Difference between revisions of "Beggs and Brill correlation"

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== Math & Physics ==
 
== Math & Physics ==
 +
=== Fluid flow energy balance ===
 
Following the law of conservation of energy the basic steady state flow equation is:
 
Following the law of conservation of energy the basic steady state flow equation is:
 
:<math> -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}}</math><ref name="BB" />
 
:<math> -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}}</math><ref name="BB" />
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:<math> \bar \rho_m = \rho_L H_L + \rho_g (1 - H_L)</math><ref name="BB" />
 
:<math> \bar \rho_m = \rho_L H_L + \rho_g (1 - H_L)</math><ref name="BB" />
  
 +
=== Flow pattern  ===
 
Colebrook–White <ref name=Colebrook/> equation for the [http://en.wikipedia.org/wiki/Darcy_friction_factor_formulae Darcy's friction factor]:
 
Colebrook–White <ref name=Colebrook/> equation for the [http://en.wikipedia.org/wiki/Darcy_friction_factor_formulae Darcy's friction factor]:
 
:<math> \frac{1}{\sqrt{f}}= -2 \log \left( \frac { \varepsilon} {3.7 D} + \frac {2.51} {\mathrm{Re} \sqrt{f}} \right)</math><ref name = Moody1944/>
 
:<math> \frac{1}{\sqrt{f}}= -2 \log \left( \frac { \varepsilon} {3.7 D} + \frac {2.51} {\mathrm{Re} \sqrt{f}} \right)</math><ref name = Moody1944/>
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=== HL  ===
  
 
Reynolds two phase number:
 
Reynolds two phase number:
 
:<math> Re = 2.2 \times 10^{-2} \frac {q_L M}{D \mu_L^{H_L} \mu_g^{(1-H_L)}}</math><ref name="HB" />
 
:<math> Re = 2.2 \times 10^{-2} \frac {q_L M}{D \mu_L^{H_L} \mu_g^{(1-H_L)}}</math><ref name="HB" />
 +
 +
=== Friction factor  ===
  
 
== Discussion  ==
 
== Discussion  ==

Revision as of 09:52, 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 pattern

Colebrook–White [2] 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)[3]

HL

Reynolds two phase number:

Re = 2.2 \times 10^{-2} \frac {q_L M}{D \mu_L^{H_L} \mu_g^{(1-H_L)}}[4]

Friction factor

Discussion

Why Hagedorn and Brown?

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

Flow Diagram

HB Block Diagram

Workflow HL

M =SG_o\ 350.52\ \frac{1}{1+WOR}+SG_w\ 350.52\ \frac{WOR}{1+WOR}+SG_g\ 0.0764\ GLR[4]
\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}[6]
\rho_g = \frac{28.967\ SG_g\ p}{z\ 10.732\ T_R} [6]
\mu_L = \mu_o \frac{1}{1 + WOR} + \mu_w \frac{WOR}{1 + WOR}[6]
\sigma_L = \sigma_o \frac{1}{1 + WOR} + \sigma_w \frac{WOR}{1 + WOR}[6]
N_L = 0.15726\ \mu_L \sqrt[4]{\frac{1}{\rho_L \sigma_L^3}}[4]
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 )[6]
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}[6]
N_{LV} = 1.938\ v_{SL}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}} [4]
N_{GV} = 1.938\ v_{SG}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}} [4]
N_{D} = 120.872\ D \sqrt{\frac{\rho_L}{\sigma_L}} [4]
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}} [7]
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} [7]
H_L = \frac{H_L}{\psi} \times \psi[4]

Modifications

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

2. Use the Griffith correlation to define the bubble flow regime[5] and calculate HL.

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, lbm/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, lbmft / lbfsec2
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, lbm/ft3
\bar \rho = integrated average density at flowing conditions, lbm/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. ↑ 1.0 1.1 1.2 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. ↑ 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. 
  3. ↑ Moody, L. F. (1944). "Friction factors for pipe flow"Paid subscription required. Transactions of the ASME. 66 (8): 671–684. 
  4. ↑ 4.0 4.1 4.2 4.3 4.4 4.5 4.6 Cite error: Invalid <ref> tag; no text was provided for refs named HB
  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. 
  6. ↑ 6.0 6.1 6.2 6.3 6.4 6.5 Lyons, W.C. (1996). Standard handbook of petroleum and natural gas engineering. 2. Houston, TX: Gulf Professional Publishing. ISBN 0-88415-643-5. 
  7. ↑ 7.0 7.1 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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