Pressure Waves in Pipelines and Impulse Pumping - NTNU

Pressure Waves in Pipelines and
Impulse Pumping
Physical Principles,
Model Development and
Numerical Simulation
Benjamin Pierre
December 2009

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63 o N, 10 o E

Abstract
The primary purpose of the work was to describe the physical principles of
impulse pumping and investigate its potential applications in petroleum engineering.
Investigations of the potential applications of impulse pumping were
based on analysis of head, flow rate and efficiency performances for pumping
of hydrocarbon fluids.
The secondary purpose of the work was to analyse pressure wave propagation
and attenuation models in fluid-filled production tubings. Additionally,
one dimensional numerical simulation methods were investigated for accurate
modelling of pressure transient events in pipelines, such as water hammer
phenomena.
Pressure wave propagation, transmission and reflection characteristics were
first studied. Finite volume methods were also studied for numerical simulation
of one-dimensional pressure transient phenomena. Accuracy of pressure
wave attenuation models was then analysed using a contradiction method,
relying on analysis of numerical simulation tools for pressure transient events.
Impulse pumping physical principles were next described using fundamentals
of pressure wave propagation. A numerical simulation tool was then developed
to reproduce impulse pumping physical principles, using a finite volume
numerical scheme.
Impulse pumping performances were modelled afterwards in terms of lifting
heights, flow rates and efficiency, based on results from the developed numerical
simulation tool. Potential applications of impulse pumping in petroleum
engineering were then analysed using the performance model.
Impulse pumping generates flow from bottomhole to wellhead using pressure
waves generated at wellhead. Fluid can be transported from bottomhole to
wellhead without theoretical lifting height limitations using impulse pumping.
Impulse pumping performances were illustrated and a range of petroleum
engineering applications was investigated.
Impulse pumping greatest efficiency is obtained for artificial lift of water from
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shallow wells. Impulse pumping performances depend on pressure wave amplitude,
wellhead pressure, lifting height, fluid compressibility and volume
occupied in the production tubing.
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Acknowledgments
I would like to express my gratitude to my supervisor, Professor Jon Steinar
Gudmundsson, for the insightful discussions, encouragement, support and
guidance throughout the work. His thirst for knowledge and his joyful willingness
to challenge have initiated countless ”thinking out loud” sessions that
have enlightened the path leading to the PhD degree.
I acknowledge Clavis Impulse Technology AS for their financial support and
for giving us access to some of their experimental data and for permission to
use them in the present thesis. For this I would like to thank Peter Grubyi,
Magomet Sagov and Jim Viktor Paulsen.
I also acknowledge the financial support from the Norwegian Research Council,
through the Petromaks program, project 196054/S60, without which this
project could not have been completed.
The support and consideration of the Technology Transfer Office at NTNU is
deeply appreciated. For this I would like to thank Erik Wold, Karl Klingsheim
and Kristin Jørstad.
I would like to thank Dr. Donna Calhoun and her colleagues at Commisariat à
l’ Energie Atomique (CEA) Saclay, for helping me with source term modelling
in CLAWPACK.
I would like to thank Professor Helena Ramos and her colleagues at the Technical
University of Lisbon, for the pleasant stay and valuable help on modelling
of unsteady friction.
The technical staff at the Department of Petroleum Engineering and Applied
Geophysics have been valuable in maintenance, repair and design of small scale
experiments that have helped in better understanding the physics of pressure
waves in pipelines. For this I would like to thank H˚akon Myhren, Roger
Over˚a, ˚Age Sivertsen and Terje Bjerkan. I would also like to thank Amir
Ghaderi, fellow PhD candidate at NTNU IPT for helping me with ultrasonic
measurements.
The cheerful help and support from the administrative staff at the Department
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of Petroleum Engineering and Applied Geophysics are deeply appreciated.
For this I would like to thank Marit Valle Raaness, Tone Sanne, Madelein
Wold, Anne Lise Brekken, Turid Halvorsen, Solveig Johnsen and Turid Oline
Uvsløkk.
Thanks to my friends at the Department of Petroleum Engineering and Applied
Geophysics for giving me an enjoyable time. I would like to thank Christian
and Ivana for always entertaining me. Pamela and Weider deserve a
special thank for always being here for me.
My family, I thank for care and support.
Mércia, no matter how far we have come, I cannot wait to see tomorrow, with
you.
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List of Contents
Abstract v
Acknowledgments vii
List of Contents ix
List of Tables xiii
List of Figures xv
Nomenclature xxv
1 Introduction 1
2 Pressure Waves in Pipelines 5
2.1 Modelling Equations and Numerical Simulation . . . . . . . . . 5
2.2 Source Term Modelling . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Local Flow History . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Quasi Two-Dimensional Models . . . . . . . . . . . . . . 9
2.2.3 Instantaneous Velocity and Acceleration . . . . . . . . . 10
2.2.4 Local and Convective Acceleration . . . . . . . . . . . . 11
2.2.5 Evaluation of Unsteady Friction Models . . . . . . . . . 12
2.2.6 Pipeline Viscoelasticity . . . . . . . . . . . . . . . . . . 13
2.3 Leak Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 Flow Measurements . . . . . . . . . . . . . . . . . . . . 14
2.3.2 Pressure Wave Characteristics . . . . . . . . . . . . . . 15
2.3.3 Pressure Signal Processing . . . . . . . . . . . . . . . . . 15
2.4 Flow Metering . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Modelling of Pressure Wave Propagation 19
3.1 Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.1 Modelling Equations . . . . . . . . . . . . . . . . . . . . 19
3.1.2 Transmission, Reflection . . . . . . . . . . . . . . . . . . 21
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3.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.1 Static Boundary Conditions . . . . . . . . . . . . . . . . 26
3.2.2 Dynamic Boundary Conditions . . . . . . . . . . . . . . 26
3.3 Pressure Wave Superposition . . . . . . . . . . . . . . . . . . . 33
3.3.1 Standing Waves . . . . . . . . . . . . . . . . . . . . . . . 39
3.4 Two-Dimensional Modelling . . . . . . . . . . . . . . . . . . . . 41
4 Acoustic Velocity in Single Phase Fluids 43
4.1 Basic Relationships . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.2 Fluid Compressibility . . . . . . . . . . . . . . . . . . . 45
4.2 Acoustic Velocities in Gases . . . . . . . . . . . . . . . . . . . . 46
4.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Acoustic Velocity in Liquid Oils . . . . . . . . . . . . . . . . . . 54
4.3.1 Acoustic Velocity in Volatile Oil . . . . . . . . . . . . . 54
4.3.2 Acoustic Velocity in Black Oil . . . . . . . . . . . . . . . 58
4.3.3 Comparison Between Volatile Oils and Black Oils . . . . 62
4.4 Acoustic Velocity in Liquid Water . . . . . . . . . . . . . . . . 62
4.4.1 Acoustic Velocity . . . . . . . . . . . . . . . . . . . . . . 62
4.4.2 Density of Water . . . . . . . . . . . . . . . . . . . . . . 64
4.4.3 Isothermal Compressibility . . . . . . . . . . . . . . . . 65
4.5 Acoustic Velocity in Fluid-Filled Pipe . . . . . . . . . . . . . . 66
4.6 Acoustic Velocity Determination . . . . . . . . . . . . . . . . . 71
5 Numerical Simulation Methods 73
5.1 Modelling Equations . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 Method of Characteristics . . . . . . . . . . . . . . . . . . . . . 76
5.4 Finite Volume Methods . . . . . . . . . . . . . . . . . . . . . . 78
5.4.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.4.2 Numerical Schemes . . . . . . . . . . . . . . . . . . . . . 79
5.4.3 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.4.4 Higher Resolution Method . . . . . . . . . . . . . . . . . 84
5.4.5 CLAWPACK . . . . . . . . . . . . . . . . . . . . . . . . 85
5.5 Numerical Scheme Selection . . . . . . . . . . . . . . . . . . . . 88
6 Pressure Wave Attenuation 91
6.1 Influence of Numerical Simulation Tool . . . . . . . . . . . . . . 91
6.1.1 Unsteady Friction Modelling Equations . . . . . . . . . 91
6.1.2 Test Case and Numerical Simulation . . . . . . . . . . . 92
6.1.3 Results of Friction Modelling . . . . . . . . . . . . . . . 93
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6.1.4 Finite Volume Method Accuracy . . . . . . . . . . . . . 96
6.1.5 Pipeline Viscoelasticity . . . . . . . . . . . . . . . . . . 97
6.2 Acoustic Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2.1 Ultrasonic Measurement . . . . . . . . . . . . . . . . . . 99
6.2.2 Experimental Work . . . . . . . . . . . . . . . . . . . . 107
6.3 Extended Acoustic Velocity Formulation . . . . . . . . . . . . . 111
7 Impulse Pumping Apparatus and Experiments 115
7.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2 Inclined Experimental Set-Up and Data . . . . . . . . . . . . . 118
7.3 Horizontal Experimental Set-Up and Data . . . . . . . . . . . . 121
8 Concept, Modelling and Simulation of Impulse Pumping 129
8.1 Concept and Modelling . . . . . . . . . . . . . . . . . . . . . . 129
8.1.1 Impulse Generator . . . . . . . . . . . . . . . . . . . . . 129
8.1.2 Pressure Wave Propagation . . . . . . . . . . . . . . . . 132
8.1.3 Pipeline Outlet . . . . . . . . . . . . . . . . . . . . . . . 133
8.1.4 Pipeline Inlet . . . . . . . . . . . . . . . . . . . . . . . . 134
8.2 Numerical Simulation Tool . . . . . . . . . . . . . . . . . . . . 135
8.3 Modelling Results . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.4 Validation of Impulse Pumping Model and Simulation Tool . . 143
9 Design, Applications and Performance of Impulse Pumping 147
9.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.1.1 Impulse Generator . . . . . . . . . . . . . . . . . . . . . 147
9.1.2 Non-Return Valves . . . . . . . . . . . . . . . . . . . . . 150
9.2 Multiphase Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . 152
9.3 Horizontal Transport of Fluids . . . . . . . . . . . . . . . . . . 153
9.3.1 In-Situ Pressure and Pressure Wave Amplitude . . . . . 153
9.3.2 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 154
9.3.3 Test Case . . . . . . . . . . . . . . . . . . . . . . . . . . 156
9.4 Artificial Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
9.4.1 Wellhead Pressure and Pressure Wave Amplitude . . . . 162
9.4.2 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 164
9.4.3 Test Case . . . . . . . . . . . . . . . . . . . . . . . . . . 165
10 Characteristic Curves of Impulse Pumping 171
10.1 Characteristic Curves . . . . . . . . . . . . . . . . . . . . . . . 171
10.2 Comparison of Artificial Lift Techniques . . . . . . . . . . . . . 174
10.2.1 Test Case Description . . . . . . . . . . . . . . . . . . . 174
10.2.2 Determination of Impulse Pumping Parameters . . . . . 175
10.2.3 Efficiency of Impulse Pumping . . . . . . . . . . . . . . 176
10.3 Possible Improvements of Impulse Pumping Efficiency . . . . . 183
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10.4 Cost of Impulse Pumping . . . . . . . . . . . . . . . . . . . . . 187
11 Discussion 189
12 Conclusions 191
References 193
A Technical Reports 201
B Pumping of Fluids Using Pressure Impulses 205
C Modelling of Pressure Wave Propagation in Pipelines: Steady-
State and Transient Friction 217
D Simulation of Rapid Pressure Transients in
Statfjord A Off-Loading Pipeline 235
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List of Tables
2.1 Theoretical Values of ̺T and ςT in Eq. 2.8 (Trikha, 1975) . . . 8
2.2 Theoretical Values of ̺V and ςV in Eq. 2.10 for Particular f ×Re
Conditions (Vardy and Hwang, 1991) . . . . . . . . . . . . . . . 9
2.3 Theoretical Values of ςC and ̺C in Eq. 2.14 (Carstens and
Roller, 1959) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1 Actuation Function Ξ for Circular Gate, Square Gate, Globe,
Needle, Butterfly and Ball Valves for Dimensionless Actuation
Time ξ (Eq. 3.20 to Eq. 3.23) (Wood and Jones, 1973, 1974) . 29
4.1 Typical Natural Gas Molar Composition (Pedersen et al., 1989) 49
4.2 Typical Volatile Oil Molar Composition, Pedersen et al. (1989) 55
4.3 Typical Black Oil Molar Composition (Pedersen et al., 1989) . 59
4.4 Structure Coefficient for Thin-Walled and Thick-Walled Pipelines,
Single-Sided Anchor or Double-Sided Anchor. Structure Fastenings
Description in Figure 4.20 . . . . . . . . . . . . . . . . 68
4.5 Typical Material Properties: Young’s Modulus and Poisson’s
Ratio (Cheremisinoff, 1996) . . . . . . . . . . . . . . . . . . . . 70
4.6 Typical Fluid Densities and Compressibilities in Petroleum Engineering
(McCain, 1990) . . . . . . . . . . . . . . . . . . . . . 70
5.1 Test Cases Description for Comparing CTCS, Lax-Wendroff
and ENO Methods . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 Minmod, Superbee, Monotonized-Centered and Van-Leer Flux
Limiter Functions (Laney, 1998) . . . . . . . . . . . . . . . . . 84
6.1 Test Case System Description and Numerical Simulation Parameters
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.2 Field A Molar Gas Composition (Jacobsen, 2008) . . . . . . . . 100
6.3 Propeller Engine Voltage and Corresponding Propeller Rotating
Speed and Linear Speed . . . . . . . . . . . . . . . . . . . . 108
7.1 Conversion between Vertical Height and Pipeline Length . . . . 118
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8.1 List of Subroutines Included in the Impulse Pumping Software 137
8.2 List of Parameters Describing Impulse Pumping . . . . . . . . . 138
8.3 Computed Flow Rate in [l min −1 ], for Different Inlet Non-Return
Valve Cracking Pressures and Pressure Wave Peak-To-Peak Amplitudes
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
9.1 Description of Test Case for Numerical Investigation of Pressure
Waveform Impact on Reachable Flow Rate . . . . . . . . . . . 150
9.2 Density, Acoustic Velocity, Bubble Point Pressure, Inlet Non-
Return Valve Opening Pressures and Pipeline In-Situ Static
Pressure for Impulse Pumping of Water, Volatile Oil and Black
Oil at 100 o C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
9.3 Minimum Production Tubing Lengths L1 in Eq. 9.14 and L2 in
Eq. 9.15 for Impulse Pumping of Water, Volatile Oil and Black
Oil at 100 o C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
9.4 Impulse Pumping Parameters for Artificial Lift of Water, Volatile
Oil and Black Oil at 100 o C, from 10000 m Deep Wellbore, 10
inches in Diameter, Neglecting Pressure Wave Attenuation . . 166
10.1 Desired Flow Rates, Wellbore Pressures and Corresponding Lifting
Heights for the Three Depressurization Phases of Statfjord
Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
10.2 Description of Test Case for Numerical Investigation of Pressure
Waveform Impact on Reachable Flow Rate . . . . . . . . . . . 184
D.1 Statfjord Oil Molar Composition at 5 ◦ C, 30 bara . . . . . . . . 238
D.2 Geometry of Case 1 . . . . . . . . . . . . . . . . . . . . . . . . 251
D.3 Geometry of Case 2 . . . . . . . . . . . . . . . . . . . . . . . . 251
D.4 Geometry of Case 3 . . . . . . . . . . . . . . . . . . . . . . . . 252
xiv

List of Figures
2.1 Typical Water Hammer Pressure Trace, Reproduced from Holmboe
and Rouleau (1967) . . . . . . . . . . . . . . . . . . . . . . 7
3.1 Acoustic Interface Schematic Representation Between Media a
and b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Influence of Acoustic Interface on Pressure Wave Propagation.
Medium a between 0 and L/2. Medium b between L/2 and L.
Za > Zb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Influence of Acoustic Interface on Pressure Wave Propagation.
Medium a between 0 and L/2. Medium b between L/2 and L.
Za < Zb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Evolution of Transmission Coefficient Against Ratio of Acoustic
Impedances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Evolution of Reflection Coefficient Against Ratio of Acoustic
Impedances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.6 Pressure and Fluid Velocity Waves Reflections From Open and
Close Boundary Conditions . . . . . . . . . . . . . . . . . . . . 27
3.7 Schematic Representation of Check Valves. a) Circular Gate
Valve. b) Square Gate Valve. c) Globe Valve. d) Needle Valve.
e) Butterfly Valve. f) Ball Valve. . . . . . . . . . . . . . . . . . 28
3.8 Valve Position During Uniform Closure . . . . . . . . . . . . . 31
3.9 Valve Position During Accelerated Closure . . . . . . . . . . . . 31
3.10 Flow Velocity Across Valve During Uniform Closure . . . . . . 32
3.11 Flow Velocity Across Valve During Accelerated Closure . . . . 32
3.12 Constructive Superposition of two Pressure Waves . . . . . . . 34
3.13 Destructive Superposition of two Pressure Waves . . . . . . . . 35
3.14 Constructive Superposition of a Pressure Wave on its own Reflection
at a Closed Boundary . . . . . . . . . . . . . . . . . . . 37
3.15 Destructive Superposition of a Pressure Wave on its own Reflection
at an Open Boundary . . . . . . . . . . . . . . . . . . . 38
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3.16 First and Third Harmonics for Pressure Standing Wave in a
Pipeline Closed at Both Ends . . . . . . . . . . . . . . . . . . . 40
3.17 First and Second Harmonics for Pressure Standing Wave in
Pipeline Open at Both Ends . . . . . . . . . . . . . . . . . . . . 40
3.18 Two-Dimensional Laminar Velocity Profile Evolution in a Fluid-
Filled Pipeline at Zero Initial Velocity During One Angular
Pressure Wave period . . . . . . . . . . . . . . . . . . . . . . . 42
3.19 Two-Dimensional Turbulent Velocity Profile Evolution in a Fluid-
Filled Pipeline at Zero Initial Velocity During One Angular
Pressure Wave period . . . . . . . . . . . . . . . . . . . . . . . 42
4.1 Typical Natural Gas Phase Envelope . . . . . . . . . . . . . . . 49
4.2 Evolution of Typical Natural Gas Density with Pressure and
Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Evolution of Typical Natural Gas Adiabatic Index with Pressure
and Temperature . . . . . . . . . . . . . . . . . . . . . . . 50
4.4 Evolution of Typical Natural Gas Z-Factor with Pressure and
Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.5 Evolution of Typical Natural Gas Isothermal Compressibility
with Pressure and Temperature . . . . . . . . . . . . . . . . . . 52
4.6 Evolution of Typical Natural Gas Acoustic Velocity with Pressure
and Temperature . . . . . . . . . . . . . . . . . . . . . . . 53
4.7 Deviation Between Eq. 4.19 and Eq. 4.20 . . . . . . . . . . . . 53
4.8 Typical Volatile Oil Phase Envelope . . . . . . . . . . . . . . . 55
4.9 Evolution of Typical Volatile Oil Density with Pressure and
Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.10 Evolution of Typical Volatile Oil Isothermal Compressibility
with Pressure and Temperature . . . . . . . . . . . . . . . . . . 56
4.11 Evolution of Acoustic Velocity in Typical Volatile Oil with Pressure
and Temperature . . . . . . . . . . . . . . . . . . . . . . . 57
4.12 Typical Black Oil Phase Envelope . . . . . . . . . . . . . . . . 59
4.13 Evolution of Typical Black Oil Density with Pressure and Temperature
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.14 Evolution of Typical Black Oil Isothermal Compressibility with
Pressure and Temperature . . . . . . . . . . . . . . . . . . . . . 60
4.15 Evolution of Acoustic Velocity in Typical Black Oil with Pressure
and Temperature . . . . . . . . . . . . . . . . . . . . . . . 61
4.16 Evolution of Acoustic Velocity in Water with Pressure and Temperature
at Zero Salinity (DelGrosso, 1974) . . . . . . . . . . . 64
4.17 Evolution of Water Density with Pressure and Temperature at
Zero Salinity (Michaelides, 1981) . . . . . . . . . . . . . . . . . 65
xvi

4.18 Evolution of Water Isothermal Compressibility with Pressure
and Temperature at Zero Salinity . . . . . . . . . . . . . . . . . 66
4.19 Water Isothermal Compressibility extrapolated from Acoustic
Velocity and Density Correlations (DelGrosso, 1974; Michaelides,
1981) and empirically correlated (Eisenberg and Kauzman, 1969) 67
4.20 Illustration of Structure Fastenings (Wylie and Streeter, 1993) 68
4.21 Acoustic Velocity in Fluid-Filled Pipelines (Wylie and Streeter,
1993) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.22 Acoustic Velocity in Fluid-Filled Non-Circular Pipelines . . . . 72
5.1 Typical Pipeline System in Pressure Transient Analysis Schematic
Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Characteristic Lines Representation in Standard X-T Plane Grid 76
5.3 Illustration of a Finite Volume Method for Updating the Cell
Average Ωn i by Fluxes at the Cell Edges . . . . . . . . . . . . . 79
5.4 Comparison of CTCS, Lax-Wendroff and ENO Numerical Schemes
on Test Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.5 Comparison of CTCS, Lax-Wendroff and ENO Numerical Schemes
on Test Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.6 Comparison of CTCS, Lax-Wendroff and ENO Numerical Schemes
on Test Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.7 Comparison of CTCS, Lax-Wendroff and ENO Numerical Schemes
on Test Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.8 Comparison of Minmod, Superbee, Monotonized Centered and
Van-Leer Flux Limiters Coupled to the Lax-Wendroff Numerical
Scheme on Test Case 1 . . . . . . . . . . . . . . . . . . . . . 86
5.9 Comparison of Minmod, Superbee, Monotonized Centered and
Van-Leer Flux Limiters Coupled to the Lax-Wendroff Numerical
Scheme on Test Case 2 . . . . . . . . . . . . . . . . . . . . .
5.10 Comparison of Minmod, Superbee, Monotonized Centered and
Van-Leer Flux Limiters Coupled to the Lax-Wendroff Numeri-
86
cal Scheme on Test Case 3 . . . . . . . . . . . . . . . . . . . . .
5.11 Comparison of Minmod, Superbee, Monotonized Centered and
Van-Leer Flux Limiters Coupled to the Lax-Wendroff Numeri-
87
cal Scheme on Test Case 4 . . . . . . . . . . . . . . . . . . . . .
5.12 Comparison of Method of Characteristics and Finite Volume
Method (Lax-Wendroff + Superbee Flux Limiter) on Standard
87
Pressure Transient Test Case . . . . . . . . . . . . . . . . . . . 89
6.1 Comparison of Method of Characteristics and Finite Volume
Method with Steady Friction Only. nX = 15. nT = 540. . . . . 94
xvii

6.2 Comparison of Method of Characteristics and Finite Volume
Method with Brunone’s Unsteady Friction Model. nX = 15.
nT = 540. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.3 Comparison of Steady Friction and Brunone’s Unsteady Friction
Models with the Method of Characteristics. nX = 15.
nT = 540. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.4 Comparison of Steady Friction and Brunone’s Unsteady Friction
Models with the Selected Finite Volume Method. nX = 15.
nT = 540. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.5 Influence of Spatial Discretization on Water Hammer Calculation
with the Selected Finite Volume Method. Detail of Pressure
Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.6 Comparison of Investigated Unsteady Friction Models Using a
Selected Finite Volume Method. nX = 1000. nT = 540. . . . . 98
6.7 Comparison of Water Hammer Simulation Using the Method of
Characteristics and a Finite Volume Method. MOC: nX = 15.
nT = 540. FVM: nX = 1000. nT = 540. . . . . . . . . . . . . . 98
6.8 Schematic Representation of an Ultrasonic Flow Meter . . . . . 100
6.9 Field A Gas Phase Envelope (Jacobsen, 2008) . . . . . . . . . . 100
6.10 Measured and Calculated Acoustic Velocities Function of Pressure
and Temperature. First Data Set . . . . . . . . . . . . . . 102
6.11 Measured and Calculated Acoustic Velocities Function of Pressure.
First Data Set . . . . . . . . . . . . . . . . . . . . . . . . 102
6.12 Measured and Calculated Acoustic Velocities Function of Temperature.
First Data Set . . . . . . . . . . . . . . . . . . . . . . 103
6.13 Measured and Calculated Acoustic Velocities Function of Flow
Velocity. First Data Set . . . . . . . . . . . . . . . . . . . . . . 103
6.14 Measured and Calculated Acoustic Velocities Function of Pressure
and Temperature. Second Data Set . . . . . . . . . . . . . 105
6.15 Measured and Calculated Acoustic Velocities Function of Pressure.
Second Data Set . . . . . . . . . . . . . . . . . . . . . . . 105
6.16 Measured and Calculated Acoustic Velocities Function of Temperature.
Second Data Set . . . . . . . . . . . . . . . . . . . . . 106
6.17 Measured and Calculated Acoustic Velocities Function of Flow
Velocity. Second Data Set . . . . . . . . . . . . . . . . . . . . . 106
6.18 Schematic Representation of Experimental Apparatus for Measuring
Acoustic Velocity in Fluid Under Rotation . . . . . . . . 107
6.19 Sound Beam Travel Times for Different Propeller Engine Voltage.
Transducer 1 . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.20 Sound Beam Travel Times for Different Propeller Engine Voltage.
Transducer 2 . . . . . . . . . . . . . . . . . . . . . . . . . 110
xviii

6.21 Sound Beam Travel Times for Different Propeller Engine Voltage.
Transducer 3 . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.22 Sound Beam Travel Times for Different Propeller Engine Voltage.
Transducer 4 . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.23 Detail of Measured and Calculated Acoustic Velocities Function
of Flow Velocity - First Data Set Figure 6.13 - and Curve Fitting112
6.24 Pressure, Fluid and Acoustic Velocity Gradients in Left-To-
Right Propagating Pressure Waves . . . . . . . . . . . . . . . . 113
7.1 Impulse Pumping Apparatus, Horizontal Configuration . . . . . 116
7.2 Impulse Pumping Apparatus, Inclined Configuration . . . . . . 116
7.3 Impulse Pumping Apparatus, Vertical Configuration . . . . . . 117
7.4 Sample of Pressure Traces at Four Locations Along the Inclined
Flow Line. ˙γ = 2.0 Hz . . . . . . . . . . . . . . . . . . . . . . . 119
7.5 Sample of Pressure Traces at Four Locations Along the Inclined
Flow Line. ˙γ = 3.0 Hz . . . . . . . . . . . . . . . . . . . . . . . 120
7.6 Maximum Pressures at Four Locations for Five Different Experiments
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.7 Sample of Piston Stroke Trace. ˙γ = 1.4 Hz, ∆l = 0.75 10 −3 m . 122
7.8 Sample of Force on the Piston Trace. ˙γ = 1.4 Hz, ∆l =
0.7510 −3 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.9 Fourier Analysis of Piston Stroke Signal. ˙γ = 1.4 Hz, ∆l =
0.7510 −3 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.10 Sample of Pressure Trace at the Outlet Non-Return Valve. ˙γ =
1.4 Hz, ∆l = 0.7510 −3 m . . . . . . . . . . . . . . . . . . . . . . 124
7.11 Sample of Pressure Trace at the Inlet Non-Return Valve. ˙γ =
1.4 Hz, ∆l = 0.7510 −3 m . . . . . . . . . . . . . . . . . . . . . . 124
7.12 Sample of Impulse Pumping Flow Rate Trace. ˙γ = 1.4 Hz,
∆l = 0.7510 −3 m . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.13 Sample of Pressure Trace at Outlet Non-Return Valve. ˙γ =
1.4 Hz, ∆l = 1.2510 −3 m . . . . . . . . . . . . . . . . . . . . . . 126
7.14 Sample of Pressure Trace at Inlet Non-Return Valve. ˙γ =
1.4 Hz, ∆l = 1.2510 −3 m . . . . . . . . . . . . . . . . . . . . . . 126
7.15 Maximum Pressure Amplitudes at Outlet Non-Return Valve
Against Impulse Generator Displacement for Different Frequencies
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.16 Maximum Pressure Amplitudes at Inlet Non-Return Valve Function
of Impulse Generator Displacement for Different Frequencies128
7.17 Experimental Flow Rates Function of Impulse Generator Displacement
for Different Frequencies . . . . . . . . . . . . . . . . 128
8.1 Impulse Pumping Apparatus, Horizontal Configuration . . . . . 130
xix

8.2 Illustration of Impulse Generator Generating Pressure Waves
in Diameter Greater than Flowline Dimensions . . . . . . . . . 131
8.3 Illustration of Piston Back-And-Forth Oscillations for Sinusoidal
Pressure Wave Generation . . . . . . . . . . . . . . . . . . . . . 132
8.4 Non-Return Valve Schematic Representations . . . . . . . . . . 134
8.5 Impulse Pumping Apparatus Applied to Artificial Lift . . . . . 136
8.6 Spatial Pressure Traces at Different Instants During Impulse
Pumping Process (In-Situ Static Pressure is Represented by 0
Pressure Line). Data Presented in Dimensionless Form. . . . . 141
8.7 Influence of Water Hammer Pressure Wave on Synthetic Pressure
Spatial Trace . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.8 Influence of Wellbore Pressure on Synthetic Pressure Spatial
Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.9 Synthetic Temporal Pressure Trace at Downhole Inlet Non-
Return Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.10 Synthetic Temporal Pressure Trace at Downhole Inlet Non-
Return Valve. ˙γ = 1.5 Hz. L = 283m . . . . . . . . . . . . . . 144
8.11 Synthetic Flow Rate for Diverse Inlet Non-Return Valve Cracking
Pressures and Pressure Wave Peak-To-Peak Amplitudes.
˙γ = 1.5 Hz. L = 283m . . . . . . . . . . . . . . . . . . . . . . . 146
9.1 Evolution of Piston Stroke for Generating Pressure Wave 100
bar in Amplitude in a 1000 m Long Pipeline . . . . . . . . . . . 148
9.2 Piston Linear Velocity for Generating Pressure Waves 100 bar
in Amplitude in a 1000 m Long Water-Filled Pipeline at Frequencies
From 1 to 8 Hz . . . . . . . . . . . . . . . . . . . . . . 149
9.3 Piston Linear Velocity for Generating Pressure Waves 100 bar in
Amplitude in a 1000 m Long Oil-Filled Pipeline at Frequencies
From 1 to 8 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.4 Comparison of Check-Valves Efficiencies for Uniform and Accelerated
Actuations . . . . . . . . . . . . . . . . . . . . . . . . 151
9.5 Water Air Mixture Acoustic Velocity Against Void Fraction at
1 bara, 10 bara and 100 bara . . . . . . . . . . . . . . . . . . . 153
9.6 Reachable Flow Rate for Horizontal Impulse Pumping of Water,
Volatile and Black Oils in 1000 m Long Pipeline, Using
Pressure Waves 100 bar in Peak-To-Peak Amplitude, 4 Hz in
Frequency, Against Difference Between Pipeline and Inlet Non-
Return Valve Opening Pressures, Neglecting Pressure Wave Attenuation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
xx

9.7 Reachable Flow Rate for Horizontal Impulse Pumping of Water,
Volatile and Black Oils in 1000 m Long Pipeline, Using
Pressure Waves 100 bar in Peak-To-Peak Amplitude, 5 Hz in
Frequency, Against Difference Between Pipeline and Inlet Non-
Return Valve Opening Pressures, Neglecting Pressure Wave Attenuation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
9.8 Inflow Efficiency for Horizontal Impulse Pumping of Water,
Volatile and Black Oils in 1000 m Long Pipeline, Using Pressure
Waves 100 bar in Peak-To-Peak Amplitude, 4 Hz in Frequency,
Against Difference Between Pipeline and Inlet Non-
Return Valve Opening Pressures, Neglecting Pressure Wave Attenuation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
9.9 Inflow Efficiency for Horizontal Impulse Pumping of Water,
Volatile and Black Oils in 1000 m Long Pipeline, Using Pressure
Waves 100 bar in Peak-To-Peak Amplitude, 5 Hz in Frequency,
Against Difference Between Pipeline and Inlet Non-
Return Valve Opening Pressures, Neglecting Pressure Wave Attenuation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
9.10 Reachable Flow Rates Against Wellbore Pressure, for Vertical
Transport of Water, Single Phase Volatile and Black Oils, at
100 o C, from a 10 km Deep Wellbore, Using Pressure Waves 4
Hz in Frequency, Neglecting Pressure Wave Attenuation . . . . 167
9.11 Reachable Flow Rates Against Wellbore Pressure, for Vertical
Transport of Water, Single Phase Volatile and Black Oils, at
100 o C, from a 10 km Deep Wellbore, Using Pressure Waves 5
Hz in Frequency, Neglecting Pressure Wave Attenuation . . . . 168
9.12 Theoretical Impulse Pumping Inflow Efficiency Against Wellbore
Pressure, for Vertical Transport of Water, Single Phase
Volatile and Black Oils at 100 o C, from a 10 km Deep Wellbore,
Using Pressure Waves 4 Hz in Frequency, Neglecting Pressure
Wave Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . 169
9.13 Theoretical Impulse Pumping Inflow Efficiency Against Wellbore
Pressure, for Vertical Transport of Water, Single Phase
Volatile and Black Oils at 100 o C, from a 10 km Deep Wellbore,
Using Pressure Waves 5 Hz in Frequency, Neglecting Pressure
Wave Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . 170
10.1 Evolution of Impulse Pumping Reachable Flow Rate Against
Production Tubing Depth . . . . . . . . . . . . . . . . . . . . . 173
10.2 Evolution of Impulse Pumping Efficiency Against Production
Tubing Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
xxi

10.3 Characteristic Curve of Impulse Pumping Using Sinusoidal Pressure
Wave, 1 Hz in Frequency, 200 bar in Peak-To-Peak Amplitude
and pHH = 101bara . . . . . . . . . . . . . . . . . . . . 174
10.4 Pressure Wave Peak-To-Peak Amplitude Against Impulse Pumping
Efficiency for Artificial Lift During Statfjord Depressurization
Phase A, Using Sinusoidal Pressure Waves 4 Hz in Frequency
(Table 10.1) . . . . . . . . . . . . . . . . . . . . . . . . 177
10.5 Wellhead Pressure Against Impulse Pumping Efficiency for Artificial
Lift During Statfjord Depressurization Phase A, Using
Sinusoidal Pressure Waves 4 Hz in Frequency (Table 10.1) . . . 177
10.6 Minimum Reachable Bottom Hole Pressure Against Impulse
Pumping Efficiency for Artificial Lift During Statfjord Depressurization
Phase A, Using Sinusoidal Pressure Waves 4 Hz in
Frequency (Table 10.1) . . . . . . . . . . . . . . . . . . . . . . . 178
10.7 Reachable Heights and Flow Rates for Depressurization of Statfjord
Field, 2748 m Deep, Using Sinusoidal Pressure Waves 412
bar in Peak-To-Peak Amplitude Against Pressure Wave Frequency178
10.8 Reachable Inflow Efficiencies and Flow Rates for Depressurization
of Statfjord Field, 2748 m Deep, Using Sinusoidal Pressure
Waves 412 bar in Peak-To-Peak Amplitude Against Pressure
Wave Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 179
10.9 Impulse Pumping Inflow Characteristics for 102 bara Wellhead
Pressure, Using Sinusoidal Pressure Waves, 1 Hz in Frequency,
52 bar in Peak-To-Peak Amplitude . . . . . . . . . . . . . . . . 181
10.10Impulse Pumping Inflow Characteristics for 104 bara Wellhead
Pressure, Using Sinusoidal Pressure Waves, 1 Hz in Frequency,
206 bar in Peak-To-Peak Amplitude . . . . . . . . . . . . . . . 181
10.11Impulse Pumping Inflow Characteristics for 207 bara Wellhead
Pressure, Using Sinusoidal Pressure Waves, 1 Hz in Frequency,
412 bar in Peak-To-Peak Amplitude . . . . . . . . . . . . . . . 182
10.12Total Impulse Pumping Characteristics for 207 bara Wellhead
Pressure, Using Sinusoidal Pressure Waves, 1 Hz in Frequency,
412 bar in Peak-To-Peak Amplitude . . . . . . . . . . . . . . . 183
10.13Spatial Traces of Impulse Pumping Using Sinusoidal Waveform 184
10.14Spatial Traces of Impulse Pumping Using Square Waveform . . 185
10.15Spatial Traces of Impulse Pumping Using Triangular Waveform 185
10.16Reachable Flow Rates Against Impulse Generator Frequency
and Pressure Wave Waveforms . . . . . . . . . . . . . . . . . . 186
10.17Total Impulse Pumping Characteristics for 207 bara Wellhead
Pressure, Using Sinusoidal Pressure Waves, 1 Hz in Frequency,
412 bar in Peak-To-Peak Amplitude, Production Tubing 1 inch,
2 inch, 4 inch, and 8 inch in Diameter . . . . . . . . . . . . . . 188
xxii

B.1 Impulse Pumping Apparatus, Horizontal Configuration . . . . . 213
B.2 Representative Values of the Reflection Coefficient . . . . . . . 214
B.3 Pressure Wave (a) After Passing a Non-return Outlet Valve, (b)
After Reflecting From a Closed Boundary, (c) After Reflecting
on an Open Boundary . . . . . . . . . . . . . . . . . . . . . . . 214
B.4 Truncated Pressure Wave Travelling (a) Downward, (b) After
Reflecting From a Non-Return Valve . . . . . . . . . . . . . . . 215
B.5 Truncated Pressure Wave Travelling After Reflecting From a
Non-Return Valve . . . . . . . . . . . . . . . . . . . . . . . . . 215
C.1 Typical Water Hammer Pressure Trace . . . . . . . . . . . . . . 229
C.2 Schematic Representation of Test Case System . . . . . . . . . 230
C.3 Comparison Between the Method of Characteristics (dotted
line) and the Selected Finite Volume Method (solid line) . . . . 230
C.4 Detail of Figure (C.3) . . . . . . . . . . . . . . . . . . . . . . . 231
C.5 Pressure Trace at Non-Return Valve . . . . . . . . . . . . . . . 232
C.6 Pressure Trace at the Valve . . . . . . . . . . . . . . . . . . . . 233
C.7 Detail of the Pressure Trace at the Valve . . . . . . . . . . . . . 234
D.1 Illustration of the Off-Loading System between the Riser Base
and the Single Path Coupler . . . . . . . . . . . . . . . . . . . . 237
D.2 Fluid Velocity Response for 6 Different Types of Valves. Uniform
Valve Closure (Wood and Jones, 1973) . . . . . . . . . . . 240
D.3 Fluid Velocity Response for 6 Different Types of Valves. Accelerated
Valve Closure (Wood and Jones, 1973) . . . . . . . . . . 240
D.4 Maximum Pressure Achieved in the Oil Off-Loading System in
3 seconds with asteel = 1000 ms and aflexible = 450 ms . . . . . 244
D.5 Cumulated Time of the Pressure Surge being over 50 barg . . . 245
D.6 Maximum Pressure Achieved in the Oil Off-Loading System in
3 s with asteel = 1000 ms and aflexible = 350 ms . . . . . . . . 246
D.7 Maximum Pressure Achieved in the Oil Off-Loading System in
3 seconds with asteel = 1100 ms and aflexible = 450 ms . . . . . 247
D.8 Illustration of the Three Cases Considered for the Modelling . . 250
xxiii

xxiv

Nomenclature
Roman Symbols
u Cross-Sectional Averaged Fluid Velocity [ms −1 ]
A Cross Section Area [m 2 ]
a Acoustic Velocity [m s −1 ]
a Acoustic Velocity [ms −1 ]
B Quadratic Matrix
C + Backward Characteristics
C + Forward Characteristics
c1
Structure Coefficient
Cp Specific Heat Capacity at Constant Pressure [J K −1 ]
cp
Cv
cv
Heat Capacity at Constant Pressure [J K −1 ]
Specific Heat Capacity at Constant Volume [J K −1 ]
Heat Capacity at Constant Volume [J K −1 ]
d Pipe Diameter [m]
dp
Piston Head Diameter [m]
DS Dissipation Function [kg m s −2 ]
E Energy [J]
e Pipe Wall Thickness [m]
f Friction Factor
fq
Quasi Steady Friction Factor
xxv

fS
Steady Friction Factor
g Gravitational Acceleration [ms −2 ]
H Vertical Lift Height [m]
h Thermal Energy [J]
I Unity Matrix
J0
J1
First Order Bessel Function
Second Order Bessel Function
K Compressibility [Pa −1 ]
k Adiabatic Index
KΓ Isothermal Compressibility [Pa −1 ] (KFluid)
KS Isentropic Compressibility [Pa −1 ]
kA Empirical Constant in Axworthy et al.’s Model
kB Empirical Constant in Brunone et al.’s Model
KG Gas Compressibility [Pa −1 ]
KO Oil Compressibility [Pa −1 ]
KW Water Compressibility [Pa −1 ]
KΓ Isothermal Compressibility [Pa −1 ] (KFluid)
kB1 Empirical Constant in Brunone et al.’s Model
KFlow Flow Structure Compressibility [Pa −1 ]
KFluid Fluid Isothermal Compressibility [Pa −1 ] (KΓ)
KPipe Pipeline Compressibility [Pa −1 ]
kRi Empirical Constant in Ramos et al.’s Model (i = 1, 2)
kRi Empirical Constant in Ramos et al.’s Model (i = 1, 2)
L Length / Depth [m]
M Molar Mass [kg mol −1 ]
Ma Mach Number
xxvi

