SPE-SAS-698

AN ACCURATE PREDICTION OF CO2 MINIMUM MISCIBILITY PRESSURE

(MMP) USING ALTERNATING CONDITIONAL EXPECTATION ALGORITHM

(ACE)

Osamah Alomair, SPE, Adel Malallah, SPE, Adel Elsharkawy, SPE, and Maqsood Iqbal, SPE, Kuwait UniversityCopyright 2011, Society of Petroleum Engineers

This paper was prepared for presentation at the 2011 SPE Saudi Arabia Section Technical Symposium and Exhibition held in AlKhobar, Saudi Arabia, 15–18 May 2011.

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been

reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum

Engineers, its officers, or members. Papers presented at the SPE meetings are subject to publication review by Editorial Committee of Society of Petroleum Engineers. Electronic reproduction,

distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not

more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and whom the paper was presented. Write Liberian, SPE, P.O. Box

833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract

Miscible gas injection nowadays becomes an imperative enhanced oil recovery (EOR) approach for increasing oil

recovery. Due to the massive cost associated with this approach a high degree of accuracy is required for

predicting the outcome of the process. Such accuracy includes, the preliminary screening parameters for gas

miscible displacement; the “minimum miscibility pressure” (MMP) and the availability of the gas.

All conventional and stat-of-the-art MMP measurement methods are either time consuming or decidedly cost

demanding processes. Therefore, in order to address the immediate industry demands a nonparametric approach

(ACE) is employed in this study to estimate an important parameter MMP. ACE algorithm correlates optimal

transforms of a set of predictors with an optimal response transform. Finally, the proposed model has produced a

maximum linear effect between these transformed variables. More than 100 MMP data points are considered both

from the relevant published literature and the experimental work. Five MMP measurements for Kuwaiti Oil are

included as a part of the testing data. The proposed model is validated using detailed statistical analysis and it

reveals that the results are more reliable than the existing correlations for pure CO2 injection to enhance oil

recovery. In addition to its accuracy, the ACE approach is more powerful, quick and can handle a huge data.

Introduction

The minimum miscibility pressure (MMP) is defined as the lowest pressure at which we have a distinct point of

maximum curvature when recovery of oil at 1.2 PV gas injected is plotted verse pressure (Johnson and Pollin;

1981). This pressure can be located graphically by the intersection of two lines that define both an immiscible and

2 [Paper Number]

miscible performance regimes on a plot of recovery versus pressure, or recovery versus composition. MMP is one

of the most important factors in the selection of candidate reservoirs for gas injection at which miscible recovery

takes place. This minimum dynamic miscibility pressure (MDMP) or simply minimum miscibility pressure, is

dependent upon the composition of the injected gas, the reservoir temperature, and the characteristics of the in

place fluid. This pressure of miscibility is independent of the nature of the porous media or of the velocity of

displacement.

Experimental Techniques for Measuring MMP

There are several experimental methods that can be used for measuring MMP for an oil-solvent system.

Traditionally, slim tube studies were conducted for this purpose (Yellig et al., 1980, Huang et al., 1994). However,

the faster and more precise rising bubble apparatus (RBA) method is becoming more common for measuring MMP

(Christiansen and Haines, 1987). This technique was first introduced by researchers at Marathon oil in 1984

(Huang et al., 1984). A comparison of the two measurement techniques has been discussed by many

investigators (Novosad et al., 1989, Elsharkway et. al., 1992, Huang and Dyer, 1993, and Srivastava et al., 1994).

A rapid experimental method for measuring MMP at low temperatures (below about 50 oC) based on the

measurement of density of the injection-gas-rich upper phase in contact with stock tank oil as a function of

pressure has been reported by Harmon and Grigg; 1986. On the other hand, flow experiments offer the most

reliable method to determine the pressure required for miscibility with N2, CO2 (both pure and impure), and

hydrocarbon gas (Glaso, 1985). Because of the complexity of the of the interactions between crude oil and gas in

a flowing system the onset of the miscibility can only be determined by comparing the relative displacement

efficiencies of controlled flow experiments (Glaso,1987). Deffrenne et. al. (1961) presented a miscible

displacement system and made the use of slim tube apparatus for enriched or vaporizing gas drives. He measured

and compared the MMP of oil using different solvents like methane, nitrogen and various other gases (Firoozabadi

and Aziz, 1986).

P-X (Pressure-Composition) diagram method needs a number of careful lab experiments using PVT cell.

Increasing amount of solvents are mixed with the oil, and Bubble Point Pressure (BPP), Dew Point Pressure

(DPP) and gas liberation curves are obtained for each mixed ratio. When the BPP and DPP curves no longer

rises with the increasing amounts of solvent added, it is assumed that FCM has been achieved. As obvious,

these tests needs accurate monitoring, and are expensive.

