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Transnational Journal of Science and Technology August 2012 edition vol. 2, No.7

64

A NEW COMPUTERIZED APPROACH TO Z-FACTOR

DETERMINATION

Kingdom K. Dune, Engr.

Orij i, Bright N, Engr.

Dept. of Petroleum Engineering, Rivers State University of Science & Tech., Port Harcourt, Nigeria

AbstractThe compressibility factor of natural gases is a parameter that is used for various engineering

purposes in the petroleum industry. The Standing and Katz correlation, among other methods, is a

widely accepted method of determining z-factor manually from charts for natural gas of either

known or unknown composition. A major setback is that, for computer-based applications, it is not

convenient to obtain z by this means. This paper highlights the limitations of the other direct z-factor

determination methods and then presents a new approach of computing z-factor based on the

Standing and Katz correlation, which eliminates the limitations observed with the other methods. A

set of z-factor equations were developed by regressing data (in different ranges of pseudo-reduced

temperatures and pressures) obtained from the Standing-Katz and Brown et al correlations using a

Visual Basic program. Results obtained from this approach were checked against those from the

other correlations and were found to be more accurate than the other methods, with average absolute

error in z less than 0.1%. A subroutine for calculating z-factor could easily be incorporated into any

window-application program written in MS Excel, MATLAB or visual Basic, using equations

developed in this approach when determining the properties of natural gas, estimating gas reserves,

sizing oil and gas separators, designing gas transmission pipelines, and pressure traverses in pipesfor multiphase flow conditions. A standard z-factor table may also be computed using the set of

equations for all ranges of pseudo-reduced temperatures and pressures.

Key Words: Compressibility, z-factor, correlation, pseudo-pressure, and pseudo-temperature


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1. Introduction

The z-factor comes into play for various engineering purposes, which include estimation of

gas reserves, design of oil and gas separators, design of pipelines for the transmission of produced

gas, among others.

A number of these tasks or procedures have been developed in such a way that makes it

necessary to employ the services of a computer in order to accomplish them in reasonable time.

Carrying out such tasks by hand would make it lengthy, tedious and as such time consuming. An

example of such tasks or procedures is the determination of pressure traverses in pipes for

multiphase flow conditions.

The Beggs and Brill method of calculating pressure traverses, for instance, is one that

requires the gas compressibility factor. This method, involving about 21 steps, is an iterative one

wherein a pressure drop is obtained at the end of each iteration using, among other data, an initial

assumed pressure drop. If the difference between the initial and calculated pressure drops is

substantial, the iteration is repeated with the calculated pressure drop in each iteration serving as the

assumed pressure drop for the next iteration. This process is continued until the difference between

the assumed and calculated pressure drops is small. Arriving at a value for the final pressure drop

typically requires a number of iterations.

What this means is that the working or operating pressure changes with each successive

iteration making it a necessity to obtain, for each iteration, all data that are pressure dependent one of

which is the gas z-factor. It is quite evident from the foregoing that manually obtaining z from the

chart and entering the value into the computer, for each iteration, is rather inconvenient as it would

undoubtedly slow down the computation process.

Programming such tasks as the Beggs and Brill method (Brown and Beggs, 1977) for

calculating pressure traverses in pipes for multiphase flow conditions cut down on the amount of

time required for the calculation. Such reduction in computation time could be increased if a means

was devised to incorporate the determination of gas compressibility factor into the program thuseliminating the need to manually obtain it from the chart for successive iterations. How can this be

accomplished?

This paper reviews existing literature and presents a new approach for determining z-factor

for computer-based applications. Three other correlations that can be programmed for

use considered in this paper are those of Hall and Yarborough, Beggs and Brill, and Drankchuk


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and Abou-Kassem. The limitations that make them unfit for use for engineering purposes requiring

precision are highlighted.

2. Background

There are various correlations available for the calculation of gas compressibility factors.

Using these correlations or equations of state (EOS), one can program the computer to solve directly

for z. The correlations or equations of state considered for such purpose are Standing & Katz (1959),

Hall and Yarborough, Beggs and Brill, and Dranchuk and Abou-Kassem.

Standing and Katz Correlation

Since z is a function of the gas pseudo-reduced temperature (T pr ) and pressure (P pr ), it is

necessary to first determine the pseudo-critical temperature (T pc) and pressure (P pc) of the gas and

subsequently use these to obtain the pseudo-reduced temperature (T pr ) and pressure (P pr ).

