An Improved CO 2-Crude Oil Minimum Miscibility Pressure Correlation

Minimum miscibility pressure (MMP), which plays an important role in miscible flooding, is a key parameter in determining whether crude oil and gas are completelymiscible. On the basis of 210 groups of CO 2 -crude oil systemminimummiscibility pressure data, an improvedCO 2 -crude oil systemminimummiscibility pressure correlationwas built bymodified conjugate gradientmethod and global optimizing method. The new correlation is a uniform empirical correlation to calculate the MMP for both thin oil and heavy oil and is expressed as a function of reservoir temperature, C 7+ molecular weight of crude oil, and mole fractions of volatile components (CH 4 and N 2 ) and intermediate components (CO 2 , H 2 S, and C 2 ∼C 6 ) of crude oil. Compared to the eleven most popular and relatively high-accuracy CO 2 -oil system MMP correlations in the previous literature by other nine groups of CO 2 -oil MMP experimental data, which have not been used to develop the new correlation, it is found that the new empirical correlation provides the best reproduction of the nine groups of CO 2 -oil MMP experimental data with a percentage average absolute relative error (%AARE) of 8% and a percentage maximum absolute relative error (%MARE) of 21%, respectively.


Introduction
CO 2 injection is one of the most effective methods to enhance oil recovery [1].Generally, the oil recovery of miscible flooding is higher than nonmiscible flooding.The minimum miscibility pressure (MMP) at which the crude oil and CO 2 become miscible is a key factor because, in general, the CO 2 is not miscible at first contact with reservoir oils but may achieve dynamic miscibility through multiple contact [2].At present, prediction of the MMP commonly contains three methods: experiment [3], empirical correlation [4], and equation of state [5,6].The slim tube test is one of the most commonly used test methods [3]; in addition, there are risingbubble apparatus (RBA) method [7], steam density method [8], multiple contact method [9], and interfacial tension vanish method [10].The experimental method is the standard method, but it needs to consume large amounts of time and money.Equation of state is precise and fast, but the miscibility function is difficult to give a clear judgment standard, because a characterization procedure of the plus-fraction must be used and such a characterization can have a huge influence on the calculated value.Thus, empirical correlation is usually used for predicting the MMP.Most the MMP empirical correlations are proposed based on the experimental data of CO 2 -oil system, while these MMP empirical correlations of CO 2 -oil system have certain constraints.
This study has two objectives.The first objective is to utilize the modified conjugate gradient and global optimization algorithm for establishing a four-parameter and improved MMP prediction model of CO 2 -oil system, which has a wider range of application, taking advantage of 210 groups of CO 2oil MMP experimental data tested by slim tube experiment in the literature.The second objective is to compare this model with the other eleven most popular and relatively highaccuracy CO 2 -oil MMP correlations presented in the previous literature by using other nine groups of CO 2 -oil MMP experimental data, which have not been used to develop the new correlation.

Experimental Section
The slim tube test has become a standard method to measure the MMP in the petroleum industry.In this study, the CO 2 -oil MMPs of three crude oil samples (i.e., oil 1, oil 2, and oil 3) are measured by using the slim tube test method.Table 1 shows the compositional analysis results of these three oil samples.It can be seen from the compositional analysis results that all these three oil samples used in this study have a large amount of volatile components (N 2 and CH 4 ) and C 7+ fraction.The molecular weights of C 7+ fraction for oil 1, oil 2, and oil 3 are measured to be 183.69g/mol, 245.36 g/mol, and 229.17 g/mol, respectively.
The slim tube apparatus used in this study is a stainless steel fine tube (length of 20 m, inner diameter of 4.4 mm, and a total pore volume of 92.75 cm 3 ) filled with the 80∼100 mesh quartz sand.Schematic diagram of the slim tube experimental apparatus is shown in Figure 1.The slim tube tests are performed on the recombined reservoir fluid with CO 2 at the given reservoir temperature.Once the slim tube is saturated with the crude oil sample, the CO 2 is introduced to displace the oil at an injection rate of 0.125 cm 3 /min.CO 2 displacement experiments are carried out at several pressures with the temperature being maintained constant at the reservoir temperature.For each test pressure, the pore volume of CO 2 injected, produced oil volume, and produced gas volume are recorded.Figure 2 plots the oil recovery factors measured at 1.20 pore volume of CO 2 injected as a function of operating pressure for oil sample 1.The acknowledged criterion for determining slim tube test to achieve miscibility is the oil recovery greater than 90% when 1.20 pore volume of CO 2 or other gases is injected, and with the displacement pressure increased, the displacement efficiency is no longer increasing [11,12].The CO 2 -oil MMP at 130 ∘ C for oil sample 1 is determined to be 20.65 MPa by pinpointing the breakpoint of the oil recovery curve (see Figure 2).By applying the same methodology as for other temperature points (110 ∘ C, 90 ∘ C, and 70 ∘ C) for oil sample 1, the CO 2 flooding minimum miscibility pressure is 20.35

