21 – 22 February 2018 Houston, TX Copyright 2018, Letton Hall Group. This paper was developed for the UPM Forum, 21 – 22 February 2018, Houston, Texas, U.S.A., and is subject to correction by the author(s). The contents of the paper may not necessarily reflect the views of the UPM Forum sponsors or administrator. Reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Letton Hall Group is prohibited. Non-commercial reproduction or distribution may be permitted, provided conspicuous acknowledgment of the UPM Forum and the author(s) is made. For more i nformation, see www.upmforum.com. Multiphase Flow in Pipes Flow Pattern Prediction by Jim Brill
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Multiphase Flow in Pipes Flow Pattern Prediction · Flow Pattern Prediction •Empirical Maps •Ros (1961 –Vertical upward flow) •Govier and Aziz (1972 –Vertical upward flow)
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21 – 22 February 2018 Houston, TX
Copyright 2018, Letton Hall Group. This paper was developed for the UPM Forum, 21 – 22 February 2018, Houston, Texas, U.S.A., and is subject to correction by the author(s). The contents of the paper may not necessarily reflect the views of the UPM Forum sponsors or administrator. Reproduction,distribution, or s torage of any part of this paper for commercial purposes without the written consent of the Letton Hall Group is prohibited. Non-commercial reproduction or dis tribution may be permitted, provided conspicuous acknowledgment of the UPM Forum and the author(s) is made. For moreinformation, see www.upmforum.com.
Multiphase Flow in Pipes Flow Pattern Prediction
by Jim Brill
Why is predicting flow patterns so important?
• Required to predict pressure gradient!• Required to size separators/slug catchers• Multiphase Metering and Pumping are sensitive to
flow pattern• Many flow assurance issues depend on flow pattern
• Distribution of paraffin deposition across pipe• Identifying where corrosion might occur• Affects heat transfer/temperature predictions
Multiphase Flow
• Two-phase Flows• Gas-Oil, Gas-Water, Oil-Water, and Oil-Solid (Sand,
Typical Taitel et al. Mechanistic Vertical Flow Pattern Map (1980)
Zhang et al. (2003) Model(Chapter 9 of Applied Multiphase Flow in Pipes and Flow
Assurance – Oil and Gas Production)
• A Unified Model for Prediction of Flow Pattern Transitions and Hydrodynamic Behavior of Gas-Liquid Pipe Flow at all Inclination Angles
• Based on Hydrodynamics of Slug Flow
Slug Flow Pattern Is Surrounded by All Other Flow Patterns
Operating Point
Typical TUFFP unified flow pattern map for horizontal flow
Typical TUFFP unified flow pattern map for vertical upward flow
Lagrangian Approach
Taylor bubble velocity vTB is taken as reference
θ
vTB
z
Variables …Af Cross sectional area occupied by liquid film Ac Cross sectional area occupied by gas coreSf Pipe perimeter wetted by liquid film Sc Pipe perimeter contacted by gas core Si Interfacial perimeter
Af
Ac
SfSi
Sc
Liquid film
Gas core
Momentum Equation for Liquid Film
Momentum Equations
Momentum Equation for Gas Core
p2 - p1( )L f
=rc(vTB - vc )(vm - vc )
L f-
t iSi +t cSc(1-HLf )A
- rcgsinq
p2 - p1( )L f
=rL vTB - v f( ) vm - v f( )
L f+
t iSi -t fS f
HLfA- rLgsinq
Momentum Equations ...
Combined Momentum Equation for the liquid film region of a Slug Unit
rL vTB - v f( ) vm - v f( ) - rc vTB - vc( ) vm - vc( )L f
+t cScAc
-t fS f
A f
+t iSi1
A f+
1
Ac
é
ëê
ù
ûú- rL - rc( )gsinq = 0
rc =rg 1- HLf - HLc( ) + rLHLc
1- HLf
where
… Eq. 1
… Eq. 2
Combined Momentum Equation for Stratified Flow, Lf = → infinity
rL (vTB - v f )(vm - v f )- rc(vTB - vc )(vm - vc )
lF+
t cScAc
-t fS f
A f
+t iSi1
A f+
1
Ac
æ
èç
ö
ø÷ - (rL - rc )gsinq = 0
Momentum Equations ...
