Robust Modeling Approach for Estimation of Compressibility Factor in

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Robust Modeling for Efficient Estimation of Compressibility Factor in Retrograde Gas Condensate Systems Mohammad M. Ghiasi, Arya Shahdi, Pezhman Barati, and Milad Arabloo Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie404269b • Publication Date (Web): 10 Jul 2014 Downloaded from http://pubs.acs.org on July 11, 2014

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Industrial & Engineering Chemistry Research

1

Robust Modeling for Efficient Estimation of Compressibility Factor

2

in Retrograde Gas Condensate Systems

3

Mohammad M. Ghiasi1, Arya Shahdi2, Pezhman Barati3, Milad Arabloo4*

4

1

National Iranian Gas Company (NIGC), South Pars Gas Complex (SPGC), Asaluyeh, Iran

5

2

Department of Petroleum Engineering, Islamic Azad University, Science and Research Branch, Tehran, Iran

6

3

Ahwaz Faculty of Petroleum Engineering, Petroleum University of Technology (PUT), Ahwaz, Iran

7

4

Young Researchers and Elites Club, North Tehran Branch, Islamic Azad University, Tehran, Iran

8 9

Abstract

10

Knowledge of the phase behavior of condensate gas systems is important for predicting reservoir

11

performance and future processing needs. In this communication, new improved models are

12

developed to calculate the gas phase and two-phase compressibility factors based on constant

13

volume depletion (CVD) analysis of the well stream effluent at any depleted state in retrograde

14

gas condensate systems. These methods are based on compositional analysis of more than 1800

15

compositions of gas condensates collected worldwide. The average absolute relative deviation

16

and correlation coefficient of the developed models from experimental gas phase and two-phase

17

compressibility factor values were about 0.73% and 0.998 and 1.30% and 0.992, respectively.

18

This study also presents an evaluation of one hundred and twenty possible methods of

19

calculating the gas compressibility factor for gas condensates. The accuracy of the new models

*

Corresponding author E-mail: [email protected] Tel. +98-917-1405706.

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has been compared to all one hundred and twenty methods. The comparison indicates that the

2

proposed models are consistent, reliable and superior to all the methods.

3

1

4

Condensate gas is considered to be a very complex reservoir fluid because of its high gas/oil

5

ratio

6

demands specific engineering methods and operations that are generally different from that for

7

either gas or oil reservoirs. The fluid PVT properties have direct influence on the performance

8

and consequently the developmental program to optimize the recovery. Thus, to optimize

9

management of gas condensate reservoirs a thorough knowledge of reservoir fluid properties and

10

economic is mandatory. Knowledge of the phase behavior of condensate gas systems is

11

important for predicting reservoir performance and future processing needs

12

important factors to be evaluated by engineers in calculation of gas flow rate through reservoir

13

rock, material balance calculations, evaluation of gas reserves, design of production equipment,

14

and planning the development of a condensate gas reservoir is the fluid compressibility factor 4-7.

15

The common sources of Z-factor values are experimental measurements, equations of state (EoS)

16

and empirical correlations. Empirical correlations, which are used to predict natural gas Z-factor,

17

are much easier and faster than equations of state. Sometimes these correlations have comparable

18

accuracy to equations of state

19

properties of dry gas reservoirs are extensive

20

condensate samples has not received enough attention. The recent development and success of

21

applying machine learning approaches to resolve various engineering complications has drawn

22

the attention to their potential applications in the petroleum industry

23

support vector machine (SVM) is gaining more popularity among researchers

Introduction

1, 2

. Development and optimization of gas condensate reservoirs to acquire more recovery

8, 9

2-4

. One of the most

. Although previous researchers on the modeling of PVT 9-15

, an investigation on the PVT properties of gas


2, 3, 7, 10, 16-20

. Nowadays,

10, 17, 18, 21-25

. The

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SVM is a new and supervised machine learning technique which works based on the statistical

2

learning theory

3

complex system studies for regression or parameter prediction, was described in Suykens and

4

Vandewalle

5

compressibility factor in dry gas systems

6

authors’ knowledge, no work has been published on the subject of modeling of retrograde gas

7

condensate for prediction of compressibility factor with this approach. Gas-condensate reservoirs

