^ new mexico institute of mining and technologybaervan.nmt.edu/publications/media/pdf/thesis/zain,...

By

zahidah Kd. zaln

Submitted in Partial Fulfillment of the

Requirement for the Degree of Master of Science

in Chemistry

^ New Mexico institute of Mining and Technology

Socorro, New Mexico

May, 1990

MPLXCATXOll OF BXNMIY CIASSXFXBR AND FACTOR ANALySXS•

XM RBFRBSBHTXHS PHA8B BBHAVIOR OF CROOB OXL

PRRC LIBRARY COPY

(i)

MsmovI

Maudotornary dlagraaB art usod to doserl^a phaaa

%t e«\i^ miim » i>wl

tonperatmres • The use of a pseudotemaxy representation

requires combining of components in an additive fashion in

order to fit them to the three vertices of an equilateral

content available from the analytical data. In this study,

binary classifier and factor analysis models are used as an

alternative representation which doscribe phase behavior in a

more comprehensive manner*

(li)

TMLB or 001R81ITS

Eflflfl

ABSTRACT ^TABLE OP CONTENTS

ivLIST OP TABLES

LIST OP FIGURES

ACKNOWLEDGEMENTS *

CHAPTER l: INTRODUCTION ^

CHAPTER 2: EXPERIMENTAL METHODS

2-1 continuous Phase Equilibrium Experiment 72-2 Gas Chromatographs "

2-2-1 Compositional Analysis "2-2-2 Recombined GC Data

CHAPTER 3: METHODS OF REPRESENTING PHASE BEHAVIOR .... 29

3-1 Ternary Representation(Currently Used Method)3-1-1 Ternary Diagram "3-1-2 Pseudoternary Diagram

3-2 Binary Classifier3-2-1 Distance Measurement From the

Centers of Gravity3-2-2 Classification by Mean Vectors <33-2-3 Limitation on the Ratio of N and D ... 463-2-4 Results of the Data Analysis 473-2-5 Interpretation and Conclusion

983-3 Factor Analysis '

3-3-1 Q-mode Factor Analysis3-3-2 Computational Procedure3-3-3 Factors and Rotation3.3.4 Results of the Data Analysis 1273-3-5 Interpretation and Conclusion

CHAPTER 4; SUMMARY AND CONCLUSION160

REFERENCES

APPENDIX A

APPENDIX B

(* . . ;;

APPENDIX C

APPENDIX D

(ill)

169

185

Table

2-1,

2-2.

2-3.

2-4.

2-5.

3-1.

3-2.

3-3.

3-4.

3-5.

3-6.

3-7.

3-8.

3-9.

3-10.

3-11.

3-12.

3-13.

3-14.

3-15.

3-16.

3-17.

3-18.

(Iv)

LIST OF TMLBS

Components of the Continuous PhaseEquilibrium Apparatus

HP 5840 Operating Conditions for Gas Analysis

HP 5880 Operating Conditions for Crude OilAnalysis

Carbon Number Versus Retention Time Window forSimulated Distillation

Report on Compositional Analysis for HP 5880 GC

Compositional Data for Upper Phase CPE 207

Result for Synthetic Oil Data

Result for Crude Oil Data

Distance Measurement for CPE 215 (d = 3)



Mean Vector Measurement for CPE 215 (d = 3)

Mean Vector Measurement for CPE 215 (d = 4) .


Distance Measurement for CPE 207 (d = 5) ....



Mean Vector Measurement for CPE 214 (d » 5) .

Distance Measurement for CPE 216 (d » 5)

Mean Vector Measurement for CPE 216 (d « 5) .

Distance Measurement for CPE 234 (d a 4) ...

Mean Vector Measurement for CPE 234 (d » 4) .

Distance Measurement for CPE 247 (d » 4) ....

Bags

9

18

19

21

28

54

55

56

65

65

66

66

67

67

70

70

73

73

76

76

79

79

82

ZAblfi

3-19.

3-20,

3-21.

3-22.

3-23.

3-24.

3-25.

3-26.

3-27.

3-28.

3-29.

3-30.

3-31.

3-32.

3-33.

3-34.

3-35.

3-36.

3-37.

3-38.

3-39.

3-40.

3-41.

3-42.

(V)

Mean Vector Measurement for CPE 247 (d « 4)

Distance Measurement for CPE 238 (d • 4)

Mean Vector Measurement for CPE 238 (d - 4)

Distance Measurement for CPE 246 (d • 4) ••

Mean Vector Measurement for CPE 246 (d • 4)

Distance Measurement for CPE 239 (d « 5)

Mean Vector Measurement for CPE 239 (d ** 5)

Distance Measurement for CPE 244 (d •> 6) ..

Mean Vector Measurement for CPE 244 (d " 6)

Distance Measurement for CPE 245 (d « 6) ..

Mean Vector Measurement for CPE 245 (d >- 6)

Result of Q-mode Factor Analysis forSynthetic Oil

Result of Q-mode Factor Analysis for Crude Oil.

Statistical Result of Q-mode Factor Analysisfor CPE 207

Rotated Factor Matrix for CPE 214










82

85

85

88

88

91

91

94

94

97

97

131

132

133

137

139

141

143

145

147

149

151

153

155

(vl)

LIST QP PICmBfl

Eiguift EAflfi

2-1. Continuous Phass Equilibrium (CPE) Apparatus .• 8

2-2. Calibration Standard for Gas Analysis byHP 5840 GC 15

2-3. Gas Analysis From CPE Exporiment by HP 5840 GC. 16

2-4. Calibration Standard for Simulated Distillationby HP 5880 GC 20

2-5. Crude Oil Chromatogram from HP 5880 GC Analysis 27(a) sample spiked with ISTD 27(b) neat crude oil sample 27

3-1. Phase Relations for Three Components In MolePercent at 160 and 2,500 psia 31

3-2. Pseudoternary Diagram Produced in CPE Experimentfor C.-C,j,-C^-C3o Synthetic Oil/CO. InjectionGas at 100 ^ and 1300 psia 35

3-3. Two-dimensional Pattern Space with PatternVector Xj 37

3-4. Two-dimensional Pattern Space with Two DistinctClusters and unknown (0) 37

3-5. Procedure for Training and Evaluation of aBinary Classifier 39

3-6. Classit nation by the Distance MeasurementsBetween the Centers of Gravity 40

3-7. Normalization of All Vectors to a ConstantLength (R) 45

3-8. Projection of Pattern Points X on aD-dimensional Sphere 45

3-9. Distance Plot for CPE 215 (d - 3) 59

3-10. Distance Plot for CPE 215 (d •* 4) 60

3-11. Distance Plot for CPE 215 (d » 5) 61

3-12. Mean Vector Plot for CPE 215 (d » 3) 62

3-13. Mean Vector Plot for CPE 215 (d «• 4) 63

(vii)

Figure

3-14. Mean Vector Plot for CPE 215 (d = 5) 64

3-15. Distance Plot for CPE 207 (d = 5) 68


•

H1

r>

Distance Plot for CPE 214 (d = 5) 71







•

OJ1

Mean Vector Plot for CPE 247 (d = 4) 81









3-33. Distance Plot for CPE 245 (d « 6) 95


3-35. Schematic Diagram of A Data Matrix 100

3-36. Degree of Correlation Between Two SamplesV and Y 104

3-37. An Example of Correlation Matrix for N Samples. 105

(Viii)

Ficmre Page

3-38. Cosine of Angle Equals Correlation CoefficientBetween Two Samples 108

3-39. The Cosine Betwieen Two Sample Vectors Determinedby the Proportions of the Variables 108

3-40. Steps In Factor Ai.dlysls 114

3-41. Scatter Diagram In Three-Dlmenslonal Ellipsoid. 120

3-42. Vectors Representing Samples with CorrespondingFactor Axes Coordinate 120

3-43. Hypothetical Unrotated Factor Loading Plot ... 126

3-44. Hypothetical Varlmax Rotated Factor LoadingPlot 126

3-45. Factor Loading Plot for Upper CPE 207 136

3-46. Factor Loading Plot for Lower CPE 207 136















(ix)

Fiqvir?







(»

AOXNOWLBDOBiaBllTS

The author wishes to exprsss hsr profound gratituds to

her advisor, Dr. Janes L. Smith, for his supervision, helpful

guidance and encouragement which enable her to complete this

thesis. Sincerest thanks are also expressed to Dr. Donald X.

Branvold and Dr. Frank Xovarik for serving on her thesis

committee.

Many thanks are extended to Dr. Anita Singh for guiding

her in using the statistical package . The author also would

like to acknowledge Eliot Boyle for initiating the conversion

of compositional data into a simple vector model.

A special thanks is extended to Mariam Saidati, Charlene

Matlock, Khazimad Mat Yusof and Zulkeffeli Mohd. Zain for

helping her in typing this thesis. Above all, the author is

deeply indebted to her husband Zairul Bakry for his helping in

the program and encouragement.

(1)

CHAPTER 1 : INTRODUCTION

Pseudoternary diagrams are frequently used to desqribe

phase behavior of COg/oil systems under a variety of

temperatures.and pressures. A pure ternary diagram of a three

component system offers a rigorous and complete descripticpn of

phase behavior. However, since most experiments involve crude

oils consisting of hundreds of components, the use of a

pseudoternary representation requires combining of compoi>ents

in an additive fashion in order to fit them to the three

vertices of an equilateral triangle. Such a procedure

dramatically masks the compositional content available from

the analytical data and the effect of each component in the

crude oil on the phase behavior cannot be observed. In this

study, other possible representations of phase behavior, yl^ich

make use of additional compositional data provided by gas

chromatographic analysis, are explored.

This study started by attempting to describe hydrocarbon

compositional data from gas chromatographic analyses of the

Continuous Phase Equilibrium (CPE) experiment performed by the

Gas Flooding and Reservoir Simulation section of the New

Mexico Petroleum Recovery Research Center (PRRC). The

intention was to represent each sample as a normalized

composition vector in multidimensional space. Each composition

was a unique vector originating from a common origin. Changes

(2)

in composition alter angles between these vectors, which givethe indication of changes in phase behavxor. By this vectorrepresentation, experimental samples are classified into twogroups: a single phase and a two phase mixture.

Pattern recognition and factor analysis are wellestablished techniques that offer excellent potential forclassification in chemical and geological studies. • Abinaryclassifier, which is one of the classification methods inpattern recognition, utilizes distance measurements from thecenter of gravity and mean vector (dot product) measurementsas a tool of classification.

in 1974, Varmuza, Rotter and Krenmayr employed bothdistance and mean vector measurements to detect type andposition of some substituents in a steroid molecule by lowresolution mass spectra.' Both of these methods were alsoutilized by woodruff, Lowry and Isenhour in 1974 to classifybinary infrared data of compounds containing C, H, Oand Natoms and a carbon content ranging from C, to For themulticategory problem of 13 classes used, a dot productcalculation produced 49.1% correct classification, while adistance measurement produced 58.7%.

Another application of pattern recognition methods isclassification of the origin of petroleum samples in

(3)

environmen'tal chemis'bxy. Oil spills can be characterizec^ by

gas chromatogreotts, infrared spectra or trace elemental

concentrations. Good results have been achieved even for

severely weathered petroleum samples.

Duewer, Kowalski and Schatzki applied a pattern

recognition technique to determine the source of an oil spill

using an elemental composition of a field sample.^ The

classification procedure was based on the comparison of the

field sample to single known source samples and to multiple

artificially weathered source samples. In 1975, Clark and Jurs

identified the type and source of petroleum samples using

fingerprint gas chromatograms and computerized pattern

recognition techniques.^ In this study, adaptive binary

pattern classifiers or dot product methods were used to place

the samples into classes and to predict unknowns. Four years

later, Clark and Jurs employed a bayesian discriminant

analysis to classify crude oils based on their gas

chromatograms taken before and after artificial weathering.^

A variety of different partitions of the data set showed the

similarities of some classes of oils and some dissimilarities

for others.

Different methods of preprocessing data prior to the

computation of a classifier can also influence the

classification of data. In 1977, seventeen preprocessing

(4)

methods had been applied to 524 low—resolution mass spectra of

steroids by Rotter and Varmuza.® The objective was to observe

the influence of Mass Spectra preprocessing on classification

by distance measurement to centers of gravity.

Factor analysis is a statistical technique used to

identify a relatively small number of factors that can be used

to represent relationships among sets of many interrelated

variables* These factors help in classifying variables or

samples. Mathematically, factor analysis approaches treat each

variable or sample as a vector and resolve it into a small

number of component vectors. Vectors may represent variables

(R-mode) or samples (Q-mode). Imbrie and Van Andel developed

the Q-mode model and applied it to two sedimentary basins.^

The main objective was to treat each heavy-mineral data as a

vector and resolve it into a small number of component

vectors.

Q-mode analysis is based on the similarity between

samples. There are several methods of measuring similarity.

Harbaugh and Demirmian (1964), employed both correlation

coefficients and distance coefficients as similarity indices

in Q-mode analysis of petrographic variations in Americus

Limestone.In 1966, Klovan applied Q-mode factor analysis to

classify sediment samples on the basis of their grain-size

distributions.^^ Two factors extracted were claimed to reflect

(5)

different types of depositional energy. McCammon (1966)

explained the use of Q-oode analysis as applied to crude oil

variations.This method was done on eight crude oil samples

which involved twenty>two variables and It effectively

classified the eight samples Into three groups.

In most casesI R-mode and Q-mode analyses are performed

on the same set of data. Hltchon, Billings and Klovan (1971)

used these methods to document flow paths and the chemical

reactions responsible for variations in the chemistry of

subsurface formation waters.*' Factor analysis is also used to

give a simple interpretation of the data matrices,

stromberg and Faschlng (1976) utilised a factor analysis to

study the relationships of trace elemental concentrations in

geological and biological data matrices.** Clusters of elements

were found which were not readily apparent from examination of

either raw data or simple correlation matrices.

The above examples illustrate the wide application of

pattern recognition and factor analysis as classifici^tion

methods. Since the primary aim is to represent the

compositional changes, both of these methods are used to

classify the single phase and two phase regions in the CPE

experiment.

The purpose of this study is to explore alternate

(6)

representations which describe phase behavior in a more

comprehensive manner than pseudotemary diagrams. Using

statistical methods such as binary classifiers and factor

analyses, all compositional data available from gas

chromatographic analyses can be incorporated into the

description of phase behavior. These analyses were applied to

four synthetic oils and seven crude oils analyzed by the CPE

experiment. These methods are compared with conventional

pseudotemary representations and the advantages and

disadvantages of each model is established.

(7)

CHAPTER 2 : EXPERIMENTAL METHODS

2-1 COHTIKUOUS PHASE EQUILIBRIUM EXPERIMENT

The Continuous Phase Equilibrium (CPE) apparatus is

designed to produce rapid measurements of viscosity, densityand composition of flowing phases in equilibrium.Theschematic diagram of the CPE apparatus is shown in Figure 2-1

and a listing of the different parts of the apparatus is givenin Table 2-1. The mixing cell is initially filled with a crude

oil at desired temperature and pressure and allowed tocirculate by means of the two pumps indicated in Figure 2-1.

Gases such as carbon dioxide, carbon dioxide/nitrogen or

carbon dioxide/methane are introduced into the mixing cell at

a controlled rate (usually 12 mL/hour). The back-pressure

regulators function to allow sample fluid to pass alternatelythrough the upper and lower sample ports and maintain acontrolled pressure in the system as the injection gas isintroduced. Fluid flowing to the back-pressure regulators from

the mixing cell pass through an oscillating tube densitometerand an oscillating quartz crystal viscometer. Two identical

sets of instmments provide real time viscosity and densitymeasurements of the upper and lower sample ports of the mixing

cell.

The fluids leaving the upper and lower back-pressure

regulators are collected separately at ambient temperature and

FIRST STAGE

Gas Injection and Mixing

(8)

SECOND STAGE

Fluid PropertyMeasurment

THIRD STAGE

Composition Measurement

Figure 2-1. Continuous Phase Equilibrium (CPE) Apparatus

(9)

Tftbla 2*1 • Coapononts of the Continuous Phas# Equilibri^ua

Apparatus

NUMBER COMPONENT

1 Ruska positive displacement motorized pump

2 134 cc mixing vessel

3 Eldex high-pressure circulating pump ( 450 cc/hr)

4 Mettler-Paar DMA 512 densitometer

5 Torsional crystal viscometer

6 Motorized back pressure regulators

7 Multi-port sample valve and sample vials

8 Air-actuated gas sample valve

9 Hewlett-Packard 5840 gas chromatograph

10 GCA 63125 wet test meter

(10)

pressure. Liquid phase is collected in sample vials for later

weighing and compositional analysis by simulated distillation.

Each sample vial is filled with liquid for one hour before

switching to the next sample vial. The separated vapor from a

given sample vial proceeds to a HP 5840 Gas Chromatograph for

on-line compositional analysis and then to a wet test meter

for measurement of volume. The vapor compositional analysis is

measured three times per sample vial. The experiment is

controlled by an HP87XM Microcomputer which:

1) reads deusitometer and viscometer output; calculates and

stores upper and lower phase densities and viscosities

data every 4 minutes.

2) advances multiport samplers at the end of a sample period

every one hour; alternates back pressure regulators

between the upper and lower phases every 3 minutes.

3) selects appropriate (upper or lower) sample streams and

sets the position of a sample switching valve in the gas

chromatograph.

4) starts gas chromatographic analysis of gas samples every

15 minutes.

5) reads and stores results of analysis.

Controlled introduction of gases into the mixing cell is

continued until phase split occurs. Before the occurrence of

the phase split, the upper and lower sample streams contain

the same single phase fluid. After the oil/injection gas

(11)

mixture enters the two phase region, the upper and lower phase

samples mostly consist of vapor and liquid phase respectively.

The occurrence of the phase split is accompanied by decrease

in both density and viscosity at the upper portal and increase

for these measurements at the lower portal.

Each filled sample vial represents one data point for

correlating viscosity and density to fluid composition. The

amounts and compositions of both the liquid and vapor

collected during a sampling period are combined to calculate

an overall composition for fluid produced during a certain

time interval.