Correlations for Predicting MMP

Correlations for predicting MMP have been proposed by a number of investigators, and are important tools for

rapid and accurate MMP calculation. Enich et al. (1988) pointed out that, ideally, any correlation should account

for each parameter known to affect the MMP, should be based on thermodynamic or physical principles that affect

the miscibility of fluids, and should be directly related to the multiple contact miscibility process. For an initial and

quick estimate, operators use correlation currently available in the literature. For screening purposes, they gave a

fair first guess depending on the data used. Moreover, they are inexpensive and can be detained by simple hand

calculation. However, the success of the correlations is usually limited to the composition range in which these

correlations were developed. The CO2 MMP correlations fall into two categories: the pure and impure CO2; while

the other category treats MMP’s of other gases.

[Paper Number] 3

In 1960, Benham et al. presented empirical curves that can estimate miscibility conditions for reservoir oils that are

displaced by rich gas, followed by some proposed equations that have been derived for predicting MMP. These

equations are a result of curve fitting Benham et al.’s data (Glaso, 1985).

Glaso presented a generalized correlation for predicting the MMP required for multi-contact miscible displacement

of reservoir fluids by hydrocarbon, CO2 or N2 gas. Glaso also showed that for hydrocarbon systems, paraffinicity

has an effect on MMP (Glaso, 1985). Several methods have been proposed to predict the CO2 dynamic miscibility

pressure of reservoir fluids from easily obtainable field data. The most widely discussed of these methods are; the

National Petroleum Council correlation, the Gulf Universities Research Consortium correlation, the method

developed by Holm and Josendal (1974) and the method developed by Yellig and Metcalfe (1980). However, none

of these correlations gives adequate emphasis to oil properties and composition and all fail to accurately predict

the miscibility pressure for variety of crude oil types. This led to investigate other techniques for predicting MMP

such as regression analysis and artificial intelligent methods.

MMP Regression Analysis

The regression analysis addresses the effect of one or more independent variables (predictors or covariates) on a

dependent variable (response). The initial stages of data analysis often involve exploratory analysis.

Unfortunately traditional multiple regression techniques are limited, since they usually require a priori assumptions

about the functional forms that relate the response and predictor variables. When the relationship between the

response and the predictor variable is unknown or inexact, linear parametric regression can yield erroneous and

even misleading results. This is the primary motivation for the use non-parametric regression techniques, which

make few assumptions about the regression surface (Friedman and Stuetzle, 1981).

The objective of fully exploring and explaining the effect of covariates on a response variable in regression analysis

is facilitated by properly transforming the independent variables. There are number of parametric transformations

for continuous variables in regression analysis. Estimating the optimal transformation is the primary motivation for

the use of non-parametric regression techniques, which make few assumptions about the regression surface

(Breiman and Friedman, 1985). Non-parametric regression techniques are based on successive refinements by

attempting to define the regression surfaces in an iterative fashion while remaining ‘data driven’ as opposed to

‘model driven’. These non-parametric regression methods can be broadly classified into those which do not

transform the response variable (such as Generalised additive models) and those which do (such as Alternating

Conditional Expectations, ACE). Moreover, the ACE algorithm can handle variables other than continuous

predictors such as categorical (ordered or unordered), integer and indicator variables (Wang and Murphy, 2004).

The present approach to estimate MMP is guided by the view that statistical methods for dealing with data that

exhibit strong linear associations are well developed; consequently, many non-standard problems are best

addressed by transforming the data to achieve increased linear association. The analysis given here also serves

to illustrate the exploratory use of the ACE algorithm to suggest expressions, and the use of R2 from the ACE

transformed variables as a benchmark. The ACE-Transformed variables exhibit substantially greater linear

association than the untransformed variables. One of the principal benefits of the ACE algorithm is that it provides

4 [Paper Number]

a theoretical standard against which more analytically appealing transformation can be judged (Veaux, et al.,

1989). The power of the ACE approach lies in its ability to recover the functional forms of variables and to uncover

complicated relationships (Wang and Murphy, 2004).

The advantage of the non-parametric regression is easy to use and can quickly provides results that reveal the

dominant independent variables and relative characteristics of the relationships (Wu et al., 2000). It can be applied

both bivariate and multivariate cases and it yields maximum correlations in transformed space (Malallah et al.,

2005). A modification of ACE algorithm with graphical (GRACE) interface was later proposed by Xue et al. (1997).

Alternating Conditional Expectation (ACE)

This study uses an algorithm (ACE) of Brieman and Friedman (1985) for estimating the transformations of a

response and a set of predictor variables in multiple regression problems in enhanced oil recovery. The name

‘alternating conditional expectations’ refers to the algorithm used to compute optimum transforms (viz. those that

minimize the summation of squares of the error). The mathematical expectation is the mean of the distribution of

a population and is denoted by μ or E(Z), where Z is the variable that describes the experiment. μ is mostly used

mostly in uni-variate statistics. The word ‘conditional’ in ACE is meant to indicate that the means of Z/Q variables

(i.e., the conditional variables) are determined. The conditions in ACE are the values of the dependent variable or

those of an independent variable. Conditional expectations can be expressed as:

E [φi (X i )Y ]

and

E [φi (X i )X i ] (1)

The proposed nonparametric approach can be applied easily for estimating the optimal transformation of different

gas injection data to obtain maximum correlation between observed variables. An ACE regression model can be

expressed as:

(2)

where θ is a function of the response (dependent) variable Y, Φi are functions of the predictors (independent)

variables Xi , i =1,2,3,… p. Thus, the ACE model replaces the problem of estimating a linear function of a p-

dimensional variable X = (X1, X2, X3, X4,…Xp) by estimating p separate one-dimensional functions, Φi, and θ using

an iterative method. These transformations are achieved by minimizing the unexplained variance of a linear

relationship between the transformed response variable and the sum of transformed predictor variables.

