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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
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)
Distance Measurement for CPE 215 (d = 4)
Distance Measurement for CPE 215 (d = 5)
Mean Vector Measurement for CPE 215 (d = 3)
Mean Vector Measurement for CPE 215 (d = 4) .
Mean Vector Measurement for CPE 215 (d = 5) .
Distance Measurement for CPE 207 (d = 5) ....
Mean Vector Measurement for CPE 207 (d = 5) .
Distance Measurement for CPE 214 (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
Rotated Factor Matrix for CPE 215
Rotated Factor Matrix for CPE 216
Rotated Factor Matrix for CPE 234
Rotated Factor Matrix for CPE 247
Rotated Factor Matrix for CPE 238
Rotated Factor Matrix for CPE 246
Rotated Factor Matrix for CPE 239
Rotated Factor Matrix for CPE 244
Rotated Factor Matrix for CPE 245
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
3-16. Mean Vector Plot for CPE 207 (d = 5) 69
•
H1
r>
Distance Plot for CPE 214 (d = 5) 71
3-18. Mean Vector Plot for CPE 214 (d = 5) 72
3-19. Distance Plot for CPE 216 (d = 5) 74
3-20. Mean Vector Plot for CPE 216 (d = 5) 75
3-21. Distance Plot for CPE 234 (d = 4) 77
3-22. Mean Vector Plot for CPE 234 (d = 4) 78
3-23. Distance Plot for CPE 247 (d = 4) 80
•
OJ1
Mean Vector Plot for CPE 247 (d = 4) 81
3-25. Distance Plot for CPE 238 (d = 4) 83
3-26. Mean Vector Plot for CPE 238 (d = 4) 84
3-27. Distance Plot for CPE 246 (d = 4) 86
3-28. Mean Vector Plot for CPE 246 (d = 4) 87
3-29. Distance Plot for CPE 239 (d = 5) 89
3-30. Mean Vector Plot for CPE 239 (d = 5) 90
3-31. Distance Plot for CPE 244 (d = 6) 92
3-32. Mean Vector Plot for CPE 244 (d = 6) 93
3-33. Distance Plot for CPE 245 (d « 6) 95
3-34. Mean Vector Plot for CPE 245 (d = 6) 96
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
3-47. Factor Loading Plot for Upper CPE 214 138
3-48. Factor Loading Plot for Lower CPE 214 138
3-49. Factor Loading Plot for Upper CPE 215 140
3-50. Factor Loading Plot for Lower CPE 215 140
3-51. Factor Loading Plot for Upper CPE 216 142
3-52. Factor Loading Plot for Lower CPE 216 142
3-53. Factor Loading Plot for Upper CPE 234 144
3-54. Factor Loading Plot for Lower CPE 234 144
3-55. Factor Loading Plot for Upper CPE 247 146
3-56. Factor Loading Plot for Lower CPE 247 146
3-57. Factor Loading Plot for Upper CPE 238 148
3-58. Factor Loading Plot for Lower CPE 238 148
3-59. Factor Loading Plot for Upper CPE 246 150
3-60. Factor Loading Plot for Lower CPE 246 150
(ix)
Fiqvir?
3-61. Factor Loading Plot for Upper CPE 239 152
3-62. Factor Loading Plot for Lower CPE 239 152
3-63. Factor Loading Plot for Upper CPE 244 154
3-64. Factor Loading Plot for Lower CPE 244 154
3-65. Factor Loading Plot for Upper CPE 245 156
3-66. Factor Loading Plot for Lower CPE 245 156
(»
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
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)
UPPER SAMPLE LOWER SAMPLE
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
UPPER SAMPLE LOWER SAMPLE
4)
SAMPLE # CLASS 1 CLASS 2 Y CLASS 1 CLASS 2 Y
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
UPPER SAMPLE LOWER SAMPLE
= 5)
SAMPLE # CLASS 1 CLASS 2 Y CLASS 1 CLASS 2 Y
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)
UPPER SAMPLE LOWER SAMPLE
SAMPLE # CLASS 1 CLASS 2 Y CLASS 1 CLASS 2 Y
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)
UPPER SAMPLE LOWER SAMPLE
SAMPLE # CLASS 1 CLASS 2 Y CLASS 1 CLASS 2 Y
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
SAMPLE # CLASS 1 CLASS 2 Y CLASS 1 CLASS 2 Y
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
Table 3-13. Mean Vector Measurement for CPE 214 (d = 5)
UPPER SAMPLE LOWER SAMPIjE
SAMPLE # CLASS 1 CLASS 2 Y CLASS 1 CLASS 2 Y
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)
Table 3-14. Distance Measurement for CPE 216 (d » 5)