Viscosities are measured by an oscillating quartz crystal

viscometer and derived from a resonance curve bandwidth

using

where:

P

M / S

f

Af

^vac

nfl AfyP \sj

Af Afvac

vac ,

= density of the fluid.

= the mass-to-surface area ratio of the crystal

= the resonant frequency

= the half conductance bandwidth

= frequencies which are measured in a vacuum

(12)

Densities are determined with a Paar DMA 512 digital

densitometer. The measuring principal of the instrument is

based on the variation of the natural frequency of a hollow

oscillator when filled with different liquids or gases.

Density measurement is based on periods and densities ofcalibrating fluids (methane and decane) which are entered in

the program prior to the experiment. A period, which is theinverse of frequency,is a calibration number given by the

densitometer. Throughout the experiment, the density of each

component in the crude oil sample is determined by the

following equation:

p = A(T2 - B)

where

T = period

(Densi ty CH^ - Densi ty ^0-^22)^ ~ (Period CH^f - (PeriodB = (period - (A)(Density CH^)

(13)

2-2 GAS CHR0MAT06RAPH

Two types of gas chromatographs are used to conduct the

analysis of fluid phases produced from the CPE experiment. A

HP 5840 gas chromatograph is directly connectea uv. "^he CPE

experiment and is used to analyze the low molecular weight

hydrocarbon gases and COg gas which evolve from the upp^r and

lower ports of the CPE apparatus. This chromatograph is

equipped with a gas sampling loop, a packed column and a

thermal conductivity detector(TCD). The 6* * 1/8" stainless

steel packed column contains a Porapak Q stationary phase

(Supelco Inc.). Porapak Q is a styrenedivinylbenzene polymer

on a 80/100 -sieve diatomaceous support. The mobile phase or

carrier gas, used to elute the sample through the column, is

helium. The TCD has the advantage of detecting COg , a major

constituent in the vapor.

A HP 5880 gas chromatograph is employed to determine

carbon number composition in the crude oil. It is configured

for direct sample injection onto a Supelcoport packed column.

The 6* * 1/8" stainless steel packed column contains a

stationary phase of 10% SP 2100 (a methyl silicone fluid) on

100/120 - sieve diatomaceous earth. Detection is done by means

of a flame ionization detector (FID) which is a universal

detector for hydrocarbons. The FID has the disadvantage of

being unable to detect COj gas. Very little COg resides in the

(14)

crude oil samples under ambient conditions.

2-2-1 C0MP08ZTI0IAL ANALYSIS

An important feature of both gas chromatographs is their

ability to raise the column's temperature at a constant and

reproducible rate. Therefore, separation is accomplished not

only by the different affinities that the solute has for the

stationary phase, but also by the varying boiling points of

the solutes. Quantitative analysis depends on the relationship

between the peak area or peak height and the amount of the

constituents. 2® All quantitation requires GO analysis of

standards with known concentrations of the components to be

analyzed. Quantitation of samples with unknown concentration

is obtained by direct comparison of peak area or height with

a standard. The HP 5840 nc is calibrated by adding a constant

volume of a gas mixture consisting of (by mole percent) 85.05%

COgr 8.21% methane, 2.00% ethane, 2.00% propane, 2.00% n-

butane and 0.74% n-pentane. Figure 2-2 is a chromatogram of

this mixture. The retention time, which is the elapsed time

from injection of the sample to the recording of the

component's peak maximum, is printed for each peak. With the

exception of COg, the order of component elution is a function

of molecular weight or carbon number. The peak area data from

the chromatogram in Figure 2-2 is directly compared with the

chromatographic area data of a gas sample of unknown

(15)

£ CH4

C,H« CO2'2"6

B— CsHs

5H12

RT rmin^ AREA

125400

AREA % MOLE % RF rMOLE % / AREA^

0.58 6.227 8.21 6.55 exp (-5)

0.71 1668000 82.822 85.05 5.10 exp (-5)

1.39 44850 2.227 2.00 4.46 exp (-5)

2.82 57690 2.865 2.00 3.47 exp (-5)

4.84 76620 3.804 2.00 2.61 exp (-5)

7.08 38450 1.909 0.74 1.92 exp (-5)

Figure 2-2. Calibration Standard for Gas Analysis

by HP 5840 6C.

'L

Z.ZB

a.96

4.50

6.747.09

7.93

8.44

9.329.69

13.36

11.28

RTf fffiAn)

0.51

0.78

1.50

2.96

4.91

7.09

AREA

193900

1616000

1987

44450

126400

108900

(16)

mem

8.067

67.232

0.083

1.849

5.259

4.531

CQMPOTONT

CH,

CO2

<^6

C3H8

C4H10

C5H,2

Figure 2-3. Gas Analysis from CPE Experiment by HP 5840 GC

S:f!

(17)

composition. Figure 2-3 is an example of gas analysis from theCPE experiment. From the calibration run (Figure 2-2) .responsefactors (mole% / area) are assigned to each component. Theseresponse factors are then used to determine gas compositionfrom chromatograms of the gases evolving from the upper andlower ports of the CPE apparatus. All gas samples are rununder the conditions indicated in Table 2-2.

An ASTM method has been established for simulatinghydrocarbon distillation with a gas chromatogrjaph. Theanalysis requires a hydrocarbon standard to correlateretention time with boiling point or carbon number. Asoftwareprogram -SIMDIS-^i ^„hich is used in the crude oil analysis,has been written to conform with a proposed ASTM standardprocedure. This program performs three main functions:(1) controls various aspects of the HP 5880 GC operations(2) calculates the data resulting from the analysis(3) stores the analysis results on a cartridge tape, which

can be retrieved or transferred to the Deo-20 or theHP87 for data calculation or long-term storage.

For simulated distillation, a calibration standard(Cj - C40, HP NO. 5080-8716, see appendix A) is run on the HP5880 GC. instrument operating conditions are given in Table 2-3. Atypical chromatogram for a calibration standard is shownin Figure 2-4. This chromatogram is divided into intervals

/*•

(18)

Table 2-2. HP 5840 Operating Conditions for Gas Analysis

Column Length, ft. 6

Column ID, in. 1/8

Stationary phase Styrenedivinylbenzene polymer

Support material Porapak Q

Support mesh size 80/100

Initial column temperature, ° C f.O

Final column temperature, ° C 240

Oven temperature program rate, ° C/min 20

Carrier gas He

Detector TCD

Detector temperature, ° C 270

Injection port temperature, ° C 300

Sample size, uL 1

(19)

Table 2-3* HP 5880 Operating Conditions for Crude Oil

Analysis

Colunm length, ft. 6

Column ID, in. 1/8

Stationaiy phase 10% SP 2100( methyl silicone fluid )

Support material Supelcoport

Support mesh size 100/120

Initial colunm temperature, ^ C 30

Final column temperature, ° C 370

Oven temperature program rate, ° C/min 15

Carrier gas He

Detector FID

Detector temperature, o C 380

Injection port temperature, ° C 370

Sample size, uL 1

START AUTO SCO

r

rrc:^cr

11.73 ^

'z c^ 13.37 Q

18

20

%̂U.1« Q

c::c:"C 18.10 ^

c:^C 19.30 ^

24

28

32c:

cj'

k' 22.ro36

1.:

2.09 C7

• '-"Cs- "-''Co

6.51

(20)

6

'11

.19 01015

17

5.48 Q10

•?.4l Q

11.06 Q

14

16

J.72 Cl

.66 Q12

OVI STOP ftUH

Figure 2-4. Calibration Standard for Simulated Distillation

by HP 5880 GO

(21)

Table 2-4. Carbon Number versus Retention Time Window

for Simulated Distillation

1840 DATA 1850 DATA 1860 DATA

CARBON# RT. (MIN) CARBON RT. (MIN) CARBON# RT. (MIN)

5 1.0 19 13.5 31 19.5

6 1.7 20 14.1 32 19.9

7 2.6 21 14.7 33 20.3

8 3.7 22 15.3 34 20.7

9 4.9 23 15.9 35 21.1

10 6.0 24 16.4 36 21.5

11 7.1 25 16.9

12 8.1 26 17.4

13 9.0 27 17.8

14 9.8 28 18.3

15 10.6 29 18.7

16 11.4 30 19.1

17 12.1

18 12.84

(22)

corresponding to Cj through C^. In the chromatogram some of

the peaks do not exist. Therefore, the retention time of peaks

not existing in the calibration standard are extrapolated as

shown by the dotted peaks. The information obtained from the

calibration standard is used to correlate retention time with

carbon number on crude oil samples. Table 2-4 gives carbon

number and retention time windows for the chromatogram of the

calibration standard in Figure 2-4.

An internal standard mixture (HP No. 5080-8723)

consisting of normal alkanes and is used in the

crude oil analysis. The purpose of this internal standard is

to serves as an integrity check of the area quantitation and

retention time reproducibility of the gas chromatograph. The

retention time data for the internal standard (ISTD) segment

(starting and end points) is determined from the calibration

standard prior to the analysis of crude oil.

The equipment and GC operating conditions are the same as

described in the ASTM D2887 method.^^ The procedure for the

crude oil analysis requires that the sample be analyzed twice.

Once where the sample is spiked with 10 - 15% of ISTD and once

vith a neat crude oil sample. The procedure is as follows:

(1) the crude oil sample (about 0.6 g) is weighed in a

standard 1.8 mL autosampler vial (Supelco cat. no. 3-

3286) and the weight is recorded to 0.0001 g.

(23)

(2) approximately 10 - 15% of the ISTD is added to the

vial; the accurate weight of the ISTD added is

recorded.

Both weights of the crude oil sample and the ISTD are

to be entered in the dialogue of the "SIMDIS" program

before the samples are analyzed by GC.

(3) the sample vial is tightly stoppered with a

septum/screw cap and the mixture is thoroughly

agitated.

(4) the samples are loaded in pairs, first sample plus

ISTD, followed immediately by a vial of crude oil

sample, into successive slots in the tray of the liquid

automatic sampler.

If the crude oil sample has a specific gravity less than

20® API, a solvent, carbon disulfide (CSg), is added to reducethe viscosity. When CSj is used, the mixture of crude oil plus

ISTD is prepared in a larger ( > 5 mL ) vial, then one-half

mixture and one-half of CSg are added to the standard 1.8 mL

vial. Approximately the same amount of CSj are added to the

crude oil sample alone in another vial. The CSg has no

detectable response to the flame ionization detector.

The area integration is done by area slice mode. The area

(24)

slice mode is the sum of detector reading over some specific

time interval (the slice width)• For the crude oil analysis,

the area slice width is 0.02 minutes. The area for each carbon

number and retention time window is compared to the total area

of C3 through and it is assumed that each hydrocarbon has

the same response factor. If the chromatogram area %

associated with the ISTD does not match the calculated weight

% of ISTD added to within 3%, the results are considered

questionable and the sample is rerun or a new sample is

prepared.

The operation of the 6C is done automatically after the

program is running. Figure 2-5 shows an example of crude oil

chromatograms; one chromatogram of sample spiked with ISTD and

the other one is a neat crude oil sample. The results of the

analysis are calculated and printed out immediately following

the chromatogram at the end of each analysis ( Table 2-5).

2-2-2 RECOMBINED OC DATA

The recombined fluid composition is calculated for each

sample from:

1) the weight of liquid collected in each sample vial

2) the volume of gas evolved from the upper and lower

section of the mixing vessel.

3) the liquid compositional analysis from the HP 5880 GC.

(25)

4) the gas compositional analysis from the HP 5840 GC.

From the liquid analysis by the HP 5880 GC, th© results

are reported as a fraction of total weight (equivalent to area

percent) of each component in the sample. Then this fraction

is used to determine the number of moles for each component by

multiplying the total weight of liquid collected in tjie sample

vial and dividing by the molecular weight of each component.

(fr. of total weight)(total weight) / (mwt. of component)

B # mole of component

For the gas analysis by HP 5840 GC, first, the peak area

of each component in a chromatogram is multiplied by the

response factor ( mole % / area ) to get the mole fraction of

each component. The response factor was previously determined

from the calibration run. The total volume of gas is measured

by a wet test meter (connected to the CPE apparatus) which is

used to calculate the total weight of each gas component. This

is done by multiply ing the total volume with the density of

each component in the gas obtained from a standard density

table.

(total volume)(density) = total weight of gas in the sample

Then the number of moles for each component is calculated as

%•r

(26)

follows,

(mole fr.)(total weight) / (mwt. of component)

a # mole of component

The final step is the addition of niimber of moles of gas and

liquid for each component in the sample and then it is

adjusted to mole fraction of the recombined composition. All

of these calculations are done by a program stored in HP 87XM

microcomputer.

oCO

d fcU)

o

o

LM U-v

(27)

Figure 2-5(a). Sample Spiked with ISTD.

Figure 2-5(b). Neat Crude Oil Sample.

Figure 2-5. crude oil Chromatogram from HP 5880 GC Analysis

(28)

Table 2-5. Report on Compositional Analysis for HP 5880 GC

flREflJi FROM C5 TO C36 OF SftMPLE. 25CPE-248 IS 71.5398

RREft OF ISTD/RREfl OF ISTD+SflMPLE ISI/(I+S> 13

C NO

5

6

7t-.

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

RRER

24482.5191923

316716

4S8626

513104

404829

428801

383383

307412

325475

325340

325617

328992

252662

311651

233191

231566

230161

213054

146356

209354

141697

13974-7

206554

140691

137292

1^535^

131891

129465

127884

129336

131773

CUM RRER

24482.5

216411

333126

1.02175E+06

1.53486E+06

1.93969E+06

2.36849E+06

2.75137E+06

3.35928E+06

3.33476E+06

3.7101E4-064.03571E+06

4.36471E+06

4.61737E+06

4.92902E+06

5.16721E+065.39878E+06

5.62894E+06

5.34699E+06

5.99335E+06

6.2Q27E+06

6.3444E+06

6.48415E+06

6.6907E+066.33139E+06

6.96868E+06

7.10404E+067.23593E+06

7.36539E+06

7.49323E+06

7.62261E+06

7.75438E+06

.185373

.170692

RRER CUM RRER?'.—

,225774 .225774

i•76993 1.99571

2.9207 4.91641

4.50604 9.42245

4.73176 14.1542

3.73327 17.8875

3.95434 21.8418

3.5355 25.3773

2.83491 23.2122

3.00148 31.2137

3.00024 34.214

3.00279 37.2167'

3.03391 40.2507

2.33001 42.5307

2.874 45.4547

2.19656 47.6512

2.13547 49.7367

2.12251 51.9092

2.01086 53.9201

1.34967 55.2697

1.93063 57.2004

1.30671 53.5071

1.28873 59.7958

1.90481 61.7006

1.29743 62.998

1.26609 64.2641

1.2482 65.5123

1.21628 66.7286

1.19391 67.9225

1.17933 69.1018

1.19272 70.2946

1.21519 71.5098

SflMPLE 27 NEXT

PRs 13:27 JUL lli 1989

(29)

CHAPTER 3 : METHODS OF REPRESEMTIMG PHASE BEHAVIOR

3-1 TERNARY REPRESENTATION (CURRENTLY USED METHOD)

The term phase is used to define any homogenous and

physically distinct part of a system which is separated from

other parts of the system by definite bounding surfaces. Two

phases that are important in the petroleum industry and in

this study are liquid and gas phase. In particular, we are

interested in phase behavior of the system; that is, the

conditions of temperature and pressure for which different

phases can exist.^ The phases which exist are identified by

their volume, viscosity and density. Reservoir fluids are

complex multicomponent mixtures of hundreds of different

hydrocarbons and some nonhydrocarbons. The exact composition

of a reservoir fluid is never known. An approximate method of

representing the phase behavior of multicomponent mixtures

utilizes the triangular diagram.The phase behavior of three

component mixtures can be represented exactly on a triangular

diagram, whereas its use for multicomponent mixtures rec[yires

that these mixtures be approximated by three pseudocomponents.

3-1-1 REPRESENTATION OF THREE COMPONENT PHASE BEHAVIOR

(TERNARY DIAGRAM)

Each corner of a triangular diagram (Figure 3-1)

represents 100% of a given component. The opposite side of the

(30)

triangle represents 0% of that component. For example, the

upper most comer of the triangle represents 100% methane (C,)

while the opposite or bottom side of the triangle represents

0% of methane. Any concentration of methane between 0 and 100%

is represented at a proportional distance between the bottom

of the triangle and the upper corner. Similarly, the lower

right comer represents 100% n-butane and the lower left

corner represents 100% decane. With this manner of specifying

component concentrations, mixtures can be plotted on the

diagram. For instance, mixture S contains 68% methane, 21% n-

butane and 11% decane. For the phase relation shown in this

figure, the mixture with overall composition represented bypoint S is a two phase mixture. This mean that if the three

components were mixed together in a pressure vessel at 2500

psia and 160 ®F in the relative proportions specified by point

S and allowed to equilibrate, two phases would result: an

equilibrium gas phase with composition Y and an equilibrium

;^Aqi;j.d pt^ase with composition X. The dashed line connectingthe equilibrium gas and liquid composition is called a tie

.lipe. Since the gas and liquid are in equilibrixim with each

other, they are fully saturated. The gas is saturated with

condensible components and therefore is at its dew point while

the liquid is saturated with vaporizable components and is at

its jpu^ble poj^nt. For the phase relation shown in this figure,

the dewpoint curve through all the dewpoint compositions joins

the bubble point curve through all the bubble point

TIE LIN

PHASEENVELOPE

100%

(31)

100% C,

X.PHASF\y *REGION*—*

EQUILIBRIUM^GAS PHASEDEWPCMNTtm^(SATURATED VAPOR)

critical POINT

POINT S

68% C

21% a

BUBBLE POINT UNE(SATURATED LIQUID)

11% C.

EQUILIBRIUMLIQUIDPHASE

Figure 3-1. Phase Relations for Three components in Mole%

at 160 "f and 2500 psia

(32)

compositions at the critical point. At the critical point, the

composition and properties of equilibrium gas and liquid

become identical. The phase boundary curve or phase envelope

separates the single phase and two phase regions of the

diagram. At the pressure and temperature of the diagram, any

system of the three components whose composition is inside the

phase envelope curve will form two phases and any system with

a composition lying outside of this curve will be in a single

phase. The single phase gas region lies above the dewpoint

curve, while the single phase liquid region lies below the

bubble point curve.

