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SPE 167579
Explicit Half Range Cosine Fourier Series Expansion For Z Factor
Lateef A. Kareem; King Fahd University of Petroleum and Minerals
Copyright 2013, Society of Petroleum Engineers This paper was
prepared for presentation at the Nigeria Annual International
Conference and Exhibition held in Lagos, Nigeria, 30 July1 August
2013. This paper was selected for presentation by an SPE program
committee following review of information contained in an abstract
submitted by the author(s). Contents of the paper have not been
reviewed by the Society of Petroleum Engineers and are subject to
correction by the author(s). The material does not necessarily
reflect any position of the Society of Petroleum Engineers, its
officers, or members. Electronic reproduction, distribution, or
storage of any part of this paper without the written consent of
the Society of Petroleum Engineers is prohibited. Permission to
reproduce in print is restricted to an abstract of not more than
300 words; illustrations may not be copied. The abstract must
contain conspicuous acknowledgment of SPE copyright.
Abstract: All natural gas processes, including production,
sweetening, drying, transportation, storage, metering and selling
requires that we possess ability to predict with great accuracy its
volume, pressure, temperature, specific heat capacity etc. Unlike
an ideal gas, a real gas does not expand and contract according the
simple pressure-volume-temperature relation. This deviation from
the ideal gas behavior is corrected with the use of z-factor. Z
factor is simply the ratio of the volume of a real gas to that
which equal mass of an ideal gas will occupy under the same
condition. Several computational methods of obtaining this factor
have been developed, but the implicit nature of these correlations
called for an explicit method of evaluating the multivariate
parameter. In this paper, over 6000 data points prepared by careful
laboratory measurement of behavior of natural gas mixture is used
to develop the model. The result was found to match the result to a
very high accuracy without over fitting. Introduction:
Compressibility factor (z-factor) of gasses is used to correct the
volume of gas estimated from the ideal gas equation to the actual
value.
It is required in all calculations involving natural gasses.
Several attempts have been made to determine the compressibility
factor as a function of the reduced property of the gasses. Result
of the experimental investigation into the z-factor-reduced
pressure-reduced temperature relation is available in form of
Standing and Kartz chat [Heidaryan et al. 2010; Sanjari and Lay
2012; Azizi et al. 2010; Londono et al. 2002; Dranchuk 1959;
Abou-kassem n.d.; Al-khamis 1995; Trube 1957; Ohirhian 2002; Jeje
and Mattar 2004; Corredor et al. 1992; Elsharkawy et al. 2000; Jr
1961; Jaeschke et al. 1991].
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2 SPE 167579
Figure 1: Plot of experimental measure of the z factor
For numerical computations, these values are fed into the
computing system and values not available are obtained by
interpolation between given values. The cost of interpolation
creates the requirement for a correlation that accurately predicts
the values on the chat. This has been attempted in two directions.
First is the data fitting without the use of an equation of state,
the second is the regression of experimental data about an equation
of state. One of the most notable members of the first group is an
explicit equation by Brills and Beggs (1974). To the second group
belong the most accurate correlations of evaluating the z-factor
till date. Examples of these correlations are Yarborough (HY),
Dranchuk, Purvis and Robinson (DPR) and Dranchuk and Abou Kassem
(DAK) correlations Brill and Beggs (BB) Compressibility Factor
1
Where 1.39 0.92 . 0.36 0.10 0.62 0.23 0.0660.86 0.037
0.3210
0.132 0.32 log 10 9 1 0.3106 0.49 0.1824
0 5 10 150.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Ppr
z
1.051.101.151.21.251.31.351.41.451.51.61.71.81.92.02.22.42.62.83.0
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SPE 167579 3
Hall and Yarborough (HY) Compressibility Factor
1 , 0.06125 . 14.76 9.76 4.58 , 90.7 242.2 42.4 ,
2.18 2.82 ,
1 0
Dranchuk and Abou Kassem (DAK) z factor
0.27
Where y is the root of the equation
1 1 0 , 0.27
,
0.3265, 1.0700, 0.5339, 0.01569, 0.05165,
0.5475, 0.7361, 0.1844, 0.1056, 0.6134,0.7210
Dranchuk, Purvis and Robinson (DPR) z factor
0.27
Where y is the root of the equation
1 1 0 ,
, , 0.27 0.31506237, 1.04670990, 0.57832720, 0.53530771,
0.61232032, 0.10488813, 0.68157001,0.68446549
Problems of associated with BB, HY, DAK and DPR: Brill and Beggs
correlation is the most successful explicit correlation for z, but
it leads to unacceptable errors at higher pressure and temperature
close to the critical temperature. While implicit correlations are
accurate at elevated temperature, they are also not very good
around the critical temperature. In addition to this, picking an
initial guess that falls within the basins of attraction of the
desired root is another problem that also leads to unwanted result
[Heidaryan et al. 2010; Sanjari and Lay 2012; Azizi et al. 2010].