8

differ from dry-gas reservoirs. In dry gas systems no phase change is occurred during the

9

pressure depletion stages and also gas stream composition remains constant during the entire life

10

of the reservoir. However, Gas condensate reservoirs exhibit complex behavior when producing

11

under dew-point pressure at constant temperature 2. Therefore, due liquid drop out from the gas

12

phase causes this dry-gas based technique couldn’t give satisfactory results. This was the primary

13

motivation behind the writing of this article. Thus, this study presents compositional models for

14

estimation of gas phase and two-phase compressibility factors of various gas condensate samples

15

based on LS-SVM modeling approach. These models are based on analysis of more than 230

16

samples of laboratory studies of constant volume depletion (CVD) for gas condensate systems.

17

The database represents a wide range of gas condensate systems obtained worldwide. The

18

performance of the developed models has been compared to simulation results of one hundred

19

twenty possible methods of calculating the gas compressibility factor for gas condensates. To

20

achieve the research purpose, the rest of this article is organized as follows. In Section 2, a brief

21

review on the existing techniques for calculation of gas compressibility factor is presented. After

22

that, the background of the proposed algorithm is discussed in Section 3. The efficiency of the

23

developed models is illustrated and compared to all other predictive techniques in subsequent

27

26

. The least square version of the SVM (LS-SVM) which widely used in

. Although there are some works on the subject of prediction of gas 10, 11

based on LS-SVM approach, to the best of



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sections. In this section also the degree of dependence of each input variable on the target value

2

is evaluated. Finally, Section 5 concludes the key findings of this paper.

3

2

4

The ratio of the real volume to the ideal volume, which is a measure of departure from ideal gas

5

low, is called the compressibility factor 28. Equations of state models are implicit in terms of the

6

compressibility factor, which means that the compressibility is calculated as a root of the

7

equation of state. Generally, Equations of state models are not very easy to use. Because most

8

mixing rules are empirical, interactions between unlike molecules are unknown, and also

9

calculations are numerous 9, 29. On the other hands, the equations of state are known to give poor

10

results for phase behavior modeling of complex hydrocarbons gas condensates especially in the

11

retrograde region. This issue is indicated in Sarkar et al. 30. It has encouraged many researchers

12

to look for other approaches 11, 13, 31. Empirical correlations 13, 14, 31-33 are one of these approaches.

13

When gas composition is available, the gas compressibility factor is estimated by predictive

14

correlations. Beggs and Brill 32, Kumar 14, Azizi, Behbahani and Isazadeh 33, Sanjari and Lay 31,

15

Bahadori et al.

16

methods use pseudo-reduced pressure and temperature (Ppr,Tpr) to estimate the Z-factor. The

17

pseudo-reduced properties of gas (Ppr,Tpr) are calculated via one of the mixing rules. Four

18

mixing rules are used to calculate pseudo-critical properties of natural gases. These mixing rules

19

are Key

20

Elsharkawy et al. 37.

21

In order to calculate the pseudo-critical properties of natural gas, one needs critical properties of

22

all components of the gas stream. The critical properties of pure components are well-known.

Compressibility factor (Z-factor) calculation

34

13

are of five well-known methods for determination of Z-factor. All of these

, Stewart-Burkhardt-Voo (SBV)

35

, Sutton-Stewart-Burkhardt-Voo (SSBV)


36

, and

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1

However, the critical properties of the plus fraction (i.e., C7+) must be calculated from

2

correlations. Several correlations are presented to calculate the pseudo-critical properties of the

3

C7+ fraction. In this study, six well-known methods including Kesler-Lee

4

Daubert

5

properties of the C7+ fraction.

6

Thus, there are on hundred and twenty methods for predicting Z-factor. Therefore, the first

7

objective of this work is to evaluate the existing methods for estimating the Z-factors of gas

8

condensate sample. The second objective of this work is to introduce new models for improved

9

prediction of gas phase as well as two-phase compressibility factor of gas condensate systems

40, 41

38

, Pedersen

39

, Riazi-

, Sim–Daubert 42, Sancet 43, and Jamialahmadi et al. 44 are used to calculate critical

10

during pressure depletion stages.