UPPER SAMPLE LOWER SAMPLE
SAMPLE # CLASS 1 CLASS 2 Y CLASS 1 CLASS 2 Y
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)
UPPER SAMPLE LOWER SAMPLE
SAMPLE # CLASS 1 CLASS 2 Y CLASS 1 CLASS 2 Y
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)
UPPER SAMPLE LOWER SAMPLE
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
Table 3-17. Mean Vector Measurement for CPE 234 (d = 4)
UPPER SAMPLE LOWER SAMPLE
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)
Table 3-18. Distance Measurement for CPE 247 (d » 4)
UPPER SAMPLE LOWER SAMPLE
SAMPLE # CLASS 1 CLASS 2 Y CLASS 1 CLASS 2 Y
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
Table 3-19. Mean Vector Measurement for CPE 247 (d = 4)
UPPER SAMPLE LOWER SAMPLE
SAMPLE # CLASS 1 CLASS 2 Y CLASS 1 CLASS 2 Y
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
CL
AS
S1
(85)
Table 3-20* Distance Measurement for CPE 238 (d • 4)
UPPER SAMPLE LOWER SAMPLE
SAMPLE # CLASS 1 CLASS 2 Y CLASS 1 CLASS 2 Y
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)
UPPER SAMPLE LOWER SAMPLE
SAMPLE # CLASS 1 CLASS 2 Y CLASS 1 CLASS 2 Y
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
UP
PE
R
LO
WE
R
•sb
'd>
'A
CL
AS
S1
PH
AS
ES
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
ES
PL
IT
OjG
OU
PP
ER
CL
AS
S2
09
a4
o
CL
AS
S1
02
0
CL
AS
S1
(88)
Table 3-22, Distance Measurement for CPE 246 (d = 4)
UPPER SAMPLE LOWER SAMPLE
SAMPLE # CLASS 1 CLASS 2 Y CLASS 1 CLASS 2 Y
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)
UPPER SAMPLE LOWER SAMPLE
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
S2
00
xo
0.8
0
0.6
0
CL
AS
S2
0.4
0
0.2
0
Figu
re3-
30.
Mea
nV
ecto
rplo
tfo
rCPE
239
(d=
5)
CL
AS
S2
UP
PE
PH
AS
ES
PL
IT
CL
AS
S1
CL
AS
S1
LO
WE
R
u>
o
(91)
Table 3-24. Distance Measurement for CPE 239 (d =
UPPER SAMPLE LOWER SAMPLE
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
CL
AS
S2
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)
Table 3-26. Distance Measurement for CPE 244 (d = 6)
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
Table 3-27. Mean Vector Measurement for CPE 244 (d = 6)
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
CL
AS
S2
KO
U1
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)
Table 3-28. Distance Measurement for CPE 245 (d = 6)
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
Table 3-29. Mean Vector Measurement for CPE 245 (d = 6)
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
(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
SAMPLE FACTOR 1 FACTOR 2
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
SAMPLE FACTOR 1 FACTOR 2
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
SAMPLE FACTOR 1 FACTOR 2
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
SAMPLE FACTOR 1 FACTOR 2
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
SAMPLE FACTOR 1 FACTOR 2
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)
Ibble 3-36. Rotated Factor Matrix for CPE 234
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.
Fact
orL
eadi
ngpl
otfo
rU
pper
CPE
234
Figu
re3-
54.
Fact
orIx
iadi
ngpl
otfo
rIj
ower
CPE
234
IIO
RiZ
ON
^lA
LF
AC
mK
IV
EK
HC
AL
FA
CrO
R2
HO
RIZ
ON
TA
LF
AC
IOR
IV
OR
IIC
AL
FA
CIt
)R2
14
14
.U Ik.
-•' U-: V. .
(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,
Rgnr
e3-5
5.Fa
ctorL
oadin
gplot
forU
pper
247
HORI
ZON
IAL
FACT
OR1
VEKt
lCAL
FACT
OR2
12
1
Fi^r
e3-
56.F
acto
rLoa
ding
plot
for
Low
erCP
E24
7
HO
RIZO
NTA
LFA
CTO
RI
VE
RII
CaL
FACT
OR
2
12
Oi
12
(147)
Ibble 3-38. Rotated Factor Matrix for CPE 238
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
Figu
re3-
57.F
actor
ladi
ngplo
tfor
Uppe
rCPE
238
IlOKI
ZONT
ALFA
CmR
1VE
R!1C
ALFA
CIOR
2
3
4 14
Figure
3-58
.Fac
torLo
ading
plotf
orLo
werC
PE23
8
hori
zont
alFA
aOR
IVE
KIIC
ALFA
CTOR
3
7 14
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
Figure
3-S9.