N Number of Sides
n Number of Moles
nT
Number of Time Steps
nX Number of Space Steps
p Pressure [Pa]
p0
pI
In-Situ Static Pressure [Pa]
Incident Pressure [Pa]
pR Reflected Pressure [Pa]
pT
pd
pu
Transmitted Pressure [Pa]
Downstream Pressure [Pa]
Upstream Pressure [Pa]
pBP Bubble Point Pressure [Pa]
pCI Inlet Non-Return Valve Cracking Pressure [Pa]
pCO Outlet Non-Return Valve Cracking Pressure [Pa]
pInlet Inlet Tank Pressure [Pa]
pLift Vertical Lift Pressure [Pa]
pMAWP Maximum Allowable Working Pressure [Pa]
pminBH Minimum Bottomhole Pressure [Pa]
pminIN Minimum Local Pressure at Inlet Non-Return Valve [Pa]
pOpening Outlet Non-Return Valve Opening Pressure [Pa]
pOutlet Outlet Tank Pressure [Pa]
pWave Pressure Wave Amplitude [Pa]
pWB Wellbore Pressure [Pa]
pWH Wellhead Pressure [Pa]
q Flow Rate [l min −1 ]
qd
Flow Rate Displaced by Piston Oscillations [l min −1 ]
xxvii

R Universal Gas Constant (R = 8.314 J mol −1 K −1 )
Re Reynolds Number (Re = ρud/µ)
s Source Term
T Travel Period [s]
t Time [s]
t0
Tv
Instant of Pressure Wave Generation [s]
Valve Full Opening Time [s]
TSim Simulated Period [s]
u Fluid Velocity [ms −1 ]
u0
uI
Initial Fluid Velocity [ms −1 ]
Incident Fluid Velocity [ms −1 ]
uR Reflected Fluid Velocity [ms −1 ]
uT Transmitted Fluid Velocity [ms −1 ]
ur
u ′ r
ux
u ′ x
Radial Fluid Velocity [ms −1 ]
Variation of Radial Fluid Velocity [ms −1 ]
Axial Fluid Velocity [ms −1 ]
Variation of Axial Fluid Velocity [ms −1 ]
uPiston Piston Linear Velocity [ms −1 ]
V Volume [m 3 ]
V0
Vd
Volume of Fluid Originally in Place [m 3 ]
Volume Displaced by Piston Oscillation [m 3 ]
X Outlet Valve Location [m]
x Axial Coordinate [m]
Y Pipe Material Young’s Modulus [Pa]
y Water Oil Mass Fraction
z Z-Factor
xxviii

Greek Symbols
αp
Thermal Expansion Coefficient [K −1 ]
β Two-Root Parameter Defining Characteristics
∆A Change in Cross Section Area [m 2 ]
∆ap Acoustic Velocity Correction for Pressure [ms −1 ]
∆aT Acoustic Velocity Correction for Temperature [ms −1 ]
∆aΣΓp Acoustic Velocity Correction for Salinity, Temperature and
Pressure [ms −1 ]
∆aΣ Acoustic Velocity Correction for Salinity [ms −1 ]
∆l Impulse Generator Stroke [m]
∆p Change in Pressure [Pa]
∆pf Frictional Pressure Drop [Pa]
δ Stoke Layer Thickness [m]
˙γ Impulse Generator Frequency [Hz]
ǫG
ǫL
Gas - Liquid Volumetric Fraction
Water - Oil Volumetric Fraction
η Efficiency [%]
Γ Temperature [K]
κ Heat Transfer Coefficient [W m −2 K −1 ]
Λ Source Term Vector
λ Wavelength [m]
µ Dynamic Viscosity [Pa s]
ν Kinematic Viscosity [m 2 s −1 ]
Ω Vector Variable
ω Angular Frequency [rad s −1 ]
∂ Partial Differential Operator
xxix

Φ Average Flux Vector
φ Theoretical Function
Π Dimensionless Angular Frequency
Ψ Constant
ρ Density [kg m −3 ]
ρG Gas Density [kg m −3 ]
ρO Oil Density [kg m −3 ]
ρW Water Density [kg m −3 ]
Σ Salinity [ppm]
τω
τµ
Wall Shear Stress [N m −2 ]
Viscous Stress [N m −2 ]
θ Inclination [rad]
θm Valve Full Opening Angle [rad]
ΥT Weighting Function in Trikha’s Model
ΥV Weighting Function in Vardy et al.’s Model
ΥZ Weighting Function in Zielke’s Model
εG
εL
̺C
Gas - Liquid Mass Fraction
Water - Oil Mass Fraction
Empirical Constant in Carstens and Roller’s Model
̺D Empirical Constant in Daily et al.’s Model
̺S
̺T
̺Z
̺V i
ςC
ςS
Empirical Constant in Safwat and Polder’s Model
Theoretical Constant in Trikha’s Model (i = 1, 2, 3)
Dimensionless Time in Zielke’s Model (̺Z = 4νt/d)
Theoretical Constant in Vardy et al.’s Model (i = 1, 2)
Velocity Profile Shape Parameter
Empirical Constant in Safwat and Polder’s Model
xxx

ςT
ςV i
Theoretical Constant in Trikha’s Model (i = 1, 2, 3)
Theoretical Constant in Vardy et al.’s Model (i = 1, 2)
�Φ Approximate Flux Function
Ξ Valve Actuation Function
ξ Valve Dimensionless Actuation Time
Superscripts
n Time Step Number
Subscripts
Γ Isothermal Condition
S Isentropic Condition
G Gas
i Cell Number
O Oil
W Water
Γ Isothermal Condition
Other Symbols
F Acceleration Due to External Forces [ms −2 ]
Ra→b Fluid Velocity Reflection Coefficient From Medium a to Medium b
Ta→b Fluid Velocity Transmission Coefficient From Medium a to Medium b
FPiston Force Applied on the Piston [N]
PGin Generated Power at Inlet Non-Return Valve [W]
PGout Generated Power at Outlet Non-Return Valve [W]
PG Generated Power [W]
PR Required Power [W]
Ra→b Reflection Coefficient From Medium a to Medium b
Ta→b Transmission Coefficient From Medium a to Medium b
xxxi

Z Acoustic Impedance [kg m −2 s −1 ]
Za
Zb
Mediam a Acoustic Impedance [kg m −2 s −1 ]
Mediam b Acoustic Impedance [kg m −2 s −1 ]
H Enthalpy [J kg −1 ]
S Entropy [J K −1 ]
U Internal Energy [J]
xxxii

Chapter 1
Introduction
Impulse pumping is a novel pumping method patented by Sagov (2004) and developed
by Clavis Impulse Technology AS. Impulse pumping has been demonstrated
for horizontal and vertical transport of liquids. However, impulse
pumping physical principles and performances in terms of head, flow rate and
efficiency have never been described nor reported in the literature before.
Analysis of physical principles and performances in terms of head, flow rate and
efficiency, are necessary to evaluate potential applications of impulse pumping.
Of particular interest are potential applications in petroleum engineering and
investigations of corresponding designs of an impulse pumping apparatus and
system.
Impulse pumping is based on pressure wave propagation, transmission and reflection
in pipelines. Hence, studying impulse pumping provides an opportunity
to better understand the phenomena of pressure waves in pipelines from
a scientific point of view. Establishing impulse pumping performances and
potential applications represents also an alternative to conventional pumping
methods, from an engineering point of view.
Conventional pumping methods are responsible for 25 % of the world electricity
demand with an average efficiency of about 60 % (Hydraulic Institute,
2004). In addition, pumping of deep wells or difficult fluids such as viscous oil
using conventional methods still remain challenges for petroleum engineers.
Therefore, studying innovative technologies offers opportunities to overcome
pumping issues with lower energy requirements.
Impulse pumping is based on pressure wave propagation in fluid-filled pipelines
that has been extensively investigated in hydraulic engineering. Of particular
interest is the considerable literature on water hammer phenomena in hydraulic
pipelines, with emphasis on propagation and attenuation modelling
that recently focused on transient friction and pipeline viscoelasticity.
1

Chapter 1. Introduction
Practical applications using pressure waves in pipelines, such as monitoring of
leakages and deposits, or flow metering, have also received particular attention
over the last twenty years. Nevertheless, state of the art one dimensional
numerical simulations still use the standard method of characteristics that
has been developed for pipeline modelling more than fifty years ago; that is,
at the beginning of the computer age.
The present thesis work analyses the principles of impulse pumping, establishes
a mechanistic model that substantiates a performance model, and explores
applications of impulse pumping in production and processing of petroleum.
In the present work, a pressure wave propagation model was investigated first
based on hydraulic studies of water hammer events. Pressure wave propagation
was then modelled in fluid-filled pipelines, with particular focus on pressure
wave characteristics during superposition, reflection and transmission at
acoustic interfaces in pipelines.
A mechanistic impulse pumping model was then derived based on pressure
wave propagation and observation of a limited number of experimental data
made available to NTNU by Clavis Impulse Technology AS. Of particular
interest was modelling of pressure wave reflections at dynamic boundaries
such as non-return valves.
Numerical simulations of pressure wave propagation using finite volume numerical
schemes were also investigated. Accuracy of finite volume methods
was investigated first. Then, results of pressure transient calculations using a
selected finite volume method and the well-established method of characteristics
were compared.
Pressure wave attenuation was investigated based on numerical simulation and
several models were compared. A new approach to pressure wave attenuation
was also investigated based on experimental data made available to NTNU
during the course of the PhD work.
The impulse pumping mechanistic model was then programmed using the
numerical simulation tool based on a finite volume numerical scheme. Results
from numerical simulations were then compared with experimental data made
available to NTNU by Clavis Impulse Technology AS.
Impulse pumping performance characteristics for a range of petroleum engineering
applications were finally investigated based on numerical simulation
of different test-cases. The thesis is divided into 10 chapters and 4 appendices
listed below:
• Chapter 2 contains literature review of basic pressure wave propagation
principles and advanced models of pressure wave attenuation mecha-
2

nisms. The literature review also describes applications based on pressure
wave propagation properties.
• Chapter 3 details modelling of pressure wave propagation, reflection and
transmission at interfaces and boundary conditions.
• Chapter 4 reviews acoustic velocity calculations in single-phase petroleum
fluids.
• Chapter 5 describes finite volume numerical methods that can be used
for one-dimensional simulation of pressure transients in pipelines, in replacement
to standard numerical methods.
• Chapter 6 analyses pressure wave attenuation models based on transient
friction and pipeline viscoelasticity, and introduces a new approach to
model pressure wave attenuation in pipelines.
• Chapter 7 describes an impulse pump apparatus and illustrates pressure
measurements obtained from experimental work during impulse pumping
of water.
• Chapter 8 defines the impulse pumping concept and presents a mechanistic
model and the corresponding numerical model and numerical
simulation tool.
• Chapter 9 defines an impulse pumping design criteria and a performance
model for horizontal and vertical transport of liquids.
• Chapter 10 analyses impulse pumping characteristic curves and investigates
the range of petroleum engineering applications where impulse
pumping is the most efficient. A real practical case is also studied to
compare impulse pumping with standard artificial lift techniques.
• Appendix A presents the list of technical reports written as part of the
PhD work.
• Appendix B contains the first scientific paper published on impulse
pumping.
• Appendix C contains a scientific publication comparing several transient
friction models using a finite volume method.
• Appendix C contains a scientific paper dealing with pressure transient
in multi section pipelines, such as platform off-loading pipelines.
3

Chapter 2
Pressure Waves in Pipelines
2.1 Modelling Equations and Numerical Simulation
”Pressure waves are disturbances that transmit energy and momentum from
one point to another through a medium without significant displacement of
matter between the two points” (Walski et al., 2003). Another way to express
this is that pressure waves are disturbances that contain information about
the propagation medium and its surroundings.
Pressure waves can be destructive or informative. On one hand, water hammer
events are commonly associated with accidents that generally result from
sudden valve actuation, or pump failures, for instance. Water hammer events
can therefore be seen as destructive. On the other hand, pressure pulses are
commonly associated with purposely generated pressure waves, for collecting
information on pipeline integrity and flow conditions, for instance. Hence,
pressure pulses can be seen as informative.
Water hammer and pressure pulses are both pressure waves. Historically, water
hammer was defined as the propagation of pressure waves in compressible
media in rigid tubes, whereas pressure pulses were defined as the propagation
of pressure waves in incompressible media in elastic tubes. Both phenomena
were then studied separately. The equivalence between water hammer and
pressure pulses was first observed by Kries (Tijsseling and Anderson, 2007).
Pressure wave propagation in general and water hammer in particular have
been extensively studied over the years. Countless scientific publications and
books are available, from pressure wave propagation to modelling of transient
friction, and to case studies. Nevertheless, pressure wave propagation
modelling can be summarized by mass, momentum and energy conservation
equations. Energy conservation is however commonly neglected due to the
rapidness of transient phenomena.
5

Chapter 2. Pressure Waves in Pipelines
After simplification, one-dimensional mass and momentum conservation equations
for studying pressure wave propagation in terms of pressure, p, and fluid
velocity, u, are, respectively (Ghidaoui et al., 2005):
∂p
+ ρa2∂u = 0 (2.1)
∂t ∂x
∂u
∂t
1 ∂p
+
ρ ∂x
−1
= fu|u| + F (2.2)
2d
where, a, defines acoustic velocity in fluid-filled pipeline, f, friction factor,
and, F, acceleration due to external forces. Initial water hammer pressure
amplitude in case of sudden valve actuation can be related to initial fluid
velocity, fluid density and acoustic velocity in the fluid-filled pipeline using
Joukowski’s equation. The theoretical equation results from simplification of
mass and momentum conservation equations. Joukowski’s equation can be
expressed by (Tijsseling and Anderson, 2007):
∆p = ρa∆u (2.3)
Numerical simulations of pressure waves and water hammer phenomena started
with the advent of computer age. The method of characteristics, based on
mass and momentum hyperbolic system characteristics received particular attention
due to its ease of programming, its low need in computer memory,
and its overall efficiency. The engineer friendly method of characteristics was
described and further detailed by Wylie and Streeter (1993).
2.2 Source Term Modelling
Recent developments of pressure wave modelling focus on momentum equation
right-hand term, called source term. The source term corresponds to pressure
wave energy dissipation mechanisms and attenuation. Pressure wave attenuation
can be observed experimentally by damping and smoothing of pressure
signals. A typical pressure wave temporal trace for a water hammer event,
resulting from sudden valve actuation, is shown in Figure 2.1.
Frictional pressure drop was the first attenuation mechanism included in pressure
wave modelling. Steady state friction factor was first calculated using the
Darcy-Weisbach equation (Moody, 1944):
fS = d 2
L u2∆pf (2.4)
where, fS, represents Darcy-Weisbach friction factor, d, pipeline diameter, L,
pipeline length, and, ∆pf, frictional pressure drop. Numerical simulation of
6

Pressure
2 4 6 8 10 12
2.2. Source Term Modelling
Time [2L/a]
Figure 2.1: Typical Water Hammer Pressure Trace, Reproduced from
Holmboe and Rouleau (1967)
water hammer events including steady state frictional pressure drop however
failed to reproduce accurately experimental data behaviour. Thus, further
modelling of pressure wave attenuation was required. Unsteady friction received
particular attention for explaining damping and smoothing.
Unsteady friction models are composed of the standard steady state frictional
pressure drop, and a frequency dependent term that is the unsteady friction
factor per se. Frequency is not explicitly included in unsteady friction models.
Nevertheless, unsteady friction factor models were derived by taking into
account that pressure waves high order harmonics attenuate faster than low order
harmonics. Unsteady friction is therefore implicitly a frequency dependent
term. Unsteady friction has first been modelled based on local flow history,
then based on flow velocity profiles, and lately based on flow accelerations.
2.2.1 Local Flow History
Evolution of past local flow history was first used for modelling unsteady
friction (Zielke, 1966, 1968). Unsteady friction was defined as a frequency
dependent pressure wave attenuation mechanism, noting that high order harmonics
attenuate faster than low frequency components. Based on experimental
observations, Zielke derived theoretically an unsteady friction factor that
considered pressure transient frequency band, and fluid flow history.
Zielke derived the dynamic component of the wall shear stress using convolution
integrals of temporal acceleration. In addition, weighting functions were
introduced to include history of flow velocity in unsteady friction modelling.
Zielke’s wall shear stress, τω, and weight functions, ΥZ, were, respectively:
τω(t) = 8ρν
� t 4ρν ∂u(τ)
u(t) +
d d 0 ∂t ΥZ (t − τ) dτ (2.5)
7

Chapter 2. Pressure Waves in Pipelines
ΥZ
�
̺Z = 4ν
⎧
� ⎪⎨ exp(−773.7091̺Z) for ̺Z > 0.02
t =
d2 0.282095̺
⎪⎩
−1/2
Z − 1.25 + 1.057855̺1/2 Z + 0.9375̺Z + ...
...0.396696̺ 3/2
Z − 0.351563̺2 Z for ̺Z < 0.02
(2.6)
Zielke developed the first unsteady friction model for laminar flow and tested
it using the method of characteristics. Calculation of Zielke’s convolution
integrals were expensive in terms of computer resources (CPU time and memory
usage). Thus, Trikha (1975) approximated Zielke’s model by simplifying
weighting functions and reducing computer costs. Trikha’s model was also
valid for laminar flow only, and was expressed by:
τω(t) = 8ρν 4ρν
u(t) +
d d (ΥT1(t) + ΥT2(t) + ΥT3(t)) (2.7)
ΥTi=1,2,3(t + ∆t) = ΥTi(t)exp −̺Ti
4ν
d
2 t
+ςTi (u (t + ∆t) − u (t)) (2.8)
where, ̺T, and, ςT, are theoretical constants. Values of ̺T and ςT are listed
in Table 2.1.
Table 2.1: Theoretical Values of ̺T and ςT in Eq. 2.8 (Trikha, 1975)
̺T ςT
i = 1 40 -8000
i = 2 8.1 -200
i = 3 1 -26.4
Both Zielke’s and Trikha’s models of frequency dependent friction were derived
for laminar flow conditions. Vardy and Hwang (1991) then derived an unsteady
friction model for a wider range of Reynolds number based on decomposition
of pipelines into concentric annuli surrounding a uniform core. Wall shear
stress and weighting functions were then expressed by:
τω(t) = 8ρν 4ρν
+
d2 d2 (ΥV 1(t) + ΥV 2(t)) (2.9)
ΥV i=1,2(t + ∆t) = ΥV i(t)exp −̺V i
4ν
d
2 t
+ςV i (u (t + ∆t) − u (t)) (2.10)
where, ̺V , and, ςV , are theoretical constants. Values of ̺V and ςV are listed
for particular values of the product of friction factor f and Reynolds number
Re in Table 2.2. Vardy and Hwang’s unsteady friction model was also less
expensive in computer resources than the original model by Zielke.
8

2.2. Source Term Modelling
Table 2.2: Theoretical Values of ̺V and ςV in Eq. 2.10 for Particular
f × Re Conditions (Vardy and Hwang, 1991)
f Re ̺V 1 ςV 1 ̺V 2 ςV 2
69.5 365 1.2 10 5 35 5 10 3
250 250 4.4 10 5 74 4.2 10 4
500 260 5.6 10 5 65 1.15 10 5
1000 350 9.8 10 5 65 4.12 10 5
2000 470 2.8 10 6 65 1.62 10 6
2.2.2 Quasi Two-Dimensional Models
Vardy and Hwang’s unsteady friction model was based on decomposition of
a pipeline into concentric annuli surrounding a uniform core (Vardy et al.,
1993). Vardy and Hwang’s model was therefore two-dimensional in essence
and mass and momentum conservation equations were solved in cylindrical
coordinates. However, Vardy and Hwang used the one-dimensional method
of characteristics to simulate their model. Of particular interest with twodimensional
modelling is the prediction of flow cross-sectional velocity profile.
Silva-Araya and Chaudry (1997) also proposed a quasi two-dimensional model
for computation of unsteady friction and prediction of flow velocity profiles.
Silva-Araya and Chaudry’s model was based on one-dimensional mass and
momentum conservation equations, but included a two-dimensional dissipation
function to account for viscous and turbulent stresses. The dissipation function
was expressed by:
DS = ∂ux
∂r
�
µ ∂ux
∂r − ρu′ xu ′ �
r
(2.11)
where, ux, defines axial flow velocity, ur, radial flow velocity, µ, dynamic
viscosity, and, ρu ′ xu ′ r, Reynolds stress tensor. Silva-Araya and Chaudry’s
model then required a turbulent model for approximating instantaneous pressure
gradients, velocity profiles and energy dissipation during transient events.
Pezzinga (1999) then developed a quasi two-dimensional model for unsteady
flow that also required modelling of turbulence.
Pezzinga based his quasi two-dimensional model on mass and momentum conservation
equations in cylindrical coordinates. Pezzinga’s model worked for
pipeline networks and calculated flow velocity profiles, thus allowing a more
accurate evaluation of unsteady flow resistances. Velocity profile predictions
permitted calculating wall shear stresses with greater accuracy. However,
Pezzinga noted that quasi two-dimensional modelling was more expensive in
9

Chapter 2. Pressure Waves in Pipelines
terms of computer resources and was barely justified by the result improvements.
Correct approximation of flow velocity profiles during transient events is nonetheless
of great importance as it is directly related to wall shear stress calculations.
Velocity profiles have been measured experimentally during transient events
by Brunone et al. (2000). Brunone et al. observed that the complexity of
experimental results were not completely reflected by available source term
modellings, even the more extended.
2.2.3 Instantaneous Velocity and Acceleration
Relations between flow structures and pressure losses were first investigated
experimentally based on turbulent flows. Daily et al. (1956) proposed a friction
factor formulation decomposed into a steady part and an unsteady part that
depends on the flow instantaneous acceleration. Daily et al.’s friction factor
was expressed by:
2d
f = fS + ̺D
u2 ∂u
∂t
(2.12)
where, ̺D, is an empirical constant equal to 0.01 and 0.62 for accelerating and
decelerating flows, respectively. Carstens and Roller (1959) proposed a similar
expression based on the power law of velocity distribution for turbulent flow
in a smooth pipe. The power law of velocity was expressed by:
u
u = (2ςC + 1)(ςC + 1)
2ς2 �
1 −
C
2r
�1/ςC (2.13)
d
where, ςC, characterizes the shape of the velocity profile and is equal to 7 for
Reynolds number less than 10 5 . Carstens and Roller’s friction factor formulation
was then expressed by:
d
f = fS + ̺C
u2 du
dt
(2.14)
Values of ̺C for corresponding values of ςC are listed in Table 2.3. Hino
et al. (1976) also studied flow velocity profiles and their relation with the
friction factor. Hino et al. proposed an empirical friction factor function of
the Stoke layer thickness for Reynolds number from 105 to 5830. The Stokes
layer thickness is the viscous boundary layer for an oscillatory flow and is
defined by::
δ =
� �1/2 2ν
ω
10
(2.15)

2.2. Source Term Modelling
Table 2.3: Theoretical Values of ςC and ̺C in Eq. 2.14 (Carstens and
Roller, 1959)
ςC ̺ C
7 0.449
8 0.391
9 0.346
10 0.310
where, δ, defines Stokes layer thickness, ν, kinematic viscosity, and, ω, angular
frequency. Hino et al.’s relation between friction and instantaneous velocity
was expressed by:
� �
4L
f = 0.1888 u
νπa
� −1/2.85
(2.16)
Hino et al.’s friction factor model was not decomposed into steady and unsteady
components. Safwat and Polder (1973) also proposed a new wall shear
stress formulation that did not explicitly include steady and unsteady components.
Safwat and Polder’s model was based on experimental work on oscillatory
flow in U-tube and was expressed by:
du
τω = ̺Su + ςS
dt
(2.17)
where, ̺S and ςS, were two empirical frequency dependent coefficients. Experimental
work on wall shear stress also led Shuy (1996) to propose a new
friction factor formulation. Shuy observed that unsteady friction increased
in decelerating flows and decreased in accelerating flows; thus contradicting
previous observations (Daily et al., 1956). Shuy’s friction factor is similar to
Eq. 2.14, with ̺C equal to -0.33 and -0.52 for accelerating and decelerating
flows, respectively.
2.2.4 Local and Convective Acceleration
Recent models of unsteady friction included local and convective accelerations
in the unsteady friction component. Local and convective acceleration were
included to account for local inertia and wall shear stress. A first model using
both accelerations was proposed by Brunone et al. (1991) and was expressed
by:
f = fS + kBd
� �
∂u
− a∂u
u|u| ∂t ∂x
11
(2.18)

Chapter 2. Pressure Waves in Pipelines
where, kB, is an empirical constant that varies between 0.03 and 0.01 (Brunone
et al., 1995), but a common value of 0.02 is normally used (Szymkiewicz and
Mitosek, 2007). Brunone et al. used the method of characteristics for simulating
water hammer events including unsteady friction. Axworthy et al. (2000)
proposed a second unsteady friction formulation for improving results from
Brunone et al.’s model. Axworthy et al.’s model was expressed by:
f = fS + kAd
� � ��
∂u u �
+ a �
∂u�
�
u|u| ∂t |u| �∂x
�
(2.19)
where, kA, is a theoretical frequency dependent relaxation time, derived from
irreversible thermodynamics. Axworthy et al. presented his model as an
improvement of Brunone et al.’s model. Of particular interest was the direction
of propagation that was considered by Axworthy et al.. Ramos et al. (2004)
also considered direction of propagation and proposed an improved unsteady
friction formulation:
f = fS + d
u|u|
�
kR1
∂u
∂t
+ kR2a u
|u|
� ��
�
�
∂u�
�
�∂x
�
(2.20)
where, kR1 and, kR2, are two empirical constants with values of 0.003 and
0.04, respectively. Brunone et al., Axworthy et al. and Ramos et al. (2004)
used the method of characteristics for simulating water hammer events with
unsteady friction modelling.
2.2.5 Evaluation of Unsteady Friction Models
Unsteady friction models based on local flow history, instantaneous velocity
and acceleration, local and convective acceleration, and quasi two-dimensional
models have been proposed over the years. Adamkowski and Lewandowski
(2006) compared several unsteady friction models and observed that local flow
history and quasi two-dimensional models were most accurate at low Reynolds
number. Modelling based on local and convective acceleration were however
more accurate at high Reynolds number.
Nixon and Ghidaoui (2007) compared several unsteady friction models in
pipeline networks, for leakage detection. Nixon and Ghidaoui concluded that
relevance of unsteady friction decreased as leakage sizes increased and could
be neglected for leak rates greater than 30 % of the initial flow rate. Nixon
and Ghidaoui also stated that introduction of unsteady friction modelling for
prediction of transient events in pipeline networks was not of particular importance
for increasing calculations accuracy.
Vitkovsky et al. (2006) also compared several unsteady friction models based
on experimental considerations. Vitkovsky et al. noticed that agreements
12

2.2. Source Term Modelling
between results from experimental work and numerical simulations were due
to the inherent dissipation of the method of characteristics. Szymkiewicz
and Mitosek (2007) concluded similarly based on mathematical study of the
modelling equations and comparing the method of characteristics to a third
order accurate finite element numerical scheme (Szymkiewicz and Mitosek,
2005).
2.2.6 Pipeline Viscoelasticity
Source term modelling focused first on unsteady friction with four different
approaches proposed in the literature: evolution of local flow history, quasi
two-dimensional model, evolution of instantaneous velocity and acceleration
and evolution of local and convective accelerations, as review above. However,
recent pressure wave propagation and attenuation models have introduced
pipeline material viscoelastic behaviour for explaining damping and smoothing
of pressure traces. Pipeline viscoelastic behaviour has received particular
attention due to the increasing use of plastic pipelines for water distribution
networks, but also for hydrocarbon transport systems.
Bergant and Tijsseling (2001) and Bergant et al. (2003a,b) investigated effects
of pipe wall viscoelasticity and noted that the single effect could not reproduced
experimental results. However, comparison of experimental data and
numerical simulations by the method of characteristics combining pipeline
viscoelasticity and unsteady friction gave satisfactory results. Pipeline wall
behaviour during transient events was decomposed into an instantaneous pipe
wall response followed by a delayed response dependent on pipeline material,
use and fatigue.
Covas (2003) and Covas et al. (2004, 2005) also investigated pipe wall viscoelastic
behaviour in plastic pipelines where unsteady friction alone could
not explain observed pressure damping. The acoustic velocity calculation was
modified to account for temporal evolution of pipeline material Young’s modulus.
Comparison between experimental data and results from numerical simulation
using the method of characteristics were in good agreement.
Pipeline viscoelasticity modelling results were obtained by using the method
of characteristics. In addition, pipeline viscoelasticity was often coupled with
unsteady friction. Numerical dissipation using the method of characteristics
was not considered by the different authors, therefore improvements offered by
pipe wall viscoelasticity modelling are questionable. Pressure wave attenuation
remains today an important field of research with no satisfactory modelling.
13

Chapter 2. Pressure Waves in Pipelines
2.3 Leak Detection
Pressure wave propagate, transmit and reflect according to elastic properties of
the propagation medium and its environment. Reflections and transmissions
of pressure waves take place at acoustic interfaces where elastic properties
change. Difference in acoustic properties can be due to changes in fluids,
flows, pipeline structure or geometry. Informative pressure waves offer then
valuable information relative to pipeline structure, integrity and flow.
2.3.1 Flow Measurements
Transient mass flow measurements were investigated by Liggett and Chen
(1994) for detecting leakages in pipeline networks. Inverse modelling was performed
from outlet flow rate measurements and calculated inlet flow rate conditions
were then compared with actual measurements. Deviations between
calculated and measured inlet flow conditions therefore indicated presence of
leaks in the network. Only detection was available from transient mass flow
measurement, and the method relied heavily on the accuracy of transient flow
modelling.
Transient flow rate measurements were also investigated by Erickson and
Twaite (1996) for detecting leakages in multiphase pipeline networks. Direct
modelling was performed from inlet flow rate measurements and calculated
outlet flow rate conditions were then compared with actual measurements. Deviations
between calculated and measured outlet flow rates indicated presence
of leaks. Erickson and Twaite’s leak detection method relied on a simplified
multiphase transient modelling, thus was not accurate.
Transient momentum balance was first investigated by Scott and Yi (1998) for
detecting, locating and dimensioning of leakages in single pipelines. Momentum
measurements presented the advantage of getting rid of unreliable inlet
flow metering and calibration uncertainties. Leakages as low as 5 % of the
outlet flow rate and more than 50 % of the outlet flow rate were located and
dimensioned for distant leaks and leaks within the first 10 % of the pipeline,
respectively.
Transient and momentum balances were among the first techniques for detecting
leakages in single pipelines and pipeline networks. Actual flow rate
measurements were used directly without signal treatment nor interpretation.
Flow measurements were used instead of more reliable pressure signals (Abhulimen
and Susu, 2004). Leak detection relied greatly on transient flow simulations,
and pressure wave properties such as acoustic velocity and reflection
characteristics were ignored.
14

2.3.2 Pressure Wave Characteristics
2.3. Leak Detection
Contractor (1965) investigated water hammer pressure waves in pipe systems
and observed wave reflections from pipeline features such as valves, orifices
and junctions. Contractor concluded that pipeline features must be treated as
boundary conditions in numerical simulations. Contractor’s investigation was
the starting point for Jonsson and Larson (1992) who analysed pressure traces
and their frequency content. Jonsson and Larson showed that propagation
characteristics contained valuable information.
Pressure wave propagation characteristics were then considered by Brunone
(1999) for detecting and locating leakages in single pipelines. A water hammer
event was purposely generated by sudden valve actuation, and was then
monitored at one end of a pipeline. Deviations between experimental pressure
measurements with and without leak were compared. Then, leakage location
was predicted from pressure wave travel time and calculated acoustic velocity
in the single phase fluid-filled pipeline.
Pressure wave propagation characteristics were also investigated by Gudmundsson
et al. (2000) for detecting, locating and monitoring deposits in multiphase
single pipelines. A water hammer event was purposely generated by sudden
valve actuation, and was then monitored at one end of the pipeline. Experimental
pressure data were then compared to synthetic pressure traces
simulated for a leak-free system. Deposits were then located and monitored
from pressure wave travel time and acoustic velocity in multiphase fluid-filled
pipeline (Dong and Gudmundsson, 1993).
Studies of pressure wave characteristics helped improving leakage detection
techniques by allowing localization and monitoring of leaks. Of particular
interest were the repeatability of transient tests for monitoring and calibrating
numerical simulations. However, leakage and deposit detection techniques
proposed by Brunone and Gudmundsson et al. relied on water hammer events
that required flowlines full stoppage while measuring. Nonetheless, both techniques
relied on simple and direct interpretation of pressure traces.
2.3.3 Pressure Signal Processing
Pressure traces were interpreted and analysed by Wang et al. (2002) for detecting,
locating and dimensioning leakages in single pipelines. A water hammer
was purposely generated and monitored. Experimental pressure traces were
then transposed in the frequency domain by Fourier transform. Leakages were
located and monitored by observing that damping of leak-free systems was independent
on harmonics whereas damping in systems with leaks depended on
frequency range. Leakages as small as 0.1 % of pipeline’s cross sectional area
15

Chapter 2. Pressure Waves in Pipelines
were detected and located using numerical simulations.
Pressure traces Fourier analysis as well as wavelet analysis were investigated by
Ferrante and Brunone (2003a,b) for detecting and locating leakages in single
pipelines. Both analysis proved to be efficient for detecting, locating and monitoring
leakages. However, modelling of pressure wave damping in fluid-filled
pipeline was questioned since leak characteristics determination depended on
numerical simulation results. Both Wang et al. and Ferrante and Brunone
used the method of characteristics for numerical simulation of pressure transients.
Leak detection and localization from numerical modelling using the method
of characteristics was investigated further by Wang and Carroll (2007). Of
particular interest were pipeline description and discretization. Wang and
Carroll observed that imprecise pipeline modelling and configuration, as well
as numerical model discretization could lead to absence of leakage detection
or greatly inaccurate calculations. The study relied also on the method of
characteristics for pressure wave modelling.
Leaks and other pipeline features effects on pressure traces were investigated
by Beck et al. (2005) using cross correlation. Effects of leakages as well as
pipeline junctions were observed from cross-correlation of reflected pressure
waves. Although being a repetition of well established pressure wave behaviours,
the paper highlighted detection of pipeline elements by simple pressure
trace physical analysis and signal treatment. The technique relied on
physical understanding of pressure wave propagation.
Lee et al. (2007) interpreted and analysed pressure pulse traces for locating
leaks in single pipelines. Pressure pulses were generated and monitored at
one end of the pipeline. Pressure traces were then transposed in frequency
domain by Fourier transform. Sharp pressure pulses of wide frequency range
were used to minimize signal distortion. Hence improved leaks localization and
monitoring. Pressure pulses did not require complete stoppage of the flowline
compared to water hammer events. Hence pressure pulses were presented as
less intrusive than water hammer for detection, localization and monitoring.
Pothof and Clemens (2008) analyzed pressure traces for detecting, locating
and monitoring gas pockets in multiphase single pipelines. A water hammer
event was purposely generated and monitored at one end of the pipeline. The
pressure trace was then transposed in frequency domain by Fourier transform.
Pothof and Clemens observed that the second harmonic in the frequency
domain indicated location and dimensions of a gas pocket. However,
the method’s poor accuracy proved to be detrimental.
Water hammer events and pressure pulses have been investigated for detect-
16

2.4. Flow Metering
ing, locating and monitoring pipeline features. Methods are divided in two
categories, the first one being a direct interpretation of pressure traces based
on physical understanding of pressure wave properties; the second being mathematical
analysis of pressure traces and correlating experimental results with
observations. Information from pressure pulses and water hammer events are
identical, however, pressure pulses do not require complete stoppage of flowlines.
2.4 Flow Metering
Pressure waves propagation velocity in fluid-filled pipeline depends on elastic
properties of the propagation medium. Acoustic velocity calculations require
fluid density and isothermal compressibility, as well as pipeline dimensions
and material Young’s modulus. Therefore, acoustic velocity measurements can
provide valuable information about pipeline structure and fluid composition.
Pressure waves propagation velocity in multiphase flowlines was investigated
by Gudmundsson et al. (1992) for metering fluid composition. Pulses were
generated at one location and four pressure transducers were monitoring pulse
propagation. Cross correlation of pressure traces at the four locations and
acoustic velocity definition in multiphase flow (Dong and Gudmundsson, 1993)
were used for calculating void fraction. The technique was repeatable and did
not require complete stoppage of the flowline.
Multiphase flow metering was further developed and applied to gas lift wells
by Gudmundsson et al. (2001). Only two pressure transducers were used for
monitoring a purposely generated water hammer event. Cross correlation and
travel time determination were then used to calculate multiphase flow void
fraction. Direct interpretation of pressure traces was used. However, water
hammer required a complete stoppage of the flowline during testing.
Pressure wave propagation velocity in gas pipelines was analysed in AGA-9
(2007). Pressure pulses are generated at one emitter and received at a distant
receiver. Pressure waves travel times were measured and wave propagation
velocities deduced from distance between transducers. Stream and counterstream
wave propagation velocities were used for calculating flowing velocities.
Large bandwidth pulses improved travel time detection at receivers.
Gas flow metering can also be performed analytically by calculating the acoustic
velocity using AGA-10 (2003) and accurate gas composition. Pressure
pulses travel times as well as flow velocities were then calculated from acoustic
velocities. Deviation between calculated and measured acoustic velocities
must be within ±0.2 % and must be verified during calibration of the apparatus.
17

Chapter 2. Pressure Waves in Pipelines
Distances between emitters and receivers are the main uncertainties in gas ultrasonic
flow metering since pressure and temperature influence pipeline materials,
thus modifying distances between transducers. However, gas ultrasonic
flow metering resolution is 0.001 ms −1 . Multiphase flow and gas flow metering
are based on acoustic velocity determination from pressure wave travel time.
Large bandwidth pressure pulses improve travel time measurements, hence
acoustic velocity determinations.
Berrebi et al. (2004) studied ultrasonic flow metering accuracy and investigated
effects of pulsating flow on measurements. Pulsating flows were generated from
dynamic installations such as pumps or valves. Pulsating flows generated oscillation
of pressure wave signals and affected travel time determination, hence
acoustic and flow velocity measurements. Berrebi et al. proposed a simple
filter for reducing ultrasonic flow measurement errors. Berrebi et al. also concluded
that no relevant modelling of pulsations effect on wave propagation was
available.
18

Chapter 3
Modelling of Pressure Wave
Propagation
3.1 Propagation
3.1.1 Modelling Equations
Rapid pressure transients can be generated by sudden valve actuation or pump
failure, for instance. The rapidness of transient phenomenon is relative to the
travel period of the pressure wavefront, or to the pressure transient wavelength.
Wavelength defines distance between wavefront and wavetail and depends on
the angular frequency, ω, and propagation velocity, a, of the transient phenomenon.
The wavelength is expressed by (Lighthill, 1978):
λ = ω
a (3.1)
2π
Water hammer describes transient events induced by sudden valve actuation.
The travel period is the difference in time between the beginning of the actuation
and the instant the wavefront has travelled back for the first time to the
valve after reflection from the end element. The pressure transient wavelength
is then twice the distance between the valve and the pipeline end element.
Water hammer phenomena are rapid if the valve actuation time is greatly
shorter than one travel period.
In an induced pressure wave event generated by piston back and forth movement,
the wavelength is the product of the acoustic velocity in the fluid-filled
pipeline, and the frequency of the piston. Wavelengths are long when greater
than the pipeline length, and short otherwise. A pressure wave can be overlooked
as a succession of two water hammer phenomena generated by opposite
valve actuations. Therefore, water hammer and pressure wave analysis can be
performed using the same modelling.
19

Chapter 3. Modelling of Pressure Wave Propagation
Pressure wave propagation in fluid-filled pipeline modelling is based on the
standard mass, momentum and energy conservation equations. Hydrocarbonfilled
pipelines are commonly several to tens of inches in diameter and can
cover hundreds of kilometers in length. Thus only one-dimensional modelling is
practically used. One-dimensional flow of a viscous and heat conducting fluid,
is characterized by the following mass, momentum and energy conservation
equations, respectively (Chevray and Mathieu, 1993):
∂ρ ∂
+ (ρu) = 0 (3.2)
∂t ∂x
∂ ∂ −∂p ∂τµ
(ρu) + (ρuu) = +
∂t ∂x ∂x ∂x + τωπd + F (3.3)
∂ ∂ ∂
(ρE) + (ρuE) =
∂t ∂x ∂x [(−p + τµ)u] − ∂h
∂x
(3.4)
where, t, represents time, x, pipeline axial direction, ρ, density, u, fluid velocity,
p, pressure, τµ, viscous stress, τω, wall shear stress, d, pipeline diameter,
E, energy, h, heat transfer, and, F, external forces. Right hand terms in the
momentum conservation equation, Eq. 3.3, define a source term, responsible
for pressure wave attenuation. Attenuation mechanisms and the source term
are attentively investigated in Chapter 6 and will not be detailed herein.
Wall shear stress, viscous stress and heat transfer in one-dimensional modelling
are expressed by, respectively (Chevray and Mathieu, 1993):
τω = −1
ρfu|u| (3.5)
2d
τµ = 4
3 µ∂u
∂x
h = −κ ∂Γ
∂x
(3.6)
(3.7)
where, µ, represents fluid dynamic viscosity, Γ, temperature, and, κ, heat
transfer coefficient. Hydrocarbon liquids heat conductivity are generally low
and rapid pressure transient processes are commonly assumed to be isothermal.
The energy conservation equation is then omitted. Shear stress is also
commonly neglected since pressure transients are longitudinal waves. The
system of equations, Eq. 3.2, Eq. 3.3 and Eq. 3.4, then becomes:
∂ρ
+ ρ∂u + u∂ρ = 0 (3.8)
∂t ∂x ∂x
ρ ∂u
∂t
−∂p 1
+ ρu∂u = − ρfu|u| + F (3.9)
∂x ∂x 2d
20