The key to solving this problem is in Fourier expansion using the
sines and cosines basis. Fourier expansion gets better as the
number of terms increases thereby eliminating the problem of over
fitting associated with explicit fits in powers of independent
variables.
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4 SPE 167579
Figure 2: Plot of Hall and Yarborough z factor chart with
convergence problem
Figure 3: A plot of Dranchuk Abou Kassem z factor chart with
convergence problem
0 5 10 150.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Ppr
Z
1.05 1.11.15 1.21.25 1.31.35 1.41.45 1.5 1.6 1.7 1.8 1.9 2 2.2
2.4 2.6 2.8 3
0 5 10 150.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Ppr
Z
1.05 1.11.15 1.21.25 1.31.35 1.41.45 1.5 1.6 1.7 1.8 1.9 2 2.2
2.4 2.6 2.8 3
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SPE 167579 5
Figure 4: A plot of Dranchuk Purvis Robinson z factor chart with
convergence problem Fourier Series: In mathematics, a Fourier
series is a decomposition of periodic functions or periodic signals
into the sum of a (possibly infinite) set of oscillating circular
functions, namely sines and cosines.
cos sin
12 , 1 cos
1 sin
The use of Fourier can be extended to non-periodic functions and
made to be exclusively sine or cosine if extended into an old or
even function respectively. Expansion of z factor: The Fourier
expansion model is developed for pseudo reduced pressure ranging
from 0 to 15 and reduced temperature from 1.05 to 3. The half range
cosine expansion was selected so as to eliminate the Gibbs
phenomenon around the boundaries at 0 and 15.
, cos Where is the half range length of the reduced pressure
15 0 15 Since the is bivariate, and is being expressed as a
Fourier series in , then all the weights of the cosine terms are
functions of And is expressed as simple degree 6 polynomial
function of reciprocal of
,
0 5 10 150.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Ppr
Z
1.05 1.11.15 1.21.25 1.31.35 1.41.45 1.5 1.6 1.7 1.8 1.9 2 2.2
2.4 2.6 2.8 3
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6 SPE 167579
Then the z-factor could be written as
, , cos But for the discrete data is forced to be equal or less
than the number of pressure point for a give pseudo reduced
temperature. There is no point taking m to 300 since it a maximum
value of 61 produced a very accurate prediction of z. Thus
taking,
cos as the variable, we can perform least square regression to
evaluate , for 0 60 and 0 6.
, , cos Validation: The explicit correlation was validated by
generating the z factor chat and performing statistical error
analysis shown in the table below . Table 1: Stastical Analysis of
the Correlation
Maximum Absolute Error
1,2,3, .
Maximum Absolute Percentage Error
100% 1,2,3, .
Average Absolute Percentage Error
100%
Root Mean Square of Absolute percentage Error
100%
Correlation of Regression
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SPE 167579 7
The error in the estimation of z factor using the explicit
correlation is also compared with error generated by the implicit
correlations.
