Difference between revisions of "Gray correlation"

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where
 
where
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:<math> \bar \rho_m = \rho_L (1-H_g) + \rho_g H_g</math>,  slip mixture density
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:<math> \bar \rho_m = \rho_L (1-H_g) + \rho_g H_g</math>,  slip mixture density <ref name= Gray />.
  
−
:<math> \rho_m = \rho_L C_L + \rho_g (1-C_L) </math>,  no-slip mixture density
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:<math> \rho_m = \rho_L C_L + \rho_g (1-C_L) </math>,  no-slip mixture density <ref name= Gray />.
  
 
Colebrook–White <ref name=Colebrook/> equation for the [http://en.wikipedia.org/wiki/Darcy_friction_factor_formulae Darcy's friction factor]:
 
Colebrook–White <ref name=Colebrook/> equation for the [http://en.wikipedia.org/wiki/Darcy_friction_factor_formulae Darcy's friction factor]:

Revision as of 13:43, 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}[2]

where

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

Colebrook–White [3] 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)[4]

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)}}[2]

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[2]
\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

1. Use watercut instead of WOR to account for the OGR=0 case.

Nomenclature

A = correlation group, dimensionless
A_p = flow area, ft2
B = correlation group, dimensionless
B = formation factor, bbl/stb
C = no-slip holdup factor, dimensionless
D = pipe diameter, ft
h = depth, ft
H = 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 = production rate, bbl/d
R = superficial liquid to gas ratio, dimensionless
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 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. ↑ 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 Gray, H. E. (1974). "Vertical Flow Correlation in Gas Wells". User manual for API 14B, Subsurface controlled safety valve sizing computer program. API. 
  2. ↑ 2.0 2.1 2.2 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. 
  3. ↑ 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. 
  4. ↑ Moody, L. F. (1944). "Friction factors for pipe flow"Paid subscription required. Transactions of the ASME. 66 (8): 671–684. 
  5. ↑ 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. ↑ 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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