3-1-2 REPRESEMTATZON OF MULTZCOMPONEMT PHASE BEHAVZOR

(PSEUDOTERNARY DZAGRAM)

The phase behavior of reservoir liquid is represented

approximately on a triangular diagram by grouping the

components of the reservoir fluid into three pseudocomponents.

In general, the three groups are low volatility, intermediate

volatility and high volatile pseudocomponents. The

representation of mixture compositions and phase behavior in

this manner is approximate since the individual components

within a pseudocomponent group have different volatilities and

will not be distributed within that group in the same way as

in the gas and liquid phases. For this reaction, the

composition and the properties of the pseudocomponent do not

(33)

remain constant for all mixtures. Also, the position of the

phase envelope curve on the triangular coordinates and the

slope of the tie lines depend on the overall mixture

composition, which cannot be defined adequately by the simple

pseudocomponent grouping.

In the CPE experiment, carbon dioxide is injected into

the homogeneous mixture of hydrocarbons at a certain

temperature and pressure. Since carbon dioxide has the

greatest solubility in low molecular weight hydrocarbons, it

will preferably extract the low molecular weight hydrocarbons

from the homogeneous mixture.At this point, phase split

occurs meaning that the original homogeneous components are

entering the two phase region where the liquid and gas phases

coexist. Figure 3-2 is a pseudoternary diagram of phase

behavior from a CPE experiment at 1300 psia and 311 K. The

diagram illustrates the compositional points starting with 0%

carbon dioxide, 68% C5 and C,q and 32% C^^ and C^q. As carbon

dioxide is injected into the mixture, the composition of the

mixture collected in upper and lower samples will follow the

compositional path until phase split occurs. After phase

split, the upper samples mostly contain gaseous components

while the lower samples are predominantly liquid components.

For each upper phase composition, there is a corresponding

lower phase composition and both are connected by a tie line.

The two points used to construct a tie line represent the COj,

mtfrnm

(34)

and compositions of the upper and lower phases

collected from the CPE experiment. The viscosity for each

compositional point is also indicated in the diagram. Ternary

diagrams for all synthetic and crude oils used in this study

are given in appendix B.

50%

C30-CI6

CPE 207

T' 31I®K (IOO®F)

Ps 8.96 MPa (I300psla)

0 5 Viscosity X10^, Pa»sor Viscosity, cp

(35)

CO2

.350--V.365-\^

.380-^ \.410^ X

.44S-Ov.480 ^ ^

Figure 3-2. Pseudoternary diagram produced in CPE experiment

for Cj-C-jQ-c^^-CjQ synthetic oil/C02 injection

gas at 100 and 1300 psia.

507

ClO"C5

(36)

3-2 BINARY CLM8ZFIBR

A binary classifier is one of tha classification methods

in pattern recognition.^^ It is used to distinguish between two

mutually exclusive classes. For instance, class 1 night

contain compounds with certain physical/chemical properties

and class 2 contains compounds with other physical/chemical

properties. The principle of a binary classifier is based on

what is called a pattern vector. A pattern vector

characterizes an event or object and then it is employed by

the binary classifier to decide if the pattern belongs to

class 1 or class 2. The basic concept of the binary classifier

is as follows: An object or an event j is described by a set

of d features X|j (i • 1 ... d) and all features of one object

form a pattern, For example, each object j is known to have

only two features (measurements) X^j and *2J' The numerical

values of the features for each object j can be represented as

a point in a two-dimensional coordinate system or pattern

space as shown in Figure 3-3. An equivalent representation is

a vector Xj rpattern vector^ from the origin to the point with

the coordinates X^j and

The hypothesis for all pattern recognition is that,

objects that have similar properties are close together in

pattern space and form a cluster. As shown in Figure 3-4, all

objects form two distinct clusters and each member of a

(37)

Figure 3-3. Two-dimensional pattern space with

pattern vector Xj.

♦ +

+ +

Figure 3-4. Two-dimensional pattern space with two

distinct clusters and unlcnown (O) •

(38)

cluster has the same property. Classification of an object (0)

whose class membership is unlcnown recpiires the determination

of the cluster to which this point belongs. To formulate a

suitable pattern space, a collection of patterns with known

class meinberships is randomly divided into two parts (see

Figure 3-5). Part 1 is used as a training set to develop a

classifier that recognizes the class membership (class 1 or

class 2) of the training set patterns. The classifier is then

tested with the patterns of the second part which is called

the prediction set. The member of the prediction set is

classified into either class 1 or class 2 by a classifier. It

is possible to extend the above two-dimensional example to

situations involving a multidimensional hyperspace. The

geometry in a d-dimensional hyperspace (d greater than 3) and

the geometry in two or three-dimensions are qualitatively the

same. The only difference is that the clustering in a d-

dimensional hyperspace is not directly visible and it is

difficult to represent graphically. However, it can be

suitably represented mathematically.

In this study, a binary classifier is used to classify

crude oil samples into two different classes: class 1 is a

group of samples before the phase split and class 2 is a group

of samples after the phase split. Therefore, each set of the

crude oil from the CPE experiment is divided into two groups

(before and after the phase split) for each upper phase and

(39)

COLLECTION OF PATTERNSWITH KNOWN

CLASS MEMBERSHIP

TRAINING SET

CLASS 1 CLASS 2

PREDICTION SET

1

TRAININGEVALUATION

f

CLASSIFIER

CLASSIFY THE PREDICllONSETINTO CLASS 1 ANDCLASS 2

Figure 3-5. Procedure for training and evaluation of a

binary classifier.

(40)

*2

_ CLASS 2

SYMMETRY PLANE

CLASS 1

Figure 3-6. Classification by the distance measurements

between the centers of gravity.

(41)

lower phase sample. The use of the binary classifier method

predicts where phase split occurs during the CPE experiment.

There are two methods used to compute the binary classifier;

distance measurements from the center of gravity and mean

vectors.*

3-2-1 DISTANCE MEASUREMEMT FROM THE CEKTER OF GRAVITY

The classification by distance measurements is ba^ed on

the center of gravity (centroid) of the compact cluster formed

by all pattern points of a certain class in the pattern space.

AS shown in Figure 3-6, both classes form compact clusters and

each of the clusters is represented by the center of gravity

*C, and "Cj. The unknown pattern is classified into that classwhich is associated with the nearest center of gravity.

Therefore, the unknown in Figure 3-6 is classified to belong

to class 1 because the distance to is shorter than that to

Cj. Both centers of gravity are separated by a symmetry planeor a decision plane. The coordinates C^t^z center

of gravity C in a d-dimensional hyperspace are calculated inthe same way as for two-dimensions. Each coordinate is the

arithmetic average of the components X| summed over all

patterns j (j = 1 ... n) of a distinct class. Therefore the

center of gravity is the mean of all patterns belonging to the

same class. The center of gravity is calculated by the

following equation,

(42)

for all dimensions 1=1 .•• d

where,

C{ a component (coordinate) i of the center of gravity

n = number of patterns in the class under

consideration

Xjj = component i of pattern with number j

The distance measurement between two points in the d-

dimensional hyperspace is

D = ^Ui-q)2 + (2)

N

where,

D = distance between center of gravity C (C,, C^, ...

Cj) and pattern point X (X,, X2, ... X^)

The unknown is classified by a decision criterion Y defined as

r = AZ? = D^-D^ (4)

d

(3)

if Y > 0 > CLASS 1

y < 0 > CLASS 2

(43)

The unknown Is classified into class 1 if Y is greater than

zero (positive) which means that the distance between the

pattern vector of the unknown to the center of gravity of

class 1 is shorter than that of class 2. On the other hand, if

Y is less than zero (negative), the unknown is classified into

class 2.

3-2-2 CLASSIFICATION BY MEAN VECTORS

This classification is based on the scalar product (dot

product) of the unknown pattern vector and the center of

gravity of each cluster. Each pattern vector point of the

center of gravity and unknown is assumed to lie on the d-

dimensional sphere with radius R (Figure 3-7). The pattern

vectors are normalized to a fixed length R by multiplication

of all vector components by a factor K (Figure 3-8) where

« - -3^ (5)xi

Xi = KXi (6)

for all dimensions i.

In this study, the radius of the sphere is taken to be 1. The

scalar product of the pattern vectors for class 1 and class 2

are calculated respectively by following equations:

(44)

= Ci . • Xi (7)

Cj . X Cji . Xi (8)

The unknown X Is assigned to that class which gives the larger

scalar product since the scalar product is inversely

proportional to the angle between the tinlcnown and the center

of gravity.

Si =• q . A" X I COS 01 (9)

where 6, = angle between and X

01 = COS"^q . X

c, X(10)

Sa = C2 . X q i \x\ cos 02 (11)

where 63 = angle between C2 and X

0, =» cos-1 q • X

a. X(12)

(45)

HYPERSPHERE

Figure 3-7• Normalization of all vectors to a constant

length (R).

*2f ^

J \J

\

m

t

/ * \\

*2 "2'""I*1 \ f

X/.. kx,*1

Figure 3-8, Projection of pattern points X on a

d-dimensional sphere.

•W4'i .">K

(46)

3-2-3 LIMITATION ON THE RATIO OF n AND d

•JUS—

The minimum requirement that is now widely accepted and

should be satisfied in all applications of pattern recognition

is based on the following rule ;

(13)

where n = number of patterns (sample)

d = number of independent features

(dimension)

If n / d is less than 3 for a binary classification, the

statistical significance of a decision plane is doubtful. In

this study, the limitation or minimum requirement of n / d is

taken in order to get reliable results. For both synthetic oil

and crude oil samples, the range of the ratio n / d is from

3.0 to 6.7 and the number of dimensions (d) is taken to be

greater than or equal to 3 (to match with the representation

of ternary diagrams).

' • m-

(47)

3-2-4 RESULT OF THE DATA ANALYSIS

A sample calculation follows:

Data ; Upper phase CPE 207 (synthetic oil)

refer to Table 3-1

Dimension : 3 (COj, Cj + C„, + Cjj)

Training sets sample 1 to 3 for class 1

sample 13 to 15 for class 2

Prediction set; sample 4

(I) Distance measurenent from tbo center of gravity

Center of gravity for class 1 (equation 1);

C1 = 1/3 (0.0 + 0.0 + 6.64) = 2.21

C2 => 1/3 (68.0 + 69.11 + 64.67) = 67.26

C3 = 1/3 (32.0 + 30.88 + 28.69) = 30.52

Center of gravity for class 2 (equation 1):

C1 = 1/3 (94.95 + 96.92 + 96.60) = 96.16

C2 = 1/3 (3.86 + 2.41 + 2.62) = 2.96

C3 = 1/3 (1.18 + 0.66 + 0.77) = 0.87

(48)

Distance measurements between sample 4 to the center of

gravity class 1 and class 2 (equation 2):

- V(32.83 - 2.21)2 (46.27 - 67.26)2 + (20.9 - 30.52)^- 38.35

Dj - v^(32.83 - 96.16)2 + (46.27 - 2.96)^ + (20.9 - 0.87)2- 79.29

By equation 4;

Y = Ad = D2 - D1 = 79.29 - 38.35 = 40.94 (positive)

Therefore seunple 4 is classified into class 1 since Y is

greater than zero.

(II) Classification by Mean Vectors

Normalize the value of X and C as follows(equation 5);

K for sample 4:

VC 32.83 )2 + ( 46.27 )2 + ( 20.9 )2 60.46

K for class 1:

K

K

V( 2.21 )2 + ( 67.26 )2 + ( 30.52 )^ 73.89

K for class 2:

V( 96.16 + ( 2.96 + ( 0.87 )' 96.21

(49)

The scalar product of the pattern vectors (equation 7 and 8);

_ ^ (2.21) (32.83) * (67.26) (46.27) -f (30.52) (20.9)^ (73.89)(60.46)

- 0.856

. (96.16) (32.83) + (2.96) (46.27) -i- (0.87) (20.9)' (96.21)(60.46)

- 0.569

Therefore sample 4 is assigned to class 1 since the scalar

product with class 1 is larger than that with class 2.

In this study, there are four synthetic and seven crude

oil data used and all the compositional data are tabulated in

Appendix C. The analysis of each compositional data by binary

classifier predicts the occurrence of the phase split. The ^

results from this analysis are tabulated in Tables 3-2 and 3-

3. Figures 3-9 to 3-34 are the distance and mean vector plots

for each sample and the data corresponding to each plot are ^

tabulated in Table 3-4 to 3-29.

(50)

3-2-5 IKTERPRSTATION AMD COHCLUSIOM

Due to the limitation on the ratio of N to D where the

ratio must be greater than or equal to 3, we were only able to

use a maximum dimension equal to 5 for synthetic oil and 6 for

crude oil experimental data. Figures 3-9, 3-10 and 3-11 are

plots for distance measurement of CPE 215 with dimension 3, 4

and 5 respectively. The comparison of these plots show that

they are very similar to each other. This observation is the

same as for mean vector plots (Figures 3-12, 3-13 and 3-14).

Therefore only one plot of distance and mean vector for each

compositional data are presented.

The values of distance and mean vector measurements are

presented in Table 3-4 to 3-29 for all samples used in this

study. For example. Table 3-4 shows a distance measurement for

CPE 215 with dimension 3. The values of distance measurement

from class 1 and class 2 for each sample vial are listed for

both upper and lower samples. Y is a decision criterion which

classify the sample vials. For instance, sample 6(upper) is

classified into class two since the distance between sample 6

to class 2 is shorter than that with class 1. Also for upper

CPE 215, it is observed that samples 1 to 4 are classified

into class 1 while sample 5 to 15 into class 2. Therefore, tho

phase split occurs at sample 5, which is the first sample

being classified into class 2.

(51)

Table 3-2 gives a sunnaary of phase split predictions for ^synthetic oil samples containing five components. The firstcolumn of this table indicates the CPE name of the syntheticoil experiment. The second, third and forth column? give the ^phase split prediction for different numbers of dimensions andnumbers of samples used in training sets. For instance, thesecond column shows that the data is combined into three -groups or dimensions (d = 3) of COj, Cj + C,^ and Cjqcomponents. The number of samples used in a training set is 3(s = 3). For CPE 214, which have a total of 17 sables, the ^first three samples are used as a training set for class 1while the last three samples for class 2. As for upper CPE214, distance measurement for the center of gravity (D)predicts the phase split at sample vial 5 and mean vectormeasurement (S) at sample vial 4. ^

The third column of Table 3-2 is divided into threeparts. Each of these parts represents different way of ^combining four groups of components. In part (I), C, and C,^composition are combined, and C,^ and Cjj are usedindividually. Part (II) combines C,^ and C,, components while ^part (III) combines C„ and C,^ components. These threedifferent combinations of synthetic oil components give a veryclose prediction on phase split by both distance and mean ^vector.

(52)

Table 3-3 presents results of phase split for crude oil

data* The components in each sample are combined as indicated

below Table 3-3. For crude oil data, comparisons are made

between totals of 2 (s » 2) and 3 (s » 3} numbers of samples

taken as a training set. By distance measurement, upper CPE

234 with d s 3 and s » 2 predicts phase split at sample vial

4, and with d » 3 and s » 3, also at sample vial 4. On the

other hand, mean vector predicts phase split at sample vial 3

with d = 3 and s = 2, and at sample vial 4 with d = 3 and s =

3. These results suggest that the number of samples used in

training set does not affect the prediction of phase split.

In the CPE experiment, the phase split is predicted by

viscosity measurements, but in the binary classifier analysis,

it is predicted directly by hydrocarbon composition. The

prediction based on viscosity is listed in the last column of

Tables 3-2 and 3-3. The results for two out of the four

synthetic oil experiments show that the binary classifier and

viscosity measurements give a very close prediction of the

phase split. Besides an approximation in experimental

analysis, a possible reason why this method does not work on

all compositional data is that the ratio of N to D for each

data is small. Therefore, the results are statistically

approaching the limits of reliability. All of the binary

classifier results for crude oil data, except for CPE 245,

correlate well with the determination of phase split using

(53)

viscosity measurement. As indicated in Table 3-3, the distancemeasurement for upper CPE 245 predicts the same phase split asby viscosity measurexftents.

AS shown in Figures 3-9 to 3-34, the points correspondingto the sample vial where phase split occurs for both upper andlower sample ports are indicated in the distance and meanvector plots. By determining the phase split, we can representthe phase behavior of each sample. For instance, in Figure 3-9, two clusters of samples which represent before and afterpLse split are labelled in this plot. Class 1representssamples before phase split and Class 2 after phase split.These plots give the same information as in the ternarydiagram.

Physically, samples in Class 1 are those that containhomogeneous or one phase mixtures which follow thecompositional path prior to phase split. After phase split,two phases coexist where the upper samples represent theequilibrium gas phase and lower samples represent theequilibrium liquid phase. Atie line can be drawn for samplesin Class 2 (two phase region) which connect samples from theupper and lower ports of the CPE apparatus.

