Multiphase Flow Technology
Agenda 29th September 2011
• Introduction to multiphase flow
• Terms & regimes
• Conservation equations
• Stratified flow
• Applications
• Slug flow
30.09.2011
1
1
Multiphase pipe flow – a key technology for oil and gas
Jan Nossen, IFE
Autumn 2011
2
What’s multiphase transportation?
• Transport of gas, oil, water, chemicals and possibly solid particles in the same pipe
• Reduces need for new production platforms
• Gather production from many wells and send to existing platform or shore
• Subsea separation and pumping/compression may be required
• More cost efficient
• Often requires chemicals to prevent corrosion and solids precipitation that can possibly restrict or stop the flow
2
3
Multiphase transportation challenges
• Capacity problems due to viscous oils, emulsions etc.
• Solids precipitation can restrict or stop the flow
• Liquid accumulation during low flow rates in gas
condensate pipelines
• Large flow transients during production ramp-up
• Slugging and other instabilities can give
problems in the receiving facilities
• Erosion/corrosion
4
Application of multiphase flow models
• Tool for system design • Piping and equipment dimensioning
• Heating and thermal insulation
• Chemical choice and dosage
• Part of system simulator • Integrated system design
• Subsea solutions
• Operator training
• Operation support – system overview
• Surveillance: Compute non-monitorable parameters - Liquid content, leak detection …
3
5
Multiphase flow research
• Improved understanding and simulation of multiphase flow • Lab experiments
• Modelling of flow phenomena
• Detailed simulations (e.g. LES/DNS)
• Oil fields with high water production • Fluid characterization, emulsion properties
• Heavy, high viscosity oils and non-Newtonian fluids (e.g. drilling fluids)
• Liquid accumulation
• Corrosion
Fundamental modelling challenge:
Turbulence models have more unknowns than equations To close the equation set, “closure laws” based on
correlating data are necessary
4
Upscaling from lab to field
• 13 parameters determine holdup(s) and pressure drop in three phase pipe flow
• To develop the closure relations, we need data
• To cover the parameter space we need, say, 5
13 ~ 10
9 data points for 5 point resolution in each
parameter
• We have ~ 200 field data points at present
• It is clearly impossible to cover the parameter space of three phase pipe flow with data
Conclusion: we need models based on
physics to extrapolate beyond lab data
Lab Field
Mechanistic model
5
9
Simulators for multiphase transportation
• One-dimensional models for three-phase gas-oil-water flow in pipes and pipe networks
• Compute phase fractions, pressure and temeprature along the pipe vs. time
• Steady state and dynamic models
• 1-D conservation equations for mass, momentum and energy
• Flow regimes: Stratified, annular, slug, bubble flow
• Empirical knowledge incorporated in phenomenological closure laws for each flow regime
10
What base knowledge is
needed? • Math
• Partial differential equations
• Fluid mechanics • Basics • Multiphase flow • Waves • Computational Fluid Dynamics
• Computer science • Programming • Applied numerical methods
• Thermodynamics/ physical chemistry
• A little statistics
1
30.09.2011
Terms & notation
A
Multiphase flow terms, notation
30.09.2011
gA
oA
wA
Gas
Oil Water
Void fraction:
Oil holdup:
Water holdup:
Total holdup: wo
ww
oo
g
AA
AA
AA
/
/
/
Ug
Uo
Uw
2
Multiphase flow terms, notation
30.09.2011
z
y
G
Name Dimension Legend . e J/kg Specific internal energy h J/kg Specific enthalpy p Pa Pressure S m Wetted length r kg/m3 Density , , g - Gas, liquid bulk, droplet fractions t Pa Shear stress yE kg/(sm3) Droplet entrainment rate yD kg/(sm3) Droplet deposition rate
4
Real stratified flow
Continuous gas
Gas bubbles in oil
Gas bubbles in water
Continuous oil
Oil drops in gas
Oil drops in water
Continuous water
Water drops in gas
Water drops in oil
Thin liquid film on upper wall wd
w
wc
od
o
oc
w
o
g
g
g
og
w
o
oc
od wd
Thin film
wc
g
3
30.09.2011
Flow regimes
Flow regimes
Depend on
• phase velocities
• phase quantity
• pipeline orientation
• flow system
