# Beggs and Brill correlation

## Brief

Beggs and Brill is an empirical two-phase flow correlation published in 1972 [1].

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

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

where

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

### Friction factor

No slip Reynolds two phase number:

$Re = 1488 \times \frac {\rho_{m,ns} v_m D} { \mu_L\ C_L + \mu_g\ (1-C_L) }$[2]

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]

Corrected two phase friction factor:

$f' = f \times e^S$[2]

where

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

and

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

with constraint:

$S = ln (2.2\ y - 1.2), when\ 1 \le y \le 1.2$[2]

## Discussion

Why Beggs and Brill?

The best correlation for the horizontal flow.
— pengtools.com

## Workflow HL

$\rho_L= \frac{62.4\ SG_o + \frac{Rs\ 0.0764\ SG_g}{5.614}}{B_o} (1-WCUT) + \frac{62.4\ SG_w}{B_w}\ WCUT$[5]
$\rho_g = \frac{28.967\ SG_g\ p}{z\ 10.732\ T_R}$[5]
$\mu_L = \mu_o (1-WCUT) + \mu_w\ WCUT$[5]
$\sigma_L = \sigma_o (1-WCUT) + \sigma_w\ WCUT$[5]
$v_{SL} = \frac{5.615 q_L}{86400 A_p} \left ( B_o (1-WCUT) + B_w\ WCUT \right )$[5]
$v_{SG} = \frac{q_g}{86400 A_p}\ \frac{14.7}{p}\ \frac{T_K}{520}\ \frac{z}{1}$[5]
$G_m = \rho_L \times v_{SL} + \rho_g \times v_{SG}$[5]
$v_m = v_{SL} + v_{SG}$
$C_L = \frac{v_{SL}}{v_m}$
$L_1 = 316\ {C_L}^{0.302}$[2]
$L_2 = 0.0009252\ {C_L}^{-2.4684}$[2]
$L_3 = 0.1\ {C_L}^{-1.4516}$[2]
$L_4 = 0.5\ {C_L}^{-6.738}$[2]
$N_{FR} = \frac{{v_m}^2}{g_c\ D}$[1]

Determine the flow pattern:

• SEGREGATED: $(C_L < 0.01\ \&\ N_{FR}< L_1)\ or\ (C_L\ge0.01\ \&\ N_{FR}[2]
• TRANSITION: $C_L \ge 0.01\ \&\ L_2 \le N_{FR} \le L_3$[2]
• INTERMITTENT: $(0.01 \le C_L <0.4\ \&\ L_3[2]
• DISTRIBUTED: $(C_L < 0.4\ \&\ N_{FR} \ge L_1)\ or\ (C_L\ge0.4\ \&\ N_{FR}>L_4)$[2]

Calculate $H_{L(0)}:$

• SEGREGATED: $H_{L(0)} = 0.98 \frac{ {C_L}^{0.4846} } { {N_{FR}}^{0.0868}}$[2]
• INTERMITTENT: $H_{L(0)} = 0.845 \frac{ {C_L}^{0.5351} } { {N_{FR}}^{0.0173}}$[2]
• DISTRIBUTED: $H_{L(0)} = 1.065\frac{ {C_L}^{0.5824} } { {N_{FR}}^{0.0609}}$[2]
with the constraint $H_L \ge C_L$[2]

$\psi = 1 + C\ (sin(1.8\ \theta) - 0.333\ (sin(1.8\ \theta))^3)$[2]
$N_{LV} = 1.938\ v_{SL}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}}$[2]

C Uphill:

• SEGREGATED: $C = (1-C_L)\ ln( 0.011 \times {N_{LV}}^{3.539}\times {C_L}^{-3.768}\times {F_{FR}}^{-1.614})$[2]
• INTERMITTENT: $C = (1-C_L)\ ln( 2.96\times {N_{LV}}^{-0.4473}\times {C_L}^{0.305}\times {F_{FR}}^{0.0978})$[2]
• DISTRIBUTED: $C=0$[2]

C Downhill:

• ALL: $C = (1-C_L)\ ln( 4.7\times {N_{LV}}^{0.1244}\times {C_L}^{-0.3692}\times {F_{FR}}^{-0.5056})$[2]
with the restriction $C \ge 0$[2]

Finally:

• SEGREGATED, INTERMITTENT, DISTRIBUTED:
$H_L = H_{L(0)} \times \psi$[2]
• TRANSITION:
$H_L = A \times H_{L(segregated)} + (1-A) \times {H_{L(intermittent)}}$[2]

where:

$A = \frac{L_3-N_{FR}}{L_3-L_2}$[2]

## Modifications

1. Force approach gas at low CL. If CL<0.001 Then f'=f.

2. Force approach to single phase fluid. If HL>1 Then HL=1.

3. Use calculated water density instead of the constant value of 62.4 lbm/ft3.

## Nomenclature

$A$ = correlation variable, dimensionless
$A_p$ = flow area, ft2
$B$ = formation factor, bbl/stb
$C$ = correlation variable, dimensionless
$C_L$ = non-slip liquid holdup factor, dimensionless
$D$ = pipe diameter, ft
$G$ = total flux weight, lbm/ft2/sec
$h$ = depth, ft
$H_L$ = liquid holdup factor, dimensionless
$H_{L(0)}$ = liquid holdup factor when flow is horizontal, dimensionless
$f$ = friction factor, dimensionless
$f'$ = corrected friction factor, dimensionless
$GLR$ = gas-liquid ratio, scf/bbl
$L_1, L_2, L_3, L_4$ = correlation variables, dimensionless
$N_FR$ = Froude number, dimensionless
$N_LV$ = liquid velocity number, dimensionless
$p$ = pressure, psia
$q_c$ = conversion constant equal to 32.174049, lbmft / lbfsec2
$q$ = flow rate, bbl/d - liquid, scf/d - gas
$Re$ = Reynolds number, dimensionless
$R_s$ = solution gas-oil ratio, scf/stb
$S$ = correlation variable, dimensionless
$SG$ = specific gravity, dimensionless
$T$ = temperature, °R or °K, follow the subscript
$v$ = velocity, ft/sec
$WCUT$ = watercut, fraction
$y$ = correlation variable, dimensionless
$z$ = gas compressibility factor, dimensionless

### Greek symbols

$\varepsilon$ = absolute roughness, ft
$\mu$ = viscosity, cp
$\rho$ = density, lbm/ft3
$\bar \rho$ = integrated average density at flowing conditions, lbm/ft3
$\sigma$ = surface tension of liquid-air interface, dynes/cm (ref. values: 72 - water, 35 - oil)
$\psi$ = inclination correction factor, dimensionless
$\theta$ = inclination angle, ° from horizontal

### Subscripts

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

## References

1. Beggs, H. D.; Brill, J. P. (May 1973). . Journal of Petroleum Technology. AIME. 255 (SPE-4007-PA).
2. Brill, J. P.; Beggs, H. D. (1991). (6 ed.). Oklahoma: U. of Tulsa Tulsa.
3. Colebrook, C. F. (1938–1939). . Journal of the Institution of Civil Engineers. London, England. 11: 133–156.
4. Moody, L. F. (1944). . Transactions of the ASME. 66 (8): 671–684.
5. Lyons, W.C. (1996). Standard handbook of petroleum and natural gas engineering. 2. Houston, TX: Gulf Professional Publishing. ISBN 0-88415-643-5.