# New explicit correlation for the compressibility factor of natural gas: linearized z-factor isotherms

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s13202-015-0209-3

- Cite this article as:
- Kareem, L.A., Iwalewa, T.M. & Al-Marhoun, M. J Petrol Explor Prod Technol (2016) 6: 481. doi:10.1007/s13202-015-0209-3

## Abstract

The compressibility factor (z-factor) of gases is a thermodynamic property used to account for the deviation of real gas behavior from that of an ideal gas. Correlations based on the equation of state are often implicit, because they require iteration and are computationally expensive. A number of explicit correlations have been derived to enhance simplicity; however, no single explicit correlation has been developed for the full range of pseudo-reduced temperatures \(\left( {1.05 \le T_{pr} \le 3} \right)\) and pseudo-reduced pressures \(\left( {0.2 \le P_{pr} \le 15} \right)\), which represents a significant research gap. This work presents a new z-factor correlation that can be expressed in linear form. On the basis of Hall and Yarborough’s implicit correlation, we developed the new correlation from 5346 experimental data points extracted from 5940 data points published in the SPE natural gas reservoir engineering textbook and created a linear z-factor chart for a full range of pseudo-reduced temperatures \((1.15 \le T_{pr} \le 3)\) and pseudo-reduced pressures \((0.2 \le P_{pr} \le 15)\).

### Keywords

Z-factor Explicit correlation Reduced temperature Reduced pressure Natural gas### List of symbols

*P*Pressure (psi)

- \(P_{pc}\)
Pseudo-critical pressure

- \(P_{pr}\)
Pseudo-reduced pressure

*T*Temperature (R)

- \(T_{pc}\)
Pseudo-critical temperature (R)

- \(T_{pr}\)
Pseudo-reduced temperature

- \(P_{pc}\)
Pseudo-critical pressure (psi)

- \(P_{pr}\)
Pseudo-reduced pressure

- \(v\)
Initial guess for iteration process

*Y*Pseudo-reduced density

*z*-factorCompressibility factor

## Introduction

The compressibility factor (z-factor) of gases 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 gases.

For generalization, the z-factor is expressed as a function of pseudo-reduced temperature and pressure (Trube 1957; Dranchuk et al. 1971; Abou-kassem and Dranchuk 1975; Sutton 1985; Heidaryan et al. 2010).

### Implicit z-factor correlations

The three most popular correlations for calculating the z-factor are implicit. The three correlations, described in the following subsections, are well known for their accuracies, almost unit correlation of regression coefficients and low maximum errors.

#### Hall and Yarborough’s correlation (Trube 1957)

*y*is the root of the following equation:

#### Dranchuk and Abou-Kassem’s correlation (Abou-kassem and Dranchuk 1975)

*y*is the root of the following equation:

#### Dranchuk, Purvis and Robinson’s Correlation (Dranchuk et al. 1971)

*y*is the root of the following equation:

These correlations are effective; however, they do not converge (or converge on wrong pseudo-reduced density values) when the temperature of the systems is close to the critical temperature. In addition, they are computationally expensive. It is these limitations that necessitated the development of the current explicit correlations.

### Explicit correlations

Explicit correlations do not require an iterative procedure. Therefore, they do not have the problem of convergence as opposed to implicit correlations. One of the best explicit correlations for evaluation of the z-factor was given by Beggs and Brills (1973). More recent ones are Heidaryan et al. (2010), Azizi et al. (2010) and Sanjari and Lay (2012) correlations. A short description of some of the explicit correlations is presented in the following subsections.

#### Brill and Beggs’ compressibility factor (1973)

#### Heidaryan, Moghdasi and Rahimi’s Correlation

Constants of Heidaryan et al.’s correlation

Constants for \(P_{pr} \le 3\) | Constants for \(P_{pr} > 3\) | |
---|---|---|

| 2.827793 | 3.252838 |

| −4.688191 × 10 | −1.306424 × 10 |

| −1.262288 | 6.449194 × 10 |

| −1.536524 | −1.518028 |

| −4.535045 | −5.391019 |

| 6.895104 × 10 | −1.379588 × 10 |

| 1.903869 × 10 | 6.600633 × 10 |

| 6.200089 × 10 | 6.120783 × 10 |

| 1.838479 | 2.317431 |

| 4.052367 × 10 | 1.632223 × 10 |

| 1.073574 | 5.660595 × 10 |

#### Azizi, Behbahani and Isazadeh’s Correlation

#### Sanjari and Nemati’s Correlation

Constants of Sanjari and Lay’s correlation

Constants for \(P_{pr} \le 3\) | Constants for \(P_{pr} > 3\) | |
---|---|---|

| 0.007698 | 0.015642 |

| 0.003839 | 0.000701 |

| −0.467212 | 2.341511 |

| 1.018801 | −0.657903 |

| 3.805723 | 8.902112 |

| −0.087361 | −1.136000 |

| 7.138305 | 3.543614 |

| 0.083440 | 0.134041 |

This correlation, however, is less efficient when compared with that of Heidaryan et al. (2010). Its regression correlation coefficient is 0.8757 and its error rate at a certain point can be as high as 90 per cent. For instance, the actual value of z from the experiment for a \(P_{pr}\) of 15 and a \(T_{pr}\) of 1.05 is 1.753, but this correlation gives a value of 3.3024. Therefore, the actual maximum error for this correlation is 104.3206 %.

To resolve the limitations in the application of the existing explicit correlations, a single correlation that is continuous over the entire range of pseudo-reduced pressure is required.