For natural gas of known composition, the pseudo-critical pressure and temperature can be

determined from Kay's mixing rule (Bradley, 1987) which gives these properties as:

P pc = yi Pci (1)

T pc = yi Tci (2)

Where P pc = pseudo-critical pressure of gas mixture, T pc = pseudo-critical temperature of gas

mixture, Pci = critical pressure of component i in the gas mixture, Tci = critical temperature of

component i in the gas mixture, and yi = mole fraction of component i in the gas mixture

For a gas whose complete analysis is not known, a correlation developed by Brown et al can be used.

This correlation, presented in graphical form, relates the pseudo critical temperatures and pressures

of naturally occurring systems with their specific gravities (Katz et al, 1959). Having determined the

T pr and P pr , z may be obtained from either the Standing-Katz (Fig. 2) or the Brown et al chart (Fig. 3)

Hall-Yarborough Equation

The equation given by Hall and Yarborough (Ikoku, 1984) is given below:

y

te P z

t

pr

212.106125.0

(3)


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where t = Tc / T, and y = the reduced density which is obtained as the solution of the equation:

This method is designed specifically to fit the Standing-Katz charts. Since the equation contains both

z and M (which is a function of z) the solution is thus arrived at by iteration using the Newton-

Raphson method.

The Beggs and Brill Correlation

The correlation by Beggs and Brill (Golan and Whitson, 1986) for the calculation of z is given

below:

D pr B CP e A A z 1

(4)

Where:

101.036.092.039.1 5.0 pr pr T T A

6

9

2

110

32.0037.0

86.0

066.023.062.0 pr

pr

pr

pr

pr pr P T

P T

P T B

pr T C log32.0132.0 , and21824.049.03106.0

10 pr pr T T

D

The Dranchuk and Abou-Kassem Equation of State

Dranchuk and Abou-Kassem (Lee and Wattenbarger, 1996) developed their equation of state

primarily to estimate the z factor with computer routines. The form of the Dranchuk and Abou-

Kassem EOS is:

pr pr pr pr pr pr pr pr T cT cT cT c z 45

3

2

211 (5)

Where pr = 0.27P pr /(zT pr ); 554

44

3

3

1

211

pr pr pr pr T AT AT AT A AT c ;

28762

pr pr pr T AT A AT c ; 2

8

1

793

pr pr pr T AT A AT c ; and 22211104 1 pr pr pr pr pr T A AT c

The constants A1 through A11 are as follows: A1 = 0.3265; A2 = -1.07; A3 = -0.5339;

04.422.2427.90

58.476.976.141

06125.0

82.218.232

232

3

43212.1 2

t

t

pr

yt t t

yt t t y

y y y yte P y


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A4 = 0.01569; A5 = -0.05165; A6 = 0.5475; A7 = -0.7361; A8 = 0.1844; A9 = 0.1056;

A10 = 0.6134; A11 = 0.721

The Dranchuk and Abou-Kassem EOS must be solved iteratively since the z factor appears

on both sides of the equation. The solution of this equation can be obtained by employing a rootsolving technique such as the Newton's method or the secant method.

Oriji (2003) while programming these methods made the following observations:

The Beggs and Brill method, while being quite accurate for certain ranges, is not applicable

when T pr < 0.92. In determining the value of the temperature dependent term A, it is necessary to

evaluate the square root of (T pr – 0.92) which would mean an imaginary root when T pr < 0.92.

Also, for some values of T pr and P pr , the temperature and pressure dependent term B, gets so

large that evaluating eB results in an overflow of values. Negative values for z were sometimes

obtained from the method for some values of T pr and P pr

The Dranchuk and Abou-Kasem method, for the most part, gave good results for z, but the EOS

involves the use of an iterative method such as the Newton's method, necessitating an

assumption before convergence would occur. Once convergence is obtained the final value is

given as the calculated z factor. It was, however, observed that in some instances different initial

or assumed values of z resulted in convergence to different values at the end of the iteration thus

resulting in different final values for z for the same set of T pr and P pr values. There were even

cases where using certain initial values for z resulted in a negative value for compressibility

factor. So despite its accuracy, this method for obtaining z factor may not be incorporated into a

design program since it is not possible to predict or determine when such erroneous values may

result.

