Difference between revisions of "Gray correlation"

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(→‎Nomenclature)
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== Nomenclature  ==
 
== Nomenclature  ==
  
−
NV velocity number
+
:<math> A </math> = correlation group, dimensionless
−
 
 
−
 
 
 
:<math> B </math> = correlation group, dimensionless
 
:<math> B </math> = correlation group, dimensionless
 
:<math> B </math> = formation factor, bbl/stb
 
:<math> B </math> = formation factor, bbl/stb
−
:<math> C </math> = coefficient for liquid viscosity number, dimensionless
 
 
:<math> D </math> = pipe diameter, ft
 
:<math> D </math> = pipe diameter, ft
 
:<math> h </math> = depth, ft
 
:<math> h </math> = depth, ft
−
:<math> H </math> = correlation group, dimensionless
+
:<math> H_g </math> = gas holdup factor, dimensionless
 
:<math> H_L </math> = liquid holdup factor, dimensionless
 
:<math> H_L </math> = liquid holdup factor, dimensionless
 
:<math> f </math> = friction factor, dimensionless
 
:<math> f </math> = friction factor, dimensionless
 
:<math> GLR </math> = gas-liquid ratio, scf/bbl
 
:<math> GLR </math> = gas-liquid ratio, scf/bbl
 
:<math> M </math> = total mass of oil, water and gas associated with 1 bbl of liquid flowing into and out of the flow string, lb<sub>m</sub>/bbl
 
:<math> M </math> = total mass of oil, water and gas associated with 1 bbl of liquid flowing into and out of the flow string, lb<sub>m</sub>/bbl
−
:<math> N_D </math> = pipe diameter number number, dimensionless
+
:<math> N_D </math> = pipe diameter number, dimensionless
−
:<math> N_GV </math> = gas velocity number, dimensionless
+
:<math> N_V </math> = velocity number, dimensionless
−
:<math> N_L </math> = liquid viscosity number, dimensionless
 
−
:<math> N_LV </math> = liquid velocity number, dimensionless
 
 
:<math> p </math> = pressure, psia
 
:<math> p </math> = pressure, psia
 
:<math> q_c </math> = conversion constant equal to 32.174, lb<sub>m</sub>ft / lb<sub>f</sub>sec<sup>2</sup>
 
:<math> q_c </math> = conversion constant equal to 32.174, lb<sub>m</sub>ft / lb<sub>f</sub>sec<sup>2</sup>
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:<math> \varepsilon </math> = absolute roughness, ft
 
:<math> \varepsilon </math> = absolute roughness, ft
−
:<math> \mu </math> = oil viscosity, cp
+
:<math> \varepsilon' </math> = pseudo wall roughness, ft
 +
:<math> \mu </math> = viscosity, cp
 
:<math> \rho </math> = density, lb<sub>m</sub>/ft<sup>3</sup>
 
:<math> \rho </math> = density, lb<sub>m</sub>/ft<sup>3</sup>
−
:<math> \bar \rho </math> = integrated average density at flowing conditions, lb<sub>m</sub>/ft<sup>2</sup>
+
:<math> \bar \rho </math> = slip mixture density, lb<sub>m</sub>/ft<sup>2</sup>
−
:<math> \sigma </math> = surface tension of liquid-air interface, dynes/cm (ref. values: 72 - water, 35 - oil)
+
:<math> \sigma </math> = surface tension of liquid-air interface, dynes/cm
−
:<math> \psi </math> = secondary correlation factor, dimensionless
 
  
 
===Subscripts===
 
===Subscripts===

Revision as of 13:24, 6 April 2017

Brief

Gray is an empirical two-phase flow correlation published in 1974 [1].

Gray is the default VLP correlation for the gas wells in the PQplot.

Math & Physics

Following the law of conservation of energy the basic steady state flow equation is:

144 \frac{\Delta p}{\Delta h} = \bar \rho_m + \rho_m \frac{f v_m^2 }{2 g_c D} + \rho_m \frac{\Delta{(\frac{v_m^2}{2g_c}})}{\Delta h}[1]

where

\bar \rho_m = \rho_L (1-H_g) + \rho_g H_g, slip mixture density
\rho_m = \rho_L C_L + \rho_g (1-C_L) , no-slip mixture density

Colebrook–White [2] equation for the Darcy's friction factor:

\frac{1}{\sqrt{f}}= -2 \log \left( \frac { \varepsilon'} {3.7 D} + \frac {2.51} {\mathrm{Re} \sqrt{f}} \right)[3]

The pseudo wall roughness:

\varepsilon' = \begin{cases} \frac{28.5}{453.592} \frac{\sigma_L}{\rho_m v_m^2}, &\mbox{if } R \geqslant 0.007 \\ \varepsilon + R \frac{\varepsilon'-\varepsilon}{0.007}, & \mbox{if } R < 0.007 \end{cases} , with the limit \varepsilon' \geqslant 2.77 \times 10^{-5}[1]

Reynolds two phase number:

Re = 2.2 \times 10^{-2} \frac {q_L M}{D \mu_L^{C_L} \mu_g^{(1-C_L)}}[4]

Discussion

Why Gray correlation?