3.1. Propagation
Flow velocities in oil and gas transport pipelines typically range from 2 ms −1
to 4 ms −1 and 20 ms −1 to 40 ms −1 , respectively. Acoustic velocities in oilfilled
and gas-filled pipelines are typically of order 850 ms −1 and 300 ms −1 ,
respectively. Therefore, the Mach number, Ma = u/a, where a is the acoustic
velocity in fluid-filled pipeline, is much less than unity in both oil and gas
transport pipelines. The acoustic velocity in fluid-filled pipeline can be defined
by (Lighthill, 1978):
a =
�
∂p
∂ρ
(3.10)
An order-of-magnitude analysis of Eq. 3.8 and Eq. 3.9 establishes that advective
terms (u ∂ ·/∂x) in both equations are negligible because of the low Mach
number. The low Mach number approximation and substitution of density by
pressure, using the acoustic velocity definition Eq. 3.10, then reduces Eq. 3.8
and Eq. 3.9 for pressure wave analysis to (Ghidaoui et al., 2005):
∂p
+ ρa2∂u = 0 (3.11)
∂t ∂x
∂u
∂t
1 ∂p
+
ρ ∂x
−1
= fu|u| + F (3.12)
2d
3.1.2 Transmission, Reflection
Pressure waves propagate, transmit and reflect in fluid-filled pipelines at the
speed of sound. Practically, the acoustic velocity in fluid-filled pipelines depends
on the fluid density and isothermal compressibility, KΓ, and on the
pipeline diameter and wall thickness, e, and the wall material Young’s modulus,
E. The acoustic velocity in a fluid-filled pipeline is expressed by (Pain,
2005):
�
�
� 1
a = � �
�
ρ KΓ + d
� (3.13)
Y e
The product of the fluid density and the wave acoustic velocity in the fluidfilled
pipeline defines the propagation medium acoustic impedance, Z (Lighthill,
1978):
Z = ρa (3.14)
Acoustic interfaces are virtual borderlines between two pipeline sections of different
acoustic impedances. Changes in impedance can originate from changes
21

Chapter 3. Modelling of Pressure Wave Propagation
in fluid density, due to pressure or temperature, or acoustic velocity, due to
pipeline geometry or material properties. Changes in acoustic impedance can
also result from a combination of both changes in density and acoustic velocity.
Reflection and transmission coefficients for a pressure wave propagating
from a medium a to a medium b, as illustrated in Figure 3.1, are, respectively
(Pain, 2005):
Ra→b = pR
pI
Ta→b = pT
pI
= Zb − Za
Zb + Za
= 2Zb
Zb + Za
pI
pR
a b
Figure 3.1: Acoustic Interface Schematic Representation Between
Media a and b
pT
(3.15)
(3.16)
The reflection coefficient is the ratio of the reflected pressure wave amplitude,
pR, and the incident pressure wave amplitude, pI. The transmission coefficient
is the ratio of the transmitted pressure wave amplitude, pT, and the incident
pressure wave amplitude. Reflection and transmission coefficients, Eq. 3.15
and Eq. 3.16, are defined as pressure ratios. However, a second set of reflection
and transmission coefficients relative to fluid velocity can also be derived and
are expressed by (Pain, 2005):
Ra→b = uR
uI
Ta→b = uT
uI
= −Ra→b
= Za
Ta→b
Zb
(3.17)
(3.18)
Influence of an acoustic interface on a square pressure wave propagating from a
high acoustic impedance medium a to a low acoustic impedance b is illustrated
in Figure 3.2. Both transmitted and reflected waves conserve square waveforms
although the transmitted pressure wave has a lower amplitude and shorter
22

3.1. Propagation
wavelength than the incident pressure wave (T < 1). The reflected pressure
wave has the same wavelength as the incident pressure wave but has a negative
amplitude (R < 0).
Changes in wavelength are due to the difference in acoustic impedance before
and after the interface. The pressure wave front propagates in medium b with
a lower impedance than medium a, therefore propagating slower in medium b
than in medium a and with a lower amplitude.
Influence of an acoustic interface on a square pressure wave propagating from a
low acoustic impedance medium a to a high acoustic impedance b is illustrated
in Figure 3.3. Both transmitted and reflected waves conserve square waveforms
although the transmitted pressure wave has a greater amplitude and longer
wavelength than the incident pressure wave (T > 1). The reflected pressure
wave has the same wavelength as the incident pressure wave but has a lower
amplitude (R < 1).
Changes in wavelength are due to the difference in acoustic impedance before
and after the interface. The pressure wave front propagates in medium b with
a greater impedance than medium a, therefore propagating faster in medium
b than in medium a and with a greater amplitude.
Evolution of transmission coefficient with the ratio of acoustic impedances
between media a and b is shown in Figure 3.4. Transmission coefficient values
range from 0 to +2, asymptotically. The transmission coefficient is 0 when
the acoustic impedance in medium b is negligible compared to the acoustic
impedance in medium a. The transmission coefficient is +2 when the acoustic
impedance in medium b is much greater compared to the acoustic impedance
in medium a.
Evolution of reflection coefficient with the ratio of acoustic impedances between
media a and b is shown in Figure 3.5. Reflection coefficient values
range from −1 to +1, asymptotically. The reflection coefficient is −1 when
the acoustic impedance in medium b is negligible compared to the acoustic
impedance in medium a. The reflection coefficient is +1 when the acoustic
impedance in medium b is much greater compared to the acoustic impedance
in medium a.
A pressure wave transmits totally from medium a to medium b when acoustic
impedances are equal, Figure 3.4. The reflection coefficient equals zero in
such case, Figure 3.5. Reflected pressure waves relative pressure can be with
or without phase change, but their amplitudes are limited by the incident
pressure wave amplitude. However, transmitted pressure waves always have
the same phase as the incident pressure wave, but their amplitude can be up
to twice the incident amplitude.
23

Chapter 3. Modelling of Pressure Wave Propagation
(p − p 0 ) / p Wave
(p − p 0 ) / p Wave
1.5
1.0
0.5
0.0
Incident →
−0.5
0 L/2 L
1.5
1.0
0.5
0.0
← Reflected
Transmitted →
−0.5
0 L/2
Distance [m]
L
Figure 3.2: Influence of Acoustic Interface on Pressure Wave
Propagation. Medium a between 0 and L/2. Medium b
between L/2 and L. Za > Zb
(p − p 0 ) / p Wave
(p − p 0 ) / p Wave
1.5
1.0
0.5
0.0
Incident →
−0.5
0 L/2 L
1.5
1.0
0.5
← Reflected
0.0
−0.5
0 L/2
Distance [m]
Transmitted →
L
Figure 3.3: Influence of Acoustic Interface on Pressure Wave
Propagation. Medium a between 0 and L/2. Medium b
between L/2 and L. Za < Zb
24

Ta→b
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
0.00
10 −3
10 −2
10 −1
10 0
Zb/Za
10 1
10 2
3.1. Propagation
Figure 3.4: Evolution of Transmission Coefficient Against Ratio of
Acoustic Impedances
Ra→b
1.00
0.75
0.50
0.25
0.00
−0.25
−0.50
−0.75
−1.00
10 −3
10 −2
10 −1
10 0
Zb/Za
Figure 3.5: Evolution of Reflection Coefficient Against Ratio of
Acoustic Impedances
25
10 1
10 2
10 3
10 3

Chapter 3. Modelling of Pressure Wave Propagation
3.2 Boundary Conditions
3.2.1 Static Boundary Conditions
Pipe walls, tanks or pumps represent static boundary conditions. Pumps and
tanks are characterized by fixed pressure values after flow establishment. Pipe
walls are characterized by a zero fluid velocity. Static boundary conditions are
categorized into open and close by the reflection coefficient value. Another way
to categorize boundary conditions is by soft wall or hard wall, corresponding
to open and close boundary, respectively.
Reflections of a square incident pressure wave at closed and open boundaries
are illustrated in Figure 3.6. A pressure wave reflects totally without phase
change at a close boundary (R = 1, Figure 3.5), or hard wall, as shown
in Figure 3.6 b). A pressure wave reflects totally with phase change at an
open boundary (R = −1, Figure 3.5), or soft wall, as shown in Figure 3.6 c).
Fluid velocity reflection coefficient, Eq. 3.17, is opposite of pressure reflection
coefficient. Thus, fluid velocity signals are phase shifted compared to pressure
signals as observed in Figure 3.6 d) and e).
The transmission coefficient for a closed boundary (R = 1, Figure 3.5) equals
+2 (Figure 3.4). However, the fluid medium does not exist beyond the boundary.
Another way to express this is that a pressure wave of the same amplitude
as the incident pressure value will reflect totally into the fluid-filled pipeline
whereas a pressure wave twice as large will propagate into the physical boundary.
Pressure waves generated in fluid-filled pipeline can propagate further into
the supporting structure and weaken mechanical parts.
3.2.2 Dynamic Boundary Conditions
Tanks and pumps represent dynamic boundary conditions during flow establishment,
or transient phases. Tanks and pumps transients are characterized
by changes in pressure and fluid velocity. Pressure or fluid velocity gradients
can be smooth such that boundary conditions are characterized by a temporal
function, or can be abrupt, thus requiring transient calculation. Check valves
are also dynamic boundaries during actuation and represent one of the main
causes for rapid pressure transients.
Water hammer pressure waves are created by check valves sudden closure or
opening. The pressure transient amplitude can be approximated by Joukowski’s
expression (Wylie and Streeter, 1993):
∆p = ρa∆u (3.19)
where, ∆p, is pressure change generated by a flow velocity change, ∆u.
26

(p − p 0 ) / p Wave
(p − p 0 ) / p Wave
a)
1.0
0.0
−1.0
Incident →
3.2. Boundary Conditions
0 L/2 L
b)
1.0
Closed Boundary
0.0
← Reflected
−1.0
c)
1.0
0.0
−1.0
0 L/2 L
Open Boundary
← Reflected
0 L/2
Distance [m]
L
(u − u 0 ) / u Wave
(u − u 0 ) / u Wave
d)
1.0
← Reflected
0.0
Closed Boundary
−1.0
−1.0
0 L/2 L
e)
1.0
Open Boundary
0.0
← Reflected
0 L/2
Distance [m]
L
Figure 3.6: Pressure and Fluid Velocity Waves Reflections From
Open and Close Boundary Conditions
a): Incident Pressure Wave.
b): Reflected Pressure Wave from a Close Boundary.
c): Reflected Pressure Wave from an Open Boundary.
d): Reflected Velocity Wave from a Close Boundary.
e): Reflected Velocity Wave from an Open Boundary.
Joukowski’s equation approximates pressure transient amplitudes for complete
closure or opening. However, partial valve actuations are also responsible for
transient phenomena. Time dependent pressure transient amplitudes can be
calculated from valve actuation functions derived from geometrical considerations
and valve actuation time (Wood and Jones, 1973).
Circular gate, square gate, globe, needle, butterfly and ball valves are studied
for uniform and accelerated actuation functions. Geometries and important
dimensions of the six check valves are represented in Figure 3.7. Actuation
functions are given in Table 3.1 where, Ξ, is the ratio of open area and total
possible flow area, and, ξ, the ratio of displacement and total stroke for circular
gate, square gate, globe and needle valves, or the opening angle for butterfly
and ball valves (Wood and Jones, 1973, 1974).
27

Chapter 3. Modelling of Pressure Wave Propagation
x
e)
c)
a)
θ
X
x
Figure 3.7: Schematic Representation of Check Valves.
a) Circular Gate Valve. b) Square Gate Valve.
c) Globe Valve. d) Needle Valve.
e) Butterfly Valve. f) Ball Valve.
X
28
X
x
B
f)
d)
b)
b
X
x
θ

3.2. Boundary Conditions
Table 3.1: Actuation Function Ξ for Circular Gate, Square Gate,
Globe, Needle, Butterfly and Ball Valves for Dimensionless
Actuation Time ξ (Eq. 3.20 to Eq. 3.23) (Wood and
Jones, 1973, 1974)
Valve Type Actuation Function
Circular Gate 1 − 2
�
arccos(ξ) − ξ
π
� 1 − ξ2 �
1 −
Square Gate
1
�
�
arccos(2ξ − 1) − (2ξ − 1) 1 − (2ξ − 1)
π
2
�
Globe ξ
Needle 2ξ − ξ 2
Butterfly
cos (ξ)
π
1 + cos(ξ)
+
2
�
arcsin(−x) + 1
�
sin (2arcsin(−x)) −
2
�
1
arcsin(x) +
π
1
2 sin(2arcsin(x))
�
�
�B �2 sin(ξ) − 1
b
where x =
1 + cos (ξ)
�
π
�
Ball 1 − cos − ξ
2
29

Chapter 3. Modelling of Pressure Wave Propagation
Uniform and accelerated actuation functions for linearly opening valves are
expressed by, respectively:
ξ = 1 − t
Tv
� �2 t
ξ = 1 −
Tv
(3.20)
(3.21)
Uniform and accelerated actuation functions for rotary opening valves are
expressed by, respectively:
� �
t
ξ = θm
(3.22)
ξ = θm
Tv
�2 � t
Tv
(3.23)
where, Tv, is valve actuation time, and, θm, complete actuation angle. Valve
uniform and accelerated closures are shown in Figure 3.8 and Figure 3.9, respectively.
For both closure functions, the circular gate valve flow area gradient
evolves slowly during the first 50 % of closure before increasing, whereas
the flow area gradient for a ball valve is important during the first 50 % of
closure before decreasing. Uniform closure of a globe valve is characterized by
a constant gradient.
Fluid velocity reductions for uniform closure of the six valves considered are
shown in Figure 3.10. The fluid velocity reduces slowly until 80 % of the fluid
initial velocity, the fluid velocity then decreases rapidly until full closure. The
fluid velocity reaches 80 % of the initial velocity at 87 %, 81 %, 76 %, 71 %,
65 % and 55 % of the actuation time for needle, circular gate, globe, square
gate, ball and butterfly valves, respectively.
Fluid velocity reductions for accelerated closure of the six valves considered
are shown in Figure 3.11. The fluid velocity reduces slowly until 80 % of the
fluid initial velocity, the fluid velocity then decreases rapidly until full closure.
The fluid velocity reaches 80 % of the initial velocity at 93 %, 90 %, 87 %,
81 %, 80 % and 74 % of the actuation time for needle, circular gate, globe,
square gate, ball and butterfly valves, respectively.
Practically, pressure waves originating from sudden valve actuations are generated
mainly during the last 20 % of valve uniform actuation, or the last
10 % of valve accelerated actuation, as observed from Figure 3.10 and Figure
3.11. Fluid velocity gradients determine also pressure waveforms. Needle
valves generate sharper waveforms than circular gate, globe, square gate, ball
and butterfly valves.
30

χ [%]
χ [%]
100
80
60
40
3.2. Boundary Conditions
20
Needle Valve
Circular Gate Valve
Square Gate Valve
Globe Valve
Ball Valve
Butterfly Valve
0
0 20 40 60 80 100
t / T [%]
v
Figure 3.8: Valve Position During Uniform Closure
100
80
60
40
20
Needle Valve
Circular Gate Valve
Square Gate Valve
Globe Valve
Ball Valve
Butterfly Valve
0
0 20 40 60 80 100
t / T [%]
v
Figure 3.9: Valve Position During Accelerated Closure
31

Chapter 3. Modelling of Pressure Wave Propagation
u(t) / u 0 [%]
100
80
60
40
20
Needle
Gate − Circular
Globe
Gate − Square
Ball
Butterfly
0
0 20 40 60 80 100
t / T [%]
v
Figure 3.10: Flow Velocity Across Valve During Uniform Closure
u(t) / u 0 [%]
100
80
60
40
20
Needle
Gate − Circular
Globe
Gate − Square
Ball
Butterfly
0
0 20 40 60 80 100
t / T [%]
v
Figure 3.11: Flow Velocity Across Valve During Accelerated Closure
32

3.3 Pressure Wave Superposition
3.3. Pressure Wave Superposition
Several pressure waves originating from multiple sources can propagate simultaneously
in a fluid-filled pipeline. Origins of multiple pressure waves in a
pipeline can be multiple sources or multiple reflections from changes in acoustic
impedance. Wave superposition can then take place in a pipeline when
several pressure waves arrive at a particular location at the exact same instant.
Wave superposition can also occur when a pressure wave superimposes
on its own reflection at boundaries.
Pressure wave superposition can be constructive or destructive (Pain, 2005).
Constructive wave superpositions take place when two waves of same phase
encounter, or at closed boundaries when a pressure wave superimposes on
its own reflection. Destructive wave superpositions occur when two waves of
opposite phase encounter, or at open boundaries when a pressure superimposes
on its own reflection. Herein presented pressure signals were obtained from
numerical simulation using a finite volume scheme detailed and selected in
Chapter 5.
Constructive wave superposition of two pressure waves of same amplitude and
phase and different waveforms is illustrated in Figure 3.12. The two pressure
waves of same phase propagate in opposite direction, a), then superimpose,
b), before continuing separately unaffected by the superposition, c). During
superposition, pressure wave amplitudes add to each other, thus creating a
pressure peak of greater amplitude than either of the pressure waves.
Destructive wave superposition of two pressure waves of same amplitude and
different waveforms and phases is illustrated in Figure 3.13. The two pressure
waves of opposite phase propagate in opposite direction, a), then superimpose,
b), before continuing separately unaffected by the superposition, c). During
superposition, pressure wave amplitudes subtract to each other, thus creating
a pressure peak of lower amplitude than either of the pressure waves.
33

Chapter 3. Modelling of Pressure Wave Propagation
(p − p 0 ) / p Wave
(p − p 0 ) / p Wave
(p − p 0 ) / p Wave
a)
2
1
0
b)
2
1
0
c)
2
1
0
→ ←
0 L/2 L
←
0 L/2 L
← →
0 L/2
Distance [m]
L
Figure 3.12: Constructive Superposition of two Pressure Waves
a): Propagation in Opposite Direction Before Superposition
b): Superposition, Additive Amplitudes
c) Propagation in Opposite Direction After Superposition
34
→

(p − p 0 ) / p Wave
(p − p 0 ) / p Wave
(p − p 0 ) / p Wave
a)
1
0
−1
b)
1
0
−1
c)
1
0
−1
→
3.3. Pressure Wave Superposition
0 L/2 L
←
0 L/2 L
←
0 L/2
Distance [m]
L
Figure 3.13: Destructive Superposition of two Pressure Waves
a: Propagation in Opposite Direction Before Superposition
b): Superposition, Subtractive Amplitudes
c): Propagation in Opposite Direction After Superposition
35
→
←
→

Chapter 3. Modelling of Pressure Wave Propagation
Constructive wave superposition of a pressure wave and its own reflection at a
closed boundary is illustrated in Figure 3.14. The pressure wave hits the closed
boundary and reflects without phase change (R = 1, Figure 3.5), Figure 3.14
b); until half the wave length has reflected and superimposes on the incident
pressure wave, Figure 3.14 c). During maximum superposition, the pressure
amplitude is twice the incident pressure wave amplitude. The pressure wave
then continues reflecting, Figure 3.14 d), until being totally reflected, Figure
3.14 e).
Destructive wave superposition of a pressure wave and its own reflection at an
open boundary is illustrated in Figure 3.15. The pressure wave hits the open
boundary and reflects with phase change (R = −1, Figure 3.5), Figure 3.15
b); until half the wave length has reflected and superimposes on the incident
pressure wave, Figure 3.15 c). During maximum superposition, the pressure
amplitude is zero as the two amplitudes cancel each other. The pressure wave
then continues reflecting, Figure 3.15 d), until being totally reflected, Figure
3.15 e).
Practically, pressure transducers must be located at distances greater than half
a wavelength from boundaries or interfaces for capturing pressure waves individually
without superposition on their own reflections. Pressure traces would
otherwise be difficult to interpret and could lead to misleading conclusions.
36

(p − p 0 ) / p Wave
(p − p 0 ) / p Wave
(p − p 0 ) / p Wave
a)
2
1
0
b)
2
1
0
e)
2
1
0
Incident →
3.3. Pressure Wave Superposition
0 L/2 L
→
←
0 L/2 L
c)
2
1
0
Closed Boundary
→←
0 L/2 L
d)
2
1
0
→
←
0 L/2 L
← Reflected
0 L/2
Distance [m]
L
Figure 3.14: Constructive Superposition of a Pressure Wave on its
own Reflection at a Closed Boundary
a): Incident Pressure Wave
b): Beginning of Pressure Wave Superposition on its Own Reflection
c): Total Superposition of Pressure Wave and its Own Reflection.
Additive Amplitudes
d): End of Pressure Wave Superposition on its Own Reflection
e): Reflected Pressure Wave
37

Chapter 3. Modelling of Pressure Wave Propagation
(p − p 0 ) / p Wave
(p − p 0 ) / p Wave
(p − p 0 ) / p Wave
a)
1
0
−1
b)
1
0
−1
e)
1
0
−1
Incident →
0 L/2 L
←
→
0 L/2 L
c)
1
0
−1
Open Boundary
→
←
0 L/2 L
d)
1
0
−1
→
←
0 L/2 L
← Reflected
0 L/2
Distance [m]
L
Figure 3.15: Destructive Superposition of a Pressure Wave on its
own Reflection at an Open Boundary
a): Incident Pressure Wave
b): Beginning of Pressure Wave Superposition on its Own Reflection
c): Total Superposition of Pressure Wave and its Own Reflection.
Subtractive Amplitudes
d): End of Pressure Wave Superposition on its Own Reflection
e): Reflected Pressure Wave
38

3.3.1 Standing Waves
3.3. Pressure Wave Superposition
Standing waves arise when incident and reflected pressure waves superimpose.
Standing wave patterns are obtained for longitudinal waves such as pressure
waves that reflect completely from static boundaries. Closed and open boundary
conditions give distinct patterns and sets of nodal points. Nodal points
are locations where superposition of incident and reflected waves are zero in
amplitude.
First and third standing wave harmonics from a pressure wave propagating in
a pipeline closed at both ends are shown in Figure 3.16. Incident and reflected
waves from closed boundaries have same phases (R = 1, Figure 3.5), thus
maximum amplitude of standing waves are twice the incident wave amplitude.
Boundaries are anti-nodal points. Another way to express this is that pressure
wave superpositions with their own reflections at boundaries generate local
pressure peaks.
First and second standing wave harmonics from a pressure wave propagating
in a pipeline open at both ends are shown in Figure 3.17. Incident and reflected
waves from open boundaries have opposite phases (R = −1, Figure
3.5), thus maximum amplitude of standing waves is the same as the incident
wave amplitude. Boundaries are nodal points. Another way to express this is
that incident pressure waves cancel their own reflection at open boundaries.
Check valves and non-return valves are neither closed nor open boundary conditions.
Therefore, no standard standing wave patterns can illustrate pressure
wave superpositions. Nodal points obtained from valve characteristics are not
fixed neither. Instead, nodal points move and are limited by the fully open
and fully closed conditions.
39

Chapter 3. Modelling of Pressure Wave Propagation
(p − p 0 ) / p Wave
2
1
0
−1
−2
n = 1
n = 3
0 L/2
x / L
L
Figure 3.16: First and Third Harmonics for Pressure Standing Wave
in a Pipeline Closed at Both Ends
(p − p 0 ) / p Wave
1
0
−1
n = 1
n = 2
0 L/2
x / L
L
Figure 3.17: First and Second Harmonics for Pressure Standing
Wave in Pipeline Open at Both Ends
40

3.4 Two-Dimensional Modelling
3.4. Two-Dimensional Modelling
Pressure transient modelling generally focuses on pressure amplitudes and onedimensional
modelling is preferred because of pipeline lengths. However, twodimensional
pressure transient modelling can also be performed and focuses
on fluid velocity by establishing velocity profiles in pipeline cross sections.
Assuming axisymmetrical homogeneous flow in a circular pipeline, where the
velocity radial component is small compared to the longitudinal component,
and where the pressure is averaged in the cross-section, mass and momentum
conservation equations are, respectively (Nathan et al., 1988):
∂p
∂t
∂p
+ ux + ρa2∂ux
∂x ∂x
∂ux
∂t +ux
∂ux
∂x
∂p
= −1
p ∂x
= 0 (3.24)
∂z
−g
∂x +µ
�
∂2ux ∂r
�
1 ∂ux
+ − 2 r ∂r
1 ∂
r ∂r
�
ru ′
xu ′ �
r (3.25)
where, u ′ xu ′ r, is Reynolds tensor. Thus, turbulence modelling is required in
two-dimensional pressure wave modelling. However, the coupling between
pressure and fluid velocity is of particular interest in two-dimensional modelling
since fluid velocity profiles can be determined during rapid pressure
events. During a pressure transient event, the flow pulsates and the velocity
profile can be described by (Uchida, 1956; Yamanaka et al., 2002)
ur = u0
⎛
⎝1 − 2
i 3/2 Π
J1
�
i3/2 �
Π2r/d
�
i3/2 ⎞
� ⎠
Π
J0
−1 ⎛
⎝1 −
J0
�
i3/2 �
Π2r/d
�
i3/2 ⎞
� ⎠ e
Π
iωt (3.26)
where, i, is the imaginary number, J0 and J1, Bessel functions of first and
second order, respectively, and Π, dimensionless angular frequency expressed
by (Uchida, 1956):
�
ω
Π = d
(3.27)
µ
Piston back and forth movements in a fluid-filled pipeline induce pressure
waves and pulsating flows. A laminar pulsating flow generated by a piston
back and forth moving piston are illustrated in Figure 3.18 and Figure 3.19 for
laminar and turbulent flow, respectively. In both cases, flow velocity changes
are driven by outer boundary regions. The fluid flow increases first, then
reaches an extreme before inverting, reaching the opposite extreme and coming
back to rest again.
41
J0

Chapter 3. Modelling of Pressure Wave Propagation
2 r / d
2 r / d
1.0
0.5
0.0
−0.5
φ = 0
−1.0
−1 0 1
1.0
0.5
0.0
−0.5
φ = π
−1.0
−1 0
u / u
x 0
1
φ = π / 4
−1 0 1
φ = 5 π / 4
−1 0 1
u x / u 0
φ = π / 2
−1 0 1
φ = 3 π / 2
−1 0 1
u x / u 0
φ = 3 π / 4
−1 0 1
φ = 7 π /4
−1 0 1
u x / u 0
Figure 3.18: Two-Dimensional Laminar Velocity Profile Evolution in
a Fluid-Filled Pipeline at Zero Initial Velocity During
One Angular Pressure Wave period
2 r / d
2 r / d
1.0
0.5
0.0
−0.5
φ = 0
−1.0
−1 0 1
1.0
0.5
0.0
−0.5
φ = π
−1.0
−1 0
u / u
x 0
1
φ = π / 4
−1 0 1
φ = 5 π / 4
−1 0 1
u x / u 0
φ = π / 2
−1 0 1
φ = 3 π / 2
−1 0 1
u x / u 0
φ = 3 π / 4
−1 0 1
φ = 7 π /4
−1 0 1
u x / u 0
Figure 3.19: Two-Dimensional Turbulent Velocity Profile Evolution
in a Fluid-Filled Pipeline at Zero Initial Velocity During
One Angular Pressure Wave period
42

Chapter 4
Acoustic Velocity in Single
Phase Fluids
4.1 Basic Relationships
4.1.1 Definition
A wave is a disturbance that transports energy in a medium. A wave propagates,
transmits and reflects according to the elastic properties of the propagation
medium. In case of a pressure wave, the disturbance amplitude is
pressure, p, and the wave propagation speed is acoustic velocity, a, in the
propagation medium. A pressure wave that propagates through a three dimensional
homogeneous medium, at a single wave speed, and that conserves
energy, can be described by the linear wave equation (Lighthill, 1978):
∂2p ∂t2 = a2 ∂2p ∂x2 (4.1)
The pressure wave propagation process described by Eq. 4.1, conserves energy,
is independent of waveform or direction of propagation. Another way to
express this is that the wave propagation process is adiabatic and reversible;
that is isentropic. Thus, the acoustic velocity relates to pressure waves characterized
by small gradients such that entropy, S, is constant. The acoustic
velocity is then expressed by (Lighthill, 1978):
��∂p �
a =
∂ρ
S
(4.2)
where, a, is acoustic velocity, p, is pressure, ρ, is density, and subscript S, denotes
isentropic conditions. Entropy is a thermodynamic function that can not
43

Chapter 4. Acoustic Velocity in Single Phase Fluids
be measured and depends on heat and temperature. Therefore, thermal conditions
can substitute entropic conditions since state variables are dependent.
Such substitution requires the enthalpy and internal energy thermodynamic
equations and are expressed by the second law of thermodynamics (Chevray
and Mathieu, 1993):
dH = ΓdS + V dp (4.3)
dU = ΓdS − pdV (4.4)
where, H, is enthalpy, Γ, temperature, V , volume and, U, internal energy. For
gases, changes in enthalpy and internal energy are also equal to the product
of changes in temperatures and heat capacities at constant pressure, cp, and
constant volume, cv, respectively. For gases, Eq. 4.3 and Eq. 4.4 can be
reformulated (Chevray and Mathieu, 1993):
dH = cpdΓ (4.5)
dU = cvdΓ (4.6)
For gases, combining Eq. 4.3 and Eq. 4.5, and Eq. 4.4 and Eq. 4.6, for
an isothermal process (dΓ = 0) and for an isentropic process (dS = 0) give,
respectively:
−V
p
−V
p
dp
dV
dp
dV
= 1 (4.7)
= cp
cv
(4.8)
The ratio of heat capacities at constant pressure, cp, and constant volume, cv,
defines a gas adiabatic index, k. A gas adiabatic index is function of pressure,
temperature and fluid composition. Of particular interest, gas adiabatic
indexes relate gas isentropic and isothermal processes such that the acoustic
velocity expression, Eq. 4.2 can be reformulated for gas:
a =
�
k
� �
∂p
∂ρ Γ
(4.9)
The above equation, Eq. 4.9, expresses the acoustic velocity in gases only.
Equivalent of Eq. 4.3 and Eq. 4.4 for liquids can not be expressed simply.
Thus, relation between isothermal and isentropic processes leading to a simple
acoustic velocity expression for liquids can not be simply derived. Nonetheless,
a common acoustic velocity expression can be formulated for both gases and
liquids when relating the isentropic partial derivative of pressure with density
in Eq. 4.2 with fluid compressibility.
44

4.1.2 Fluid Compressibility
4.1. Basic Relationships
Fluid compressibility is a measure of relative changes in volume for changes in
pressure. Because density equals the ratio of mass and volume occupied, and
that a fluid mass does not change with pressure, fluid compressibility can also
be expressed as a measure of changes in density under pressure constraints.
Fluid compressibility, K, depends on fluid composition, pressure, temperature
and thermodynamic process associated, and is expressed by (Leighton, 1994):
K = 1
ρ
� �
∂ρ
∂p
(4.10)
Combining Eq. 4.2 and Eq. 4.10 for an isentropic process, a fluid acoustic
velocity is then defined by, (Lighthill, 1978):
a =
� 1
ρKS
(4.11)
For gases, semi-empirical equations of state provide explicit relationships between
pressure and density. Therefore changes in density with pressure as
expressed by the partial derivative of density with pressure can be evaluated
theoretically. In addition, isothermal and isentropic processes for gases are
related by the adiabatic index k. The acoustic velocity expression, Eq. 4.11
can then be reformulated for gases by (Lighthill, 1978):
a =
�
k
ρKΓ
(4.12)
For liquids, empirical correlations provide relationships between isothermal
and isentropic compressibilities, as well as tables of isothermal compressibilities.
For liquids, isothermal and isentropic compressibilities are related by
(Stallard et al., 1969):
KΓ = KS + Γα2 p
ρCp
(4.13)
where, Cp, represents specific heat capacity at constant pressure, and, αp,
liquid thermal expansion coefficient at constant pressure. Liquid isothermal
compressibility is commonly approximated by isentropic compressibility (Stallard
et al., 1969). For example, at 25 o C, water thermal expansion coefficient
and specific heat capacity are 69 10 −6 K −1 and 4.18 10 −3 J kg −1 K −1 , respectively
(Eisenberg and Kauzman, 1969). The thermal expansion correction is
45

Chapter 4. Acoustic Velocity in Single Phase Fluids
then 4.35 10 −12 Pa −1 whereas water isothermal compressibility is 435 10 −12
Pa −1 (McCain, 1990).
Thermal expansion correction term is negligible in Eq. 4.13 and uncertainties
have low impact on isothermal compressibilities calculations (Stallard et al.,
1969). Thus, acoustic velocity in liquids can be calculated using an expression
similar to gases with k = 1. Acoustic velocity in liquids is then expressed by
(Leighton, 1994):
�
1
a =
(4.14)
ρKΓ
Acoustic velocities in both gases and liquids are generally calculated using
Eq. 4.12 with k = 1 for liquids. Acoustic velocity then depends on density,
pressure and temperature. Isothermal compressibilities can be derived from
semi-empirical equations of state for gases, while empirical correlations are
necessary to evaluate liquid compressibilities.
4.2 Acoustic Velocities in Gases
4.2.1 Derivation
Acoustic velocity in gases, Eq. 4.12, depends on density, pressure and temperature.
Gases thermodynamical properties can be theoretically derived from
semi-empirical equations of state that relate state variables. Each equation of
state focuses on different characteristics and corresponds to different assumptions.
Commonly used equations of state for gases are the ideal gas law and
the real gas law. Peng-Robinson equation of state is also used even though
other equations of state are reported better for gas.
The ideal gas law correlates density, pressure and temperature for a perfect
gas that assumes infinitely small molecules without intermolecular forces. The
ideal gas law predicts physical and thermodynamical properties of gases close
to atmospheric pressure at moderate temperatures with 95 % accuracy. The
ideal gas law relates pressure and temperature with the volume occupied by n
molecules, using a universal gas constant, R. The ideal gas law is (Katz and
Lee, 1990):
pV = nRΓ (4.15)
The real gas law correlates density, pressure and temperature for a natural
gas that considers molecule sizes and intermolecular forces using a correction
factor, z. The z-factor is an empirical function of pressure, temperature and
gas composition. The real gas law predicts physical and thermodynamical
46

4.2. Acoustic Velocities in Gases
properties of gases at pressures greater than atmospheric pressure and a wider
range of temperature than the ideal gas law. The real gas law is (Katz and
Lee, 1990):
pV = znRΓ (4.16)
The ideal gas law and real gas law can be reformulated by substituting volume
by density. The ideal gas law can be obtained from the real gas law for z = 1,
so properties will be derived from the real gas law only. The number of moles,
n, is the ratio of a gas mass, m, and its molecular mass, M; and density is the
ratio of mass and volume. After substitution, the real gas law becomes:
p = zρ R
Γ (4.17)
M
The isothermal compressibility required partial derivative of density with pressure
at constant temperature can be obtained directly from Eq. 4.17. Both
density and z-factor are functions of pressure and temperature such that the
isothermal compressibility of a gas following the real gas law becomes:
KΓ = 1
� �
1 ∂z
− (4.18)
p z ∂p
Γ
The first term in Eq. 4.18 is the ideal gas isothermal compressibility since
the z-factor is constant and equals unity. The second term in Eq. 4.18 is
then a correction factor that accounts for the changes in molecule sizes and
molecular forces with pressure at constant temperature. Acoustic velocities in
natural gases can be approximated by neglecting the second term in Eq. 4.18.
However, the true acoustic velocity can be obtained from combining Eq. 4.12
and Eq. 4.18:
�
�
� k
a = � � � � � (4.19)
� 1 1 ∂z
ρ −
p z ∂p
Γ
The above acoustic velocity formula, Eq. 4.19, is to be compared with a
standard calculation provided by the American Gas Association (AGA-10,
2003). AGA is a trade organization that promotes the use of natural gas and
that provides standards for calculating natural gas properties such as z-factor
(AGA-8, 1992), or acoustic velocity (AGA-10, 2003). In the latter report,
acoustic velocity is derived from thermodynamical relations and is expressed
by (Savidge et al., 1988):
��cp �� �� � � �
RΓ ∂z
a =
z + ρ
(4.20)
cv M ∂ρ Γ
47

Chapter 4. Acoustic Velocity in Single Phase Fluids
Acoustic velocity expressions for natural gas, Eq. 4.19 and Eq. 4.20, are
different. The first expressions is directly deduced from the real gas equation
of state, whereas the second expression includes a high accuracy equation
of state for the z-factor. The present study relies on values computed with
the Peng-Robinson fluid package included in HYSYS, a process simulation
software. Both formulations are herein numerically compared.
The Peng-Robinson equation of state is a cubic, two-constant equation of
state, of the van der Walls type (Peng and Robinson, 1976). The Peng-
Robinson equation of state is commonly used in petroleum engineering for
its accurate prediction of vapor pressure, liquid densities and phase equilibria
for multi-component fluids. The Peng-Robinson equation of state is particularly
accurate near the critical point (Katz and Lee, 1990).
4.2.2 Example
A typical natural gas molar composition is given in Table 4.1 (Pedersen et al.,
1989). The corresponding phase envelope is shown in Figure 4.1. Two phases
existence envelope range from -155 o C to -20 o C, in temperature, and from 1
bara to 80 bara, in pressure. Gas density, adiabatic index and z-factor, are
obtained using the Peng-Robinson fluid package included in HYSYS. First
order partial derivatives of z-factor with pressure or density in Eq. 4.18, Eq.
4.19 and Eq. 4.20, are approximated by finite differences.
Evolution of natural gas density with pressure and temperature is shown in
Figure 4.2. The gas density increases with pressure and decreases with temperature.
At 25 o C, the gas density increases from 0.75 kg m −3 at 1 bara, to
42.72 kg m −3 at 50 bara, and to 95.92 kg m −3 at 100 bara. At 100 bara, the
density decreases from 95.92 kg m −3 at 25 o C, to 82.11 kg m −3 at 50 o C, and
to 65.30 kg m −3 at 100 o C.
Evolution of natural gas adiabatic index with pressure and temperature is
shown in Figure 4.3. The gas adiabatic index increases with pressure and
decreases with temperature. At 25 o C, the gas adiabatic index increases from
1.28 at 1 bara, to 1.49 at 50 bara, and to 1.77 at 100 bara. At 100 bara, the
gas adiabatic index decreases from 1.77 at 25 o C, to 1.59 at 50 o C, and to 1.42
at 100 o C.
Evolution of natural gas z-factor with pressure and temperature is shown
in Figure 4.4. The gas z-factor decreases with pressure and increases with
temperature. At 25 o C, the z-factor decreases from 1 at 1 bara, to 0.87 at 50
bara, and to 0.77 at 100 bara. At 100 bara, the z-factor increases from 0.77 at
25 o C, to 0.83 at 50 o C, and to 0.91 at 100 o C. The isothermal compressibility,
KΓ, can be calculated from the pressure and the gradient in z-factor.
48

4.2. Acoustic Velocities in Gases
Table 4.1: Typical Natural Gas Molar Composition (Pedersen et al.,
1989)
Pressure [bara]
100
80
60
40
20
Component Mole Fraction
N2 0.30
CO2 1.10
C1 90.00
C2 4.90
C3 1.90
C4 (i + n) 1.10
C5 (i + n) 0.40
C6 (i + n) 0.30
Critical Point
Bubble Point
Dew Point
0
−160 −140 −120 −100 −80 −60 −40 −20 0
Temperature [ o C]
Figure 4.1: Typical Natural Gas Phase Envelope
Composition given in Table 4.1
Gas Properties Calculated Using Peng-Ronbinson Equation of State
49

Chapter 4. Acoustic Velocity in Single Phase Fluids
Density [kg m −3 ]
160
140
120
100
80
60
40
20
Γ = 25 o C
Γ = 50 o C
Γ = 100 o C
0
0 50 100 150
Pressure [bara]
Figure 4.2: Evolution of Typical Natural Gas Density with Pressure
and Temperature
Adiabatic Index
2.0
1.8
1.6
1.4
1.2
Γ = 25 o C
Γ = 50 o C
Γ = 100 o C
1.0
0 50 100 150
Pressure [bara]
Figure 4.3: Evolution of Typical Natural Gas Adiabatic Index with
Pressure and Temperature
Composition given in Table 4.1
Gas Properties Calculated Using Peng-Ronbinson Equation of State
50

4.2. Acoustic Velocities in Gases
Evolution of natural gas isothermal compressibility with pressure and temperature
is shown in Figure 4.5. The gas isothermal compressibility decreases
with pressure and temperature. At 25 o C, the gas compressibility decreases
from 10 10 −6 Pa −1 , to 0.227 10 −6 Pa −1 at 50 bara, and to 0.117 10 −6 Pa −1
at 100 bara. At 100 bara, the isothermal compressibility goes from 0.117 10 −6
Pa −1 at 25 o C, to 0.111 10 −6 Pa −1 at 50 o C, and to 0.105 10 −6 Pa −1 at 100 o C.
The greater the pressure, the stiffer the gas becomes.
Evolution of natural gas acoustic velocity with pressure and temperature, computed
from Eq. 4.19 and Eq. 4.20 are shown in Figure 4.6. In both cases,
the acoustic velocity increases with temperature, and decreases with pressure
down to a minimum, before increasing again. At 25 o C, the acoustic velocity
goes from 414 ms −1 at 1 bara, to 392 ms −1 at 50 bara, and to 397 ms −1 at
100 bara. At 100 bara, the acoustic velocity increases from 397 ms −1 at 25 o C
to 418 ms −1 at 50 o C, and to 455 ms −1 at 100 o C.
Acoustic velocities calculated with Eq. 4.19 and reference Eq. 4.20 are almost
identical. However, results deviate slightly from each other, as observed in
Figure 4.7. Deviations decrease with temperature, and increase with pressure
up to a maximum before decreasing again. A maximum deviation of 0.1 % is
obtained at 25 o C, 70 bara which is reasonable for most of the cases, except
for ultrasonic measurements.
51