(54)

Table 3-1i Compositional Data for Upper Phase CPE 207

MOLE%

Sample if COi Cs Cio Ci6 C30 C5+C10 C16+C30

1 0.00 14.00 54.00 19.00 13.00 68.00 32.00

2 0.00 21.84 47.27 18.42 12.46 69.11 30.88

3 6.64 20.62 44.05 17.17 11.52 64.67 28.69

4 32.83 13.98 32.29 12.57 8.33 46.27 20.90

5 56.53 8.49 21.20 8.24 5.54 29.69 13.78

6 73.01 5.46 13.04 5.08 3.41 18.5 8.49

7 75.53 4.24 12.27 4.77 3.19 16.51 7.96

8 81.36 3.20 9.32 3.65 2.45 12.52 6.10

9 84.19 3.00 7.79 3.02 2.00 10.79 5.02

10 94.76 1.38 2.72 0.81 0.33 4.10 1.14

11 95.21 1.24 2.53 0.74 0.28 3.77 1.02

m 12 95.89 1.10 2.14 0.63 0.23 3.24 0.86

13 94.95 0.93 2.93 0.87 0.31 3.86 1.18

14 96.92 0.74 1.67 0.50 0.16 2.41 0.66

15 96.60 0.68 1.94 0.59 0.18 2.62 0.77

(55)

Table 3-2i Result for Synthetic Oil Data

d=3, 5=3 d =4, s=3I d=:5, s=3 CPE'

CPE DATA (I) (II) (HI) PHASE

D S D S D S D S D S SPLIT

207 (upper) 5 5 5 5 5 5 5 5 5 5 8

207 (lower) 5 5 5 5 5 4 5 5 5 4 8

214 (upper) 5 4 5 4 5 4 5 4 5 4 6

214 Qower) 4 4 4 4 4 4 4 4 4 4 5

215 (upper) 5 5 5 5 5 5 5 5 5 4 5

215 (lower) 4 4 4 4 4 4 4 4 4 4 5

216 (upper) 5 5 5 5 5 5 5 5 5 4 9

216 (lower) 5 5 5 5 5 5 5 5 5 5 9

Note:

D: distance measurement from the center of gravity.S: mean vector.

d: number of dimension

s: number of samples in the training set.d=3, s=3: CO2, €5+Cio, C16+C30ds=4, s=3 (I): CO2, C5+C10, C16, C30

(II): CO2, C5, Cio, C16+C30(ni): CO2, C5, C10+C16, C30

d=5, s=3: CO2, C5, Cio, C16, C30# based on viscosity measurement byCPE experiment

(56)

Table 3-3. Result for Crude Oil Data

CPE DATA

234 (upper)*

234 (lower)*

247 (upper)

247 (lower)

238 (upper)

238 (lower)

246 (upper)

246 (lower)

239 (upper)

239 (lower)

244 (upper)

244 (lower)

d=:3,s = 2 d=3,s=3 d=s4,s=2 d=4,s=s3

245 (upper) 8

245 (lower) 4

CPE^PHASE

SPLIT

(57)

Continue Tsible 3-3

CPE DATAd=5 ,s=2 d = 5 ys=3 d = 6 ,s=2 d=6 ,s=3

CPE'

D S D S D S D SPHASE

SPLIT

239 (upper) 7 6 7 7 — - - - 7

239 (lower) 6 6 6 6 - — - - 7

244 (upper) 5 5 5 5 5 5 5 5 5

244 (lower) 5 5 5 5 5 5 5 5 5

245 (upper) 8 5 8 5 8 5 8 5 8

245 (lower) 4 4 5 4 4 4 5 4 8

Note: d=3, s=2&3: CO2, Ci—C12, C13-C37d = 4, s=2&3: CO2, C1-C12, C13-C25, C26-C37+d = 5, s= 2&3: CO2, Ci-Cp, C10-C19, C20-C29, C30-C37+d = 6, s= 2&3i C02t Ci—C7, Cg—Ci4, C15—C22» C23—C29, 030-^037+

d = 3, s=2&3: CO2, C4-C12, C13-C37+* d=4, s=2&3: CO2, C4-C12, Ci3-C24» C2S-C37+*based on viscosity measurement by CPE experiment

•'m-

(58)

DI8TAKCE PLOTS AMD TABLES

MEAN VECTOR PLOTS AKD TABLES

3

13

&

12

0

11

0

90

CL

AS

S2

so

20

Figu

re3-

9.D

ista

nce

plot

forC

PE21

5(d

=3

)

CL

AS

S! L

O

PH

AS

ES

PL

IT

'S

'iS

'4b

'gb

'dD

tBT-

'ab

CL

AS

S1

UP

PE

R

CL

AS

S2

13

0h

CL

AS

S2

Figu

re3-

10.D

ista

nce

plot

for

CPE

215

(d

=4

)

CL

AS

S1

LO

WE

R

PH

AS

ES

PU

T

'db

'^

tB-

CL

AS

S1

UP

PE

R

<n

o

Figu

re3-

11.D

ista

nce

plot

for

CPE

215

(d

=5

)

12

t^

CL

AS

S1

PH

AS

ES

PL

IT

CL

AS

S2

o\

UP

PE

R

LO

WE

R

tio

CL

AS

S1

Figu

re3-

12.M

ean

Vfe

ctorp

lotf

orC

PE21

5(d

=3

)

CL

AS

S2

LO

WE

R

UP

PE

R

PH

AS

ES

PL

IT

CL

AS

S2

0.4

CL

AS

S1

02

oS"

o5

CL

AS

S1

0.8

0

0.6

0

CL

AS

S2

0.4

0

02

a

Figu

re3-

13.M

ean

Vecto

rplo

tfor

CPE

215

(d=

4)

CL

AS

S2

PH

AS

ES

PL

IT

CL

AS

S1

CL

AS

S1

LO

WE

R

o%

u>

Figu

re3-

14.

Mea

nV

ecto

rpl

otfo

rCPE

215

(d=

5)

0^

CL

AS

S2

LO

WE

R

o.G

a

PH

AS

ES

PL

ITU

PP

ER

CL

AS

S2

a\

0.4

0

CL

AS

S1

02

0

"o!4

CL

AS

S1

(65)

Table 3-4. Distance Measurement for CPE 215 (d » 3)

UPPER SAMPLE LOWER SAMPLE

UPLE # CLASS 1 CLASS 2 Y CLASS 1 CLASS 2 Y

1 5.835 123.916 1 6.697 83.834 1

2 5.670 123.770 1 6.128 83.188 1

3 11.484 106.624 1 12.499 64.757 1

4 43.334 74.774 1 46.772 30.566 2

5 61.989 56.122 2 65.876 11.629 2

6 95.642 22.514 2 77.119 2.025 2

7 117.579 0.543 2 79.380 3.469 2

8 117.785 0.337 2 80.913 4.137 2

9 117.911 0.203 2 77.051 1.417 2

10 117.986 0.134 2 77.103 1.332 2

11 117.862 0.260 2 77.394 0.972 2

12 117.986 0.125 2 77.451 0.727 2

13 117.932 0.182 2 77.464 0.445 2

14 118.152 0.062 2 77.654 0.428 2

15 118.224 0.124 2 76.639 0.816 2

Table 3-5. Distance Measurement for CPE 215 (d = 4)


SAMPLE # CLASS 1 CLASS 2 Y CLASS 1 CLASS 2 Y

1 5.825 121.910 1 6.729 82.784 1

2 5.554 121.674 1 5.833 81.849 1

3 11.345 104.792 1 12.310 63.856 1

4 42.667 73.462 1 45.926 30.280 2

5 61.027 55.103 2 64.721 11.606 2

6 93.900 22.262 2 75.830 1.717 2

7 115.595 0.543 2 77.893 2.925 2

8 115.801 0.337 2 79.574 3.776 2

9 115.929 0.203 2 75.819 1.260 2

10 116.002 0.133 2 75.884 1.136 2

11 115.878 0.260 2 76.205 0.802 2

12 116.005 0.125 2 76.286 0.597 2

13 115.950 0.182 2 76.328 0.366 2

14 116.177 0.062 2 76.539 0.401 2

15 116.246 0.123 2 75.602 0.711 2

(66)

Table 3-6. Distance Measurement for CPE 215 (d « 5)UPPER SAMPLE LOWER SAMPLE

SAMPLE #

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

CLASS 1 CLASS 2 Y CLASS 1 CLASS 2

5.5115.612

11.06940.63258.063

90.551110.004

110.171110.269110.342110.229110.316110.274110.443110.470

115.864115.935

99.400

69.77452.345

20.3190.532

0.3470.239

0.220

0.270

0.136

0.184

0.056

0.164

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

6.2696.038

12.10343.88861.79372.00974.59475.51871.76871.79071.940

71.92171.87871.96370.895

77.73177.432

59.620

27.84210.270

2.2444.872

4.477

1.538

1.4251.019

0.7490.520

0.399

0.849

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

Table 3-7. Mean Vector Measurement for CPE 215 (d 3)UPPER SAMPLE LOWER SAMPLE

SAMPLE # CLASS 1 CLASS 2 Y CLASS 1 CLASS 2

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0.998

0.988

0.989

0.798

0.588

0.238

0.075

0.074

0.073

0.0720.073

0.072

0.073

0.071

0.071

0.008

0.008

0.2150.654

0.849

0.986

1.000

1.000

1.0001.0001.0001.000

1.000

1.0001.000

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

0.9970.9970.9870.7580.540

0.412

0.390

0.372

0.412

0.4110.4070.406

0.406

0.403

0.413

0.341

0.343

0.5490.904

0.988

1.000

0.999

0.999

1.0001.0001.0001.000

1.000

1.000

1.000

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

(67)

Table 3-8. Mean Vector Measurement for CPE 215 (d


4)


1 0.998 0.008 1 0.997 0.321 1

2 0.998 0.008 1 0.997 0.321 1

3 0.988 0.225 1 0.986 0.540 1

4 0.785 0.677 1 0.745 0.905 2

5 0.572 0.861 2 0.525 0.988 2

6 0.237 0.986 2 0.400 1.000 2

7 0.079 1.000 2 0.380 1.000 2

8 0.077 1.000 2 0.361 0.999 2

9 0.076 1.000 2 0.398 1.000 2

10 0.076 1.000 2 0.397 1.000 2

11 0.077 1.000 2 0.393 1.000 2

12 0.076 1.000 2 0.391 1.000 2

13 0.076 1.000 2 0.389 1.000 2

14 0.075 1.000 2 0.387 1.000 2

15 0.074 1.000 2 0.395 1.000 2

Table 3-9. Mean Vector Measurement for CPE 215 (d


= 5)


1 0.997 0.004 1 0.996 0.290 1

2 0.997 0.003 1 0.996 0.291 1

3 0.983 0.262 1 0.979 0.552 1

4 0.733 0.738 2 0.690 0.928 2

5 0.517 0.896 2 0.472 0.993 2

6 0.194 0.993 2 0.366 0.999 2

7 0.084 1.000 2 0.336 0.998 2

8 0.083 1.000 2 0.332 0.999 2

9 0.083 1.000 2 0.370 1.000 2

10 0.083 1.000 2 0.369 1.000 2

11 0.083 1.000 2 0.369 1.000 2

12 0.083 1.000 2 0.369 1.000 2

13 0.083 1.000 2 0.370 1.000 2

14 0.083 1.000 2 0.370 1.000 2

15 0.083 1.000 2 0.380 1.000 2

10

0-

CL

AS

S1

CL

AS

S2

60

-P

HA

SE

SP

UT

Figu

re3-

15.D

istan

cepl

otfo

rCPE

207

(d=

5)

UP

PE

R

LO

WE

R

CL

AS

S1

CL

AS

S2

a\

09

>3

Figu

re3-

16.M

ean

Vec

torp

lot

for

CPE

207

((J

=5

)

1.2

0

LO

WE

R

CL

AS

S2

PH

AS

ES

PL

IT0

.80

UP

PE

R

CL

AS

S2

0.6

0a\

0.4

0

CL

AS

S1

0:2

o!8

CL

AS

S1

(70)

Table 3-10. Distance Measurement for CPE 207 (d « 5)



1 7.755 112.277 1 4.771 93.356 1

2 3.931 110.400 1 3.598 93.474 1

3 6.619 102.764 1 7.664 84.026 1

4 35.627 72.793 1 40.858 49.541 1

5 62.803 45.578 2 57.774 32.558 2

6 81.825 26.559 2 74.595 15.854 2

7 84.616 23.770 2 83.555 7.217 2

8 91.366 17.021 2 88.112 3.576 2

9 94.656 13.725 2 92.888 4.061 2

10 106.797 1.623 2 93.580 4.173 2

11 107.308 1.113 2 93.125 3.468 2

12 108.111 0.417 2 92.709 2.889 2

13 106.939 1.448 2 90.861 0.90^ 2

14 109.293 0.933 2 90.231 0.021 2

15 108.893 0.520 2 89.570 0.907 2

Table 3-11. Mean Vector Measurement for CPE 207 (d = 5)



1 0.993 0.025 1 0.997 0.142 1

2 0.998 0.025 1 0.999 0.140 1

3 0.994 0.149 1 0.993 0.267 1

4 0.784 0.670 1 0.730 0.802 2

5 0.439 0.925 2 0.512 0.936 2

6 0.245 0.983 2 0.323 0.989 2

7 0.223 0.987 2 0.240 0.998 2

8 0.170 0.994 2 0.206 0.999 2

9 0.146 0.997 2 0.167 0.999 2

10 0.072 1.000 2 0.162 0.999 2

11 0.069 1.000 2 0.164 1.000 2

12 0.065 1.000 2 0.165 1.000 2

13 0.073 1.000 2 0.177 1.000 2

14 0.059 1.000 2 0.181 1.000 2

15 0.061 1.000 2 0.184 1.000 2

CL

AS

S2

60

Figu

re3-

17.

Dis

tanc

epl

otfo

rC

PE21

4(d

=5

)

CL

AS

S1

UP

PE

R

LO

WE

R

PH

AS

ES

PL

IT

CL

AS

S1

CL

AS

S2

lio

*

Figu

re3-

18.M

ean

Vec

tor

plot

for

CPE

214

(d=

5)

LO

WE

R

CL

AS

S2

0.8

0

0.6

0

PH

AS

ES

PL

ITU

PP

ER

CL

AS

S2

lo

0.4

&

CL

AS

S1

0.2

&

02

"oU

o!6

CL

AS

S1

(73)

Table 3-12. Distance Measurement for CPE 214 (d • 5)

UPPER DATA LOWER DATA


1 7.476 115.106 1 11.900 83.857 1

2 6.776 114.541 1 11.738 83.629 1

3 14.103 93.776 1 23.541 48.503 1

4 47.440 60.367 1 41.173 30.915 2

5 62.808 44.995 2 53.477 18.653 2

6 70.643 37.160 2 58.495 13.623 2

7 92.924 14.888 2 67.497 4.978 2

8 107.301 0.847 2 69.810 3.010 2

9 107.132 0.790 2 73.242 2.420 2

10 107.292 0.639 2 71.491 1.767 2

11 107.458 0.463 2 71.681 1.579 2

12 107.352 0.509 2 72.754 1.637 2

13 107.459 0.384 2 72.077 1.080 2

14 107.676 0.269 2 73.363 1.698 2

15 107.699 0.158 2 71.846 0.511 2

16 107.830 0.031 2 70.884 1.110 2

17 107.870 0.144 2 73.222 1.317 2


UPPER SAMPLE LOWER SAMPIjE


1 0.994 0.008 1 0.983 0.228 1

2 0.994 0.008 1 0.983 0.229 1

3 0.970 0.351 1 0.904 0.754 1

4 0.644 0.833 2 0.722 0.923 2

5 0.463 0.933 2 0.584 0.977 2

6 0.384 0.961 2 0.531 0.989 2

7 0.199 0.996 2 0.443 0.999 2

8 0.114 1.000 2 0.421 0.999 2

9 0.115 1.000 2 0.391 1.000 2

10 0.114 1.000 2 0.407 1.000 2

11 0.114 1.000 2 0.405 1.000 2

12 0.115 1.000 2 0.396 1.000 2

13 0.114 1.000 2 0.A02 1.000 2

14 0.113 1.000 2 0.391 1.000 2

15 0.113 1.000 2 0.404 1.000 2

16 0.113 1.000 2 0.412 1.000 2

17 0.113 1.000 2 0.390 1.000 2

Figu

re3-

19.D

ista

nce

plot

for

CPE

216

(d

=5

)

CL

AS

S1

CL

AS

S2

UP

PE

R

LO

WE

R

PH

AS

ES

PU

T

6'

lb'

2to

'3

b'

JO'

ebA

'7b

'

CL

AS

S1

3

Figu

re3-

20.M

ean

Vec

torp

lotf

orC

PE21

6(d

=5

)

0.8

0

LO

WE

R

0.6

0U

PP

ER

PH

AS

ES

PL

IT

CL

AS

S2

0.4

0

02

0C

LA

SS

1

ole

'

CL

AS

S1

(76)




1 3.570 112.378 1 3.329 103.810 1

2 3.346 112.119 1 9.356 104.784 1

3 6.425 102.787 1 10.904 95.163 1

4 42.171 67.014 1 39.262 61.788 1

5 59.214 49.956 2 63.976 37.172 2

6 71.214 37.974 2 74.802 26.383 2

7 79.672 29.527 2 79.948 21.193 2

8 86.215 23.002 2 82.228 19.084 2

9 87.452 21.700 2 90.532 10.733 2

10 94.514 14.664 2 91.620 9.599 2

11 95.619 13.506 2 94.396 6.972 2

12 105.100 3.997 2 93.943 7.074 2

13 106.902 2.217 2 92.812 8.231 2

14 108.134 0.977 2 91.738 9.382 2

15 108.528 0.560 2 90.795 10.^54 2

16 109.138 0.055 2 105.683 4.745 2

17 109.592 0.509 2 106.521 5.735 2

Table 3-15 Mean Vector Measurement for CPE 216 (d = 5)



1 0.999 0.022 1 0.999 0.069 1

2 0.999 0.022 1 0.990 0.069 1

3 0.995 0.162 1 0.984 0.179 1

4 0.714 0.745 2 0.751 0.733 2

5 0.495 0.899 2 0.433 0.940 2

6 0.356 0.956 2 0.313 0.976 2

7 0.272 0.987 2 0.265 0.986 2

8 0.215 0.988 2 0.225 0.992 2

9 0.206 0.990 2 0.175 0.997 2

10 0.153 0.996 2 0.168 0.998 2

11 0.146 0.997 2 0.146 0.999 2

12 0.089 1.000 2 0.150 0.999 2

13 0.078 1.000 2 0.156 0.999 2

14 0.071 1.000 2 0.162 0.998 2

15 0.069 1.000 2 0.168 0.998 2

16 0.066 1.000 2 0.076 1.000 2

17 0.064 1.000 2 0.075 1.000 2

3}

>

Figu

re3-

21.D

ista

nce

plot

forC

PE23

4(d

=4

)