For a given dataset consisting of a response variable Y and predictor variables X1, X2, X3, X4, … Xp, the ACE

algorithm starts out by defining arbitrary measurable mean transformations θ(Y), Φ1(X1), Φ2(X2), Φ3(X3), … Φp(Xp).

The error variance (ε2) that is not explained by a regression of the transformed dependent variable on the sum of

transformed independent variables is (under the constraint, E(θ2(Y) =1)

(3)

θ (Y )=α +∑i=1

p

φi (X i )+ ε

ε 2 (θ , φ1 , φ2 , φ3 , .. .φp ) = E {[θ (Y ) −∑i=1

p

φi (X i ) ]}2

[Paper Number] 5

The minimization of (ε2) with respect to Φ1(X1), Φ2(X2), Φ3(X3), … Φp(Xp) and θ(Y) is carried out through a series of

single–function minimizations, resulting in the following equations:

(4)

(5)

The two basic mathematical operations involved here is conditional expectation and iterative minimization, hence

the name alternating conditional expectations. The final Φi (Xi), i=1, 2, 3 … p, and θ(Y) after the minimization are

estimates of the optimal transformations Φi* (Xi), i=1, 2, 3 … p, and θ*(Y). In the transformed space, the response

and predictor variables are related as follows:

(6)

Where e* (misfit) is the error not captured by the use of the ACE transformations and is assumed to have a normal

distribution with zero mean. These optimal ACE transformations are derived solely from the given data and do not

require a priori assumptions of any functional form for the response or predictor variables and thus provide a

powerful tool for exploratory data analysis. The dependent variable for any data point is calculated as:

(7)

The calculation involves n forward transformations of X1, X2, X3, X4, … Xp to Φ1(X1), Φ2(X2), Φ3(X3), … Φp(Xp) and

a backward transformation:

(8)

Investigating the Factors Affecting MMP

Generally, MMP increases steadily with increasing temperature, and oils with higher density and molecular weight

have a higher MMP. It has been reported that even small impurities, can significantly affect the miscibility pressure

(Glaso, 1987). Alston et. al. (1985) documented the fact that the achievement of miscibility is strongly related to

reservoir temperature and oil composition, particularly C5+ molecular weight. Holm and Josendal (1974) found that

MMP was only affected by the type of hydrocarbons present in the range C5 to C30 fractions of the crude oil. Yellig

and Metcalfe (1980) found little significance of C7+ properties of the oil on the CO2 MMP. Alston et. al. (1985) have

shown that the reservoir oil volatile and intermediate fractions can significantly affect the MMP when their ratios

depart from unity (Glaso, 1985). This also explained the effects of both solution gas (live oil systems) and impurity

of CO2 sources (Alston, et. al., 1985). James et al (1981) presented an empirical correlation which predicted the

φ i (X i ) = E [θ (Y ) −∑j≠i

p

φ j (X j ) /X i]θ (Y ) =

E [∑i=1

p

φ i (X i ) /Y ]‖E [∑

i=1

p

φ i (X i ) /Y ]‖

θ¿ (Y ) =∑i=1

p

φi¿ (X i ) +e

¿

Y = θ¿−1

∑i=1

p

φi¿ (X i )

θ¿−1

∑i=1

p

φi¿ (X i )

6 [Paper Number]

MMP for a wide variety of live oils and stock oils with both pure and diluted CO2. This correlation, requiring only the

oil gravity, molecular weight, reservoir temperature and injection gas composition, showed substantially better

agreement with experiment. Many correlations relating the MMP to the physical properties of the oil and the

displacing gas have been proposed to facilitate screening procedures and to gain insight into the miscible

displacement process (Alston et. al., 1985; Orr and Silva, 1987; Rathmell et al.; 1971)

To study the effect of these parameters on tested data, several sensitivity analyses were conducted. Figure 1

show the relationship between the independent variables and MMP for the data used in this study. The correlation

coefficient of each independent variable is shown. It is clear that the temperature is the most dominant factor.

Methane has the same proportional effect where as C6 and C7+ has adverse effect on MMP.