Facto
rLoa
dingp
lotfor
Uppe
rCPE
246
IIORI
ZOm
ALFA
CTOR
1VE
KHCA
LFAC
IOR
2
13
12
Figu
re3-
60.F
actor
Load
ingplo
tforl
^wer
CPE2
46HO
RIZO
INTIA
LFAC
TOR
IVC
RHCA
LFA
CTOR
2
13
U1
O
(151)
Table 3-40. Rotated Factor Matrix for CPE 239
UPPER SAMPLE
SAMPLE FACTOR 1 FACTOR 2
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.
Facto
rLoa
dingp
lotfor
Uppe
rCPE
IIORI
ZONI
ALfa
ctor
IVE
RnCA
LFAC
TOR
26
89 1
2 15
23
9Fig
ure3-6
2.Fa
ctorL
oadin
gplot
forLo
werC
PE23
9ho
rizon
talF
AaOR
1VE
RTIIC
ALFA
CTOR
2
8 9 II 15
M Ul
to
(153)
Ibble 3-41. Rotated Factor Matrix for CPE 244
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
re3-
63.F
acto
rL
oadi
ngpl
otfo
rUpp
erC
PE24
4
IIO
RIZ
OW
AL
FA
CrO
RI
VE
Rfl
ICA
LF
AC
IOR
2
4
6 20
Figu
re3-
64.F
acto
rl^
oadi
ngpl
otfo
rIjo
wer
CPE
244
HO
RIZ
ON
TA
LFA
CTO
RI
VC
KFI
CA
LFA
CI'O
R2
20 8 S
U1
(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.
Facto
rLoa
dingp
lotfor
Uppe
rCPE
245
llaRI
Z<W
IAI.I
ACIX
)RI
VERI
ICAL
FACT
OR2
20
Figure
3-66.
Facto
rLoad
ingplo
tforL
ower
CPE2
45H0
RI20
N1AL
FACT
ORI
VEKH
CALF
ACTX
JR2
20
18
57
M U1
o\
(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
0.00,14.00,54.00,19.00,13.000.00,14.35,52.34,18.95,14.3517.63,13.09,42.27,15.51,11.5046.49,8.40,27.61,9.98,7.5159. 75, 6. 22, 20. 78,7.58, 5. 6766.58,4.91,17.50,6.38,4.6485.61,2.91,7.09,2.52,1.8797.97, 1. 32,0. 60,0.08,0.0497.88,1.13,0.84,0.11,0. 0498.02,1.08,0.77,0.10,0.0398. 17, 0. 99, 0.72, 0. 08, 0. 0398.10,0.95,0.83,0.09,0.0298. 20, 0. 88, 0.80, 0. 09, 0. 0398.37,0.90,0.65,0.07,0.0198. 41, 0. 80, 0. 70,0. 07, 0.0298.54,0.68,0.69,0.07,0.0298.60,0.57,0.74,0. 08,0.02
U21617,5
0.00,14.00,54.00,19.00,13.000.00,12.44,53.07,19.77,14.727.69,13.46,48.02,17.76,13.0738.18,10.30,31.03,11.79,8.7052.90,8.47,23.51,8.64,6.4863.18,7.24,17.98,6.64,4.9670.44,6.25,14.10,5.31,3.9076.07,5.53,11.15,4.20,3.0677.26,4.81,10.89,4.06,2.9983.31,4.03,7.64,2.89,2.1384. 35,3.34,7.35,2.83,2.1292.68,1.88,3.57,1.20,0.6794.21,1.76, 2.71,0. 86,0.4695.28,1.48,2.17,0.71,0.3695.65,1.28,2.08, 0.66,0.3296.17,1.19,1.79,0.57,0.2796. 57, 1. 08, 1.60, 0. 52, 0. 24
L21417,5