Chapter 4. Acoustic Velocity in Single Phase Fluids
z − Factor
1.00
0.95
0.90
0.85
0.80
0.75
Γ = 100 o C
Γ = 50 o C
Γ = 25
0.70
0 50 100 150
Pressure [bara]
o C
Figure 4.4: Evolution of Typical Natural Gas Z-Factor with Pressure
and Temperature
Compressibility [10 −6 Pa −1 ]
12
10
8
6
4
2
Γ = 25 o C
Γ = 50 o C
Γ = 100 o C
0
0 50 100 150
Pressure [bara]
Figure 4.5: Evolution of Typical Natural Gas Isothermal
Compressibility with Pressure and Temperature
Composition given in Table 4.1
Gas Properties Calculated Using Peng-Ronbinson Equation of State
52

Acoustic Velocity [m s −1 ]
475
450
425
400
4.2. Acoustic Velocities in Gases
Γ = 100 o C
Γ = 50 o C
Γ = 25
375
0 50 100 150
Pressure [bara]
o C
Definition
AGA 10
Figure 4.6: Evolution of Typical Natural Gas Acoustic Velocity with
Pressure and Temperature
Error [%]
0.10
0.08
0.06
0.04
0.02
Γ = 25 o C
Γ = 50 o C
Γ = 100 o C
0.00
0 50 100 150
Pressure [bara]
Figure 4.7: Deviation Between Eq. 4.19 and Eq. 4.20
Composition given in Table 4.1
Gas Properties Calculated Using Peng-Ronbinson Equation of State
53

Chapter 4. Acoustic Velocity in Single Phase Fluids
4.3 Acoustic Velocity in Liquid Oils
4.3.1 Acoustic Velocity in Volatile Oil
A typical volatile oil composition molar composition is given in Table 4.2 and
the corresponding phase envelope is shown in Figure 4.8. Two phases existence
envelope range from -150 o C to 250 o C, in temperature, and from 0 bara to 250
bara, in pressure. Volatile oil density is obtained using the Peng-Robinson
fluid package included in HYSYS. Isothermal compressibility is approximated
by finite difference (Eq. 4.10) and neglecting thermal expansion (Eq. 4.13).
Evolution of volatile oil density with pressure and temperature is shown in
Figure 4.9. The density increases with pressure and decreases with temperature.
At 25 o C, the density increases from 721 kg m −3 , to 727 kg m −3 , and
to 732 kg m −3 at 1 bara, 50 bara and 100 bara, respectively. At 100 bara,
the density decreases from 732 kg m −3 , to 720 kg m −3 , and to 695 kg m −3 at
25 o C, 50 o C and 100 o C, respectively.
Evolution of volatile oil isothermal compressibility with pressure and temperature
is shown in Figure 4.10. The isothermal compressibility decreases
with pressure and increases with temperature. At 25 o C, the isothermal compressibility
decreases from 1.67 10 −9 Pa −1 , to 1.56 10 −9 Pa −1 , and to 1.46
10 −9 Pa −1 at 1 bara, 50 bara and 100 bara, respectively. At 100 bara, the
isothermal compressibility increases from 1.46 10 −9 Pa −1 , to 1.60 10 −9 Pa −1 ,
and to 1.98 10 −9 Pa −1 at 25 o C, 50 o C and 100 o C, respectively.
Evolution of acoustic velocity with pressure and temperature, in volatile oil,
calculated with Eq. 4.14, is shown in Figure 4.11. The acoustic velocity
increases with pressure and decreases with temperature. At 25 o C, the acoustic
velocity increases from 911 ms −1 , to 940 ms −1 , and to 968 ms −1 at 1 bara, 50
bara and 100 bara, respectively. At 100 bara, the acoustic velocity decreases
from 968 ms −1 , to 930 ms −1 , and to 853 ms −1 at 25 o C, 50 o C and 100 o C,
respectively.
54

4.3. Acoustic Velocity in Liquid Oils
Table 4.2: Typical Volatile Oil Molar Composition, Pedersen et al.
(1989)
Component Mole Fraction Component Mole Fraction
N2 1.67 CO2 2.18
C1 60.51 C7 2.45
C2 7.52 C8 2.41
C3 4.74 C9 1.69
C4 (i + n) 4.12 C10 1.42
C5 (i + n) 2.97 C11 1.02
C6 (i + n) 1.99 C12 (12+) 5.31
Pressure [bara]
300
250
200
150
100
50
Critical point
Bubble Point
Dew Point
0
−200 −100 0 100
Temperature [
200 300 400
o C]
Figure 4.8: Typical Volatile Oil Phase Envelope
Composition given in Table 4.2 from Pedersen et al. (1989)
55

Chapter 4. Acoustic Velocity in Single Phase Fluids
Density [kg m −3 ]
760
740
720
700
680
Γ = 25 o C
Γ = 50 o C
660
0 20 40 60 80
Γ = 100
100
Pressure [bara]
o C
Figure 4.9: Evolution of Typical Volatile Oil Density with Pressure
and Temperature
Compressibility [10 −9 Pa −1 ]
2.40
2.20
2.00
1.80
1.60
1.40
1.20
Γ = 100 o C
Γ = 50 o C
1.00
0
Γ = 25
20 40 60 80 100
Pressure [bara]
o C
Figure 4.10: Evolution of Typical Volatile Oil Isothermal
Compressibility with Pressure and Temperature
Composition given in Table 4.2 from Pedersen et al. (1989)
Volatile Oil Properties Calculated Using Peng-Robinson Equation of State
56

Acoustic Velocity [m s −1 ]
1150
1100
1050
1000
950
900
850
800
Γ = 25 o C
Γ = 50 o C
Γ = 100 o C
4.3. Acoustic Velocity in Liquid Oils
750
0 20 40 60 80 100
Pressure [bara]
Figure 4.11: Evolution of Acoustic Velocity in Typical Volatile Oil
with Pressure and Temperature
Composition given in Table 4.2 from Pedersen et al. (1989)
Volatile Oil Properties Calculated Using Peng-Robinson Equation of State
57

Chapter 4. Acoustic Velocity in Single Phase Fluids
4.3.2 Acoustic Velocity in Black Oil
A typical black oil composition molar composition is given in Table 4.3 and the
corresponding phase envelope is shown in Figure 4.12. Two phases existence
envelope range from -150 o C to 400 o C, in temperature, and from 0 bara to 170
bara, in pressure. Volatile oil density is obtained using the Peng-Robinson
fluid package included in HYSYS. First order partial derivatives in Eq. 4.10
is approximated by finite difference and thermal expansion in Eq. 4.13 is
neglected.
Evolution of black oil density with pressure and temperature is shown in Figure
4.13. The density increases with pressure and decreases with temperature. At
25 o C, the density increases from 747 kg m −3 at 1 bara, to 751 kg m −3 at 50
bara, and to 756 kg m −3 at 100 bara. At 100 bara, the density decreases from
756 kg m −3 at 25 o C, to 742 kg m −3 at 50 bara, and to 717 kg m −3 at 100 o C.
Evolution of black oil isothermal compressibility with pressure and temperature
is shown in Figure 4.14. The isothermal compressibility decreases with
pressure and increases with temperature. At 25 o C, the isothermal compressibility
decreases from 1.35 10 −9 Pa −1 at 1 bara, to 1.27 10 −9 Pa −1 at 50 bara,
and to 1.19 10 −9 Pa −1 at 100 bara. At 100 bara, the isothermal compressibility
increases from 1.19 10 −9 Pa −1 at 25 o C, to 1.28 10 −9 Pa −1 at 50 o C, and to
1.46 10 −9 Pa −1 at 100 o C.
Evolution of acoustic velocity with pressure and temperature, in black oil,
calculated with Eq. 4.14, is shown in Figure 4.15. The acoustic velocity
increases with pressure and decreases with temperature. At 25 o C, the acoustic
velocity increases from 994 ms −1 at 1 bara, to 1024 ms −1 at 50 bara, and
to 1053 ms −1 at 100 bara. At 100 bara, the acoustic velocity decreases from
1053 ms −1 at 25 o C, to 1027 ms −1 at 50 o C, and to 977 ms −1 at 100 o C.
58

4.3. Acoustic Velocity in Liquid Oils
Table 4.3: Typical Black Oil Molar Composition (Pedersen et al.,
1989)
Component Mole Fraction Component Mole Fraction
N2 0.67 CO2 2.11
C1 34.93 C11 1.72
C2 7.00 C12 1.74
C3 7.82 C13 1.74
C4 (i + n) 5.48 C14 1.35
C5 (i + n) 3.80 C15 1.34
C6 (i + n) 3.04 C16 1.06
C7 4.39 C17 1.02
C8 4.71 C18 1.00
C9 3.21 C19 0.90
C10 1.79 C20 (20+) 9.18
Pressure [bara]
300
250
200
150
100
50
Critical Point
Bubble Point
Dew Point
0
−200 −100 0 100
Temperature [
200 300 400
o C]
Figure 4.12: Typical Black Oil Phase Envelope
Composition given in Table 4.3 from Pedersen et al. (1989)
59

Chapter 4. Acoustic Velocity in Single Phase Fluids
Density [kg m −3 ]
760
740
720
700
680
Γ = 25 o C
Γ = 50 o C
660
0 20 40 60 80
Γ = 100
100
Pressure [bara]
o C
Figure 4.13: Evolution of Typical Black Oil Density with Pressure
and Temperature
Compressibility [10 −9 Pa −1 ]
2.40
2.20
2.00
1.80
1.60
1.40
1.20
Γ = 100 o C
Γ = 50 o C
Γ = 25 o C
1.00
0 20 40 60 80 100
Pressure [bara]
Figure 4.14: Evolution of Typical Black Oil Isothermal
Compressibility with Pressure and Temperature
Composition given in Table 4.3 from Pedersen et al. (1989)
Oil Properties Calculated Using Peng-Ronbinson Equation of State
60

Acoustic Velocity [m s −1 ]
1150
1100
1050
1000
950
900
850
800
4.3. Acoustic Velocity in Liquid Oils
Γ = 25 o C
Γ = 50 o C
750
0 20 40 60 80
Γ = 100
100
Pressure [bara]
o C
Figure 4.15: Evolution of Acoustic Velocity in Typical Black Oil with
Pressure and Temperature
Composition given in Table 4.3 from Pedersen et al. (1989)
Oil Properties Calculated Using Peng-Ronbinson Equation of State
61

Chapter 4. Acoustic Velocity in Single Phase Fluids
4.3.3 Comparison Between Volatile Oils and Black Oils
Volatile oils contain more light molecules (C1 to C6) than black oils as observed
from the molar compositions, Tables 4.2 and 4.3. Therefore, black oils are
denser and less compressible than volatile oils. At 50 o C, 50 bara, black oil
and volatile oil densities are 738 kg m −3 and 714 kg m −3 , respectively (Figure
4.9 and Figure 4.13). At 50 o C, 50 bara, black oil and volatile oil isothermal
compressibilities are 1.36 10 −9 Pa −1 and 1.73 10 −9 Pa −1 , respectively (Figure
4.10 and Figure 4.14).
Black oils greater densities and compressibilities induce greater acoustic velocities
than in volatile oils. At 50 o C, 50 bara, acoustic velocities in black and
volatile oils are 998 ms −1 and 900 ms −1 , respectively (Figures 4.11 and 4.15).
Currently, no theory can predict accurate acoustic velocities in oils, mainly due
to the diversity of hydrocarbons. However, empirical correlations can predict
acoustic velocities function of molecular weight within 3 % accuracy (Wang
et al., 1990).
Critical temperatures are lower for volatile oils than for black oils. The studied
volatile oil and black oil critical temperatures are 170 o C and 356 o C, respectively
(Figure 4.8 and Figure 4.12). Nevertheless, critical pressures are greater
for volatile oils than for black oils. The studied volatile oil and black oil critical
pressures are 170 bara and 107 bara, respectively (Figure 4.8 and Figure
4.12). Volatile oil critical pressures are generally close to reservoir conditions.
4.4 Acoustic Velocity in Liquid Water
4.4.1 Acoustic Velocity
Unlike liquid oils, water is a single molecule fluid. Thus, thorough experimental
work results are available through steam tables for instance. Density
and acoustic velocity are two parameters extensively investigated. Accurate
empirical correlations for the acoustic velocity in natural water relate wave
propagation speeds to pressure, p, temperature, Γ, and salinity, Σ. One accurate
correlation has been proposed by DelGrosso (1974):
a(Σ, Γ, p) = a0,0,0 + ∆aΣ + ∆aΓ + ∆ap + ∆aΣΓp
(4.21)
where, a0,0,0, is the acoustic velocity reference value at zero salinity, Σ = 0 ppm,
zero temperature, Γ = 0 o C and atmospheric pressure. Functions ∆aΣ, ∆aΓ,
∆ap and ∆aΣΓp, are correction operators for salinity, temperature, pressure
and interaction between these three parameters, respectively. Correction functions
are (DelGrosso, 1974):
62

∆aΣ = 0.132952290781 10 1 Σ
+ 0.128955756844 10 −3 Σ 2
∆aΓ = 0.501109398873 10 1 (Γ − 273.15)
− 0.550946843172 10 −1 (Γ − 273.15) 2
+ 0.221535969240 10 −3 (Γ − 273.15) 3
∆ap = 0.156059257041 10 0 Ψp
+ 0.244998688441 10 −4 (Ψp) 2
− 0.883392332513 10 −8 (Ψp) 3
∆aΣΓp = −0.127562783426 10 −1 (Γ − 273.15)Σ
+ 0.635191613389 10 −2 (Γ − 273.15)Ψp
+ 0.265484716608 10 −7 ((Γ − 273.15)Ψp) 2
− 0.159349479045 10 −5 (Γ − 273.15)(Ψp) 2
+ 0.522116437235 10 −9 (Γ − 273.15)(Ψp) 3
− 0.438031096213 10 −6 (Γ − 273.15) 3 Ψp
− 0.161674495909 10 −8 (ΣΨp) 2
+ 0.968403156410 10 −4 (Γ − 273.15) 2 Σ
+ 0.485639620015 10 −5 (Γ − 273.15)ΣΨp
4.4. Acoustic Velocity in Liquid Water
(4.22)
(4.23)
(4.24)
− 0.340597039004 10 −3 (Γ − 273.15)ΣΨp (4.25)
where, Ψ, is constant equal to 1.019716213 used for converting pressures from
bara to kg cm −2 . Evolution of acoustic velocity with pressure and temperature,
in water at zero salinity, is shown in Figure 4.16. The acoustic velocity
increases with pressure and temperature. At 1 bara, the acoustic velocity increases
from 1426 ms −1 at 5 o C to 1497 ms −1 at 25 o C, and to1543 ms −1 at
50 o C. At 25 o C, the acoustic velocity increases from 1497 ms −1 to 1509 ms −1
at 40 bara, and to 1534 ms −1 at 120 bara.
63

Chapter 4. Acoustic Velocity in Single Phase Fluids
Acoustic Velocity [m s −1 ]
1600
1575
1550
1525
1500
1475
1450
1425
1400
0 10 20 30 40 50
Temperature [ o p = 120 bara
p = 80 bara
p = 40 bara
p = 20 bara
p = 10 bara
p = 1 bara
C]
Figure 4.16: Evolution of Acoustic Velocity in Water with Pressure
and Temperature at Zero Salinity (DelGrosso, 1974)
4.4.2 Density of Water
Water density is thoroughly tabulated function of pressure, temperature and
salinity. Accurate knowledge of water density is of particular interest for predicting
geothermal system performances, for instance. Based on steam tables,
empirical correlations define density as a function of temperature and molality
of salt in pure water (Michaelides, 1981). Density prediction involves calculations
of enthalpy, temperature elevation and specific volume. Calculations
and results are detailed in Michaelides (1981).
Evolution of water density with pressure and temperature, using Michaelides
correlation and using the ASME Steam Fluid package included in HYSYS,
is shown in Figure 4.17. Density decreases with temperature and increases
with pressure. At 1 bara, the density calculated using Michaelides correlation
decreases from 999.7 ms −1 at 5 o C, to 996.9 ms −1 at 25 o C, and to 989.8
ms −1 at 50 o C. At 25 o C, the density calculated using Michaelides correlation
increases from 997.3 ms −1 at 1 bara, to 998.6 ms −1 at 40 bara, and to 1003
ms −1 at 120 bara.
Densities predicted by Michaelides correlation and by the ASME Steam fluid
package included in HYSYS are similar at low pressures and low temperatures,
Figure 4.17. However, the maximum discrepancy is less than 0.10 % and is
64

Density [kg m −3 ]
1006
1004
1002
1000
998
996
4.4. Acoustic Velocity in Liquid Water
994
992
990
988
986
0 10 20 30 40 50
Temperature [ o p = 120 bara
p = 80 bara
p = 40 bara
p = 20 bara
p = 10 bara
p = 1 bara
HYSYS
Michaelides
C]
Figure 4.17: Evolution of Water Density with Pressure and
Temperature at Zero Salinity (Michaelides, 1981)
therefore acceptable since empirical correlations rely on experimental data
with inaccuracy greater than 0.1 %.
4.4.3 Isothermal Compressibility
Evolution of water isothermal compressibility with pressure and temperature
can be deduced from acoustic velocity and water density evolution with pressure
and temperature using Eq. 4.14.
Evolution of water isothermal compressibility with pressure and temperature
is shown in Figure 4.18. Water isothermal compressibility decreases with
temperature and pressure. At 1 bara, the isothermal compressibility decreases
from 0.492 10 −9 Pa −1 at 5 o C, to 0.448 10 −9 Pa −1 at 25 o C, and to
0.425 10 −9 Pa −1 at 50 o C. At 25 o C, the isothermal compressibility decreases
from 0.425 10 −9 Pa −1 at 1 bara, to 0.440 10 −9 Pa −1 at 40 bara, and to 0.424
10 −9 Pa −1 at 120 bara. A value of 0.435 10 −9 Pa −1 (3 10 −6 psi −1 ) is a commonly
admitted value for water compressibility (Dake, 1978).
The water isothermal compressibility calcaulted above at atmospheric pressure
can be compared to empirical correlations (Eisenberg and Kauzman, 1969).
The comparison of the extrapolated isothermal compressibility from acoustic
velocity and density and empirically correlated isothermal compressibility
65

Chapter 4. Acoustic Velocity in Single Phase Fluids
Compressibility [10 −9 Pa −1 ]
0.52
0.50
0.48
0.46
0.44
0.42
0.40
p = 1 bara
p = 10 bara
p = 20 bara
p = 40 bara
p = 80 bara
p = 120 bara
0.38
0 10 20 30 40 50
Temperature [ o C]
Figure 4.18: Evolution of Water Isothermal Compressibility with
Pressure and Temperature at Zero Salinity
is shown in Figure 4.19. Deduced and empirical isothermal compressibilities
coincide at low temperature but diverge at high temperature. At 50 o C, calculated
isothermal compressibility is 0.425 10 −9 Pa −1 and empirically correlated
is 0.442 10 −9 Pa −1 .
4.5 Acoustic Velocity in Fluid-Filled Pipe
Pressure wave propagation velocity in fluid-filled pipeline depends on fluid
elastic properties, as well as pipeline geometry and spatial description. Pressure
waves used in seismic exploration are generated at sea-surface, propagate
through sea and Earth. In such case, the medium is described as infinite and
acoustic velocity calculations follow aforementioned formulas. However, pressure
waves in petroleum engineering propagate in pipelines. Acoustic velocity
in liquid-filled pipeline can be calculated using (Wylie and Streeter, 1993):
�
�
� 1
a = � �
�
ρ KΓ + 1
� (4.26)
∆A
A ∆p
where, A, is pipeline cross-sectional area, ∆A, is change in cross-sectional
area for change in pressure, ∆p. Changes in cross-sectional area for a given
66

Compressibility [10 −9 Pa −1 ]
0.515
0.490
0.465
0.440
4.5. Acoustic Velocity in Fluid-Filled Pipe
ASME Steam Tables
Eisenberg Empirical Correlation
0.415
0 10 20 30 40 50
Temperature [ o C]
Figure 4.19: Water Isothermal Compressibility extrapolated from
Acoustic Velocity and Density Correlations (DelGrosso,
1974; Michaelides, 1981) and empirically correlated
(Eisenberg and Kauzman, 1969)
pressure gradient included in Eq. 4.26 mirrors Eq. 4.10 and defines the inverse
of a pipeline compressibility. The greater changes in pressure, the greater a
pipeline cross-sectional deformation.
Changes in cross-sectional area depend on a pipeline material, geometry and
anchoring. A pipeline can be thin or thick-walled, circular or non-circular
and can be anchored at one side only, or at both sides. In the latter case,
the pipeline can be restricted in axial movement or include expansion joints,
Figure 4.20. The acoustic velocity for a circular pipeline, when considering
wall thickness and anchoring can be expressed in a general form by (Wylie
and Streeter, 1993):
�
�
� 1
a = � �
�
ρ KΓ + d
� (4.27)
c1
e Y
where, d, is diameter, e, wall-thickness, Y , pipeline material Young’s modulus,
and, c1, structure coefficient that depends on the pipeline fastenings. Structure
coefficients are presented in Table 4.4.
67

Chapter 4. Acoustic Velocity in Single Phase Fluids
3
2
1
Figure 4.20: Illustration of Structure Fastenings (Wylie and Streeter,
1993)
Type 1: Pipe Anchored at One End Only.
Type 2: Pipe Anchored at Both Ends.
Type 3: Pipe Anchored at Both Ends with Expansion Joints.
Table 4.4: Structure Coefficient for Thin-Walled and Thick-Walled
Pipelines, Single-Sided Anchor or Double-Sided Anchor.
Structure Fastenings Description in Figure 4.20
Structure Thin-Walled Thick-Walled
Fastening Pipeline Pipeline
Type c1 c1 c2
1 1 − µ
2
2 1 − µ2
c2 + d
d + e
�
1 − µ
2
c2 + D � 1 − µ 2�
d + e
3 1 c2 + d
d + e
68
�
2e
d
(1 + µ)

4.5. Acoustic Velocity in Fluid-Filled Pipe
Pipeline are typically thin-walled, anchored at both ends and include expansion
joints. Acoustic velocity calculations then follow Korteweg’s formulation
(Tijsseling and Anderson, 2007)
�
�
� 1
a = � �
�
ρ KΓ + d
� (4.28)
Y e
Evolution of acoustic velocity with pipeline compressibility for three different
fluids and two different pipeline materials is shown in Figure 4.21. The three
considered fluids are water, a typical oil and a typical gas. Studied fluid densities
and isothermal compressibilities are listed in Table 4.6 (McCain, 1990).
Listed density and compressibility values are in accordance with previously
calculated values. The two considered pipeline materials are steel and PVC.
Studied materials Young’s moduli and Poisson’s ratio are listed in Table 4.5
(Cheremisinoff, 1996).
Acoustic velocity calculations using Eq. 4.28 are for liquids only. However,
the same equation can be used for gas when the z-factor value is neglected.
Another way to express this is that Eq. 4.28 can be used for gas when pipeline
compressibility effect is much greater than the z-factor effect on the acoustic
velocity calculation. In addition, calculated acoustic velocities are used only
for representation of pipeline compressibility effect.
The acoustic velocity increases with a pipeline wall thickness and internal
diameter ratio, and with Young’s modulus, as shown in Figure 4.21. For a
water-filled thin PVC pipeline, the acoustic velocity increases from 358 ms −1
at 5 % e/d ratio, to 919 ms −1 at 50 % e/d ratio, and to 1107 ms −1 at 100
% e/d ratio. The acoustic velocity increases with a pipeline material Young’s
modulus as shown in Figure 4.21. For a 50 % e/d ratio, the acoustic velocity
increases from 919 ms −1 for a water-filled thin PVC pipeline, to 1477 ms −1
for a water-filled thin steel pipeline.
Acoustic velocities for thin-walled and thick-walled steel pipelines are similar
for all fluids, but are distinct for PVC pipelines, as shown in Figure 4.21. For
a 50 % e/d ratio and an oil-filled PVC pipeline, calculated acoustic velocities
are 616 ms −1 using the thin-walled formula, Eq. 4.28, and 699 ms −1 using
the thick-walled formula, Eq. 4.27 and Table 4.4. For a 50 % e/d ratio
and a water-filled PVC pipeline, calculated acoustic velocities are 711 ms −1
using the thin-walled formula, Eq. 4.28, and 919 ms −1 using the thick-walled
formula, Eq. 4.27 and Table 4.4.
69

Chapter 4. Acoustic Velocity in Single Phase Fluids
Acoustic Velocity [m s −1 ]
1500
1250
1000
750
500
Water in Steel
Oil in Steel
Gas in Steel
Water in PVC
250
Oil in PVC
Gas in PVC
Thin−Walled Pipeline
Thick−Walled Pipeline
0
0 25 50 75 100
Wall Thickness / Diameter [%]
Figure 4.21: Acoustic Velocity in Fluid-Filled Pipelines (Wylie and
Streeter, 1993)
Table 4.5: Typical Material Properties: Young’s Modulus and
Poisson’s Ratio (Cheremisinoff, 1996)
Material Young’s Modulus [Pa] Poisson’s Ratio
Steel 200 10 9 0.30
PVC 2.8 10 9 0.41
Table 4.6: Typical Fluid Densities and Compressibilities in
Petroleum Engineering (McCain, 1990)
Fluid Density [kg m −3 ] Compressibility [Pa −1 ]
Water 1030 0.435 10 −9
Oil 760 1.98 10 −9
Gas 60 188 10 −9
70

4.6. Acoustic Velocity Determination
Acoustic velocity values are determined either by fluid compressibility or by
pipeline compressibility. The lowest compressibility determines the acoustic
velocity. Water compressibility is 100 times greater than a steel pipeline compressibility.
Hence, acoustic velocity in water-filled pipeline is limited by the
pipeline compressibility. On the other hand, gas compressibility are commonly
100 times lower than a PVC pipeline compressibility. Thus, acoustic velocity
in gas-filled pipeline is limited by the gas compressibility. Oil and steel compressibilities
are of the same order of magnitude, therefore acoustic velocity
in oil-filled pipeline is greatly dependent on the ration e/d.
Fluid transport can also take place in non-circular pipelines. Such pipelines are
not usual in petroleum engineering, however the acoustic velocity formulation
for non-circular pipelines is given for completeness. For a non-circular pipeline,
changes in cross-sectional area with pressure depend on the number of sides,
N, making the polygonal shape. Changes in cross-sectional area to be included
in the acoustic velocity formulation, Eq. 4.26, is (Thorley, 1991):
1 ∆A d
=
A ∆p Y e
⎛
⎜
tan
⎜
⎝1 +
4
� �
180
N
15
� d
e
� 2
⎞
⎟
⎠
(4.29)
Acoustic velocities for circular, square, hexagonal and octagonal water-filled
steel pipelines, are shown in Figure 4.22. The acoustic velocity increases with
the number of sides making the polygonal cross-section. For a 50 % d/e
ratio, the acoustic velocity increases from 151 ms −1 for a square pipe, to 427
ms −1 for an hexagonal pipe, and to 713 ms −1 for an octagonal pipeline. The
acoustic velocity in a water- filled steel circular pipeline is 1191 ms −1 for a 50
% e/d ratio.
4.6 Acoustic Velocity Determination
The acoustic velocity in a fluid-filled pipeline depends on fluid density and
isothermal compressibility, and on pipeline geometry and material. Acoustic
velocity in a fluid-filled pipeline can be calculated using Eq. 4.28 or Eq. 4.19,
for instance. But the acoustic velocity can also be measured experimentally
from pressure data recorded by two transducers set apart along a pipeline
during a pressure wave propagation experiment.
Previously defined acoustic velocities are phase velocities while measured acoustic
velocities are group velocities. Phase velocity refers to a theoretical single
frequency wave, whereas experimental pressure waves consist of a broad range
71

Chapter 4. Acoustic Velocity in Single Phase Fluids
Acoustic Velocity [m s −1 ]
1500
1250
1000
750
500
250
Circular Pipe
4 − Sides
6 − Sides
8 − Sides
0
0 25 50 75 100
Diameter / Wall Thickness [%]
Figure 4.22: Acoustic Velocity in Fluid-Filled Non-Circular Pipelines
of frequencies that define a group of single frequency waves. Frequency dependent
acoustic velocities can however be determined from group velocity and
are expressed by (Tijsseling and Anderson, 2007):
�
� ⎛ �
�
�
a(ω) = a�2
⎝1 + 1 +
� 32ν
dω
� 2
⎞
⎠
−1
(4.30)
where, ω, represents circular frequency, ν, kinematic viscosity, a(ω), single
frequency phase velocity, and, a, measured group velocity. The measured
group velocity must be comparable with the calculated value using Eq. 4.28.
When fluid compressibilities are of the same order of magnitude as pipeline
compressibilities, then the thick-walled pipeline formulation should be used,
Table 4.4.
72

Chapter 5
Numerical Simulation
Methods
5.1 Modelling Equations
Analysis of pressure wave propagation in fluid-filled pipelines is performed
using the fundamental conservation equations. Heat transfer is commonly neglected
in pressure transient analysis. Thus the energy conservation equation
is excluded from the system of equations to be solved. One-dimensional mass
and momentum equations for viscous fluids are then, respectively (Chevray
and Mathieu, 1993):
∂ρ ∂
+ (ρu) = 0 (5.1)
∂t ∂x
∂ ∂
(ρu) +
∂t ∂x (ρu2 ) = −∂p ∂
+
∂x ∂x
�
4
3 µ∂u
�
+
∂x
4τω
πd
(5.2)
where, p, represents pressure, u, fluid velocity, ρ, fluid density, µ, fluid dynamic
viscosity, and, τω, wall shear stress. Wall shear stress includes frictional pressure
drop and is conventionally expressed by the Darcy-Weisbach formulation
(Ghidaoui et al., 2005). Simplifications of the above system of two equations,
Eq. 5.1 and Eq. 5.2 are obtained from order-of-magnitude analysis, neglecting
viscosity and restating the system in terms of pressure and fluid velocity
variables. The system of equations then becomes (Wylie and Streeter, 1993):
1
ρa2 ∂p ∂u
+ = 0 (5.3)
∂t ∂x
∂u
∂t
1 ∂p
+
ρ ∂x
−1 f
=
2 d u2 − gsin(θ) (5.4)
73

Chapter 5. Numerical Simulation Methods
where, f, represents the Darcy-Weisbach friction factor, d, pipeline diameter,
a, pressure wave propagation velocity, and, θ, pipeline inclination from the
horizontal. The wave propagation velocity can be expressed by (Lighthill,
1978):
�
∂p
a =
(5.5)
∂ρ
The above system of two first-order partial differential equations, Eq. 5.3 and
Eq. 5.4 can be restated in a conservative form and in a matrix form. Eq. 5.3
and Eq. 5.4 in conservative form become (Laney, 1998):
∂Ω
∂t
+ ∂Φ(Ω)
∂x
= Λ (5.6)
consisting of the vector variable, Ω, the flux vector, Φ, and the source term
vector, Λ:
� � � � � �
p ρa2u 0
Ω = Φ(Ω) = Λ =
(5.7)
u
p s
Matrix form of Eq. 5.3 and Eq. 5.4 is expressed by (Laney, 1998):
I ∂Ω
∂t
− B∂Ω
∂x
= Λ (5.8)
where, Ω and Λ are vectors defined in Eq. 5.7, I, represents the unity matrix,
and B is a quadratic matrix defined by:
� �
0 ρa2 B =
(5.9)
1 0
Explicit analytical solutions of the system of two first-order partial differential
equations, Eq. 5.3 and Eq. 5.4, or Eq. 5.6, or Eq. 5.8, are rarely possible.
However, simple assumptions can lead to simplifications of the two equations
to be solved, and first approximations can be postulated. Nevertheless, several
numerical methods can be used to perform more complete transient analysis.
74

5.2 Analytical Solutions
5.2. Analytical Solutions
Rigid water column is an analytical method for calculating pressure values and
transient times in simplified water-hammer problems. Simplifications assume
fully compressible fluid, fully rigid pipeline walls and a long transient generation.
The transient generation is relative to a pressure wave travel period
in a fluid-filled pipeline. Pressure transients are rapid if the generation time
is much less than the travel period. Pressure transients are considered long
otherwise. The travel period is defined by (Wylie and Streeter, 1993):
T = 2L
a
(5.10)
where, L, represents pipeline length. A schematic representation of a pipeline
system typically used in pressure transient analysis is shown in Figure 5.1.
Figure 5.1: Typical Pipeline System in Pressure Transient Analysis
Schematic Representation
The system is composed of a pipeline connected to a tank at one end, and to
a quick-acting valve at the other end. The valve closure generates a pressure
transient that propagates back and forth in the fluid-filled pipeline. Using rigid
water column approximations, the time to reach a new steady-state condition
after transient initiation, and the steady state pressure are (Thorley, 1991):
t = −ρLu
∆p
pu = pd + 1 L
ρf
2 d u20 L
d
(5.11)
(5.12)
where, pu is the upstream pressure and pd, the downstream pressure. Information
provided by the rigid water column approximations are insufficient in
conventional pressure transient analysis since multiple transmissions and reflections
can take place along the fluid-filled pipeline. However, the analytical
solution is practical for a first estimation.
75

Chapter 5. Numerical Simulation Methods
A graphical method was the first method to solve the complete system of equations
Eq. 5.3 and Eq. 5.4. Characteristics were calculated and represented
in a spatial-temporal graph (x-t plane). The graphical method transformed
into the method of characteristics when computer calculation permitted to
compute characteristics more efficiently than hand-written methods.
5.3 Method of Characteristics
The linear hyperbolic system of two first order partial differential equations
formed by Eq. 5.3 and Eq. 5.4 relates pressure, p, and fluid velocity, u. The
linear hyperbolic system is solved in the discretized x-t plane. Grid points
spatial and temporal distances are ∆x and ∆t, respectively. The grid can be
rectangular, diamond-shaped staggered, or composed by the linear hyperbolic
system characteristics. The rectangular double grid (Figure 5.2) is the most
commonly used (Wylie and Streeter, 1993).
∆t
t
∆x
C + C −
Figure 5.2: Characteristic Lines Representation in Standard X-T
Plane Grid
The hyperbolic system has two distinct real characteristics. Characteristic
lines in the x-t plane form a new system of coordinates where the hyperbolic
system Eq. 5.6 transforms into four first-order ordinary differential equations
that can be solved numerically by finite difference approximations. Characteristics
are easily obtained from the eigenvalues of the system in its matrix
form, Eq. 5.8, and are calculated by (Wylie and Streeter, 1993):
|I − βB| = 0 (5.13)
where, β, is a parameter that has two roots. Values of β define the characteristics:
dx
= β (5.14)
dt
76
x

5.3. Method of Characteristics
Characteristics of Eq. 5.3 and Eq. 5.4 are then ±a. The source term, s, does
not influence the characteristics of the system. Characteristic lines in the x-t
plane are lines of constant slope that define a new coordinate system. In the
new coordinate system, the two first-order partial differential equations transform
into four first-order ordinary differential equations that can be grouped
in two characteristic paths, (Wylie and Streeter, 1993):
C + ⎧
⎪⎨
dp
+ adu = −as
= dt dt
⎪⎩
dx
= +a
dt
C − ⎧
⎪⎨
dp
− adu = as
= dt dt
⎪⎩
dx
= −a
dt
(5.15)
(5.16)
where, C + , represents the forward characteristic path, and, C − , the backward
characteristics path. Each group of equations can be solved into the characteristic
system of coordinates such that pressure and fluid velocity values
can be calculated at each intersection of two characteristic lines. A typical
representation of a x-t plane and the corresponding system characteristics for
a simple pipeline system problem is shown in Figure 5.2.
The Courant condition defines the maximum distances between grid points.
The Courant condition is expressed by:
a∆t ≤ ∆x (5.17)
Characteristic lines intersections and x-t plane nodes do not necessarily coincide
as represented in Figure 5.2. Hence, pressure and fluid velocity values at
x-t plane nodes must be interpolated (Vardy, 2008). Interpolations decrease
calculation accuracy. A moving x-t plane grid can be used instead, where ∆x
and ∆t become δx and δt, and vary with time and space. However, adaptive
grid methods are more expensive in computer resources and time, and mesh
refinement close to boundaries is also necessary.
Boundary conditions such as pumps, tanks or valves, require pressure and
fluid velocity values for the problem to be solvable. A problem is underdefined
when hyper-static. Another way to express this is that a problem is
defined when the number of parameters equals the number of equations and
sufficient initial conditions are known. Interpolations are generally required
at boundaries since characteristic lines intersections barely coincide with grid
nodes, even using an adaptive grid method.
77

Chapter 5. Numerical Simulation Methods
The method of characteristics offers an engineer-friendly method for computing
pressure and fluid velocity values in a pipeline during pressure transient
propagation. The method of characteristics is first order in both space and
time because of finite difference approximations of first-order ordinary differential
equations. Nonetheless, the method of characteristics provide fast
solution with an acceptable accuracy that can be improved by grid refinement
and high order interpolations.
Numerical errors and large deviations from experimental data are often reported
in pressure transient analysis. Of particular interest is the non-physical
numerical attenuation due to interpolation to obtain pressure and fluid velocity
values at grid points. Despite its easiness and efficiency to compute results
in pipeline systems, research tends to focus on different numerical techniques
and schemes such as finite volume methods (Soares et al., 2008).
5.4 Finite Volume Methods
5.4.1 Description
Finite volume methods are based on the integral form of Eq. 5.6 in order to
use grid cells instead of grid points. Total integrals of vector variable Ω are
now evaluated over each grid cells. Grid cell Q values evolve with time by the
flux over the edges of each grid cells. Flux functions and grid cells averaging
methods define several numerical schemes. The integral form of Eq. 5.6 is
(LeVeque, 2002):
�
d
�
Ω(x, t)dx = Φ
dt ζi
ϕ(x i− 1
2
� �
, t) − Φ
ϕ(x i+ 1
2
�
, t)
(5.18)
where, ζ, is the spatial domain, and the above integration defines the average
value over the ith cell. The spatial domain is divided into grid cells (or finite
volumes) of length:
∆x = x 1
i+ − x 1
i− 2 2
(5.19)
The vector variable Ω evolves with space and time. The cell average at each
position and each time depends on the flux from the adjacent cells as illustrated
in Figure 5.3. A cell average value after incrementing the time step is expressed
by (LeVeque, 2002):
Ω n+1
i
= Ωni − ∆t
� �
�Φ n
Ωi , Ω
∆x
n � �
i+1 − Φ� n
Ωi−1, Ω n� i
�
78
(5.20)

tn+1
tn
Φ n
i− 1
2
Ω n+1
i
Φ n
i+ 1
2
Ω n i−1 Ω n i Ω n i+1
5.4. Finite Volume Methods
Figure 5.3: Illustration of a Finite Volume Method for Updating the
Cell Average Ωn i by Fluxes at the Cell Edges
where, i, represents the space index, n, the time index, and, � Φ, a numerical
flux function that approximates the average flux function Φ. Different finite
volume numerical schemes correspond to different formulation of � Φ in Eq. 5.20.
Available finite volume methods try to include a Riemann solver for capturing
wave propagation. The Riemann problem occurs at a grid edge where the data
is piecewise constant with a single jump discontinuity. The Riemann problem
is expressed by (Laney, 1998):
Ω(xi, tn) =
�
Ωl if x < 0
Ωr if x > 0
(5.21)
where, Ωl and Ωr, are solutions just before and just after the grid node x = 0,
respectively.
5.4.2 Numerical Schemes
Finite volume numerical schemes differ from one another by the flux average
approximation function, � Φ. Three different numerical schemes are herein studied
based on three different flux averaging techniques. The investigated numerical
scheme are Central Time Central Space (CTCS), Lax-Wendroff (LW),
and Essentially Non-Oscillatory (ENO). The three schemes are explicit and
second-order accurate in both space and time (Laney, 1998). The three selected
schemes represent three different mathematical concept.
The CTCS method is a conservative numerical scheme based on finite difference
approximations of first-order derivatives. The CTCS scheme is also
referred to as the Leapfrog method. The CTCS numerical scheme locates
shocks accurately since it is a conservative scheme. However, shock shapes are
not correctly predicted because of the odd-even time and space decoupling.
79

