Difference between revisions of "Beggs and Brill correlation"

Revision as of 11:44, 26 May 2017

Brief

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

It does distinguish between the 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

Following the law of conservation of energy the basic steady state flow equation is:

-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}}

^[1]

where

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

^[1]

Flow patterns

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

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

DISRIBUTED: $(C_L < 0.4\ \&\ N_{FR} \ge L_1)\ or\ (C_L\ge0.4\ \&\ N_{FR}>L_4)$ ^[2]

Liquid Holdup H_L

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

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 < y < 1.2

^[2]

Discussion

Why Hagedorn and Brown?

One of the consistently best correlations ...
— Michael Economides et al^[5]

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

^[6]

\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}

^[7]

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

^[7]

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

^[7]

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

^[7]

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

^[6]

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

^[5]

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

^[7]

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}

^[7]

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

^[6]

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

^[6]

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

^[6]

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

^[5]

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

^[8]

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

^[5]

\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}

^[8]

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

^[6]

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]

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

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]

DISRIBUTED: $C=0$ ^[2]

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]

Modifications

1. Use the no-slip holdup when the original empirical correlation predicts a liquid holdup H_L less than the no-slip holdup ^[5].

2. Use the Griffith correlation to define the bubble flow regime^[5] 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

References

↑ ^1.0 ^1.1 ^1.2 Beggs, H. D.; Brill, J. P. (May 1973). "A Study of Two-Phase Flow in Inclined Pipes". Journal of Petroleum Technology. AIME. 255 (SPE-4007-PA).
↑ ^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 Cite error: Invalid <ref> tag; no text was provided for refs named BB1991
↑ Colebrook, C. F. (1938–1939). "Turbulent Flow in Pipes, With Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws". Journal of the Institution of Civil Engineers. London, England. 11: 133–156.
↑ Moody, L. F. (1944). "Friction factors for pipe flow". Transactions of the ASME. 66 (8): 671–684.
↑ ^5.0 ^5.1 ^5.2 ^5.3 ^5.4 ^5.5 Economides, M.J.; Hill, A.D.; Economides, C.E.; Zhu, D. (2013). Petroleum Production Systems (2 ed.). Westford, Massachusetts: Prentice Hall. ISBN 978-0-13-703158-0.
↑ ^6.0 ^6.1 ^6.2 ^6.3 ^6.4 ^6.5 Cite error: Invalid <ref> tag; no text was provided for refs named HB
↑ ^7.0 ^7.1 ^7.2 ^7.3 ^7.4 ^7.5 Lyons, W.C. (1996). Standard handbook of petroleum and natural gas engineering. 2. Houston, TX: Gulf Professional Publishing. ISBN 0-88415-643-5.
↑ ^8.0 ^8.1 Trina, S. (2010). An integrated horizontal and vertical flow simulation with application to wax precipitation (Master of Engineering Thesis). Canada: Memorial University of Newfoundland.

[BB-1] 1.0 ^1.1 ^1.2 Beggs, H. D.; Brill, J. P. (May 1973). "A Study of Two-Phase Flow in Inclined Pipes". Journal of Petroleum Technology. AIME. 255 (SPE-4007-PA).

[BB1991-2] 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 Cite error: Invalid <ref> tag; no text was provided for refs named BB1991

[Colebrook-3] Colebrook, C. F. (1938–1939). "Turbulent Flow in Pipes, With Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws". Journal of the Institution of Civil Engineers. London, England. 11: 133–156.

[Moody1944-4] Moody, L. F. (1944). "Friction factors for pipe flow". Transactions of the ASME. 66 (8): 671–684.

[Economides-5] 5.0 ^5.1 ^5.2 ^5.3 ^5.4 ^5.5 Economides, M.J.; Hill, A.D.; Economides, C.E.; Zhu, D. (2013). Petroleum Production Systems (2 ed.). Westford, Massachusetts: Prentice Hall. ISBN 978-0-13-703158-0.

[HB-6] 6.0 ^6.1 ^6.2 ^6.3 ^6.4 ^6.5 Cite error: Invalid <ref> tag; no text was provided for refs named HB

[Lyons-7] 7.0 ^7.1 ^7.2 ^7.3 ^7.4 ^7.5 Lyons, W.C. (1996). Standard handbook of petroleum and natural gas engineering. 2. Houston, TX: Gulf Professional Publishing. ISBN 0-88415-643-5.

[Trina-8] 8.0 ^8.1 Trina, S. (2010). An integrated horizontal and vertical flow simulation with application to wax precipitation (Master of Engineering Thesis). Canada: Memorial University of Newfoundland.

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@@ Line 17: / Line 17: @@
 === Flow patterns  ===
+Froude number:
+<math> H_{FR} = \frac{{v_m}^2}{g_c\ D}</math>
 SEGREGATED: <math> (C_L < 0.01\ \&\ N_{FR}< L_1)\ or\ (C_L\ge0.01\ \&\ N_{FR}<L_2)</math><ref name="BB1991" />

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Difference between revisions of "Beggs and Brill correlation"

Revision as of 11:44, 26 May 2017

Contents

Brief

Math & Physics

Fluid flow energy balance

Flow patterns

Liquid Holdup H_L

Friction factor

Discussion

Flow Diagram

Workflow H_L

Modifications

Nomenclature

Greek symbols

Subscripts

References

Navigation

Categories

Products

Difference between revisions of "Beggs and Brill correlation"

Revision as of 11:44, 26 May 2017

Contents

Brief

Math & Physics

Fluid flow energy balance

Flow patterns

Liquid Holdup HL

Friction factor

Discussion

Flow Diagram

Workflow HL

Modifications

Nomenclature

Greek symbols

Subscripts

References

Liquid Holdup H_L

Workflow H_L