the core increases, resulting in the seepage resistance during EHA displacement; the oil relative permeability decreases, and the water relative permeability increases.(3) Increasing core permeability makes the iso-permeability point shift to the left.As the permeability of the core increases, the relative permeabilities of the water and oil phases increase and decrease, respectively, which causes the iso-permeability point to shift to the left.

Influence of Crude Oil Viscosity on Relative Permeability
The concentration of the EHA solution was fixed at 1600 mg/L and the core permeability K was fixed at 2000 mD; the relative permeability curves under different crude oil viscosity were obtained, and they can be found in Figure 6  By comparing the EHA flooding relative permeability curves under different core permeability conditions, the following regularities are found: (1) The bigger the core permeability is, the smaller the irreducible water saturation is, the smaller the residual oil saturation becomes, and the wider the two-phase seepage zone.With the rise of core permeability, the average size of the pore throat increases and it is easier for crude oil to enter the pore throat, which makes the irreducible water saturation decrease.With the rise of pore throat size, the shear effect of EHA solution in the core reduces and the aggregate structure is more complete, so the enhanced oil displacement efficiency is improved and the residual oil saturation is reduced.Since the saturation of irreducible water and residual oil decreases with the core permeability, the span interval of the two phases becomes wider.(2) The bigger the permeability is, the higher the water's relative permeability is and the lower the oil's relative permeability becomes.With the rise of permeability, the average pore throat size of the core increases, resulting in the seepage resistance during EHA displacement; the oil relative permeability decreases, and the water relative permeability increases.(3) Increasing core permeability makes the iso-permeability point shift to the left.As the permeability of the core increases, the relative permeabilities of the water and oil phases increase and decrease, respectively, which causes the iso-permeability point to shift to the left.

Influence of Crude Oil Viscosity on Relative Permeability
The concentration of the EHA solution was fixed at 1600 mg/L and the core permeability K was fixed at 2000 mD; the relative permeability curves under different crude oil viscosity were obtained, and they can be found in Figure 6 and Table 6.By comparing the relative permeability curves of EHA displacement at diverse crude oil viscosities, the following rules are recognized: (1) The bigger the viscosity of crude oil, the smaller the irreducible water saturation, the bigger the residual oil saturation, and the narrower the two-phase flow area.When the crude oil viscosity is high, more water can be discharged in the process of saturated oil, so the irreducible water saturation is smaller; similarly, in the process of water flooding, the higher the crude oil viscosity is, the more difficult it is to displace it from the pores, so the residual oil saturation is higher and the span of the two-phase flow area is smaller.(2) When the crude oil viscosity ascends, both water and oil relative permeability decrease under the same water saturation.The bigger the crude oil viscosity is, the worse the flowability in porous media becomes, resulting in the descending of oil relative permeability.Meanwhile, the oil occupies more rock surface during the saturation process, which makes it difficult for the water phase to flow in the pores during the displacement process, so the relative permeability of the water phase decreases.(3) Increasing oil viscosity makes the iso-permeability point deviate to the left.As the oil viscosity rises, the saturation of irreducible water decreases while the saturation of residual oil increases.Meanwhile, the oil occupies more rock surface, resulting in its hydrophilicity being weakened, so the iso-permeability point shifts to the left.

Conclusions
(1) Considering the dynamic changes of the new displacement agent viscosity and the heavy oil viscosity under the actual displacement state, a method of iso-permeability point trail method was established to solve the problem of calculating the relative permeability of the new amphiphilic-polymer displacement by means of the mathematics method and Python programming.This novel method combines the J.B.N. method, Corey's classical model, the exponential expression of relative permeability, and oil-water fractional flow equation.The constantly changing viscosity values of the agent are not needed.The fitted curve is more reasonable, feasible, and smooth.It provides the foundation for efficient numerical simulation in the future.(2) The relative permeability curves for waterflooding (0 mg/L EHA solution) and the new amphiphilic-polymer flooding with different concentrations (800, 1200, 1600 mg/L EHA solution) were measured and analyzed.Compared with water flooding, under the same water saturation, the water and oil relative permeabilities decrease and increase, respectively, and the iso-permeability point shifts to the right.The irreducible water saturation does not change significantly, but the saturation of residual oil declines, and the two-phase seepage zone expands.And with the increase in EHA concentration, the trend of the above phenomenon is more significant.The residual oil saturation decreases by 0.032 on average for each 400 mg/L increase in EHA concentration.(3) The relative permeability curves of the new amphiphilic polymer (1600 mg/L EHA solution) with the same concentration were measured and analyzed for different core permeabilities (500, 2000, 5000 mD).The results show that when the core permeability increases, the water's relative permeability increases simultaneously, while the oil's relative permeability declines.Meanwhile, the iso-permeability point shifts to the left, the saturation of irreducible water and residual oil decreases, and the range of the corresponding two-phase region becomes wider.The mean left shift value of the iso-permeability point is 0.007 and the mean decrease value of residual oil saturation is 0.020 for every 1000 mD increase in core permeability.(4) The relative permeability of the new displacement agent (1600 mg/L EHA solution) with the same concentration was measured and analyzed for different oil dynamic viscosities (150, 250, 500, 1000 mPa•s).The conclusion is that the oil's relative permeability decreases with the increase of oil viscosity.When the value of the water relative permeability rises, the iso-permeability point shifts to the left, the saturation of irreducible water and residual oil decreases, and the corresponding two-phase region becomes narrower.The average left shift of the iso-permeability point is 0.011 and the average increase in residual oil saturation is 0.020 for every 100 mPa•s increase in crude oil dynamic viscosity.

16 Figure 1 .
Figure 1.Program framework for measuring the relative permeability of amphiphilic polymer displacement by the iso-permeability point trial method.

Figure 1 .
Figure 1.Program framework for measuring the relative permeability of amphiphilic polymer displacement by the iso-permeability point trial method.

( 7 )
Sample centrifugation.Use a centrifuge to separate the produced liquid at high speed in order to obtain a clear oil-water interface and measure the oil-water content.(8) Input the experimental data into the Python program and calculate the relative permeability curve automatically.Energies 2023, 16, x FOR PEER REVIEW 7 of 16

Figure 2 .( 5 )
Figure 2. Flow chart of displacement experiment.(5) Saturated oil: after water permeability is measured, close the water injection pipeline, open the oil injection pipeline, keep the thermostatic chamber at 65 °C, and saturate the core with oil under simulated reservoir conditions.(6) Oil displacement experiment:

Table 2 .
Composition of injected water.

Table 3 .
Experimental parameters of relative permeability curve.

Table 4 .
Experimental results with different concentrations of EHA solution.

Table 4 .
Experimental results with different concentrations of EHA solution.

Table 5 .
Experimental results of EHA displacement with different core permeabilities.

Table 5 .
Experimental results of EHA displacement with different core permeabilities.

Table 6 .
Experimental results of EHA displacement with different crude viscosities.

Table 6 .
Experimental results of EHA displacement with different crude viscosities.