0
Combined Momentum Equation for Annular Flow, Sc=0
Momentum Equations ...
rL (vTB - v f )(vm - v f ) - rc(vTB - vc )(vm - vc )
lF+
t cScAc
-t fS f
A f
+t iSi1
A f+
1
Ac
æ
èç
ö
ø÷ - (rL - rc )gsinq = 0
0 + 0
… Eq. 3
Flow Pattern Transitions
Operating Point
Typical TUFFP unified flow pattern map for horizontal flow
Typical TUFFP unified flow pattern map for vertical upward flow
Stratified to Annular Flow
Based on concept that as the superficial gas velocity increases, the liquid film climbs up the pipe wall. When the wetted wall Fraction, , reaches 0.9 transition to annular flow is reached.
Thus, a “closure relationship” is required to predict .
Slug to Stratified and/or Annular Flow Boundary(Main difference between Unified and other Mechanistic Models)
Requires trial and error solution of combined momentum equation for the film region of a slug unit or for the combined momentum equation for stratified or annular flow.
Given vSg, find a value of vSL that gives the same value of HLf from these two (or three) equations. This is a difficult calculation, and Torres found an alternative that is new in the unified model.
Torres Alternative
Kinematics – the branch of mechanics treating the motion of bodies in the abstract, and without reference to the action of forces (or momentum balances).
Torres Alternative is based on an equation (HLfk) that is derived from liquid mass balances in the film region and the slug unit.
• Continuity equation for liquid mass flow rate entering and leaving the liquid film region.
• Equations for liquid and gas mass flow rates over a slug unit.• Definition of entrainment fraction to account for liquid entrained in
gas core of film region. • Defines the minimum HLf for slug flow to exist.
Torres Alternative for Slug to Stratified and/or Annular Flow Boundary
Note that these are developed from multiphase flow experiments and may be sensitive to pipe diameter, liquid viscosity, inclination angle, etc.
Example Problem 1 – Determine whether slug flow or stratified/annular flow exists, assuming smooth vertical pipe and no liquid entrainment (fE=0, HLc=0). Note that stratified flow cannot exist since the pipe is vertical.
Determine if annular flow exists.
Solve the annular flow combined momentum equation (Eq. 3) iteratively for HLf (or δL)
t fS f
A f+t iSi
1
A f+
1
Ac
é
ëê
ù
ûú- rL - rc( )gsinq = 0 … (Eq. 3)
Variables …Af Cross sectional area occupied by liquid film Ac Cross sectional area occupied by gas coreSf Pipe perimeter wetted by liquid film Sc Pipe perimeter contacted by gas core Si Interfacial perimeter
Af
Ac
SfSi
Sc
Liquid film
Gas core
Trigonometric Calculations
A = πd2/4 = (3.14)(0.089)2/4 = 0.00622 m2
Sf = πd = (3.14)(0.089) = 0.2796 m
Assume a film thickness δL = 0.005578 m
HLf =4d L d -d L( )
d2=
4 0.005578 ( ) 0.089 - 0.005578 ( )0.089( )
2= 0.23498
Si = p d - 2d L( ) = p 0.089 - 2( ) 0.005578 ( )éë ùû = 0.24455 m
S f = pd = p(0.089) = 0.2796 m
v f =vSL
HLf
=0.1( )
0.23498( )= 0.42557 m / s
Calculate velocities and densities
vc =vSg
1- HLf
=3.0
1- 0.23498= 3.92147 m / s
rc = rg = 115 kg /m3
rc =rg 1- HLf - HLc( ) + rLHLc
1- HLf
… Eq. 2
fi = fc 1+ 300d Ld
æ
èçö
ø÷
dc = d - 2d L = 0.089 - 2( ) 0.005578( ) = 0.07784 m NRec=
rgvcdc
mg=
115( ) 3.92147( ) 0.07784( )0.00002( )
= 1,755,172
(Turbulent flow)
fc = 0.184NRec
-0.2 = 0.184( ) 1,755,172( )-0.2
= 0.01037
fi = 0.01037( ) 1+ 3000.005578
0.089
æ
èçö
ø÷é
ëê
ù
ûú = 0.20535Wallis, 1969
t i =firc vc - v f( )
2
8=
0.20535( ) 115( ) 3.92147 - 0.42557( )2
8= 36.076 Pa
Calculate Interfacial Shear Stress
Calculate Film Shear Stress
NRe f=
rLv f
mL
4A f
S f + Si( )
é
ë
êê
ù
û
úú
=890( ) 0.42557( )
0.005( )4 0.00146( )
0.2796 + 0.24455( )
é
ëê
ù
ûú = 844.01
f f =64
NRe f
=64
844.01= 0.07583
t f =f frLv f
2
8=
0.07583( ) 890( ) 0.42557( )2
8= 1.52786 Pa
Laminar Flow
Is Annular Flow Momentum Equation (Eq. 3) Satisfied?
