Difference between revisions of "Gray correlation"

Latest revision as of 09:10, 6 December 2018

Brief

Gray is an empirical two-phase flow correlation published in 1974 ^[1].

Gray is the default VLP correlation for the gas wells in the PQplot.

Gray in PQplot Vs Prosper & Kappa

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 + \rho_m \frac{f v_m^2 }{2 g_c D} + \rho_m \frac{\Delta{(\frac{v_m^2}{2g_c}})}{\Delta h}

where

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

, slip mixture density ^[1].

\rho_m = \rho_L C_L + \rho_g (1-C_L)

, no-slip mixture density ^[1].

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]

The pseudo wall roughness:

\varepsilon' = \begin{cases} \frac{28.5}{453.592} \frac{\sigma_L}{\rho_m v_m^2}, &\mbox{if } R \geqslant 0.007 \\ \varepsilon + R \frac{\varepsilon'-\varepsilon}{0.007}, & \mbox{if } R < 0.007 \end{cases}

, with the limit

\varepsilon' \geqslant 2.77 \times 10^{-5}

^[1]

Reynolds two phase number:

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

^[4]

Discussion

Why Gray correlation?

The Gray correlation was found to be the best of several initially tested ...
— Nitesh Kumar l^[5]

Workflow H_g & C_L

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= 62.4\ SG_o \frac{1}{1+WOR} + 62.4\ SG_w\ \frac{WOR}{1 + WOR}

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

^[6]

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_g \times 10^6}{86400 A_p}\ \frac{14.7}{p}\ \frac{T_K}{520}\ \frac{z}{1}

C_L = \frac{v_{SL}}{v_{SG}+v_{SL}}

v_m = v_{SL} + v_{SG}

\rho_m = \rho_L C_L + \rho_g (1-C_L)

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

^[6]

\sigma_L = \frac{\sigma_o\ q_o + 0.617\ \sigma_w\ q_w}{q_o + 0.617\ q_w}

^[1]

N_V = 453.592\ \frac{{\rho_m}^2 {v_m}^4}{g_c \sigma_L (\rho_L - \rho_g)}

^[1]

N_D = 453.592\ \frac{g_c (\rho_L - \rho_g) D^2}{\sigma_L }

^[1]

R = \frac{v_{SL}}{v_{SG}}

^[1]

B = 0.0814 \left ( 1 - 0.554\ \ln \left (1 + \frac{730 R}{R+1} \right ) \right )

^[1]

A = -2.2314 \left ( N_V \left (1 + \frac{205}{N_D} \right ) \right )^B

^[1]

H_g = \frac{1-e^A}{R+1}

^[1]

Modifications

1. Use Fanning correlation for the dry gas (WGR=0 and OGR=0 case).

2. Use watercut instead of WOR to account for the OGR=0 case.

3. If the relative roughness: $\frac{\varepsilon'}{D} > 0.05$ use 0.05 in the Moody Diagram ^[3].

4. If H_L can't be calculated then H_L = C_L.

Nomenclature

A

= correlation group, dimensionless

A_p

= flow area, ft2

B

= correlation group, dimensionless

B

= formation factor, bbl/stb

C

= no-slip holdup factor, dimensionless

D

= pipe diameter, ft

h

= depth, ft

H

= 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_V

= velocity number, dimensionless

p

= pressure, psia

q_c

= conversion constant equal to 32.174049, lb_mft / lb_fsec²

q

= production rate, bbl/d

R

= superficial liquid to gas ratio, dimensionless

Re

= Reynolds number, dimensionless

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

\varepsilon'

= pseudo wall roughness, ft

\mu

= viscosity, cp

\rho

= density, lb_m/ft³

\bar \rho

= slip density, lb_m/ft²

\sigma

= surface tension of liquid-air interface, dynes/cm

Subscripts

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

References

↑ ^1.00 ^1.01 ^1.02 ^1.03 ^1.04 ^1.05 ^1.06 ^1.07 ^1.08 ^1.09 ^1.10 Gray, H. E. (1974). "Vertical Flow Correlation in Gas Wells". User manual for API 14B, Subsurface controlled safety valve sizing computer program. API.
↑ 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.
↑ ^3.0 ^3.1 Moody, L. F. (1944). "Friction factors for pipe flow". Transactions of the ASME. 66 (8): 671–684.
↑ ^4.0 ^4.1 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.
↑ Kumar, N.; Lea, J. F. (January 1, 2005). "Improvements for Flow Correlations for Gas Wells Experiencing Liquid Loading" (SPE-92049-MS).
↑ ^6.0 ^6.1 ^6.2 Lyons, W.C. (1996). Standard handbook of petroleum and natural gas engineering. 2. Houston, TX: Gulf Professional Publishing. ISBN 0-88415-643-5.