Therefore, another method was sought that would give values for the gas compressibility factor

without the limitations highlighted above.

3. Theoretical Development

This approach is based on the Standing and Katz method which is generally accepted as the

industry standard and were developed from data collected on methane and natural gases (Bradley,

1987). In addition to the Standing-Katz charts, the charts by Brown et al for low-pressure systems

were also used.


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The compressibility factor charts are essentially curves with the gas compressibility factor, z,

being a function of the pseudo-reduced pressure P pr . These curves appear on the charts for various

values of T pr the pseudo-reduced temperature.

In this method, pseudo-reduced pressure values were selected and regressed with

corresponding z values obtained from the charts to give equations that expressed z as a function of

P pr . This regression process had to be carried out for each pseudo-reduced temperature value on the

chart.

In a bid to ensure that the regression process gave rise to equations that were as accurate and

reliable as possible, two regression exercises were carried out for each set of values. These two

different exercises were carried out in such a way that they yielded two different equations – one

linear and the other quadratic. Both equations expressed z in terms of P pr . The equations were of the

form:

z = A(P pr ) + B (6)

z = A(P pr )2 + B(P pr ) + C (7)

Where A, B, C are constants

These two equations were subsequently tested over a range of P pr values and the equation

that gave z values that were more in agreement with values obtained from the charts was adopted. In

some cases the linear equation proved more accurate while in others the quadratic was more

accurate. Table 1 shows a sample data and the resultant regression equation.

In many cases, however, it was not possible to obtain an equation that gave accurate results

for the entire range of P pr values. Thus to ensure that the computer generates accurate values for z, it

was necessary in such cases to divide the range of P pr values into several sub-ranges and then obtain

equations for these sub-ranges. In these instances also, two equations – one linear and the other

quadratic – were obtained and tested for each of these sub-ranges. The more accurate and reliable

equations were adopted (See Table 2).

There were instances where it was not possible to obtain satisfactory equations that give

accurate z-factor for certain ranges of P pr values. In these cases, a number of values of P pr along with

the corresponding z-factor values were selected for interpolation. Using these predetermined values


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one can obtain accurate z values for this range of P pr via interpolation. A summary of the equations

derived from the regression processes and by means of which z values may be obtained is found in

Table 3

As was the case for the z-factor, various values of specific gravity were regressed with the

corresponding values of pseudo-critical temperature and pressure respectively setting the specific

gravity as the independent variable. This process yielded two (2) equations of the form:

X pc = A (g) 2

+ B (g) + C (8)

Where X pc = pseudo-critical constant (temperature or pressure), g = specific gravity of natural gas

and A, B, C are constants

Dune and Oriji (2007) regressed data obtained from the Standing and Katz method and

obtained the following equations:

T pc = 158.01 + 342.12(g) – 16.04(g)2

(9)

P pc = 688.634 – 21.983(g) – 13.886(g)2

(10)

4. The Computer Program

Using the equations in Table 3 in conjunction with equations (9) and (10), a Visual Basic

program (Siler and Spotts, 1998), to compute compressibility factor (z), was developed. For a gas of

unknown composition, the program accepts the gas specific gravity as an input parameter and with

this computes the gas pseudo-critical temperature and pressure from equations (9) and (10). If,

however the gas composition is known, equations (1) and (2) are utilised to compute T pc and P pc

after the gas composition has been entered as required. With the T pr and P pr values, z can be got

using the appropriate equation from Table 3. For those range of P pr values for which an adequate

equation could not be found, a routine that interpolates between predetermined values of P pr and z, to

give the required z value, was incorporated into the program.

The interpolation routine was extended to cover the entire process of determining z so that

even if the pseudo-reduced temperature value is not explicitly incorporated into the program, it is

still able to give a value for z. For example, if the input data results in a T pr value of 1.45 and a P pr


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value of 1.2, the program obtains a value for z by determining values for z at T pr = 1.4 and T pr = 1.5

when P pr =1.2 and then interpolating between these values to give the appropriate value for z.

5. Results and Discussion

Results obtained from the computer program were analyzed to ascertain their level ofaccuracy. In order to determine the accuracy of the compressibility factor values obtained from this

method, z values obtained from it were compared in Table 4 with those obtained from the

correlations by Hall and Yarborough, Beggs and Brill, Dranchuk and Abou-Kassem and manual

reading of the appropriate compressibility factor charts (Standing and Katz) for several pseudo-

reduced temperature and pressure values.