## New explicit z-factor correlation

A new explicit z-factor is developed as a multi-stage correlation based on Hall and Yarborough’s implicit correlation. The implicit correlation was rearranged to return a value of *y* using an approximate value of *z*. The *y*-values on the right side of the expression were replaced by \(\left( {A_{1} P_{pr} /z} \right)\). Non-linear regression was performed using the derived model. The resulting correlation for reduced density is given in Eq. 15, while Eq. 14 provides an extra iteration to bring the results closer to those obtained by Hall and Yarborough. The constants for these two equations are shown in

Constants of the new correlation

Constants | |||
---|---|---|---|

| 0.317842 |
| −1.966847 |

| 0.382216 |
| 21.0581 |

| −7.768354 |
| −27.0246 |

| 14.290531 |
| 16.23 |

| 0.000002 |
| 207.783 |

| −0.004693 |
| −488.161 |

| 0.096254 |
| 176.29 |

| 0.166720 |
| 1.88453 |

| 0.966910 |
| 3.05921 |

| 0.063069 |

### Linearized z-factor isotherms

*M*are chosen to be a function of reduced temperature and a regression analysis performed to extend the applicability of Eq. 17 to cover the ranges \(1.15 \le T_{pr} \le 3\) and \(0.2 \le P_{pr} \le 15\). For this range of values of reduced temperature and pressure,

*M*is given by

*B*and

*C*maintain the same definition as in Eq. 16. Hence, a graph of \(zC\) against \(P_{pr}^{2} \left( {1 - \frac{B}{{C^{2} }}} \right)\) gives straight lines passing through the point \(\left( {P_{pr}^{2} \left( {1 - \frac{B}{{C^{2} }}} \right) = 0, zC = 1} \right)\) with slopes

*M*. A plot of the straight line form of the z-factor is shown in Fig. 6.

## Results and discussion

Statistical detail of the constants of the new correlation

Constants | Estimated values | 95 % confidence interval |
| Correlation regression coefficient |
---|---|---|---|---|

| 0.317842 | {0.316999, 0.318686} | 5.9718379920 × 10 | 0.999864 |

| 0.382216 | {0.379011, 0.385422} | 2.53091273354 × 10 | |

| −7.76835 | {−7.95029, −7.58642} | 4.4950402736 × 10 | |

| 14.2905 | {14.071, 814.5093} | 5.1169104227 × 10 | |

| 2.18363 × 10 | {2.13411, 2.23314} × 10 | 8.8734666722 × 10 | |

| −0.00469257 | {−0.00476861, −0.00461652} | 4.4906844765 × 10 | |

| 0.0962541 | {0.095078, 0.0974301} | 2.43824633475 × 10 | |

| 0.16672 | {0.160731, 0.17271} | 2.01035053289 × 10 | |

| 0.96691 | {0.964676, 0.969145} | 2.17145789885 × 10 | |

| 0.063069 | {0.0624199, 0.0637180} | 2.4388910420 × 10 | 0.999945 |

| −1.966847 | {−1.97194278, −1.96175122} | 2.8734273447 × 10 | |

| 21.0581 | {20.7208, 21.3954} | 1.17296332188 × 10 | |

| −27.0246 | {−28.0223, −26.0269} | 2.14835618371 × 10 | |

| 16.23 | {15.5168, 16.9432} | 4.28626039700 × 10 | |

| 207.783 | {203.66, 211.906} | 9.1427403949 × 10 | |

| −488.161 | {−499.149, −477.174} | 1.48225067124 × 10 | |

| 176.29 | {169.521, 183.06} | 1.40366406588 × 10 | |

| 1.88453 | {1.87601, 1.89306} | 4.6430138718 × 10 | |

| 3.05921 | {3.03467, 3.08375} | 5.5399223318 × 10 |

Comparison of the explicit correlations with the experimental data

Models | Maximum absolute error | Coefficient of regression | Maximum percentage error | Average percentage error | Root mean square of percentage error |
---|---|---|---|---|---|

Sanjari and Lay (2012) | 0.7664 | 0.94946 | 45.5651 | 3.7463 | 7.3258 |

Heidaryan et al. (2010) | 0.0220 | 0.99963 | 3.71630 | 0.4876 | 0.7369 |

Azizi et al. (2010) | 0.3543 | 0.87240 | 60.0251 | 13.5907 | 15.7493 |

Equation 14 | 0.0270 | 0.99972 | 5.9976 | 0.4379 | 0.6929 |

Equation 16 | 0.0396 | 0.99899 | 8.7970 | 0.8267 | 1.2430 |

### Example calculations

With Figs. 1 and 5, evaluate and compare the compressibility factor of a 0.7 gravity gas at 2000 psig and 150 °F.

#### Solution 1

#### Solution 2

## Conclusions

A simple accurate correlation for evaluating the z-factor that can be linearized has been developed. This correlation performs excellently in the ranges \(1.15 \le T_{pr} \le 3\) and \(0.2 \le P_{pr} \le 15\). It is simple and single-valued. A noteworthy advancement is that the new correlation is continuous over the full range of pseudo-reduced pressures \((0.2 \le P_{pr} \le 15)\). This will widen its applicability to include cases such as the evaluation of natural gas compressibility, in which the derivative of the compressibility factor with respect to the pseudo-reduced pressure is required. For the range outside the coverage of this correlation, implicit correlations can be applied; however, this new explicit correlation can be used to provide an initial guess to speed up the iteration process.

## Copyright information

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