Chapter 5. Numerical Simulation Methods
The CTCS numerical scheme is centered, linear, and expressed by (Laney,
1998):
Ω n+1
i
= Ωn−1
i
∆t � n
− 2 Φ(Ω
∆x
i+1) − Φ(Ω n i−1) �
(5.22)
The Lax-Wendroff numerical scheme is a linear, centered method, based on
decomposition of Eq. 5.6 in Taylor series. Several other methods are derived
from the Lax-Wendroff technique of Taylor series decomposition by changing
flux averaging. The Lax-Wendroff numerical scheme in its conservative form
is expressed by (Laney, 1998):
Ω n+1
i
ˆΦ n
i+ 1
2
a n
i+ 1
2
= Ωn i − ∆t
�
ˆΦ n
i+ 1 −
2
ˆ Φ n
i− 1
�
2
∆x
= 1 � n
Φ(Ω
2
i+1) + Φ(Ω n i ) � − ∆t
2∆x an
i+ 1
� n
Φ(Ωi+1) − Φ(Ω
2
n i ) �
⎧
⎪⎨ Φ(Ω
=
⎪⎩
n i+1 ) + Φ(Ωni )
Ωn i+1 − Ωn for Ω
i
n i �= Ωn i+1
Φ(Ω n i ) for Ωn i = Ωn i+1
(5.23)
The ENO numerical scheme is a piecewise-polynomial reconstruction-evolution
method based on polynomial interpolation. The ENO scheme reconstructs
solutions from cell-integral averages. The numerical scheme is a high resolution
method of second order but less stable than the Lax-Wendroff scheme for
instance. The ENO numerical scheme is not detailed here but can be found
in standard textbooks such as (Laney, 1998).
5.4.3 Test Cases
Four test cases were used to assess strengths and weaknesses of the three
studied numerical schemes. The four test cases descriptions are given in Table
5.1. The first test case illustrates phase and amplitude error on a completely
smooth solution. The second test case illustrates progressive contact smearing
and dispersion with two jump discontinuities. The third test case is similar
to the second but illustrates the importance of the number of grid points.
The fourth test case illustrates sonic point importance in the Burger equation
expressed by (Laney, 1998):
∂ϕ
∂t
+ ∂
∂x
� 1
2 ϕ2
�
80
(5.24)

Table 5.1: Test Cases Description for Comparing CTCS,
Lax-Wendroff and ENO Methods
Case Equation
5.4. Finite Volume Methods
Initial Conditions Grid
ϕ(x,0) Points
1
−sin(πx)
40
2
∂ϕ ∂ϕ
+ = 0
∂t ∂x ⎧
⎪⎨ 1
⎪⎩
for |x| < 1
0
3
for 1
3
4
�
∂ϕ ∂ 1
+
∂t ∂x 2
< |x| ≤ 1
3
600
ϕ2
�
= 0 40
Comparison of the three numerical schemes on test case 1 is shown in Figure
5.4. The three numerical schemes capture the sinusoidal shape. Lax-Wendroff
and CTCS calculated solutions are almost superimposed. Both CTCS and
Lax-Wendroff predict the sinusoidal amplitude correctly but the solutions lag
behind the exact solution. The ENO method predicts phase correctly but the
sinusoidal wave is smeared at both extremes.
Comparison of the three numerical schemes on test case 2 is shown in Figure
5.5. Both CTCS and Lax-Wendroff methods locate shocks properly. However,
predicted solutions by CTCS and Lax-Wendroff numerical schemes undershoot
and overshoot the exact solution, and both result behaviours are greatly oscillatory.
The ENO method captures the square wave shape properly and locates
shocks correctly with moderate smearing.
Comparison of the three numerical schemes on test case 3 is shown in Figure
5.6. Increasing the number of grid points improves prediction of shock
locations. However, both CTCS and Lax-Wendroff methods exhibit a highly
oscillatory behaviour that overshoots and undershoots the exact solution. On
the other hand, the ENO numerical scheme smearing of the square wave observed
in Figure 5.5 is now reduced.
Comparison of the three numerical schemes on test case 4 is shown in Figure
5.7. The CTCS numerical scheme fails to predict correct location or amplitude
of the exact solution. The Lax-wendroff locates accurately the shock and the
expansion fan but still undershoots and overshoots the exact solution, especially
close to the sonic point. The ENO scheme predicts accurately location
and amplitude of the expansion fan and the shock.
81

Chapter 5. Numerical Simulation Methods
1.00
0.75
0.50
0.25
0.00
−0.25
−0.50
−0.75
−1.00
Exact Solution
CTCS
Lax−Wendroff
ENO
−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00
Figure 5.4: Comparison of CTCS, Lax-Wendroff and ENO Numerical
Schemes on Test Case 1
1.25
1.00
0.75
0.50
0.25
0.00
Exact Solution
CTCS
Lax−Wendroff
ENO
−0.25
−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00
Figure 5.5: Comparison of CTCS, Lax-Wendroff and ENO Numerical
Schemes on Test Case 2
82

1.25
1.00
0.75
0.50
0.25
0.00
5.4. Finite Volume Methods
Exact Solution
CTCS
Lax−Wendroff
ENO
−0.25
−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00
Figure 5.6: Comparison of CTCS, Lax-Wendroff and ENO Numerical
Schemes on Test Case 3
1.25
1.00
0.75
0.50
0.25
0.00
Exact Solution
CTCS
Lax−Wendroff
ENO
−0.25
−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00
Figure 5.7: Comparison of CTCS, Lax-Wendroff and ENO Numerical
Schemes on Test Case 4
83

Chapter 5. Numerical Simulation Methods
The three second-order accurate methods in both space and time do not perform
identically. The ENO scheme performs better globally than both CTCS
and Lax-Wendroff. Lax-Wendroff numerical scheme captures shocks accurately
but presents artificial oscillations. Nevertheless, the Lax-Wendroff numerical
scheme is of particular interest because flux limiter functions can be
added to increase the resolution of the method.
5.4.4 Higher Resolution Method
Flux limiter coupled to second-order accurate numerical schemes increases
the resolution of the methods. The Lax-Wendroff numerical results can be
improved by adding a flux limiter to the flux function. The numerical scheme
is still second-order accurate in both space and time, but the resolution is
increased, especially close to shocks. The Lax-Wendroff numerical scheme
coupled to a flux limiter function is expressed by (Laney, 1998):
Ω n+1
i
= Ωni − ∆t
∆x ˆ �
φ(θ) ˆF n
i+ 1 −
2
ˆ F n
i− 1
�
2
(5.25)
where, ˆ φ, represents a flux-limiter function, and, ˆ Φ, has been defined in
Eq. 5.23. Four flux limiter functions are herein studied: minmod, superbee,
monotonized-centered and Van-Leer flux limiter. The four flux limiter
functions are given in Table 5.2. Comparison of the four flux limiter functions
was performed using the four test cases described previously.
Table 5.2: Minmod, Superbee, Monotonized-Centered and Van-Leer
Flux Limiter Functions (Laney, 1998)
Flux Limiter Function
Minmod
Superbee
ˆ φ(θ) = max (0, min(1, θ))
ˆ φ(θ) = max (0, min(1, 2θ), min(2, θ))
Monotonized-Centered ˆ � � ��
1 + θ
φ(θ) = max 0, min , 2, 2θ
2
Van-Leer
84
ˆ φ(θ) = θ + |θ|
1 + |θ|

5.4. Finite Volume Methods
Comparison of the four flux limiters on test case 1 is shown in Figure 5.8.
All four flux limiter functions improve the phase prediction compare to the
standard Lax-Wendroff numerical scheme. However, all functions smear the
sinusoidal shape. Nevertheless, the Superbee flux limiter smearing is negligible
while both minmod and Van-Leer functions smearing are 25 % of the signal
amplitude.
Comparison of the four flux limiters on test case 2 is shown in Figure 5.9. All
four flux limiter functions improve shock locations and reduce Lax-Wendroff
numerical scheme oscillatory behaviour. Both minmod and Van-Leer flux
limiter functions underestimate slightly the square wave amplitude, while Superbee
and monotonized-centered functions capture the wave amplitude accurately.
Comparison of the four flux limiters on test case 3 is shown in Figure 5.10.
Increasing the number of grid points improves prediction of shock locations
compare to test case 2. The square wave amplitude is now also well-captured
by all four flux limiter functions. However, increasing the number of grid
points increase computational time and necessary resources.
Comparison of the four flux limiters on test case 4 is shown in Figure 5.11.
All four flux limiter functions improve shock and expansion fan locations,
but most importantly, the wave shape is now captured without undershooting
or overshooting close to sonic points compare to the standard Lax-Wendroff
method. All four flux limiter functions give superimposed solutions.
Coupling a flux limiter function to the standard Lax-Wendroff numerical
scheme improves shock and expansion fan locations at convex corners, and better
captures wave shape. Increasing the number of grid points globally helps
improving the accuracy of the solution. The Superbee flux limiter function
performs globally better than the three other flux limiter functions studied,
as seen in Figures 5.8 to 5.11.
5.4.5 CLAWPACK
CLAWPACK - Conservation LAW PACKage - is a software package of wave
propagation methods for hyperbolic equations (LeVeque, 2002). Several numerical
schemes are included in form of subroutines. Available numerical
schemes include CTCS, Lax-Wendroff and the four flux limiter functions herein
studied. Of particular interest, the Lax-Wendroff numerical scheme with the
Superbee flux limiter was used in the rest of the thesis for pressure wave simulation.
85

Chapter 5. Numerical Simulation Methods
1.00
0.75
0.50
0.25
0.00
−0.25
−0.50
−0.75
−1.00
Exact Solution
Lax−Wendroff
LW + Minmod
LW + Superbee
LW + MC
LW + Van−Leer
−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00
Figure 5.8: Comparison of Minmod, Superbee, Monotonized
Centered and Van-Leer Flux Limiters Coupled to the
Lax-Wendroff Numerical Scheme on Test Case 1
1.25
1.00
0.75
0.50
0.25
0.00
Exact Solution
Lax−Wendroff
LW + Minmod
LW + Superbee
LW + MC
LW + Van−Leer
−0.25
−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00
Figure 5.9: Comparison of Minmod, Superbee, Monotonized
Centered and Van-Leer Flux Limiters Coupled to the
Lax-Wendroff Numerical Scheme on Test Case 2
86

1.25
1.00
0.75
0.50
0.25
0.00
5.4. Finite Volume Methods
Exact Solution
Lax−Wendroff
LW + Minmod
LW + Superbee
LW + MC
LW + Van−Leer
−0.25
−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00
Figure 5.10: Comparison of Minmod, Superbee, Monotonized
Centered and Van-Leer Flux Limiters Coupled to the
Lax-Wendroff Numerical Scheme on Test Case 3
1.25
1.00
0.75
0.50
0.25
0.00
Exact Solution
Lax−Wendroff
LW + Minmod
LW + Superbee
LW + MC
LW + Van−Leer
−0.25
−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00
Figure 5.11: Comparison of Minmod, Superbee, Monotonized
Centered and Van-Leer Flux Limiters Coupled to the
Lax-Wendroff Numerical Scheme on Test Case 4
87

Chapter 5. Numerical Simulation Methods
5.5 Numerical Scheme Selection
Selection of a numerical scheme is performed using a standard test case composed
of a 100 meter long pipeline, connected to a 50 bara constant pressure
tank on the left-hand side, and to a quick-acting valve on the right side (Figure
5.1). Water flows from the tank into the pipeline and flows at 2 ms −1
constant velocity. The acoustic velocity in the fluid-filled pipeline is fixed at
1435 ms −1 .
The test case is modelled using both the method of characteristics and the
Lax-Wendroff scheme coupled to Superbee flux limiter. Frictional pressure
drop and heat transfer are not considered. Another way to express this is that
the source term Λ in Eq. 5.8 equals zero. The valve is suddenly actuated such
that a rapid pressure transient is created at the valve and propagates through
the pipeline. 4 seconds are simulated for observing pressure waves reflections.
The method of characteristics models the test case with 1000 space grid points
and 30873 space steps for ensuring a Courant number equal to one for limiting
numerical dissipation. The Lax-Wendroff numerical scheme coupled to Superbee
flux limiter included in CLAWPACK, models the test case with 1000 grid
points and 1000 space steps. Each time step is iterated to ensure a Courant
number equal to one.
The modelling results by the method of characteristics and the finite volume
method are showed in Figure 5.12. The pressure trace obtained from both
methods are identical, confirming that both method are equivalent for sourceless
pressure transient modelling. Therefore, only the Lax-Wendroff numerical
scheme coupled to Superbee flux limiter is used in the following of the thesis.
88

Pressure [bara]
100
75
50
25
5.5. Numerical Scheme Selection
MOC
FVM
0
0.0 0.5 1.0 1.5 2.0
Time [s]
2.5 3.0 3.5 4.0
Figure 5.12: Comparison of Method of Characteristics and Finite
Volume Method (Lax-Wendroff + Superbee Flux
Limiter) on Standard Pressure Transient Test Case
89

Chapter 6
Pressure Wave Attenuation
6.1 Influence of Numerical Simulation Tool
6.1.1 Unsteady Friction Modelling Equations
Pressure wave modelling is based on mass and momentum conservation equations
where the source term is responsible for smoothing and damping of pressure
waves. Steady state frictional pressure drop was the first attenuation
mechanism included, and state of the art pressure wave modelling now includes
unsteady friction factors in addition. Various unsteady friction models
have been proposed since the first model by Zielke (1966). Of particular interest
are unsteady friction models proposed by Brunone et al. (1995) and Ramos
et al. (2004).
Brunone et al.’s and Ramos et al.’s unsteady friction models decompose the
friction factor in two parts. The first part is the quasi steady friction factor, fq,
and is commonly calculated using Darcy-Weisbach friction factor formulation,
for instance. The second part takes into account local and instantaneous
convective acceleration of flow velocities. Brunone et al. (1995) and Ramos
et al. (2004) unsteady friction models are, respectively:
f = fq + kB1d
� �
∂u
− a∂u
(6.1)
u|u| ∂t ∂x
f = fq + d
u|u|
�
kR1
∂u
∂t
u
+ kR2
|u| a
� ��
�
�
∂u�
�
�∂x
�
(6.2)
where, kB1, kR1 and kR2, are empirical constants that can be tuned for matching
experimental data with numerical simulations. Standard values are 0.02,
0.003 and 0.04, for kB1, kR1 and kR2, respectively. Simulations of water hammer
events including unsteady friction modelling were performed by Brunone
et al. and Ramos et al. using the method of characteristics.
91

Chapter 6. Pressure Wave Attenuation
6.1.2 Test Case and Numerical Simulation
A water hammer test case was simulated using the method of characteristics
and the previously selected finite volume scheme. The method of characteristics
is included in a commercial software, Bentley Hammer V 8i, (Haestad
Methods Solution Center, 2008). Whereas the finite volume method is included
in an in house numerical simulation tool for pressure transient simulations
in single pipelines. The finite volume method is based on the Lax-
Wendroff numerical scheme coupled with Superbee flux limiter function (Chapter
5).
The water hammer test case includes a steel horizontal pipeline, 1000 m in
length, 10 inches in diameter, and liquid-water-filled. The fluid density is
1000 kg m −3 , and the acoustic velocity in the fluid-filled pipeline is set at 1200
ms −1 . The pipeline is connected to a 50 bara constant pressure reservoir on its
left-hand side, and to a quick acting check valve on its right-hand side. Water
was initially flowing at 2 ms −1 before valve actuation, and the steady friction
factor was set to 0.01. The valve actuation is assumed ideal and instantaneous.
The water hammer test case is first simulated using the method of characteristics
included in Bentley Hammer V 8i commercial software. The Courant
number, a dt/dx, was set to 1 to limit the method of characteristis inherent
numerical dissipation. The water hammer test case is simulated using
steady state friction factor only, and then including Brunone et al.’s unsteady
friction factor. Ramos et al.’s unsteady friction factor is not included in
Bentley Hammer V 8i commercial software.
The water hammer test case was then simulated using the selected finite volume
method. The Courant number was also set to 1 and other simulation
parameters from Bentley Hammer V 8i commercial software were used as set
parameters in the more flexible in house finite volume method numerical code.
Of particular interest, 15 spatial nodes were used. Comparison between the
two numerical methods is then performed based on numerical scheme only.
Complete test case description and numerical simulation parameters are summarized
in Table 6.1.
Table 6.1: Test Case System Description and Numerical Simulation
Parameters
L [m] d [m] p0 [bara] u0 [ms −1 ] a [ms −1 ] nX nT TSim [s]
100 0.254 50 2 1200 15 540 30
92

6.1.3 Results of Friction Modelling
6.1. Influence of Numerical Simulation Tool
Simulation results using the method of characteristics and the selected finite
volume method, including steady friction only, are shown in Figure 6.1.
Both numerical method predicts correctly the first pressure transient amplitude
according to the Joukowski equation. However, predicted pressure trace
is square-shaped with MOC and round-shaped with FVM. Predicted pressure
amplitudes also decrease faster with FVM than with MOC. In addition, pressure
wave propagation velocity is accurately with MOC but under-estimated
with FVM.
Simulation results using the method of characteristics and the selected finite
volume method, including Brunone et al.’s unsteady friction factor, are shown
in Figure 6.2. Both numerical methods predict correctly the first pressure
transient amplitude according to the Joukowski equation. However, predicted
pressure traces are square-shaped with MOC and round-shaped with FVM.
Pressure amplitudes are first lower with MOC until 15 s, then pressure amplitudes
decrease faster with FVM. Pressure wave propagation velocity is again
accurately predicted with MOC but under-estimated with FVM.
Simulation results using the method of characteristics, and including steady
friction factor, and Brunone et al.’s unsteady friction factor, are repeated in
Figure 6.3. As previously observed, pressure traces are square-shaped and
pressure wave propagation velocities are accurately predicted. Both models
predict the same first pressure wave amplitude according to the Joukowski’s
equation. However, pressure amplitudes decrease faster with Brunone et al.’s
unsteady friction modelling as reported in the literature (Brunone et al., 1995).
Simulation results using the selected finite volume method, and including
steady friction factor, and Brunone et al.’s unsteady friction factor, are repeated
in Figure 6.4. As previously observed, pressure traces are roundshaped
and pressure wave propagation velocity are under-estimated. Both
models predict the same pressure trace, that is shape, travel time, damping
and smoothing. Another way to express this is that Brunone et al.’s unsteady
friction model does not improve water hammer calculation when simulated
with FVM.
Simulation results from the method of characteristics show that unsteady friction
increases pressure damping as claimed by Brunone et al. (1995). However,
simulation results from the finite volume method show that unsteady friction
does not increase pressure damping compare to steady state friction. But,
simulation results from the finite volume method show that the pressure wave
propagation velocity is under-estimated, thus indicating poor numerical resolution
and meshing for the finite volume method. Finite volume meshing can
however be adjusted.
93

Chapter 6. Pressure Wave Attenuation
Pressure [bara]
80
70
60
50
40
30
MOC Steady
FVM Steady
20
0 5 10 15
Time [s]
20 25 30
Figure 6.1: Comparison of Method of Characteristics and Finite
Volume Method with Steady Friction Only. nX = 15.
nT = 540.
Pressure [bara]
80
70
60
50
40
30
MOC Unsteady
FVM Unsteady
20
0 5 10 15
Time [s]
20 25 30
Figure 6.2: Comparison of Method of Characteristics and Finite
Volume Method with Brunone’s Unsteady Friction
Model. nX = 15. nT = 540.
94

Pressure [bara]
80
70
60
50
40
30
6.1. Influence of Numerical Simulation Tool
MOC Steady
MOC Unsteady
20
0 5 10 15
Time [s]
20 25 30
Figure 6.3: Comparison of Steady Friction and Brunone’s Unsteady
Friction Models with the Method of Characteristics.
nX = 15. nT = 540.
Pressure [bara]
80
70
60
50
40
30
FVM Steady
FVM Unsteady
20
0 5 10 15
Time [s]
20 25 30
Figure 6.4: Comparison of Steady Friction and Brunone’s Unsteady
Friction Models with the Selected Finite Volume Method.
nX = 15. nT = 540.
95

Chapter 6. Pressure Wave Attenuation
6.1.4 Finite Volume Method Accuracy
The water hammer test case can be simulated using the selected finite volume
method and increasing numerical resolution and spatial meshing. Simulation
results using the finite volume method and steady state friction factor are
shown for increasing number of spatial nodes in Figure 6.5. The pressure
trace becomes sharper with increasing number of spatial nodes, and propagation
velocity are correctly captured for nX = 1000, therefore increasing the
simulation accuracy. Another way to express this is that the selected finite
volume method accuracy increases with the number of spatial nodes.
Pressure [bara]
70
68
66
64
62
60
58
n X = 1000
n X = 200
n X = 100
n X = 50
n X = 15
56
19.5 20.0 20.5 21.0
Time [s]
21.5 22.0 22.5
Figure 6.5: Influence of Spatial Discretization on Water Hammer
Calculation with the Selected Finite Volume Method.
Detail of Pressure Trace
Simulation results using the selected finite volume method, and including
steady friction factor, Brunone et al.’s and Ramos et al.’s unsteady friction
factors, with nX = 1000, are shown in Figure 6.6. The three simulated pressure
traces are merged, thus indicating that unsteady friction models do not
increase pressure wave damping. In simple words, unsteady friction models
do not improve water hammer prediction compare to steady state friction. In
addition, no pressure trace smoothing is observed.
Simulation results using the method of characteristics, including steady state
and Brunone et al.’s unsteady friction factors, and using the selected finite
volume method are repeated in Figure 6.7. Both numerical schemes predict
96

6.1. Influence of Numerical Simulation Tool
accurately the first pressure transient amplitude and pressure wave propagation
velocity. However, the method of characteristics predicts faster damping
than the finite volume method, thus illustrating that the method of characteristics
inherent dissipation is non-physical.
Unsteady friction factor terms increase convective acceleration. However, convective
terms are physically negligible in accordance with the low Mach number
approximation. Unsteady friction factors proposed by Brunone et al. (1995)
and Ramos et al. (2004) are therefore artifacts that increase numerical dissipation
and that by coincidence mimics pressure wave attenuation mechanisms.
In addition, unsteady friction factor models do not change the modelling equations
hyperbolic nature, and therefore can not reproduce pressure trace damping.
Such result is in accordance with Szymkiewicz and Mitosek (2007).
The selected finite volume method is therefore more accurate and appropriate
than the method of characteristics for simulating pressure wave transients,
provided that spatial discretization is sufficient enough. In addition, water
hammer simulation using a finite volume method indicates that current one
dimensional pressure wave attenuation mechanisms are inaccurate and do not
capture the physics of the phenomenon. New models need therefore to be
investigated, modelled, and simulated.
6.1.5 Pipeline Viscoelasticity
Pipeline material viscoelastic behaviour has received much attention in the
past two decades due to the increasing use of plastic pipelines. Pipeline viscoelasticity
models relate to pipeline material deformation properties during
pressure transient events. Pipeline materials Young’s moduli change with time
during pressure transient event. Pipeline materials first have an instantaneous
response to a pressure transient, then a delayed response that depends on material
characteristics and fatigue (Covas, 2003).
Viscoelasticity modelling uses pipeline decomposition in Kelvin-Voigt elements
that can be represented by a purely viscous damper and a purely elastic
string. The number of elements, as well as retardation time and tensile moduli
of each element are modelling parameters. Numerical schemes inherent
dissipation and characteristics need therefore to be taken into consideration
(Weinerowska-Bords, 2007).
Pipeline viscoelasticity modelling has been simulated using the method of
characteristics. Hence, the method of characteristics inherent numerical dissipation
also induces greater pressure wave damping than in practice. Thus,
pipeline viscoelasticity modelling results and accuracy are questionable. In
addition, pipeline viscoelasticity is negligible in steel pipelines, thus pressure
wave attenuation in steel pipelines is still inaccurately estimated.
97

Chapter 6. Pressure Wave Attenuation
Pressure [bara]
80
70
60
50
40
30
Steady Friction
Brunone
Ramos
20
0 5 10 15
Time [s]
20 25 30
Figure 6.6: Comparison of Investigated Unsteady Friction Models
Using a Selected Finite Volume Method. nX = 1000.
nT = 540.
Pressure [bara]
80
70
60
50
40
30
MOC Steady
MOC Unsteady
FVM Unsteady
20
0 5 10 15
Time [s]
20 25 30
Figure 6.7: Comparison of Water Hammer Simulation Using the
Method of Characteristics and a Finite Volume Method.
MOC: nX = 15. nT = 540. FVM: nX = 1000. nT = 540.
98

6.2 Acoustic Velocity
6.2. Acoustic Velocity
Current pressure wave attenuation mechanisms focus on the modelling equations
source term, and viscoelasticity models focus on including pipeline elastic
properties in acoustic velocity calculations. Current pressure wave propagation
velocity models include fluids and pipeline properties. Acoustic velocity
calculations then require fluid density and isothermal compressibility, and
pipeline compressibility that depends on pipeline dimensions, materials and
fastenings. The acoustic velocity of a pressure wave in fluid-filled pipeline
therefore be expressed by:
a =
�
1
ρ (KFluid + KPipe)
(6.3)
where, ρ, is fluid density, KFluid, fluid isothermal compressibility, and, KPipe,
pipeline compressibility. The influence of every component of the acoustic
velocity calculation have been detailed in Chapter 4. It is also noted that
the state of the fluid is accounted by density and isothermal compressibility
changes. However, flow structures, such as turbulent eddies, are not considered
by current calculation models.
6.2.1 Ultrasonic Measurement
Pressure wave propagation velocity in gas flowlines can be obtained experimentally
using ultrasonic flow meters. A schematic representation of an ultrasonic
flow meter is shown in Figure 6.8. An emitter generates a sonic beam that
propagates to a receiver. Flow velocity and acoustic velocity are then calculated
from pressure wave travel time between emitter and receiver. Ultrasonic
measurements are performed with great accuracy and therefore provide valuable
data.
Three different ultrasonic measurement data sets from one 24 inches FMC
apparatus, class 600, located on a natural gas transport pipeline at Field A,
were made available to NTNU by Jacobsen (2008). Gas composition from field
A is given in Table 6.2, and the gas phase envelope is shown in Figure 6.9.
Data sets include pressure and temperature conditions, and measured flow
and acoustic velocities. Gas properties are calculated from gas composition,
pressure and temperature conditions using the Peng-Robinson fluid package
included in Hysys.
99

Chapter 6. Pressure Wave Attenuation
Emitter
δ
Receiver
Figure 6.8: Schematic Representation of an Ultrasonic Flow Meter
Table 6.2: Field A Molar Gas Composition (Jacobsen, 2008)
Pressure [bara]
Component C1 C2 C3 iC4 nC4
% 85.823 8.271 2.335 0.233 0.255
Component iC5 nC5 C6 CO2 N2
% 0.029 0.029 0.038 1.825 1.150
70
60
50
40
30
20
10
Critical Point
Bubble Point
Dew Point
0
−160 −140 −120 −100 −80 −60 −40 −20
Temperature [ o C]
Figure 6.9: Field A Gas Phase Envelope (Jacobsen, 2008)
100

6.2. Acoustic Velocity
At experimental pressure and temperature conditions, averaged density and
dynamic viscosity are 55.6 kg m −3 and 1.24 10 −2 cP, respectively. Theoretically
calculated values of density, adiabatic index and z-factor are then used
to calculate acoustic velocity in the gas using:
�
�
� k
a = � � � �
� 1 1 ∂z
ρ −
p z ∂p
Γ
� (6.4)
Evolutions of measured and calculated acoustic velocities with pressure and
temperature are shown in Figure 6.10. Calculated values are about 2 ms −1
lower in average than measured values. Curve shapes are however similar, with
approximately constant acoustic velocity for decreasing pressures at constant
temperature, and for decreasing temperatures at constant pressure. Differences
between measured and calculated values can be explained by the use of
the Peng-Robinson equation of state for predicting gas properties.
Evolution of measured and calculated acoustic velocities with pressure and
temperature are also shown separately in Figure 6.11 and Figure 6.12, respectively.
Measured acoustic velocity decreases from 374 ms −1 at 57.98 bara
to 373.7 ms −1 at 58.12 bara. Whereas calculated acoustic velocity oscillates
slightly around 372 ms −1 for the range of measured pressures. Measured
acoustic velocity increases from 374 ms −1 at 8.87 o C, to 374.5 ms −1 at 9.07 o C.
Whereas calculated values increase from 371.8 ms −1 to 372.3 ms −1 for the
range of measured temperatures.
Measured and calculated acoustic velocities evolution with gas flow velocity
is shown in Figure 6.13. Calculated values are slightly oscillating around 372
ms −1 , whereas measured acoustic velocities increase from 373.7 ms −1 at 1
ms −1 to 374.5 ms −1 at 20 ms −1 . Acoustic velocity dependence on flow
velocity is observed experimentally but unexpected from a modelling point of
view, and is not taken into consideration by current models (AGA-10, 2003).
Two other data set were analysed similarly. Both data sets are similar and
in the same range of pressure and temperature, therefore only one data set is
herein presented. Measured and calculated acoustic velocities evolution with
pressure and temperature is shown in Figure 6.14. Calculated values are 2
ms −1 lower than measured values, in average. The large acoustic velocity
drop at 4.5 o C is however not predicted. Curve shapes are however globally
similar. Differences between measured and calculated values can again be
explained by the use of the Peng-Robinson equation of state for predicting gas
properties.
101

Chapter 6. Pressure Wave Attenuation
Acoustic Velocity [m s −1 ]
373.5
373.0
372.5
372.0
371.5
9.00
8.95
8.90
Temperature [ o C]
8.85
8.80
57.95
58.00
58.05
Measured
Calculated
58.10
Pressure [bara]
58.15
Figure 6.10: Measured and Calculated Acoustic Velocities Function
of Pressure and Temperature. First Data Set
Acoustic Velocity [m s −1 ]
375.0
374.5
374.0
373.5
373.0
372.5
372.0
Measured
Calculated
371.5
57.95 58.00 58.05
Pressure [bara]
58.10 58.15
Figure 6.11: Measured and Calculated Acoustic Velocities Function
of Pressure. First Data Set
102

Acoustic Velocity [m s −1 ]
375.0
374.5
374.0
373.5
373.0
372.5
6.2. Acoustic Velocity
372.0
371.5
8.80 8.85 8.90 8.95
Temperature [
9.00 9.05 9.10
o Measured
Calculated
C]
Figure 6.12: Measured and Calculated Acoustic Velocities Function
of Temperature. First Data Set
Acoustic Velocity [m s −1 ]
375.0
374.5
374.0
373.5
373.0
372.5
372.0
Measured
Calculated
371.5
0 5 10 15 20 25
Flow Velocity [m s −1 ]
Figure 6.13: Measured and Calculated Acoustic Velocities Function
of Flow Velocity. First Data Set
103

Chapter 6. Pressure Wave Attenuation
Measured and calculated acoustic velocities evolution with pressure and temperature
are also shown separately in Figure 6.15 and Figure 6.16, respectively.
Measured and calculated acoustic velocities decrease linearly between
50.5 bara and 56 bara from 373 ms −1 to 371 ms −1 , and from 371 ms −1 to
369 ms −1 , respectively. Measured and calculated acoustic velocities increase
linearly between 3.9 o C and 5.3 o C from 369 ms −1 to 371 ms −1 , and from
367 ms −1 to 369 ms −1 , respectively. In both evolutions, the large acoustic
velocity drop observed is not predicted by calculations.
Measured and calculated acoustic velocities evolution with gas flow velocity
is shown in Figure 6.17. Both measured and calculated acoustic velocities
decrease with flow velocity between 3 ms −1 and 21 ms −1 , from 372 ms −1 to
369 ms −1 , and from 369 ms −1 to 367 ms −1 , respectively. Measured acoustic
velocity is therefore not dependent upon gas flow velocity as opposed to the
first data set. Measured acoustic velocity from the third data set is also
independent of gas flow velocity.
Measured acoustic velocity changed with gas flow velocity in the three ultrasonic
measurement sets provided. The acoustic velocity evolution could not
be predicted by theoretical calculations and pressure and temperature adjustments
in the first case, whereas acoustic velocity evolution could be calculated
theoretically in the two other cases. However, it is noticeable that temperature
is about 8 o C in the first case and only 4 o C in the two other cases. Pressure
values are in the same range for all three cases.
Decrease of acoustic velocity with flow velocity at low temperature can be
explained by the presence of liquid droplets. Liquid droplets can form in gasfilled
pipeline at low temperature, and stick to the pipeline wall. Increase in
gas flow velocity can then overcome wall shear stress for droplets and acoustic
velocity is then measured in a two phase fluid. Small amount of droplets can
decrease dramatically acoustic velocity in gas-filled pipelines. Thus explaining
the decrease in acoustic velocity with gas flow velocity.
On the other hand, increase in acoustic velocity is unexpected and has not been
observed previously. Increase in velocity can also be observed as contrary to
what could be expected since the system could have gradually emptied for
adhered liquid during the testing. Thus, the observed phenomenon calls for
confirmation and additional experimental work.
104

Acoustic Velocity [m s −1 ]
374
372
370
368
366
364
362
5.5
5.0
4.5
Temperature [ o C]
4.0
3.5
50
52
6.2. Acoustic Velocity
54
Measured
Calculated
56
Pressure [bara]
Figure 6.14: Measured and Calculated Acoustic Velocities Function
of Pressure and Temperature. Second Data Set
Acoustic Velocity [m s −1 ]
374
372
370
368
366
364
Measured
Calculated
362
50 51 52 53 54 55 56 57
Pressure [bara]
Figure 6.15: Measured and Calculated Acoustic Velocities Function
of Pressure. Second Data Set
105
58

Chapter 6. Pressure Wave Attenuation
Acoustic Velocity [m s −1 ]
374
372
370
368
366
364
362
3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4
Temperature [ o Measured
Calculated
C]
Figure 6.16: Measured and Calculated Acoustic Velocities Function
of Temperature. Second Data Set
Acoustic Velocity [m s −1 ]
374
372
370
368
366
364
Measured
Calculated
362
0 5 10 15 20 25
Flow Velocity [m s −1 ]
Figure 6.17: Measured and Calculated Acoustic Velocities Function
of Flow Velocity. Second Data Set
106

6.2.2 Experimental Work
6.2. Acoustic Velocity
A small scale experiment has been set-up at NTNU to observe the hypothesis
validity of acoustic velocity being dependent on flow velocity. A vertical plastic
tube, 60 cm in depth, and 3.9 cm in diameter, is filled with tap water at
atmospheric pressure and room temperature. A small diameter propeller is
located at the surface to rotate the fluid in the water column. Eight emittertransmitter
transducers are located equidistantly along the water column. A
schematic representation of the apparatus is shown in Figure 6.18.
600 mm
1
2
3
4
5
6
7
8
39 mm
Figure 6.18: Schematic Representation of Experimental Apparatus
for Measuring Acoustic Velocity in Fluid Under
Rotation
107
30 mm
60 mm
95 mm

Chapter 6. Pressure Wave Attenuation
Transducers measured sound beam travel times in the fluid with a 500 kHz
sampling frequency. The first transducer is located 95 mm from the surface,
then transducers are equidistant, 60 mm apart from each others. The bottom
of the propeller is located 30 mm from the surface. The experimental tube
is flexible with an average diameter of 39 mm. However, distances between
emitters and receivers are not accurately determined and can evolve, therefore
results are presented in form of travel time and are relative to each transducer.
The propeller rotating velocity and direction are controlled by adjusting the
supplied voltage. The rotating speed is limited by vibrations of the propeller
engine and its support. Maximum velocity is therefore reached for 4.5 V .
The rotating speed increases linearly with voltage, and equals 1280 rpm at 5
V , without charge. The propeller radius is 19 mm. Voltage, corresponding
rotating velocities and linear velocities are listed in Table 6.3. Rotating speed
during experiment are slightly lower due to the resistant torque induced by
water.
Table 6.3: Propeller Engine Voltage and Corresponding Propeller
Rotating Speed and Linear Speed
Voltage [V ] 0 2.5 3.5 4.5
Rotating Speed [rpm] 0 640 896 1152
Linear Speed [ms −1 ] 0 1.27 1.78 2.29
Pressure traces from transducers 1 for four different propeller engine voltages
are shown in Figure 6.19. Travel times are measured for 0 V amplitude. Travel
times are similar and equal 31.6 µs for 0 V and 2.5 V propeller engine voltages.
Then, travel times equal 31.4 µs for both 3.5 V and 4.5 V propeller engine
voltages. Decrease in travel times indicate that pressure waves propagate
faster when the fluid rotating velocity increases. For a 39 mm diameter, the
maximum increase in pressure wave propagation velocity equals 8 ms −1 .
Pressure traces from transducer 2 for four different propeller engine voltages
are shown in Figure 6.20. Travel times are similar and equal 31.1 µs for 0 V
and 2.5 V propeller engine voltages. Then travel times decrease to 30.86 µs
and to 30.84 µs for 3.5 V and 4.5 V , respectively. Decrease in travel times
indicate once again that pressure wave propagate faster when the fluid rotating
velocity increases. For a 39 mm diameter, the maximum increase in pressure
wave propagation velocity equals 11 ms −1 .
Pressure traces from transducer 3 for four different propeller engine voltages
are shown in Figure 6.21. Travel times are similar and equal 31.93 µs for 0
V and 2.5 V propeller engine voltages. Then travel times decrease to 31.9
µs for both 3.5 V and 4.5 V . Decrease in travel times indicate once again
108

6.2. Acoustic Velocity
that pressure wave propagate faster when the fluid rotating velocity increases.
For a 39 mm diameter, the maximum increase in pressure wave propagation
velocity equals 2 ms −1 .
Pressure traces from transducer 4 for four different propeller engine voltages
are shown in Figure 6.22. Travel times are approximately equal to 31.72 µs
in the four different cases. Travel times recorded from transducers 5, 6, 7 and
8 are equal independently of the propeller rotating speed and are not shown
herein. Such observation indicates that the pressure wave propagation velocity
changes with fluid velocity.
The propeller maximum linear velocity equals 2.29 ms −1 whereas a difference
in pressure wave propagation velocity approximating 10 ms −1 is observed.
Thus indicating that fluid velocity affects acoustic velocity. Fluid velocity
changes due to the propeller are greatest the closest to the propeller. However,
the propeller proximity to the surface can induce some air bubbles during
the rotation, hence explaining a greater maximum increase in pressure wave
propagation in transducer 2 than in transducer 1.
Experimental inaccuracies due to vibrations, inappropriate control of propeller
rotating velocity and fluid properties induce that obtained results are only
indicative and not conclusive. However, experimental results tend to confirm
that acoustic velocity depends on flow conditions and structure.
Amplitude [V]
0.03
0.02
0.01
0.00
−0.01
V = 4.5 V
V = 3.5 V
V = 2.5 V
V = 0.0 V
−0.02
31.0 31.2 31.4 31.6 31.8 32.0
Time [µs]
Figure 6.19: Sound Beam Travel Times for Different Propeller
Engine Voltage. Transducer 1
109

Chapter 6. Pressure Wave Attenuation
Amplitude [V]
0.03
0.02
0.01
0.00
−0.01
V = 4.5 V
V = 3.5 V
V = 2.5 V
V = 0.0 V
−0.02
30.6 30.8 31.0 31.2
Time [µs]
Figure 6.20: Sound Beam Travel Times for Different Propeller
Engine Voltage. Transducer 2
Amplitude [V]
0.03
0.02
0.01
0.00
−0.01
V = 4.5 V
V = 3.5 V
V = 2.5 V
V = 0.0 V
−0.02
31.8 31.9 32.0 32.1
Time [µs]
Figure 6.21: Sound Beam Travel Times for Different Propeller
Engine Voltage. Transducer 3
110

Amplitude [V]
0.03
0.02
0.01
0.00
−0.01
V = 4.5 V
V = 3.5 V
V = 2.5 V
V = 0.0 V
6.3. Extended Acoustic Velocity Formulation
−0.02
31.5 31.6 31.7
Time [µs]
31.8 31.9
Figure 6.22: Sound Beam Travel Times for Different Propeller
Engine Voltage. Transducer 4
6.3 Extended Acoustic Velocity Formulation
Acoustic velocity of pressure wave propagating in fluid-filled pipeline depends
on fluid and pipeline elastic properties, and experimental observations indicate
that flow velocity is also an important parameter. Based on the decomposition
of acoustic velocity formulation in fluid isothermal compressibility and pipeline
compressibility, a new expression is proposed for determining acoustic velocity
in single phase fluid-filled pipeline. The acoustic velocity can now be expressed
by:
�
1
a =
(6.5)
ρ (KFluid + KPipe − KFlow)
where, ρ, is fluid density, KFluid, fluid isothermal compressibility, KPipe,
pipeline compressibility, and, KFlow, flow structure compressibility. Compressibilities
are expressed in Pa −1 , that is inverse of pressure. No appropriate
flow structure compressibility expression has yet been found. Nonetheless,
flow structure compressibility should decrease when fluid velocity increases.
Another way to express this is that the fluid becomes stiffer when flowing
velocity increases.
111

Chapter 6. Pressure Wave Attenuation
Based on the first ultrasonic measurement data set (Figure 6.13), a fluid structure
compressibility in the range 10 9 Pa −1 is calculated for a flow velocity of
8 ms −1 and a Reynolds number about 21 10 9 . Curve fitting of acoustic velocity
function of flow velocity with an exponential function is shown in Figure
6.23. The curve fitting matches experimental data with a linear regression
coefficient of 0.9973, and is expressed by:
a = 0.010506 exp(0.26739u) + 373.75 (6.6)
Acoustic Velocity [m s −1 ]
374.4
374.3
374.2
374.1
374.0
373.9
373.8
Experiment
Fit
373.7
0 2 4 6 8 10 12 14 16
Flow Velocity [m s −1 ]
Figure 6.23: Detail of Measured and Calculated Acoustic Velocities
Function of Flow Velocity - First Data Set Figure 6.13 -
and Curve Fitting
Acoustic velocity dependence on flow velocity can be responsible for pressure
wave attenuation. Pressure and flow velocity are dependent variables, and
pressure waves are coupled with fluid velocity waves. Another way to express
this is that fluid velocity changes as a pressure wave propagates. Oscillations of
fluid velocity have been investigated by Yamanaka et al. (2002) and illustrated
in Chapter 3. Changes in pressure and fluid velocity along a pressure wave
then induce acoustic velocity gradients along the pressure wave.
112