Figure 1: Effect of different independent variables on MMP

Table1: Data range used for the input variables

H2S

CO

2

N2

C1

C2

C3

C4

C5

C6

C7+

MC

5+

MC

7+

T (

F)

MM

P

(psi

g)

Max

17.5

6

24.0

0

16.4

4

76.4

3

23.1

6

18.4

0

11.2

3

9.49

16.0

0

73.6

1

262

286

265

3705

Min

0.00

1

0.00

1

0.00

1

2.56

0.01

0.89

0.28

0.25

0.79

5.70

132

151

71

1101

Development of ACE MMP Model

[Paper Number] 7

As discussed earlier, MMP is a function of temperature, crude oil composition and composition of the solvent. To

understand the in-situ crude oil composition impact on MMP, the functional form of MMP Model is:

(9)

HCCOMP = Mole Fraction of hydrocarbons (C1, C2, C3, C4, C5, C6, and C7+)

NHCCOMP = Mole Fraction of non-hydrocarbons (H2S, CO2, N2)

T = Temperature

MC5+ = Molecular weight of Pentane Plus

MC7+ = Molecular weight of Heptane Plus

The data set used in this study consisted of 113 MMP measurements (pure CO2) taken from worldwide gas

injection projects from the published literature. The ranges of variables and MMPs used for this study are shown in

Table 1. The collected data cover a wide range of API gravities (13 – 58 oAPI), reservoir pressures and

temperatures. The data were divided into two sets. The training set consisted of 96 points and a testing set of 17,

which were randomly selected from the total set of data. Out of 17 testing points, 5 were taken from MMP

measurements of Kuwait oil fields. All the Kuwaiti crude oil samples were thoroughly studied in Kuwait University

PVT Lab. Both, the detailed compositional analyses and minimum miscibility pressure measurements were

experimentally determined. The ACE algorithm provides a nonparametric optimization of the dependent (MMP)

and independent variables (HCCOMP, NHCCOMP, T, MC5+, MC7+); it does not provide a computational model for these

variables. However, the optimal data transforms can be fitted by simple polynomials that can be used to predict

the dependent variable. The default polynomial is of degree two but for any improvement the degree can be

increased. To find the maximum/optimal correlation, the ACE algorithm has the capability of using the independent

and dependent variables in their actual space or in the logarithmic space. After testing all possible combinations of

the independent and dependent variables in either logarithmic or actual space, the following suit of polynomials is

considered;

MMP=f (HCCOMP , NHCCOMP , T , MC 5+ , MC7+ )

8 [Paper Number]

Op

tim

al T

ran

sfo

rm P

oly

no

mia

ls

ln_

C1_T

r= 1

.391

2E

-02x

2 +

3.2

202E

-02

x -

1.86

11E

-01

ln_

C2_T

r= 5

.579

5E

-04x

6 -

6.7

876E

-03x

5 +

1.0

678E

-02x

4 +

1.2

248E

-01x

3 -

4.57

24E

-01x

2 +

5.2

793

E-0

1x -

2.0

734

E-0

1

ln_

C3_T

r= 1

.638

1E

-02x

4 -

1.3

052E

-01x

3 +

2.6

607E

-01x

2 -

1.20

19E

-01

x -

4.33

56E

-02

ln_

C4_T

r= -

5.19

12E

-02

x6 +

2.7

982

E-0

1x5

- 3.

7272

E-0

1x4

- 2.

4664

E-0

1x3

+ 7

.168

2E

-01x

2 -

2.7

774E

-01x

- 1

.348

5E-0

2

ln_

C5_T

r= -

1.96

74E

-04

x4 -

2.7

402E

-02x

3 +

1.8

143E

-03x

2 +

1.4

054E

-01x

- 7

.217

4E-0

2

ln_

C6_T

r= -

5.46

86E

-02

x5 +

2.6

628

E-0

1x4

- 3.

6496

E-0

1x3

+ 7

.699

8E-0

2x2

- 8

.360

4E

-04x

+ 1

.119

4E-0

1

ln_

C7+

Tr=

-2.

445

9E-0

2x2

+ 3

.889

5E-0

2x +

2.1

657E

-01

ln_

CO

2_T

r= 9

.402

9E

-06x

6 -

3.2

025E

-05x

5 -

1.0

969

E-0

3x4

- 8.

125

3E-0

4x3

+ 2

.016

6E-0

2x2

+ 1

.327

8E-0

2x -

3.2

906E

-02

ln_

H2S

_Tr=

3.0

640

E-0

4x4

+ 2

.774

7E

-03x

3 +

6.3

899E

-03

x2 -

4.0

849E

-03x

- 7

.933

7E-0

2

ln_

N2_T

r= -

3.32

52E

-04

x4 +

6.1

601

E-0

4x3

+ 8

.446

2E

-03x

2 -

2.5

561E

-02x

- 8

.438

1E-0

3

Mw

5+_T

r= 1

.905

6E

-08x

4 -

1.5

287E

-05x

3 +

4.5

136E

-03x

2 -

5.77

89E

-01

x +

2.6

867E

+01

Mw

7+_T

r= 1

.563

4E

-05x

2 -

1.5

754E

-03x

- 4

.418

0E-0

1

Tem

p_T

r= -

7.97

31E

-06x

2 +

2.1

085E

-02x

- 2

.830

6E+

00

MM

P=

2.5

923E

+0

1 S

umT

r2 +

6.5

136

E+

02 S

umT

r +

2.0

097E

+03

Var

iab

le

C1

C2

C3

C4

C5

C6

C7+

CO

2

H2S

N2

Mw

5+

Mw

7+

Tem

p

MM

P

[Paper Number] 9

The sum of all these optimal transformed independent variables is;