11

3

12

3.1 Least square support vector machine (LS-SVM) modeling

13

The SVM is a supervised learning model which was proposed by Vapnik45 in 1995. SVMs use

14

the spirit of the structural risk minimization principle

15

binary linear classifier 47. Due to the specific formulation of the SVM algorithm, it gives a sparse

16

solution and both linear and nonlinear regressions can be performed

17

drawback of the SVM is its higher computational burden because of required constrained

18

optimization programing 47. In 1999, Suykens and Vandewalle 27 presented a modification to the

19

traditional SVM called least-squares SVM (LS-SVM) so as to facilitate the solution of the

20

original SVM framework. LS-SVM appears to offer advantages similar to those of SVM, but its

21

great advantage is that LS-SVM applies a set of linear equations (linear programming), instead

22

of quadratic programming problems in order to reduce the complexity of optimization process 27.

Methodology

45, 46

. SVM is a form of non-probabilistic


23

. However the major


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Considering the problem of approximating a given dataset {( x1 , y1 ), ( x2 , y2 ),...(x N , y N )} with a

2

nonlinear function: f ( x) = 〈 w, Φ ( x)〉 + b

(1)

3

where 〈.,.〉 represent dot product; Φ (x) represents the nonlinear function that performs linear

4

regression; b and w are bias terms and weight vector, respectively. In LS-SVM for function

5

estimation, the optimization problem is formulated as 47:

min J(w,e) = w ,b ,e

1 2 1 N 2 w + γ ∑ ek 2 2 k =1

s.t. yk = 〈 w, Φ( xk )〉 + b + ek

(2)

k = 1,..., N

(3)

6

where ek ∈ R are error variables; γ ≥ 0 is a regularization constant. The Lagrangian of the

7

problem is defined by 47:

LLS −SVM = 8

10

(4)

with Lagrange multipliers α k ∈ R . The situation for optimally are determined by 47:  ∂LLS − SVM  ∂w   ∂LLS − SVM  ∂b  ∂L  LS − SVM  ∂ek  ∂LLS − SVM   ∂α k

9

1 2 1 N 2 N w + γ ∑ ek − ∑α k {〈 w, Φ( xk )〉 + b + ek − y k } 2 2 k =1 k =1

N

= 0 → w = ∑α k Φ( xk ) k =1

N

= 0 → ∑α k = 0 k =1

(5)

= 0 → α k = γ ek = 0 → 〈 w, Φ ( xk )〉 + b + ek − y k = 0

By specifying Y = [ y1 ;...; y N ] , 1v = [1;...;1] , α = [α1 ;...;α N ] and eliminating ek and w , following equations are obtained 47:


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 b  0 0 1TN    =   −1 1N Ω + γ I N  α  Y 

1

(6)

where I N is an N × N identity matrix, and Ω ∈ R N × N is the kernel matrix presented by Ω kl = Φ ( x k ) Φ ( x l ) = K ( x k , x l ),

k,l = 1,...N

(7)

2

For LS-SVM, there are several kernel functions

3

used kernel functions which is defined as:17, 18, 21.

K ( x k , xl ) = exp(− x k − xl

2

27, 47

. However, RBF is one of the most widely

(8)

/σ 2 )

4

where, σ 2 is the squared variance of the Gaussian function.

5

3.2 Designing LS-SVM models for representation of Z-factors of gas condensate

6

samples

7

To develop LS-SVM models for accurate prediction of gas phase and two phase gas condensate

8

deviation factor, temperature (T), hydrocarbon (C1-C7+) and non-hydrocarbon fluid compositions

9

(CO2, H2S, N2), and characteristics of the heptanes-plus fraction (SGC7+, MWC7+) were assumed

10

as correlating parameters:

Z gas phase = f 1 (T , C1 − C 7 + , CO 2 , H 2 S , N 2 , SG C 7 + , MWC 7 + )

(9)

Z two - phase = f 2 (T , C1 − C 7 + , CO2 , H 2 S , N 2 , SGC 7 + , MWC 7 + )

(10)

11

3.3 Gas condensate PVT data

12

Compositions, gas phase and two phase compressibility factor of more than 230 gas condensate