Table 4: Compressibility Factor, Z, for Various Methods.

T pr P pr This

Approach

Hall-

Yarborough

Beggs &

Brill

Dranchuk &

Abou-Kassem

Standing &

Katzs

0.75 0.048 0.9512 1.0069 N/A 0.9546 0.951*

1.30 0.020 0.9950 0.8840 0.9977 0.9969 0.9967*

1.62 0.065 0.9970 1.0005 0.9959 0.9950 0.9956*

1.20 1.500 0.6740 0.1605 0.6759 0.6532 0.6730*

1.80 0.900 0.9562 1.0036 0.9599 0.9550 0.9700

2.00 1.200 0.9718 1.0041 0.9690 0.9621 0.9700

2.60 13.000 1.2693 1.3987 0.8222 1.2732 1.0500

2.88 6.000 1.0615 1.1436 -0.4410 1.0607 1.0620

1.76 3.500 0.8776 0.9739 0.8727 0.8818 0.8768

1.25 10.200 1.1824 1.0038 N/A 1.1818 1.1825

*Values were obtained from z-factor charts developed by Brown et al (Bradley, 1987; Fig 3)

The extent to which results from this approach and the other methods deviated from those

obtained from the Standing-Katz method for a given T pr of 1.20 can be seen from Table 5. A plot of

absolute error versus pseudo reduced pressure for the different methods were made (Fig. 1). A look

at Table 5 as well as the plot (Fig. 1) reveals that the results from this approach are actually more

accurate and in harmony with those obtained from the Standing-Katz and Brown et al methods thanthose from the other correlations considered.

A measure of the degree of accuracy of the various methods is seen when one considers the

error in computing z-factor from the various methods. These errors are computed with the results

obtained from the Standing-Katz and Brown et al charts serving as the reference. Whereas errors

greater than 1% were obtainable (in some cases) with the 3 correlations considered, the error


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associated with this approach do not exceed 0.2%. Indeed, the average error associated with this

approach, for the T pr and P pr values indicated, is less than 0.1%. This is much better than the

averages of the other correlations which are in excess of 0.5%. Thus utilizing z values from this

approach yields results that are accurate, reliable and acceptable.

Table 5: Absolute Error in z, % Deviation from Standing & Katz for Tpr = 1.20

P pr

z-factor Absolute Error in z, %

Hall-

Yarboro

ugh

Beg

gs &

Brill

Dranchuk

&

Abou-

Kasem

This

Appro

ach

Standi

ng

&

Katz

Hall-

Yarboro

ugh

Beggs

&

Brill

Dranchu

k-

Abou-

Kasem

This

Appro

ach

0.0

48

0.9934 0.99

23

0.9904` 0.9897 0.9899

*

0.3536 0.2424 0.0505 0.0202

0.5 0.8934 0.90

26

0.8951 0.9000 0.900* 0.7333 0.2889 0.5444 0.000

0.9 0.7999 0.81

27

0.8027 0.8160 0.815* 1.8528 0.2822 1.5092 0.1227

1.5 0.6576 0.67

59

0.6532 0.6740 0.673* 2.2883 0.4309 2.9421 0.1486

2.5 0.5218 0.49

78

0.5181 0.5199 0.520 0.3462 4.2692 0.3654 0.0192

3.5 0.5618 0.56

05

0.5632 0.5661 0.567 0.9171 1.1464 0.6702 0.1587

6.0 0.7906 0.80

53

0.7937 0.7897 0.790 0.0759 1.9367 0.4684 0.0380

8.0 0.9848 0.99

86

0.9870 0.9902 0.989 0.4247 0.9707 0.2022 0.1213


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10.

2

1.1963 1.20

93

1.1959 1.1952 1.195 0.1088 1.1967 0.0753 0.0167

13.

0

1.4602 N/A 1.4550 1.4565 1.458 0.1509 N/A 0.2058 0.1029

15.