(10)

The predicted / calculated MMP will be:

(11)

The best combination that yields the highest correlation coefficient (R2), the lowest average absolute relative error

(AARE), the lowest average relative error (ARE), and the lowest standard deviation (SD) is:

(12)

or

(13)

Data Analysis and Results Discussion

In the proposed nonparametric ACE model for MMP estimation, there are 13 predictors. All these are tested both in

real and logarithmic space and the maximum optimal correlation coefficients (more than 90 %) are obtained. All

mole fractions (hydrocarbon and non-hydrocarbon) are found more optimal to their respective transforms in the log

space whereas temperature and both plus fraction molecular weights are having maximal correlation effect in the

real space.

Hydrocarbon Optimal Transforms

A suite of Figure 2 (a, b, c, d, e, f and g) is ACE defined optimal transforms in the logarithmic space. Different kinds

of trends are observed.

Light hydrocarbon gases (C1 and C2) show optimal transforms. ACE defined ethane transform is in logarithmic

space with a polynomial of degree six. Methane is showing a pattern.

Intermediate hydrocarbons (C3, C4, and C5) transforms are relatively better defined in the logarithmic space. All the

polynomials are of higher degrees. There is a pattern is observed in butane and pentane.

Heavy end hydrocarbons (C6 and C7+) transforms are determined with polynomials of 5 & 2 respectively.

SUM=∑i=1

13

p i

MMP = pO−1 (SUM )

MMP=a2 SUM2+a1SUM

1+a0

MMP=25 . 923 SUM 2+651 .360 SUM 1+2009 . 7

10 [Paper Number]

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.0 1.0 2.0 3.0 4.0 5.0

_ln_C1

_ln

-C1_

Tr

Figure 2 (a)

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

-5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0

_ln_C2

_ln

-C2_

Tr

Figure 2 (b)

Figure 2 (c)

-0.20

-0.15

-0.10

-0.05

0.00

0.05

-0.5 0.5 1.5 2.5 3.5

_ln_C3

_ln

-C3_

Tr

[Paper Number] 11

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

_ln_C4

_ln

-C4_

Tr

Figure 2 (d)

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5

_ln_C5

_ln

-C5_

Tr

Figure 2 (e)

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

_ln_C6

_ln

-C6_

Tr

Figure 2 (f)

12 [Paper Number]

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

_ln_C7+

_ln

-C7+

_Tr

Figure 2 (g)

Non-hydrocarbon Optimal Transforms

Non-hydrocarbon gases (CO2, H2S and N2) optimal transforms, determined by ACE are very well defined in Figure

3 (a, b, and c) in the logarithmic space. There are some patterns observed of all the three gases.

-0.10

-0.05

0.00

0.05

0.10

0.15

-8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0

_ln_CO2

_ln

-CO

2_Tr

Figure 3 (a)

[Paper Number] 13

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

-8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0

_ln_H2S

_ln

-H2S

_Tr

Figure 3 (b)

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

-8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0

_ln_N2

_ln

-N2_

Tr

Figure 3 (c)

Real Space Optimal Transforms

Both plus fraction molecular weights (Mw5+ and Mw7+) and temperature transforms are defined in the real space by

ACE. Heptane plus ACE transform is showing some trend. Temperature transform is defined in real space

optimally and a second degree polynomial determined by ACE is showing a good trend in Figure 4(c).

14 [Paper Number]

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0 50 100 150 200 250 300

Mw5+

Mw5+

_Tr

Figure 4(a)

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0 50 100 150 200 250 300 350

Mw7+

Mw7+

_Tr

Figure 4 (b)

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

0 50 100 150 200 250 300

Temp

Temp

_Tr

Figure 4 (c)

[Paper Number] 15

Figure 5 shows the optimal transform of the dependent (response) variable. This is also tested in both (real and

logarithmic) spaces and finally it is defined in the real space. Coefficients of this fit polynomial will be incorporated

in the final SUM equation to estimate MMP.

0

500

1000

1500

2000

2500

3000

3500

4000

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

MMP_Tr

MMP

Figure 5: Optimal Transformation of MMP by ACE.

Finally, all these predictors’ transforms both from real and logarithmic space are added up and correlate with

transform of MMP. In figure 6, an excellent correlation is obtained with R2 value of 0.98014. This proves the

incredible power of ACE algorithm.

