Nanometer-Scale CO 2 -Shale Oil Minimum Miscibility Pressure Calculations Based on Modified PR-EOS

,


Introduction
With the continuous development of the social economy in China, the total demand for energy is also constantly increasing.According to the latest data in 2021, the total oil imports reached 512.98 million tons in China, and the dependence on foreign oil reached 72.05% [1].Therefore, the issue of energy security cannot be ignored.In addition, many oilfields have entered the production decline stage.And the crude oil recovered with existing technology is of poor quality and has low recovery.To alleviate the crisis of energy shortage and achieve self-sufficiency in crude oil production, it is essential to develop unconventional reservoir resources with richer reserves [2,3].
The shale oil reserves in China are extremely considerable.And the total crude oil in place is as high as 47.64 billion tons, while the total technically recoverable reserves are 15.97 billion tons.It is dominated by continental shale oil which includes medium-low maturity shale oil and medium-high maturity shale oil [4].However, the exploitation of shale oil reservoirs is particularly difficult.The reason is that the porosity and permeability of shale oil reservoirs are low, and the brittle minerals and clay minerals in reservoirs are well developed.Besides, the heterogeneity in shale oil reservoirs is strong [5].The methods to exploit the shale oil reservoirs mainly include the multiple-stage fracturing technology for horizontal wells, the exploitation technology by continuous gas injection, the exploitation technology by cyclic gas injection, the gas injection huff and puff technology, and so on [5].The CO 2 injection huff and puff technology has a better effect on improving shale oil recovery [6].Exploiting the shale oil reservoirs by injecting CO 2 can not only increase the total production of shale oil but also realize the rational use and geological storage of CO 2 .Meanwhile, it is also an important technology to achieve carbon capture, utilization, and storage (CCUS).Liu and Rui suggested that applying CO 2 flooding in oilfields can be an effective method for reducing CO 2 emissions by storing the CO 2 in oil reservoirs [7].The storage-driven CO 2 EOR can even realize netzero or even negative CO 2 emissions [8].The addition of certain cosolvents, such as propanol or dimethyl ether, can enhance the solubility of CO 2 in oil, thereby improving the effectiveness of CO 2 -enhanced oil recovery (CO 2 -EOR) [9].It is an innovation to solve the greenhouse effect problems and protect the ecological environment [10][11][12].The mechanisms of CO 2 injection to enhance oil recovery mainly include the following: (1) reducing the viscosity of crude oil and improving the fluidity of crude oil; (2) making the volume of crude oil expand and part of the shale oil break away from the shackles of the formation to increase the internal kinetic energy; (3) replacing the adsorbed shale oil through equimolecular replaceability and extracting the light hydrocarbon components to reduce the remaining oil saturation; (4) reducing the oil and water interfacial tension (IFT) and the resistance of oil displacement; (5) making CO 2 and shale oil realize miscible and reducing the flow resistance of the remaining oil in the millimeter to nanometer pores; and (6) carbonated formation water can be dissolved to eliminate scaling and blockage and improve the formation permeability; (7) CO 2 dissolved gas flooding, which can not only supply the formation energy but also promote the flow of shale oil in pores and fractures [13][14][15][16].Among them, the miscibility of CO 2 and shale oil has an important impact on the improvement of shale oil recovery.When the formation pressure is greater than the minimum miscibility pressure (MMP) of the oil and CO 2 , the oil and CO 2 miscibility makes the interfacial tension between oil and CO 2 turn zero.Therefore, the shale oil in the nanopores can be largely extracted.And the efficient exploitation of shale oil reservoirs is realized.
MMP is a crucial parameter to verify whether CO 2 and crude oil are miscible or not.It is significant to obtain the oil-CO 2 MMP in the micro-nanoscale [17][18][19][20].The methods for determining the MMP primarily involve experimental methods and theoretical calculation methods [21].The slim tube experiment is recognized as the most reliable standard test method [22,23].It can simulate not only the oil and gas displacement in one-dimensional space but also the complex system of three phases.However, this experimental method also has some disadvantages that cannot be ignored.Firstly, this method is time-consuming and costly [24,25].Compared with the slim tube experiment, the vanishing interfacial tension (VIT) technique is simple in operation and less timeconsuming.At the same time, the method is more rigorous, and the measured MMP is more accurate [26].However, it is not suitable for multicontact miscible systems, and it would be affected by subjective factors [27].The multicontact experiment method has the advantages of high efficiency, high precision, and low cost.Meanwhile, the MMP measured by the multicontact experiment method is generally under the condition of a single condensing drive or vaporizing