CL

AS

S1

CL

AS

S2

UP

PE

RP

HA

SE

SP

LIT

LO

WE

R

CL

AS

S2

6'

lb'

db'

db'

»iiS

oiio

i5)

CL

AS

S1

Figu

re3-

22.M

ean

Vecto

rplo

tfor

CPE

234

(d=

4)

LO

WE

R

CL

AS

S2

o.e

&

UP

PE

R

PH

AS

ES

PL

IT

0.6

0

CL

AS

S2

09

0.4

a

CL

AS

S1

02

0

02

0:4

CL

AS

S1

(79)

Table 3-16. Distance Measurement for CPE 234 (d => 4)


SAMPLE # CLASS 1 CLASS 2 y CLASS 1 CLASS 2 y

1 30.624 109.988 1 32.408 70.784 1

2 4,785 77.786 1 1.751 37.291 1

3 28.142 52.631 1 30.797 10.230 2

4 41.956 38.124 2 44.025 9.010 2

5 70.876 9.066 2 50.757 14.158 2

6 81.878 2.355 2 54.184 15.955 2

7 79.447 0.660 2 50.798 13.438 2

8 80.370 0.796 2 49.995 12.619 2

9 81.252 1.592 2 49.030 11.554 2

10 84.459 4.833 2 48.732 10.889 2

11 81.643 1.882 2 44.230 6.050 2

12 77.807 2.279 2 39.744 1.122 2

13 80.610 0.750 2 40.229 1.608 2

14 81.373 1.531 2 36.287 2.660 2



SAMPLE # CLASS 1 CLASS 2 y CLASS 1 CLASS 2 y

1 0.877 0.059 1 0.860 0.385 1

2 0.996 0.555 1 0.999 0.814 1

3 0.885 0.852 1 0.867 0.988 2

4 0.774 0.943 2 0.770 0.993 2

5 0.573 0.998 2 0.720 0.987 2

6 0.508 1.000 2 0.682 0.983 2

7 0.524 1.000 2 0.713 0.988 2

8 0.518 1.000 2 0.718 0.989 29 0.513 1.000 2 0.723 0.990 2

10 0.494 0.999 2 0.731 0.993 211 0.511 1.000 2 0.761 0.998 212 0.536 1.000 2 0.790 1.000 2

13 0.518 1.000 2 0.788 1.000 2

14 0.513 1.000 2 0.817 0.999 2

Figu

re3-

23.D

ista

nce

plot

forC

PE24

7(d

=4

)

CL

AS

S1

CL

AS

S2

PH

AS

ES

PL

ITU

PP

ER

LO

WE

CL

AS

S2

CL

AS

S1

00

o

0.8

0

0.G

0

CL

AS

S2

0.4

0

02

0

Figu

re3-

24.M

ean

Vec

torp

lotf

orC

PE24

7(d

=4

)

CL

AS

S2

o!4

'o!

6

CL

AS

S1

PH

AS

ES

PL

IT

LO

WE

R

UP

PE

R

(82)




1 8.742 112.319 1 5.161 89.746 12 6.942 110.496 1 5.775 90.514 13 15.606 88.000 1 10.878 74.072 14 55.443 48.945 2 36.629 48.711 15 80.624 24.429 2 63.089 22.786 26 90.439 13.299 2 71.098 14.388 27 93.834 9.844 2 77.919 8.493 28 97.296 6.341 2 79.999 5.718 29 99.080 4.535 2 82.725 2.697 2

10 101.857 1.764 2 85.736 1.694 211 103.668 0.113 2 81.137 4.215 212 105.284 1.690 2 87.467 2.867 2




1 0.996 0.139 1 0.998 0.330 12 0.998 0.162 1 0.999 0.331 13 0.982 0.411 1 0.993 0.488 14 0.716 0.840 2 0.893 0.753 15 0.454 0.967 2 0.650 0.947 26 0.350 0.992 2 0.555 0.979 27 0.316 0.996 2 0.477 0.993 28 0.284 0.998 2 0.446 0.997 29 0.268 0.999 2 0.410 0.999 2

10 0.245 1.000 2 0.374 1.000 211 0.229 1.000 2 0.416 0.999 212 0.217 1.000 2 0.353 1.000 2

-3

J'-

--

r

Figu

re3-

25.D

ista

nce

plot

for

CPE

238

(d=

4)

CL

AS

S!

PH

AS

ES

PL

IT

CL

AS

S2

UP

PE

R

CL

AS

S2

LO

WE

R

A

CL

AS

S1

lio

09

U>

Figu

re3-

26.M

ean

Vecto

rplo

tfor

CPE

238

(d=4

)

CL

AS

S2

LO

WE

R

PH

AS

ES

PU

T

UP

PE

CL

AS

S2

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(85)

Table 3-20* Distance Measurement for CPE 238 (d • 4)



1 6.400 109.513 1 6.637 84.932 12 7.062 110.175 1 6.673 85.030 1

3 13.434 89.696 1 13.268 65.774 14 51.802 51.337 2 32.188 47.036 15 72.250 30.877 2 48.461 30.784 26 80.312 22.933 2 62.891 17.256 27 90.804 12.386 2 65.009 14.665 2

8 90.412 12.800 2 68.808 10.572 29 93.461 9.732 2 67.525 11.297 2

10 96.545 6.624 2 69.952 9.149 211 98.691 4.469 2 69.445 9.101 2

12 101.676 1.462 2 74.740 4.137 213 103.037 0.098 2 73.587 5.558 214 104.669 1.555 2 87.438 9.686 2

Table 3*21. Mean Vector Measurement for CPE 238 (d » 4)



1 0.998 0.168 1 0.997 0.388 12 0.997 0.157 1 0.997 0.339 13 0.986 0.396 1 0.985 0.559 14 0.730 0.836 2 0.898 0.765 15 0.514 0.955 2 0.758 0.902 26 0.441 0.976 2 0.606 0.971 2

7 0.341 0.994 . 2 0.578 0.979 28 0.345 0.993 2 0.531 0.989 29 0.317 0.996 2 0.541 0.988 2

10 0.290 0.998 2 0.510 0.993 211 0.272 0.999 2 0.511 0.993 212 0.247 1.000 2 0.446 0.999 213 0.236 1.000 2 0.457 0.998 214 0.223 1.000 2 0.335 0.996 2

CL

AS

S2

tso

11

0

10

0

90

80

70

GO

SO

40

30

20

10

"iC

Figu

re3-

27.

Dis

tanc

epl

otfo

rC

PE24

6(i

]-4

)

CL

AS

S1

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PE

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R

•sb

'd>

'A

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AS

S1

PH

AS

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PL

IT

00

'ilo

i2a

Figu

re3-

28.M

ean

\fect

orpl

otfo

rCPE

246

(d=:

4)

CL

AS

S2

0.8

0L

OW

ER

PH

AS

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PL

IT

OjG

OU

PP

ER

CL

AS

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09

a4

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02

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(88)

Table 3-22, Distance Measurement for CPE 246 (d = 4)



1 28.413 120.180 1 11.554 90.304 12 20.668 111.632 1 10.527 89.405 13 48.518 44.133 2 21.970 56.992 14 70.241 22.086 2 45.189 33.805 25 77.857 14.309 2 54.956 23.982 26 85.064 6.907 2 68.453 10.813 27 88.059 3.841 2 72.212 7.521 28 89.439 2.430 2 76.304 3.622 29 85.297 6.523 2 75.701 3.515 2

10 88.608 3.181 2 75.593 3.340 211 89.586 2.188 2 76.722 2.181 212 92.397 0.631 2 80.592 1.752 213 93.322 1.560 2 79.349 0.765 2

Table 3-23. Mean Vector Measurement for CPE 246 (d » 4)


lMPLE # CLASS 1 CLASS 2 Y CLASS 1 CLASS 2 Y

1 0.934 0.048 1 0.990 0.258 12 0.954 0.118 1 0.993 0.276 13 0.751 0.902 2 0.948 0.662 14 0.567 0.981 2 0.762 0.893 25 0.506 0.993 2 0.655 0.951 26 0.450 0.998 2 0.509 0.991 27 0.428 1.000 2 0.474 0.995 28 0.418 1.000 2 0.426 0.999 29 0.445 0.999 2 0.428 0.999 2

10 0.422 1.000 2 0.423 0.999 211 0.415 1.000 2 0.413 1.000 212 0.397 1.000 2 0.376 1.000 213 0.391 1.000 2 0.384 1.000 2

Figure

3-29.

Dista

ncep

lotfor

CPE2

39(tJ

=5

CL

AS

S1

CL

AS

S2

UP

PE

RPH

ASE

SPL

IT

LO

WE

R

da

CL

AS

S1

CL

AS

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00

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0.8

0

0.6

0

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AS

S2

0.4

0

0.2

0

Figu

re3-

30.

Mea

nV

ecto

rplo

tfo

rCPE

239

(d=

5)

CL

AS

S2

UP

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AS

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IT

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S1

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AS

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R

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o

(91)

Table 3-24. Distance Measurement for CPE 239 (d =


Y CLASS 1 CLASS 2

5)

SAMPLE # CLASS 1 CLASS 2

1 3.688 94.5752 3.421 93.9233 6.929 84.2634 19.568 71.8575 37.361 54.3876 44.009 47.2167 52.337 39.0258 64.518 27.2439 70.903 20.755

10 76.555 15.25111 80.650 10.74712 83.610 7.43613 87.725 3.26814 91.294 0.40615 93.710 2.865

2

2

2

2

2

2

2

2

2

3.942

3.931

7.775

20.599

29.687

44.301

48.370

51.14055.900

58.632

60.069

62.506

64.588

65.699

65.585

69.03768.646

58.288

48.002

38.905

25.409

19.11318.00211.7889.957

7.072

3.081

1.753

0.702

1.190

2

2

2

2

2

2

2

2

2

2

Table 3-25. Mean Vector Measurement for CPE 239 (d = 5)UPPER SAMPLE

SAMPLE #

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

CLASS 1 CLASS 2

0.999 0.2690.999 0.2680.996 0.3880.966 0.5380.866 0.7340.810 0.8050.730 0.8730.603 0.9420.532 0.9680.471 0.9830.422 0.9920.385 0.9970.341 0.9990.303 1.0000.279 1.000

2

2

2

2

2

2

2

2

2

LOWER SAMPLE

CLASS 1 CLASS 2 Y

0.999 0.496 10.999 0.496 10.995 0.612 10.959 0.731 10.913 0.815 10.799 0.920 20.755 0.952 20.727 0.959 20.668 0.982 20.636 0.988 20.613 0.994 20.574 0.999 20.548 1.000 20.528 1.000 20.528 1.000 2

Figu

re3-

31.D

ista

nce

plot

forC

PE24

4(

d=

6)

CL

AS

S1

CL

AS

S2

UP

PE

R

LO

W

PH

AS

ES

PL

IT

CL

AS

S1

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4^

'1<

0

u>

to

}

Figure

3-32.

Mean

Vecto

rplot

forCP

E24

4(d=

6

0.8

0C

LA

SS

2L

OW

ER

PH

AS

ES

0.6

0U

PP

ER

CL

AS

S2

0.4

0

0.2

0

CL

AS

S1

(94)


SAMPLE #

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

UPPER SAMPLECLASS 1 CLASS 2

4.4162.222

5.76526.143

62.65271.90390.40584.49794.55897.48899.696

101.591103.986104.408105.525106.202107.295107.756108.206108.654

111.716110.216102.73682.06846.503

36.35618.787

24.53814.15311.0308.672

6.788

4.273

3.856

2.7082.032

0.919

0.470

0.206

0.503

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

LOWER SAMPLECLASS 1 CLASS 2

7.230

5.564

12.776

34.328

52.125

64.538

62.593

63.355

70.342

73.02064.546

72.872

76.277

69.380

84.768

84.041

80.922

81.773

85.228

75.592

87.88586.27168.286

47.30335.04127.457

20.327

18.50914.00112.27316.3868.737

4.902

13.5187.6895.361

1.749

1.848

5.780

7.421

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2


SAMPLE #

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

UPPER SAMPLECLASS 1 CLASS 2

0.999

0.999

0.9970.911

0.5310.3890.2450.3020.203

0.175

0.1550.1410.1210.1180.109

0.1040.0960.0940.0890.088

0.056

0.059

0.162

0.4940.888

0.9520.9860.9750.993

0.996

0.9980.9991.0001.0001.000

1.0001.000

1.000

1.000

1.000

2

2

2

2

LOWER SAMPLECLASS 1 CLASS 2

0.995

0.997

0.981

0.841

0.671

0.546

0.520

0.502

0.434

0.402

0.470

0.378

0.337

0.379

0.275

0.2700.285

0.275

0.254

0.305

0.183

0.205

0.458

0.747

0.871

0.926

0.968

0.980

0.985

0.985

0.978

0.994

0.998

0.989

0.998

0.999

1.000

1.000

0.999

0.998

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

Figu

re3-

33.D

istan

cepl

otfo

rCPE

245

(d=

6)

CL

AS

S!

CL

AS

S

PH

AS

ES

PL

IT

UP

PE

R

LO

WE

CL

AS

S1

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AS

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0.8

0

0.6

0

CL

AS

S2

0.4

0

0.2

0

I0

2

figu

re3-

34.M

ean

Vec

tor

plot

for

CPE

245

(d=

6)

CL

AS

S2

PH

AS

ES

PL

Il

0:4

•0:

6

CL

AS

S1

UP

PE

R

CL

AS

S1

LO

WE

R

(97)


SAMPLE #

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

UPPER SAMPLE LOWER SAMPLECLASS 1 CLASS 2 Y CLASS 1 CLASS 2 y

7.426 104.849 1 6.956 65.258 11.397 97.398 1 3.397 62.628 16.349 92.224 1 9.822 50.113 1

24.033 74.123 1 29.546 30.530 139.292 58.859 1 57.435 27.101 242.817 55.315 1 39.350 21.451 244.927 53.350 1 51.512 11.478 290.023 8.785 2 54.913 8.926 292.578 6.157 2 74.671 20.589 293.256 5.299 2 64.171 9.068 293.985 4.429 2 64.890 8.781 294.777 3.612 2 66.136 9.492 296.085 2.465 2 63.718 6.996 296.073 2.266 2 65.163 7.991 296.276 2.005 2 65.450 7.840 296.966 1.356 2 63.106 4.879 297.318 0.860 2 55.111 4.607 297.453 0.704 2 56.615 4.657 298.432 0.351 2 60.452 2.208 298.502 0.429 2 60.902 2.509 2


SAMPLE #

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

UPPER SAMPLE LOWER SAMPLECLASS 1 CLASS 2 Y CLASS 1 CLASS 2 Y

0.988 0.440 1 0.988 0.440 10.997 0.491 1 0.997 0.491 10.976 0.711 1 0.976 0.711 10.823 0.922 2 0.823 0.922 20.540 0.924 2 0.540 0.924 20.734 0.965 2 0.734 0.965 20.632 0.988 2 0.632 0.988 20.603 0.992 2 0.603 0.992 20.450 0.977 2 0.450 0.977 20.527 0.994 2 0.527 0.994 20.521 0.995 2 0.521 0.995 20.512 0.995 2 0.512 0.995 20.527 0.997 2 0.527 0.997 20.517 0.996 2 0.517 0.966 20.513 0.997 2 0.513 0.997 20.525 0.999 2 0.525 0.999 20.576 0.999 2 0.576 0.999 20.592 0.998 2 0.592 0.998 20.536 1.000 2 0.536 1.000 20.533 0.999 2 0.533 0.999 2

(98)

nCIOR MOOXSZS

Factor analysis is a statistical technique used to

identify a relatively small number of factors that can be used

to represent relationships among sets of many interrelated

variables. It is also defined as a generic term that describes

a variety of mathematical procedures applicable to the

analysis of data matrices.^ Mathematically, a factor refers

to one of a number of things that when multiplied together

yield a product. Factor analysis was initially devised by

psychologists in the early 1900's. Since then there have Ipeen

many advances in the technique and applications have spread

beyond psychology into other areas of study. In the late

1950*s the method was first applied to geologic problems.

Geologists are commonly faced with problems where a large

number of properties are measured or described on a large

number of things.* The "things" may, for example, be rocks and

the "properties** may be the €Utto\ints of various minerals making

up the roclcs. If these data are arranged in tabular form such

that each rock represents a row of the teUsle and each mineral

species a column, then the resulting chart of ntimbers is

referred to as a data matrix. This is an analogy to the crude

oil compositional data where a row represents the sample

number and a column represents the mole percent of each carbon

component in the crude oil. In general, a matrix is a tedale of

(99)

numbers with so many rows and so many columns. The rows of a

data matrix represent "things" or sample number called sample

and the columns represent property called variable. Figure 3-

35 shows a schematic diagram of a data matrix. In the

terminology of matrix algebra, an entire matrix is symbolized

by a capital letter. "X" is used to symbolize any data matrix

in this study. The size of the matrix is specified by a double

subscript notation, thus X,,refers to a table of numbers with

N rows and m columns. For instance, if 20 samples have been

analyzed for 38 carbons, the resulting data matrix is

symbolized as Xjo^jg. Analysis of such a data matrix may pose a

considerable problem if it contains many numbers. Therefore

the objective of factor analysis is to simplify by determining

if relationships among a set of variables may be reduced to a

smaller number of underlying fundamental variables or

"factors**. Two common methods of factor analysis are principal

component and maximum likelihood analyses. In this study, only

principal component analysis is performed on the compositional

data.

In dealing with a data matrix, there are two distinct yet

interrelated factor analytic modes; R-mode and Q-mode. If the

primary purpose of the investigation is to understand the

inter-relationships among the variables, then the analysis is

said to be an R^mode problem. On the other hand, if the

primary purpose is to determine interrelationships among the

'•vT

(100)

Variable 1 Variable! Variables ... Itoiable m

Sample 1 Xu Xu Xu • • • Xl^m

Sample 2 X2.1 X22 X23 • • • X2jn

Sample 3

•

X3.1

•

X33

•

X«

•

X3,ni

• •

•

•

Sample N

•

«

Xn4

•

•

•

•

Xn4

• •

• •

• • •

Figure 3-35. A scheoiatic diagram of a data matrix

AV' ..