Figure 5: A plot of z factor chart using the explicit Fourier
correlation
Figure 6: A Comparison of Explicit Fourier correlation and the
measured z
0 5 10 150.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Ppr
z
1.051.101.151.21.251.31.351.41.451.51.61.71.81.92.02.22.42.62.83.0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
z-exp
z-es
t
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8 SPE 167579
Figure 7: A Comparison of Hall and Yabourough correlation and
the measured z
Figure 8: A Comparison of Dranchuk & Abou Kaseem correlation
and the measured z
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
z-exp
z-es
t
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
z-exp
z-es
t
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SPE 167579 9
Figure 9: A Comparison of Dranchuk, Purvis & Robinson
correlation and the measured z
The error plot shown in figure 10 shows that the explicit
Fourier z factor correlation generated are closer to zero than all
other correlations hence the far higher correlation of regression
coefficient. See table 2
Figure 10: Error plot for Hall & Yaborough, Dranchuk &
Abou Kassem, Dranchuk, Purvis & Robinson, and Fourier z factor
correlations
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
z-exp
z-es
t
0 1000 2000 3000 4000 5000 6000-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
Hall and YaboroughDranchuk & Abou KaseemDranchuk, Purvis
& RobinsonFourier z
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10 SPE 167579
Recommendation The Fourier series of a function has been known
to always converge to the mean of the value of the function at the
point of discontinuity. And by extension, it means the integral of
the Fourier series also converges to the integral of the function,
however the same is not always true for the derivative of the
Fourier series and the derivative of the function.
12
Hence, this correlation should not be used in the evaluation of
the Isothermal compressibility of the gas. Isothermal
compressibility is the change in volume per unit volume per unit
change in pressure
1
Using the real gas equation
, ,
1 1 1 1 1 In terms of reduced properties
1 1
1 1
,,
But because
1 1 , ,
Conclusion The Fourier series expansion of z factor is a very
accurate mean of evaluating the z factor explicitly. The explicit z
factor so developed has a better performance in terms of
correlation of regression, root mean square of error and maximum
absolute error as shown in the table below.
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SPE 167579 11
Table 2: Summary of comparison of the correlations
Maximum Absolute Error
Maximum Absolute Percentage Error
Hall and Yarborough 0.0221722 5.22538578 Dranchuk & Abou
Kaseem 0.0496129 13.6272523 Dranchuk, Purvis & Robinson
0.0494820 13.1811496
Fourier z 0.0166618 2.944261
Average Absolute Percentage Error
Root mean square percentage error
Hall and Yarborough 0.3129010 0.543093764 Dranchuk & Abou
Kaseem 0.3589575 0.752813852 Dranchuk, Purvis & Robinson
0.4410269 0.812348227
Fourier z 0.1409741 0.275287118
Coefficient of regression
Hall and Yarborough 0.9998811487 Dranchuk & Abou Kaseem
0.9998350091 Dranchuk, Purvis & Robinson 0.9997621456
Fourier z 0.9999724440 Nomenclature
Gas compressibility Pseudo reduced compressibility Number of
data point Universal Gas constant Pressure Pseudo critical pressure
Pseudo reduced pressure
Temperature Pseudo critical temperature Pseudo reduced
temperature Pseudo critical pressure Pseudo reduced pressure
Initial guess for iteration process Pseudo reduced density
Compressibility factor
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12 SPE 167579
Reference
ABOU-KASSEM, P.M.D.J.H., EQUATIONS OF STATE.
AL-KHAMIS, M.N., 1995. Evaluation of Correlations for Natural
Gas Compressibility Factors by.
AZIZI, N., BEHBAHANI, R. AND ISAZADEH, M. A., 2010. An efficient
correlation for calculating compressibility factor of natural
gases. Journal of Natural Gas Chemistry, 19(6), pp.642645.
CORREDOR, J.H., PIPER, L.D., TEXAS, A., MCCAIN, W.D. AND
HOLDITCH, S.A., 1992. Compressibility Factors for Naturally
Occurring Petroleum Gases.