6.3. Extended Acoustic Velocity Formulation
Acoustic velocity gradients along a pressure wave then induce acoustic impedance
gradients along the pressure wave. Acoustic impedance gradients are responsible
for pressure wave transmission and reflection. Thus, acoustic impedance
gradients along a pressure wave induce multiple transmission and reflection of
the pressure wave itself as it propagates. Pressure, fluid velocity and acoustic
velocity gradients for left-to-right propagating pressure waves are illustrated
in Figure 6.24.
∆p or ∆u or ∆a
a)
1
0
−1
a2 > a1
⇒
a2 < a1
Distance / Wavelength
b)
1
0
−1
a2 < a1
⇒
a2 > a1
Distance / Wavelength
Figure 6.24: Pressure, Fluid and Acoustic Velocity Gradients in
Left-To-Right Propagating Pressure Waves
Left-to-right propagating positive pressure waves attenuate because the transmission
coefficient is greater than 1, T < 1, Figure 6.24a). Reflected rightto-left
propagating wave will be negative, R < 0, in such case. Left-to-right
propagating negative pressure waves do not attenuate because the transmission
coefficient is greater than 1, T > 1, Figure 6.24b), thus inducing a greater
pressure wave negative amplitude. Reflected right-to-left propagating wave
will be negative, R > 0, in such case.
Pressure wave attenuation due to gradients of acoustic impedance along the
pressure wave itself can be explained in case of positive pressure pulse. Of particular
interest is pressure wave attenuation during total reflection at boundary
conditions. A reflected pressure wave superposition on itself during reflection
generates a pressure wave double in amplitude for a closed boundary, and cancels
itself for a soft boundary. Acoustic impedance gradients along the wave
113

Chapter 6. Pressure Wave Attenuation
are therefore greater during reflection than during propagation. Another way
to express this is that a pressure wave would attenuate more during reflection
than during regular propagation process.
However, negative pressure waves attenuation can not be explained in the
same way. In simple words, the proposed explanation needs to be extensively
investigated based on more detailed and accurate experimental results, both
involving positive and negative pressure pulses, that are pressure waves that
increase and decrease locally and temporarily the in-situ static pressure, respectively.
114

Chapter 7
Impulse Pumping Apparatus
and Experiments
7.1 Apparatus
Impulse pumping is a new concept for transport of fluid in pipelines, patented
by Sagov (2004), and developed by Clavis Impulse Technology AS. In impulse
pumping, the fluid source can be located far away from the external energy
driven mechanism, and the fluid sink can be located anywhere along the flowline.
Impulse pumping is based on the dynamics of pressure wave propagation
while conventional pumps, such as centrifugal equipments, increase the
pipeline in-situ static pressure for transporting fluids.
An impulse pumping apparatus comprises an impulse generator, an outlet nonreturn
valve, an inlet non-return valve, and a fluid-filled flowline. Inlet and
outlet non-return valves connect the fluid-filled flowline to fluid source and sink
tanks, respectively. Inlet and outlet non-return valves are located at distance
L and X of the impulse generator, respectively. The impulse generator is the
only component that requires an external energy supply.
Impulse pumping can be applied to horizontal, inclined and vertical situations,
schematically represented in Figure 7.1, Figure 7.2, and figure 7.3, respectively.
In vertical situation, impulse pumping defines a new artificial lift technique
where fluid is pumped from downhole to wellhead. The impulse generator
is located at wellhead, thus the only external energy requirement is at the
wellhead. The inlet and outlet non-return valves are located at distance L
and X from the impulse generator, respectively.
115

Chapter 7. Impulse Pumping Apparatus and Experiments
In principle, the impulse generator produces a pressure wave that propagates
in the fluid-filled pipeline. Pressure waves then actuate distant outlet nonreturn
valve such that fluid enters the flowline from the inlet tank. Pressure
waves also actuate outlet non-return valve such that fluid exits the flowline into
the outlet non-return tan. Therefore, pressure wave propagation generates a
pumping action where fluid flows from an inlet tank to an outlet tank through
a fluid-filled pipeline.
1
X
3
2
Figure 7.1: Impulse Pumping Apparatus, Horizontal Configuration
1: Impulse Generator
2: Outlet Non-Return Valve
3: Outlet Tank
4: Inlet Non-Return Valve
5: Inlet Tank
1
2
3
X
Figure 7.2: Impulse Pumping Apparatus, Inclined Configuration
116
L
L
4
4
5
5

L
X
5
4
1
2
3
7.1. Apparatus
Figure 7.3: Impulse Pumping Apparatus, Vertical Configuration
1: Impulse Generator
2: Outlet Non-Return Valve
3: Outlet Tank
4: Inlet Non-Return Valve
5: Inlet Tank
117

Chapter 7. Impulse Pumping Apparatus and Experiments
7.2 Inclined Experimental Set-Up and Data
Laboratory experimental data from an early impulse pump prototype for inclined
pumping were obtained from Clavis Impulse Technology AS. An inclined
liquid-water-filled pipeline, 75 m in length, 27 m in height, was connected to
an impulse generator at its upper point, and to a source tank at its lower part,
as shown in Figure 7.2. A sink tank was located nearby the impulse generator.
Inlet and outlet non-return valves separated the flowline from source and sink
tanks, respectively. Fluid was initially at rest and atmospheric temperature.
Experiments were performed at impulse generator frequencies from 2 Hz to 4
Hz, with 0.5 Hz increments. The impulse generator prototype used for vertical
experiments was different from the equipment used for horizontal experiments.
In addition, impulse generator frequency was the only user-set parameter.
Experimental data sets include pressure traces at the impulse generator, at
13 m depth, at 21 m depth, and 27 m depth, that is at the inlet non-return
valve. Corresponding pipeline lengths are listed in Table 7.1. Pressure signals
were acquired with a 800 Hz sampling frequency.
Table 7.1: Conversion between Vertical Height and Pipeline Length
Height [m] Length [m]
13 36.3
21 58.6
27 75
Pressure traces at four locations for a 2 Hz impulse generator frequency are
shown in Figure 7.4. Pressure signals are cyclic, with minimum pressure approximately
0 bara. Maximum pressure amplitudes increase with depth. Maximum
pressures are 1.8 bara, 3.3 bara, 4.4 bara and 5 bara at 0 m, 13 m, 21
m and 27 m depth, respectively. Increase in pressure with depth due to gravity
is approximately 1 bara per 10 meters. Pressure waves attenuation with
distance is not directly observed.
Pressure traces at 13 m, 21 m and 27 m, are divided in three parts. The
pressure approximately at 0 bara first increases rapidly to reach a maximum
value. The pressure then goes down before reaching a new maximum value.
The short decline in pressure is not observed close to the impulse generator,
therefore indicates that the decrease in pressure is induced by an element
further along the pipeline, such as pipeline components or non-return valves.
118

Pressure [bara]
6
5
4
3
2
1
0
27 meter deep
21 meter deep
13 meter deep
0 meter deep
7.2. Inclined Experimental Set-Up and Data
−1
9.6 9.8 10 10.2
Time [s]
10.4 10.6
Figure 7.4: Sample of Pressure Traces at Four Locations Along the
Inclined Flow Line. ˙γ = 2.0 Hz
Pressure traces at four locations for a 3 Hz impulse generator frequency are
shown in Figure 7.5. Pressure signals are cyclic, with minimum pressure approximately
0 bara. Maximum pressure amplitudes increase with depth. Maximum
pressures are 2.5 bara, 7.4 bara, 11.2 bara and 12 bara at 0 m, 13 m,
21 m and 27 m depth, respectively. Increase in pressure with depth due to
gravity is approximately 1 bara per 10 meters. Observed increase in pressure
amplitudes about 10 bara per 30 meters.
Pressure maximum at four locations along the inclined pipeline are shown
against impulse generator frequency in Figure 7.6. Maximum pressure values
increase with impulse generator frequency. At the impulse generator, the
maximum pressure increases from 2 bara at 2 Hz, to 3 bara at 3 Hz, and
to 5.5 bara at 4 Hz. At the inlet non-return valve, the maximum pressure
increases from 4.5 bara at 2 Hz, to 12.9 bara at 3 Hz, and to 13.7 bara at 4
Hz.
Maximum pressures at 13 m depth first increase with frequency, then decreases
slightly before stabilizing around 8 bara. Frequency spectra of pressure signals
are not centered on the user-set impulse generator frequency. Another way
to express this is that measured frequency are in disagreements with user set
impulse generator frequency reported in data logs. Such observation indicates
that the impulse generator frequency calibration was incorrectly performed.
However, pressure measurements are not affected by such inaccuracy.
119

Chapter 7. Impulse Pumping Apparatus and Experiments
Pressure [bara]
14
12
10
8
6
4
2
0
27 meter deep
21 meter deep
13 meter deep
0 meter deep
−2
9.6 9.8 10 10.2
Time [s]
10.4 10.6
Figure 7.5: Sample of Pressure Traces at Four Locations Along the
Inclined Flow Line. ˙γ = 3.0 Hz
Pressure [bara]
14
12
10
8
6
4
2
27 meter deep
21 meter deep
13 meter deep
0 meter deep
0
2 2.5 3
Frequency [Hz]
3.5 4
Figure 7.6: Maximum Pressures at Four Locations for Five Different
Experiments
120

7.3. Horizontal Experimental Set-Up and Data
7.3 Horizontal Experimental Set-Up and Data
Laboratory experiments with an impulse pump prototype were carried out
in March 2007, by NTNU, at Clavis Impulse Technology AS in Hokksund
(Norway). An horizontal liquid-water-filled pipeline, 283 m in length, 1 inch
in diameter, and 1 10 −3 m in wall thickness, was connected to an impulse
generator at one end, and to a source tank at the other end. A sink tank was
approximately located 1 m from the impulse generator. Inlet and outlet nonreturn
valves separated the flowline from source and sink tanks, respectively.
Fluid was initially at rest and room temperature.
Experiments were performed at impulse generator frequencies from 1.3 Hz
to 1.7 Hz, with 0.1 Hz steps, and piston displacements from 0.5 10 −3 m to
1.5 10 −3 m, with 0.25 10 −3 m steps. Experimental data sets include impulse
generator force and piston displacement signals, flow rates measured at the
outlet, and pressure traces at the outlet non-return valve, and at the inlet
non-return valve. Data were acquired with a 1200 Hz sampling frequency.
Data herein shown for presentation purposes were obtained from a particular
experiment at 1.4 Hz and 0.75 10 −3 m piston frequency and displacement,
respectively. Other data corresponding to the range of frequencies and piston
displacements tested during experimental work, exhibited similar properties
as data shown hereunder. Therefore only one data set is presented in the
following of the chapter.
The impulse generator comprises a back and forth moving piston and a freetranslating
disk. The back and forth moving piston is mechanically actuated
at user set frequency and stroke. Piston displacements are transmitted to the
free translating disk through a compressible fluid. Displacement ratio between
the piston and the disk was measured prior to the experimental work and was
equal to 12. Such configuration allowed the mechanically actuated piston to
be separated from the flowline, hence facilitating operation and maintenance
during experimental work.
The back and forth moving piston was fitted with a stroke gauge. The sinusoidal
piston displacement, 9 10 −3 m in peak to peak amplitude for this
particular experiment, is shown in Figure 7.7. The resulting free-translating
disk displacement is then 0.75 10 −3 m in amplitude. The back and forth moving
piston was also fitted with a force gauge. The force signal is a sinusoid
truncated on its upper part. The truncated sinusoidal force on the piston signal,
2.5 10 3 N in amplitude for this particular experiment, is shown in Figure
7.8.
121

Chapter 7. Impulse Pumping Apparatus and Experiments
Stroke [10 −3 m]
4
2
0
−2
−4
−6
−8
75 76 77 78 79 80
Time [s]
Figure 7.7: Sample of Piston Stroke Trace. ˙γ = 1.4 Hz,
∆l = 0.75 10 −3 m
Force [10 3 N]
3.0
2.5
2.0
1.5
1.0
0.5
0.0
75 76 77 78 79 80
Time [s]
Figure 7.8: Sample of Force on the Piston Trace. ˙γ = 1.4 Hz,
∆l = 0.7510 −3 m
122

Normalized Amplitude
1.0
0.8
0.6
0.4
0.2
7.3. Horizontal Experimental Set-Up and Data
0.0
0 0.5 1
Frequency [Hz]
1.5 2
Figure 7.9: Fourier Analysis of Piston Stroke Signal. ˙γ = 1.4 Hz,
∆l = 0.7510 −3 m
Frequency spectrum of the stroke signal using Fourier transform contains a
single 1.4 Hz frequency, as shown in Figure 7.9. Hence, experimental data
correspond to user-set impulse generator frequency and displacement parameters.
A first pressure transducer was located in the fluid-filled pipeline just upstream
of the outlet non-return valve. The outlet pressure signal is a sinusoid
truncated on its upper part. The truncated sinusoidal pressure signal, approximately
4 bara in amplitude for this particular experiment, is shown in
Figure 7.10. Minimum and maximum pressure values at the outlet are 0.75
bara and 4.4 bara, approximately. The truncated upper part of the signal
contains pressure oscillations that correspond to oscillations observed on the
force signal, Figure 7.8.
A second pressure transducer was located in the fluid-filled pipeline just downstream
of the inlet non-return valve. The inlet pressure signal is cyclic, with
maximum pressure 8.4 bara in amplitude, as shown in Figure 7.11. Minimum
pressure values are -0.1 bara, indicating that pressure transducer calibration
was inaccurate and pressure data are reported with an offset. The inlet nonreturn
valve is located further from the impulse generator than the outlet
non-return valve. However, pressure amplitudes observed are greater at the
inlet non-return valve than at the outlet non-return valve.
123

Chapter 7. Impulse Pumping Apparatus and Experiments
Pressure [bara]
10
8
6
4
2
0
75 76 77 78 79 80
Time [s]
Figure 7.10: Sample of Pressure Trace at the Outlet Non-Return
Valve. ˙γ = 1.4 Hz, ∆l = 0.7510 −3 m
Pressure [bara]
10
8
6
4
2
0
75 76 77 78 79 80
Time [s]
Figure 7.11: Sample of Pressure Trace at the Inlet Non-Return
Valve. ˙γ = 1.4 Hz, ∆l = 0.7510 −3 m
124

Flow Rate [l min −1 ]
1.95
1.90
1.85
1.80
1.75
7.3. Horizontal Experimental Set-Up and Data
1.70
75 76 77 78 79 80
Time [s]
Figure 7.12: Sample of Impulse Pumping Flow Rate Trace.
˙γ = 1.4 Hz, ∆l = 0.7510 −3 m
An electromagnetic flowmeter was located at the outlet downstream of the
non-return valve. Flow rate data are shown in Figure 7.12. Presented flow
rate data are averaged over 50 points for clarity. The measured flow rate
fluctuates around a 1.81 l min −1 ±0.1 l min −1 average value, indicating that
impulse pumping induces pulsating flow.
Outlet and inlet pressure traces for 1.4 Hz and 1.25 10 −3 m impulse generator
frequency and displacement, are shown in Figure 7.13 and Figure 7.14,
respectively. The outlet maximum pressure is again approximately 4.4 bara
while the maximum inlet pressure is now 10 bara, thus confirming that the
pressure amplitudes are greater at the inlet non-return valve than at the outlet
non-return valve close to the impulse generator.
Maximum pressure evolution at the outlet non-return valve, function of the
impulse generator displacement for different frequencies is shown in Figure
7.15. Maximum pressure values at the outlet non-return valve increase with
displacement and decrease with frequency. At 1.6 Hz frequency, maximum
pressure amplitude increases from 1.75 bara at 0.5 10 −3 m, to 4.3 bara at 1
10 −3 m, to 5.4 bara at 1.75 10 −3 m. At 0.75 10 −3 m displacement, maximum
pressure amplitude decreases from 5 bara at 1.3 Hz, to 4 bara at 1.5 Hz, and
to 3 bara at 1.6 Hz.
125

Chapter 7. Impulse Pumping Apparatus and Experiments
Pressure [bara]
10
8
6
4
2
0
248 249 250 251 252 253
Time [s]
Figure 7.13: Sample of Pressure Trace at Outlet Non-Return Valve.
˙γ = 1.4 Hz, ∆l = 1.2510 −3 m
Pressure [bara]
10
8
6
4
2
0
248 249 250 251 252 253
Time [s]
Figure 7.14: Sample of Pressure Trace at Inlet Non-Return Valve.
˙γ = 1.4 Hz, ∆l = 1.2510 −3 m
126

Pressure [bara]
12
10
8
6
4
2
γ = 1.6 Hz
γ = 1.5 Hz
γ = 1.4 Hz
γ = 1.3 Hz
7.3. Horizontal Experimental Set-Up and Data
0
0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2
Stroke [10 −3 m]
Figure 7.15: Maximum Pressure Amplitudes at Outlet Non-Return
Valve Against Impulse Generator Displacement for
Different Frequencies
Maximum pressure evolution at the inlet non-return valve, function of the
impulse generator displacement for different frequencies is shown in Figure
7.16. Maximum pressure values at the outlet non-return valve increase with
displacement and decrease with frequency. At 1.5 Hz frequency, maximum
pressure amplitude increases from 7.5 bara at 0.5 10 −3 m, to 10.8 bara at 1
10 −3 m, to 11.2 bara at 1.75 10 −3 m. At 0.75 10 −3 m displacement, maximum
pressure amplitude decreases from 8.9 bara at 1.3 Hz, to 7.8 bara at 1.6 Hz.
Flow rate evolutions for the set of experimental data, from 1.3 Hz to 1.6 Hz
and from 0.5 10 −3 m to 1.5 10 −3 m impulse generator frequencies and displacements,
ares shown in Figure 7.17. The flow rate increases with impulse
generator displacement and frequency. For a 1 10 −3 m impulse generator displacement,
the flow rate increases from 3.13 l min −1 at 1.3 Hz, to 3.29 l min −1
at 1.4 Hz, and to 3.83 l min −1 at 1.6 Hz. For a 1.6 Hz impulse generator frequency,
the flow rate increases from 1.1 l min −1 at 0.5 10 −3 m to 4.87 l min −1
at 1.25 10 −3 m.
The 1.5 Hz experimental flow rate lies below the other experimental flow rates,
with a drop at 1.75 10 −3 ms −1 (or 1.25 10 −3 m impulse generator displacement).
The 1.5 Hz experiment disparity with other experimental flow rates
can indicate experimental measurement errors or be a characteristics linked
to impulse pumping.
127

Chapter 7. Impulse Pumping Apparatus and Experiments
Pressure [bara]
12
10
8
6
4
2
γ = 1.6 Hz
γ = 1.5 Hz
γ = 1.4 Hz
γ = 1.3 Hz
0
0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2
Stroke [10 −3 m]
Figure 7.16: Maximum Pressure Amplitudes at Inlet Non-Return
Valve Function of Impulse Generator Displacement for
Different Frequencies
Flow Rate [l min −1 ]
6
5
4
3
2
1
γ = 1.6 Hz
γ = 1.5 Hz
γ = 1.4 Hz
γ = 1.3 Hz
0
0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2
Stroke [10 −3 m]
Figure 7.17: Experimental Flow Rates Function of Impulse
Generator Displacement for Different Frequencies
128

Chapter 8
Concept, Modelling and
Simulation of Impulse
Pumping
8.1 Concept and Modelling
Impulse pumping is a new pumping concept based on pressure wave propagation
(Sagov, 2004). An impulse pumping apparatus comprises an impulse
generator, an outlet non-return valve and an inlet non-return valve that connect
a fluid-filled pipeline to outlet and inlet tanks, respectively. A schematic
representation of a horizontal impulse pumping apparatus is shown in Figure
8.1, where inlet and outlet non-return valves are located at distance L and X
from the impulse generator, respectively.
8.1.1 Impulse Generator
The impulse generator oscillates back and forth to produce a pressure wave
that propagates in the fluid-filled pipeline on top of the in-situ static pressure.
Forward oscillations generate pressure pulses that increase temporarily the
local pressure. Whereas backward oscillations generate pressure pulses that
decrease temporarily the local pressure. Forward and backward oscillations
then generate positive and negative pressure pulses relative to local in-situ
static pressures, respectively.
The impulse generator can be seen as a back and forth moving piston. The
oscillation function defines a pressure waveform that depends on frequency
and stroke length. For illustration purposes, a sinusoidal pressure waveform
similar to the experimental piston stroke signal is modelled (Figure 7.7). The
impulse generator behaves as a moving boundary condition whose movement is
129

Chapter 8. Concept, Modelling and Simulation of Impulse Pumping
1
X
3
2
Figure 8.1: Impulse Pumping Apparatus, Horizontal Configuration
1: Impulse Generator
2: Outlet Non-Return Valve
3: Outlet Tank
4: Inlet Non-Return Valve
5: Inlet Tank
modelled by a time dependent fluid velocity. The impulse generator boundary
condition can be expressed by:
L
u (0, t) = pWave
ρa sin(ω (t − t0)) (8.1)
where, u, is fluid velocity, pWave, pressure wave amplitude, ρ, fluid density, a,
acoustic velocity in fluid-filled pipeline, ω, angular frequency, and, t0, initial
time of pressure wave generation. The generated pressure wave then propagates
on top of the in-situ static pressure. At maximum pressure amplitude
(sin(ω (t − t0)) = 1), the above equation Eq. 8.1 is similar to the Joukowski
equation:
pWave = ρau (8.2)
Back and forth piston oscillations induce changes in volume in the fluid-filled
pipeline, hence changes in pressure that propagate in the form of a pressure
wave. A pressure wave can consequently be decomposed in countless infinitesimal
displacements of the impulse generator. Changes in pressure and impulse
generator displacements, thus changes in volume, are related by fluid compressibility
(Leighton, 1994):
K = −1 ∂V
V0 ∂p
130
4
5
(8.3)

8.1. Concept and Modelling
where, K, represents fluid compressibility, and, V0, volume of fluid originally
in place. For infinitesimal changes in volume, the first order partial derivative
of volume with pressure included in the fluid compressibility definition Eq. 8.3
can be approximated by:
K ≈ 1
V0
∆V
∆p
(8.4)
Back-and-forth oscillations of the impulse generator can be modelled by displacements
of a piston. Pressure waves can be generated directly in the fluidfilled
pipeline, L in length, and d in diameter, or in a connected pipeline of
greater diameter dp, as illustrated in Figure 8.2. The fluid compressibility (Eq.
8.3) can then be integrated over a piston stroke ∆l:
K =
� dp
d
� 2 1
L
� ∆l
0
∂x
∂p
φ dp
Figure 8.2: Illustration of Impulse Generator Generating Pressure
Waves in Diameter Greater than Flowline Dimensions
φ d
(8.5)
The piston stroke required for generating a pressure wave, pWave in amplitude,
can now be calculated using Eq. 8.5 assuming fluid compressibility changes
negligible with pressure:
� �2 d
∆l = KL
(8.6)
dp
pWave
The piston stroke in Eq. 8.6 is calculated from a central position of the
piston. Another way to express this is that a pressure 2pWave in peak-to-peak
amplitude can be decomposed in one ∆l forward movement, followed by two
∆l backward movements and one ∆l forward movement, as illustrated for a
sinusoidal pressure wave in Figure 8.3. Pressure wave generation therefore
requires four piston strokes per wavelength. Hence, the piston linear velocity,
uPiston, corresponding to oscillation frequency, ˙γ, can now be expressed by:
uPiston = 4∆l ˙γ (8.7)
131

Chapter 8. Concept, Modelling and Simulation of Impulse Pumping
(p − p 0 ) / p Wave
1.0
0.8
0.6
0.4
0.2
0.0
−0.2
−0.4
−0.6
−0.8
↑
Forward
Backward
↓
−1.0
0.0 0.2 0.4 0.6 0.8 1.0
Distance / Wavelength
↓
Backward
Forward
Figure 8.3: Illustration of Piston Back-And-Forth Oscillations for
Sinusoidal Pressure Wave Generation
One wavelength decomposition in 4 piston displacements is theoretical and is
herein used in order to relate piston stroke length and velocity with pressure
wave peak-to-peak semi amplitude. Practically, only two phases make for one
pressure wave cycle: a 2 ∆l forward movement followed by a 2 ∆l backward
displacement.
8.1.2 Pressure Wave Propagation
Generated pressure waves propagate on top of the in-situ static pressure in
the fluid-filled pipeline. Acoustic velocity in hydrocarbon pipelines are in the
range 300 ms −1 for gases, and 750 ms −1 for single phase liquid oils. Fluid is
initially at rest and back and forth impulse generator oscillations induce low
velocity pulsating flow. The corresponding Mach number is then very low.
Hence, assumptions commonly used for water hammer modelling can also be
applied for impulse pumping modelling. Mass and momentum conservation
equations are then, respectively (Ghidaoui et al., 2005):
∂p
+ ρa2∂u = 0 (8.8)
∂t ∂x
∂u
∂t
1 ∂p
+
ρ ∂x
−1
= fu|u| + F (8.9)
2d
132
↑

8.1. Concept and Modelling
where, f, is friction factor, and, F, acceleration due to external forces. Unsteady
friction and pipeline viscoelasticity are omitted for simplicity and due
to the uncertainty of their validity. Therefore, steady-state frictional pressure
drop is the only pressure wave attenuation mechanism included in modelling
of impulse pumping. Pressure wave amplitudes are then overestimated and
waveform changes not accurately modelled. Nonetheless, herein modelling of
impulse pumping uses sinusoidal pressure waves, that is without steep gradients
in amplitude. Thus, changes in pressure waveform are not of importance.
Pressure waves propagation velocity is the acoustic velocity in the fluid-filled
pipeline. The acoustic velocity depends on fluid density, ρ, and isothermal
compressibility, KΓ, and pipeline diameter, d, wall-thickness, e, and material
Young’s modulus, Y . The fluid-filled pipeline is fastened at one end by the
impulse generator and at the other end by the inlet tank. Acoustic velocity
for a fluid-filled thin-walled pipeline is then expressed by (Wylie and Streeter,
1993):
�
�
� 1
a = � �
�
ρ KΓ + d
� (8.10)
Y e
8.1.3 Pipeline Outlet
A generated pressure wave then propagates from the impulse generator to
the outlet non-return valve. The outlet non-return valve can open when the
local pressure in the fluid-filled pipeline, p0, is greater than the valve opening
pressure. The outlet non-return valve opening pressure is defined as the sum
of the outlet tank pressure, pOutlet, and the valve mechanical resistance, or
cracking pressure, pCO. A schematic representation of the outlet non-return
valve is shown in Figure 8.4(a).
Positive pressure pulses increase temporarily the local pressure in the fluidfilled
pipeline at the outlet non-return valve. The outlet non-return valve can
then open and fluid flows from the fluid-filled pipeline into the outlet tank.
The outlet non-return valve opening then limits the pressure inside the fluidfilled
pipeline by truncating the positive pressure pulse upper part. The outlet
non-return valve opening condition, thus the maximum local pressure inside
the fluid-filled pipeline can be expressed by:
p0 (X) > pOutlet + pCO
(8.11)
The fluid initially at rest in the fluid-filled pipeline at the outlet non-return
valve starts flowing out when the valve opens. Consequently, a negative water
hammer pressure wave propagates inside the fluid-filled pipeline. The local
133

Chapter 8. Concept, Modelling and Simulation of Impulse Pumping
p0 (L)
pOutlet
p0 (X)
pCO
(a) Outlet Non-Return Valve
pCI
(b) Inlet Non-Return Valve
pInlet
Figure 8.4: Non-Return Valve Schematic Representations
pressure in the fluid-filled pipeline at the valve then decreases, thus counteracting
the positive pressure pulse opening action. Local pressure fluctuations
induce non-return valve vibrations that generate pressure oscillations of low
amplitude and high frequency.
The outlet non-return valve closes when the inequality Eq. 8.11 is no longer
verified. Truncated pressure waves then propagate from the outlet non-return
valve to the distant inlet non-return valve.
8.1.4 Pipeline Inlet
The inlet non-return valve can open when the local pressure in the fluid-filled
pipeline, p0 (L), is lower than the valve opening pressure. The inlet valve
opening pressure is defined as the inlet tank pressure, pInlet, decreased by the
valve cracking pressure, pCI. A schematic representation of an inlet non-return
valve is shown in Figure 8.4(b).
134

8.2. Numerical Simulation Tool
Negative pressure pulses decrease temporarily the local pressure in the fluidfilled
pipeline at the inlet non-return valve. The inlet non-return valve can
then open and fluid flows from the inlet tank into the fluid-filled pipeline.
Negative pressure pulses then reflect totally with phase change at an open
boundary condition. Another way to express this is that negative pressure
pulses reflect as positive pressure pulses at an open boundary condition. The
inlet non-return valve opening condition can be expressed by:
p0 (L) < pInlet − pCI
(8.12)
The fluid initially at rest in the fluid-filled pipeline at the inlet non-return valve
starts flowing when the valve opens. A positive water hammer pressure wave
therefore propagates inside the fluid-filled pipeline. The local pressure in the
fluid-filled pipeline at the valve then increases, thus counteracting the negative
pressure pulse opening action. Local pressure fluctuations induce non-return
valve vibrations that generate pressure oscillations of low amplitude and high
frequency.
The inlet non-return valve closes when the inequality Eq. 8.12 is no longer
valid. Positive pressure pulses increase temporarily the local pressure in the
fluid-filled pipeline at the inlet non-return valve. The valve then remains
closed and positive pressure pulses reflect totally at a close boundary. Another
way to express this is that positive pressure pulses reflect as positive pressure
pulses. Pressure waves with truncated positive pressure pulses and reflected
phase-changed negative pressure pulses then propagate back to the impulse
generator.
In simple words, a pressure wave that includes a positive pulse and a negative
pulse propagates from the impulse generator in the fluid-filled pipeline. The
positive pressure pulse actuates the outlet non-return valve and fluid flows
into the outlet tank. The negative pressure pulse actuates the inlet nonreturn
valve and fluid flows from the inlet tank. Thus, pressure waves produce
a cyclic pumping action characterized by a pulsating flow.
8.2 Numerical Simulation Tool
Impulse pumping is based on pressure wave propagation that is modelled using
the one dimensional hyperbolic system of mass and momentum conservation
equations. The common low Mach number approximation used in pressure
transient modelling is also applied in modelling of impulse pumping. Impulse
pumping process can then be simulated using the Lax-Wendroff numerical
scheme coupled with the Superbee flux limiter selected in Chapter 5. Simulated
output variables are then fluid velocity and pressure at every grid points
and time steps.
135

Chapter 8. Concept, Modelling and Simulation of Impulse Pumping
A numerical simulation tool was developed based on a Riemann solver with
the Lax-Wendroff numerical scheme and the Superbee flux limiter included
in CLAWPACK. Specific subroutines are implemented to reproduce impulse
pumping apparatus characteristics and behaviour, taking into account fluid
properties and system configuration, dimensions and geometry. The developed
software is intended for petroleum engineers and is therefore best described in
artificial lift applications, that is vertical well as illustrated in Figure 8.5.
3
2
1
X XF
Formation
Figure 8.5: Impulse Pumping Apparatus Applied to Artificial Lift
1: Impulse Generator
2: Outlet Non-Return Valve
3: Outlet Tank
136
L

8.2. Numerical Simulation Tool
Specific subroutines were developed around the Riemann solver to reproduce
impulse pumping characteristics for diverse applications. Each apparatus component
workings as function of pressure and fluid velocity are programmed
according to the developed model. In addition, dedicated subroutines are implemented
to compute fluid properties such as pressure gradient inside the
production tubing, or flow rate, and impulse pumping efficiency. Subroutine
names and short descriptions are listed in Table 8.1.
Table 8.1: List of Subroutines Included in the Impulse Pumping
Software
Subroutine Name Description
fluid Fluid Properties
acoustic Acoustic Properties
pulse Impulse Generator
outlet Outlet Valve Condition
inlet Inlet Valve Condition
flowgrad Inlet Flow Gradient
claw1 CLAWPACK Main Routine
attenuation Pressure Wave Attenuation
rp1 Riemann Solver
out1 Output Results
flowrate Impulse Pumping Flow Rate
efficiency Impulse Pump Efficiency
The ”rp1” routine is the Riemann solver included in CLAWPACK and corresponds
to pressure wave propagation modelling while the ”attenuation” subroutine
integrates the source term. Subroutines ”inlet” and ”outlet” follow
inlet and outlet non-return valve workings, respectively. Subroutines ”fluid”
and ”acoustic” calculate pressure, density and acoustic velocity gradients inside
the production tubing. Subroutines are called successively by the main
routine ”claw1” that interfaces user-set parameters and numerical calculations.
Simulated processes depend on user-set fluid properties, system configuration,
application characteristics and impulse pumping parameters. Fluid properties
are informed at impulse pumping pressure and temperature conditions.
Properties such as density, viscosity or heat capacities can be obtained from
hydrocarbon molar composition using the Peng-Robinson fluid package included
in Hysys. Isothermal compressibilities are calculated by the ratio of
density and pressure gradients at constant temperature.
137

Chapter 8. Concept, Modelling and Simulation of Impulse Pumping
System configuration includes pipeline length, diameter, wall thickness and
roughness, material Young’s modulus, non-return valve locations, types and
cracking pressures. Application characteristics are wellbore pressure and pressure
loss coefficient, and well index productivity. In addition, spatial and
temporal meshes are user-set parameters of the numerical simulation tool.
User-set parameters for the system and application description, and computational
parameters, are listed in Table 8.2.
Table 8.2: List of Parameters Describing Impulse Pumping
Characteristic Variable Unit
Annulus Fluid Level XF [m]
Fluid Isothermal Compressibility KΓ [Pa −1 ]
Fluid Density ρ [kg m −3 ]
Fluid Heat Transfer Coefficient κ [W m −2 K −1 ]
Fluid Pressure p [bara]
Fluid Temperature Γ [ o C]
Fluid Viscosity ν [ o C]
Impulse Generator Frequency ˙γ [Hz]
Impulse Generator Stroke ∆l [m]
Inlet Valve Cracking Pressure pCI [bar]
Inlet Valve Opening Time topeninlet [s]
Inlet Valve Type InletType
Outlet Tank Pressure pOutlet [bara]
Outlet Valve Cracking Pressure pCO [bar]
Outlet Valve Location X [m]
Outlet Valve Opening Time topenoutlet [s]
Outlet Valve Type OutletType
Pipeline Diameter d [m]
Pipeline Length L [m]
Pipeline Material Young’s Modulus Y [Pa]
Pipeline Wall Roughness k
Pipeline Wall Thickness e [m]
Wellbore Pressure pWB [bara]
Wellbore Pressure Loss Coefficient kv
Well Productivity Index PI
Pressure wave propagation characteristics are defined from fluid properties
and system configuration. Acoustic velocity in fluid-filled pipeline is calculated
from fluid and pipeline properties using Eq. 8.10. Acoustic velocity
dependence on pressure and temperature is considered by calculating pressure
and temperature gradients in the production tubing from gravity and heat
138

8.3. Modelling Results
transfer, respectively. Pressure wave attenuation is calculated using steady
friction factor including pipeline wall roughness effects.
Application characteristics and impulse pumping parameters also affect the
pumping action. Application characteristics are wellbore pressure and pressure
loss coefficient, and well productivity index. Impulse pumping parameters
are impulse generator frequency and stroke. Defining parameters, application
description and impulse pumping characteristics are input parameters of the
simulation tool and are listed in Table 8.2. Simulation parameters are number
of grid points and simulation period of pressure wave propagation process.
Valve actuation functions investigated in Chapter 3 are included in subroutines
inlet and outlet. However, a needle valve corresponds to the experimental setup.
Simulation tool output parameters are pressure and fluid velocity along
the pipeline during the simulated period. Such outputs allow for visualization
of impulse pumping process. The reachable flow rate is also an output of the
program.
8.3 Modelling Results
The impulse pumping concept and modelling can now be illustrated and investigated
using the impulse pumping simulation tool. Herein presented illustrations
were obtained from simulations of a 100 meter long vertical liquid-waterfilled
pipeline, 1 inch in diameter, and neglecting pressure wave attenuation.
Simulated pressure waves are 2.5 Hz in frequency, and their amplitudes equal
wellhead pressure. Liquid water density and acoustic velocity are 1000 kg m −3
and 1400 ms −1 , respectively. Pressure waves are then 560 m in wavelength.
Spatial traces of a single pressure wave during impulse pumping action are
synthetically illustrated in Figure 8.6. A left-to-right truncated sinusoidal
pressure wave, 2 pWave in peak-to-peak amplitude before outlet non-return
valve, propagates to the downhole closed inlet non-return valve, Figure 8.6 (a).
The inlet-non-return valve is assumed to open as soon as the local pressure
at the valve in the fluid-filled pipeline becomes lower than the in-situ static
pressure, represented by the 0 pressure line in Figure 8.6.
The negative pressure pulse then reaches the boundary and decreases temporarily
the local pressure at the inlet non-return valve. The valve then opens
and the negative pulse reflects with phase change from an open boundary
condition. Another way to express this is that the negative pressure pulse
becomes positive after reflection at the open inlet non-return valve. The negative
pressure pulse then superimposes on its own phase-changed reflection and
cancels it temporarily, Figure 8.6 (b).
139

Chapter 8. Concept, Modelling and Simulation of Impulse Pumping
The inlet non-return valve closes as soon as the local pressure in the fluidfilled
pipeline equals the in-situ static pressure. Another way to express this is
that the inlet non-return valve closes after total reflection of the negative pressure
pulse. The right-to-left moving reflected phase-changed negative pressure
pulse then superimposes on the left-to-right moving positive pressure pulse,
Figure 8.6(c). The local pressure amplitude is then temporarily the sum of
the two pressure pulses amplitudes. For the illustrated case, the pressure
amplitude is temporarily about twice the original pressure pulse amplitude.
The positive pressure pulse then reflects totally at the closed inlet non-return
valve; that is, without phase change. Another way to express this is that the
positive pressure pulse reflects as a positive pressure pulse of same amplitude.
After complete reflection at the closed inlet non-return valve, the right-to-left
moving pressure wave then propagates from downhole back to wellhead, Figure
8.6 (d). The pumping action occurs only while the inlet non-return valve is
open. Thus, impulse pumping induces pulsating flows.
The water hammer pressure wave due to the inlet non-return valve opening
was not represented for simplification. The water hammer effect can nonetheless
be considered as illustrated in Figure 8.7. Maximum pressure waves when
considering and neglecting the water hammer effect are obtained at the phasechanged
reflected negative pressure pulse, and are 1.12 pWave and pWave, respectively.
The water hammer pressure wave then changes the phase-changed
reflected negative pressure pulse waveform and increases its amplitude.
Wellbore and downhole pressures were first assumed to be equal, and the inlet
non-return valve was assumed to open instantaneously without mechanical
resistance for simplification. However, the valve opening pressure can be lower
than the downhole pressure. As the negative pulse reflects without phase
changed at a closed inlet non-return valve, onto itself, to decrease the local
pressure further, there comes a point where the downhole pressure is less than
the wellbore pressure. The inlet non-return valve will then open and fluid will
flow into the pipeline.
Influence of wellbore static pressure on the spatial trace of a single pressure
wave after reflection at the inlet non-return valve is illustrated in Figure 8.8.
Maximum amplitudes, reported from the partially phase-changed reflected
negative pressure pulse, are 1.12 pWave, 0.75 pWave, 0.25 pWave and -0.25
pWave, for 0, 0.5 pWave, pWave and 1.5 pWave pressure differences between
downhole and wellbore, respectively. The lower the wellbore static pressure,
the less time the inlet non-return valve opens and the less fluid flows into the
production tubing.
140

(p − p 0 ) / p Wave
(p − p 0 ) / p Wave
(p − p 0 ) / p Wave
(p − p 0 ) / p Wave
a)
2.0
1.5
1.0
0.5
0.0
−0.5
8.3. Modelling Results
−1.0
0 0.2 0.4 0.6 0.8 1
b)
2.0
1.5
1.0
0.5
0.0
−0.5
−1.0
0 0.2 0.4 0.6 0.8 1
c)
2.0
1.5
1.0
0.5
0.0
−0.5
−1.0
0 0.2 0.4 0.6 0.8 1
d)
2.0
1.5
1.0
0.5
0.0
−0.5
−1.0
0 0.2 0.4 0.6 0.8 1
Distance / Length
Figure 8.6: Spatial Pressure Traces at Different Instants During
Impulse Pumping Process (In-Situ Static Pressure is
Represented by 0 Pressure Line). Data Presented in
Dimensionless Form.
a): Left-to-Right Moving Truncated Sinusoidal Pressure Wave.
b): Superposition of Negative Pressure Pulse with its Own Phase-Changed
Reflection at Open Valve.
c): Superposition of Positive Pulse with Phase-Changed Reflected Pressure
Pulse.
d): Right-to-Left Moving Truncated Partially Phase-Changed Pressure
Wave.
141

Chapter 8. Concept, Modelling and Simulation of Impulse Pumping
(p − p 0 ) / p Wave
1.25
1.00
0.75
0.50
0.25
0.00
−0.25
−0.50
−0.75
−1.00
With Water Hammer
No Water Hammer
−1.25
0 0.2 0.4 0.6 0.8 1
Distance / Length
Figure 8.7: Influence of Water Hammer Pressure Wave on Synthetic
Pressure Spatial Trace
(p − p 0 ) / p Wave
1.25
1.00
0.75
0.50
0.25
0.00
−0.25
−0.50
−0.75
p Opening = 0
p Opening = −0.5 p Wave
p Opening = −1.0 p Wave
−1.00
p = −1.5 p
Opening Wave
p = −2.0 p
Opening Wave
−1.25
0 0.2 0.4 0.6 0.8 1
Distance / Length
Figure 8.8: Influence of Wellbore Pressure on Synthetic Pressure
Spatial Trace
142