13

samples were applied for this study. For each of samples a series of CVD tests have been

14

reported. All the data have been collected from open literature. The data bank represents a wide 7 ACS Paragon Plus Environment


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1

range of gas condensate systems obtained worldwide. For each gas condensate composition, the

2

data include experimentally measured gas gravity, two-phase and gas phase Z-factor at a stage

3

pressure, and temperature reservoir for the produced gas. It also measures the amount of the

4

produced gas, compositional analysis of the produced gas from methane to heptane plus, and

5

molecular weight and specific gravity of the heptane plus fraction. The data cover a wide range

6

of reservoir conditions, temperature ranges from 75 to 418 °F and pressure ranges from 260 to

7

12,814 psia. A complete description of the data bank is given in Table 1. The data cover a wide

8

range of gas condensates from very lean gases to very rich gases. A part of the data employed for

9

modeling purpose is provided in the Supporting Information as Tables S-1.

10

3.4 Data processing and computation

11

Machine learning, a subdivision of artificial intelligence, concerns the development and study of

12

systems that can train from data. The main core of machine learning deals with representation

13

and generalization. Representation of data instances and predictions made by the model

14

evaluated on these instances are part of all machine learning systems. Generalization is the

15

property that the model will perform accurate forecasts on unseen data instances; the conditions

16

under which this can be guaranteed are a key subject for practical use. With the aim of

17

developing effective LS-SVM based models, the available database is randomly divided into

18

three sub-data sets consisting of the “training” set, “validating” set and the “test” set. The first

19

part known (i.e., training set) is used for construction and training of the model. The second part

20

(validation set) is used for selecting optimal parameters of the LS-SVM model and also to avoid

21

the over-fitting problems. The task of remaining data, i.e. test set, is to evaluate the capability of

22

proposed model for prediction of unseen data during the model development process. It should

23

be noted that the division of database into three mentioned sections is performed randomly. 8 ACS Paragon Plus Environment

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However, in distribution of the data into these subdata sets, several divisions are implemented to

2

for determination of optimal division of data sets for use in LS-SVM model. To this end, the

3

effect of altering the ratio of training/validation/test data sets on the performance of the model.

4

Four split ratios of 50/20/20 (train/validation/test percent), 60/20/20, 70/15/15, and 80/10/10 on

5

the model performance were investigated. Results showed that the split ratio of 70/15/15 was

6

more appropriate than other forms and this ratio was selected as optimum form.

7

The mean square error (MSE) between the developed model results and corresponding

8

experimental values reported in the literature, as defined by Equation (11), was considered as

9

objective function during model computation:

MSE =

1 ∑ (t i − oi ) 2 n i

(11)

10

in which, t and o are target value and estimated value, respectively.

11

In this study, we have used the LS-SVM algorithm developed by Pelckmans et al. 48 and Suykens

12

and Vandewalle 27.

13

4

14

4.1 Evaluation of the available approaches

15

The capability of one hundred and twenty feasible methods in predicting both gas phase and two-

16

phase compressibility factors has been evaluated as follows: first, both critical temperature and

17

critical pressure of the heptane plus content of each system have been computed by employing

18

one of the aforesaid correlations for heptane plus characterization. Second, by using a mixing

19

rule, the pseudo-critical temperature and pseudo-critical pressure for each system have been

Results and discussion



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calculated. Then, the pseudo-reduced properties of the system have been computed. Finally, the

2

gas compressibility factors have been calculated via the aforementioned correlations for

3

compressibility factor. Some statistical parameters including average absolute relative deviation

4

(AARD), average relative deviation (ARD), root mean square error (RMSE), and correlation

5

coefficient (R-value) have been used to evaluate the accuracy and reliability of the existing

6

methods for predicting the gas compressibility factors.

7

The summary of obtained results for one hundred and twenty feasible approaches that could be

8

used for predicting the compressibility factor of gas phase have been given in Tables 2 to 5.