• gas/liquid
• liquid/liquid
30.09.2011
oil
gas
IFE Lab data
4
Flow regimes: Near horizontal
Gas/liquid regimes:
• Stratified
• Annular
Slug
• Bubble
Liquid/liquid regimes:
• Stratified (separated)
• Partly separated/dispersed
• Dispersed oil/water flow
Flow regimes: Near vertical Gas/liquid regimes:
• (a-b) bubble flow
• (c-d) slug/churn flow
• (e) annular flow
Liquid/liquid regimes:
Dispersed
.:.:
.: .:.:
.:
.:.:
.:
.:.:
.:
.:.:
.: .:.: .:
.:.:
.:
.:.:
.:
.:.:
.:
.:.:
.:
a b c d e
5
Transition criteria
30.09.2011
Large Wave Stratified
Slug
Bubble Liquid velocity
Gas velocity
Flow regime transitions
• The most important flow regime transition in pipelines
is from stratified flow to slug flow
• Two conditions must be fulfilled for slug flow to exist:
• Stratified flow must be unstable (Kelvin-Helmholtz instability)
• Slugs that are formed must be able to grow (Minimum slip)
• The Kelvin-Helmholtz criterion tells that the stratified
flow region gets smaller with increasing pressure
• Experimental data show that the slug flow region also
gets smaller with increasing pressure
• For high pressure we get a region of large wave flow
in between stratified and slug
6
30.09.2011
In between
slugs are stable
stratified flow is stable
USL
USG
neither is stable
1
Conservation equations
in multiphase pipe flow
Jan Nossen, IFE
Autumn 2011
Introduction
• Basic conservation equations: Mass, momentum, energy
• The conservation equations are formally similar for all
flow regimes (flow patterns)
• The differences between the flow regimes manifest
themselves in different terms (closure relations) for
• Wall and interfacial friction factors
• Dispersion of other phases as droplets and bubbles
• Momentum transfer between phases due to mass exchange
• Thermodynamics enter conservation equations thru
• Boundary conditions
• Source terms
• Mass transfer
2
Nomenclature
Name Dimension Legend . A m2 Pipe cross sectional area e J/kg Specific internal energy G kg/(m2s2) Gravity term h J/kg Specific enthalpy p Pa Pressure S m Wetted length U m/s Velocity Y m Vertical coordinate (wrt. gravity) z m Axial coordinate kg/m3 Density
- Gas, liquid bulk, droplet fractions Pa Shear stress
E kg/(sm3) Droplet entrainment rate
D kg/(sm3) Droplet deposition rate
Conservation equations for 3-phase
stratified flow
• 3 momentum equations • Gas layer with oil and water droplets
• Oil layer with gas bubbles and water droplets
• Water layer with gas bubbles and oil droplets
• 9 mass equations • 3 continuous phases
• 6 dispersed phases
• 9 energy equations • Alternatively, 1 mixture energy equation
3
Mass equations for simplified case
No bubbles/droplets in liquid
• Gas:
• Oil Bulk:
• Oil Droplets:
• Similarly for the water film and droplets
01
)( ggg UAzAt
DoEoooooo UAzAt
1)(
DoEodhhhh UAzAt
1)(
Droplet
entrainment
Droplet
deposition
Momentum equations
g
gogogg
gggg GA
S
A
S
z
pUA
zAU
t
21)(
oowowgogooo
ooooooo GA
S
A
S
A
S
z
pUA
zAU
t
21
wowowww
wwwwwww GA
S
A
S
z
pUA
zAU
t
21
4
Holdup equation for fully developed
steady stratified flow • The momentum equations for fully developed steady
stratified two-phase flow can be reduced to
• By eliminating the pressure gradient, one nonlinear
algebraic equation is obtained for the gas fraction:
GIIGG G
A
S
A
S
z
p0
LIILL G
A
S
A
S
z
p)1(0
0)1( LIILL
GIIGG G
A
S
A
SG
A
S
A
S
1
1
Stratified flow models
Jan Nossen, IFE
Autumn 2011
Introduction
• The different flow require different terms (closure
relations) in momentum equations for
• Wall and interfacial friction factors
• Dispersion of phases as drops and bubbles
• Momentum transfer between phases due to phase change
• The last point is closely connected with
thermodynamics and will not be treated here
30.09.2011
2
3
Pure stratified flow – volume fractions
Gas
Oil
Water
Void fraction:
Oil holdup:
Water holdup:
Total holdup: wo
ww
oo
g
AA
AA
AA
/
/
/
gA
oA
wA
4
Shear stresses in pipe flow
• Single phase flow:
• Laminar flow: Hagen-Poiseuille
• Turbulent flow: Håland
(Explicit approximation to
Colebrook-White formula)
DU
A
SU
A
S
z
p
U
1
2
1
42
1
8
1
22
2
Re/64 /Re UD
11.1
107.3Re
9.6log8.1
1
D
3
5