drive [28].But the real miscibility of gas flooding is usually achieved by the combination of condensing and vaporizing drives.Therefore, this method cannot accurately measure the MMP [29].For the rising bubble method, it is low-cost, has a short experimental cycle, and has reliable results [30].However, this method can only measure the MMP of oil-gas miscibility under the condition of the vaporizing drive.So it is not suitable for the measure of MMP, whose miscible mechanism is condensing drive or condensing-vaporizing drive [31].The steam density method is low in expenditure and short in the experimental cycle [32].Nevertheless, it is generally carried out at a low temperature, and the measured MMP in the experiment needs to be converted into the value at the real formation temperature.There may be some errors in the experimental results by using this method [33].
In order to find a faster way to determine MMP, scholars obtain empirical correlations by fitting the relevant experimental data.A large number of empirical correlations have been proposed, such as the Glaso correlations, Johnson-Pollin correlations, and Alston correlations [34][35][36].Compared with the experimental method, the empirical correlation method is simple, convenient, and low-cost.However, the empirical correlation is only applicable to the calculation of MMP under specific reservoir conditions.Once the target reservoir conditions change, the calculation results may have large errors.So, this method is not a general theoretical calculation method.Compared with the experimental method, the method of characteristics (MOC) can quickly give the predicted value of MMP, which does not depend on the movement of the oil phase and gas phase.And the two phases can fully contact to be miscible when using MOC [30].It is also applicable to pure CO 2 and crude oil, which helps researchers understand the characteristics of gas flooding.However, this method is only suitable for pure CO 2 and crude oil.And the calculation process is complex, and it is difficult to converge to the correct key tie line [37,38].In addition, there are some potential problems, which result in the low prediction accuracy of MMP in the negative flash calculation [38].The one-dimension slim tube compositional simulation method can reflect the oil-gas flow state in porous media by using the numerical simulation method to simulate the slim tube experiment.This method can simulate complex three-phase systems.There may be a deviation in MMP due to the influence of numerical dispersion effects [39].Although using the high-order method can mitigate the dispersion effect, it cannot completely remove its influence.Furthermore, the instability of the three-phase equilibrium calculation and the fuzzy permeability model also make the predicted MMP inaccurate.The multiple mixing cell (MMC) method is a method that simulates the contact and mixture of oil and gas.Initially, two cells involve the injection gas and the reservoir oil at a fixed temperature and pressure.Then, it obtains new oil and gas components after two-phase miscibility equilibrium, and the new oil and gas components contact and mix separately again with the original gas and oil components.Finally, constantly repeat this process until all the key tie lines are found and converge to the specified tolerance.This method has two common models, one is the AHMADI model and another is the JAU-BERT model, as shown in Figures 1 and 2 [40,41].The MMC model can predict the MMP of two-phase systems 2 Geofluids or multiphase complex systems simply, quickly, and stably.
The calculated results are in good agreement with the measured values of the slim tube experiment.It is suitable for the system with the combination of condensing and vaporizing drives, and it considers the influence of asphaltene precipitation and low-temperature conditions on MMP [42,43].Therefore, the predicted value of MMP by MMC is more accurate than other theoretical calculation methods.However, the MMC also has some problems.For example, it is incorrect to judge the initial value, and the solution is not only one.
In conclusion, the experimental method generally has a long period and a high cost, while the theoretical calculation method is relatively faster and more accurate.However, most of these two types of methods currently obtain the MMP under bulk conditions, and there are few solutions for the MMP of the oil and gas two phases at the nanoscale.Therefore, a new, efficient, and accurate method to determine the MMP is proposed.In this paper, based on the modified Peng-Robinson equation of state (PR-EOS), a flash calculation model is established which takes into account the influence of the confinement effect at the nanoscale.Then the flash calculation model is used to judge whether the oil-gas two phases have reached an equilibrium state so as to calculate the interfacial tension between the two phases.Afterward, we use the extrapolation method to obtain the MMP of the oil and gas system at the nanoscale.Finally, the reliability of the model is verified by comparing the MMP obtained by the MMC method with the MMP obtained in this paper.In addition, the effects of confinement effect, reservoir temperature, hydrocarbon type, and nanopore size on the MMP are emphatically studied.