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Sum_Tr_Indep

MMP_

Tr

Figure 6: (MMP_Tr vs. Sum of Optimal Transformation of independent variables) by ACE

(Optimal Regression)

All transformed independent variables (predictors) and the response (MMP) are found numerically. The best linear

regression between them is shown in the last figure-6, and the equation of this match is;

16 [Paper Number]

MMP= 25.923 SUMTr2 + 651.360 SUMTr1 + 2009.7

Table 2 presents the statistical analysis between the results predicted by ACE model and that by published

correlations. It is inferred from the analysis that ACE is a powerful tool and shows high accuracy as compared to

other correlations.

Table 2: Statistical Analysis for MMP (Training Data)

Model AARE (%)ARE

(%)

SD

RMSESSE (%)

R

SSE SSR

ACE 4.68 -0.66 139.20 1.86E+06 0.956 0.907

Alston (1985) 14.40 -1.94 518.96 2.59E+07 0.394 1.921

Cronqst (1978) 12.97 -11.26 484.53 2.25E+07 0.471 2.069

Yell-Met (1980) 15.58 14.89 391.89 1.47E+07 0.654 1.088

Orr-Jen (1984) 15.36 2.63 394.70 1.50E+07 0.649 1.936

Glaso (1985) 7.99 5.39 248.03 5.91E+06 0.861 0.967

Validation of ACE Model for MMP

To check the validity/credibility of ACE model and to check its predictive capability for MMP, all the derived

polynomials of variables (both predictors and Response) were examined using testing data of 17 points. Five of

these are the experimental measurements made for Kuwaiti oil fields.

0

1000

2000

3000

4000

0 1000 2000 3000 4000

Measured (EXP) MMP

Cal

cula

ted

(A

CE

) M

MP

Figure 7: Cross-plot of MMP Testing data.

[Paper Number] 17

There is a good match between the experimentally measured and ACE estimated values for MMP. It is shown in

figure 7. This proves the validity of our proposed ACE model. A detailed statistical analysis with different statistical

tools is explained in table 3 shown below.

Table 3: Statistical Analysis for MMP (Testing Data)

ModelAARE

(%)

ARE

(%)

SD

RMSESSE (%)

R

SSE SSR

ACE 2.84 10.22 3324 1.88E+10 0.818 0.783

Alston (1985) -0.03 22.65 12394 2.61E+11 0.628 2.583

Cronqst (1978) -13.75 15.56 11572 2.28E+11 0.858 2.326

Yell-Met (1980) 15.93 15.93 9359 1.49E+11 0.842 1.054

Orr-Jen (1984) 3.84 15.06 9426 1.51E+11 0.781 1.897

Glaso (1985) 2.92 8.54 5923 5.96E+10 0.833 1.387

For comparison purposes, the same testing data was also applied to the available (pure CO2) MMP correlations.

The overall performance of ACE for predicting MMP values is better and convincing.

Conclusions

A nonparametric model to predict MMP is developed based on 96 data points. The proposed ACE model is shown

to be more accurate than the existing conventional regression correlations. This model is able to predict MMP for

pure CO2 as a function of thirteen independent variables (all possible factors affecting MMP). The model has

certain advantages:

The approach solves the general problem of establishing the linearity assumption required in regression

analysis, so that the relationship between response and independent variables can be best described and

existence of non-linear relationship can be explored and uncovered.

An examination of these results can give the data analyst insight into the relationships between these

variables, and suggest if transformations are required.

The ACE plot is very useful for understanding complicated relationships and it is an indispensable tool for

effective use of the ACE results.

It provides a straightforward method for identifying functional relationships between dependent and

independent variables.

Although ACE provides a largely automated approach to estimating optimal transformations, it does not mean that

the ACE results should be trusted blindly and used dogmatically, additional information and experience of the data

analyst remain important. It should be emphasised that the success of the ACE algorithm, like other modern

statistical methods, relies on the quality of the data and underlying association between the response and

independent variables.

18 [Paper Number]

Acknowledgement

The authors would like to thank Kuwait University for supporting this project through the Research Grant GE 01/07.

Nomenclature

AARE = average absolute relative error, %

ARE = average relative error, %

C1 = methane mole fraction

C2 = ethane mole fraction

C3 = propane mole fraction

C4 = butane mole fraction

C5 = pentane mole fraction

C6 = hexane mole fraction

C7+ = heptane plus mole fraction

CO2 = carbon dioxide mole fraction

E = mathematical expectation

e* = misfit (error)

f = function

HCCOMP = a group, mole fractions of hydrocarbon composition

H2S = hydrogen sulphide mole fraction

Ln = natural log

MC5+ = Molecular wt of Pentane Plus

MC7+ = Molecular wt of Heptane Plus

MMP = minimum miscibility pressure (psig)

NHCCOMP = a group, mole fractions of non-hydrocarbon composition

N2 = nitrogen mole fraction

R = correlation coefficient, %

RE = relative error

R2 = ratio of data variability

RMSE = Root Mean Square Error

SD = standard deviation of the errors

SSE = sum of squares of the errors

SSR = regression sum of squares

SST = total sum of squares

T = temperature (F)

Tr = transform

References

[Paper Number] 19

Adel Malallah et al.: “Accurate Estimation of the World Crude Oil Properties Using Graphical Alternating

Conditional Expectations,”

Alcocer, C. F. and Menzie, D. E.: “EOR by Nitrogen Injection: The Effect of Increased Temperature and the amount

of gas in solution,” paper SPE 12709 presented at SPE/DOE 1984 EOR Symposium, Tulsa, April 15-18.

