Gray correlation

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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.

Gray in PQplot Vs Prosper & Kappa

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}

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 [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= 62.4\ SG_o \frac{1}{1+WOR} + 62.4\ SG_w\ \frac{WOR}{1 + WOR}
 \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 Fanning correlation for the dry gas (WGR=0 and OGR=0 case).

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

3. If the relative roughness:  \frac{\varepsilon'}{D} > 0.05 use 0.05 in the Moody Diagram [3].

4. If HL can't be calculated then HL = CL.

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 32.174049, lbmft / lbfsec2
 q = production rate, bbl/d
 R = superficial liquid to gas ratio, dimensionless
 Re = Reynolds number, dimensionless
 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. 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. 3.0 3.1 Moody, L. F. (1944). "Friction factors for pipe flow"Paid subscription required. Transactions of the ASME. 66 (8): 671–684. 
  4. 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. 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 Lyons, W.C. (1996). Standard handbook of petroleum and natural gas engineering. 2. Houston, TX: Gulf Professional Publishing. ISBN 0-88415-643-5.