# An Improved CO2-Crude Oil Minimum Miscibility Pressure Correlation

Authors: Hao Zhang; Dali Hou; Kai Li
Journal: Journal of Chemistry
 (2015)

---

## 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.


---

## Body

## 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
M
C
7
+
183.69
245.36
229.17The 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 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 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)P
m
,
m
i
n
,
p
u
r
e
=
0.11027
1.8
T
+
32
0.744206
+
0.0011038
M
C
5
+
0.0015279
x
V
O
L
(i) The tested oil gravity ranged from 23.7 to 44.8°API(ii) The tested T ranged from 21.67 to 120.8°C (iii) The tested experimental MMP ranged from 7.4 to 34.5 MPa
Lee [13]
(b)P
m
,
m
i
n
,
p
u
r
e
=
7.3924
×
1
0
2.772
-
[
1519
/
(
492
+
1.8
T
)
]
(i) Based on equating MMP with CO2 vapor pressure when T < CO2 critical temperature, while using the corresponding correlation when T
⩾ CO2 critical temperature(ii) If MMP < P
b,  P
b is taken as MMP
Yelling-Metcalfe [14]
(c)P
m
,
m
i
n
,
p
u
r
e
=
12.6472
+
0.01553
(
1.8
T
+
32
)
+
1.24192
×
1
0
-
4
1.8
T
+
32
2
-
716.9427
(
1.8
T
+
32
)
Limitations: 35.8 ≤T < 88.9°C(i) If MMP < P
b,  P
b is taken as MMP
Orr-Jensen [15]
(d)P
m
,
m
i
n
,
p
u
r
e
=
0.101386
e
x
p
10.91
-
2105
255.372
+
0.5556
1.8
T
+
32
(i) Based on extrapolated vapor pressure (EVP) method(ii) Used to estimate the MMP for low temperature reservoirs (T
<
49°C)
Glaso [16]
Whenx
M
E
D
′ < 18 mol%,(e1)  P
m
,
m
i
n
,
p
u
r
e
=
5.58657
-
2.3477
×
1
0
-
2
M
C
7
+
+
1.1725
×
1
0
-
11
M
C
7
+
3.73
e
x
p
786.8
M
C
7
+
-
1.058
(
1.8
T
+
32
) When x
M
E
D
′ > 18%,(e2)  P
m
,
m
i
n
,
p
u
r
e
=
20.33
-
2.3477
×
1
0
-
2
M
C
7
+
+
1.1725
×
1
0
-
11
M
C
7
+
3.73
e
x
p
786.8
M
C
7
+
-
1.058
(
1.8
T
+
32
)
-
0.836
×
x
M
E
D
′
Considers the effect of intermediates (C2–C6) only when x
M
E
D
′ (C2–C6) < 18 mol%
Alston [17]
WhenP
b
≥
0.345 MPa(f1)  P
m
,
m
i
n
,
p
u
r
e
=
6.0536
×
1
0
-
6
1.8
T
+
32
1.06
M
C
5
+
1.78
x
V
O
L
x
M
E
D
0.136 When P
b
<
0.345 MPa(f2)  P
m
,
m
i
n
,
p
u
r
e
=
6.0536
×
1
0
-
6
1.8
T
+
32
1.06
M
C
5
+
1.78
If MMP <P
b,  P
b is taken as MMP
Emera-Sarma [18]
WhenP
b
≥
0.345 MPa(g1)  P
m
,
m
i
n
,
p
u
r
e
=
5.0093
×
1
0
-
5
1.8
T
+
32
1.164
M
C
5
+
1.2785
x
V
O
L
x
M
E
D
0.1073 When P
b
<
0.345 MPa(g2)  P
m
,
m
i
n
,
p
u
r
e
=
5.0093
×
1
0
-
5
1.8
T
+
32
1.164
M
C
5
+
1.2785
Limitations: 40.8 <T < 112.2°C; 8.28 < P
m
,
m
i
n
,
p
u
r
e < 30.2 MPa; 166.2 < M
C
5
+ < 267.5 g/mol
Yuan [19]
(h)P
m
,
m
i
n
,
p
u
r
e
=
a
1
+
a
2
M
C
7
+
-
a
3
x
M
E
D
′
+
a
4
+
a
5
M
C
7
+
+
a
6
x
M
E
D
′
M
C
7
+
2
(
1.8
T
+
32
)
+
a
7
+
a
8
M
C
7
+
-
a
9
M
C
7
+
2
-
a
10
x
M
E
D
′
1.8
T
+
32
2
a
1
=
-
9.8912; a
2
=
4.5588
×
10
-
2; a
3
=
-
3.1012
×
