Dranchuk correlation

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Brief

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

Math & Physics

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

 z =  1-z+
\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}}

Discussion

Why the Dranchuk correlation?

It's classics!
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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. 1.0 1.1 Standing, M. B.; Katz, D. L. (December 1942). "Density of Natural Gases"Free registration required. Transactions of the AIME. Society of Petroleum Engineers. 146 (SPE-942140-G). 
  2. 2.0 2.1 2.2 Dranchuk, P. M.; Abou-Kassem, H. (July 1975). "Calculation of Z Factors For Natural Gases Using Equations of State"Free registration required. The Journal of Canadian Petroleum. 14 (PETSOC-75-03-03).