8.4. Validation of Impulse Pumping Model and Simulation Tool
Experimental data are temporal traces from pressure transducers at particular
locations. Synthetic temporal pressure traces at the inlet non-return valve
for two different opening pressures are shown in Figure 8.9. Maximum pressure
amplitudes for both inlet non-return valve opening pressures are 2 pWave,
whereas minimum pressure amplitude is approximately downhole pressure and
zero absolute pressure for 0 and pWave opening pressures, respectively. Another
way to express this is that minimum pressure amplitudes are downhole
pressure decreased by opening pressures.
(p − p 0 ) / p Wave
2.0
1.5
1.0
0.5
0.0
−0.5
−1.0
p Opening = 0
−1.5
0 0.2 0.4 0.6
p = −1 p
Opening Wave
0.8 1
t / T
0
Figure 8.9: Synthetic Temporal Pressure Trace at Downhole Inlet
Non-Return Valve
8.4 Validation of Impulse Pumping Model and
Simulation Tool
Model validation was performed by comparing experimental pressure and flow
rate measurements with results from numerical simulations. Synthetic temporal
pressure traces were computed at the inlet non-return valve and compared
with experimental data. The experimental set-up was a horizontal steel
pipeline, 283 meter in length, 1 inch in diameter, and 1 mm in wall thickness.
Experimental fluid was tap water at atmospheric temperature, that contained
a small amount of air bubbles. Therefore, density and acoustic velocity in the
fluid-filled pipeline are assumed 1000 kg m −3 , and 1200 ms −1 , respectively.
143

Chapter 8. Concept, Modelling and Simulation of Impulse Pumping
In-situ static pressure in the fluid-filled pipeline and the inlet tank were assumed
3 bara and 1 bara, respectively. Impulse generator frequency was assumed
1.5 Hz. The outlet non-return valve cracking pressure was assumed
4.2 bara. Several simulations with different inlet non-return valve cracking
pressure and pressure wave peak-to-peak amplitude are used to model experimental
uncertainties related to these two parameters. Steady frictional
pressure loss is the only pressure wave attenuation mechanism included in the
modelling and the simulations. Numerical simulations are run for one second
of a single pressure wave propagation.
Pressure values are calculated at grid points and the temporal trace at the inlet
non-return valve is shown in Figure 8.10 for 0.2 bar inlet non-return cracking
pressure, and 4.5 bar pressure wave peak-to-peak amplitude. A maximum
pressure value of 14 bara is observed. The pressure peak is due to superposition
of multiple pressure wave; namely, the positive pressure pulse superposition
with its own reflection, and the phase-changed reflected negative pressure pulse
reflected at the closed boundary impulse generator.
Pressure [bara]
16
14
12
10
8
6
4
2
0
−2
0 0.2 0.4 0.6 0.8 1
Time [s]
Figure 8.10: Synthetic Temporal Pressure Trace at Downhole Inlet
Non-Return Valve. ˙γ = 1.5 Hz. L = 283m
144

8.4. Validation of Impulse Pumping Model and Simulation Tool
Synthetic maximum pressure values are overestimated due to the lack of accurate
modelling of pressure wave attenuation. In addition, more oscillations are
expected at the maximum pressure peak due to the outlet non-return valve
oscillations. However, such oscillations were idealized in the modelling. Nevertheless,
synthetic pressure data are in good agreement with experimental
pressure signals, Figure 7.11. Pressure traces are therefore correctly predicted
by the impulse pumping model and the numerical simulation tool developed.
Computed flow rates for the various inlet non-return valve cracking pressure
and pressure wave amplitude conditions are shown in Table 8.3 and illustrated
in Figure 8.11. The reachable flow rate depends greatly on both the inlet nonreturn
valve cracking pressure and the pressure wave peak-to-peak amplitude.
The maximum flow rate of 2.97 l min −1 is reached for a 4.5 bar pressure wave
peak-to-peak amplitude, and 0.4 bar inlet non-return valve cracking pressure.
Flow rate fluctuations are explained by the water hammer pressure wave generated
during inlet non-return valve opening.
The water hammer pressure wave induced by inlet non-return valve opening
counteracts the negative pressure pulse opening action. A low amplitude water
hammer is then required to limit pressure oscillations at the inlet non-return
valve. However, a low flow rate arises from a low amplitude water hammer
pressure wave. Therefore a compromise must be reached to maximize the
reachable flow rate during impulse pumping. Such compromise must be analysed
for every individual applications.
The computed flow rate varies between 2 and 3 l min −1 for the inlet non-return
valve cracking pressures and pressure wave peak-to-peak amplitude simulated.
Simulated results are therefore in the same range as experimental flow rate
between 1 and 4 l min −1 for the same impulse generator frequency, Figure
7.17. Thus, impulse pumping is correctly modelled, and simulated results are
close to experimental measurements.
145

Chapter 8. Concept, Modelling and Simulation of Impulse Pumping
Table 8.3: Computed Flow Rate in [l min −1 ], for Different Inlet
Non-Return Valve Cracking Pressures and Pressure Wave
Peak-To-Peak Amplitudes
Flow Rate [l min −1 ]
3.00
2.75
2.50
2.25
pCI[bar]
2pWave [bar]
4.5 5.0 5.5
0.1 2.19 2.53 2.34
0.2 2.52 2.47 2.38
0.4 2.97 2.47 2.63
p CI = 0.1 bar
p CI = 0.2 bar
p CI = 0.4 bar
2.0
4.50 4.75 5.00
p [bar]
Wave
5.25 5.50
Figure 8.11: Synthetic Flow Rate for Diverse Inlet Non-Return Valve
Cracking Pressures and Pressure Wave Peak-To-Peak
Amplitudes. ˙γ = 1.5 Hz. L = 283m
146

Chapter 9
Design, Applications and
Performance of Impulse
Pumping
9.1 Design
9.1.1 Impulse Generator
Impulse pumping is based on pressure wave propagation induced by backand-forth
oscillations of an impulse generator that can be modelled by piston
displacements. A direct relation can therefore be derived between piston displacements
and pressure wave amplitude (Eq. 8.6). A second equation can
also relate piston velocity and pressure wave amplitude (Eq. 8.7). Impulse
generator characteristics can consequently be derived theoretically according
to impulse pumping conditions.
Evolutions of piston stroke for generating pressure changes, 100 bar in amplitude,
that is a 200 bar peak-to-peak pressure wave, in 1000 m long water-filled
and oil-filled pipelines are shown in Figure 9.1 against ratio of piston head
and pipeline diameters (dp/d in Figure 8.2). Investigated water and oil compressibilities
are 0.435 10 −9 Pa −1 and 1.98 10 −9 Pa −1 , respectively (McCain,
1990). Calculated strokes in water-filled pipeline decrease from 4.35 m, to
0.27 m and to 6.8 10 −3 m for dp/d ratios equal to 1, 4 and 8, respectively.
Calculated strokes in an oil-filled pipeline decrease from 19.8 m, to 1.24 m and
to 0.31 m for dp/d ratios equal to 1, 4 and 8, respectively.
Evolutions of piston linear velocities in a water-filled pipeline for piston frequencies
between 1 Hz and 8 Hz are shown in Figure 9.2. Velocity decreases
with increase in ratio dp/d. At 1 Hz, velocity decreases from 17.4 ms −1 to
1.1 ms −1 , and to 0.27 ms −1 for dp/d ratios equal to 1, 4 and 8, respectively.
147

Chapter 9. Design, Applications and Performance of Impulse Pumping
Stroke [m]
10 4
10 3
10 2
10 1
10 0
10 −1
Oil
Water
10
0 2 4 6 8 10
−2
d / d
p
Figure 9.1: Evolution of Piston Stroke for Generating Pressure Wave
100 bar in Amplitude in a 1000 m Long Pipeline
Piston velocity increases with frequency. For a ratio dp/d equal to 4, velocity
increases from 1.1 ms −1 , to 2.2 ms −1 , and to 8.7 ms −1 for 1 Hz, 2 Hz, and
8 Hz frequencies, respectively.
Evolutions of piston linear velocities in an oil-filled pipeline for piston frequencies
between 1 Hz and 8 Hz are shown in Figure 9.3. Velocity decreases with
increase in ratio dp/d. At 1 Hz, velocity decreases from 79.2 ms −1 to 4.95
ms −1 , and to 1.24 ms −1 for dp/d ratios equal to 1, 4 and 8, respectively.
Piston velocity increases with frequency. For a ratio dp/d equal to 4, velocity
increases from 4.95 ms −1 , to 9.9 ms −1 , and to 39.6 ms −1 for 1 Hz, 2 Hz,
and 8 Hz frequencies, respectively.
Generation of pressure wave using pistons is limited due to reachable strokes
and linear velocities. Low frequency pressure waves require lower piston velocity.
Great dp/d ratios also induce shorter strokes and lower piston velocities.
148

Piston Velocity [m s −1 ]
10 5
10 4
10 3
10 2
10 1
10 0
8 Hz
4 Hz
2 Hz
1 Hz
10
0 2 4 6 8 10
−1
d / d
p
Figure 9.2: Piston Linear Velocity for Generating Pressure Waves
100 bar in Amplitude in a 1000 m Long Water-Filled
Pipeline at Frequencies From 1 to 8 Hz
Piston Velocity [m s −1 ]
10 5
10 4
10 3
10 2
10 1
10 0
8 Hz
4 Hz
2 Hz
1 Hz
10
0 2 4 6 8 10
−1
d / d
p
Figure 9.3: Piston Linear Velocity for Generating Pressure Waves
100 bar in Amplitude in a 1000 m Long Oil-Filled
Pipeline at Frequencies From 1 to 8 Hz
149
9.1. Design

Chapter 9. Design, Applications and Performance of Impulse Pumping
9.1.2 Non-Return Valves
Reachable flow rates theoretically depend on the inlet non-return valve type.
Check valves actuation functions and flow rate evolutions with actuation are
detailed in Chapter 3 and are used herein to evaluate flow rates against nonreturn
valve type. Needle valves are the only non-return valves investigated
in Chapter 3. However, a flap valve with the same actuation function as a
butterfly valve, Figure 3.7 e), but with a different mechanical design is also
investigated.
Reachable flow rates are calculated using needle and butterfly non-return valve
types. Calculations are based on numerical simulations of impulse pumping of
water from a 1000 m deep wellbore at 75 bara constant pressure (corresponding
to 235 m lifting height) using sinusoidal pressure waves, 5 Hz in frequency, 200
bar in peak-to-peak amplitude (Table 9.1). No noticeable differences can be
observed between numerical simulation with needle valve model and butterfly
valve model. Reachable flow rates depend neither on the inlet non-return
valve type or time for full opening, nor on the inlet non-return valve actuation
pattern, uniform or accelerated.
Table 9.1: Description of Test Case for Numerical Investigation of
Pressure Waveform Impact on Reachable Flow Rate
L [m] d [inch] pWH [bara] pWB [bara] pWave [bara]
1000 10 100 75 100
˙γ [Hz] TSim [s] a ρ Valve Type
5 1.5 1200 1000 Needle
nX
nT
10000 1000
Practically, reachable flow rates do not depend on inlet non-return valve type.
Nevertheless, inlet and outlet non-return valve types can be selected based
on mechanical efficiency. Non-return valve mechanical efficiency can be calculated
using actuation functions and flow rate evolutions studied in Chapter 3,
by comparing flow evolutions during real and instantaneous valve actuation.
Mechanical efficiency calculations herein assume symmetry between opening
and closure functions, no mechanical resistance and same actuation time for
every non-return valves and also check-valves investigated.
Mechanical efficiencies for uniform and accelerated actuations for the six checkvalves
considered are shown in Figure 9.4. Uniform actuation efficiency is
approximately 2 % greater than accelerated actuation for every check-valve
investigated. Mechanical efficiencies for uniform actuation are 91.3 %, 87.6 %,
84.6 %, 80.6 %, 76.8 %, and 68.4 % for needle valve, circular gate valve, globe
150

Efficiency [%]
100
90
80
70
60
50
40
30
20
10
0
Uniform
Accelerated
Needle Circular Globe Square Ball Butterfly
Valve Type
9.1. Design
Figure 9.4: Comparison of Check-Valves Efficiencies for Uniform and
Accelerated Actuations
valve, square gate valve, ball valve and butterfly valve, respectively. For nonreturn
valves only, needle valves are more efficient mechanically than butterfly
valves.
Impulse pumping is theoretically applicable for handling difficult fluids such
as highly viscous or non-Newtonian fluids. Non-return valve opening is theoretically
slowed down by fluid viscosity. However, non-return valve closure is
also slowed down by fluid viscosity since non-return valve actuation is herein
assumed symmetric. Highly viscous fluids are therefore assumed handled as
standard fluids in impulse pumping applications.
151

Chapter 9. Design, Applications and Performance of Impulse Pumping
9.2 Multiphase Fluid Flow
Impulse pumping is theoretically applicable to single phase and two-phase
flows; that is gas-liquid, and liquid-liquid. Gas-Liquid two-phase flows are
herein characterized by small amount of gas in liquid that decrease pressure
wave propagation velocity and increase pressure wave attenuation, as compared
to single phase fluids. Pressure wave propagation velocity can be calculated
by (Dong and Gudmundsson, 1993):
a =
⎛
⎜
⎝
(εGcpG + (1 − εG)(εLcpW + (1 − εL)cpO))
(εGcvG + (1 − εG)(εLcvW + (1 − εL)cvO))
(ǫGρG + (1 − ǫG)(ǫLρW + (1 − ǫL)ρO))
(ǫGKΓG + (1 − ǫG) (ǫLKΓW + (1 − ǫL)KΓO))
⎞
⎟
⎠
1/2
(9.1)
where, εG, defines gas-liquid mass fraction, εL, water-oil mass fraction, ǫG,
gas-liquid volumetric fraction, ǫL, water-oil volumetric fraction, ρ, density,
KΓ, isothermal compressibility, cp and cv, heat capacity at constant pressure
and constant volume, respectively, and indexes G, W and O, gas, water and
oil, respectively.
Evolution of acoustic velocity with void fraction is illustrated for a waterair
mixture at 1 bara, 10 bara and 100 bara in Figure 9.5. Acoustic velocity
decreases dramatically for small amount of gas bubbles present in the mixture.
At 1 bara, acoustic velocity decreases from 1516 ms −1 to 141 ms −1 and to
52 ms −1 for 0, 0.01 and 0.08 void fractions. Acoustic velocity increases with
pressure. At 0.5 void fraction, air-water mixture acoustic velocity increases
from 28.2 ms −1 to 73.9 ms −1 , and to 223.7 ms −1 for 1 bara, 10 bara and
100 bara, respectively.
Small amount of gas bubbles can generate cavitation that can be detrimental
and damage irreversibly equipments such as non-return valves. Cavitation
should therefore be avoided during impulse pumping. Above bubble point
pressures, hydrocarbon fluids are single phase in nature. Thus, local pressures
inside the pipeline should not be lowered below bubble point pressure during
impulse pumping, particularly during propagation of the negative pulse of a
pressure wave, or during reflection of the negative part at a closed inlet nonreturn
valve. Non-return valves are idealized and Net Positive Suction Head
(NPSH) is not considered.
The phase envelope corresponding to the volatile oil composition given in Table
4.2 is illustrated in Figure 4.8. The phase envelope corresponding to the black
oil composition given in Table 4.3 is illustrated in Figure 4.12. At 100 o C,
bubble point pressures are 240 bara and 145 bara for volatile and black oil,
152

9.3. Horizontal Transport of Fluids
respectively. Maintaining local pressure inside a pipeline above bubble point
pressure ensures single phase flow without cavitation.
Impulse pumping of gas-liquid two-phase flow can be managed without causing
irreversible damages by maintaining inside local pressures above bubble point
pressures. Water is easier than volatile or black oils to handle since bubble
point pressure is below atmospheric pressure.
Impulse pumping of liquid-liquid two-phase flows can also be obtained without
causing irreversible damages to the production-tubing.
Acoustic Velocity [m s −1 ]
1600
1400
1200
1000
800
600
400
200
100 bara
10 bara
1 bara
0
0 0.2 0.4 0.6 0.8 1
Void Fraction
Figure 9.5: Water Air Mixture Acoustic Velocity Against Void
Fraction at 1 bara, 10 bara and 100 bara
9.3 Horizontal Transport of Fluids
9.3.1 In-Situ Pressure and Pressure Wave Amplitude
Horizontal or quasi horizontal transport of hydrocarbons and water are common
in industry. Pipeline pressures must be kept above fluid bubble point
pressures in order to avoid cavitation during impulse pumping of multiphase
fluids (gas - liquid). In-situ local pressure must consequently be selected for
local pressure not to be lowered further than bubble point pressures, even during
propagation of negative pressure pulses, or during reflection of negative
pressure pulses at a closed inlet non-return valve.
153

Chapter 9. Design, Applications and Performance of Impulse Pumping
Reachable minimum pressures in a horizontal pipeline during impulse pumping
can take place at the inlet non-return valve during reflection of a negative
pressure pulse at a closed boundary. For pressure waves 2 pWave in peakto-peak
amplitude, maximum decrease in pressure during superposition of
a negative pressure pulse on its own reflection equals 2 pWave; that is, the
pressure wave amplitude. The minimum local pressure then becomes:
pminIN = p0 (L) − 2pWave
(9.2)
where, p0 (L), is in-situ local static pressure at the inlet non-return valve. The
minimum local pressure at the inlet non-return valve ensures no flow from the
inlet inside the production tubing. Another way to express this is that the
minimum pressure will never be reached if pumping of liquids is obtained.
In-situ local pressure and pressure wave amplitude must therefore be selected
to ensure inlet non-return valve opening and avoid cavitation inside the horizontal
pipeline. For instance, at 100 o C, horizontal impulse pumping of volatile
oil can be obtained with pipeline in-situ static pressure of 300 bara, pressure
wave peak-to-peak amplitude of 120 bar, and inlet non-return valve opening
pressure greater than 240 bara. At 100 o C, horizontal impulse pumping of
black oil can be obtained with pipeline in-situ static pressure of 300 bara,
pressure wave peak-to-peak amplitude of 310 bar, and inlet non-return valve
opening pressure greater than 145 bara.
Horizontal impulse pumping of water can be obtained for almost any pipeline
in-situ static pressure and pressure wave peak-to-peak amplitudes lower than
two times the in-situ static pressure in the pipeline.
9.3.2 Efficiency
Impulse pumping efficiency in horizontal pipeline or quasi horizontal pipeline
can be calculated directly from impulse generator characteristics and reachable
flow rates. The impulse generator is the only part of an impulse pumping apparatus
that requires external energy supply. Hence, impulse generator characteristics
and generated pressure wave amplitude define the required power.
Modelling an impulse generator by a back and forth oscillating piston, the
required power can be calculated by:
PRequired = FPistonuPiston
(9.3)
where, FPiston, represents force applied on the moving piston, and, uPiston,
piston linear velocity. The force on the piston during one cycle directly relates
to the pressure wave amplitude:
FPiston = πd 2 ppWave
154
(9.4)

9.3. Horizontal Transport of Fluids
Using the piston velocity calculated in Eq. 8.7, Eq. 9.3 becomes:
PR = 4πKL˙γ (d pWave) 2
(9.5)
where, L, is pipeline length, ˙γ, piston oscillation frequency, and, d, pipeline
diameter. Power generated by impulse pumping can be calculated by reachable
flow rate and frictional pressure drop in the pipeline.
Generated power definition is then:
PG = qpf
(9.6)
where, q, characterizes reachable flow rate, and, pf, frictional pressure drop
defined by:
pf = 1 L
f
2 d ρu2
(9.7)
where, f, is friction factor. Power generated at the inlet non-return valve by
impulse pumping can then be simplified as:
PGin = 8 L
π2f d5ρq3 (9.8)
Impulse pumping inflow efficiency for horizontal or quasi horizontal pipelines is
the ratio of generated and required powers. Impulse pumping inflow efficiency
can after simplifications be expressed by:
η = PGin
PR
= 2
π 3
ρfq 3
K ˙γd 7 (pWave) 2
(9.9)
Inflow efficiency of impulse pumping for horizontal or quasi horizontal transport
of fluids depends on fluid density and compressibility, impulse generator
frequency, pressure wave amplitude and reachable flow rate. Pressure wave
attenuation is taken into account by the steady state friction factor, f. Another
way to express this is that estimated efficiency is optimistic since steady
friction only does not model accurately pressure wave attenuation (Chapter
6).
155

Chapter 9. Design, Applications and Performance of Impulse Pumping
9.3.3 Test Case
Impulse pumping of water, volatile and black oils at 100 o C, is considered in
a 1000 m long pipeline, connected to a tank at one end, and to an impulse
generator at the other end. In-situ static pressures are set to 100 bar above
bubble point pressures and inlet non-return valve opening pressures are increased
from 10 bar above bubble point pressure to 1 bar below in-situ static
pressure. Pressure waves 200 bar in peak-to-peak amplitude are used in each
case. Pressure conditions for water, volatile oil and black oil are listed in Table
9.2.
Reachable flow rates against pressure gradient between pipeline and inlet nonreturn
valve opening pressure for pressure waves 4 Hz and 5 Hz in frequency,
and neglecting pressure wave attenuation, are shown in Figure 9.6 and Figure
9.7, respectively. Reachable flow rates using pressure waves 4 Hz or 5 Hz
in frequency are the greatest with volatile oil, and the lowest with water.
Reachable flow rates globally decrease with increase in pressure difference
between pipeline in-situ static pressure and inlet tank.
Reachable flow rates of volatile oil using pressure waves 4 Hz in frequency and
neglecting pressure wave attenuation are 95 m 3 min −1 , 75 m 3 min −1 and 33
m 3 min −1 for pressure differences between pipeline and inlet tank of 1 bara,
40 bara and 90 bara, respectively. Reachable flow rates of volatile oil, black oil
and water, using pressure waves 4 Hz in frequency, for pipeline in-situ static
pressure about 45 bar greater than inlet tank pressure, are 75 m 3 min −1 , 22.2
m 3 min −1 and 10.4 m 3 min −1 , respectively.
Reachable flow rates of volatile oil using pressures 5 Hz in frequency and
neglecting pressure wave attenuation are 60.7 m 3 min −1 , 45.7 m 3 min −1 and
36 m 3 min −1 for pressure differences between pipeline and inlet tank of 1 bara,
40 bara and 90 bara, respectively. Reachable flow rates of volatile oil, black oil
and water, using pressure waves 4 Hz in frequency, for pipeline in-situ static
pressure about 45 bar greater than inlet tank pressure, are 45.7 m 3 min −1 ,
23.4 m 3 min −1 and 10.2 m 3 min −1 , respectively.
Differences between reachable flow rates using pressure waves 4 Hz and 5
Hz in frequency arise from inlet non-return valve oscillations. Water hammer
pressure waves induced by inlet non-return valve opening superimpose on
top of negative pressure pulses that open the inlet non-return valve. Hence
counteract the opening action and are responsible for pressure wave oscillations.
Reachable flow rates are therefore not predictable easily without help
of numerical simulations.
Impulse pumping inflow efficiencies calculated using Eq. 9.9 against pressure
differences between pipeline and inlet tank, assuming a 0.01 friction factor, are
156

9.3. Horizontal Transport of Fluids
Table 9.2: Density, Acoustic Velocity, Bubble Point Pressure, Inlet
Non-Return Valve Opening Pressures and Pipeline In-Situ
Static Pressure for Impulse Pumping of Water, Volatile
Oil and Black Oil at 100 o C
Based on Data From Figure 4.8, Figure 4.12 and McCain (1990)
Fluid
ρ a pBP pOpening p0
[kg m −3 ] [ms −1 ] [bara] [bara] [bara]
Water 1000 1400 1 11 101
Volatile Oil 700 900 240 250 340
Black Oil 720 1000 145 155 245
shown for pressure waves 4 Hz and 5 Hz in frequency in Figure 9.8 and Figure
9.9, respectively. Impulse pumping inflow efficiency is greater for low pressure
difference between pipeline and inlet tank. However, maximum reachable impulse
pumping efficiency for the simulated test case is 3.25 % and is obtained
for pumping of volatile oil. Impulse pumping efficiencies of water and black
oils oscillate between 0.1 and 0.5 %.
Horizontal transport of fluids using impulse pumping can be obtained with
low inflow efficiency. In addition, impulse pumping efficiency must be considered
jointly with additional equipment efficiencies for increasing inlet tank
pressure, if necessary. In-situ static pressure inside the fluid-filled pipeline can
be regulated by the outlet non-return valve.
Impulse pumping can be of interest for horizontal transport of fluids despite
its low efficiency. Of particular interest is the possibility of separating the
impulse generator mechanical part from the main flow line, hence facilitating
maintenance. Of interest is also the high pressure fluid exiting the pipeline.
Such high pressure can help further in processing of petroleum.
157

Chapter 9. Design, Applications and Performance of Impulse Pumping
Flow Rate [10 3 l min −1 ]
100
90
80
70
60
50
40
30
20
10
Volatile Oil
Black Oil
Water
0
0 20 40 60 80 100
p − p [bara]
0 Opening
Figure 9.6: Reachable Flow Rate for Horizontal Impulse Pumping of
Water, Volatile and Black Oils in 1000 m Long Pipeline,
Using Pressure Waves 100 bar in Peak-To-Peak
Amplitude, 4 Hz in Frequency, Against Difference
Between Pipeline and Inlet Non-Return Valve Opening
Pressures, Neglecting Pressure Wave Attenuation
158

Flow Rate [10 3 l min −1 ]
100
90
80
70
60
50
40
30
20
10
9.3. Horizontal Transport of Fluids
Volatile Oil
Black Oil
Water
0
0 20 40 60 80 100
p − p [bara]
0 Opening
Figure 9.7: Reachable Flow Rate for Horizontal Impulse Pumping of
Water, Volatile and Black Oils in 1000 m Long Pipeline,
Using Pressure Waves 100 bar in Peak-To-Peak
Amplitude, 5 Hz in Frequency, Against Difference
Between Pipeline and Inlet Non-Return Valve Opening
Pressures, Neglecting Pressure Wave Attenuation
159

Chapter 9. Design, Applications and Performance of Impulse Pumping
Efficiency [%]
3.5
3.0
2.5
2.0
1.5
1.0
0.5
Volatile Oil
Black Oil
Water
0.0
0 20 40 60 80 100
p − p [bara]
0 Opening
Figure 9.8: Inflow Efficiency for Horizontal Impulse Pumping of
Water, Volatile and Black Oils in 1000 m Long Pipeline,
Using Pressure Waves 100 bar in Peak-To-Peak
Amplitude, 4 Hz in Frequency, Against Difference
Between Pipeline and Inlet Non-Return Valve Opening
Pressures, Neglecting Pressure Wave Attenuation
160

Efficiency [%]
3.5
3.0
2.5
2.0
1.5
1.0
0.5
9.3. Horizontal Transport of Fluids
Volatile Oil
Black Oil
Water
0.0
0 20 40 60 80 100
p − p [bara]
0 Opening
Figure 9.9: Inflow Efficiency for Horizontal Impulse Pumping of
Water, Volatile and Black Oils in 1000 m Long Pipeline,
Using Pressure Waves 100 bar in Peak-To-Peak
Amplitude, 5 Hz in Frequency, Against Difference
Between Pipeline and Inlet Non-Return Valve Opening
Pressures, Neglecting Pressure Wave Attenuation
161

Chapter 9. Design, Applications and Performance of Impulse Pumping
9.4 Artificial Lift
9.4.1 Wellhead Pressure and Pressure Wave Amplitude
Artificial lift is theoretically possible with an impulse pump located at the wellhead
since the vertical lift depends directly on the wellhead pressure. Minimum
reachable pressures are again bubble point pressures in order to avoid cavitation.
In addition, maximum pressure are limited to a maximum allowable
working pressure, pMAWP, of 6000 psia (414 bara), standardized by ASME
B31 (2003). Superposition of pressure waves and in-situ static wellhead pressure
must therefore be maintained between the bubble point pressure and the
maximum allowable working pressure.
In vertical production tubings, minimum pressures occur at wellhead during
propagation of negative pressure pulses, and at bottomhole during reflection of
negative pressure pulses at a closed inlet non-return valve. For pressure waves
2 pWave in peak-to-peak amplitude, maximum decrease in pressure during
superposition of a negative pressure pulse on its own reflection equals 2 pWave;
that is the pressure wave amplitude. The minimum bottomhole local pressure
can then be expressed by:
pminBH = pWH + ρgL − 2pWave
(9.10)
where, pWH, is wellhead pressure, g, gravitational constant, and, L, production
tubing depth. The minimum bottomhole pressure at the inlet non-return valve
ensures no inflow from the wellbore. Another way to express this is that the
minimum bottomhole pressure will never be reached if a pumping action is
obtained.
Wellhead pressures and pressure wave amplitude must therefore be selected to
ensure inlet non-return valve opening and avoid cavitation inside the pipeline.
The wellbore pressure must also be greater than the bubble point pressure.
Optimised wellhead pressure and pressure wave peak-to-peak amplitude can
then be calculated based on bubble point pressure and maximum allowable
working pressure. The optimised wellhead pressure can be expressed by:
pWH = pMAWP + pBP
2
(9.11)
The corresponding optimised pressure wave peak-to-peak amplitude is then:
2pWave = pMAWP − pBP
162
(9.12)

9.4. Artificial Lift
Pumping of fluids from wellbore to wellhead in petroleum production is required
if the production tubing bottomhole pressure is lower than the gravitational
pressure induced by the equivalent ocean water column. In addition,
minimum bottomhole pressure must be greater than the fluid bubble point
pressure at considered temperature conditions. Minimum bottomhole pressure
lower and upper limits can then be expressed by:
pBP < pminBH < ρWgL (9.13)
where, ρW, is water density. Both inequalities in Eq. 9.13 express minimum
production tubing depth conditions. After simplification, the minimum production
tubing depth expressed by the first inequality in Eq. 9.13 becomes:
L1 > pMAWP − pBP
2
1
ρg
(9.14)
where, ρ, represents fluid density. After simplification, the minimum production
tubing depth expressed by the second inequality in Eq. 9.13 becomes:
L2 > −pMAWP + 3pBP
2
1
(ρW − ρ)g
(9.15)
Production tubing depths are then limited by the greater value between L1
and L2. Minimum production tubing depths for impulse pumping of water,
volatile oil and black oil at 100 o C are listed in Table 9.3. Only L1 can be
calculated for water. Production tubing depths are limited by L2 value of
5181 m and L1 value of 1911 m for impulse pumping of volatile and black oil,
respectively. Impulse pumping in production tubings shorter than minimum
calculated values can not be obtained without inducing cavitation that can
cause damages to the equipment.
Minimum production tubing depths are evaluated based on optimised wellhead
pressure and pressure wave peak-to-peak amplitude. However, wellhead
pressure and pressure wave peak-to-peak amplitude can take other values. Of
particular interest is impulse pumping of water with low bubble point pressure
where wellhead pressure can be decreased dramatically and allow impulse
pumping from short depths. For instance, a 10 bara wellhead pressure and
a 20 bar pressure wave peak-to-peak amplitude can be generated for impulse
pumping of water from a 100 m deep water-filled wellbore.
163

Chapter 9. Design, Applications and Performance of Impulse Pumping
Table 9.3: Minimum Production Tubing Lengths L1 in Eq. 9.14 and
L2 in Eq. 9.15 for Impulse Pumping of Water, Volatile Oil
and Black Oil at 100 o C
9.4.2 Efficiency
Fluid
pBP
[bara]
ρ
[kg m
L1 L2
−3 ] [m] [m]
Water 1 1000 2110 ×
Volatile Oil 240 700 1274 5181
Black Oil 145 720 1911 364
Impulse pumping efficiency in vertical production tubings can be calculated
directly from impulse generator characteristics and reachable flow rates. The
impulse generator is the only part of an impulse pumping apparatus that
requires external energy supply. Hence, impulse generator characteristics and
generated pressure wave amplitude define the required power. Modelling an
impulse generator by a back and forth oscillating piston, the required power
can be calculated by:
PR = 4πKL˙γd 2 p 2 Wave
(9.16)
Power generated by impulse pumping applied to artificial lift can be defined
by flow rate and reached vertical lift. Generated power at the inlet non-return
valve can then be expressed by:
PGin = qpLift
(9.17)
where the vertical lift pressure, pLift, can directly be calculated from the
difference in pressure between the wellbore and the production tubing depth
equivalent water column. The vertical lift pressure can then be expressed by:
pLift = ρgL − pWB
(9.18)
After simplifications, inflow efficiency of impulse pumping applied to vertical
lift can be calculated by:
η = PGin
PR
= q (ρgL − pWB)
4πKL˙γd 2 p 2 Wave
164
(9.19)

9.4. Artificial Lift
Inflow efficiency of impulse pumping applied to vertical lift depends on wellbore
pressure, fluid density and compressibility, pressure wave peak-to-peak
amplitude, production tubing depth and reachable flow rate. Of particular
interest is the lack of parameter representing directly pressure wave attenuation
in the previous equation Eq. 9.19 compared to impulse pumping inflow
efficiency for horizontal transport of fluids Eq. 9.9. Pressure wave attenuation
is only represented by the reachable flow rate.
9.4.3 Test Case
Reachable flow rates and impulse pumping inflow efficiencies are calculated for
artificial lift of hydrocarbons from a 10 km deep wellbore through a production
tubing 10 inches in diameter. Impulse pumping parameters for artificial lift
of water, volatile oil and black oil are listed in Table 9.4. Reachable flow rates
are simulated using sinusoidal pressure waves 4 Hz and 5 Hz in frequency,
and neglecting pressure wave attenuation.
Evolution of reachable flow rates against wellbore pressure for sinusoidal pressure
waves 4 Hz and 5 Hz in frequency, neglecting pressure wave attenuation,
are shown in Figure 9.10 and Figure 9.11, respectively. Reachable flow rates
using pressure waves 4 Hz and 5 Hz in frequency are the greatest for black oil.
Reachable flow rates are greater for water than for volatile oil until approximately
930 bara wellbore pressure, then reachable flow rates are the lowest
with water. Reachable flow rates increase with wellbore pressure.
Reachable flow rates of black oil using sinusoidal pressure waves, 4 Hz in
frequency and neglecting pressure wave attenuation, are 36 m 3 min −1 , 92.7
m 3 min −1 and 155 m 3 min −1 for 875 bara, 925 bara and 975 bara wellbore
pressures, respectively. Reachable flow rates of black oil, volatile oil and water,
using sinusoidal pressure waves, 4 Hz in frequency, for a 950 bara wellbore
pressure are 119 m 3 min −1 , 41.7 m 3 min −1 and 20.2 m 3 min −1 , respectively.
Reachable flow rates of black oil using sinusoidal pressure waves, 5 Hz in
frequency and neglecting pressure wave attenuation, are 27.1 m 3 min −1 , 55.2
m 3 min −1 and 156 m 3 min −1 for 875 bara, 925 bara and 975 bara wellbore
pressures, respectively. Reachable flow rates of black oil, volatile oil and water
using sinusoidal pressure waves, 5 Hz in frequency, for a 950 bara wellbore
pressure are 112 m 3 min −1 , 34.1 m 3 min −1 and 56 m 3 min −1 , respectively.
Differences between reachable flow rates using pressure waves 4 Hz and 5 Hz
in frequency arise from inlet non-return valve oscillations.
Evolution of impulse pumping inflow efficiency for sinusoidal pressure waves 4
Hz and 5 Hz in frequency, neglecting pressure wave attenuation, are shown in
Figure 9.12 and Figure 9.13, respectively. Impulse pumping inflow efficiency
using sinusoidal pressure waves 4 Hz and 5 Hz in frequency is the greatest
165

Chapter 9. Design, Applications and Performance of Impulse Pumping
for artificial lift of black oil. Impulse pumping inflow efficiency is greater for
water than for volatile oil until approximately 930 bara wellbore pressure, then
impulse pumping inflow efficiency is the lowest for water.
Impulse pumping inflow efficiencies for artificial lift of black oil, using sinusoidal
pressure waves 4 Hz in frequency and neglecting pressure wave attenuation
are 0.74 %, 1.0 % and 0.18 % for 875 bara, 925 bara and 975 bara
wellbore pressures, respectively. Impulse pumping inflow efficiencies for artificial
lift of black oil, volatile oil and water, using sinusoidal pressure waves 4
Hz in frequency, for a 950 bara wellbore pressure are 0.72 %, 0.34 % and 0.17
%, respectively.
Impulse pumping inflow efficiencies for artificial lift of black oil, using sinusoidal
pressure waves 5 Hz in frequency and neglecting pressure wave attenuation
are 0.45 %, 0.48 % and 0.15 % for 875 bara, 925 bara and 975 bara
wellbore pressures, respectively. Impulse pumping inflow efficiencies for artificial
lift of black oil, volatile oil and water, using sinusoidal pressure waves 4
Hz in frequency, for a 95 bara wellbore pressure are 0.54 %, 0.27 % and 0.08
%, respectively.
Artificial lift of hydrocarbons using impulse pumping is obtained with a very
low inflow efficiency due to the required power to generate pressure waves using
a back and forth moving piston. In addition, impulse pumping of hydrocarbon
requires single phase fluids in order to avoid cavitation that could result in
irreversible damages to the apparatus. Wellbore pressures required to maintain
fluids in single phase state are very seldom in practice. Artificial increase in
wellbore pressure is possible, yet goes against the advantage of having energy
supply requirement close to wellhead only with impulse pumping. Impulse
pumping applied to artificial lift is consequently possible, but not practical.
Table 9.4: Impulse Pumping Parameters for Artificial Lift of Water,
Volatile Oil and Black Oil at 100 o C, from 10000 m Deep
Wellbore, 10 inches in Diameter, Neglecting Pressure
Wave Attenuation
ρ pWH 2 pWave p min(BH)
[kg m −3 ] [bara] [bar] [bara]
Water
1000 207 412 775
Volatile Oil
700 327.5 175 839
Black Oil
720 280 270 716
166

Flow Rate [10 3 l min −1 ]
160
140
120
100
80
60
40
20
Black Oil
Water
Volatile Oil
9.4. Artificial Lift
0
850 875 900 925 950 975 1000
Wellbore Pressure [bara]
Figure 9.10: Reachable Flow Rates Against Wellbore Pressure, for
Vertical Transport of Water, Single Phase Volatile and
Black Oils, at 100 o C, from a 10 km Deep Wellbore,
Using Pressure Waves 4 Hz in Frequency, Neglecting
Pressure Wave Attenuation
167

Chapter 9. Design, Applications and Performance of Impulse Pumping
Flow Rate [10 3 l min −1 ]
160
140
120
100
80
60
40
20
Black Oil
Water
Volatile Oil
0
850 875 900 925 950 975 1000
Wellbore Pressure [bara]
Figure 9.11: Reachable Flow Rates Against Wellbore Pressure, for
Vertical Transport of Water, Single Phase Volatile and
Black Oils, at 100 o C, from a 10 km Deep Wellbore,
Using Pressure Waves 5 Hz in Frequency, Neglecting
Pressure Wave Attenuation
168

Efficiency [%]
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Black Oil
Water
Volatile Oil
9.4. Artificial Lift
0.0
850 875 900 925 950 975 1000
Wellbore Pressure [bara]
Figure 9.12: Theoretical Impulse Pumping Inflow Efficiency Against
Wellbore Pressure, for Vertical Transport of Water,
Single Phase Volatile and Black Oils at 100 o C, from a
10 km Deep Wellbore, Using Pressure Waves 4 Hz in
Frequency, Neglecting Pressure Wave Attenuation
169

Chapter 9. Design, Applications and Performance of Impulse Pumping
Efficiency [%]
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Black Oil
Water
Volatile Oil
0.0
850 875 900 925 950 975 1000
Wellbore Pressure [bara]
Figure 9.13: Theoretical Impulse Pumping Inflow Efficiency Against
Wellbore Pressure, for Vertical Transport of Water,
Single Phase Volatile and Black Oils at 100 o C, from a
10 km Deep Wellbore, Using Pressure Waves 5 Hz in
Frequency, Neglecting Pressure Wave Attenuation
170

Chapter 10
Characteristic Curves of
Impulse Pumping
10.1 Characteristic Curves
Pump characteristic curves illustrate performances of an equipment in terms of
reachable vertical lift and efficiency against flow rate. Characteristic curves of
impulse pumping applied to artificial lift depend on production tubing depth,
wellhead pressure and pressure wave peak-to-peak amplitude. Characteristic
curves are herein presented for particular test cases that illustrate impulse
pumping of aquifer from 10 m deep to 5000 m deep.
Impulse pumping of aquifer, 1000 kg m −3 in density, using a production tubing
0.124 m in diameter is herein investigated through two test cases. The first
test case is characterized by a 21 bara wellhead pressure, a 40 bar pressure
wave peak-to-peak amplitude and a constant 70 m lifting height. The second
test case is characterized by a 207 bara wellhead pressure, a 412 bar pressure
wave peak-to-peak amplitude and a constant 700 m lifting height. Sinusoidal
pressure waves, 1 Hz in frequency are used in both cases.
Characteristic curves for heights between 10 m to 1000 m and between 1000 m
and 5000 m are obtained using the first and the second test case, respectively.
Reachable flow rates are simulated using the impulse pumping simulation tool
described in Chapter 8. Required power and inlet generated power are calculated
using Eq. 9.16 and Eq. 9.17, respectively. Power generated at the outlet
non-return valve due to outflow of high pressure liquid is added to the inlet
generated power to calculate the total efficiency. The generated power at the
outlet non-return valve can be expressed by:
PGout = q(pWH − pOutlet) (10.1)
171

