Journal of Chemistry
Volume 2015 (2015), Article ID 175940, 10 pages
http://dx.doi.org/10.1155/2015/175940

Research Article

An Improved CO₂-Crude Oil Minimum Miscibility Pressure Correlation

Hao Zhang,1,2 Dali Hou,¹ and Kai Li¹

¹School of Energy, Chengdu University of Technology, Chengdu, Sichuan 610059, China
²State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Chengdu University of Technology, Chengdu, Sichuan 610059, China

Received 25 August 2015; Accepted 11 October 2015

Academic Editor: Tomokazu Yoshimura

Copyright © 2015 Hao Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Minimum miscibility pressure (MMP), which plays an important role in miscible flooding, is a key parameter in determining whether crude oil and gas are completely miscible. On the basis of 210 groups of CO₂-crude oil system minimum miscibility pressure data, an improved CO₂-crude oil system minimum miscibility pressure correlation was built by modified conjugate gradient method 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, C₇₊ molecular weight of crude oil, and mole fractions of volatile components (CH₄ and N₂) and intermediate components (CO₂, H₂S, and C₂~C₆) of crude oil. Compared to the eleven most popular and relatively high-accuracy CO₂-oil system MMP correlations in the previous literature by other nine groups of CO₂-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 CO₂-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.

1. Introduction

CO₂ 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 CO₂ become miscible is a key factor because, in general, the CO₂ 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 rising-bubble 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 CO₂-oil system, while these MMP empirical correlations of CO₂-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 CO₂-oil system, which has a wider range of application, taking advantage of 210 groups of CO₂-oil 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 high-accuracy CO₂-oil MMP correlations presented in the previous literature by using other nine groups of CO₂-oil MMP experimental data, which have not been used to develop the new correlation.

2. Experimental Section

The slim tube test has become a standard method to measure the MMP in the petroleum industry. In this study, the CO₂-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 (N₂ and CH₄) and C₇₊ fraction. The molecular weights of C₇₊ fraction for oil 1, oil 2, and oil 3 are measured to be 183.69 g/mol, 245.36 g/mol, and 229.17 g/mol, respectively.

Table 1: Compositional analysis results of three oil samples with C₇₊ molecular weight in mole percentage.

Component
Oil sample 1
Oil sample 2
Oil sample 3

CO₂
2.076
0.889
0.38

N₂
0.887
1.962
0.56

C₁
31.744
14.96
13.21

C₂
8.217
8.545
2.55

C₃
7.86
5.563
3.66

IC₄
1.756
0.434
1.42

NC₄
3.222
1.072
3.78

IC₅
1.954
0.067
1.68

NC₅
1.531
0.165
2.74

FC₆
4.366
0
8.26

C₇₊
36.387
66.343
61.79

Total
100.00
100.00
100.00

183.69
245.36
229.17

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 cm³) filled with the 80100 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 CO₂ at the given reservoir temperature. Once the slim tube is saturated with the crude oil sample, the CO₂ is introduced to displace the oil at an injection rate of 0.125 cm³/min.

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

CO₂ 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 CO₂ injected, produced oil volume, and produced gas volume are recorded. Figure 2 plots the oil recovery factors measured at 1.20 pore volume of CO₂ 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 CO₂ or other gases is injected, and with the displacement pressure increased, the displacement efficiency is no longer increasing [11, 12]. The CO₂-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 CO₂ flooding minimum miscibility pressure is 20.35 MPa, 19.95 MPa, and 19.3.8 MPa, respectively. As for oil sample 2 and oil sample 3 at 74.8°C and 89.7°C formation, the CO₂ flooding minimum miscibility pressure is 26.80 MPa and 22.65 MPa, respectively. A conservative error of 3% can be applied due to its complexity. Figures 2 and 3 indicate that crude oil recovery increases with injecting pressure and CO₂-crude oil minimum miscible increases with temperature.

Figure 2: Variation of the oil recovery measured at 1.20 pore volume of CO₂ injected at various injection pressure for oil sample 1.

Figure 3: Variation of oil sample miscibility pressure with temperature.