The Gray correlation was found to be the best of several initially tested ...
— Nitesh Kumar l[5]

Workflow Hg & CL

M =SG_o\ 350.52\ \frac{1}{1+WOR}+SG_w\ 350.52\ \frac{WOR}{1+WOR}+SG_g\ 0.0764\ GLR[4]
\rho_L= \frac{62.4\ SG_o + \frac{Rs\ 0.0764\ SG_g}{5.614}}{B_o} \frac{1}{1+WOR} + 62.4\ SG_w\ \frac{WOR}{1 + WOR}[6]
\rho_g = \frac{28.967\ SG_g\ p}{z\ 10.732\ T_R} [6]
v_{SL} = \frac{5.615 q_L}{86400 A_p} \left ( B_o \frac{1}{1+WOR} + B_w \frac{WOR}{1+WOR} \right )[6]
v_{SG} = \frac{q_g \times 10^6}{86400 A_p}\ \frac{14.7}{p}\ \frac{T_K}{520}\ \frac{z}{1}
C_L = \frac{v_{SL}}{v_{SG}+v_{SL}}
v_m = v_{SL} + v_{SG}
\rho_m = \rho_L C_L + \rho_g (1-C_L)
\mu_L = \mu_o \frac{1}{1 + WOR} + \mu_w \frac{WOR}{1 + WOR}[6]
\sigma_L = \frac{\sigma_o\ q_o + 0.617\ \sigma_w\ q_w}{q_o + 0.617\ q_w} [1]
N_V = 453.592\ \frac{{\rho_m}^2 {v_m}^4}{g_c \sigma_L (\rho_L - \rho_g)} [1]
N_D = 453.592\ \frac{g_c (\rho_L - \rho_g) D^2}{\sigma_L } [1]
R = \frac{v_{SL}}{v_{SG}} [1]
B = 0.0814 \left ( 1 - 0.554\ \ln \left (1 + \frac{730 R}{R+1} \right ) \right ) [1]
A = -2.2314 \left ( N_V \left (1 + \frac{205}{N_D} \right ) \right )^B [1]
H_g = \frac{1-e^A}{R+1}[1]

Modifications

use watercut instead of WOR

Nomenclature

A = correlation group, dimensionless
B = correlation group, dimensionless
B = formation factor, bbl/stb
D = pipe diameter, ft
h = depth, ft
H_g = gas holdup factor, dimensionless
H_L = liquid holdup factor, dimensionless
f = friction factor, dimensionless
GLR = gas-liquid ratio, scf/bbl
M = total mass of oil, water and gas associated with 1 bbl of liquid flowing into and out of the flow string, lbm/bbl
N_D = pipe diameter number, dimensionless
N_V = velocity number, dimensionless
p = pressure, psia
q_c = conversion constant equal to 32.174, lbmft / lbfsec2
q_L = total liquid production rate, bbl/d
Re = Reynolds number, dimensionless
R_s = solution gas-oil ratio, scf/stb
SG = specific gravity, dimensionless
T = temperature, °R or °K, follow the subscript
v = velocity, ft/sec
WOR = water-oil ratio, bbl/bbl
z = gas compressibility factor, dimensionless

Greek symbols

\varepsilon = absolute roughness, ft
\varepsilon' = pseudo wall roughness, ft
\mu = viscosity, cp
\rho = density, lbm/ft3
\bar \rho = slip mixture density, lbm/ft2
\sigma = surface tension of liquid-air interface, dynes/cm

Subscripts

g = gas
K = °K
L = liquid
m = gas/liquid mixture
o = oil
R = °R
SL = superficial liquid
SG = superficial gas
w = water

References

  1. ↑ Jump up to: 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Gray, H. E. (1974). "Vertical Flow Correlation in Gas Wells". User manual for API 14B, Subsurface controlled safety valve sizing computer program. API. 
  2. Jump up ↑ Colebrook, C. F. (1938–1939). "Turbulent Flow in Pipes, With Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws"Paid subscription required. Journal of the Institution of Civil Engineers. London, England. 11: 133–156. 
  3. Jump up ↑ Moody, L. F. (1944). "Friction factors for pipe flow"Paid subscription required. Transactions of the ASME. 66 (8): 671–684. 
  4. ↑ Jump up to: 4.0 4.1 Hagedorn, A. R.; Brown, K. E. (1965). "Experimental study of pressure gradients occurring during continuous two-phase flow in small-diameter vertical conduits". Journal of Petroleum Technology. 17(04): 475–484. 
  5. Jump up ↑ Kumar, N.; Lea, J. F. (January 1, 2005). "Improvements for Flow Correlations for Gas Wells Experiencing Liquid Loading"Free registration required (SPE-92049-MS). 
  6. ↑ Jump up to: 6.0 6.1 6.2 6.3 Lyons, W.C. (1996). Standard handbook of petroleum and natural gas engineering. 2. Houston, TX: Gulf Professional Publishing. ISBN 0-88415-643-5. 
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