Difference between revisions of "Hagedorn and Brown correlation"

Revision as of 12:34, 6 April 2017

Brief

Hagedorn and Brown is an empirical two-phase flow correlation published in 1965 ^[1].

It doesn't distinguish between the flow regimes.

The heart of the Hagedorn and Brown method is a correlation for the liquid holdup H_L ^[2].

Hagedorn and Brown is a default VLP correlation for the oil wells in the PQplot.

Math & Physics

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

144 \frac{\Delta p}{\Delta h} = \bar \rho_m + \frac{f q_L^2 M^2}{2.9652 \times 10^{11} D^5 \bar \rho_m} + \bar \rho_m \frac{\Delta{(\frac{v_m^2}{2g_c}})}{\Delta h}

^[1]

where

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

^[1]

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]

Reynolds two phase number:

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

^[1]

Discussion

Why Hagedorn and Brown?

One of the consistently best correlations ...
— Michael Economides et al^[2]

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

^[1]

\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}

^[5]

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

^[5]

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

^[5]

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

^[5]

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

^[1]

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

^[2]

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

^[5]

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}

^[5]

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

^[1]

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

^[1]

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

^[1]

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

^[2]

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

^[6]

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

^[2]

\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}

^[6]

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

^[1]

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].

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

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

Nomenclature

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 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 32.174, lb_mft / lb_fsec²

q_L

= 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

= oil 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 ^1.3 ^1.4 ^1.5 ^1.6 ^1.7 ^1.8 ^1.9 Hagedorn, A. R.; Brown, K. E. (1965). "Experimental study of pressure gradients occurring during continuous two-phase flow in small-diameter vertical conduits". Journal of Petroleum Technology. 17(04): 475–484.
↑ ^2.0 ^2.1 ^2.2 ^2.3 ^2.4 ^2.5 ^2.6 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.
↑ 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 Lyons, W.C. (1996). Standard handbook of petroleum and natural gas engineering. 2. Houston, TX: Gulf Professional Publishing. ISBN 0-88415-643-5.
↑ ^6.0 ^6.1 Trina, S. (2010). An integrated horizontal and vertical flow simulation with application to wax precipitation (Master of Engineering Thesis). Canada: Memorial University of Newfoundland.

[HB-1] 1.0 ^1.1 ^1.2 ^1.3 ^1.4 ^1.5 ^1.6 ^1.7 ^1.8 ^1.9 Hagedorn, A. R.; Brown, K. E. (1965). "Experimental study of pressure gradients occurring during continuous two-phase flow in small-diameter vertical conduits". Journal of Petroleum Technology. 17(04): 475–484.

[Economides-2] 2.0 ^2.1 ^2.2 ^2.3 ^2.4 ^2.5 ^2.6 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.

[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.

[Lyons-5] 5.0 ^5.1 ^5.2 ^5.3 ^5.4 ^5.5 Lyons, W.C. (1996). Standard handbook of petroleum and natural gas engineering. 2. Houston, TX: Gulf Professional Publishing. ISBN 0-88415-643-5.

[Trina-6] 6.0 ^6.1 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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[2]

[3]

[4]

[5]

[6]

@@ Line 7: / Line 7: @@
 The heart of the [[Hagedorn and Brown]] method is a correlation for the liquid holdup H<sub>L</sub> <ref name=Economides />.
-[[Hagedorn and Brown]]  is a default correlation for oil well in the [[:Category:PqPlot|PQplot]].
+[[Hagedorn and Brown]]  is a default [[VLP]] correlation for the oil wells in the [[:Category:PqPlot|PQplot]].
 == Math & Physics ==

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Difference between revisions of "Hagedorn and Brown correlation"

Revision as of 12:34, 6 April 2017

Contents

Brief

Math & Physics

Discussion

Flow Diagram

Workflow H_L

Modifications

Nomenclature

Greek symbols

Subscripts

References

Navigation

Categories

Products

Difference between revisions of "Hagedorn and Brown correlation"

Revision as of 12:34, 6 April 2017

Contents

Brief

Math & Physics

Discussion

Flow Diagram

Workflow HL

Modifications

Nomenclature

Greek symbols

Subscripts

References

Workflow H_L