Alston, R. B., Kokolis, O. P. and James, C. F. (1985). CO2 Minimum Miscibility Pressure: A Correlation for Impure

CO2. SPE 11959.

Ballard, J. R. and Smith, L. R.:” Reservoir Engineering, design of a low pressure Rich Gas Miscible flood Project, ”

J. Pet. Tech. (May 1972) 599-605.

Brieman, L., and Friedman, J. H.: “Estimating optimal transformations for multiple regressions and correlation,” j.

Am. Stat. Assoc. 1985, 580-619.

C. H. Wu et al.: “Non-Parametric regression and neural-network infill drilling recovery models for carbonate

reservoirs,” Computer and Geosciences 26 (2000) 975-987.

Christian, L. D. et al. : “Planning a Tertiary Oil-Recovery Project for Jay/LEC Field Unit,” JPT (Aug. 1981) 1535-44.

 

Christiansen, R. L. and Haines, H. K. (1987). Rapid Measurement of Minimum Miscibility Pressure with the Rising-

Bubble Apparatus. SPE 13114.

Cook, A. B., Walter, C. J., and Spencer, G. C.:” Realistic K-values of C7+ Hydrocarbons for Calculating Oil

Vaporization during Gas Cycling at High Pressure”. J. Pet. Technol. 1969, 901-915.

Deffrenne, P. et al.:” The determination of pressures of miscibility,” paper SPE 116 presented at the 1961 SPE

Annual Meeting, Dallas, Oct. 8-16.

Dulao Wang and Michael Murphy: “Estimating Optimal Transformations for multiple Regression Using the ACE

Algorithm,” Journal of data Science 2 (2004), 329-346.

Elsharkay, A. M., Poettmann, F. H. and Christiansen, R. L.: “Measuring Minimum Miscibility Pressure: Slim-Tube or

Rising Bubble Method,” paper SPE/DOE 24114, presented at the 1992 eighth Symposium on EOR, Tulsa, April

22-24.

Enich et al.: “A thermodynamic correlation for the MMP in CO2 flooding of petroleum reservoirs,” SPERE, February,

81-92. 1988.

20 [Paper Number]

Firoozabadi, A. and Aziz, K. (1986). Analysis and Correlation of Nitrogen and Lean-Gas Miscibility Pressure. SPE

13669.

Gasem, K.A.M., Dickson, K.B., Shaver, R.D., and Robinson Jr., R.L.: “Experimental phase densities and interfacial

tensions for a CO2/Synthetic-oil and a CO2/reservoir-oil system,” SPE Reserv. Eng. Aug. 1993, 170–174.

Glaso, O. (1985). Generalized Minimum Miscibility Pressure Correlation. SPE 12893.

Glaso, O. (1987). Miscible Displacement: Recovery Tests with Nitrogen. SPE 17378.

Harmon, R. A. and Grigg, R. B. (1986). Vapor Density Measurement for Estimating Minimum Miscibility Pressure.

SPE 15403.

Høier, L. “Miscibility Variations in Compositionally Grading Petroleum Reservoirs,” Ph.D. Dissertation,” The

Norwegian University of Science and Technology”, 1997.

Holm, L. W. and Josendal, V. A.,” Mechanisms of Oil Displacement by Carbon Dioxide,” (1974), SPE 4736.

Huang, S. S. and Dyer. S. B.: “Miscible Displacement in the Weyburn Reservoir: A Laboratory Study,” JCPT (1993)

32, No. 7, 42-5.

Huang, S.S. et al.: ‘‘A Laboratory Miscible Displacement Study for the Recovery of Saskatchewan’s Crude Oil,’’ J.

Cdn. Pet. Tech.,April 1994, 33- 43.

Jaubert, J. N., Arras, L., Neau, E., and Avaulle´e, L. Properly Defining the Classical Vaporizing and Condensing

Mechanisms When a Gas Is Injected Into a Crude Oil. Ind. Eng. Chem. Res. 1998, 37, 4860.

Jensen, F. and Michelsen, M. L. Calculation of First Contact and Multiple Contact Minimum Miscibility Pressure. In

Situ 1990, 1-17.

Jessen, K., and Michelsen, M. L. Minimum Miscibility Pressure Calculation. IFP seminar: Production of reservoir

fluids in frontier conditions, Rueil-Malmaison, Dec 4-5, 1997.

Johnson, J.P. and Pollin, J.S. (1981). Measurement and Correlation of CO2 Miscibility Pressures SPE/DOE 9790.