10
-
1; a
4
=
1.4748
×
10
-
2
a
5
=
8.0441
×
10
-
4;  a
6
=
5.6303
× 101;  a
7
=
-
8.4516
×
10
-
4; a
8
=
8.8825
×
10
-
6
a
9
=
-
2.7684
×
10
-
8; a
10
=
-
6.3830
×
10
-
6
Limitations: 21.7 <T < 148°C;  P
m
,
m
i
n
,
p
u
r
e < 70 MPa;139 < M
C
7
+ < 319 g/mol
Shokir [20]
(i1)P
m
,
m
i
n
,
p
u
r
e
=
-
0.068616
Z
3
+
0.31733
Z
2
+
4.9804
Z
+
13.432 (i2)  Z
=
∑
i
=
1
4
Z
i (i3)  Z
i
=
a
i
+
b
i
x
i
+
c
i
x
i
2
+
d
i
x
i
3
a
1
=
-
2.9182; a
2
=
-
3.1227
×
10
-
1; a
3
=
-
4.9485
×
10
-
2; a
4
=
25.430
b
1
=
7
.
5340
×
10
-
2; b
2
=
-
7.9169
×
10
-
3; b
3
=
4.2165
×
10
-
2; b
4
=
-
3.9750
×
10
-
1
c
1
=
-
5.5996
×
10
-
4; c
2
=
1.3644
×
10
-
3; c
3
=
-
2.7853
×
10
-
3; c
4
=
1.9860
×
10
-
3
d
1
=
2.3660
×
10
-
6; d
2
=
-
1.3721
×
10
-
5; d
3
=
3.5551
×
10
-
5; d
4
=
-
3.1604
×
10
-
6
Limitations: 32.2 <T < 112.2°C; 6.9 < P
m
,
m
i
n
,
p
u
r
e <30.28 MPa; 185 < M
C
5
+ < 268 g/mol
Chen [21]
(j)P
m
,
m
i
n
,
p
u
r
e
=
3.9673
×
1
0
-
2
T
0.8293
M
C
7
+
0.5382
x
C
1
+
N
2
0.1018
x
C
2
-
C
6
-
0.2316
Limitations: 32.2 <T < 118.3°C; 6.9 < P
m
,
m
i
n
,
p
u
r
e < 28.17 MPa; 185 < M
C
7
+< 249 g/mol
Ju [22]
(k1)P
m
,
m
i
n
,
i
m
p
u
r
e
=
-
0.04562
S
3
+
0.33399
S
2
+
4.9811
S
+
13.569 (k2)  S
=
∑
i
=
1
8
S
i (k3)  S
i
=
a
i
+
b
i
x
i
+
c
i
x
i
2
+
d
i
x
i
3 The detailed parameters refer to [18]
Limitations:P
m
,
m
i
n
,
p
u
r
e < 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:(1)
P
m
,
m
i
n
,
p
u
r
e
=
a
ln
⁡
1.8
T
+
32
b
ln
⁡
M
C
7
+
c
1
+
x
V
O
L
x
M
E
D
′
d
.On the basis of Emera-Sarma model, reservoir temperature,M
C
7
+ 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 ln
⁡
(
1.8
×
T
+
32
) is used to suppress temperature effect on the hydrocarbon gas-oil MMP when the reservoir temperature is relatively high. The reason why M
C
7
+ instead of M
C
5
+ is used in the correlation is partially because M
C
7
+ is a routine measurement item in a typical compositional analysis report, while M
C
5
+ normally need to be calculated from M
C
7
+. In addition, it is found in this study that the use of M
C
7
+, rather than M
C
5
+, even leads to a slightly better performance of (1) in terms of the correlation coefficient R
2. Meanwhile, M
C
7
+ is replaced by ln
⁡
(
M
C
7
+
) to reduce the influence of M
C
7
+ on the MMP of CO2-oil system when M
C
7
+ is larger. And x
V
O
L
/
x
M
E
D
′ is replaced by 1
+
x
V
O
L
/
x
M
E
D
′ to avoid the fact that x
V
O
L
/
x
M
E
D
′ 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 parametersm
=
m
1
,
m
2
,
m
3
,
m
4. 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°C~191.97°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 m
=
m
1
,
m
2
,
m
3
,
m
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 R
2
=
0.9488 (Figure 4).Table 3
Regression parameters.