Main Factors Influencing the MMP.
Reviewing published MMP slim tube test data and previously presented empirical models indicates the existence of the following [21,22]: (1) The MMP of CO 2 -oil system is determined by the reservoir temperature, the components in the injected gas, and the components and properties of oil.
(2) On the constant condition of the components in the injected gas and the components and properties of oil, the MMP increases with increasing the reservoir temperature.
(3) On the constant condition of the components in the injected gas and the reservoir temperature, the higher the content of C 2 ∼C 6 and the lower the molecular weight in the crude oil, the smaller MMP.On the contrary, the more the heavy components in the crude oil are, the less favorable it will be for miscibility.
(4) On the constant condition of the reservoir temperature and the components and properties of oil, the MMP decreases with increasing the content of intermediate components (CO 2 , H 2 S, and C 2 ∼C 6 ) and increases with increasing the content of volatile components (CH 4 and N 2 ) in the injected gas.
The paper is focused on building an improved MMP model of pure CO 2 -oil system, so the influence of injection gas components on MMP has been taken into account.Based on the shortages of the above empirical formula in Table 2 and the sensitive factors proposed in Section 3.2 influencing the MMP, we selected the four sensitive factors including reservoir temperature, relative molecular weight of C 7+ , the volatile components (CH 4 and N 2 ), and intermediate components (CO 2 , H 2 S, and C 2 ∼C 6 ) of crude oil to develop an improved MMP prediction correlation with four parameters by using the modified conjugate gradient and global optimization algorithm regression theory.