3. Building of MMP Predicting Model

3.1. Existing Methods

Over the years, several empirical correlations have been developed for determining the MMP of CO₂-oil system. The most popular and relatively high-accuracy correlations applied for prediction of CO₂-oil MMP are those developed by Cronquist [4], Lee [13], Yelling-Metcalfe [14], Orr-Jensen [15], Glaso [16], Alston [17], Emera-Sarma [18], Yuan [19], Shokir [20], Chen [21], and Ju [22]. Table 2 shows the expression of the above correlations and its application restrictions.

Table 2: Correlations for CO₂-oil MMP.

Author
Published empirical correlation
Remarks

Cronquist [4]
(a)  
(i) The tested oil gravity ranged from 23.7 to 44.8°API 
(ii) The tested ranged from 21.67 to 120.8°C 
(iii) The tested experimental MMP ranged from 7.4 to 34.5 MPa

Lee [13]
(b)  
(i) Based on equating MMP with CO₂ vapor pressure when < CO₂ critical temperature, while using the corresponding correlation when ⩾ CO₂ critical temperature
(ii) If MMP < , is taken as MMP

Yelling-Metcalfe [14]
(c)  
Limitations: 35.8 ≤ T < 88.9°C
(i) If MMP < , is taken as MMP

Orr-Jensen [15]
(d)  
(i) Based on extrapolated vapor pressure (EVP) method
(ii) Used to estimate the MMP for low temperature reservoirs (°C)

Glaso [16]
When < 18 mol%,
(e1)   
When > 18%,
(e2)  
Considers the effect of intermediates (C₂–C₆) only when (C₂–C₆) < 18 mol%

Alston [17]
When  MPa
(f1)   
When  MPa
(f2)  
If MMP < , is taken as MMP

Emera-Sarma [18]
When  MPa
(g1)   
When  MPa
(g2)  
Limitations: 40.8 < T < 112.2°C; 8.28 < < 30.2 MPa; 166.2 < < 267.5 g/mol

Yuan [19]
(h)    

; ; ;  
;   10¹;  ;  
;
Limitations: 21.7 < T < 148°C;  < 70 MPa;
139 < < 319 g/mol

Shokir [20]
(i1)   
(i2)   
(i3)   
; ; ;  
; ; ;  
; ; ;  
; ; ;
Limitations: 32.2 < T < 112.2°C; 6.9 < <30.28 MPa; 185 < < 268 g/mol

Chen [21]
(j)  
Limitations: 32.2 < T < 118.3°C; 6.9 < < 28.17 MPa; 185 < < 249 g/mol

Ju [22]
(k1)   
(k2)   
(k3)   
The detailed parameters refer to [18]
Limitations: < 40 MPa

3.2. 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 CO₂-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 C₂~C₆ 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 (CO₂, H₂S, and C₂~C₆) and increases with increasing the content of volatile components (CH₄ and N₂) in the injected gas.

The paper is focused on building an improved MMP model of pure CO₂-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 C₇₊, the volatile components (CH₄ and N₂), and intermediate components (CO₂, H₂S, and C₂~C₆) 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.

3.3. Mathematical Model

The determined MMP of CO₂-crude 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 CO₂-oil system.

And the prediction model, based on Emera-Sarma model, also consists of four affecting factors (reservoir temperature, C₇₊ molecular weight of crude oil, mole fractions of volatile components (CH₄ and N₂), and mole fractions of intermediate components (CO₂, H₂S, and C₂~C₆) in the crude oil) and four parameters; the following improved correlation was developed:

On the basis of Emera-Sarma model, reservoir temperature, in crude oil, mole content of volatile component (CH₄ and N₂), and mole content ratio of intermediate components (CO₂, H₂S, and C₂~C₆) in crude oil were modified. The term is used to suppress temperature effect on the hydrocarbon gas-oil MMP when the reservoir temperature is relatively high. The reason why instead of is used in the correlation is partially because is a routine measurement item in a typical compositional analysis report, while normally need to be calculated from . In addition, it is found in this study that the use of , rather than , even leads to a slightly better performance of (1) in terms of the correlation coefficient . Meanwhile, is replaced by to reduce the influence of on the MMP of CO₂-oil system when is larger. And is replaced by to avoid the fact that approaches to zero because of too fewer volatile components in heavy oil which result in great differences between the parameters.