Shear stresses in channel flow
• Pressure drop in channel flow calculated using
hydraulic diameter concept
• Hydraulic diameter = Diameter of ”equivalent”
circular pipe
• Good approximation in most cases
• Not for geometries with narrow corners (i.e. triangle)
• Pressure drop:
HHH
H
UDD
S
AD
DU
z
p
Re)/(Re,4
1
2
1 2
6
Hydraulic diameters: Traditional method
• The gas is regarded as flowing in a closed channel
• The liquid is regarded as flowing in an open channel
• For three-phase gas-oil-water, the choices are less
obvious and demonstrate limitations of the concept
HoD
HgDGas
Liquid
4
7
Gas wall friction factor
• Simplest approach: Hydraulic diameter concept with
closed channel
• Gas velocity profile is generally asymmetrical in wavy
flow, leading to increase of shear stresses
• Friction factor can be corrected for waves
8
Liquid wall friction factor
• Smooth stratified flow (no waves): Hydraulic
diameter approach gives good results
• Wavy stratified flow: Hydraulic diameter approach
gives poor results because velocity profile is
modified by waves (Espedal 1998)
• Several correlations proposed in literature
• Biberg (2007) proposed a model computing friction
factors by matching turbulent velocity profiles in both
phases at interface
5
9
Interfacial friction factor
• Many different correlations in literature
• Most based on low pressure air/water experiments
• Poor extrapolation properties to high pressure
• The model of Biberg (2007) gives the interfacial
shear stress from the continuity of gas and liquid
velocities at the interface
10
Real stratified flow
Continuous gas
Gas bubbles in oil
Gas bubbles in water
Continuous oil
Oil drops in gas
Oil drops in water
Continuous water
Water drops in gas
Water drops in oil wd
w
wc
od
o
oc
w
o
g
o
w
o
oc
od wd
Thin film
wc
g
6
11
Dispersions in stratified flow
• drop transport (entrainment) in the gas
• Gas bubbles in the liquid layer
• Oil- water dispersions
12
drop transport in gas
• At high gas flow rates, drops are torn off the interface
and flow along with the gas (entrainment)
• Equilibrium is achieved between drop entrainment
from the liquid and deposition of drops to the liquid
• More than 50% of the volume flow of liquid can be in
the form of drops
• There is some correlation between the drop
entrainment and the Weber number
•
7
13
Gas bubbles in the liquid
• Gas is entrained as bubbles into a liquid layer at the
gas-liquid interface
• Known from hydraulic engineering (rivers, dams)
• Occurs as result of turbulence and breaking waves
• Equilibrium gas fraction determined by
available kinetic energy
• Significant gas entrainment occurs above a geometry
dependent critical Froude number
Oil-water dispersions
• Oil can be dispersed into water as drops,
or water into oil
• Dispersions can be stabilized by shear or chemicals
(surfactants)
• A dispersion of small drops stabilized by surfactants
is called an emulsion
• Important for pressure drop • Dispersed drops increase pressure drop in most cases
because drops behave like solid particles
• In some cases dispersed drops can decrease pressure drop
because drops are deformed by shear stress
• Phase inversion (Transition between water drops in oil and oil
drops in water) can give large pressure drop
8
Oil-water dispersions
• Very different flow regime
and pressure drop as
function of water fraction
(water cut) for different oils
• Surface chemistry affects
drop breakup and
coalescence – interfaces
can be rigid or flexible
9/30/2011
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0 % 10 % 20 % 30 % 40 % 50 % 60 % 70 % 80 % 90 % 100 %
Watercut [%]
Rela
tiv
e p
ressu
re g
ra
die
nt
[-]
Model oil - water
Crude oil - waterDW
DWDW
DW
ST
DW
DWDW
DO
STST
DWDW
STSTST
DODO
STSTST
DO
DO
DO
Two-phase oil-water flow
Umix = 1.75 m/s, From Utvik et al (1999)
ST = stratified, DO = Dispersed oil continuous,
DW = Dispersed water continuous
Emulsion viscosity
• In oil-water dispersions the apparent viscosity increases dramatically towards phase inversion
• Einstein developed formula for the apparent viscosity for a suspension of hard spheres in liquid:
• Later workers extended Einstein’s formula to more realistic systems, notably Pal and Rhodes (1999)
2
50.1
pure
mixture
WC
inv
= drop concentration
9
17
Summary
• Stratified models need to include models for:
• Flow geometry
• Wall and interfacial friction factors
• drops in gas phase
• Gas bubbles in liquid layer
• Oil drops in water and water drops in oil
• Hydraulic diameters are often used for modelling
friction factors but fails for wavy flow
• Friction factors can be modelled more accurately
from detailed models of velocity profiles
• Drops and bubbles influenced by surface chemistry
1
30.09.2011
Jan Nossen, IFE
Autumn 2011
Multiphase flow – applications
Acknowledgments
• Terje Sira (IFE)
• Zheng Gang Xu (SPT Group)
• Magnus Nordsveen (Statoil)
• SINTEF Petroleum
• IFE
30.09.2011
2
Multiphase Flow Main Challenges
• Pressure loss
• Liquid management
• Pipeline diameter selection
• Liquid inventory control
• Normal operation
• Water accumulation
• Rate changes, shut-down and Restart
• Pigging
• Sizing of process equipment, e.g. separators/slug catchers
• Prediction and control of slugging
Stratified flow in horizontal gas condensate pipe
• A small stream of condensate on top of a small stream of water
• The pressure gradient drives the gas which drives the condensate
which drives the water
• Typical values: Gas velocity Ug = 3 m/s, liquid velocity Ul = 1 m/s
Liquid holdup h = 0.01
Superficial velocities:
Usg = 3 m/s Usl = hUl = 1 cm/s
• Liquid transport modified by droplets in gas and water droplets in
condensate/condensate droplets in water
p1 p2
Ug
Uh Uw
Forces on
gas Forces on
condensate
Forces on
water
3
Steady state pressure drop and holdup
• Below a certain production rate, pressure gradient
and holdup start building up in the uphill sections
30/09/2011
Liq
uid
in
ven
tory
Production rate
Friction dominated flow
Gravity dominated flow
Pre
ssu
re d
rop
Pressure drop
Liquid inventory
1-phase pressure drop
Pipe diameter selection
• Too small diameter:
• Large pressure drop due to friction at high production rates
• Too large diameter:
• Too low velocity
• High holdup and flow instabilities at low production rates
• Large liquid surges during production start-up or ramp-up
• Possibly even severe slugging
• More expensive pipelines
• In 1-phase flow you can just make the pipe big enough
• In multiphase flow you have to balance capacity needs
with need to avoid liquid accumulation and instabilities
30.09.2011
4
Liquid surge during ramp-up
30.09.2011
Liq
uid
in
ven
tory
Production rate Q1 Q2
Liquid flowing from pipeline into slug catcher
when increasing rate from Q1 to Q2
PIPELINE WATER HOLDUP PROFILE
0
5
10
15
20
0 20000 40000 60000 80000
Pipeline Distance
Wate
r H
old
up
(%
)
0
400
800
1200
1600
Elevation (m)
Elevation
Water
Holdup
Liquid accumulation and water separation in low points
• Increased liquid
accumulation
and pressure drop
• Large water slugs
disturb process
• Corrosion
• WATER
Potential problems in multiphase flow
5
Potential problems in multiphase flow
• SHUT-IN/RESTART and RATE CHANGES • Liquid redistributes due to gravity during shut-in
• On startup, liquid in dips can exit the pipeline as large slugs as flow is ramped up
B-Gas and Liquid Outlet Flow
A-Liquid Distribution After Shutdown
shutdown
Flo
wra
te
gas
liquid
Troll gas: Onshore slug-catcher
6
Potential problems in multiphase flow
• PIGGING • Push a “pig” device through the pipe to
• Push out excess liquid and/or wax on the pipe wall
• Inspect the pipe for corrosion and wax using an instrumented pig
• Pigging the line can create a large liquid slug ahead of the pig
• The pigging operation can be optimized using simulations
A: Slug build-up
B. front arrival
C. slug surface
D. Pig arrivalTime
Flo
wra
te
gas
liquidA B C D
Potential problems in multiphase flow
• SEVERE SLUGGING
• A: Low spots fills with liquid and
flow is blocked
• B: Pressure builds up behind the
blockage
• C&D: When pressure becomes
high enough, gas blows liquid
out of the low spot as a slug
A. Slug formation
B.Slug production
C. Gas penetration
D. Gas blow-down
– Severe slugs can cause large pressure swings
and liquid surges out of pipeline.