• >

(101)

samples, then the analysis is refexxed to as o-mode. In many

cases both R* and Q-mode analyses are performed on the same

data matrix. In this paper, only Q-mode factor analysis is

discussed since the purpose of this study is to separate

samples instead of variables in order to observe the phase

split.

3-3-1 Q-MODB FACTOR AHALYSZS

The objective of the Q-mode analysis is to find groups of

samples that are similar to one another in terms of their

total composition. The raw data cannot be analyzed directly by

Q-mode factor analysis, instead it utilizes a similarity

matrix in order to solve the problem. Once the similarity

matrix is computed, the Q-mode analysis method of solution can

be performed on this similarity matrix.

There are three sdLmilarity indices commonly used in Q-

mode analysis:

1) cgrrglatign

The Pearson product moment correlation coefficient has

been used to indicate the degree of relationship between two

samples. If X, auu Xj are any two rows of the data matrix then.

(102)

m

a " — (1)jn 0

measures the degree of relationship between the two samples,

where,

3^ and xj " mean values of sample X, and Xj*ik *jk • data matrix for row i and row j with

corresponding k varisO^le

m - number of variable

The correlation coefficient may be visualized as the degree to

which observations of two samples approach a straight line

when plotted as points on a X-Y diagram (refer to Figure 3-

36) • The correlation coefficient may vary from -l to +1.

Either -1 or +1 indicates that the points form a straight line

(Figure 3-36A), whereas zero indicates complete dissociation

or no correlation (Figure 3-36C). Values between zero and +l

"1 indicate the degree of correlation between two samples

(Figure 3-36B). The origin of these graphs may be shifted

without changing the configuration of the points. If the mean

value of a sample is subtracted from every value of the sample

as in equation (1), the results are called a deviate seorg*.

The resulting numbers show how far from the mean each variable

is. This also results in shifting the origin of the sample to

the mean value. If we define X| and Xj as two samples in

deviate form, then the correlation formula becomes

(103)

m

E XiiXjk- I *•' — (2)

yti""-'where

*fie *jk • <iata matrix of row i and row j with

corresponding k variable

n a number of variable

Atable of correlation coefficients forms a matrix consistingof coefficients in rows and columns in which the number of

rows (or columns) is equal to the number of samples, and the

matrix is symmetrical about the diagonal containing ones.Figure 3-37 shows an example of a correlation matrix for N

number of samples.

2) Distance Coefficlanf.

The Distance coefficient method forms an alternative wayof expressing the degree of similarity between samples.^" Theyare based on the distance between two samples in a rectangular

coordinate system in m-dimensional space. The closer the two

points representing two samples are together, the greater the

similarity, and vice versa. Generally, distance coefficients

should be calculated from data in which the values are either

all negative, and in which the absolute values

do not exceed 1.0. Transformation of raw data are generally

necessary to satisfy these conditions. A simple linear

(104)

A. Perfect linear relofionshipr s 1.0

B. Correlationr»-0.9

• •

• X

C. Complete dissociationr«0.0

Figure 3-36. Degree of correlation between two samplesX and Y

R =

(105)

1 r12

121

. rij\r

. r2N

Figure 3-37. An example of correlation matrix for Nsamples

(106)

transfonnation is to divide each sample by the highestobserved absolute value for that sample. Distance coefficientsbetween two samples i and j nay be defined by the formula:

m

dy - 1.0 - ^ ,3)

Where,

dfj = distance coefficient between any two rows i

and j

*ik and Xjit = any two rows i and j with corresponding kvariable

N » niunber of sample

n » number of variable

Maximum distance yields a coefficient of zero, minimumdistance yields a coefficient of i and negative values areavoided.

3) Coefficient of Proporfc^ftpai

imbrie and Purdy (1962) define an index of proportionalsimilarity which indicate that the degree of similaritybetween two samples may be evaluated in relation to theproportions of their constituents.» For any two samples, i andj (which are row vectors of the data matrix), the coefficientof proportional similarity, gggAng thgtq, is determined from:

(107)

m

•IJ -coaQ^i = (4)

Where,

*ik *jk ~ rows i and j with corresponding

k variable

m B number of variable

This equation computes the cosine of the angle between the two

row vectors in m-dimensional space. The value of cos 0 ranges

from 0 to 1; registering zero for two samples having a

complete dissimilarity (vectors at 90° ) and unity for two

samples having perfect similarity or identical proportions

(collinear vectors) assuming that all values in the data

matrix are positive.

Comparison between equation (2) and (4) shows that both

of these equations are exactly equivalent. Thus the

correlation coefficient between any two samples is also the

cosine of the angle between the two vectors representing the

samples in m-dimensional space. Figure 3-38 shows the relation

between cosine of angle and correlation coefficient between

two samples. A geometrical interpretation of cos0 for two

variables is given in Figure 3-39. Note that the sample 1 and

3 contain variables in the proportion 2:1 with cos e„ = i.o

m

2

i

r •0.00

(108)

r« 0.707 r«l.00r«-0,50

Figure 3-38. Cosine of angle equals correlation ooefncientbetween sample l and 2.

. J 1 Lt 2 3 4

VARIABLE 1

Figure 3-39. The cosine between two sample vectors

determined by the proportions of the variables

(109)

indicating the collinearity or absolute similarity.

In this study, only the coefficient of proportional

similarity is used as a method of forming a similarity matrix.

The similarities between all possible pairs of samples are

calculated and arranged in a square, symmetric, similarity

matrix This matrix contains all the information

concerning the interrelations between the N samples under

study. Q-mode analysis begins at this point where the

objective is to find new, hypothetical samples whose

compositions are linear combinations of those of the original

samples. The new seunples, or Q-mode factors, can be conceived

of as being composite factors, combinations of which can be

used to reconstruct the original samples. The primary purpose

of Q-mode factor analysis is to determine p linear

combinations of the original N samples that describe the

variad^les or chemical compositions without significant loss of

information (assuming p « N). The following equation

summarizes the underlying rationale of factor analysis:

W, = a„ F, + a,2 Fg + ... + a^p Fp + a, (5)

In words, the equation states that any sample in standardized

form W,, consists of a linear combination of p common factors

plus a unique factor (Ef) • In the factor model, the F*s refer

to hypothetical samples called factors. It is assximed that

(110)

each of these p factors is involved in the delineation of two

or more samples, thus the factors are said to be common to

several samples and p is assumed to be less than N (number of

seunples)• The a*s in equation (5) are the constants used to

combine the p factors so that the factors can predict the

value of Wf In factor analysis, the a*s are termed loadings

and the F*s factor scores. The unique factor represents the

part of W| that cannot be explained by the common factors and

they are assumed to be uncorrelated with each other and with

the common factors.

The factor model contains n such equations; one for each

sample. For a particular variable k, equations (5) becomes:

® ^Ik ^12 ^2k ••• "*• ^\p ^pk ®«k

The values for the a *s do not change from variable to

variable, but the values of the F's do change from variable to

variable. An excellent way to view the F's is to think of them

as new samples that are linear combinations of the old

samples. As such, each variable can contain a different amount

of each one of these new samples.

The equation for the variance of a sample in standard

form is given by

where.

(Ill)

" h " ft m ' '

X|̂ » data matrix for k coliimn

Xj = the mean value for j row

m = number of variable

''kj " standard form of row j with corresponding column k

The variance described the scatter of samples about the mean

where Xj represents the mean of the j sample. Due to the

standardization process, the variance of Wj Is equal to one.In terms of the factor model, the variance may be written as:

EVo/ - M—

. ..... VSV . VSV (a)JB m m '

^ ^ ® Jn **} m

There are two simplifying restrictions which may now be

Imposed.

1) The factors must be In standard form (having mean equal

to zero and variance equal to 1).

2) The factors must be uncorrelated (the correlation

coefficient equal to zero).

(112)

The first constraint makes every term of the form

to one since this is the variance of the factor. The second

constraint makes every term of the form equal to

zero* The entire equation becomes:

° + ^2j + . . . + ^pj + 3^ (9)

In equation (9), the total variance of a sample is to be made

up of the sum of the squared a •s and the total variance

consists of two parts.

1) That due to the common factors which is termed as the

conmunalitv symbolized hj^.

= aij * aij + ... + (lo)

2) That due to the unique factor which is equal to 1 - hj^

and by definition is that part of the variance of sample

j that is not shared by any of the other samples.

The algebraic notation of the factor model is very ciunbersome

and not readily comprehended. Matrix notation allows an easier

representation of the model.

data matrix can be transformed to the

standardized version The total data matrix is considered

to be derivable from the product of two other matrices, matrix

F and A or

(113)

W = AF* (11)

where F' is the transpose of F (refer to Appendix D for matrix

operation) •

W can further be considered as the sum of two matrices C and

E; C containing **true** measures, E containing error measures:

W « C + E (12)

Both F and A can be partitioned into two components, a common

variance part and an error (unique) part. Thus

W - Ac F'c + Ay F\ (13)

In factor analysis, only matrix C, which represents the true

measure, becomes the main interest. It is sufficient to find

the solution for F^ and A^ only. E can always be obtained from

E « Z - C.

As indicated in Figure 3-40, there are four general steps

in a Q-mode factor analysis.^ The first step is the

computation of a similarity matrix of raw data by any of the

similarity indices. The resulting squared, symmetric data

matrix is then used in factor extraction (second step). In

this step, the number of factors necessary to represent the

data and the method of calculating them are determined. The

third step which is rotation is focused on transforming the

(114)

COMPUTE SIMnOFRAV

-ARTTYMAnUX(TDAIA

1

FACTOR EXTOACnON( SPECIFICAnON OF METHOD AND

NUMBER OF FACTORS )

ROTAnON

COMPUTE FACTOR SCORE

Figure 3-40. Steps in a Q-mode factor analysis

(115)

factors to make then more interpretable. In the last step,scores for each factor can be computed for each case. Thesescores can then be used In a variety of other analyses.

3-3-2 COMPUTATZOHAL PSOCSOUlUB

using the cos 0 measure of similarity, the followingequations reveal the necessary steps in the analysis.

instead of working out cos 6,, in one operation, we maydo this in two steps. First we define

^m

Xh§(14)

Where,

^ "" 1# • • • , N

••• , n

Xn - data matrix for row i with corresponding column km - number of variables

N • number of samples

That is, dividing every element in a row by the square root ofthe row sum of squares normalizes the data matrix so that:

a

ik " 1 for i =

Then:

(116)

a

coae,j - (15)

It Is, however, more convenient to use matrix notation. Let

*n.« •'s data natrix. Form the diagonal matrix D whoseprincipal diagonal contains the square root of the row vectorlengths of x. That is.

m

^ for j - (16)

Then w- d-'/2 X (17)

where Dis an Mby Mdiagonal matrix of the row sum of squaresof X as calculated by equation (16). This operation ensures

that every row vector of wis of unit length. The similaritymatrix is computed from

S » WW» - X x»

The basic factor equation is

W- AF* or W» - PA» (19)

where,

A • factor loadings matrix

P - factor score matrix

W « row normalized data matrix

The relationships between W, s. A, and F are given by

S = WW« = AF'FA' (20)

(1X7)

The condition that the factors will be uncorrelated is

on P; that is, p will be orthQn»^«i

P'P - 1 (21)

where I is the identity matrix. Thus equation (20) becomes

S - AA« (22)

A constraint that the p matrix be orthonormal is imposedbecause there is an infinite number of pairs of matrices F andAwhose product will yield w. This means that factors will bein standard form and furthermore, they will be uncorrelated.If these factors are considered to be new samples then thisimplies that the new samples have no mutual correlation amongthem.

The constraint is imposed that

A*A - A (23)where A is the diagonal matrix of eiaenv»i.,A« of thesimilarity matrix S, or

O'SO - A (24)where u contains the eioanvaetQi-« associated with A.

U is a square orthonormal matrix so that :

U'U 1 00* - I (25)

The following matrix manipulation provides the solution

(a) Pre-multiply (24) by o

UU'SU = UA

SO - UA (26)

(X18)

(b) Post-multiply (26) by U

SUU* « UAU'

S - 0AU« (27J(c) Because s is a square symmetric matrix with the Gramian

property (positive semi-definiteness) then,

S - OAO' - o A'/2 01 (28)

Compare equation (22) and equation (28)

8 • AA' - U A''® A"2

therefore,

A " OA''̂ or A' " A''̂ U* (29)

The end result of the matrix manipulations is equation (29).This simply means that the desired matrix of factor loadingsis the matrix of eigenvectors of s, scaled by the square rootsof the corresponding eigenvalue. The matrix of factor score

(P) is obtained by the following matrix manipulation:

W» - FA'

W«A - FA'A

W'A - FA

hence, F - W A A"' (30)

Essentially, eigenvalues ar«» the roots of a series of

derivative equations set up so as to maximize the

variance and retain orthogonality of the factors. Physically,the eigenvectors merely represent the positions of the axes of

the ellipsoid (or hyper-ellipsoid) shown on Figure 3-41. This

(119)

figure represents the swarm of data points involving three

samples. The eigenvalues are proportional to the lengths of

these axes. The largest eigenvalue and its corresponding

eigenvector represent the major axis of the ellipsoid. It is

important to note that the data points show maximum spreadalong this axis, that is, the variance of the data points is

at a maximum. The second largest eigenvalue and its

eigenvector represent the largest minor axis. The axis is at

right angles to the major axis and the data points are seen to

have the second largest amount of variance along this

direction. The same reasoning applies to the remainingeigenvalues and eigenvectors. What is accomplished in usingeigenvectors is to create a new frame of reference for the

data points. Rather than using the old set of samples as

reference axes, the eigenvectors are used instead. These have

the property that they are located along directions of maximum

variance and are uncorrelated.

In this study, Q-xnode factor analysis is performed on

each of eleven samples. (Four synthetic oil saraples and seven

crude oil samples) • The main purpose is to separate each

sample into two groups of before and after phase split. The

changes in compositional data implies the separation of these

two groups. Therefore by performing the Q-mode analysis, we

are able to predict the sample number where the phase split

occur and separate the samples into two clusters.

(120)

.V

Figure 3-41. Scatter diagram In three-dimensional ellipsoid

I

2 Factor I

Figure 3-42. Vectors representing samples with corresponding

factor axes coordinate

(121)

In this study, the method of extraction chosen is a principacomponent and the number of factors extracted are two. Th.rotation method used is a varimax rotation which is discussecin the following section.

3-3-3 FACTORS AND ROTATION

The matrix of fagtgr lOfldlnqp, A, has Nrows and pcolumns. The rjjHa correspond to the origin;.! t^esalimna are the faffitSES. Each column has been scaled so thatthe sum Of squared elements in the column is equivalent to theamount of original variance accounted for by that factor. Theelements in a column may be considered as the coefficients ofa linear equation relating the samples to the factor - messence, they give the recipe for the factor. Therefore, thecolumns of the Amatrix can be used to give some physicalmeaning to the factors, a row of the Amatrix shows how thevariance of a sample is distributed among the factors,interrelationships between samples can be determined by acomparison of their rows in the Amatrix. As pointed out inequation (lo), the sum of the factor loadings squared in a rowof Ais an expression of the amount of variance of a sampleaccounted for by the p factors. This Is termed thegPtWUMPal 11-y. The communality attached to each row of the Amatrix gives an appreciation of how well each sample isexplained by the p factors considered. Another valid view of

(122)

an element in A is that it represents the correlation or

similarity between a sample and a factor. Becausecorrelations or similarities are angular measures, the rowelements actually represent the cosines between a sample andthe p reference factor axes. Groupings of samples and trends

between them often yield important clues as to the physicalsignificance of the factors.

We may represent the coefficient or factor loadingbetween each sample and factor axes in vector form. For

instance, Figure 3-42 shows vectors representing factor

loadings for four samples with factor axes I and II. Factor

I coincides with vector 2 and factor II coincides with vector

4. On the other hand, each of vector 3 and 1 has both loadingon factor I and II.

The matrix of factor scores, F, will not be discussed in

detail here because we do not calculate them in this study. In

general, the matrix of factor scores, f, consists of m rows

and p columns where mis the number of variables and p equalsthe number of common faetora. Each column is in standard form

with zero mean and unit variance, and there is zero

correlation between coltimns. Because the factors are linear

combinations of the original samples, they can themselves be

considered as new samples.

(123)

Although the factor matrix obtained in the extraction

indicates the relationship between the factors and individual

samples, it is usually difficult to identify meaningful

factors based on this matrix. Often the samples and factors do

not appear correlated in any interpretable pattern. Since one

of the goals of factor analysis is to identify factors that

are substantively meaningful (in the sense that they summarize

sets of closely related samples), the rotation of factor

analysis attempts to transform the initial matrix into one

that is easier to interpret.

An attempt is made to achieve what is termed simple

structure. by which is meant that the factor axes are located

in positions such that:^^

1) For each factor only a relatively few samples will have

high loadings, and the remainder will have small

loadings•

2) Each sample will have loadings on only a few of the

factors•

3) For any given pair of factors, a number of samples will

have small loadings on both factors.

4) For any given pair of factors, some of the samples will

have high loadings on the second factor but not on the

first.

5) For any given pair of factors, very few of the samples

will have high loadings on both.

(124)

These conditions attempt to place the factor axes In more

meaningful positions so that they will be highly correlated

with some of the original samples. A large number of

rotational methods have been designed, but only varlmax

rotation will be discussed here.

An approximation to simple structure, designed by Kaiser

(1958), uses a rigid rotation procedure. This means thgit the

orthogonal principal component factors will be rigidly rotated

and maintained orthogonal. Kaiser*s approach Is to find a new

set of positions for the principal factors such that the

variance of the factor loadings on each factor Is a maximum.