The objective optimization function contains four parameters . The CO₂-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°C191.97°C and C₇₊ molecular weights range from 130 g/mol to 402.7 g/mol [2, 20–37]. 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 CO₂-oil MMP database is used to determine the tuned coefficients 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 (Figure 4).

Table 3: Regression parameters.

Parameters
Value

8.3397 × 10⁻⁵

3.9774

3.3179

1.7461 × 10⁻¹

Figure 4: Resulted CO₂-oil MMP from the new correlation versus the experimental measurements.

Step 1. Given the constant , pick the initial point , , and place .

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 and [22]. In this paper, , in which is the transposed conjugate gradient.

Step 3. Step length is determined by 1D linear search.

Step 4. Place , in which is the conjugate gradient search direction and is determined as follows:

in which . It can be inferred that , 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 , , 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.

Step 5. is determined, and place . Process turns to Step 2.

Finally, the modified MMP correlation of CO₂-crude oil is as follows:

Compared with the other 11 correlations in Table 2, the correlation has broader application (pressure range: 070 MPa, temperature range: 21.67191.97°C, and relative molecular weight of C₇₊: 130402.7 g/mol).

4. 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 CO₂-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.

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

T (°C)

Experimental data (MMP) (MPa)
Reference

60.00
149.690
136.470
39.370
46.160
24.680
11.138
[39]

80.00
149.690
136.470
39.370
46.160
24.680
14.152
[39]

137.22
149.690
136.470
39.370
46.160
24.680
18.379
[39]

100.00
151.740
138.530
26.760
37.950
13.530
14.634
[40]

80.00
170.080
160.590
2.630
7.100
12.150
16.062
[41]

130.00
183.690
165.262
23.131
30.982
32.631
20.650
Current work

74.80
245.690
229.085
16.501
16.733
16.922
26.800
Current work

89.70
229.170
211.213
11.790
24.470
13.770
22.630
Current work

53.00
205.740
182.800
12.933
25.333
18.706
13.090
[42]

Table 5: Comparison of predicted MMPs for nine oil samples by the correlation proposed in this study and other eleven literature correlations.

Exp. (MMP)
(MPa)
Correlation proposed in this study
Cronquist’s correlation
Lee’s correlation
Yelling-Metcalfe’s correlation

MMP (MPa)
ARE, %
MMP (MPa)
ARE, %
MMP (MPa)
ARE, %
MMP (MPa)
ARE, %

11.138
10.594
4.881
14.062
26.251
13.055
17.214
12.135
8.947

14.152
12.619
10.833
13.693
3.241
18.143
28.201
15.154
7.080

18.379
17.548
4.522
24.043
30.816
28.968
57.614
24.077
31.004

14.634
14.243
2.673
15.050
2.844
24.340
66.325
18.139
23.954

16.062
15.387
4.203
14.645
8.820
20.143
25.408
15.154
5.654

20.650
19.990
3.196
25.722
24.561
35.810
73.416
22.870
10.752

26.800
22.551
15.853
22.616
15.613
16.716
37.627
14.381
46.338

22.630
17.864
21.060
16.765
25.919
26.007
14.921
16.594
26.673

13.090
12.399
5.278
12.419
5.129
11.525
11.959
11.014
15.860

AARE, %

8.055

15.910

36.965

19.585

MAARE, %

21.060

30.816

73.416

46.338

Exp. (MMP)
(MPa)
Orr-Jensen’s correlation
Glaso’s correlation
Alston’s correlation
Emera-Sarma’s correlation