– Severe slugging requires a dynamic model
to predict and control
7
Potential problems in multiphase flow
• LONG SLUGS
• In hydrodynamic slug flow we
have a random distribution of
slug lengths
• Some slugs can be very long,
creating problems (filling slug
catcher)
• Long slugs difficult to predict
Slug duration distribution
From Xu et al (1997)
Po
pu
lati
on
De
nsit
y
10
20
30
40
50
60
70
80
90
100
110
120
35%
30%
25%
20%
Olga Data
Test Data
15%
10%
5%
0%
Slug Duration (s)
One-dimensional multiphase
pipe flow simulators
• One dimensional models for
multiphase flow of gas, oil and water in
wells, pipelines and networks
• Steady state and dynamic models
• 1-D conservation equations for mass,
momentum and energy
• Experimental experience used in
developing closure relations
• Wall and interfacial friction factors
• Transport of drops and bubbles
• Testing against field data important for
validation
8
15
Description of a case to be
simulated
Pipe geometry Fluid properties Boundary
conditions
Pipe elevation profile
Pipe inner diameter
Internal wall
roughness
Wall thickness
Wall material
properties
Mass fractions
Densities
Viscosities
Surface and interfacial
tensions
Heat conductivities
Specific enthalpies
Specific heat
capacities
Boundary conditions
at pipe ends (Flow,
pressure,
temperature)
External temperature
Thermal properties of
surroundings (water,
air, soil)
CFD models for multiphase flow
• Standard CFD commercial codes are widely used for
simulating laminar and turbulent single phase flow
• Multiphase flow give us several new challenges
• Where is the interface?
• Large scale interfaces (stratified flow), small scale interfaces
(bubbly flow) or both at the same time
• Continuous phases typically described by Euler methods
(i.e. Reynolds averaged Navier-Stokes equations)
• Dispersed phases typically described by either Lagrange
methods (tracking each bubble/drop/particle) or Euler methods
• More difficult when a phase is both continuous and dispersed
30/09/2011
9
Eulerian CFD methods for multiphase flow
• Diffuse interface methods
• Volume of Fluid (VOF) method
• Computes volume fraction of each phase in each control volume
• Level Set method
• Interface given as zero level of an auxiliary function
• Phase Field method
• Extra conservation equation for «phase field» across interface
• Sharp interface methods
• Front tracking – explicit interface tracking
30/09/2011
1D versus CFD multiphase flow models
CFD models
• High resolution
• Fewer closure relations
• Depending on resolution
• Scales not resolved must
be modelled
• Slow or VERY slow
• Impractical for pipelines
• Can be good for
equipment
30/09/2011
1D models
• Low resolution
• Many closure relations
• Friction factors
• Entrainment/deposition
• Drop/bubble properties
• Fast
• Only tool today for long
pipelines
10
Why are CFD models impractical for pipelines?
• Long high pressure gas condensate pipeline
• 1m = 102 cm diameter, 100 km = 107 cm long
• Stratified wavy flow
• Bulk flow varies over O(1 cm) length
• 1 cm grid resolution => N ~ 102102107 = 1011 grid cells
• Bubbles, drops and other features will have sub-grid
scale and need to be modelled
• Huge computing times – but uncertain accuracy
30/09/2011
1
Fundamentals of slug flow
Jan Nossen, IFE
Autumn 2011
What is slug flow?