[Gray-1] 1.00 ^1.01 ^1.02 ^1.03 ^1.04 ^1.05 ^1.06 ^1.07 ^1.08 ^1.09 ^1.10 Gray, H. E. (1974). "Vertical Flow Correlation in Gas Wells". User manual for API 14B, Subsurface controlled safety valve sizing computer program. API.

[Colebrook-2] 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.

[Moody1944-3] 3.0 ^3.1 Moody, L. F. (1944). "Friction factors for pipe flow". Transactions of the ASME. 66 (8): 671–684.

[HB-4] 4.0 ^4.1 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.

[Kumar-5] Kumar, N.; Lea, J. F. (January 1, 2005). "Improvements for Flow Correlations for Gas Wells Experiencing Liquid Loading" (SPE-92049-MS).

[Lyons-6] 6.0 ^6.1 ^6.2 Lyons, W.C. (1996). Standard handbook of petroleum and natural gas engineering. 2. Houston, TX: Gulf Professional Publishing. ISBN 0-88415-643-5.

[1]

[2]

[3]

[4]

[5]

[6]

@@ Line 2: / Line 2: @@
 == Brief ==
-[[Gray]] is an empirical two-phase flow correlation published in '''1974''' <ref name= Gray />.
+[[Gray correlation|Gray]] is an empirical two-phase flow correlation published in '''1974''' <ref name= Gray />.
-[[Gray]] is the default [[VLP]] correlation for the '''gas wells''' in the [[:Category:PqPlot|PQplot]].
+[[Gray correlation|Gray]] is the default [[VLP]] correlation for the '''gas wells''' in the [[PQplot]].
+[[File: GRAY.png|thumb|500px|link=https://www.pengtools.com/pqPlot?paramsToken=d14638acea57a4523b4553153a5dcb5a|Gray in PQplot Vs Prosper & Kappa |right]]
 == Math & Physics ==
 Following the law of conservation of energy the basic steady state flow equation is:
-:<math> 144 \frac{\Delta p}{\Delta h} =  \bar \rho_m + \rho_m \frac{f v_m^2 }{2 g_c D} + \rho_m \frac{\Delta{(\frac{v_m^2}{2g_c}})}{\Delta h}</math><ref name="Gray" />
+:<math> 144 \frac{\Delta p}{\Delta h} =  \bar \rho_m + \rho_m \frac{f v_m^2 }{2 g_c D} + \rho_m \frac{\Delta{(\frac{v_m^2}{2g_c}})}{\Delta h}</math>
 where
-:<math> \bar \rho_m = \rho_L (1-H_g) + \rho_g H_g</math>,  slip mixture density
+:<math> \bar \rho_m = \rho_L (1-H_g) + \rho_g H_g</math>,  slip mixture density <ref name= Gray />.
-:<math> \rho_m = \rho_L C_L + \rho_g (1-C_L) </math>,  no-slip mixture density
+:<math> \rho_m = \rho_L C_L + \rho_g (1-C_L) </math>,  no-slip mixture density <ref name= Gray />.
 Colebrook–White <ref name=Colebrook/> equation for the [http://en.wikipedia.org/wiki/Darcy_friction_factor_formulae Darcy's friction factor]:
@@ Line 29: / Line 31: @@
 == Discussion  ==
-Why [[Gray]] correlation?
+Why [[Gray correlation|Gray]] correlation?
 {{Quote| text = The Gray correlation was found to be the best of several initially tested ... | source = Nitesh Kumar l<ref name= Kumar />}}
@@ Line 37: / Line 39: @@
 :<math> M =SG_o\ 350.52\ \frac{1}{1+WOR}+SG_w\ 350.52\ \frac{WOR}{1+WOR}+SG_g\ 0.0764\ GLR</math><ref name="HB" />
-:<math> \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}</math><ref name= Lyons/>
+:<math> \rho_L= 62.4\ SG_o \frac{1}{1+WOR} + 62.4\ SG_w\ \frac{WOR}{1 + WOR}</math>
 :<math> \rho_g = \frac{28.967\ SG_g\ p}{z\ 10.732\ T_R} </math><ref name= Lyons/>
@@ Line 69: / Line 71: @@
 == Modifications  ==
-use watercut instead of WOR
+.  Use [[Fanning correlation]] for the dry gas ([[WGR]]=0 and [[OGR]]=0 case).
+. Use [[WCUT| watercut]] instead of [[WOR]] to account for the [[OGR]]=0 case.
+. If the relative roughness: <math> \frac{\varepsilon'}{D} > 0.05 </math> use 0.05 in the Moody Diagram <ref name = Moody1944/>.