Parameters
Value
a
8.3397 × 10−5
b
3.9774
c
3.3179
d
1.7461 × 10−1Figure 4
Resulted CO2-oil MMP from the new correlation versus the experimental measurements.Step 1.
Given the constantσ
1
∈
(
0
,
1
/
2
)
,
σ
1
<
σ
2
<
1
,
ε
>
0, pick the initial point m
0
∈
R
4, d
0
=
-
g
0, and place k
=
0.Step 2.
Ifg
k
<
ε, algorithm stops and m
k in f
(
m
) is to be obtained; otherwise, algorithm turns to Step 3. g
k, the conjugate gradient of f
(
m
) at m
k, represents g
k
=
β
k
×
f
(
m
k
), in which g
k is the norm of g
k and β
k is the parameter. Generally, two expression forms include β
k
F
R
=
g
k
2
/
g
k
-
1
2 and β
k
P
R
=
g
k
T
g
k
-
g
k
-
1
/
g
k
-
1
2 [22]. In this paper, β
k
=
m
a
x
0
,
β
k
F
R
+
m
i
n
0
,
g
k
T
g
k
-
1
/
g
k
-
1
T
d
k
-
1, in which g
k
T is the transposed conjugate gradient.Step 3.
Step lengthα
k is determined by 1D linear search.Step 4.
Placem
k
+
1
=
m
k
+
α
k
d
k, in which d
k is the conjugate gradient search direction and is determined as follows:(2)
d
k
=
-
g
k
k
=
0
,
-
θ
k
g
k
+
β
k
d
k
-
1
k
≥
1
,
in whichθ
k
=
1
+
g
k
T
d
k
-
1
/
g
k
2
β
k. It can be inferred that d
k
=
-
g
k
-
g
k
T
d
k
-
1
/
g
k
2
β
k
g
k
+
β
k
d
k
-
1, where g
k
T is to be multiplied at both sides to obtain the following expression:(3)
g
k
T
d
k
=
-
g
k
2
-
g
k
T
d
k
-
1
g
k
2
β
k
g
k
2
+
β
k
g
k
T
d
k
-
1
=
-
g
k
2
<
0
.It is obvious thatg
k
T
d
k is always less than 0 and g
k
T is greater than 0, which results in downward search direction. Moreover, if β
k
≠
0
,
-
g
k
T
g
k
-
1
/
g
k
-
1
T
d
k
-
1
≥
0, β
k
=
β
k
F
R, the modified conjugate gradient method is FR conjugate gradient method [38]. Otherwise, combining correlation ((c), see Table 2), we can draw that (4)
β
k
=
β
k
F
R
+
g
k
T
g
k
-
1
g
k
-
1
T
d
k
-
1
=
g
k
2
g
k
-
1
2
-
g
k
T
g
k
-
1
g
k
-
1
2
=
g
k
T
g
k
-
g
k
-
1
g
k
-
1
2
=
β
k
P
R
.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.
d
k is determined, and place k
=
k
+
1. Process turns to Step 2.Finally, the modified MMP correlation of CO2-crude oil is as follows:(5)
f
m
=
P
m
,
m
i
n
,
p
u
r
e
=
8.3397
×
1
0
-
5
ln
⁡
1.8
T
+
32
3.9774
ln
⁡
M
C
7
+
3.3179
1
+
x
V
O
L
x
M
E
D
′
0.17461
.Compared with the other 11 correlations in Table2, the correlation has broader application (pressure range: 0~70 MPa, temperature range: 21.67~191.97°C, and relative molecular weight of C7+: 130~402.7 g/mol).