… Eq. 3t fS f
A f+t iSi
1
A f+
1
Ac
é
ëê
ù
ûú- rL - rc( )gsinq = 0
-1.52786( ) 0.2796( )
0.00146( )+ 36.076( ) 0.24455( )
1
0.00146( )+
1
0.00476( )
é
ëê
ù
ûú - 890 -115( ) 9.8( )sin 0( )
= - 292.60 + 7,896.17 - 7,595.00
= 8.57 » 0
Although the combined momentum equation has not quite converged, a very slight change in the assumed value for film thickness or HLf would result in convergence.
ùû = 1.2( ) 3.1( )+ 0.35 9.8( ) 0.089( ) = 4.047 m / s
Determine HLfk from Eq. 4 with fE = 0.0
HLfk =HLLS vTB - vm( ) + vSLéë ùû
vTB=
0.807( ) 4.047 - 3.1( ) + 0.1éë ùû
4.047= 0.2135
Since HLf from the annular flow combined momentum balance equation (0.23498), is greater than HLfk (0.2135) for slug flow to occur, the flow pattern is slug.
Calculate Kinematic Film Liquid Holdup, HLfk
Solutions to Eq. 3 were also obtained for two other values of vSg (2.0 and 4.0 m/s) and the results are plotted below. The transition from stratified/annular flow to slug flow clearly occurs when HLf from Eqs. 3 and 4 (and Eq. 1) are the same at a vSg
of approximately 3.35 m/s.
Note that this is Example 9.3 of the new SPE Textbook. See the Textbook for calculation of the
Pressure Gradient for slug flow.
Example Problem 2 – Determine whether slug flow or stratified/annular flow exists, assuming smooth vertical pipe and liquid entrainment from Oliemans correlation. All other variables are the same as in Example 1.
Since some of the liquid is now entrained in the gas core, there is less available for the liquid film.
Torres Alternative for Slug to Stratified and/or Annular Flow Boundary
• Kinematic film holdup = minimum HLf for slugs to exist.• If HLf solution from combined momentum equation for stratified
flow or annular flow > HLfk, then slug flow exists.• At the transition, HLfk = HLf from all combined momentum equations.
… Eq. 4
Determine if annular flow exists.Solve the annular flow combined momentum equation (Eq. 3)
iteratively for HLf (or δL)
Assume a film thickness δL = 0.00289 m. The annular flow combined momentum equation is satisfied with this assumption.
Then, HLf = 0.1257 (vs 0.23498 in Example 1)
and HLfk = 0.1930.
Since HLf < HLfk then slug flow cannot exist and the flow pattern is annular.
Solutions to the annular flow combined momentum equation were also obtained for two other values of vSg (2.0 and 4.0 m/s) and the results are plotted below. The transition from stratified/annular flow to slug flow clearly occurs when HLf from Eqs. 3 and 4 (and Eq. 1) are the same at a vSg of approximately 2.5 m/s.
ST/Ann Flow
Slug Flow
Clearly, closure relationships can have a significant impact on predicting flow pattern and pressure gradient. For this reason, much of the TUFFP research program is now devoted to developing improved closure relationships for various pipe diameters, inclination angles, oil viscosities, wetting angle at pipe wall in stratified flow, etc.