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Table of Contents
1 Introduction 
1.1 Multi-phase flow assurance
1.1.1 General
1.1.2 Nuclear reactor multi-phase
models
1.1.3 Multi-phase flow in the
petroleum industry
1.2 Two-phase flow
1.2.1 Flow regimes in horizontal
pipes
1.2.2 Slugging
1.2.3 Flow regimes in vertical pipes
1.2.4 Flow regime maps
1.2.5 Flow in concentric and
eccentric annulus
1.3 Three and four-phase flow
1.3.1 Types of three-phase and quasi
four-phase flow
1.3.2 Three-phase flow regimes
1.4 Typical flow assurance tasks
1.5 Some definitions
1.5.1 General
1.5.2 Volume fraction, holdup and
water cut
1.5.3 Superficial velocity
1.5.4 Mixture velocity and density
1.5.5 Various sorts of pipes
2 Conservation equations
2.1 Introduction
2.2 Mass conservation
2.2.1 Comparing single-phase and
multi-phase mass conservation
2.2.2 Mass conservation for well
mixed phases
2.3 Multi-phase momentum conservation
2.3.1 Main equations
2.3.2 Pressure differences between
phases due to elevation differences
2.3.3 Summarizing the forces between
phases
2.3.4 Comparing single- and
multi-phase momentum conservation
2.4 Energy conservation
2.4.1 Comparing single-phase and
multi-phase energy conservation
2.5 Mass transfer between phases with equal pressures
2.6 Comments on the conservation equations
2.6.1 Averaging
2.6.2 Closure relationships
3 Two-Fluid Model
3.1 Problem definition
3.2 Mass conservation
3.3 Momentum conservation
3.4 Gas and liquid pressure difference in stratified flow
3.5 Friction in stratified flow
3.6 Steady-state incompressible flow solution
3.6.1 The model
3.6.2 Solution method
3.7 Steady-state compressible flow solution
3.8 Fully transient simulation model
3.9 The drift-flux model
3.10 Ignoring inertia in the momentum equations
3.11 Incompressible transient model
4 Three-fluid model
4.1 General
4.2 Mass conservation
4.3 Momentum conservation
4.4 Energy equation
4.5 Fluid properties
5 Friction, deposition and entrainment
5.1 Friction between gas core and liquid film
5.1.1 General about friction
5.1.2 The friction model
5.1.3 The Darcy-Weisbach friction
factor for the liquid film-gas interface
5.1.4 Friction between the liquid
film and the wall
5.2 Droplet gas friction and dynamic response time
5.3 Droplet liquid friction forces
5.3.1 Introduction
5.3.2 Zaichik and Alipchenkov’s
eddy-droplet interaction time model
5.3.3 Droplet-liquid film friction
modeled as if the droplets were a continuum
5.4 Droplet deposition
5.5 Liquid film entrainment
5.6 Droplet size
5.6.1 Maximum stable droplet diameter
due to average velocity difference
5.6.2 Maximum stable droplet diameter
due to turbulence
5.6.3 Average droplet diameter
6 Solving the two-phase three-fluid
equations
6.1 Steady-state incompressible isothermal flow
6.2 Comparing with measurements
6.3 Steady-state compressible flow
6.4 Transient three-fluid two-phase annular flow model
7 Gas-liquid slug flow
7.1 Slug mechanisms
7.2 Empirical slug period correlations
7.2.1 Slug frequency and slug length
7.2.2 Slug fractions
7.2.3 Taylor-bubble and slug bubble
velocities
7.3 Slug train friction
7.4 Dynamic slug simulation
8 Including boiling and condensation
8.1 Extending the three-fluid two-phase model
8.2 Mass conservation
8.3 Momentum conservation
8.3.1 Main equations
8.3.2 Some comments on interface
velocity
8.4 Energy equation
8.5 Pressure equation
8.6 Mass transfer from liquid (film and droplets) to gas
8.7 Slip between gas and droplets in annular flow
8.8 Droplet deposition in annular flow
8.8.1 The Wallis-correlation
8.8.2 The Oliemans, Pots, and
Trope-correlation
8.8.3 The Ishii and Mishima-correlation
