Gray correlation

Brief

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

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:

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

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 equation for the Darcy's friction factor:

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

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 $\varepsilon' \geqslant 2.77 \times 10^{-5}$

Reynolds two phase number:

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

Discussion

Why Gray correlation?

The Gray correlation was found to be the best of several initially tested ...

— Nitesh Kumar l

Workflow H_g & C_L

$M =SG_o\ 350.52\ \frac{1}{1+WOR}+SG_w\ 350.52\ \frac{WOR}{1+WOR}+SG_g\ 0.0764\ GLR$

$\rho_L= \frac{62.4\ SG_o + \frac{Rs\ 0.0764\ SG_g}{5.614}}{B_o} \frac{1}{1+WOR} + 62.4\ SG_w\ \frac{WOR}{1 + WOR}$

$\rho_g = \frac{28.967\ SG_g\ p}{z\ 10.732\ T_R}$

$v_{SL} = \frac{5.615 q_L}{86400 A_p} \left ( B_o \frac{1}{1+WOR} + B_w \frac{WOR}{1+WOR} \right )$

$v_{SG} = \frac{q_g \times 10^6}{86400 A_p}\ \frac{14.7}{p}\ \frac{T_K}{520}\ \frac{z}{1}$

$C_L = \frac{v_{SL}}{v_{SG}+v_{SL}}$

$v_m = v_{SL} + v_{SG}$

$\rho_m = \rho_L C_L + \rho_g (1-C_L)$

$\mu_L = \mu_o \frac{1}{1 + WOR} + \mu_w \frac{WOR}{1 + WOR}$

$\sigma_L = \frac{\sigma_o\ q_o + 0.617\ \sigma_w\ q_w}{q_o + 0.617\ q_w}$

$N_V = 453.592\ \frac{{\rho_m}^2 {v_m}^4}{g_c \sigma_L (\rho_L - \rho_g)}$

$N_D = 453.592\ \frac{g_c (\rho_L - \rho_g) D^2}{\sigma_L }$

$R = \frac{v_{SL}}{v_{SG}}$

$B = 0.0814 \left ( 1 - 0.554\ \ln \left (1 + \frac{730 R}{R+1} \right ) \right )$

$A = -2.2314 \left ( N_V \left (1 + \frac{205}{N_D} \right ) \right )^B$

$H_g = \frac{1-e^A}{R+1}$

Modifications

• Use watercut instead of WOR to allow for 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, lb_m/bbl

$N_D$ = pipe diameter number, dimensionless

$N_V$ = velocity number, dimensionless

$p$ = pressure, psia

$q_c$ = conversion constant equal to 32.174, lb_mft / lb_fsec²

$q$ = 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, lb_m/ft³

$\bar \rho$ = slip density, lb_m/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