Beggs and Brill correlation

Brief

Beggs and Brill is an empirical two-phase flow correlation published in 1972 .

It distinguish between 4 flow regimes.

Beggs and Brill is the default VLP correlation in sPipe.

Beggs and Brill in sPipe Vs GAP

Math & Physics

Fluid flow energy balance

$-144 \frac{\Delta p}{\Delta z} = \frac{sin(\theta)\ \bar \rho_m + \frac{f'\ G_m\ v_m}{2\ g_c\ D}}{1- \bar \rho_m\ \frac{v_m\ v_{SG}}{g_c\ p}}$

where

$\bar \rho_m = \rho_L H_L + \rho_g (1 - H_L)$

Flow patterns

Froude number:

$H_{FR} = \frac{{v_m}^2}{g_c\ D}$

SEGREGATED: $(C_L < 0.01\ \&\ N_{FR}< L_1)\ or\ (C_L\ge0.01\ \&\ N_{FR}<L_2)$

TRANSITION: $C_L \ge 0.01\ \&\ L_2 \le N_{FR} \le L_3$

INTERMITTENT: $(0.01 \le C_L <0.4\ \&\ L_3<N_{FR}\le L_1)\ or\ ( C_L\ge0.4\ \&\ L_3<N_{FR} \le L_4)$

DISTRIBUTED: $(C_L < 0.4\ \&\ N_{FR} \ge L_1)\ or\ (C_L\ge0.4\ \&\ N_{FR}>L_4)$

Liquid Holdup H_L

SEGREGATED, INTERMITTENT, DISRIBUTED: $H_L = H_{L(0)} \times \psi$

TRANSITION: $H_L = A \times H_{L(segregated)} + (1-A) \times {H_{L(intermittent)}}$

where:

$A = \frac{L_3-N_{FR}}{L_3-L_2}$

Friction factor

No slip Reynolds two phase number:

$Re = 1488 \times \frac {\rho_{m,ns} v_m D} { {\mu_L}^{C_L} {\mu_g}^{1-C_L} }$

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

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

Corrected two phase friction factor:

$f' = f \times e^S$

where

$S = \frac{ ln(y)}{ -0.0523 + 3.182\ ln(y) - 0.8725\ ln(y)^2 + 0.01853\ ln(y)^ 4}$

and

$y = \frac{ C_L} { {H_L}^2 }$

with constraint:

$S = ln (2.2\ y - 1.2), when\ 1 < y < 1.2$

Discussion

Why Hagedorn and Brown?

One of the consistently best correlations ...

— Michael Economides et al

Flow Diagram

Workflow H_L

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

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

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

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

$\sigma_L = \sigma_o \frac{1}{1 + WOR} + \sigma_w \frac{WOR}{1 + WOR}$

$N_L = 0.15726\ \mu_L \sqrt[4]{\frac{1}{\rho_L \sigma_L^3}}$

$CN_L = 0.061\ N_L^3 - 0.0929\ N_L^2 + 0.0505\ N_L + 0.0019$

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

$v_{SG} = \frac{q_L \left ( GLR-R_s \left( \frac{1}{1+WOR}\right) \right )}{86400 A_p}\ \frac{14.7}{p}\ \frac{T_K}{520}\ \frac{z}{1}$

$N_{LV} = 1.938\ v_{SL}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}}$

$N_{GV} = 1.938\ v_{SG}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}}$

$N_{D} = 120.872\ D \sqrt{\frac{\rho_L}{\sigma_L}}$

$H = \frac{N_{LV}}{N_{GV}^{0.575}}\ \left ( \frac{p}{14.7} \right )^{0.1} \frac{CN_L}{N_D}$

$\frac{H_L}{\psi} = \sqrt{\frac{0.0047+1123.32 H + 729489.64H^2}{1+1097.1566 H + 722153.97 H^2}}$

$B = \frac{N_{GV} N_{LV}^{0.38}}{N_{D}^{2.14}}$

$\psi = \begin{cases} 27170 B^3 - 317.52 B^2 + 0.5472 B + 0.9999, &\mbox{if } B \le 0.025 \\ -533.33 B^2 + 58.524 B + 0.1171, & \mbox{if }B > 0.025 \\ 2.5714 B +1.5962, & \mbox{if }B > 0.055 \end{cases}$

$H_L = \frac{H_L}{\psi} \times \psi$

SEGREGATED: $H_{L(0)} = 0.98 \frac{ {C_L}^{0.4846} } { {N_{FR}}^{0.0868}}$

INTERMITTENT: $H_{L(0)} = 0.845 \frac{ {C_L}^{0.5351} } { {N_{FR}}^{0.0173}}$

DISRIBUTED: $H_{L(0)} = 1.065\frac{ {C_L}^{0.5824} } { {N_{FR}}^{0.0609}}$

with the constraint $H_L \ge C_L$

$\psi = 1 + C\ (sin(1.8\ \theta) - 0.333\ (sin(1.8\ \theta))^3)$

Uphill

SEGREGATED: $C = (1-C_L)\ ln( 0.011 \times {N_{LV}}^{3.539}\times {C_L}^{-3.768}\times {F_{FR}}^{-1.614})$

INTERMITTENT: $C = (1-C_L)\ ln( 2.96\times {N_{LV}}^{-0.4473}\times {C_L}^{0.305}\times {F_{FR}}^{0.0978})$

DISRIBUTED: $C=0$

Downhill

ALL: $C = (1-C_L)\ ln( 4.7\times {N_{LV}}^{0.1244}\times {C_L}^{-0.3692}\times {F_{FR}}^{-0.5056})$

Modifications

1. Use the no-slip holdup when the original empirical correlation predicts a liquid holdup H_L less than the no-slip holdup .

2. Use the Griffith correlation to define the bubble flow regime and calculate H_L.

3. Use watercut instead of WOR to account for the watercut = 100%.

Nomenclature

$A_p$ = flow area, ft2

$B$ = correlation group, dimensionless

$B$ = formation factor, bbl/stb

$C$ = coefficient for liquid viscosity number, dimensionless

$D$ = pipe diameter, ft

$h$ = depth, ft

$H$ = correlation group, 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, lb_m/bbl

$N_D$ = pipe diameter number, dimensionless

$N_GV$ = gas velocity number, dimensionless

$N_L$ = liquid viscosity number, dimensionless

$N_LV$ = liquid velocity number, dimensionless

$p$ = pressure, psia

$q_c$ = conversion constant equal to 32.174049, lb_mft / lb_fsec²

$q$ = 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

$\mu$ = viscosity, cp

$\rho$ = density, lb_m/ft³

$\bar \rho$ = integrated average density at flowing conditions, lb_m/ft²

$\sigma$ = surface tension of liquid-air interface, dynes/cm (ref. values: 72 - water, 35 - oil)

$\psi$ = secondary correlation factor, dimensionless

Subscripts

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