## 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)P
m
,
m
i
n
,
p
u
r
e
=
0.11027
1.8
T
+
32
0.744206
+
0.0011038
M
C
5
+
0.0015279
x
V
O
L
(i) The tested oil gravity ranged from 23.7 to 44.8°API(ii) The tested T ranged from 21.67 to 120.8°C (iii) The tested experimental MMP ranged from 7.4 to 34.5 MPa
Lee [13]
(b)P
m
,
m
i
n
,
p
u
r
e
=
7.3924
×
1
0
2.772
-
[
1519
/
(
492
+
1.8
T
)
]
(i) Based on equating MMP with CO2 vapor pressure when T < CO2 critical temperature, while using the corresponding correlation when T
⩾ CO2 critical temperature(ii) If MMP < P
b,  P
b is taken as MMP
Yelling-Metcalfe [14]
(c)P
m
,
m
i
n
,
p
u
r
e
=
12.6472
+
0.01553
(
1.8
T
+
32
)
+
1.24192
×
1
0
-
4
1.8
T
+
32
2
-
716.9427
(
1.8
T
+
32
)
Limitations: 35.8 ≤T < 88.9°C(i) If MMP < P
b,  P
b is taken as MMP
Orr-Jensen [15]
(d)P
m
,
m
i
n
,
p
u
r
e
=
0.101386
e
x
p
10.91
-
2105
255.372
+
0.5556
1.8
T
+
32
(i) Based on extrapolated vapor pressure (EVP) method(ii) Used to estimate the MMP for low temperature reservoirs (T
<
49°C)
Glaso [16]
Whenx
M
E
D
′ < 18 mol%,(e1)  P
m
,
m
i
n
,
p
u
r
e
=
5.58657
-
2.3477
×
1
0
-
2
M
C
7
+
+
1.1725
×
1
0
-
11
M
C
7
+
3.73
e
x
p
786.8
M
C
7
+
-
1.058
(
1.8
T
+
32
) When x
M
E
D
′ > 18%,(e2)  P
m
,
m
i
n
,
p
u
r
e
=
20.33
-
2.3477
×
1
0
-
2
M
C
7
+
+
1.1725
×
1
0
-
11
M
C
7
+
3.73
e
x
p
786.8
M
C
7
+
-
1.058
(
1.8
T
+
32
)
-
0.836
×
x
M
E
D
′
Considers the effect of intermediates (C2–C6) only when x
M
E
D
′ (C2–C6) < 18 mol%
Alston [17]
WhenP
b
≥
0.345 MPa(f1)  P
m
,
m
i
n
,
p
u
r
e
=
6.0536
×
1
0
-
6
1.8
T
+
32
1.06
M
C
5
+
1.78
x
V
O
L
x
M
E
D
0.136 When P
b
<
0.345 MPa(f2)  P
m
,
m
i
n
,
p
u
r
e
=
6.0536
×
1
0
-
6
1.8
T
+
32
1.06
M
C
5
+
1.78
If MMP <P
b,  P
b is taken as MMP
Emera-Sarma [18]
WhenP
b
≥
0.345 MPa(g1)  P
m
,
m
i
n
,
p
u
r
e
=
5.0093
×
1
0
-
5
1.8
T
+
32
1.164
M
C
5
+
1.2785
x
V
O
L
x
M
E
D
0.1073 When P
b
<
0.345 MPa(g2)  P
m
,
m
i
n
,
p
u
r
e
=
5.0093
×
1
0
-
5
1.8
T
+
32
1.164
M
C
5
+
1.2785
Limitations: 40.8 <T < 112.2°C; 8.28 < P
m
,
m
i
n
,
p
u
r
e < 30.2 MPa; 166.2 < M
C
5
+ < 267.5 g/mol
Yuan [19]
(h)P
m
,
m
i
n
,
p
u
r
e
=
a
1
+
a
2
M
C
7
+
-
a
3
x
M
E
D
′
+
a
4
+
a
5
M
C
7
+
+
a
6
x
M
E
D
′
M
C
7
+
2
(
1.8
T
+
32
)
+
a
7
+
a
8
M
C
7
+
-
a
9
M
C
7
+
2
-
a
10
x
M
E
D
′
1.8
T
+
32
2
a
1
=
-
9.8912; a
2
=
4.5588
×
10
-
2; a
3
=
-
3.1012
×
10
-
1; a
4
=
1.4748
×
10
-
2
a
5
=
8.0441
×
10
-
4;  a
6
=
5.6303
× 101;  a
7
=
-
8.4516
×
10
-
4; a
8
=
8.8825
×
10
-
6
a
9
=
-
2.7684
×
10
-
8; a
10
=
-
6.3830
×
10
-
6
Limitations: 21.7 <T < 148°C;  P
m
,
m
i
n
,
p
u
r
e < 70 MPa;139 < M
C
7
+ < 319 g/mol
Shokir [20]
(i1)P
m
,
m
i
n
,
p
u
r
e
=
-
0.068616
Z
3
+
0.31733
Z
2
+
4.9804
Z
+
13.432 (i2)  Z
=
∑
i
=
1
4
Z
i (i3)  Z
i
=
a
i
+
b
i
x
i
+
c
i
x
i
2
+
d
i
x
i
3
a
1
=
-
2.9182; a
2
=
-
3.1227
×
10
-
1; a
3
=
-
4.9485
×
10
-
2; a
4
=
25.430
b
1
=
7
.