0.00,14.00,54.00,19.00,13.000.00,12.66,53.44,19*35,14.5529.95,10.15,36.68,13.30,9.9245.09,8.11,28.64,10.35,7.8155. 73, 6. 36,23. 22, 40, 6. 3060.12,5.28,21.15,7.70,5.7567.83,4.28,17.01,6^18,4.6769.80,3.96,15.89,5^88,4.4872.73,3.51,14.28,5.35,4.1371.26,3.39,15.16,5.77,4.4271.43,3.23,15.09,5.80,4.4572.34,3.02,14.56,5.71,4.3771.77,2.75,14.92,5.92,4.6072.85,2.58,14.24,5.82,4.5171.55,2.36,14.94,6.27,4.8970.68,2.09,15.27,6.70,5.2572.57,1.76,13.87,6.53,5.27
L216
17,5
0.00,14.00,54.00,19.00,13.00O.00,6.41,55.55,21.82,16.225.96,20.72,44.83,16.38,12.1035.41,10.56,32.93,12.12,8.9956.49,8.48,21.37,7.82,5.8365.77,7.20,16.36,6.13,4.5470.32,6.10,14.33,5.34,3.9171.21,5.33,11.79,4.40,3.2679.42,4.83,9.54,3.56,2.6680.42,4.39,9.21,3.43,2.5582.69,4.31,7.63,3.01,2.3682.36,2.95,7.89,3.58,3.2281.28,2.79,8.08,3.99,3.8680.24,2.46,8.33,4. :^2,4.5579.35,2.29,8.46,4.74,5.1792.64, 1.34,3. 13, l.<48, 1.4193.57,1.16,3.39,1.17,0.71
(171)
U 23414,3S
0.00,0.00,4.80,5.81,11.21,,1.53,1.50,1.38,1.31,1.20,
3.71,8.17,6.47,4.26,4.60,3.13,3.73,2.74,2.52,1.89,2.290.77,1.40,0.7,0.68,0.66,0.64,0.62,0.90,0.58,0.85,0.58,
0.30,0.59, 12. 38
26.50 0.38 4. 08, 4. 84, 8. 01, 7. 01, 5. 80, 4. 33, 3. 02, ^ m 27,2. 27,2. 64, 1. 35, 1. 73,1. 34,1. 63
,1.13 1.06 0. 98, 0. 93, 0. 83, 0. 55, 0. 99, 0. 50, 0. 48, 0. 47,0. 46,0. 44, 0. 64, 0. 41,0. 60,0. 41
,0.21 0.42 0. 78
49.62 0.84 3. 28, 3. 87, 5.53, 4. 72, 3. 92, 3. 02, 1. 33, 2. 15,1. 43, 1. 74, 1. 28, 1. 18,0.86,1, 07
,0.74 0,70 0. bS , 0. 61, 0. 56, 0. 36, 0. 65, 0. 33, 0. 32, 0. 31,0. 30,0. 23, 0. 42, 0. 27,0. 40,0. 27
,0. 14 0.27 5. 73
60.33 1.00 2. 67, 3. 33, 4. 20, 3. 54, 2. 96, 2. 28, 1. 50, 1. 63,1. 13,1. 32, 0. 97, 0. 83,0. 67,0. 81
,0.56 0.53 0. 43, 0. 46, 0. 42, 0. 27, 0. 43, 0. 25, 0. 24, 0. 23,0. 23,0. 22, 0. 32, 0. 20,0. 30,0. 20
,0.10 0.21 •4. 37
85.63 0.77 1. 20, 1. 80, 1. 60, 1. 51, 1. 22, 0. 84, 0. 54, 0. 56,0. 38,0. 43, 0. 30. 0. 27,0. 20,0. 24
,0.16 0.15 0. 13, 0. 12, 0. 11, 0. 07, 0. 13, 0. 06, 0. 06, 0. 06,0. 06,0. 05, 0. 08, 0. 05,0. 07,0. 05
,0.02 0.05 1. 04
94.73 0.23 0. 33, 1. 03, 0. 67, 0. 77, 0. 62, 0. 20, 0. 13, 0. 13,0. 03,0. 10, 0. 07, 0. 06,0. 05,0. 06
,0.04 0.04 0. 03, 0. 03, 0. 03, 0. 02, 0. 03, 0. 02, 0. 01, 0. 01,0. 01,0. 01, 0. 02, 0. 01,0. 02,0. 01
,0.01 0.01 0. 25
32.86 0.40 0. 32, 1. 58, 0. SI, 0. 95, 0. 80, 0. 42, 0. 26, 0. 26,0. 16,0. 17, 0. 11, 0. 10,0. 07,0. 07