9

Similarly, the comparison results for estimating the two-phase compressibility factors have been

10

tabulated in Tables 6 to 9. With accordance to the error analysis results tabulated in Tables 2-5, it

11

could be concluded that utilizing the Elsharkawy

12

correlation for characterizing the heptane plus fraction and Azizi et al. 33 correlation for Z-factor

13

prediction will contributes to obtain better results in the case of estimating the gas phase

14

compressibility factor with AARD=2.031, RMSE=0.031, and R=0.9522. The selection was

15

based on AARD value. On the basis of the results summarized in Tables 6-9, the best way to

16

calculate the two-phase compressibility factor is employing Azizi et al.

17

employing SSBV

18

with AARD=7.967, RMSE=0.100, and R=0.7038. Figures 1-2 demonstrate the graphical

19

evaluation of the performance of using Azizi et al.

20

mixing rule coupled with Kesler-Lee

21

as best approach to calculate the gas phase compressibility factor. Indeed, Figure 1 shows the

22

outputs of the previously mentioned method vs. corresponding experimental values. As can be

23

seen from Figure 1, R-value is equal to 0.9522. Figure 2 shows the relative error deviations from

36

mixing rule, and Sancet

38

43

37

mixing rule coupled with Kesler-Lee

33

38

correlation (by

correlation for characterizing the C7+ fraction)

33

correlation by employing Elsharkawy

37

correlation for characterizing the heptane plus fraction,


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1

the experimental values. However, owing to the high relative errors followed from employing

2

this approach, still an accurate and reliable approach is in demand for the application of interest.

3

In the case of two-phase compressibility factor, the graphical evaluation of accuracy of using

4

SSBV

5

pseudo-critical pressure and pseudo-critical temperature, characterizing the heptane plus fraction,

6

and Z-factor, respectively, is depicted in Figures 3-4. These figures clearly illustrate that the

7

abovementioned algorithm, which is more accurate than all other investigated methods for

8

predicting two-phase compressibility factor, could not be utilized successfully to obtain

9

satisfactory results. Moreover, selected approach has a tendency of having a large error in the

10

two-phase Z-factors at low pressures. Especially for Z-factors below 1.0, selected approach

11

significantly overestimates the two-phase deviation factor. Based on the presented figures and

12

plots it is concluded at this point that the existing methods for calculation of Z-factors for

13

retrograde gas condensate systems have huge error. This necessitates the development of new

14

models for improved prediction of these important parameters.

15

4.2 Developed LS-SVM models

16

Optimized values of the presented LS-SVM model for predicting the gas phase compressibility

17

factor including γ and σ 2 are found to be 90.75049 and 4.14847, respectively. The γ and σ 2

18

values of the proposed LS-SVM model for estimation of the two-phase compressibility factor are

19

equal to 262.9459 and 5.4021, respectively. The tuning parameters of the developed models have

20

been obtained by employing coupled simulated annealing (CSA)

21

LS-SVM toolbox

36

mixing rule, Sancet

48

43

correlation, and Azizi et al.

33

correlation for calculating the

49

optimization technique. The

has been equipped with this optimizer. CSA automatically starts to find the



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model’s parameters once the modeling process starts. In contrast to other optimization

2

techniques, CSA is not slow and can escape from local minima 49.

3

Comparisons between the results of the proposed model and the corresponding experimental gas

4

phase deviation factor values are convincingly illustrated in Figures 5-7. Furthermore, the

5

statistical parameters of the proposed model including correlation coefficients (R-value), average

6

relative deviation (ARD), average absolute relative deviations (AARD), and root mean square

7

errors (RMSE) are reported in Table 10. All the data applied in this study were used “as

8

received” basis from the literature. However, the effect of experimental data uncertainty on the

9

prediction accuracy cannot be ignored. Ignoring the experimental uncertainty/error, developed

10

LS-SVM model for prediction of gas phase compressibility factor is more accurate than the

11

existing methods. In order to illustrate the prediction accuracy of the developed model, three of

12

the best performing approaches have been selected. The prediction error of these methods have

13

been calculated and reported in Table 11. On the basis of the results summarized in this table, it

14

is clear that the best method is LS-SVM model with AARD= 0.726%, RMSE=0.012, and R-

15

value=0.9979.