That Is, when the value of V In the following expression Is

maximized, simple structure should be obtained,

Where,

P / u \4 p n

(31)

bjij = the loading of sample j on factor p on the new,

rotated factor axes

N a number of sample

p » number of factor

hj^ = total variance of sample j due to common

factors

(125)

The process can be readily understood in terns of matrix

algebra.^ Given the N by p matrix of principal factor loadtnog

A, the objective is to transform it to a N by p matrix of

varimax factor loading B such that B will satisfy equation

(31). In matrix terms, this can be accomplished by

B - AT (32)

where

COS ((> -sin <i>sin <|> cos <|>

^ is the angle of rotation required to yield a maximum value

of V in equation (31) and Is determined by an iterative

process. The matrix B contains the loadings of the original N

samples on the p rotated factors and can be interpreted in the

same way as the A matrix. Figure 3-43 and 3-44 shows the

hypothetical unrotated factor loading plot and varimax rotated

factor loading plot respectively. For the varimax rotated

factor loading plot, if a rotation has achieved a simple

structure, clusters of samples will occur near the ends of

the axes and at their intersection. Samples at the end of an

axis are those that have high loadings on only that factor.

Samples near the origin of the plot have small loadings on

both factors. Samples that are not near the axes are explained

by both factors. If a simple structure has been achieved,

there will be few, if any, variables with large loadings on

more than one factor.

li

(126)

HOUZONTAL riCTOII I VnTZOO. rACTOK 2

u

3 11

8 •*9

10

Figure 3-43. Hypothetical unrotated factor loadix^g plot

MRIZSKTAL riCTOR 1 VOtneAL nCTOII a

X1X IC

• XIXXXIXXX

3

XX 11

ft

X 7XX

•

4 112 X14

XXX1

2

:s

XX

14XX

9

XXXXXX

Figure 3-44. Hypothetical Varimax rotated factor loading

plot

(127)

3-3-4 RESULT OP THE DATA MtKLYBXB

In this study, the SPSS-X Statistical Package Software Is

used to perform a Q-node factor analysis on the synthetic oil

and crude oil samples.^ Since this software Is originallydesigned for R-mode analysis, we modify the program subcommand

to make It suitable for Q-mode factor analysis. The similaritymatrix Is fed Into the program, Instead of raw data, as anInput and this similarity matrix Input Is used In the

principal component extraction to calculate factor loading,rotated factor loading, communallty, eigenvalues and

percentage of variance. Also a rotated factor loading plot Is

graphed at the end of analysis. An example of statistical

results are given in Tables 3-32 and Figures 3-45 and 3-46 for

upper and lower CPE 207 sample.

The goal of factor analysis is to identify the not-

dlrectly observable factors based on a set of observable

samples, in most cases, the factors used for characterizing aset of samples are not known in advance but are determined byfactor analysis. However, in this study, we already know that

the samples can be characterized into two groups of before and

after phase split. Therefore, only two factors are extracted.

Each sample is expressed as a linear combination of factors.

Table 3-32 is a factor matrix which contains the coefficients

that relate the samples to the factors. This table shows that

(128)

saapl. 1for upper CPB 207 can be expressed as (equation (5))'

samplel - (--^6787) FACTORl * (.25114) FACm)R2f

where the coefficients in front of factor l an<3 factor 2 ar.both factor loadings. CoauDunality, which neasures the degree,of relationship among samples, is calculated by taking the sunOf the squared factor loadings for each sample. Therefore, thecommunality for sample l is given by (equation (10)),

("•96787)' + (.25114)^ • 0.99985

All samples and factors are expressed in standardized formwith mean of oand standard deviation of i. since there are 15samples in upper CPE 207 and each is standardized to havevariance of 1, the total variance is 15. The total varianceexplained by each factor is listed in the column labeled -Eigenvalue. The next column contains the percentage of thetotal variance attributable to each factor. For example, thelinear combination formed by factor l has a variance of 13.76which is 91.7% of the total variance of 15. The last column,the cumulative percentage, indicates the percentage ofvariance attributable to each factor.

The rotated factor matrix and the corresponding rotated

(129)

factor loading plot for all other sanples are given in Tables

3-33 to 3-42 and Figures 3-47 to 3-66 respectively. In order

to make results easier to read, only the factor loading values

greater than 0.5 are printed and the samples are sorted

according to the decreasing order of factor loading values.

The results on the prediction of the phase split from Q-mode

factor analysis are tabulated in Tables 3-30 and 3-31.

3-3-5 ZMTBRPRBTATZOM AMD C0HCLU8Z0M

Each row of a factor matrix contains the coefficients

used to express a standardized sample in terms of the factors.

These coefficients are called factor loadings, since they

indicate how much weight is assigned to each factor. Factors

with large coefficients (in absolute value) are related to the

specified sample. For example. Factor 1 (in Table 3-32) for

upper CPE 207 is the factor with high loading for samples 1

through 15, except sample 4, and factor 2 with sample 4. This

result indicates a breakthrough occurs at sample 5. In the

phase behavior study, this breakthrough is due to the phase

split. Therefore, for upper CPE 207, it is concluded that

samples 5 through 15 are in one group while samples 1 through

4 are in another group. Physically, samples 1 through 4

represent before phase split and samples 5 through 15 after

phase split. The same conclusion is made on the rotated factor

matrix result.

(130)

Tables 3-32 and 3-33 compare the phase split prediction

by an unrotated and rotated factor matrix. In most cases, both

of these factor matrices give a similar result. The comparison

of a factor matrix with a binary classifier and viscosity

measurement on the phase split prediction are also made. The

results are well correlated for these three methods. It is

concluded that the Q-mode factor analysis definitely can be

used to predict the occurrence of phase split and hence the

phase behavior of the complex hydrocarbons. The representation

of phase behavior by this method can be observed by examining

the rotated factor loading plot for each sample. For example,

Figure 3-45 shows the factor loading plot for upper CPE 207.

As indicated in the plot, sample 4 is very close to factor 2

axes, while sample 5 is close to the factor 1 axes. This means

that phase split occurs at sample 5. Samples 1 through 4

represent one phase region which follows a compositional path

as in pseudotemary diagrams. The two phase region, where

liquid and gas phases coexist, is represented by samples 5

through 15. The upper sample in the two phase region is the

equilibrium gas phase and the lower sample is the equilibrium

liquid phase.

(131)

Table 3-30. Result of Q-mode Factor Analysis

for Synthetic oil Data

— v.'....,jj ;; iH'-'.'

CPE DATA ^ACTORMATRIX

207 Cupper) 5

207 (lower) 5

214 (upper) 4

214 (lower) 4

215 (upper) 5

215 (lower) 4

216 (upper) 5

216 (lower) 5

ROTATEDFACTORMATRIX

BINARY CPECLASSIFIER ( BY VISCOSITY)

(132)

Table 3-31. Result of Q-mode Factor Analysis

for Crude oil Data

1CPE DATA FACTORMATRIX

ROTATEDFACTORMATRIX

BINARYCLASSIFIER

CPE 1(BY VISCOSITY)

2 4 3 5 11 234 (lower) 2 2 4 5 Ij 247 (upper) 4 4 5 4 1j 247 (lower) 4 4 4 4 11238 (upper) 4 4 5 5 j1 238 (lower) 4 4 4 5 11 246 (upper) 3 3 4 3 j1 246 (lower) 3 3 3 3 11239 (upper) 6 5 6 7 J1 239 (lower) 6 6 7 7 j

4 5 5 5 11 244 (lower) 5 4 5 5 I1245 (upper) 4 4 8^5 8 jI 245 (lower) 4 4 5/4 8 I

(133)

-Rble 3-32. Statistical Result of Q-mode Factor Analysis for CPE 207.UPPER SAMPLE

FACTOR MATRIX

SAMPLE FACTOR 1 FACTOR 2

14 .99808 .0619515 .99803 .0627512 .99795 .0640011 .99785 .0654910 .99779 .0663913 .99779 .066439 .99565 .093138 .99472 .102647 .99199 .126346 .99050 .137521 -.96787 .251142 -.96703 .254665 .95853 .284983 -.95502 .29652

4 -.36933 .92930

SAMPLE COMMUNALm

1 .999852 .999993 .999994 1.000005 1.000006 1.000007 1.000008 1.000009 1.00000

10 1.0000011 1.0000012 1.0000013 1.0000014 1.0000015 1.00000

final STAnsncs

FACTOR EIGENVALUE PCTOFVAR CUMPCT

13.760101.23972

91.7

8.391.7

100.0

(134)

Continue Thble 3-32

ROTATED FACTOR MATKDC

SAMPLE

5

6

7

8

9

FACTOR 1 FACTOR 2

.99955 -.02984

.98369 -.17989.98159 -.19097.97676 -.21433.97467 -.22366.96833 -.24968.96832 -.24972.96809 -.25060.96772 -.25204.96740 -.25325.96720 -.25403

-.84037 .54187-.83847 .54495-.81394 .58094

-.05942 .99823

LOWER SAMPLE

FACTOR MAlllDC

SAMPLE FACTOR 1 FACTOR 210 .99991 .0133911 .99990 .0138412 .99990 .014359 .99988 .01564

13 .99983 .0182014 .99981 .0192615 .99979 .020468 .99957 .029267 .99905 .043466 .99649 .083665 .96933 .245751 -.95218 .305452 -.95157 .307413 -.93428 .35653

4 .45792 .88899

(135)

SAMPLE COMMUNAUTY

Continue Ikble 3-32

final STAHSTTCS

FACTOR EIGENVALUE PCTOFVAR CUMPCT1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

.99996

.99999

.99999

.99999

1.00000

1.000001.00000

1.00000

1.00000

1.00000

1.00000

1.00000

1.00000

1.00000

1.00000

13.82262

1.17730

rotated factor matrix


3

2

1

10

11

12

9

13

14

15

8

7

6

5

-1.00000

-.99864

-.99852

.92947

»92931

.92912

.92864

.92768

.92729

.92684

.92350

.91795

.90123

.81808

.11100

.00013

-.05194

-.05398

.36890.36931

.36979

.37098

.37336

.37435

.37546

.38360

.39669

.43334

.57510

.99382

92.2

7.892.2

100.0

Figu

re3-

45.F

acto

rL

oadi

ngp

liO

RIZ

OK

rAL

FA

CT

OR

23

tfo

rU

pp

erC

PE

207

VE

Rn

CA

LF

AC

IOK

2

9 IS

Figu

re3-

46.F

acto

rle

adin

gp

llO

RIZ

OK

rAL

FA

CT

OR

tfo

rlu

>w

erC

PE

20

7

VE

R:n

CA

LF

AC

rOR

2

4

6 IS

M CJ

a\

(137)

Table 3-33. Rotated Factor Matrix for CPE 214

UPPER SAMPLE

SAMPLE FACTOR I FACTOR 2

3 -.97357

2 -.921921 -.92179

16 .73089 .6825017 .73085 .6825414 .73082 .6825715 .73077 .68262

8 .73063 .6827811 .73060 .6828113 .73045 .6829610 .73037 .6830512 .73037 .68305

9 .73015 .68328

4 .990405 .54761 .836746 .61669 .787217 .70533 .70888

LOWER SAMPLE


5

4

6

7

8

16

10

11

15

13

12

9

14

17

2

1

.99457

.99114

.98847

.97800

.97561

.97434

.97395

.97377

.97350

.97336

.97272

.97229

.97212

.97194

-.80752

-.80720-.58983

-.59028

-.99978

Figu

re3-

47.F

acto

rLoa

ding

plot

forU

pper

CPE

214

llO

KIZ

ON

-rA

LF

AC

TO

RI

VE

KO

CA

LFA

CT

OR

2

4

17

)>

Figu

re3-

48.F

acto

rL

oadi

ngpl

otfo

rL

ower

CP

E21

4

HO

RIZ

OK

IAL

FA

CIO

RI

VO

KH

CA

LF

AC

FO

R2

17 6

CJ

00

(139)

Thble 3-34. Rotated Factor Matrix for CPE 215

UPPER SAMPLE


14 .99626

10 .99626

15 .99625

9 .99625

12 .99625

13 .99624

11 .99624

8 .99624

7 .99622

6 .99099

2 -.96864

1 -.96862

3 -.93745

5 .90156

4 .99912

LOWER SAMPLE


4 .93526

5 .80873 .5881715 .78131 .62414

9 .77925 .62672

10 .77917 .6268113 .77895 .6270812 .77893 .6271111 .77890 .6271514 .77884 .62722

6 .77866 .627447 .77245 .635078 .77108 .63674

3 -.901301 -.55073 -.834682 -.55113 -.83441

Figu

re3-

49.F

acto

rLoa

ding

plot

forU

pper

CPE

215

IIORI

ZON^

fAL

FACT

OR1

VER

HCA

LFA

CTOR

2

6 IS

Figu

re3-

50.F

acto

rLo

adin

gpl

otfo

rLo

wer

CPE

215

HO

RIZO

NTA

LFA

CTO

R1

VEK

IICA

LFA

CTO

R2

15 5

•th

O

(141)

Ibble 3-35. Rotated Factor Matrix for CPE 216

UPPER SAMPLE


2 -.99893

1 -.998903 -.99483

17 .95592

16 .9556715 .9553214 .95512

13 .9543912 .9532311 .9461010 .945139 .936828 .935267 .923696 .899435 .81753

4

.57589

.99823

LOWER SAMPLE

SAMPLE FACrOR 1

17 .9977316 .99771

11 .9958212 .9957913 .9956114 .9954315 .9952310 .995079 .994778 .992537 .990086 .986052 -.972821 -.971185 .96720

3 -.95766

FACTOR 2

.99645

i' •t

Figu

re3-

52.F

acto

rLoa

ding

plot

forL

ower

CPE

216

IlO

RIZ

ON

TA

LF

AC

TO

VE

KII

CA

LFA

CI

OR

2

67 15

17

Figu

re3-

51.F

actor

Ixmdin

gplot

forU

pper

CPE

216

HORI

ZONT

ALFA

CTOR

IVC

RIIC

ALFA

CTOR

2

6 7 11 17

H*

lO

(143)


UPPER SAMPLE

SAMPLE FACTOR 1 FACTO

1 -.94075

0 .77694 .62956

6 .77544 .63141

11 .77499 .63197

9 .77499 .6319714 .77448 .6325913 .77435 .63275

8 .77420 .632937 .77376 .63347

12 .77296 .63445

5 .76866 .639654 .73662 .67627

2 .919573 .69447 .71933

LOWER SAMPLE

SAMPLE FACTOR 1

2

3

14

4

12

13

9

11

8

5

7

10

6

.90459

.74508

.72463

.72347

.72022

.72002

.71691

.71622

.71585

.71579

.71512

.71216

.70806

FACTOR 2

.66697

.68911

.69035

.69372

.69394

.69716

.69786

.69825

.69831

.69899

.70201

.70615

-.89944

Figu

re3-

53.

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pper

CPE

234

Figu

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54.

Fact

orIx

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ower

CPE

234

IIO

RiZ

ON

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(143)

Figure 3-37. Rotated Factor Matrix for CPE 247

UPPER SAMPLE

factor 1SAMPLE factor 2

4 .990205 .949176 .935677 .932688 .930479 .92905

10 .9277311 .9266012 .92602

1 -.75394 -.656912 -.73626 -.67669

3-.98249

SAMPLE

4

5

6

7

8

9

10

11

12

3

2

1

lower sample

FACTOR 1 FACTOR 2

.98374

.80082

.76742

.75944

.75346

.74823

.74457

.74299

.74120

.59891

.64114

.65058

.65749

.66344

.66754

.66930.67128

-.96582

-.87872

'.87590

.•ixw; *♦*«(• .^uvwAiWa,

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247

HORI

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FACT

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7

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(147)


UPPER SAMPLE

SAMPLE FACTOR 1

5 .999866 .999668 .999337 .999299 .99921

10 .9991111 .9990212 .9989113 .9988614 .99881

4 .998062 -.956731 -.95332

3

FACTOR 2

.99884

LOWER SAMPLE

SAMPLE FACTOR14 .9957912 .9945413 .9943010 .9932711 .99321

8 .992549 .992337 .991066 .990255 .981722 -.980881 -.975854 .94150

3

FACTOR 2

.99561

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actor

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238

IlOKI

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3-58

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CTOR

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09

(149)

TUile 3-39. Rotated Factor Matrix for CPE 246

UPPER SAMPLE

VMPLE FACTOR 1 FACTOR 2

3 .81999 .570314 .79288 .609155 .78509 .61936 •V

6 .77873 .627357 .77659 .629979 .77582 .630848 .77548 .63130 . "V

10 .77515 .63167 V

11 .77494 .6319412 .77435 .6326713 .77415 .63292

2 -.59175 -.804241 -.63555 -.77053

LOWER SAMPLE

SAMPLE FACTOR 1 FACTOR

3 .950734 .79220 .610075 .75951 .650116 .73881 .673527 .73333 .679458 .72971 .683399 .72916 .68398

10 .72814 .6850811 .72751 .6857513 .72626 .6871612 .72549 .68795

2 -.940621 -.60067 -.75417

r

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IIORI

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CTOR

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(151)

Table 3-40. Rotated Factor Matrix for CPE 239

UPPER SAMPLE


4 -.99631

3 -.973071 -.938202 -.93094

15 .72102 .6928814 .71484 .69926

6 .988857 .919915 .910428 .57013 .821559 .62395 .78146

10 .64999 .7599411 .67747 .7355312 .69371 .7202413 .70543 .70876

LOWER SAMPLE

SAMPLE FACTOR 1 FACT<

7 .997566 .991618 .986489 .97624

10 .9690911 .9607612 .9547313 .9523214 .9506815 .94748

1 -.75815 .651502 -.73396 .67915

4 .972545 .58320 .812153 -.61917 .78525

Figure

3-61.