,0.05 0.04 0. 03, 0. 03, 0. 02, 0. 01, 0. 02, 0. 01, 0. 01, 0. 01,0. 01,0. 01, 0. 01, 0. 01,0. 01,0. 01
,0.00 0.01 0. 14
33.60 0.28 0. 43, 1. 34, 0. 94, 0. 85, 0. 72, 0. 36, 0. 22, 0. 22,0. 14,0. 13, 0. 10, 0. 08,0. 06,0. 06
,0.04 0.03 0. 03, 0. 02, 0. 02, 0. 01, 0. 02, 0. 01, 0. 01, 0. 01,0. 01,0. 01, 0. 01, 0. 01,0. 01,0. 01
,0.00 0.01 0. 12
34.31 0.31 0. 44, 1. 40, 0. 68, 0. 75, 0. 55, 0. 33, 0. 20, 0. 20,0. 13,0. 14, 0. 03, 0. 07,0. 05,0. 06
,0.03 0.03 0. 02, 0. 02, 0. 02, 0. 01, 0. 02, 0. 01, 0. 01, 0. 01,0. 01,0. 01, 0. 01, 0. 00,0. 01,0. 00
,0.00 0.01 0. 08
36.84 0. 13 0. 19, 0. 85, 0. 41, 0. 42, 0. 45, 0. 14, 0. 03, 0. 03,0. 06,0. 06, 0. 04, 0. 03,0. 02,0. 03
,0.02 0.01 0. 01, 0. 01, 0. 01, 0. 00, 0. 01, 0, 00, 0. 00, 0. 00,0. 00,0. 00, 0. 00, 0. 00,0. 00,0. 00
,0.00 0.00 0. 05
34.64 0.25 0. 35, 1. 40, 0. 75, 0. 83, 0. 65, 0. 22, 0. 14, 0. 15,0. 03,0. 10, 0. 06, 0. 05,0. 04,0. 04
,0.02 0.02 0. 02, 0. 02, 0. 01, 0. 01, 0. 01, 0. 00, 0. 00, 0. 00,0. 00,0. 00, 0. 00, 0. 00,0. 00,0. 00
,0.00 0.00 0. 08
31.63 1.03 1. 48, 2. 53, 0. 78, 0. 75, 0. 50, 0. 23, 0. 13, 0. 16,0. 03,0. 10, 0. 07, 0. 06,0. 04,0. 04
,0.03 0.02 0. 02, 0. 02, 0. 01, 0. 01, 0. 01, 0. 01, 0. 00, 0. 00,0. 00,0. 00, 0. 01, 0. 00,0. 01,0. 00
,0.00 0.00 0. 03
33.85 0.28 0. 55, 2. 07, 0. 77, 0. 82, 0. 53, 0. 20, 0. 13, 0. 14,0. 08,0. 03, 0. 06, 0. 05,0. 03,0. 04
,0.02 0.02 0. 01, 0. 01, 0. 01, 0. 01, 0. 01, 0. 00, 0. 00, 0. 00,0. 00,0. 00, 0. 00, 0. 00,0. 00,0. 00
,0.00 0.00 0. 13
34.46 0. 11 0. 37, 1. 42, 0. 99, 0. 83, 0. 35, 0. 16, 0. 10, 0. 11,0. 07,0. 07, 0. 05, 0. 04,0. 03,0. 03
,0.02 0.01 0. 01, 0. 01, 0. 01, 0. 00, 0. 01, 0. 00, 0. 00, 0. 00,0. 00,0. 00, 0. 00, 0. 00,0. 00,0. 00
,0.00 0.00 0. 10
(172)
14,33 L234
0.00,0.00,4.16,3.63,10.79,10.01,8.49,6.39,4,39.4.68.3 "B 3 7-•'> B-> •> '-7 i oo|o!37'o'76,'n?4S '̂̂ '̂~9«10,1.34,4.06,4.77,7.32,6.83.5.77,4,39,2«92.2-12 2 IS 2 48 i rp i to i •oa t k*
10.'2S,'0.'31,'7.'64'"* ^ "•!o:el:V.S:!:!®:o:S:o: 3^,o:l3;!:":S:S;klV, i;ll; klf;k,0.16,0.32,4.8164.89,0.79,2.33,3.13,3.73,3.33,2.79,2.03,1.37,1.46,1.02,1.16,0.88,0.80,0.60,0.73
I?* 22,0.21,0.20,0.20,0.31,0.20,0.31,0.22,0.12,0.24,3.37
»*=3.0-SS.O.98,0. 74,0.68,0.31,0.61'n'?n'n'tS'2*;?Z' >8«0* '7,0.17,0.26,0.17,0.26,0.18fUtt lUfU* CBm 02
1.09,1.00,0.73,0.910.27,0.23,0.23,0.37,0.24,0.36,0.24
