# Dranchuk correlation

## Dranchuk gas compressibility factor correlation

Dranchuk correlation is the fitting equation of the classic Standing and Katz [1] gas compressibility factor correlation.

## Math & Physics

$z = 1 + \left(A_1 +\frac{A_2}{T_{pr}} +\frac{A_3}{T^3_{pr}} +\frac{A_4}{T^4_{pr}} +\frac{A_5}{T^5_{pr}} \right)\ \rho_r+ \left(A_6 +\frac{A_7}{T_{pr}} +\frac{A_8}{T^2_{pr}} \right)\ \rho^2_r -A_9\ \left(\frac{A_7}{T_{pr}}+\frac{A_8}{T^2_{pr}}\right) \rho^5_r +A_{10}\ \left(1+A_{11}\ \rho^2_r\right)\ \frac{\rho^2_r}{T^3_{pr}} \ e^{(-A_{11}\ \rho^2_r)}$[2]

where:

$\rho_r = \frac{0.27\ P_{pr}}{{z\ T_{pr}}}$
$P_{pr} = \frac{P}{P_{pc}}$
$T_{pr} = \frac{T}{T_{pc}}$

A1 = 0.3265
A2 = –1.0700
A3 = –0.5339
A4 = 0.01569
A5 = –0.05165
A6 = 0.5475
A7 = –0.7361
A8 = 0.1844
A9 = 0.1056
A10 = 0.6134
A11 = 0.7210

## Discussion

Why the Dranchuk correlation?

It's classics!
— www.pengtools.com

## Workflow

To solve the Dranchuk equation use the iterative secant method.

To find the pseudo critical properties from the gas specific gravity [1]:

$P_{pc} = ( 4.6+0.1\ SG_g-0.258\ SG^2_g ) \times 10.1325 \times 14.7$
$T_{pc} = ( 99.3+180\ SG_g-6.94\ SG^2_g ) \times 1.8$

## Application range

$0.2 \le P_{pr} < 30 ; 1.0 < T_{pr} \le 3.0$[2]

and

$P_{pr} < 1.0 ; 0.7 < T_{pr} \le 1.0$[2]

## Nomenclature

$A_1..A_{11}$ = coefficients
$\rho_r$ = reduced density, dimensionless
$P$ = pressure, psia
$P_{pc}$ = pseudo critical pressure, psia
$P_{pr}$ = pseudoreduced pressure, dimensionless
$SG_g$ = gas specific gravity, dimensionless
$T$ = temperature, °R
$T_{pc}$ = pseudo critical temperature, °R
$T_{pr}$ = pseudoreduced temperature, dimensionless
$z$ = gas compressibility factor, dimensionless

## References

1. Standing, M. B.; Katz, D. L. (December 1942). . Transactions of the AIME. Society of Petroleum Engineers. 146 (SPE-942140-G).
2. Dranchuk, P. M.; Abou-Kassem, H. (July 1975). . The Journal of Canadian Petroleum. 14 (PETSOC-75-03-03).