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