,Q.13,0. ^««, 3. 23
'S'S'0.99,1.13, 0.87,0.80,0.61, 0.74' 23* 22,0.21,0.21,0.31,0.20,0.29,0.20
,0.10,0.21,3.99
0.33,0.21,0.31,0.21r^*11,O.22,4.17
' 27,0.26, 0.23,0.37,0. 24,0.34,0. 24fO«IZfO* 24f4» 33
32« 0.31,0.31, 0.29, 0.42,0.27,0.40,0.26,0.lo,0.^o,O.2o
*ol'S*qT'S'51'r22*-27,1.73,1.31, 1.23,0.96,1. IB'0*16'0*3^*0*46 '
'S*?o'a'3o'5 2' 0.46,0.43,0.43,0.61,0.39,0.38,0.38fO*lBfQ« oBf 0» 49
^2,1.48,1.91,1.48,1.39,1.10,1.3646,0.43, 0.43,0.62,0.40,0.39,0.38
,U.l7,0.3B,0. 89
°?'no'?"rtB'JI'ff'̂ *lf*2"® '̂®*°'*'®'®®'̂ *^°''*®''2-2®'''®3'2*»2.»*64,l.3S,1.23,1.31f O. 21, O. 42, O. 99
»
14,38— - r ^ ^
S.7e,14.40,11.23,1.41,0.38,1
(173)
U238
m- m f ^ m f -m- m
24,0.12,0.13,0.08,0.08,0.0£01,0.00,0.00,0.00,0.00,0.00
(176)
. Uib
34,0.32
e,0.10,0.10,0.03,0.05,0.14,1.8748.13,22.33,1.75,2.16,2.08,0.10,1.69,3,0.74,0.47,0.55,0.33,0.37,0.34,0.33,3,0.15,0. 14,0. 14,0.0'7.n.20,2.3750.33,20.73,1.75,2.52,2.10,0.33,1.S3.3,0.66,0.43,0.51,0.37,0.35.0.32,0.31,
14.0.14,0.07,0.20,2,1 iTA -»e» 4 i*..
1.
.24,0.24.0.
.03,1.03,1.09,0.73,0.
.14,0.20,0.20,0.12,0.73,0.83,0.58,0.59,0.512,0.12,0.12,0.11,0.1w f ^ w f ^ ^ f m • mm f -w m
5.08,2.07,1.13,1.22,0.85,0.0.30,0.20,0.28,0.30,0.18,0.
.69,1.13,1.23,0.31, 1.
.23,0.40,0.41,0.25,0.
26,0.94,1.01,0.333,0.33,0.31,0.4
20,38 U244
,0.00
fl77>
. V. VV, V. vv, V. V. vu, u. O, OO, O. 0
. p, 0.35,0.15,0. 09,0.08,0.08,0. OS, 0.03,0. 03.00,0,00,0.00,0.00,0.00,0.00,0.00,0,00,0,0000,0,00
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3.
•I.
A«n
trix
Isre
ctaoR
ular
char
tof
numb
ers.
Ana
trix
issy
mbo
l.ize
Uby
aca
pital
lette
rand
itssiz
eis
shown
bytwo
subs
crip
ts;th
efi
rst
refe
rri«R
toth
emm
ber
ofro
ws,
the
seco
ndth
enu
nber
ofco
lumns
.Th
usre
pres
ents
the
natri
xA
with
rrow
san
dc
coluH
uis.
Any
mnnb
erIn
theM
trix
ister
med
anele
men
t.Th
us.
ajj
isthe
eleme
ntof
Ain
thel-t
hrow
andj-t
hco
luan.
Two'
matr
ices
are
said
tobe
equa
lif
alle
lemen
tsco
rresp
ond
exac
tly.
Th
atis
.A
-B
ifa
Ijb.
jfo
ral
li
and
j.Ili
ctr»
as,K
.scof
„«,
trix
is.,„
o,|,c
rin
the
ro«
»nj
culu
ms
arc-
into
rclia
iijcd
.It
iss,i
ri,oI
i,od
byan
apos
tropU
o.T
hus
A*is
the
tran
spos
eo
fm
atri
xA
.S
peci
alty
pes
of
mat
rice
sin
clud
e:
'̂g'̂t
anpu
larm
atrix
.Ha
smo
rerow
sth
anco
lumns
orvi
ceve
rsa.