Modified PR-EOS.
The equation of state is a mixture of theory and experience, which is used to express the P-V-T phase behavior of the fluid.It can not only accurately predict the equilibrium state of oil and gas in two phases but also analyze and determine the changes in the two-phase conditions, especially the change in the critical point trend.Scholars have done a lot of researches on the equation of state, mainly including the Redlich-Kwong equation of state (RK-EOS), the Peng-Robinson equation of state (PR-EOS), and the modified PR-EOS [21,44].
In this paper, the modified PR-EOS is used to establish the flash calculation model in nanopores [44].And the modified PR-EOS is shown as follows: where where P is the system pressure; T is the system temperature; R is the universal gas constant; ν is the molar volume; a and b are the gravitational coefficient and the repulsion coefficient separately; T c and P c are the critical temperature and the critical pressure in bulk phase separately; T r is the reduced temperature; ω is the acentric factor; α is the adjustable temperature function.However, considering the large capillary pressure and confinement effect in nanopores, the critical temperature and the critical pressure of the fluid in nanopores are calculated according to the equations which are related to the Lennard-Jones diameter (σ LJ ) and the pore radius in this paper [45].
where T cp is the critical temperature in nanopores; P cp is the critical pressure in nanopores; r p is the nanopore radius.

Calculation Derivation of MMP.
The flash calculation model based on the modified PR-EOS determines whether the oil and gas phases in the nanosystem reach an equilibrium state according to the equality of the fugacity of the two phases.After the oil-gas two phases are in equilibrium, 3 Geofluids the interfacial tension between the two phases is calculated using the following formulas.When the extrapolated interfacial tension is zero, the corresponding pressure is the MMP of the CO 2 -oil system in nanopores.At the same time, considering that the Newton iteration method is involved in many parts of the calculation process and artificial calculation is more complex and time-consuming, so in this paper, the MATLAB software to realize the calculation of the flash calculation model was used.The flow chart of the flash calculation model based on the modified PR-EOS for phase properties estimate the Macleod-Sugden equation for interfacial tension calculation, and the MMP calculation in nanopores is shown in Figure 3.In addition, the relevant calculation formulas and equations can be found in the supporting information.
According to the research requirement, we need to set the following parameters before the calculation: initial capillary pressure (P cap ) is 0 atm; oil and gas contact angle (θ) is 30 °; universal gas constant (R) is 82.06 atm•cm 3 /(mol•K); calculation error is 0.0005.

Results and Discussion
3.1.Model Validation.In order to verify the reliability of the proposed model, the MMP of the CO 2 -octane system is calculated and compared with the MMC model at 373 K and 5 nm pore radius.The relationships between IFTs and pressures are shown in Figure 4.The interfacial tension decreases with the increase of pressure, which shows a good linear relationship.The fitting result is as follows: where P is the pressure of the system and IFT is the interfacial tension of oil and gas phases.The MMP of the CO 2octane system can be obtained by extrapolating to the corresponding pressure where the IFT is zero.According to the linear relationship, the MMP of this system can be predicted to be 9.64 Mpa.For the MMC model, the calculated MMP is around 9.7 MPa (as shown in Figure 5).The relative error between the calculated MMP in this research and the MMP calculated by the MMC model is 0.62%.It seems that our proposed model is feasible to predict the MMP at the nanoscale, which has high reliability and good prediction.
Besides, the estimated IFT in this research is also compared with that of the experimental IFT data in reference [17], as shown in Figure 6.It is found that the simulated results are close to those of the reference [17], with an average absolute relative deviation (AARD) of about 7.2%.