Mathematical Model.
The determined MMP of CO 2crude oil system is the result of multiple factors interaction.Therefore, we should take full account of the sensitive factors in Section 3.2 and then maximize and utilize the experimental data.However, when the independent variable and dependent variable uncertainty or error is larger, prediction results by the traditional least squares linear regression method are very low.Thus, the optimization and regression algorithm can solve the problem well.In this paper we use the modified conjugate gradient and global optimization algorithm to establish a prediction model for the MMP of CO 2 -oil system.
And the prediction model, based on Emera-Sarma model, also consists of four affecting factors (reservoir temperature, C 7+ molecular weight of crude oil, mole fractions of volatile components (CH 4 and N 2 ), and mole fractions of intermediate components (CO 2 , H 2 S, and C 2 ∼C 6 ) in the crude oil) and four parameters; the following improved correlation was developed: On the basis of Emera-Sarma model, reservoir temperature,  C 7+ in crude oil, mole content of volatile component (CH 4 and N 2 ), and mole content ratio of intermediate components (CO 2 , H 2 S, and C 2 ∼C 6 ) in crude oil were modified.The term ln(1.8 ×  + 32) is used to suppress temperature effect on the hydrocarbon gas-oil MMP when the reservoir temperature is relatively high.The reason why  C 7+ instead of  C 5+ is used in the correlation is partially because  C 7+ is a routine measurement item in a typical compositional analysis The detailed parameters refer to [18] Limitations:  ,min,pure < 40 MPa The objective optimization function contains four parameters ( = ( 1 ,  2 ,  3 ,  4 )).The CO 2 -oil MMP database used in this study includes a total of 210 MMP measurements from the literature, among which the temperature has a range of 21.67 ∘ C∼191.97 ∘ C and C 7+ molecular weights range from 130 g/mol to 402.7 g/mol [2,[20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37].In addition, it should be noted that 176 out of the 210 measurements are obtained from overseas data in the literature, while the remaining 34 measurements are obtained from domestic data in the literature.The CO 2 -oil MMP database is used to determine the tuned coefficients ( = ( 1 ,  2 ,  3 ,  4 )) in (1) by regression fitting using the modified conjugate gradient and global optimization algorithm.The regression fitting has been conducted by using the Matlab programming.The tuned coefficients are given in Table 3 and (1) generates a fit with  2 = 0.9488 (Figure 4).
Step 2. If ‖  ‖ < , algorithm stops and   in () is to be obtained; otherwise, algorithm turns to Step 3.   , the conjugate gradient of () at   , represents   =   ×(  ), in which ‖  ‖ is the norm of   and   is the parameter.Generally, two expression forms include Step 3.
Step length   is determined by 1D linear search.
Step 4. Place  +1 =   +     , in which   is the conjugate gradient search direction and is determined as follows: in which where    is to be multiplied at both sides to obtain the following expression: It is obvious that      is always less than 0 and    is greater than 0, which results in downward search direction.Moreover, if   ̸ = 0, −    −1 /  −1  −1 ≥ 0,   =  FR  , the modified conjugate gradient method is FR conjugate gradient method [38].Otherwise, combining correlation ((c), see Table 2), we can draw that It is called FR conjugate gradient method.It is indicated that the FR conjugate gradient method takes in excellent global convergence of FR algorithm and excellent numerical result of PR algorithm.
Finally, the modified MMP correlation of CO 2 -crude oil is as follows: Compared with the other 11 correlations in Table 2, the correlation has broader application (pressure range: 0∼70 MPa, temperature range: 21.67∼191.97∘ C, and relative molecular weight of C 7+ : 130∼402.7 g/mol).

Calculation Results and Analysis
Generally, the absolute error (6), the absolute relative error (7), and the average absolute relative error (8) are used to express the deviation between the calculated MMP by the empirical correlation and optimize the most appropriate empirical correlation for predicting the MMP of CO 2 -oil system: A new correlation validation is performed with more MMP data (Table 4).These MMP data have not been used to develop the new correlation.The comparative results of the calculated MMP by the correlation proposed in this study and the other eleven most popular and relatively high-accuracy correlations presented in the previous literatures are shown in Figure 5 and Table 5.The average absolute relative errors (AARE) for the correlation proposed in this study, Cronquist's correlation, Lee's correlation, Yelling-Metcalfe's correlation, Orr-Jensen's correlation, Glaso's correlation, Alston's correlation, Emera-Sarma's correlation, Yuan's correlation, Shokir's correlation, Chen's correlation, and Ju's correlation are 8%, 16%, 37%, 20%, 32%, 19%, 20%, 13%, 27%, 21%, 14%, and 29%, respectively.The maximum absolute relative errors (MARE) for the proposed correlation in this study, Cronquist's correlation, Lee's correlation, Yelling-Metcalfe's correlation, Orr-Jensen's correlation, Glaso's correlation, Alston's correlation, Emera-Sarma's correlation, Yuan's correlation, Shokir's correlation, Chen's correlation, and Ju's correlation are 21%, 31%, 73%, 46%, 78%, 50%, 39%, 25%, 58%, 52%, 28%, and 57%, respectively.These results indicate that the proposed correlation in this study is significantly more precise than the other correlations.The results of the calculated MMP by the correlation proposed in this study, the measured MMP by slim tube test, and the absolute error (AE) are shown in Figures 5 and 6.From Table 5, it is clearly seen that the absolute errors (AE) of the calculated MMP by the model proposed in this study of many oil samples are less than 1.5 MPa, which are very close to the experimental data.