Koch, H. A. Jr. and Hutchinson, C. A. Jr.:”Miscible Displacements of Reservoir Oil using Flue Gas, ” JPT (Jan.

1958) 7-19; Trans., AIME, 213.

Kuo, S. S. Prediction of Miscibility for the Enriched-Gas Drive Process. SPE 1985, 14152, 1-12.

[Paper Number] 21

Lee, J. I. and Reitzel, G. A.: “High pressure, Dry Gas Miscible Flood-Brazeau River Nisku Oil Pools,” JPT (Nov.

1982) 2503-2509.

Marcelo Blanco et al.: “Modelling of an environmental parameter by use of the alternating conditional expectation

method,” Chemometrics and Intelligent Laboratory Systems 46 (1999) 31-39.

McNeese, C. R.: “High Pressure Gas Process and the use of Flue gas,” paper presented at the 1963 ACS Spring

Meeting, Los Angeles, March 31-April 4.

Meltzer, B. D., Hurdle, J. M., and Cassingham, R. W.: “An efficient Gas Displacement Project – Rayleigh Field,

Mississippi,” JPT (May 1965) 509-14.

Metcalfe, R. S., Fussel, D. D., and Shelton, J. L.:”A Multi-cell Equilibrium Separation Model for the Study of Multiple

Contact Miscibility in Rich Gas Drives”. SPEJ 1973, 147-155.

Neau, E., Avaulle´e, L., and Jaubert, J. N.: A New Algorithm for Enhanced Oil Recovery Calculations. Fluid Phase

Equilib. 1996, 117, 265-272.

Novosad, Z., Sibbald, L. R. and Costain, T. G.: “Design of miscible solvents for a rich gas drive-comparison of slim

tube tests with rising bubble tests,” paper 89-40-3, presented at 1989 CIM 40th Annual Technical Meeting, Banff,

May 28-31.

Orr Jr., F.M., and Jensen, C.M., 1984.: ” Interpretation of pressure composition phase diagrams for CO2/crude-oil

systems”. Soc. Pet. Eng. J. 10, 485–497.

Orr, F. M. and Silva, M. K. Effect of Oil Composition on Minimum Miscibility Pressure-Part 2: Correlation, Soc. Pet.

Eng. Res. Eng. 1987, 27 (6), 479-491.

Pedersen, K. S., Fjellerup J., and Thomassen, P.:” Studies of Gas Injection into Oil Reservoirs by a Cell to Cell

Simulation Model”. SPE 1986, 15599, 1-8.

Peterson, A. V.: “Optimal Recovery Experiments with N2 and CO2,” Pet. Eng. Intl. (Nov. 1978) 40-50.

Rathmell, J.J., Stalkup, F. I., and Hassinger, R. C.:”A laboratory investigation of miscible displacement by CO2, ”

paper SPE 3483 presented at the SPE 1971 Annual Fall Meeting, New Orleans, Oct. 3-6.

Richard d. De Veaux et al.: “ACE Guide Transformation method for estimation of the coefficient of soil-water

receptivity,” Technometrics, Feb 1989, Vol. 31, No. 1.

22 [Paper Number]

Rushing, M. D. et al.:”Nitrogen may be used for Miscible Displacement in Oil Reservoirs,” JPT (Dec. 1978)

1715-16.

Srivastava, R.K. et al.: “A comparison of minimum miscibility pressure determinations for Weyburn CO2 solvent

design,” paper no-94-50 presented at the 45th Annual Technical Meeting, Calgary, June 12-15, 1994.

Stalkup, F. I. Displacement Behavior of the Condensing/ Vaporizing Gas Drive Process. SPE 1987, 16715, 171-

182.

Wang, Y., and Orr, F. M.: ” Analytical Calculation of Minimum Miscibility Pressure”. Fluid Phase Equilib. 1997, 139,

101-124.

Whitson, C. and Michelsen, M. L. The Negative Flash. Fluid Phase Equilib. 1989, 53, 51-7.

Whitson, C. H. Trends in Modeling Reservoir Fluids. Seventh International Conference on Fluid Properties and

Phase Equilibria for Chemical Process Design, Snowmass, CO, June 18- 23, 1995.

Xue, G., Datta-Gupta, A., Valko P. and Blasingame T.:”Optimal Transformations for Multiple Regression:

Application to Permeability Estimation from Well Logs,” SPE Formation Evaluation, (June 1997) 85-94.

Yellig, W. F., and Metcalfe, R. S. (1980) .:” Determination and Prediction of CO2 Minimum Miscibility Pressures”.

SPE 7477

Zhou, DD., and Orr, F. M. Jr.: “An analysis of Rising Bubble experiments to determine MMPs,” paper SPE 30786

presented at the 1995 SPE Annual Technical Conference and Exhibition, Dallas, Oct. 22-25.

Zick, A. A. A Combined Condensing/Vaporizing Mechanism in the Displacement of Oil by Enriched Gases. SPE

1986, 15493, 1-11.