5340
×
10
-
2; b
2
=
-
7.9169
×
10
-
3; b
3
=
4.2165
×
10
-
2; b
4
=
-
3.9750
×
10
-
1
c
1
=
-
5.5996
×
10
-
4; c
2
=
1.3644
×
10
-
3; c
3
=
-
2.7853
×
10
-
3; c
4
=
1.9860
×
10
-
3
d
1
=
2.3660
×
10
-
6; d
2
=
-
1.3721
×
10
-
5; d
3
=
3.5551
×
10
-
5; d
4
=
-
3.1604
×
10
-
6
Limitations: 32.2 <T < 112.2°C; 6.9 < P
m
,
m
i
n
,
p
u
r
e <30.28 MPa; 185 < M
C
5
+ < 268 g/mol
Chen [21]
(j)P
m
,
m
i
n
,
p
u
r
e
=
3.9673
×
1
0
-
2
T
0.8293
M
C
7
+
0.5382
x
C
1
+
N
2
0.1018
x
C
2
-
C
6
-
0.2316
Limitations: 32.2 <T < 118.3°C; 6.9 < P
m
,
m
i
n
,
p
u
r
e < 28.17 MPa; 185 < M
C
7
+< 249 g/mol
Ju [22]
(k1)P
m
,
m
i
n
,
i
m
p
u
r
e
=
-
0.04562
S
3
+
0.33399
S
2
+
4.9811
S
+
13.569 (k2)  S
=
∑
i
=
1
8
S
i (k3)  S
i
=
a
i
+
b
i
x
i
+
c
i
x
i
2
+
d
i
x
i
3 The detailed parameters refer to [18]
Limitations:P
m
,
m
i
n
,
p
u
r
e < 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:(1)
P
m
,
m
i
n
,
p
u
r
e
=
a
ln
⁡
1.8
T
+
32
b
ln
⁡
M
C
7
+
c
1
+
x
V
O
L
x
M
E
D
′
d
.On the basis of Emera-Sarma model, reservoir temperature,M
C
7
+ 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 ln
⁡
(
1.8
×
T
+
32
) is used to suppress temperature effect on the hydrocarbon gas-oil MMP when the reservoir temperature is relatively high. The reason why M
C
7
+ instead of M
C
5
+ is used in the correlation is partially because M
C
7
+ is a routine measurement item in a typical compositional analysis report, while M
C
5
+ normally need to be calculated from M
C
7
+. In addition, it is found in this study that the use of M
C
7
+, rather than M
C
5
+, even leads to a slightly better performance of (1) in terms of the correlation coefficient R
2. Meanwhile, M
C
7
+ is replaced by ln
⁡
(
M
C
7
+
) to reduce the influence of M
C
7
+ on the MMP of CO2-oil system when M
C
7
+ is larger. And x
V
O
L
/
x
M
E
D
′ is replaced by 1
+
x
V
O
L
/
x
M
E
D
′ to avoid the fact that x
V
O
L
/
x
M
E
D
′ 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 parametersm
=
m
1
,
m
2
,
m
3
,
m
4. 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°C~191.97°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 m
=
m
1
,
m
2
,
m
3
,
m
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 R
2
=
0.9488 (Figure 4).Table 3
Regression parameters.