+. If H<sub>L</sub> can't be calculated then H<sub>L</sub> = C<sub>L</sub>.
 == Nomenclature  ==
-NV velocity number
+:<math> A </math> = correlation group, dimensionless
+:<math> A_p </math> = flow area, ft2
+:<math> B </math> = correlation group, dimensionless
+:<math> B </math> = formation factor, bbl/stb
+:<math> C </math> = no-slip holdup factor, dimensionless
+:<math> D </math> = pipe diameter, ft
+:<math> h </math> = depth, ft
+:<math> H </math> = holdup factor, dimensionless
+:<math> f </math> = friction factor, dimensionless
+:<math> GLR </math> = gas-liquid ratio, scf/bbl
+:<math> M </math> = total mass of oil, water and gas associated with 1 bbl of liquid flowing into and out of the flow string, lb<sub>m</sub>/bbl
+:<math> N_D </math> = pipe diameter number, dimensionless
+:<math> N_V </math> = velocity number, dimensionless
+:<math> p </math> = pressure, psia
+:<math> q_c </math> = conversion constant equal to 32.174049, lb<sub>m</sub>ft / lb<sub>f</sub>sec<sup>2</sup>
+:<math> q </math> = production rate, bbl/d
+:<math> R </math> = superficial liquid to gas ratio, dimensionless
+:<math> Re </math> = Reynolds number, dimensionless
+:<math> SG </math> = specific gravity, dimensionless
+:<math> T </math> = temperature, °R or °K, follow the subscript
+:<math> v </math> = velocity, ft/sec
+:<math> WOR </math> = water-oil ratio, bbl/bbl
+:<math> z </math> = gas compressibility factor, dimensionless
+===Greek symbols===
+:<math> \varepsilon </math> = absolute roughness, ft
+:<math> \varepsilon' </math> = pseudo wall roughness, ft
+:<math> \mu </math> = viscosity, cp
+:<math> \rho </math> = density, lb<sub>m</sub>/ft<sup>3</sup>
+:<math> \bar \rho </math> = slip density, lb<sub>m</sub>/ft<sup>2</sup>
+:<math> \sigma </math> = surface tension of liquid-air interface, dynes/cm
+===Subscripts===
+g = gas<BR/>
+K = °K<BR/>
+L = liquid<BR/>
+m = gas/liquid mixture<BR/>
+o = oil<BR/>
+R = °R<BR/>
+SL = superficial liquid<BR/>
+SG = superficial gas<BR/>
+w = water<BR/>
 == References ==
@@ Line 131: / Line 182: @@
 }}</ref>
-<ref name=Turner>{{cite journal
+<ref name=Kumar>{{cite journal
   |first1=N. |last1=Kumar
   |first2=J. F. |last2=Lea
   |title=Improvements for Flow Correlations for Gas Wells Experiencing Liquid Loading
- |journal=Journal of Petroleum Technology
   |number=SPE-92049-MS
   |date=January 1, 2005
   |url=https://www.onepetro.org/conference-paper/SPE-92049-MS
-  |url-access= subscription
+  |url-access=registration
 }}</ref>
@@ Line 145: / Line 195: @@
 [[Category:pengtools]]
-[[Category:pqPlot]]
+[[Category:PQplot]]
+{{#seo:
+|title=Gray correlation
+|titlemode= replace
+|keywords=Gray correlation
+|description=Gray correlation is an empirical two-phase flow correlation published in 1974.
+}}

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Difference between revisions of "Gray correlation"

Latest revision as of 09:10, 6 December 2018

Contents

Brief

Math & Physics

Discussion

Workflow H_g & C_L

Modifications

Nomenclature

Greek symbols

Subscripts

References

Navigation

Categories

Products

Difference between revisions of "Gray correlation"

Latest revision as of 09:10, 6 December 2018

Contents

Brief

Math & Physics

Discussion

Workflow Hg & CL

Modifications

Nomenclature

Greek symbols

Subscripts

References

Workflow H_g & C_L