Conclusions
(1) Four sensitive factors are determined for affecting the MMP of CO 2 -oil system, which includes the reservoir temperature, C 7+ molecular weight of oil, mole fractions of volatile components (CH 4 and N 2 ), and mole fractions of intermediate components (CO 2 , H 2 S, and C 2 ∼C 6 ) of oil.Based on the above sensitive factors, a four-parameter and improved MMP prediction model of CO 2 -oil system is (2) The nine groups of CO 2 -oil MMP experimental data, which have not been used to develop the new correlation, were calculated by the empirical correlation proposed in this study and other eleven most popular and relatively highaccuracy empirical correlation presented in the literature to validate the new correlation.It can be seen from the comparative results that the accuracy of the empirical correlation proposed in this study is significantly more precise than the other eleven most popular and relatively high-accuracy empirical correlations presented in the literature.The range of the absolute error is less than 1.5 MPa, which corresponds to the requirement of engineering design of CO 2 displacement.

2 Figure 1 :
Figure 1: Schematic diagram of the slim tube experimental apparatus.

Figure 2 :Figure 3 :
Figure 2: Variation of the oil recovery measured at 1.20 pore volume of CO 2 injected at various injection pressure for oil sample 1.

Figure 4 :
Figure 4: Resulted CO 2 -oil MMP from the new correlation versus the experimental measurements.

Figure 6 :
Figure 6: The MMPs of the system of nine CO 2 -oil samples and their errors predicted by the correlation proposed in this paper.

Table 1 :
Compositional analysis results of three oil samples with C 7+ molecular weight in mole percentage.

Table 2 :
Correlations for CO

Table 3 :
Regression parameters.report, while  C 5+ normally need to be calculated from  C 7+ .In addition, it is found in this study that the use of  C 7+ , rather than  C 5+ , even leads to a slightly better performance of (1) in terms of the correlation coefficient  2 .Meanwhile,  C 7+ is replaced by ln( C 7+ ) to reduce the influence of  C 7+ on the MMP of CO 2 -oil system when  C 7+ is larger.And  VOL /  MED is replaced by (1 +  VOL /  MED ) to avoid the fact that  VOL /  MED approaches to zero because of too fewer volatile components in heavy oil which result in great differences between the parameters.

Table 4 :
The nine oil samples components, properties, and MMP data for the new correlation validation.

Table 5 :
Comparison of predicted MMPs for nine oil samples by the correlation proposed in this study and other eleven literature correlations.
Nomenclature C 5+ : M o l ecula rw e i gh to fC 5+ in the crude oil, g/mol : Reservoir temperature, ∘ C  C 7+ : M o l ecula rw e i gh to fC 7+ in the crude oil, g/mol  VOL : Mole fraction of volatile components (CH 4 + N 2 ) in the crude oil, mol%  MED : Mole fraction of intermediate components (CO 2 , H 2 S, and C 2 ∼C 4 ) in the crude oil, mol%   MED : Mole fraction of intermediate components (CO 2 , H 2 S, and C 2 ∼C 6 ) in the crude oil, mol%  ,min,pure : Minimum miscibility pressure by pure CO 2 injection, MPa  ,min,impure : Minimum miscibility pressure by impure CO 2 injection, MPa  1 : Reservoir temperature, ∘ C  2 : Mole fraction of volatile components (CH 4 + N 2 ) in the crude oil, mol%  3 : Mole fraction of intermediate components (CO 2 , H 2 S, and C 2 ∼C 6 ) in the crude oil, mol%  4 : M o l e c u l a rw e i g h to fC 5+ in the crude oil, g/mol  5 : Mole fraction of volatile components (C 1 ) in the injection gas, mol%  6 : Mole fraction of intermediate components (C 2 ∼C 4 ) in the injection gas, mol%  7 : Mole fraction of volatile components (N 2 ) in the injection gas, mol%  8 : Mole fraction of volatile components (H 2 S) in the injection gas, mol%  C 1 +N 2 : Mole fraction of volatile components (CH 4 + N 2 ) in the crude oil, mol%  C 2 −C 6 : Mole fraction of intermediate components (C 2 ∼C 6 ) in the crude oil, mol% AE: Absolute error %ARE: Percentage absolute relative error %AARE: Percentage average absolute relative error %MARE: Percentage maximum average absolute relative error.