Confinement
Effect on MMP at the Nanoscale.The influence of the confinement effect on MMP at the nanoscale is shown in Figure 7.The MMPs of the CO 2 -octane system increase with increasing temperature for both at the nanoscale and in the bulk phase.At the nanoscale, the MMP increases with temperature initially, and then the growth rate for MMP slowed down when the temperature is over 343.15 K.However, for the bulk phase, the MMP increases rapidly as the temperature increases.Besides, it shows that the MMP in nanopores is generally lower than that in the bulk phase.It seems that the influence of the confinement effect on MMP at the nanoscale cannot be ignored.When the temperature is low (shown in Figure 8), the MMP difference in the bulk phase and at the nanoscale is small, and the relative error is less than 10%.With the temperature rising, the difference between the MMP at the nanoscale and the MMP in the bulk phase is increasing.When the temperature is 372.53K, the relative error reaches more than 30%.Therefore, it is speculated that the confinement effect has a stronger influence on MMP in the nanosystem at higher temperatures.
In the process of CO 2 injection displacement in a shale reservoir, the intermolecular interaction force at the nanoscale is strong and cannot be ignored.The adsorption layer is formed on the surface of nanopores, which have a great influence on the properties of fluid [46].The physical properties of the fluid in nanopores are often different from the measured values in the laboratory.The influence of the confinement effect at the nanoscale can make the critical temperature, critical pressure, interfacial tension, and other properties of the fluid change greatly.It is generally believed that the bubble pressure and interfacial tension in nanopores are lower than those in the bulk phase.And the smaller the nanopore size, the more significant the difference with the bulk phase.Therefore, the MMP of CO 2 and crude oil at the nanoscale is different from Figure 2: MMC model established by JAUBERT [41].
that in the bulk phase due to the confinement effect in nanopores.The measured MMP at the nanoscale is generally smaller than that in the bulk phase.

Effect of Temperature on MMP.
The effect of temperature on MMPs for CO 2 and different alkane systems is shown in Figure 9.The simulation results show that the MMP increases with the rising temperature.For all of the alkane systems, the MMP increases first, and then the increasing rate slows down.This phenomenon is particularly obvious in CO 2 -butane and CO 2 -hexane systems.For CO 2octane and CO 2 -decane systems, the MMP is more sensitive to the higher temperature compared to the alkane with a lower carbon atom number.
The influences of reservoir temperature on MMP are mainly reflected in the changes in the P-V-T properties of crude oil, such as the density, viscosity, and dissolved gasoil ratio.Most scholars have found that the MMP increases when the temperature rises [21,23].The MMP is in proportion to the reservoir temperature.The density of CO 2 becomes smaller with the continuous increase in temperature when other conditions in the reservoir remain the same, which results in the weak solubility of CO 2 , and less CO 2 is dissolved in crude oil.At high temperature, the CO 2 extraction amount of the light hydrocarbon components decreases, which can also lead to an increase in MMP.Liu et al. explained the effect of reservoir temperature and pressure on the oil-gas interfacial tension at the microlevel [47].The reservoir temperature and pressure affect the two-phase interfacial tension by changing CO 2 density and oil-gas thermodynamic parameters, which ulteriorly affect the MMP.