• Long bubbles (Taylor bubbles) alternate with more or
less aerated liquid plugs, called slugs
• The slug front (bubble tail) acts like a hydraulic jump
• Gas bubbles are entrained into the slug front and
transported backward relative to the front
• The liquid below the bubble can also contain bubbles
2
What is slug flow (II)
• Mechanisms in slug flow
• From Hale et al. (2000)
Gas entrained
by slug body.
Gas released
from slug body.
Aerated liquid "shed"
from faster moving slug. Liquid film "scooped up"
by faster moving slug body.
Gas released
from film.
Direction of slug body and film.
What is slug flow (III)
• Behind the slug front there is normally a mixing zone
with strong turbulence
• Increased wall shear stress (friction)
• Gas entrainment
• The gas is nearly homogeneously distributed
in the mixing zone
• The gas is rising towards the
top of the pipe behind the
mixing zone at low velocities
3
Severe slugging (terrain slugging)
30.09.2011
• Occurs in dip geometry at low
flow velocity
• Liquid slows down and blocks
gas from passing the bend
• Pressure builds up and starts
pushing out liquid
• Long bubble expands and blows
rapidly out on top
• Cycle repeats
• Gives large pressure and flow
oscillations, disturbing process
• Can be controlled by choking
(valve with small opening) Drawing: ABB
Horizontal and vertical slug flow
• Long, near horizontal pipes:
• Relatively long slugs (> 30 diameters)
• Vertical pipes:
• Shorter slugs (typically < 20 diameters)
• Falling liquid around the Taylor bubble
• In vertical flow Taylor bubbles are only stable for
pipes of diameter less than about 10 cm
• In larger diameter pipes we normally get chaotic flow
(churn flow) instead of slug flow
• Slugs developed in a long pipe can survive a short
riser, but will die out in a long, large diameter riser
4
Flow regime transitions
• The most important flow regime transition in pipelines
is from stratified flow to slug flow
• Two conditions must be fulfilled for slug flow to exist:
• Stratified flow must be unstable (Kelvin-Helmholtz instability)
• Slugs that are formed must be able to grow (Minimum slip)
• The Kelvin-Helmholtz criterion tells that the stratified
flow region gets smaller with increasing pressure
• Experimental data show that the slug flow region also
gets smaller with increasing pressure
• For high pressure we get a region of large wave flow
in between stratified and slug
30.09.2011
In between
slugs are stable
stratified flow is stable
USL
USG
neither is stable
5
Slug formation
• Complex phenomenon, not fully
understood
• Studies at Imperial College,
Hale et al. (2001)
• ”Slug precursor” formed on top
of long wave formed after
previous slug
• Most ”slug precursors” collapse,
while a few survive and grow
into long slugs
• Other mechanisms as well
Slug growth
• Most new, short slugs collapse because there isn’t
enough liquid ahead of them to grow on
• Liquid left by a collapsed slug is picked up by the
next one
• Slug frequency gradually decaying along the pipe
• Terrain effects from ups and downs important in long
pipelines
6
History of slug flow modelling
• Theoretical modelling since the 1970s
• Traditional concept: Unit Cell Model
• Pipe discretized into control volumes
• Fully developed flow assumed in each control volume
• Development of each individual slug ignored
• The unit cell model cannot predict slug length
• More modern approach: Slug tracking, where slugs
are tracked from they are formed until they vanish
• Main challenge: Model slug formation and growth
Unit cell model for slug flow
• Most common concept for slug flow modelling
• Infinite train of identical slug and Taylor bubbles
• Fully developed dispersed bubble flow assumed in slug
• Sometimes extended to include increased friction behind the slug front
• Fully developed stratified/annular flow assumed
in Taylor bubble zone
• Sometimes extended to include developing flow (slug tail profile)
• Has also been extended to gas-oil-water 3-phase flow
7
Unit cell model: Assumptions
• The flow is assumed steady and periodical in a frame of
reference moving with the pattern velocity
• In other words the flow is assumed locally fully
developed