0

1.6452 N/A 1.6358 1.6432 1.643 0.1339 N/A 0.4382 0.0122

Average Absolute Error, % 0.6714 1.196 0.6792 0.0691

*Values were obtained from z-factor charts developed by Brown et al4 (Fig 3)

Figure 1: Absolute Error in z, % Deviation from Standing & Katz for Tpr = 1.20

Graph of Error in z-Factor Vs Ppr

0

0.75

1.5

2.25

3

3.75

4.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Ppr

A b s o l u t e E r r o r ( %

Hall-Yarborough

Beggs & Brill

Dranchuk & Abou-Kassem

This Approach


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6. Conclusion

A comparison of results from this approach with those obtained from other methods shows

that the results from this approach are more reliable and accurate. The method is therefore fit for use

for all computer-based applications that require the gas compressibility factor.

A look at the summary of equations obtained and utilized by this method reveals that the

equations for the various ranges of T pr and P pr are numerous. Programming these equations for

use is therefore recommended as the most practical way of using this method.

References:

Bradley, H. B.: Petroleum Engineering Handbook, SPE, Richardson, TX, chap. 12, chap 20, 1987.

Brown, K. E. and Beggs, H. D.: The Technology of Artificial Lift Methods Vol. 1, PennWell Books,

Tulsa, OK, 1977, p 85.

Dune, K. K. & Oriji, B. N.: “Alternative correlation for the computation of critical temperature &

pressure.” Global Journal of Engineering Research, Calabar, vol. 5, no. 1, 2007, pp 69-74,

Golan, M. and Whitson, C. H.: Well Performance, D. Reidel Publishing Co., Dordrecht, Holland,

1986, pp. 17 – 21.

Ikoku, C. U.: Natural Gas Production Engineering , John Wesley and Sons, N.Y., 1984.

Katz, D. L. et al: Handbook of Natural Gas Engineering , McGraw-Hill Books Co., N.Y., 1959.

Lee, J. and Wattenbarger, R. A.: Gas Reservoir Engineering , SPE, Richardson, TX, 1996, pp. 6 – 7.

Oriji, B. N.: Separator Design: A Computerized Approach, B.Tech Thesis, Rivers State University

of Science and Technology, 2003 pp 14 – 20.

Siler, B. and Spotts, J.: Special Edition Using Visual Basic 6, Que, 1998.

Nomenclature

e -Euler's constant 2.7182


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-gas specific gravity

M -Molar mass of gas

P -pressure, psia

Pc -critical pressure, psia

Pci -critical pressure of ith component in gas mixture, psia

P pc -pseudo-critical pressure, psia

P pr -pseudo-reduced pressure

g -density of gas, lbm/ft3

pr -pseudo-reduced density

T -temperature,o

R

Tc -critical temperature,oR

Tci -critical temperature of ith component in gas mixture,oR

T pc -pseud0-critical temperature,oR

T pr -pseudo-reduced temperature

yI -mole fraction of ith component in the gas mixture

z -gas compressibility factor

Table 1: Regression Data Obtained From Brown et al z Factor Chart

T pr = 0.9 0 P pr 0.07

Selected P pr Corresponding z

0.010 0.9947

0.020 0.98930.035 0.9814

0.050 0.9732

0.060 0.9678

Resultant Equation: z = - 0.537647 (P pr ) + 1.000098


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Table 2: Regression Data Obtained From Standing-Katz z Factor Chart for T pr = 2.2

10 P pr 15 4 P pr 10

Ppr z Ppr Z

10.5

12.0

13.0

14.0

15.0

1.17

1.238

1.280

1.322

1.361

5

7

8

9

10

0.988

1.045

1.077

1.113

1.156

Resultant equation: Resultant equation:

z = 0.04109 P pr + 0.745537 921786.0003488.0001988.0 2 pr pr P P z


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Figure 2: Compressibility Factor Chart of Natural Gases After Standing and Katz2

Fig 3: Gas Compressibility at Low Reduced Pressures After Brown et al4

Table 3: Summary of Equations used to evaluate compressibility factor, z

T pr Range of P pr Equation for z

0.60 0 ≤ P pr ≤ 0.016 000075.1527273.3 pr P z

0.65 0 ≤ P pr ≤ 0.036 19.1 pr P z

0.70 0 ≤ P pr ≤ 0.07 9998172.03610345.1 pr P

0.75 0 P pr 0.07 000017.1017619.1 pr P z

0.8 0 P pr 0.07 99994.0815.0 pr P z

0.8 0.07 < P pr 0.27 929185.0611208.0641854.0 2

pr pr P P z

0.85 0 P pr 0.07 99995.065923.0 pr P z

0.90 0 P pr 0.07 000098.1537647.0 pr P z

0.90 0.07 < P pr 0.54 Interpolate Using Predetermined

0.95 0 P pr 0.07 000105.1459742.0 pr P z


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1.0 0 P pr 0.07 000112.13818375.0 pr P z