Squa
rem
atrix
.Ha
sth
esam
enu
mbe
rof
rows
asco
lum
ns.
(b)
(c)
Squa
resy
mm
etric
mat
rix.
Asq
uare
mat
rixsu
chth
ata
a.j
for
all
valu
esof
ian
dj.
The
lower
left
trian
gula
rpa
rtof
the
matr
ixbe
lowth
edi
agon
alis
am
irror
imag
eof
the
uppe
rri
ght
tria
ngul
arpo
rtio
n.(d
)Di
agon
alm
atrix
.On
lythe
eleme
ntsin
the
prin
cipal
dia-
Bona
lar
eno
nzer
oan
dal
lot
her
elem
ents
are
zero
.Th
atis
ajj
0wh
enI
»j
but
a.^»
0wh
en1
/j
(e)
(f)
(r)
(h)
Iden
tity
matr
ix.
Adi
agon
alm
atrix
whos
edi
agon
alele
men
tso
ileq
ual
1.
Col
""'
"iih
nro
wIm
ton
lyoa
cco
lumn.
Bo.v
ecto
r.A
.utr
i..it
hn
colu
.ns
but
only
ono
row.
*"•
•"ri
xK
ithon
ero
wan
don
eco
luan
.M
:.trice
scan
l.ead
ded
toge
ther
(ors
ubtra
cted)
only
ifthe
yar
esi
ieco
ii.|.a
tibIe
.th
atIs
.eac
h.a
trix
must
have
the
sane
numb
ero
fro
ws
and
colu
mn
s.
MJit
lon
orsu
btra
ctio
nis
done
onan
elem
ent
bycle
ment
basis
.Th
usthe
elem
ents
ofC
inC
.A
.B
are
equa
lto
the
sums
ofco
rres|»
ndin
|.ele
men
tsof
A.m
l«:
c.
a.,
.b.
.fo
ral
li
ami
j.U
»J
I.M
;ttr
lxH
cft
nit
ion
A•
43
I
72
4
15
9
28
6
2.T
ran
siio
se
'47
12
32
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96.
Ty
pes
of
>l;
itri
ces
"sr
86
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recta
ng
ula
r
'20
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05
0
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1.
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go
nal
4.»
titr
ixA
dd
itio
n
S3
1
02
5
78
1
43
6
10
00
01
00
00
10
00
01
iden
tity
Siz
eo
fA
is4
by3;
A 43
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isth
etr
ans|
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seo
fA
abov
e.
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8'
12
4
70
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sq
uare
-1
n
3 0 4 1
co
lum
nv
e^
or
Ca
A♦
B
26
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43
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sq
uare
sy
mm
etr
ic
[35
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1]ro
wv
ecto
r
79
1
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6
98
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is O H X
00
Ul
5.
Hi.l
ti,.li
c.«lo
nof
«.rlc
esca
non
lybe
pcrfo
t-ej
Ift|,o
„,rt,e
rof
col,«
,.of
„,c,„
.fac
t.rU
e<,u»
l,o
th.m
..b.r
ofro
.sof
rhopo
st-f^
cor.
I,.C
.^
All
elem
ent
ofC
isde
fined
asfo
llow
s;
6.
7.
'u*I
,"it
•"iJ
"here
.is
then.«
bero
fcolu
m,o
fAand
thenn
nber
ofro
«of
B.A
roHof
cIs
prod
uced
by«i
ltlpl
yln«
the
firs
tro
uof
AU«
osea
chco
rn™of
«.Th
UIs
r.pe,
t«l
for
..ery
r«.„
fA.m
.lltil
l-C.
wit
rix
isi'O
Mpl
elc.
Hie
.Inor
prod
uctM
Knt
Isde
fined
asC
.A'A
.c
conta
insthe
s.«s
ofsq
uares
and
cros
spr
oduc
tsof
thero
.sof
A.
The
trace
ofa
squa
reut
rUI,
the
su.o
fIt
sdi
acon
alele
nent
sTl,
ew
trlx
analo
tofs
calar
divi
sion
Is«!
co.nl
ished
byIn
ver-
siun.
IfA
Isa
squa
re.a
trU
and
AB
.B-A
.I
then
BIs
said
tobe
theIn
verse
ofA.
The
notat
ion
A*'i
sco
Mon
lyu,
«lto
de-
no
leth
ein
ver
seo
fA
.
Findin
gthe
Invers
eof
.M
trUI,
arat
herc
o.vl
lcat«
lpro-
cedu
rean
dthe
reade
rIs
refe
rred
toan
y|oo
dtex
ton
«trU
alg
eb
rafo
rd
eta
ils.