DRANCHUK, R.A.P.D.B.R.P.M., 1959. GENERALIZED COMPRESSIBILITY
FACTOR TABLES. , (1).
ELSHARKAWY, A., HASHEM, Y. AND ALIKHAN, A., 2000.
Compressibility Factor for Gas Condensates. Proceedings of SPE
Permian Basin Oil and Gas Recovery Conference.
HEIDARYAN, E., MOGHADASI, J. AND RAHIMI, M., 2010. New
correlations to predict natural gas viscosity and compressibility
factor. Journal of Petroleum Science and Engineering, 73(1-2),
pp.6772.
JAESCHKE, M. ET AL., 1991. Accurate Prediction of
Compressibility ~ actors by the GERG Virial Equation. , (August),
pp.343350.
JEJE, O. AND MATTAR, L., 2004. Comparison of Correlations for
Viscosity of Sour Natural Gas. Proceedings of Canadian
International Petroleum Conference, pp.19.
JR, L.E.R., 1961. Simplified Graphical Method of Determining Gas
Compressibility Factors.
LONDONO, F.E., ARCHER, R.A., BLASINGAME, T.A. AND TEXAS, A.,
2002. SPE 75721 Simplified Correlations for Hydrocarbon Gas
Viscosity and Gas Density Validation and Correlation of Behavior
Using a Large-Scale Database.
OHIRHIAN, P.U., 2002. SPE 30332 Calculation of the Pseudo
Reduction Compressibility of Natural Gases. , (4), pp.116.
SANJARI, E. AND LAY, E.N., 2012. An accurate empirical
correlation for predicting natural gas compressibility factors.
Journal of Natural Gas Chemistry, 21(2), pp.184188.
TRUBE, A., 1957. Compressibility of Natural Gases. Journal of
Petroleum Technology, 9(1).
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SPE 167579 13
Appendix 1. The value of ,
0 1 2 3 4 5 6
0 1.544 -3.629 12.312 -19.352 10.642 2.667 -3.204 1 -1.264
12.865 -58.207 134.949 -171.040 110.811 -28.750 2 0.050 0.822
-7.984 25.384 -34.801 21.883 -5.294 3 0.479 -6.793 36.040 -95.710
135.666 -97.241 27.582 4 -0.082 0.689 -2.470 5.482 -7.824 7.087
-2.809 5 -0.354 2.998 -10.415 19.561 -22.036 14.754 -4.457 6 -0.491
5.531 -25.626 62.664 -85.018 60.625 -17.623 7 -0.604 6.988 -33.131
81.932 -111.266 78.530 -22.404 8 -0.315 3.895 -19.415 49.888
-69.432 49.593 -14.155 9 -0.174 2.029 -9.658 23.708 -31.418 21.175
-5.609
10 -0.231 2.568 -11.390 25.656 -30.628 18.189 -4.107 11 -0.015
-0.065 1.418 -6.395 12.869 -12.233 4.468 12 0.121 -1.658 9.068
-25.374 38.482 -30.007 9.412 13 0.413 -4.918 23.767 -59.655 82.006
-58.526 16.947 14 0.598 -6.806 31.529 -76.099 100.928 -69.692
19.570 15 0.527 -6.080 28.431 -68.964 91.522 -63.011 17.590 16
0.644 -7.153 32.281 -75.731 97.404 -65.101 17.665 17 0.471 -5.161
22.911 -52.810 66.637 -43.618 11.567 18 0.204 -2.240 9.903 -22.543
27.829 -17.614 4.451 19 0.135 -1.410 5.850 -12.281 13.645 -7.483
1.532 20 -0.121 1.436 -6.938 17.503 -24.295 17.609 -5.210 21 -0.287
3.209 -14.658 35.003 -46.109 31.790 -8.964 22 -0.253 2.880 -13.370
32.406 -43.222 30.089 -8.543 23 -0.333 3.727 -17.001 40.426 -52.865