1.0 1.0 < P pr 1.2 892978.3451697.6817158.2 2

pr pr P P z

1.0 1.2 < P pr 1.6 27348.0215878.0131016.0 2

pr pr P P z

1.05 0 P pr 0.8 997714.0291905.0083333.0 2

pr pr P P z


1.05 1.6 < P pr 2.0 61.0465.015.0 2

pr pr P P z

1.05 2.0 < P pr < 4.0 033131.0121228.0001027.0 2

pr pr P P z

1.05 4.0 P pr < 7.0 026571.0149143.0002286.0 2

pr pr P P z

1.05 7 P pr 15 115012.0116867.0000515.0 2

pr pr P P z

1.10 0 P pr 0.07 000131.1275172.0 pr P z


1.10 1.6 < P pr < 2.3 21503.1884788.0230864.0 2

pr pr P P z

1.10 2.3 P pr 3.5 355425.0050591.0026257.0 2

pr pr P P z

1.10 3.5 < P pr < 7.0 086866.0120452.00007.0 2

pr pr P z

1.10 7 P pr 15 189738.0102012.0 pr P z

1.2 0 P pr 0.07 000147.1216207.0 pr P z


1.2 1.6 < P pr < 3.4 Interpolate Using Predetermined1.2 3.4 P pr < 5.5 313023.0062909.0002686.0

2 pr pr P P z

1.2 5.5 P pr 8.0 210439.0093761.0000464.0 2

pr pr P P z

1.2 8 P pr 15 243254.009333.0 pr P z

1.3 0 P pr 1.6 Interpolate Using Predetermined

1.3 1.6 < P pr < 3.8 Interpolate Using Predetermined

1.3 3.8 P pr < 6.0 59884.0026463.0010114.0 2

pr pr P P z

1.3 6 P pr 15 28228.0086983.0 pr P z

1.40 0 P pr 0.07 000008.112459.0 pr P z

1.40 0.07 < P pr 1.6 000687.1121092.0009738.0 2

pr pr P P

1.40 1.6 < P pr < 4.0 0735735.1221064.003299.0 2

pr pr P P z

1.40 4.0 P pr < 6.0 76088.0056954.0011371.0 2

pr pr P P z

1.40 6 P pr 15 361643.0077786.0 pr P z

1.50 0.0 P pr 1.6 004459.110381.0013426.0 2

pr pr P P

1.50 1.6 < P pr 3.0 013774.1127019.0015808.0 2

pr pr P P z


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1.50 3.0 < P pr < 5.0 994725.0128492.0018409.0 2

pr pr P P z

1.50 5.0 P pr < 8.0 647786.0015229.0003429.0 2

pr pr P P z

1.50 8.0 P pr 11.5 404538.0071736.0 pr P z

1.50 11.5 < P pr 15 4254.00711.0 pr P z

1.60 0 P pr 0.07 000132.1067933.0 pr P z

1.60 0.07 < P pr 1.6 999739.0074544.0008579.0 2

pr pr P P z

1.60 1.6 < P pr 3.6 006528.10975.0012337.0 2

pr pr P P z

1.60 3.6 < P pr


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2.4 4.0 P pr < 9.5 931105.0009525.00013045.0 2

pr pr P P z

2.4 9.5 P pr 15 7169.0048873.0000458.0 2

pr pr P P z

2.6 0.0 P pr < 4.0 001055.1007958.0002286.0 2

pr pr P P z

2.6 4.0 P pr < 10.0 949436.0010063.0001098.0 2

pr pr P P z

3.0 0.0 P pr 3.0 99927.0006209.0000699.0 2

pr pr P P z

3.0 3.0 < P pr < 10.0 01047.10000815.0001576.0 2

pr pr P P z

3.0 10.0 P pr 15 845286.0031414.0000071.0 2

pr pr P P z

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File Asli Beggs and Brill