^'t
rix
Mil
ltin
il
B
•A
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6.
7.
31
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aent
7-
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1♦
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2
(pos
t-fa
ctor
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re-f
acto
r)(p
rodu
ct]
P1
3
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1.
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"42
6"
13
21
06
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66
10.
"ajo
rp
rod
uct
nom
ent
^tc
thatt
hetra
ceof
both
produ
ctsI.
.qu.
,jfa
trix
inve
rsio
n.Gi
ven
thesq
uare
matr
ixA
Itn
fln.
1M
•«»
HM
trix
A,
It
tsn
ccess
ary
tofin
da
ajtr
ixB
such
that
AB>
|or
re]
lAj
(I]
Expa
ndin
gth
eab
ove
yiel
ds
•ll"
!!»
ii''u
"Zl'
ll"l
l'*!?
etc.
Bhen
h-sare
detem
ined
soas
tosa
tisfy
this
seto
fsl«
.ltan
eous
e.p,at
ions
then
B.A"
',,he
^
♦aj
^b^j
.. •
•♦
"ln'
'nl
•1
.0
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2*'22
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0
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2•
1.0
H 00
OS
8.
9.
Asq
uare
aatr
ixQ
issa
idto
beor
thoR
onal
If
Q'Q
an
uliu
ri;
I)is
ad
iag
on
alK
ttri
x.
Asq
uare
aatr
ixQ
issa
idto
beor
thcn
orm
alif
Q'Q
*=
I
The
Rank
ofa
Matr
ix.
The
rank
ofa
«atri
xma
ybe
defin
ed,
inte
r«s
ofits
colu
mor
roMve
ctor
s,as
the
numb
erof
linea
rlyin
-dc
penJ
ent
row,
orco
luan
.ve
ctor
spr
esen
tin
the
BKitr
i*.
Ano
ther
view
ofra
nkis
asfo
lio«s
.A
mat
rixX
can
beex
pres
sed
asthe
produ
ctof
twom
atrice
swh
oseco
snon
Crde
ris
r.If
Xca
nnot
beex
pres
sed
asthe
prod
uct
ofan
ypa
irof
.atr
ices
wUh
aco
nson
orde
rle
sstha
nr.
then
the
rank
ofX
isr.
Itca
nbe
appr
eciat
edfro
.th
isth
ata
very
large
matr
ix.ay
have
alow
rank
and
thus
l.cux
pres
sabl
eas
the
prod
uct
oftw
osm
alle
rtrices. Th
isis
the
basis
ofpr
actic
ally
all
«ilti
varia
te.e
thod
sof
dat
aan
aly
sis.
8.
9.
"•c.
:]•"""
QIs
orth
ogon
al
p>.S
LO-8
66O.
SJ
Qis
ortl
iono
rmal
Ran
ko
fa
Matr
w
63
12
84
16
J2
62
4
[.'3
CJC
..*]
Q*q
•0
0'
.I
«"r-
rrr:
:.;"
•'-
•<
The
rank
of*
,s,
for
ono
linea
rlyIn
depe
nden
tve
ctor
;al
lco
luTOs
(or
rows)
are
mltl
ples
ofea
cho
ther.
[21
43
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63
12-
48
416
6.
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624
^
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S"A
>1
53
-S25
IS.
21
0S
15
3
.S2
0IS
rank
ofA
Is2
for
ther
ear
etw
o'iK
oarly
indep
ende
ntire
ctorsi
col-
•«»
Ian
d2
are
Mltl
ples
ofea
cl.O
ther.
"Hie
rank
ofA
is3.
00
vj
'Mi;
cnv
alu
esan
JE
igen
vrg
ror.
filve
na
real
squa
resy
MM
tric
ki(
*ai
i.lvt
fct<
»rs
IIsi
ii'li
that
All
«X
u
or
All
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lls
0
or
(A«
AI)
u>
0
(U]
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-(a
i)
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h.„
rJer
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tcJ.It
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,,.
fc..
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11:!:
:""'"
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AQ•
QA
or
Q'A
CJ
or
A-
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A'
which
Is«f
.rr«I
t..s
.h.b
asic
.f.fc
.
Ifthe
reare
.no
nzero
eigen
.al^s
.the
basic
struc
ture
sue.
S«ts
thatt
uoSK
ll•at
rlces
conta
inthe
»««,
|„f„r
«,io
„do
e.
•1
n
A0
«
-
UA
A
.
M 09
09