36.065 -10.032 24 -0.333 3.780 -17.419 41.683 -54.635 37.233
-10.318 25 -0.283 3.180 -14.521 34.460 -44.828 30.327 -8.342 26
-0.273 3.031 -13.633 31.836 -40.717 27.070 -7.315 27 -0.173 1.895
-8.419 19.415 -24.504 16.052 -4.265 28 -0.148 1.592 -6.962 15.800
-19.602 12.604 -3.281 29 -0.101 1.071 -4.573 10.069 -12.052 7.429
-1.838 30 0.086 -0.932 4.120 -9.538 12.228 -8.239 2.282 31 0.140
-1.590 7.328 -17.588 23.180 -15.912 4.450 32 0.172 -1.922 8.756
-20.816 27.243 -18.608 5.185 33 0.213 -2.356 10.599 -24.871 32.129
-21.676 5.970 34 0.272 -3.006 13.523 -31.680 40.773 -27.343 7.469
35 0.190 -2.114 9.599 -22.727 29.586 -20.072 5.546 36 0.167 -1.853
8.353 -19.597 25.263 -16.976 4.649 37 0.084 -0.960 4.451 -10.760
14.280 -9.856 2.764 38 0.120 -1.299 5.739 -13.201 16.690 -11.001
2.955 39 0.043 -0.476 2.148 -5.025 6.428 -4.264 1.146 40 0.051
-0.526 2.209 -4.801 5.702 -3.504 0.869
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14 SPE 167579
41 -0.043 0.485 -2.237 5.345 -6.986 4.750 -1.317 42 -0.070 0.751
-3.305 7.617 -9.691 6.462 -1.766 43 -0.041 0.475 -2.244 5.523
-7.457 5.241 -1.500 44 -0.107 1.199 -5.465 12.946 -16.825 11.388
-3.140 45 -0.083 0.915 -4.112 9.656 -12.489 8.440 -2.329 46 -0.061
0.686 -3.148 7.532 -9.895 6.771 -1.887 47 -0.033 0.354 -1.588 3.756
-4.930 3.402 -0.962 48 0.007 -0.047 0.090 0.077 -0.464 0.551 -0.215
49 -0.005 0.074 -0.437 1.250 -1.870 1.409 -0.421 50 -0.025 0.265
-1.164 2.657 -3.316 2.144 -0.561 51 0.008 -0.106 0.559 -1.506 2.180
-1.616 0.481 52 0.044 -0.477 2.093 -4.794 6.054 -4.001 1.082 53
0.045 -0.494 2.207 -5.166 6.679 -4.518 1.249 54 0.069 -0.757 3.373
-7.849 10.061 -6.736 1.841 55 0.069 -0.783 3.595 -8.575 11.204
-7.609 2.101 56 0.045 -0.493 2.209 -5.179 6.702 -4.537 1.255 57
0.012 -0.147 0.742 -1.931 2.734 -2.000 0.591 58 0.065 -0.709 3.160
-7.351 9.409 -6.284 1.712 59 0.048 -0.533 2.401 -5.622 7.224 -4.830
1.315 60 -0.031 0.323 -1.362 2.958 -3.491 2.124 -0.521
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SPE 167579 15
Appendix 2. Matlab code for evaluating z factor given the
specific gravity, temperature and pressure function z = fourierz
(s,T,P) %% Compressibility Factor (z) % Author : Kareem Lateef
Adewale % Date : 28 February 2013
%---------------------------------------- % s is the gas gravity, T
is the temperature in Fahrenheit and P is %the pressure in psia
%---------------------------------------- % Solving the problem Tc
= 170.491+307.344*s; %pseudo critical temperature Pc =
709.604-58.718*s; %pseudo critical pressure Tr = (T+460)/Tc;
%pseudo reduced temperature Pr = P/Pc; %pseudo reduced pressure t =
Tr^-1; load ('H.mat'); C = H'; c = C(:); A = zeros(size(c)); for h
= 1:61 for i = 1:7 m = i + (h-1)*7; A(m) =
(t^(i-1))*cos((h-1)*pi*Pr/15); end end z = sum(c.*A);