Parameters
Value
a
8.3397 × 10−5
b
3.9774
c
3.3179
d
1.7461 × 10−1Figure 4
Resulted CO2-oil MMP from the new correlation versus the experimental measurements.Step 1.
Given the constantσ
1
∈
(
0
,
1
/
2
)
,
σ
1
<
σ
2
<
1
,
ε
>
0, pick the initial point m
0
∈
R
4, d
0
=
-
g
0, and place k
=
0.Step 2.
Ifg
k
<
ε, algorithm stops and m
k in f
(
m
) is to be obtained; otherwise, algorithm turns to Step 3. g
k, the conjugate gradient of f
(
m
) at m
k, represents g
k
=
β
k
×
f
(
m
k
), in which g
k is the norm of g
k and β
k is the parameter. Generally, two expression forms include β
k
F
R
=
g
k
2
/
g
k
-
1
2 and β
k
P
R
=
g
k
T
g
k
-
g
k
-
1
/
g
k
-
1
2 [22]. In this paper, β
k
=
m
a
x
0
,
β
k
F
R
+
m
i
n
0
,
g
k
T
g
k
-
1
/
g
k
-
1
T
d
k
-
1, in which g
k
T is the transposed conjugate gradient.Step 3.
Step lengthα
k is determined by 1D linear search.Step 4.
Placem
k
+
1
=
m
k
+
α
k
d
k, in which d
k is the conjugate gradient search direction and is determined as follows:(2)
d
k
=
-
g
k
k
=
0
,
-
θ
k
g
k
+
β
k
d
k
-
1
k
≥
1
,
in whichθ
k
=
1
+
g
k
T
d
k
-
1
/
g
k
2
β
k. It can be inferred that d
k
=
-
g
k
-
g
k
T
d
k
-
1
/
g
k
2
β
k
g
k
+
β
k
d
k
-
1, where g
k
T is to be multiplied at both sides to obtain the following expression:(3)
g
k
T
d
k
=
-
g
k
2
-
g
k
T
d
k
-
1
g
k
2
β
k
g
k
2
+
β
k
g
k
T
d
k
-
1
=
-
g
k
2
<
0
.It is obvious thatg
k
T
d
k is always less than 0 and g
k
T is greater than 0, which results in downward search direction. Moreover, if β
k
≠
0
,
-
g
k
T
g
k
-
1
/
g
k
-
1
T
d
k
-
1
≥
0, β
k
=
β
k
F
R, the modified conjugate gradient method is FR conjugate gradient method [38]. Otherwise, combining correlation ((c), see Table 2), we can draw that (4)
β
k
=
β
k
F
R
+
g
k
T
g
k
-
1
g
k
-
1
T
d
k
-
1
=
g
k
2
g
k
-
1
2
-
g
k
T
g
k
-
1
g
k
-
1
2
=
g
k
T
g
k
-
g
k
-
1
g
k
-
1
2
=
β
k
P
R
.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.
d
k is determined, and place k
=
k
+
1. Process turns to Step 2.Finally, the modified MMP correlation of CO2-crude oil is as follows:(5)
f
m
=
P
m
,
m
i
n
,
p
u
r
e
=
8.3397
×
1
0
-
5
ln
⁡
1.8
T
+
32
3.9774
ln
⁡
M
C
7
+
3.3179
1
+
x
V
O
L
x
M
E
D
′
0.17461
.Compared with the other 11 correlations in Table2, the correlation has broader application (pressure range: 0~70 MPa, temperature range: 21.67~191.97°C, and relative molecular weight of C7+: 130~402.7 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:(6)
A
E
=
C
a
l
-
E
x
p
,
(7)
A
R
E
%
=
∑
i
=
1
N
C
a
l
-
E
x
p
E
x
p
i
×
100
,
(8)
A
A
R
E
%
=
1
N
∑
i
=
1
N
C
a
l
-
E
x
p
E
x
p
i
×
100
.A new correlation validation is performed with more MMP data (Table4). 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)
M
C
7
+
M
C
5
+
x
M
E
D
x
M
E
D
′
x
V
O
L
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
57.428Figure 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.



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*Source: 175940-2015-11-10.xml*