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IIORI

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(153)


SAMPLE3

2

1

20

19

18

17

16

15

14

13

12

11

10

9

7

8

6

5

UPPER SAMPLEfactor 1

-.99742

-.98673

-.98237

.89531

.89514

.89494

.89476

.89432

.89400

.89347

.89311

.89191

.89100

.88976

.88783

.88518

.88058

.87171

.84982

FACTOR 2

.52703

.98246

LOWER SAMPLESAMPLE FACTOR 1 FACTOR

4 .994795 .963736 .945897 .936538 .933119 .92745

11 .9270310 .9248812 .9200613 .9173314 .9173015 .9141317 .9134320 .9133116 .9131618 .9127919 .91206

1 -.71863 -.69535

3-.98742

2 -.70402 -.71016

Figu

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PE24

4

IIO

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OW

AL

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6 20

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244

HO

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(155)

Tible 3-42. Rotated Factor Matrix for CPE 245

UPPER SAMPLESAMPLE FACTOR 1 FACTOR 2

4 .999995 .933736 .922147 .916878 .85215 .523299 .85042 .52610

10 -84997 .5268311 .84932 .5278812 .84891 .5285414 .84840 .5293613 .84837 .5294115 .84813 .5297916 .84761 .5306117 .84751 .5307818 .84740 .5309519 -84688 .5317820 .84687 .53180

3 -.98253 i2 -.910071 -.57452 -.81822

LOWER SAMPLESAMPLE FACTOR 1 FACTOR 2

4 .999236 .99541 i7 .991958 .99065

18 .9903317 .9895019 .9881620 .9878810 .9878513 .98769

11 .9876116 .9875414 .98734 i12 .98722 'IS .987199 .985965 .95263

1 -.86258 -.505372 -.82177 -.56979

^ -.99964

Figure

3-«5.

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245

llaRI

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ICAL

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57

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(157)

CHAPTER 4 t SUMMARY AMD CONCLUSION

This lihesis presents the represen'tatlon of phase behavior

of several COg/oil mixtures using binary classifier and factor

analysis models. The binary classifier performed on synthetic

oil and crude oil experiments, gives a very close prediction

of the phase split by both distance between the center of

gravity and mean vector measurements. This observation

indicates that both of these binary computations are equallygood in representing phase behavior of synthetic oil or crude

oil. Three different combinations of five components in the

synthetic oil also give a very close prediction of the phase

crude oil experiments, a comparison is made

between totals of 2 and 3 numbers of samples taken as a

training set. The results suggest that the number of samplesused in a training set do not affect the prediction of phase

The phase split predicted by the factor analysis model

is in good agreement with the binary classifier model.

In thto CPE experiment, the phase split is predicted by

viscosity measurement, but in the binary classifier and factor

analysis models, it is predicted directly by hydrocarbon

composition. The results for two out of the four synthetic oil

experiments show that prediction of the phase split using thebinary classifier and factor analysis models give a very close

prediction which is consistent with phase split as determined

(158)

by viscosity measurements. For crude oil experiments, all of

the results , except CPE 245, obtained from these models

correlate well with that obtained by viscosity.

CPE 207 and CPE 216 phase splits do not match with phase

splits obtained by viscosity. It is speculated that viscosity

measurements are approximated since the changes in viscosity

are not constant throughout the experiment. However, the phase

split predicted by binary classifier and factor analysis do

correspond to the maximum changes in the viscosity of these

compositional data.

This study can be extended by applying binary classifier

and factor analysis on more synthetic oil data. Also in the

factor analysis, factor scores can be calculated to analyze

the relationship among components in crude oil. In mo^t CPE

experiments, only 12 to 20 samples were collected.

Statistically, the validity of the binary classifier results

would be improved if a larger sample population in propprtioii

to number of components were used. For example, compositional

analysis every 10 minutes instead of hourly.

The main conclusions obtained from this study are as

follows:

1) The phase behavior of multicomponent mixtures is

approximated by three pseudocomponents. Therefore the effect

(159)

of each component on the phase behavior cannot be observed

directly.

2) The CPE experiment predicts the phase split by viscosity

mesurement. The phase split predicted by both binary

classifier and factor analysis are directly obtained from

changes in the compositional data.

3) In most cases, the prediction of phase split by binary

classifier and factor analysis correlate well with that by

viscosity measurement. It is concluded that phase behavior can

be explained directly by compositional data.

4) The factor analysis model has an advantage over the binary

classifier model and the ternary diagram because it makes use

of all compositional data available from gas chromatographic

analysis.

5) The phase behavior representation by the binary classifier

is better than the ternary diagram because it can extend the

number of grouping component to the minimum limit of n/d

ratio.

(160)

REFERENCES

1. Isenhour T.L.; Kowalskl B.R.; Jurs P.C., 1974,**Applications of Pattern Recognition to Chemistry". CRCCritical Reviews in Anal. Chem., p. 1-44.

2. Shoenfeld P.S.; DeVoe J.R., 1976, "Statistical andMathematical Methods in Analytical Chemistry". Anal.Chem., V. 48, no. 5, p. 403R.

2. Varmuza K.; Rotter H.; Krenmayr P., 1974,"Interpretation of Steroid Mass Spectra with PatternRecognition Methods". Chromatographia, v. 7, no. 9,p. 522.

4. Woodruff, H.B.; Lowry S.R.; Isenhour T.L., 1975, "AComparison of Two Discriminant Functions for ClassifyingBinary Infrared Data". Appl. Spect.,v. 29, no. 3,p.226.

5. Duewer D.L; Kowalskl B.R.; Schatzki T.F., 1975, "SourceIdentification of Oil Spills by Pattern Recognition:Analysis of Natural Elemental Composition". Anal. Chem.,V. 47, p. 1573.

6. Clark H.A.; Jurs P.C., 1975, "Qualitative Determinationof Petroleum Sample Type from Gas Chromatograms UsingPattern Recognition Techniques". Anal.Chem., v. 47,no.3,p.374.

7. Clark H.A.; Jurs P.C., 1979, "Clacsification of CrudeOil Gas Chromatograms by Pattern RecognitionTechniques". Anal. Chem., v. 51, p. 616.

8. Rotter H.; Varmuza K., 1977, "Computer-AidedInterpretation of Steroid Mass Spectra by PatternRecognition Methods: Influence of Mass SpectralPreprocessing on Classification by Distance Measurementto Centers of Gravity". Anal. Chim. Acta., v. 95, p.25-32.

9. Imbrie, J.; Van Andel, T.H., 1964, "Vector Analysis ofHeavy-Mineral Data". Geol. Soc. Amer. Bull., v. 75,p. 1131-1156.

10. Harbaugh, J.W.; Demirmen, F., 1964, "Application ofFactor Analysis to Petrographic Variations of AmericansLimestone (Lower Permian), Kansas and Oklahoma". Kan.Geol. Survey Dist., Pub. 15.

11. Klovan, J.E., 1966, "The Use of Factor Analysis inDetermining Depositional Environments From Grain-Size

(161)

Distributions". Jour. Sed. Petrology, v.36, no.l, p.115-125.

12. McCammon, R.B., 1966, "Principal Component Analysis andIts Application in Large-scale Correlation Studies".Jour. Geol., V. 74, no. 5, pt.2, p. 721-733.

13. Hitchon, B.; Billings, G.K.; Klovan, J.E.,1971,"Geochemistry and Origin of Formation Waters in theWestern Canada Sedimentary Basin-III. FactorsControlling Chemical Composition". Geochim. etCosmochim. Acta, v. 35, p. 567-598.

14. stroiDberg E.W.; Fasching J.L., 1976, "The Application ofCluster Analysis to Trace Elemental Concentrations inGeological and Biological Matrices". National Bureau ofStandards Special Publication 422.

15. Orr F.M.; Silva M.K.; Lien C.L., 1980, "LaboratoryExperiments to Evaluate Field Prospects for COgFlooding". SPE 9534, presented at SPE Meeting, WestVirginia, Nov. 5-7.

16. Orr F.M.; Silva M.L., 1983, "Equilibrium Mixtures ofCO,/Hydrocarbon Systems, Part 1: Measurement by aContinuous Multiple Contact Experiment". Soc. Pet. Eng.J. (April), p. 272-280.

17. Hutter C.E.; Franklin J.C., 1984, "Operation and Controlof The Continuous Multiple Contact Experiment by TheHP87XM Microcomputer". PRRC Report 84-7, New MexicoPetroleum Recovery Research Center.

18. Kovarik F.S.; Taylor M.A., 1987, "Viscosity Measurementsof High-Pressure COg/Hydrocarbon Mixtures". AICHE AnnualMeeting, Nov. 15-20, New York.

19. Taylor M.A.; Heller J.P.; Hutter C., 1987, "Design andApplication of the Torsional Crystal Quartz Viscometerfor The Continuous Phase Equilibrium Apparatus". PRRCReport 87- 2, New Mexico Petroleum Recovery ResearchCenter.

20. Debbrecht F.J, 1985, "Qualitative and QuantitativeAnalysis by Gas Chromatography". Modern Practice of GasChromatography, Second Edition (Edited by Grob R.L.).John Wiley and Sons, New York, p. 359.

21. Lien C.L.; 1981, The Program for Crude Oil AnalysisBased on the Proposed ASTM Methods. New Mexico PetroleumRecovery Research Center.

(162)

22• ASTM Standards, Part 25, 1976, "Proposed Test Method forBoiling Point Range Distribution of Crude Petroleum byGas Chromatographyi*. American Society for Testing andMaterials, Philadelphia, PA.

23. McCain W.D., 1973, •• Changes of state**. The Propertiesof Petroleum Fluids. Pennwell Books Publishing Company,Tulsa, Oklahoma, p. 44.

24. Stalkup F.I., 1984, "Principles of Phase Behavior andMiscibilityi*. Miscible Displacement. SPE MonographSeries, Henry L. Doherty Series, v. 8, p. 6.

25. Orr F.M.; Yu A.D.; Lien C.L., 1980, "Phase Behavior ofCO. and Crude oil in Low Temperature Reservoirs". spE8813, presented at the First Joint SPE/DOE Symposium onEnhanced Oil Recovery, Tulsa, April 20-23.

26. Varmuza K., 1980, "Pattern Recognition in AnalyticalChemistry". Anal. Chim. Acta., v. 122, p. 227-240.

27. Klovan, J.E., 1975, "R- and Q-mode Factor Analysis".Concepts in Geostatistics. (Edited by McCammon R.B.).Springer-Verlag, New York, p. 21.

28. Imbrie J.; Purdy E.G., 1962, "Classification of ModernBahamian Carbonate Sediments", Classification ofCarbonate Rocks. A Symposium, Mem. 1, Amer. Assoc.Petroleum Geol., p. 253-272.

29. Joreskog K.G.; Klovan J.E.; Reyment R.A., 1976, "BasicMathematical and Statistical Concepts". GeologicalFactor Analysis. Elsevier Scientific Publishing Company,N.Y., p. 32.

30. SPSSX User's Guide, 3rd Edition, 1988, "Factor". SPSSInc., XL., p. 480.

31. Kaiser, H.F., 1958, "The Varimax Criterion for AnalyticRotation in Factor Analysis". Psychometrika, v. 23,p. 187-200.

(163)

APPENDIX A : CALIBRATION STANDARD (SIMULATED DISTILLATION)

FORMULA MATERIALS WEIGHT %

CsHij n-Pentane 8.3C6H,4 n-Hexane 4.4C7Hi« n-Heptane 4.6CgHis n-Octane 4.7

C9H20 n*Nonane 4.8

C10H22 n-Decane 9.7

C11H24 n-Undecane 4.9C12H26 n-Dodecane 19.9C14H30 n-Tetradecane 10.2C15H32 n-Pentadecane 5.1C16H34 n-Hexadecane 10.2C17H36 n-Heptadecane 5.2CisHaa n->Octadecane 2.2C20H42 n-Eicosane 1.3C24H50 n-lfetracosane 0.9C28H58 n-Octacosane 0.9C32H66 n-Dotriacontane 0.9C36H74 n-Hexatriacontane 0.9C40H82 n-Tetracontane 0.9

(164)

APPENDIX B : TERNARY DIAGRAMS FOR ALL EXFERIHENTAL DATA

CO 2o.oss

->0.041 (CENTI POISE)

50 7o

0.430

0.962

0.400

0.90S

0.936

0.920

0.272

0.288

0—0.264

O—0.926

o—0.992

O—0.410

0—0.492 ICENTIPOtSE)

507«

C30"C|6 synthetic OIL. 190®F. 2000 psia (CPE 214)

C30-CI6

CPE 215

T= 339®K (I50*F)

Ps 10.35 MPo (ISOOpsio)OS Viscosity x lO' .Po-s

or Viscosity, cp

.96S

1.019

CIO-C5

50%

C30-C16

CPE 216T«3I0.8*K (IOO'F)P«10.35 MPo {ISOOpslo)o» Viscosity X 10' .Po's

or Viscosity, cp

WASSON-STOI09®F 1500 psiaCPE 234

✓.235

9—922

p-~.998

P^l.48

50%

C5-C10

Ca-C12

• Vir.v.:•: ' '

CPE 238

Texaco OH 1 CO

1370 psia. 105T

Density, Viscosity

0.86t. 0.

0.856, 0.

0.852, 0.775

0.845, 0.704

0.838,

0.827, 0

0.811,0.41

0.790, 0.332

Cr*

TEXACO

(166)

C02

1.667, 0.082

0.670. 0.082

0.674, 0.081

1.685, 0.085

0.688, 0.090

1.691, 0.101

1.694,0.110

1.700, 0.113

704, 0.221

©-/-0.783. 0.502

0.777,0.611

O 0.771, 0.768

0.763, 1.048

u

CI-CI2

CPE 244

Oxy Ott B 4- CO21587 psia, 130T

Density, Viscosity

0.832. 0.664

0.830. 0.634

0.815, 0.525

0.809, 0.4

0.794. 0.42

0.788, 0.397-

CPE 245

Amoco oa B -I- CO.

2350 psia, 165T

Density, Viscosity

0.871, 0.922

0.869, Oj

0863, 0.815

0.859. 0.807

0.856, 0.748-0.851. 0.678

0.849, 00845, 0.61

O840. 0.572

0.838. 0.544

0.830. 0.495

0.748. 0.692

0.745, 0.964

0.531, 0.0560

1.535, 0.0570

0.538, 0.058

0.540, 0,059

0.542, 0.061

0.549,0.073

554, 0.075

0.560, 0.077

0.567, 0.085

1.573, 0.097

0.506, 0.121

O 0.759, 0.538

O 0.758, 0.605

© 0.753, 0.687

1.571. 0.048

0.575. 0.051

.581, 0.057

0.583. 0.061

•0.586, 0.065

0.591, 0.069

0.594, 0.074

1.596. 0.081

€>r-0.829. 0.654

O-pO.827, 0.982

©-rO.826. 1.150

0—0826, 1.276

0^0.823, 1.7410.822. 1.748

Or—0.821. 1.753

1.598, 0.085

1.601, 0.098

.617, 0.106

CPE 246

Amoco OH A+ COg1200 psia. t06T

0«wtty, Viscosity

(0.871. 0.924)(0.069.0.856)

{0.804, 0.

(0.863, a7f4)(0.657, 0.

(0.850, 0.61

(0.846, 0.626)(0.836, 0.779)

(0.828, 0.872)

Ct»

CPE 247

Texaco 03 2 + COg1700 psta, 116^DwsJty, Viscosity

0.856, 0.787

0.852.0.0.844. 0.642

0.838. 0.572

0.828, 0.5130.816, 0.456

0.800, 0.404

(168)

0.811, 1.924

7/

0.325, 0.0312

0.335, 0.0316

0.348, 0.0317

0.355, 0.0319

0.366, 0.0321

0.374, 0.0360

0.383, 0.0410

1.406, 0.0484

0.417. 0.0507

O 0.822, 1.052

0.813, 1.424

u

0.674, 0.069

0.676, 0.069

•0.678. 0.070

0.688. 0.071

0.683. 0.073

-0.688. 0.078

0.701, 0.082

0.782. 0.380

®-0.761. 0.419

O—-0.749. 0.530

-^0.745. 0683^—0.744, 0.770

(169)

APPENDIX C : EXPERIMENTAL DATA

**** Example for upper 207 (U207) experimental data15,5 = total samples, total components

Composition is in mole%, with total of 100% in eachsample.

U207

15,5

0.0,14.0,54.0,19.0,13.00.0,21.84,47.27,18.42,12.466.64,20.62,44.05,17.17,11.5232.83, 13.98,32.29,12.57,8.3356. 53, 8. 49, 21.20,8.24, 5.5473.01,5.46,13.04,5.08,3.4175.53,4.24,12.27,4.77,3.1981.36,3.20,9.32,3.65,2.4584.19,3.0,7.79,3.02,2.094.76,1.38,2.72,0.81,0.3395.21,1.24,2.53,0.74,0.2895.89,1.1,2.14,0.63,0.2394.95,0.93,2.93,0.87,0.3196.92,0.74,1.67,0.50,0. 1696. 60, 0. 68, 1.94,0.59,0.18

U215

15,5

0.00,14.00,54.00,19.00,13.000.00,13.26,54.23,18-77,13.7313.71,12.84,45.32,16.02,12.1239.18,8.97,31.91,11.34,8.6054.08,6.93,23.76,8.67,6.5681.35,6.29,7.53,2.75,2.0898.72,0.99,0.23,0.04,0.0298.88,0.89,0.20,0.02,0.0198.97,0.82,0.17,0.02,0.0199.03,0.82,0.13,0.01,0.0098. 94,0. 84, 0.20, 0. 02, 0.0099.03,0.74,0.20,0.02,0.0198.99,0.77,0.21,0.02,0.0199.15,0.63,0.17,0.03,0.0299.21,0.51,0.26,0.02,0.00

L207

15,5

21,11.82,4.01,2.7806,9.24,3.28,2.2644,8.89,3.42,2.5318,8.90,3.73,2.9412,8.84,3.98,3.2600.9.36.4.62.3.94

15,5> w, w

0.00,14.00,34.00,19.00,13.0.00,12.73,33.20,19.46,14.15.00,12.77,44.03,IS.02,1242.40,a.73,29.79,10.83,7.857.Sn_C-7il CA n e

(170)

U21417,5

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U21617,5

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L21417,5

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L216

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(171)

U 23414,3S

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M 09

09

accepted on behalf of the facultyof the Institute by the following coniinittee:

Adviser ~

TL./ ^ X.. .

-it Api^i I ^10Date

^ new mexico institute of mining and technologybaervan.nmt.edu/publications/media/pdf/thesis/zain,...

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