**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 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

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

5.5 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

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

8.5 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

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

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

12.5 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

18.4 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**