• The slug front and the bubble nose are assumed to
have the same velocity
UGB
ULB
UB
UGS, ULS
BU
BU
Slug collapse in downward flow
• For low mixture velocities we normally get slug flow
uphill and stratified flow downhill
• The unit cell model assumes local equilibrium
independent of upstream and downstream conditions
• The model will therefore often predict that slugs
vanish immediately when they reach a hilltop
• In reality slugs can often survive for a considerable
distance in a downhill if slug collapse is slow
• Slug tracking or similar techniques are necessary to
study how slugs develop in a hilly terrain pipeline
8
Unit cell model: Continuity (I)
• Total volume flux (mixture velocity) is
• Total mass flow is constant along the pipe
(in a stationary frame of reference)
• If we assume constant densities, the mixture velocity
is also constant along the pipe
• Volume flux in the slug:
• Volume flux in the bubble:
UGB
ULB
UB
UGS, ULS
SLSGM UUU
MLBBGBB
MLSSGSS
UUU
UUU
)1(
)1(
Unit cell model: Continuity (II)
• In a frame of reference moving with the fronts,
continuity of each phase across the bubble nose gives
UGB
ULB
UB
UGS, ULS
))(1())(1(
)()(
BLBBBLSS
BGBBBGSS
UUUU
UUUU
9
Closure relations for the unit cell
• Bubble nose velocity
• Void fraction (gas volume fraction) in slugs
• Velocity difference between gas and liquid in slugs
• Friction factors in bubble zone and in slug
• Extra pressure drop behind slug front
UGB
ULB
UB
UGS, ULS
The bubble nose velocity
• The bubble nose velocity is approximately linear in
the mixture velocity:
• The distribution coefficient C0 is a function of the
velocity profile in the slug
• C0 ≈ 2 for laminar flow in the slug
• C0 ≈ 1.2 for fully turbulent flow in the slug
• Smooth transition between laminar and turbulent flow
• The drift velocity U0 is a function of inclination angle,
densities, liquid viscosity and pipe diameter
30.09.2011
00 UUCU MB
10
The gas fraction in slugs
• The gas fraction in the slug depends on
• Velocity distribution, pipe diameter, inclination, densities,
viscosities, surface tension, surface rheology
• The surface rheology and its influence is little known
• Gas fraction in slugs is complex and difficult to model
• Empirical correlations mostly used. Typically
• Some simple mechanistic models exist
• Predicts gas entrainment at front and the gas distribution in the slug
• Typically poor extrapolation properties
),( MSS U
The gas distribution in slugs
• Gas is entrained into the slug at the front
• The gas entrainment can be increased by gas in the
stratified layer below the bubble
• The gas distribution in the slugs is governed by
forces on the bubbles
• An inhomogeneous bubble distribution can give rise
to a distribution slip (gas and liquid are distributed
differently in regions of low and high velocity)
• This affects the average gas fraction in the slug
11
The gas distribution in slugs (II)
• The gas is strongly skewed towards the top of the
pipe, Nydal (1991)
The flow in the bubble zone
• The momentum equations for fully developed flow in
the bubble zone read
• By eliminating the pressure gradient, one nonlinear
algebraic equation is obtained for the gas fraction:
GBIBIBGBGB
B GA
S
A
S
z
p0
LBIBIBLBLB
B GA
S
A
S
z
p)1(0
0)1( LBIBIBLBLB
BGBIBIBGBGB
B GA
S
A
SG
A
S
A
S
12
The flow in the bubble zone (II)
• To solve the holdup equation, the velocities UGS and
ULS have to be computed for a given
• The continuity equations differ from pure stratified flow
• For stratified flow,
• For the stratified flow in the bubble region,
B
)1/(;/ SLLSGG UUUU
)(1
1
)(
BLS
B
SBLB
BGS
B
SBGB
UUUU
UUUU
Three-phas gas-oil-water slug flow
• Low velocities: Oil and water separated (stratified)
• High velocities: Oil and water dispersed (mixed)
• Moderate velocities: Oil and water mixed in slug and
separated in bubble zone (because ULS > ULB)
• Mixing of oil and water can increase frictional
pressure drop (increased mixture viscosity)
13
Summary
• Qualitative description of slug flow
• Horizontal and vertical slug flow
• Transition from stratified to slug flow
• Slug formation, growth and collapse
• The unit cell model for gas-liquid slug flow
• Gas-oil-water slug flow – effect of mixing