16

Table 12 reports some important statistical quality measures of the developed model for

17

prediction of two-phase compressibility factor. Results of error analysis reported in Tables 6-9

18

and Table 12 reveal that the new model based on LS-SVM framework is the best method with

19

AARD= 1.302%, RMSE=0.052, and R-value=0.9921. Moreover, comparisons between the

20

results of the developed model and the corresponding experimental two-phase compressibility

21

factor values are illustrated in Figures 8-10. A tight cloud of points about 45o line for all of the

22

data indicate the robustness of the developed model. Similar to the previous section, the

23

prediction accuracy of three of the best performing approaches has been investigated (see Table 12 ACS Paragon Plus Environment

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11). Based on the results reported in Table 11, it is obvious that the developed LS-SVM model is

2

the best model with AARD= 1.302%, RMSE=0.052, and R-value=0.9921.

3

In addition, to make a make a comparison between Figures 4 and 9, it is evident that that

4

proposed LS-SVM model has the smaller error range and least scatter around zero error line.

5

These results illustrate that the developed LS-SVM modeling approach has better and reliable

6

performance compared to existing techniques for improved representation of behavior of

7

retrograde gas condensate during depletion states.

8

Unlike the SVM, the LS-SVM model requires only two adjustable parameters which makes it an

9

ideal engineering tool in practical modeling works with limited data available 24. Considering the

10

above, the LS-SVM model predictions normally do not fail for the aforementioned systems.

11

4.3 Case study

12

The ability of the new method for calculating the gas condensate compressibility factor as a

13

function of changing pressure has been investigated for two retrograde gas sample provided from

14

Elsharkawy and Foda 6. The composition and other properties of these two samples during the

15

depletion stages are reported in Table 13. It should be notated that the studied gas sample no. 1 is

16

a highly sour retrograde gas sample which contains significant amount of hydrogen sulfide

17

(H2S=28.16%) while the gas sample no. 2 is a sweet gas condensate sample. It is interesting to

18

see the performance of the developed LS-SVM model for prediction of compressibility factors of

19

very sour as well as sweet samples. After carefully examining all 120 methods, it was found that

20

SSBV

21

fraction and Bahadori et al.

22

calculation of Z-factor of case studied gas condensate samples. Figures 11(a) and 11(b) show the

36

mixing rule coupled with Sancet 13

43

correlation for characterizing the heptane plus

correlation for Z-factor calculation was the best approach for



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Page 14 of 57

1

comparison between the experimental and predicted compressibility factors by applying

2

Bahadori et al. 13 and proposed LS-SVM model for both of the gas samples. As shown in Fig. 11,

3

the developed LS-SVM model presented in this study is more accurate than Bahadori et al.

4

correlation. Moreover, Table 14 compares some key statistical quality measures of developed

5

LS-SVM model versus selected method. It is concluded at this point that that developed LS-

6

SVM model is much more accurate than selected approach for determination of retrograde gas

7

condensate compressibility factor for both sweet and sour samples.

8

It is interesting to evaluate the data base applied in the case study, since any uncertainty affects

9

the prediction capability of the model. To this end, leverage value statistical approach

50

13

was

10

implemented in this study. The recognition of the suspended data is undertaken through

11

sketching the Williams plot. The calculation procedure of this technique includes determination

12

of the residual values between LS-SVM predicted Z-factor and experimental value for all data

13

sets and a matrix referred to as Hat matrix

14

calculate the Hat matrix (H): 51-54:

51

. The following equation is generally used to

H = X ( X T X ) −1 X T

(12)

15

where, X represents a two-dimensional matrix having m rows (demonstrating total number of

16

hired data) and n columns (demonstrating number of model parameters) and T symbolizes the

17

operator of transpose

18

region of the problem. Recognition of the doubtful data or outliers is graphically carried out

19

through sketching the Williams plot based on the determined H values from Eq. (12)

20

plot shows the relation existing between H values and standardized residuals

21

leverage (H*) is typically fixed at 3(n+1)/m. Morover, leverage value of 3 is generally considered

22

as the cut-off value to accept the measurements within the range of ± 3

53, 54

. Elements on the main diagonal of Hat matrix signify the feasible


53-55

53

53, 54

. This

. A warning

. Those data which

Page 15 of 57

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1

violate the acceptable range (points are located in Residual>3 or Residual

Robust Modeling Approach for Estimation of Compressibility Factor in

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