MMP (MPa)
ARE, %
MMP (MPa)
ARE, %
MMP (MPa)
ARE, %
MMP (MPa)
ARE, %

11.138
10.002
10.196
16.679
49.746
6.756
39.342
9.362
15.945

14.152
14.306
1.087
19.472
37.593
8.611
39.155
12.214
13.695

18.379
32.845
78.712
18.464
0.462
14.032
23.650
13.738
25.251

14.634
19.691
34.556
13.128
10.290
10.462
28.512
12.071
17.513

16.062
14.306
10.934
14.973
6.779
15.941
0.751
13.032
18.863

20.650
29.963
45.097
22.617
9.525
20.944
1.423
18.150
12.105

26.800
13.087
51.170
22.328
16.687
24.636
8.074
27.222
1.575

22.630
16.777
25.863
26.327
16.337
15.512
31.454
23.042
1.820

13.090
8.734
33.279
16.029
22.452
11.524
11.963
12.102
7.550

AARE, %

32.321

18.874

20.481

12.702

MAARE, %

78.712

49.746

39.342

25.251

Exp. (MMP)
(MPa)
Yuan’s correlation
Shokir’s correlation
Chen’s correlation
Ju’s correlation

MMP (MPa)
ARE, %
MMP (MPa)
ARE, %
MMP (MPa)
ARE, %
MMP (MPa)
ARE, %

11.138
12.775
14.699
12.549
12.669
11.002
1.218
7.091
36.332

14.152
18.512
30.809
15.783
11.522
13.967
1.309
9.539
32.598

18.379
13.735
25.266
27.908
51.845
21.849
18.878
19.767
7.554

14.634
8.304
43.255
15.366
5.004
15.543
6.210
11.280
22.921

16.062
14.568
9.302
15.191
5.420
19.961
24.274
13.697
14.727

20.650
17.164
16.879
26.111
26.445
24.313
17.739
21.215
2.734

26.800
20.033
25.250
20.132
24.881
19.328
27.879
11.409
57.428

22.630
18.113
19.960
15.509
31.468
17.374
23.226
11.615
48.674

13.090
20.738
58.425
10.647
18.664
11.973
8.531
7.577
42.116

AARE, %

27.094

20.880

14.363

29.454

MAARE, %

58.425

51.845

27.879

57.428

Figure 5: The resulted nine-oil-sample MMP from the new correlation proposed in this study versus the calculated nine-oil-sample MMP from 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.

Figure 6: The MMPs of the system of nine CO₂-oil samples and their errors predicted by the correlation proposed in this paper.

5. Conclusions

(1) Four sensitive factors are determined for affecting the MMP of CO₂-oil system, which includes the reservoir temperature, C₇₊ molecular weight of oil, mole fractions of volatile components (CH₄ and N₂), and mole fractions of intermediate components (CO₂, H₂S, and C₂~C₆) of oil. Based on the above sensitive factors, a four-parameter and improved MMP prediction model of CO₂-oil system is established by using the modified conjugate gradient and the global optimization algorithm.

(2) The nine groups of CO₂-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 high-accuracy 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 CO₂ displacement.

Nomenclature

: | Molecular weight of C₅₊ in the crude oil, g/mol
: | Reservoir temperature, °C
: | Molecular weight of C₇₊ in the crude oil, g/mol
: | Mole fraction of volatile components () in the crude oil, mol%
: | Mole fraction of intermediate components (CO₂, H₂S, and C₂~C₄) in the crude oil, mol%
: | Mole fraction of intermediate components (CO₂, H₂S, and C₂~C₆) in the crude oil, mol%
: | Minimum miscibility pressure by pure CO₂ injection, MPa
: | Minimum miscibility pressure by impure CO₂ injection, MPa
: | Reservoir temperature, °C
: | Mole fraction of volatile components (CH₄ + N₂) in the crude oil, mol%
: | Mole fraction of intermediate components (CO₂, H₂S, and C₂~C₆) in the crude oil, mol%
: | Molecular weight of C₅₊ in the crude oil, g/mol
: | Mole fraction of volatile components (C₁) in the injection gas, mol%
: | Mole fraction of intermediate components (C₂~C₄) in the injection gas, mol%
: | Mole fraction of volatile components (N₂) in the injection gas, mol%
: | Mole fraction of volatile components (H₂S) in the injection gas, mol%
: | Mole fraction of volatile components () in the crude oil, mol%
: | Mole fraction of intermediate components (C₂~C₆) 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.

Conflict of Interests

Here, all the authors solemnly declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by a Grant from the National Natural Science Foundation of China (no. 51374044).

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