8.8.4 The Sawant, Ishii, and
Mori-correlation
8.9 Dispersed bubble flow
8.10 Slug flow
9 Improved slug flow modeling
9.1 Introduction
9.2 Governing equations
9.3 Friction model
9.4 Slug bubble entrainment and release
9.4.1 Slug bubble velocity
9.4.2 Bubbles entering and leaving
the liquid slug
9.4.3 Film and slug front/tail
velocities
9.5 Model validity and results
10 Multi-phase flow heat exchange
10.1 Introduction
10.2 Classical, simplified mixture correlations
10.3 Improved correlations for all
flow regimes in horizontal two-phase flow
10.4 Flow regime-dependent approximation for horizontal
flow
10.5 Flow-regime dependent two-phase correlations for
inclined pipes
10.6 Dispersed bubble flow
10.7 Stratified flow
10.8 Slug flow
11 Flow regime determination
11.1 The Beggs & Brill flow regime map
11.2 The Taitel & Duckler horizontal flow model
11.3 Flow regimes in vertical flow
11.3.1 Bubble to slug transition
11.3.2 Transition to dispersed-bubble
flow
11.3.3 Slug to churn transition
11.3.4 Transition to annular flow
11.4 Flow regimes in inclined pipes
11.4.1 Bubble to slug transition
11.4.2 Transition to dispersed-bubble
flow
11.4.3 Intermittent to annular
transition
11.4.4 Slug to churn transition
11.4.5 Downward inclination
11.5 The minimum-slip flow regime criterion
12 Numerical solution methods
12.1 Some essentials about numerical methods
12.1.1 Some problems with higher
order methods
12.1.2 Using Taylor-expansion to
approximate
12.1.3 Truncation error, order,
stability, consistency, and convergence
12.1.4 Implicit integration methods
12.1.5 Combining explicit and
implicit methods
12.2 Some essentials about hyperbolic equations
12.3 Solving systems of hyperbolic equations
12.3.1 Flux-vector splitting
12.3.2 Lax-Friedrich’s method
12.4 Hyperbolic equations with source terms
12.5 Selecting discretization methods
12.6 Improved TR-BDF2 method
12.7 Semi-implicit methods
12.8 Newton-Rapson and Newton-Krylov iteration
12.8.1 The problem with Newton-Rapson
iteration for large systems
12.8.2 Creating the Jacobian with
fewer function calls
12.8.3 Some problems with
Newton-iteration
12.8.4 Avoiding the Jacobian using
Newton-Krylov iteration
13 Two-phase liquid-liquid flow
13.1 General
13.2 Emulsion viscosity
13.3 Phase inversion criteria
13.4 Stratified flow friction modeling
14 Two-phase liquid-solid flow
14.1 General about liquid-solid flow
14.2 The building up of solids in the
pipeline
14.3 Minimum transport velocity
15 Three-phase gas-liquid-liquid flow
15.1 Introduction
15.2 Main equations
15.3 Three-layer stratified flow
15.4 Incompressible steady-state slug flow model
15.5 Combining the different flow regimes into a unified
model
16 Three-phase gas-liquid-solid flow
16.1 Introduction
16.2 Models and correlations
17 Fluid properties
17.1 General
17.2 Equations of state
17.3 Other properties for equation closure
17.3.1 Enthalpy
17.3.2 Internal energy
17.3.3 Entropy
17.3.4 Heat capacity
17.3.5 Joule-Thompson coefficient
17.3.6 Speed of sound
17.3.7 Viscosity and thermal
conductivity
17.3.8 Interfacial surface tension
18 Deposits and pipe damage
18.1 Introduction
18.2 Hydrates
18.2.1 General
18.2.2 Hydrate blockage prevention
18.2.3 Hydrate formation rate
prediction
18.3 Waxes
18.4 Asphaltenes
18.5 Scales
18.6 Corrosion, erosion, and cavitation
18.6.1 General
18.6.2 Corrosion simulation models
18.7 Heavy oil and emulsions
19 Various subjects
19.1 Multi-phase flowmeters and flow estimators
19.2 Gas lift
19.2.1 General
19.2.2 Oil & water-producing well
with gas lift: Simulation example
19.3 Slug catchers
Suggested reading
Nomenclature