Effect of Hydrocarbon Type on MMP.
The effects of hydrocarbon type on MMP at different temperatures are shown in Figure 10.As the carbon atom number in alkanes increases, the MMP gets enhanced.Therefore, crude oil which has more light hydrocarbon components is easier to reach the miscible state in actual oil fields.In addition, it is found that the relationship between MMP and the number of carbon atoms is close to a straight line.Through linear fitting, it is found that the fitting degrees at different temperatures are different, but they are all close to 1. So, it is inferred that MMP has a linear relationship with the number of carbon atoms in n-alkanes.
Due to the diversity of the crude oil components and properties, the displacement mechanism of CO 2 and crude     found that the interfacial tension is positively correlated with the number of carbon atoms in the CO 2 and single-component oil phase system through the interfacial tension experiment [47].The larger the carbon atom number, the greater the interfacial tension and the larger the MMP.In actual oil reservoirs, the variation of MMP is mainly related to the content of C 2 -C 6 in crude oil.The MMP increased with the decrease of C 2 -C 6 content, while it is positively correlated with the content of heavy components [48,49].This is mainly determined by the relationship between CO 2 extraction and the miscibility principle.During the contact between CO 2 and crude oil, the light components in the crude oil are continuously extracted and enriched in the CO 2 phase.After reaching a certain level, the miscibility of CO 2 and crude oil is realized.
As the light components in the remaining oil continue to decrease, the pressure required for miscibility will also increase, which makes it difficult for CO 2 to be miscible with the remaining oil.11.It shows that the MMP has a positive correlation with the nanopore radius.At small nanopore size, the MMP increases rapidly with the rising nanopore radius.The MMP reduction in small nanopores is caused primarily by the critical point shift due to the confinement effect [46].The  7 Geofluids interactions between molecules and nanopore walls are decreased with increasing nanopore size, which significantly affects the interface between vapor and liquid phases and interfacial free energy.This results in a rapid increase in IFT and a rapid decrease in MMP at small nanopores.
However, the increasing trend of MMP begins to slow down gradually when the pore radius grows to a certain value.It can be inferred that when the nanopore radius is getting larger, the MMP of the CO 2 -oil system in nanopores should be closer to the MMP in the bulk phase.Sun and Li found that MMP remains constant and is not affected by the confinement effect when the pore size is greater than 10 nm [46].It is inferred that molecule-level phenomena no longer play an important role in affecting the phase behavior of CO 2 and crude oil within the nanopores (such as adsorption and nonhomogeneous density distributions in the nanopore).The distribution of the pore radius is the key factor when calculating the MMP in nanopores.It indicates that the average pore radius cannot be directly used to calculate MMP as the MMP (especially for larger pore radius) is no longer linear with the pore radius.At large nanopore size, Bao et al. also found that the MMP of the CO 2 -decane system has almost no significant difference from the MMP by using micro-nanofluidic technology [50].With the rising nanopore size, the MMP at the nanoscale and the MMP in the bulk phase are getting closer.

Conclusions
All major conclusions from this work are shown in the following section: (1) An MMP calculation method is proposed through the extrapolation of IFT for the CO 2 -oil system based on modified PR-EOS at the nanoscale.The relative error between the calculated MMP in this research and the MMC method is 0.62% at 373 K and a 5 nm pore radius.It indicates that the proposed method is efficient to predict the MMP accurately with less calculation (2) The calculated MMP at the nanoscale is generally smaller than that in the bulk phase.At the same time, the confinement effect has a stronger influence on the MMP in the nanopores at a higher temperature (3) At the same reservoir temperature, MMP is positively correlated with the carbon atom number of n-alkanes.Therefore, crude oil which has more light hydrocarbon components is easier to reach the miscible state (4) The MMP has a positive correlation with the nanopore radius.When the nanopore radius is getting larger, the MMP of the CO 2 -oil system in nanopores would be closer to the MMP in the bulk phase.In addition, the average pore radius cannot be directly used to calculate MMP as the MMP (especially for larger pore radius) is no longer linear with the pore radius

Figure 7 :Figure 8 :
Figure 7: The calculated MMPs of the CO 2 -octane system at the pore radius of 5 nm and the experimental MMPs in the bulk phase at T = 311:15 K to 372.53 K.

Figure 9 :
Figure 9: The effect of temperature on MMP of CO 2 and pure alkane system (C 4 H 10 , C 6 H 14 , C 8 H 18 , and C 10 H 22 ) at T = 290:15 K to 373.15 K with 5 nm pore radius.
Tang et al. alsofound that the more molar components of C 2 -C 6 in crude oil components, the lower the MMP.Therefore, reservoirs with high light component content should be selected for CO 2 injection flooding, where CO 2 and crude oil can be miscible at low formation pressure.During CO 2 flooding, the adjustment of the injectionproduction scheme in time is the key to improving the oil recovery by monitoring and analyzing the miscible state in the reservoir.The bottom-hole flow pressure and injection pressure should be appropriately increased if the MMP of CO 2 and crude oil increases.Therefore, it is necessary to dynamically calculate and analyze the MMP of the reservoir to clarify the miscible conditions in the reservoir.