Difference between revisions of "Beggs and Brill correlation"

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Latest revision as of 18:10, 3 November 2018

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

Flow Diagram

HB Block Diagram

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}<L_2)[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<N_{FR}\le L_1)\ or\ ( C_L\ge0.4\ \&\ L_3<N_{FR} \le L_4)[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:

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. ↑ 1.0 1.1 1.2 1.3 Beggs, H. D.; Brill, J. P. (May 1973). "A Study of Two-Phase Flow in Inclined Pipes"Paid subscription required. Journal of Petroleum Technology. AIME. 255 (SPE-4007-PA). 
  2. ↑ 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 Brill, J. P.; Beggs, H. D. (1991). Two-Phase Flow In PipesPaid subscription required (6 ed.). Oklahoma: U. of Tulsa Tulsa. 
  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. ↑ 5.0 5.1 5.2 5.3 5.4 5.5 5.6 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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