Chapter 10. Characteristic Curves of Impulse Pumping
where, pWH, defines wellhead pressure, q, reachable flow rate, and, pOutlet,
outlet tank pressure. The total efficiency can therefore be expressed by:
η = PGin + PGout
PR
(10.2)
Evolutions of reachable flow rate against production tubing depth for both test
cases are shown in Figure 10.1. Reachable flow rates increase with production
tubing depth for both test cases. Reachable flow rate increases from 151
l min −1 at 100 m depth, to 508 l min −1 at 600 m depth, and to 598 l min −1
at 800 m depth for the first test case. Reachable flow rate increases from 956
l min −1 at 100 m depth, to 4010 l min −1 at 1750 m depth, and to 5870 l min −1
at 3750 m depth for the second test case. Reachable flow rate fluctuations are
due to inlet non-return valve oscillations.
Evolutions of impulse pumping total efficiency against production tubing depth
for both test cases are shown in Figure 10.2 using a 1 bara outlet pressure.
Impulse pumping total efficiency decreases when production tubing depth increases.
Impulse pumping total efficiency decreases from 20.1 % at 100 m
depth, to 11.3 % at 600 m depth, and to 10 % at 800 m depth for the first
test case. Impulse pumping total efficiency decreases when production tubing
depth increases. Impulse pumping total efficiency oscillates from 1.2 % at 1000
m depth, to 2.9 % at 1750 m depth, and to 1.8 % at 3750 m depth for the
second test case.
Impulse pumping is therefore most efficient for artificial lift of shallow water.
Performance of impulse pumping for artificial lift of shallow water can be
illustrated by its characteristic curve. The characteristic curve is obtained
using sinusoidal pressure waves, 1 Hz in frequency, 200 bar in peak-to-peak
amplitude, propagating in a production tubing, 0.124 m in diameter, with
101 bara wellhead pressure, and 1 bara wellbore pressure. The corresponding
characteristic curve is shown in Figure 10.3.
Increase in lifting height induces decrease in reachable flow rate and total
impulse impulse pumping efficiency. Reachable flow rates decreases from 1170
l min −1 at 90 m, to 63.1 l min −1 at 990 m. Impulse pumping total efficiency
decreases from 25.2 % at 90 m, to 0.25 % at 990 m. Oscillations in reachable
flow rate, hence in impulse pumping total efficiency, are due to fluctuations
of pressure in the vicinity of the inlet non-return valve. Impulse pumping
greatest efficiency is then obtained for artificial lift of shallow water.
172

Flow Rate [l min −1 ]
6000
5000
4000
3000
2000
1000
Case 1
Case 2
10.1. Characteristic Curves
0
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Depth [m]
Figure 10.1: Evolution of Impulse Pumping Reachable Flow Rate
Against Production Tubing Depth
Efficiency [%]
25.0
22.5
20.0
17.5
15.0
12.5
10.0
7.5
5.0
2.5
Case 1
Case 2
0.0
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Depth [m]
Figure 10.2: Evolution of Impulse Pumping Efficiency Against
Production Tubing Depth
173

Chapter 10. Characteristic Curves of Impulse Pumping
Height [m]
1000
800
600
400
200
Head
Efficiency
0
0 200 400 600 800 1000 1200
Flow Rate [l min −1 0 200 400 600 800 1000 1200
]
0
0 200 400 600 800 1000 1200
0
Figure 10.3: Characteristic Curve of Impulse Pumping Using
Sinusoidal Pressure Wave, 1 Hz in Frequency, 200 bar
in Peak-To-Peak Amplitude and pHH = 101bara
10.2 Comparison of Artificial Lift Techniques
Jet pumps, electrical submersible pumps (ESP), and gas lift, are three standard
techniques used for artificial lift of water and hydrocarbons in production
tubing. Impulse pumping provides a new artificial lift technique that can be
compared to standard artificial lift by considering a real test case application.
The considered test case is the depressurization of the Statfjord oil field (Boge
et al., 2005).
10.2.1 Test Case Description
The Statfjord field, located on the Norwegian continental shelf in the Tampen
area, is the largest producing oil field in Europe. Oil production from Statfjord
has been optimised over the years by pressure maintenance, drilling of deviated
and horizontal wells, continued infill drilling, water alternating gas injection.
The Statfjord field oil production is to be shifted to mainly gas production by
depressurizing the field. Depressurization is accelerated by pumping aquifer
and injected water from the reservoir. Gas will then be liberated from the
remaining oil and migrate towards the top of the reservoir.
174
30
24
18
12
6
Efficiency [%]

10.2. Comparison of Artificial Lift Techniques
Table 10.1: Desired Flow Rates, Wellbore Pressures and
Corresponding Lifting Heights for the Three
Depressurization Phases of Statfjord Field
Case Stage pWB q [Sm 3 s −1 ] q [Sm 3 min −1 ] H [m]
A Early 200 bara 0.0579 3.747 709.3
B Middle 100 bara 0.0434 2.604 1728.6
C Late 70 bara 0.0289 1.734 2034.4
The depressurization cycle can be divided in three cases that correspond to
different phase of the process. The three stages are early, middle and late
stage of depressurization, and are characterised by three different reservoir
pressures. Wellbore pressures are assumed equal to reservoir pressure for simplifications.
Water is then pumped from wellbore to wellhead using a 2748
m production tubing, 0.124 m in diameter (Pedersen, June 2007). Wellbore
pressures corresponding to the three depressurization stages investigated, and
required flow rates are listed in Table 10.1.
10.2.2 Determination of Impulse Pumping Parameters
Pressure wave peak-to-peak amplitude can be calculated for all three depressurization
phased function of impulse pumping efficiency using Eq. 9.19 and
test case parameters. Pressure wave peak-to-peak amplitudes are calculated
using sinusoidal pressure waves, 4 Hz in frequency for depressurization phase
A, corresponding to a wellbore pressure of 200 bara. Evolutions of pressure
wave peak-to-peak amplitude, wellhead pressure and minimum reachable bottomhole
pressure against impulse pumping efficiency for various flow rate conditions
are shown in Figure 10.4, Figure 10.5 and Figure 10.6, respectively.
Pressure wave peak-to-peak amplitude decreases with increase in impulse
pumping efficiency. For the required flow rate (Table 10.1), pressure wave
peak-to-peak amplitude decreases from 228.7 bar, to 72.3 bar, and to 22.9 bar
for 1 %, 10 % and 100 % impulse pumping efficiency, respectively. Pressure
wave peak-to-peak amplitude increases with flow rate. At 50 % impulse pumping
efficiency, pressure wave peak-to-peak amplitude increases from 32.4 bar,
to 45.8 bar, to 64.7 bar and to 91.5 bar for qRequired, 2 qRequired, 4 qRequired
and 8 qRequired flow rates, respectively.
Wellhead pressure decreases with increase in impulse pumping efficiency. For
the required flow rate, wellhead pressure decreases from 114.4 bara, to 36.2
bara, and to 11.4 bara for 1 %, 10 % and 100 % impulse pumping efficiency,
respectively. Wellhead pressure increases with flow rate. At 50 % impulse
pumping efficiency, wellhead pressure increases from 16.2 bara, to 22.9 bara,
175

Chapter 10. Characteristic Curves of Impulse Pumping
to 32.4 bara and to 45.8 bara for qRequired, 2 qRequired, 4 qRequired and 8 qRequired
flow rates, respectively.
Minimum reachable bottomhole pressure increases with impulse pumping efficiency.
For the required flow rate (Table 10.1), minimum reachable bottomhole
pressure increases from 155.2 bara, to 233.4 bara, and to 258.1 bara for 1 %,
10 % and 100 % impulse pumping efficiency, respectively. Minimum reachable
bottomhole pressure decreases with flow rate. At 50 % impulse pumping efficiency,
minimum reachable bottomhole pressure decreases from 253.4 bara, to
246.7 bara, to 237.2 bara and to 223.8 bara for qRequired, 2 qRequired, 4 qRequired
and 8 qRequired flow rates, respectively.
Impulse pumping is applicable to Statfjord depressurization phase A either
with a very low efficiency or with a flow rate several times greater than the
required flow rate. A 20 % impulse pumping efficiency can theoretically be
reached by pumping at 8 times the required flow rates. The required flow
rate can theoretically be increased by decreasing the number of wells used for
artificial lift while keeping the field total flow rate the same. Required flow rate
must be compared with reachable flow rate in order to assess the applicability
of impulse pumping.
Impulse pumping reachable heights with 207 bara wellhead pressure, using
sinusoidal pressure waves 412 bar in peak-to-peak amplitude, are shown for
different frequencies in Figure 10.7. Reachable heights decrease with increase
in flow rate. For pressure waves 1 Hz in frequency, height decrease following
a straight line from 2238 m to 0 m for 0 m 3 min −1 and 5.37 m 3 min −1 flow
rates, respectively. Height is almost independent of pressure wave frequency,
except at high flow regime where inlet non-return valve oscillations induce
irregular flow rates.
10.2.3 Efficiency of Impulse Pumping
Inflow efficiency of impulse pumping using sinusoidal pressure waves, 1 Hz in
frequency, 412 bar in peak-to-peak amplitude, with 207 bara wellhead pressure,
are shown for different frequencies in Figure 10.8. Impulse pumping efficiency
first increase with flow rates and then decrease. For pressure waves 1 Hz
in frequency, impulse pumping efficiency increases from 0.25 % to 0.53 %,
and then decreases to 0.16 % for 0.25 m 3 min −1 , 2.63 m 3 min −1 and 4.91
m 3 min −1 flow rates, respectively. Impulse pumping efficiency decreases with
increase in pressure wave frequency. For about 2.63 m 3 min −1 reachable flow
rate, impulse pumping decreases from 0.53 % to 0.27 %, and to 0.13 % for
pressure waves 1 Hz, 2 Hz and 4 Hz in frequency, respectively.
176

2 p Wave [bar]
700
600
500
400
300
200
100
10.2. Comparison of Artificial Lift Techniques
q = 8 * q Requested
q = 4 * q Requested
q = 2 * q Requested
q = 1 * q Requested
0
0 20 40 60 80 100
Efficiency [%]
Figure 10.4: Pressure Wave Peak-To-Peak Amplitude Against
Impulse Pumping Efficiency for Artificial Lift During
Statfjord Depressurization Phase A, Using Sinusoidal
Pressure Waves 4 Hz in Frequency (Table 10.1)
Wellhead Pressure [bara]
400
350
300
250
200
150
100
50
q = 8 * q Requested
q = 4 * q Requested
q = 2 * q Requested
q = 1 * q Requested
0
0 20 40 60 80 100
Efficiency [%]
Figure 10.5: Wellhead Pressure Against Impulse Pumping Efficiency
for Artificial Lift During Statfjord Depressurization
Phase A, Using Sinusoidal Pressure Waves 4 Hz in
Frequency (Table 10.1)
177

Chapter 10. Characteristic Curves of Impulse Pumping
Minimum Bottomhole Pressure [bara]
300
250
200
150
100
50
0
q = 1 * q
Requested
q = 2 * q
Requested
−50
q = 4 * q
Requested
q = 8 * q
Requested
−100
0 20 40 60 80 100
Efficiency [%]
Figure 10.6: Minimum Reachable Bottom Hole Pressure Against
Impulse Pumping Efficiency for Artificial Lift During
Statfjord Depressurization Phase A, Using Sinusoidal
Pressure Waves 4 Hz in Frequency (Table 10.1)
Head [m]
2500
2000
1500
1000
500
1 Hz
2 Hz
4 Hz
8 Hz
0
0 1000 2000 3000 4000 5000 6000
Flow Rate [l min −1 ]
Figure 10.7: Reachable Heights and Flow Rates for Depressurization
of Statfjord Field, 2748 m Deep, Using Sinusoidal
Pressure Waves 412 bar in Peak-To-Peak Amplitude
Against Pressure Wave Frequency
178

Efficiency [%]
1.2
1.0
0.8
0.6
0.4
0.2
10.2. Comparison of Artificial Lift Techniques
1 Hz
2 Hz
4 Hz
8 Hz
0.0
0 1000 2000 3000 4000 5000 6000
Flow Rate [l min −1 ]
Figure 10.8: Reachable Inflow Efficiencies and Flow Rates for
Depressurization of Statfjord Field, 2748 m Deep, Using
Sinusoidal Pressure Waves 412 bar in Peak-To-Peak
Amplitude Against Pressure Wave Frequency
Reachable lifting heights are almost independent of pressure wave frequency.
However, impulse pumping efficiency is directly proportional to the inverse of
pressure wave frequency. Consequently, low pressure wave frequencies are preferred
since lifting heights remain unchanged, and impulse pumping efficiency
is then maximized.
Volume of water displaced by the back and forth moving piston during one
pressure wave can be calculated by:
Vd = π d2p ∆l (10.3)
4
where, ∆l, represents piston stroke and is calculated using Eq. 8.6. After
simplifictaions, the displaced flow rate can be expressed by:
qd = π d2
4 KL˙γpWave
(10.4)
The volume of water displaced by the back and forth moving piston, in a
production tubing 0.124 m in diameter, 2748 m in depth, using pressure
179

Chapter 10. Characteristic Curves of Impulse Pumping
waves 1 Hz in frequency, 412 bar in peak-to-peak amplitude then equals 17.8
m 3 min −1 ; that is more than the volume of fluid pumped. Therefore, impulse
pumping is not a volumetric pumping method.
Impulse pumping inflow characteristics that correspond to inflow efficiency
only are obtained from numerical simulations, for selected wellhead pressures
and pressure wave peak-to-peak amplitudes, using sinusoidal pressure waves,
1 Hz in frequency. Impulse pumping inflow characteristics for 52 bara, 104
bara and 207 bara wellhead pressures are shown in Figure 10.9, Figure 10.10
and Figure 10.11, respectively. Corresponding pressure wave peak-to-peak
amplitudes are 102 bar, 206 bar and 410 bar, respectively.
Maximum reachable lifting heights increase with wellhead pressure. Maximum
reachable heights using sinusoidal pressure waves, 1 Hz in frequency, 102 bar,
206 bar and 410 bar in peak-to-peak amplitude, are 709 m, 1219 m and 2238
m, respectively. Lifting heights decrease with flow rates. Maximum reachable
flow rates are, 2.42 m 3 min −1 , 3.68 m 3 min −1 and 5.37 m 3 min −1 , for 52 bara,
104 bara and 207 bara wellhead pressures, respectively.
Maximum impulse pumping efficiency decreases with increase in wellhead pressure.
Maximum impulse pumping efficiency calculated using sinusoidal pressure
waves, 1 Hz in frequency, 102 bar in peak-to-peak amplitude, is 0.99 %,
obtained for 1.82 m 3 min −1 flow rate and 199.6 m head (Figure 10.9). Maximum
impulse pumping efficiency calculated using sinusoidal pressure waves,
1 Hz in frequency, 102 bar in peak-to-peak amplitude, is 0.69 %, obtained for
1.45 m 3 min −1 flow rate and 709.3 m lifting height (Figure 10.10). Maximum
impulse pumping efficiency calculated using sinusoidal pressure waves, 1 Hz
in frequency, 207 bar in peak-to-peak amplitude, is 0.69 %, obtained for 2.63
m 3 min −1 flow rate and 1219 m lifting height (Figure 10.11).
180

Head [m]
2500
2000
1500
1000
500
10.2. Comparison of Artificial Lift Techniques
Head
Efficiency
0.4
0.2
0
0 1000 2000 3000 4000 5000 6000
Flow Rate [l min −1 0 1000 2000 3000 4000 5000 6000
]
0.0
0 1000 2000 3000 4000 5000 6000
0.0
Figure 10.9: Impulse Pumping Inflow Characteristics for 102 bara
Wellhead Pressure, Using Sinusoidal Pressure Waves, 1
Hz in Frequency, 52 bar in Peak-To-Peak Amplitude
Head [m]
2500
2000
1500
1000
500
Head
Efficiency
0.4
0.2
0
0 1000 2000 3000 4000 5000 6000
Flow Rate [l min −1 0 1000 2000 3000 4000 5000 6000
]
0.0
0 1000 2000 3000 4000 5000 6000
0.0
Figure 10.10: Impulse Pumping Inflow Characteristics for 104 bara
Wellhead Pressure, Using Sinusoidal Pressure Waves, 1
Hz in Frequency, 206 bar in Peak-To-Peak Amplitude
181
Efficiency [%]
Efficiency [%]

Chapter 10. Characteristic Curves of Impulse Pumping
Head [m]
2500
2000
1500
1000
500
Head
Efficiency
2.0
1.6
1.2
0.8
0.5
0
0 1000 2000 3000 4000 5000 6000
Flow Rate [l min −1 0 1000 2000 3000 4000 5000 6000
]
0.0
0 1000 2000 3000 4000 5000 6000
0.0
Figure 10.11: Impulse Pumping Inflow Characteristics for 207 bara
Wellhead Pressure, Using Sinusoidal Pressure Waves, 1
Hz in Frequency, 412 bar in Peak-To-Peak Amplitude
Impulse pumping of aquifer with vertical lift of 709.3 m, 1728.6 m, and 2034.4
m is theoretically possible using sinusoidal pressure waves, 1 Hz in frequency,
412 bar in peak-to-peak amplitude with a 207 bara wellhead pressure (Figure
10.11). Reachable flow rates are approximately 3.81 m 3 min −1 , 1.43 m 3 min −1
and 0.65 m 3 min −1 , respectively. Corresponding efficiencies are 0.45 %, 0.41
% and 0.24 %, respectively. Calculated flow rate for phase A is approximately
the requested flow rate, whereas calculated flow rates for phases B and C are
lower than requested (Table 10.1).
Impulse pumping can consequently be applied to Statfjord depressurization
plan. Impulse pumping greatest inflow efficiency is obtained using 207 bara
wellhead pressure, and sinusoidal pressure waves, 1 Hz in frequency, 412 bar
in peak-to-peak amplitude. Total efficiency of impulse pumping with such
values can then be calculated using Eq. 10.2 with a 1 bara outlet pressure,
and reachable flow rates calculated previously.
Impulse pumping total efficiency for phase A of depressurization of Statfjord
is illustrated in Figure 10.12. Impulse pumping total efficiency increases with
reachable flow rate, thus with decrease in lifting height. Impulse pumping
total efficiency increases from 0.5 % at 1983 m, to 1.4 % at 1219 m, and to
1.9 % at 5.9 m. Impulse pumping is theoretically applicable for the three
182
Efficiency [%]

Head [m]
2500
2000
1500
1000
500
10.3. Possible Improvements of Impulse Pumping Efficiency
Head
Efficiency
2.0
1.6
1.2
0.8
0.5
0
0 1000 2000 3000 4000 5000 6000
Flow Rate [l min −1 0 1000 2000 3000 4000 5000 6000
]
0.0
0 1000 2000 3000 4000 5000 6000
0.0
Figure 10.12: Total Impulse Pumping Characteristics for 207 bara
Wellhead Pressure, Using Sinusoidal Pressure Waves, 1
Hz in Frequency, 412 bar in Peak-To-Peak Amplitude
depressurization phases of the Statfjord field. Nevertheless, the very low total
efficiency makes impulse pumping impractical for deep water.
10.3 Possible Improvements of Impulse Pumping
Efficiency
Impulse pumping reachable flow rates and efficiency calculations assume sinusoidal
pressure waves. However, other pressure wave waveforms are theoretically
possible. Sinusoidal, triangular and square waveforms are herein
compared in terms of reachable flow rates during impulse pumping of water
from a 75 bara constant pressure wellbore, 1000 m deep, using a water-filled
production tubing, 10 inches in diameter. Test case characteristics and numerical
simulation parameters are listed in Table 10.2.
Spatial traces of sinusoidal, square and triangular pressure waves propagating
from wellhead to bottomhole are shown in Figure 10.13a), Figure 10.14a),
and Figure 10.15a), respectively. Spatial traces of sinusoidal, square and triangular
pressure waves propagating back from bottomhole to wellhead after
impulse pumping action are shown in Figure 10.13b), Figure 10.14b), and Figure
10.15b), respectively. Negative pressure pulses are reflected with a partial
183
Efficiency [%]

Chapter 10. Characteristic Curves of Impulse Pumping
phase change, and a waveform change, due to the inlet non-return valve opening.
Table 10.2: Description of Test Case for Numerical Investigation of
Pressure Waveform Impact on Reachable Flow Rate
L [m] d [inch] pWH [bara] pWB [bara] pWave [bara]
1000 10 100 75 100
˙γ [Hz] TSim [s] a ρ Valve Type
5 1.5 1200 1000 Needle
nX
nT
10000 1000
Depth [m]
a)
0
100
200
300
400
500
600
700
800
900
1000
0 100 200 300
Pressure [bara]
b)
0
100
200
300
400
500
600
700
800
900
1000
0 100 200 300
Pressure [bara]
Figure 10.13: Spatial Traces of Impulse Pumping Using Sinusoidal
Waveform
a) Propagating From Wellhead to Bottomhole
b) Propagating from Bottomhole to Wellhead
184

Depth [m]
a)
0
100
200
300
400
500
600
700
800
900
10.3. Possible Improvements of Impulse Pumping Efficiency
1000
0 100 200 300
Pressure [bara]
b)
0
100
200
300
400
500
600
700
800
900
1000
0 100 200 300
Pressure [bara]
Figure 10.14: Spatial Traces of Impulse Pumping Using Square
Waveform
Depth [m]
a)
0
100
200
300
400
500
600
700
800
900
1000
0 100 200 300
Pressure [bara]
b)
0
100
200
300
400
500
600
700
800
900
1000
0 100 200 300
Pressure [bara]
Figure 10.15: Spatial Traces of Impulse Pumping Using Triangular
Waveform
a) Propagating From Wellhead to Bottomhole
b) Propagating from Bottomhole to Wellhead
185

Chapter 10. Characteristic Curves of Impulse Pumping
Flow Rate [10 3 l min −1 ]
20
18
16
14
12
10
8
6
4
Square
Sinus
Triangle
2
0 2 4 6 8 10
Frequency [Hz]
Figure 10.16: Reachable Flow Rates Against Impulse Generator
Frequency and Pressure Wave Waveforms
Test Case Characteristics Described in Table 10.2
Evolutions of flow rate using sinusoidal, square and triangular waveforms, for
frequency between 1 Hz and 8 Hz are shown in Figure 10.16. Reachable
flow rates are the greatest with square pressure waves and the lowest with
triangular pressure waves, except at 1 Hz where reachable flow rate is 30.2
m 3 min −1 and 80.3 m 3 min −1 for sinusoidal and triangular pressure waves, respectively.
Reachable flow rates with square pressure waves are 189 m 3 min −1 ,
187 m 3 min −1 , 185 m 3 min −1 and 176 m 3 min −1 , at 1 Hz, 2 Hz, 4 Hz and 8
Hz, respectively.
Evolution of reachable flow rate against impulse generator frequency for sinusoidal
pressure waves shows that maximum flow rates are obtained for
steep wavefronts. Reachable flow rates with sinusoidal pressure waves are
30.2 m 3 min −1 , 103 m 3 min −1 , 150 m 3 min −1 and 149 m 3 min −1 , at 1 Hz,
2 Hz, 4 Hz and 8 Hz, respectively. Greater impulse generator frequencies
induce steeper sinusoidal pressure wavefronts. Another way to express this
is that sinusoidal pressure wavefronts are more moderate when wavelength
increases.
A drop in reachable flow rate for the considered test case is observed for
impulse pumping with sinusoidal pressure waves 5 Hz in frequency in Figure
10.16. Reachable flow rate goes from to 150 m 3 min −1 at 4 Hz, to 80
m 3 min −1 at 6 Hz and to 150 m 3 min −1 at 8 Hz. Such drop in reachable
186

10.4. Cost of Impulse Pumping
flow rate is observed only with sinusoidal pressure waves and can be explained
by oscillations of the inlet non-return valve when the inflow water hammer
pressure wave counteracts the negative pressure pulse opening effect.
Greater impulse generator frequency and square shaped pressure waves can
give rise to greater reachable flow rates. Hence, to greater impulse pumping
efficiency. Different waveforms can be obtained using a back-and-forth oscillating
piston as an impulse generator by changing the piston velocity command.
Ideal square velocity command are not attainable in practice. However, velocity
commands approaching square shaped wavefronts can be implemented
practically.
10.4 Cost of Impulse Pumping
Efficiency is only one criteria for selecting artificial lift technique. Pedersen
(June 2007) concluded that ESP were the most efficient artificial lift technique
for the three depressurization phases of the Statfjord field, with efficiency of
38 % for phase A, and 44 % for phases B and C. Jet pump efficiencies for the
same conditions were 31 % for phase A, 26 % for phase B and 24 % for phase
C. However, ESP apparatus need to be shifted every two years on average,
whereas jet pumps can operate for longer periods without shifting out. In
addition, ESP apparatus are more expensive to buy than jet pumps.
Capital and operational expenditures must be considered during selection of
an artificial lift technique. Capital expenditures, CAPEX, relate to buying
costs whereas operational expenditures, OPEX, relate to maintenance and
shifting of apparatus. OPEX related to impulse pumping are lower than with
ESP or jet pumps since external energy supply requirements are located at
wellhead only. In addition, fluid does not go through the impulse generator;
that is, shifting of impulse generator can be performed without shutting the
production tubing or changing pressure conditions.
Impulse pumping greatest efficiency is obtained for artificial lift of shallow
water. Impulse pumping applied to deep water is theoretically possible despite
its low efficiency. Nevertheless, impulse pumping low efficiency can be
compensated by low OPEX and especially advantages related to maintenance.
Additionally, cost analysis can help compromising on impulse pumping design.
Production tubing parameter is an important impulse pumping design criteria.
Impulse pumping flow rate is directly proportional to production tubing
diameter, but efficiency is proportional to flow rate, and inversely proportional
to production tubing square. Impulse pumping total efficiency for phase A of
depressurization of Statfjord studied above is illustrated for production tubing
1 inch, 2 inch, 4 inch, and 8 inch in diameter in Figure 10.17.
187

Chapter 10. Characteristic Curves of Impulse Pumping
Efficiency [%]
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
d = 1 inch
d = 2 inch
d = 4 inch
d = 8 inch
0.0
0 20 40 60 80 100
Flow Rate [10 3 l min −1 ]
Figure 10.17: Total Impulse Pumping Characteristics for 207 bara
Wellhead Pressure, Using Sinusoidal Pressure Waves, 1
Hz in Frequency, 412 bar in Peak-To-Peak Amplitude,
Production Tubing 1 inch, 2 inch, 4 inch, and 8 inch
in Diameter
Maximum flow rate increases linearly from 1100 l min −1 , to 8800 l min −1
for production tubing 1 inch and 8 inch in diameter, respectively. Whereas
impulse pumping total efficiency decreases from 0.9 % to 0.11 %, for production
tubing 1 inch and 8 inch in diameter, respectively. A compromise between flow
rate and efficiency must therefore be reached, and cost analysis can provide a
selection criteria.
188

Chapter 11
Discussion
Impulse pumping physical principles are based on well established pressure
wave propagation, transmission and reflection characteristics. The description
of impulse pumping physical principles consequently constitutes a solid
mechanistic model. However, a number of uncertainties arise from modelling
details in general, and numerical simulation in particular.
Inlet and outlet non-return valve mechanical characteristics were assumed from
geometrical considerations and static pressure on the upstream. However, the
inlet non-return valve actuation result from reflection of a negative pressure
pulse and induces a positive water hammer pressure wave. Another way to
express this is that the inlet non-return valve upstream pressure is dynamic
and depends on the pressure gradient across the valve.
Non-return valve mechanical characteristics uncertainties influence impulse
pumping flow rate estimations. Impulse pumping flow rates are estimated
from numerical simulation results, based on uncertain non-return valve mechanical
characteristics. However, comparisons between numerical simulation
and experimental data for horizontal transport of water performed with an
unknown valve design, proved to be in good agreement.
Pressure wave attenuation is another uncertainty that influences impulse pumping
flow rate estimations. Impulse pumping flow rate depends on negative pressure
pulse amplitude at bottomhole; that is, after propagation in the fluid-filled
production tubing. However, numerical simulation was performed including
steady state friction only since other attenuation models are inaccurate. Another
way to express this is that estimated flow rate values are optimistic.
Impulse pumping critical parameters are pressure wave amplitude, wellhead
pressure, lifting height, fluid compressibility and volume occupied by the fluid.
However, the fluid volume influences differently the flow rate and the impulse
pumping efficiency. Increase in diameter induces a greater flow rate, but the
189

Chapter 11. Discussion
greater the volume, the greater the energy required to generate a pressure wave
at the wellhead. Impulse pumping efficiency is proportional to the inverse of
the square diameter.
Production tubing diameter is an important impulse pumping design criteria.
However, a compromise must be found between flow rate and efficiency. Such
compromise depends on every situation where impulse pumping is applied,
and represents also an economical criteria.
Comparison between experimental data and results from numerical simulations
were in good agreement. Nevertheless, comparisons were based on a
limited number of experimental data for horizontal transport of water. More
experimental work are therefore necessary for artificial lift situations. Experimental
work is also required to investigate pressure wave attenuation in
production tubings.
A new approach to pressure wave attenuation based on acoustic velocity
changes with fluid velocity within a pressure wave has been proposed. However,
uncertainties linked to the proposed new approach must be assessed both
experimentally and using numerical simulations.
190

Chapter 12
Conclusions
Impulse pumping generates flow from bottomhole to wellhead using pressure
waves generated at wellhead. A positive pressure pulse first actuates a near
wellhead non-return valve that induces outflow from the production tubing.
A negative pressure pulse then actuates a bottomhole inlet non-return valve
that induces inflow to the production tubing. Impulse pumping defines a new
pumping method in addition to dynamic and volumetric pumping methods.
Artificial lift of single phase liquids is theoretically possible without lifting
height limitation using impulse pumping. The lifting height is directly proportional
to the negative pressure wave amplitude that depends on wellhead
pressure. Thus, a one bar increase in negative peak pressure that can be induced
by a one bara increase in wellhead pressure, can generate a 10 m water
lifting height increase.
Impulse pumping greatest efficiency is obtained for artificial lift of shallow water.
An estimated 25 % efficiency can be reached for 100 m water lifting height
according to the presented performance model. Impulse pumping performance
depends on pressure wave amplitude, wellhead pressure, lifting height, fluid
compressibility and volume occupied in the production tubing. The only effect
of frequency is on impulse pumping efficiency.
Impulse pumping of two-phase oil and gas fluids, or single phase live oils, are
not recommended due to cavitation. Impulse pumping of two-phase water
and oil fluids, or dead oils, are possible, yet with low efficiency. Impulse
pumping of deep water wells is also possible, yet with low efficiency due to
power requirements for generating pressure waves in nearly incompressible
fluids.
One-dimensional pressure wave attenuation models based on transient friction
and pipeline viscoelasticity are inaccurate. Modelling of transient friction are
not physically correct and matches between experimental data and results
191

Chapter 12. Conclusions
from numerical simulations using the method of characteristics are purely
coincidental.
Finite volume methods can simulate pressure transient events with greater
accuracy than the well established method of characteristics. Finite volume
methods have less inherent numerical dissipation than the method of characteristics.
Pressure signals and flow velocities during impulse pumping can
therefore be simulated using a finite volume numerical scheme.
192

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200

Appendix A
Technical Reports
Seven technical reports were written as part of the thesis work. A list of titles
and tables of content is given below:
• Propagation of Pressure Waves in Liquid-Filled Pipes (Confidential),
December 2007
1. Introduction
2. Experimental Apparatus
3. Experimental Data
4. Analysis of Inclined Set-Up Data
5. Analysis of Horizontal Set-Up Data
6. Impulse Propagation
7. Standing Wave
8. Flow Rate
9. Discussion
• Simulation of Valve Conditions Using CLAWPACK (Confidential),
January 2008
1. Introduction
2. One Dimensional Wave Equation
3. Numerical Schemes
4. Standard Test-Cases
5. Numerical Scheme for Valve Model
6. Conclusions
• Numerical Schemes, Compressibility and Pulsating Flow (Confidential),
August 2008
201

Appendix A
1. Introduction
2. Further Work on Numerical Schemes
3. Modelling of Fluid Exit Point
4. Velocity Profile from Piston Movement
5. Conclusions
• Water Hammer Simulation of Pressure in the Failed Flexible Hose for
Off-Loading Oil at Statfjord A, October 2008
1. Introduction
2. Wave Transmission at an Interface
3. Simulation Cases
4. Conclusions
• Parameters Affecting the Speed of Sound in Fluids (Confidential),
December 2008
1. Introduction
2. Speed of Sound Definition
3. Experimental Work
4. Ultrasonic Meter Data Analysis
5. Extended Speed of Sound Definition
6. Conclusions
• Horizontal Impulse Pump Performance (Confidential), January 2009
1. Introduction
2. Impulse Pumping
3. Impulse Pump Performance
4. Conclusions
• Vertical Impulse Pump Modelling (Confidential), February 2009
1. Introduction
2. Impulse Pumping
3. Non-Return Valves
4. Vertical Impulse Pumping Principles
5. Pressure and Pressure Gradients
6. Vertical Impulse Pumping Cases
202

7. Examples
8. Conclusions
Appendix A
• Impulse Pumping Design, Model and Simulation (Confidential), December
2009
1. Introduction
2. Impulse Generator
3. Outlet Non-Return Valve
4. Inlet Non-Return Valve
5. Characteristic Curves
6. Test Case
7. Conclusions
203

204

Appendix B
Pumping of Fluids Using
Pressure Impulses
Presented at SPE EUROPEC, Amsterdam , June 2009. Paper SPE 120896
reformatted for the present thesis.
Abstract
Impulse pumping is a new method to pump fluids. The method is described
briefly and then numerical simulations were carried out to illustrate the nature
of impulse pumping. The simulations are based on the linear ideal wave equation
and the numerical approach is based on a second-order Godunov scheme.
The work illustrates how sinusoidal pressure waves are reflected at open and
closed boundaries, representing open and closed non-return valves in impulse
pumping. An impulse pumping example was presented for 40 m lifting of water.
It was shown how the wellhead pressure used determines the lifting depth
and flow rate achievable; the greater the wellhead pressure, the greater the lifting
and rate. In impulse pumping the wellhead pressure is controlled by the
cracking pressure of a non-return outlet valve. Fluid enters the production
tubing through a non-return inlet valve downhole. The stroke length of a pressure
wave generator determines the wellhead pressures achievable in impulse
pumping. It was suggested that the water hammer equation can be used to
estimate the flow rate in impulse pumping.
Introduction
The pumping of fluids is an important unit operation worldwide, including
the petroleum industry. Volumetric and dynamic pumps find different appli-
205

Appendix B
cations. They have in common to be located where fluids are pumped from.
In addition to location, the pumping of difficult fluids is a challenge. The
efficiency and reliability of pumping systems is of considerable importance
(Hydraulic Institute 2008).
A novel pumping concept has been proposed by Sagov (2004), here called
impulse pumping. The actuating unit of an impulse pumping system does not
need to be located where the fluids are pumped from. Instead, an inlet nonreturn
valve is placed at the distant-end of a pipeline or production tubing. In
the case of oil and water wells, the inlet non-return valve is located downhole,
while the actuating unit and associated power source (motor) are place at the
wellhead.
According to the description of impulse pumping by Sagov (2004), a sinusoidal
pressure wave is generated at the wellhead. In an example provided, the
frequency range was from 0.5 to 3.5 Hz and a typical stroke length of the
actuating unit was given as 5 mm. The pressure values reported were in
the range 0 to 10 bara. These are typical values presented by Sagov (2004)
to illustrate one possible embodiment of an impulse pumping system. Other
wave forms are possible, other frequencies are possible and other stroke lengths
are possible.
While further experimental data are not available, it is of interest to consider
the theoretical basis of impulse pumping using established pressure wave
physics and numerical simulations. In this paper the impulse pumping method
will be described based on the information provided by Sagov (2004). Furthermore,
the propagation (transmission and reflection) of pressure waves in
an impulse pumping system will be modelled. Simple models to estimate the
pressure wave amplitude and flow rate are also suggested.
Pressure Wave Propagation
A simplified representation of an impulse pump is shown in Figure 1 based
on the original drawing provided by Sagov (2004). At the left-hand-side an
impulse generator (also called an oscillator) with a back-and-forth moving element
generates a continuous sinusoidal pressure wave, which is superimposed
on the in-situ static pressure. Nearby a non-return outlet valve is located.
The cracking (opening) pressure of the valve will be high and can be adjusted.
In fact, this valve controls the wellhead pressure. At the distant end (righthand-side
of drawing) of the pipe a non-return inlet valve is located. This
valve needs to have a low and fixed cracking pressure. Cracking pressure is
the pressure difference required to open a non-return valve.
The sinusoidal pressure wave propagates at the speed of sound from the im-
206

Appendix B
pulse generator to the outlet valve. In this paper the wave pressure higher
(above) than the in-situ static pressure will be referred to as a positive pressure
and the pressure lower (below) than the in-situ static pressure will be
referred to as a negative pressure. When a positive pressure wave arrives at
the non-return outlet valve, the valve will open and close at the set cracking
pressure. The pressure wave moves further in the direction of the distant end.
When a negative pressure wave arrives at the non-return inlet valve at the
distant end, the valve will open and fluid enters the impulse pumping system.
The nature of pressure waves in fluids has been described by Pain (2005). The
propagation (transmission and reflection) of pressure waves depends on the
acoustic impedance:
z = ρa (B.1)
where, ρ, is fluid density and, a, the acoustic velocity. The reflection, R, and
transmission, T, coefficients for a pressure wave at an impedance interface are,
respectively:
R = pR
pI
T = pT
pI
= z2 − z1
z2 + z1
= 2z2
z2 + z1
(B.2)
(B.3)
For the nomenclature used a pressure wave arrives from medium 1 and is
reflected back into medium 1 and transmitted into medium 2. The subscript,
I, stands for incident.
Representative values of the reflection coefficient, R, are illustrated in Figure
2. The coefficient is plotted against the speed of sound ratio a2/a1 from 0 to 1.
When the ratio is 0 the coefficient R = −1 and the boundary condition can be
described as open, and a pressure wave has total reflection and phase change.
When the speed of sound ratio is 1 the coefficient R = 0 and the boundary
condition can be described as non-existing; that is, the same situation as an
open pipe. Not shown in Figure 2 is when the reflection coefficient R = 1
where the speed of sound ratio tends to infinity and the boundary condition
can be described as closed.
Pressure Wave Simulation
The linear ideal wave equation was used to simulate the propagation of pressure
waves in a fluid in a pipe. The simulation method was based on a secondorder
Godunov numerical scheme implemented in CLAWPACK (LeVque 2002).
207

Appendix B
The typical time step was 1 s and the typical space step was 0.20 m such that
the Courant number was in the range 0.95 and 1. In the simulations the fluid
density was set to 1000 kg m −3 and the speed of sound was set to 1400 ms −1 .
Attenuation mechanisms are not included in the linear ideal wave equation.
Such a simplification has no practical significance for the present work; the
focus of the work has been on illustrating basic principles.
Simulations were carried out to illustrate the nature of reflections from closed
and open boundaries in a horizontal pipe. A closed boundary represents a
solid boundary such as a closed valve. An open boundary correspondingly
represents an open valve. The grid chosen was 1 m long and a sinusoidal wave
10 bar positive and 10 bar negative from the pressure wave generator was
used. In the modelling the positive pressure wave was truncated by 2.5 bar to
illustrate the effect of an outlet non-return valve. The simulation results are
shown in Figure 3.
Figure 3a illustrates a pressure wave travelling from left-to-right after having
passed an outlet non-return valve. Two cases of reflections are illustrated
in (b) and (c). Figure 3b illustrates the shape of the same pressure wave
after having reflected from a closed boundary, moving from right-to-left in the
figure. It shows that the pressure wave reflects without phase change. Figure
3c illustrates the shape of the same pressure wave after reflecting from an
open boundary, again moving from right-to-left in the figure. It shows that
the pressure wave reflects with phase change.
Simulations were then carried out to illustrate the nature of reflections at open
and closed boundaries in a vertical pipe. The grid chosen was 100 m long
(deep) and a static pressure of 10 bara at the pressure generator. In other
words, 10 bara wellhead pressure. A single superimposed sinusoidal pressure
wave 10 bar positive and 10 bar negative was chosen. Because the simulations
were carried out for illustration purposes, a high pressure wave frequency was
used for convenience. The simulation results are shown in Figure 4 at two
times/locations for a propagating pressure wave.
Figure 4a illustrates a single truncated pressure wave travelling downward.
It is superimposed on the sum of the wellhead pressure and the hydrostatic
pressure. Figure 4b illustrates the same pressure wave after reflection from
an open non-return inlet valve at 100 m depth (downhole). In the numerical
simulations the non-return inlet valve can be modelled as opening immediately
when the pressure wave starts to become negative. It can also be modelled
as opening at different percentages of the negative pressure wave. In the case
illustrated in Figure 4b the valve was modelled to open when the negative
pressure wave reached 25 percent of its maximum value of 10 bara. The valve
was modelled to close on the reverse cycle of the negative pressure wave, again
208

at the 25 percent value.
Appendix B
The above simulations say nothing about the pumping action of an impulse
pump. The simulations just illustrate how a pressure wave reflects from a
non-return inlet valve when closed and when open, irrespective of how and
why it opens.
Downhole Pumping Action
In the text above the nature of pressure wave reflections at open and closed
boundaries were illustrated; an open boundary represents an open valve and
a closed boundary represents a closed valve. What remains to be illustrated is
how the downhole non-return inlet valve allows pumping action to take place.
Fluid will flow from the wellbore into the production tubing when the outside
pressure is higher than the inside pressure. A non-return inlet valve should
preferably have a low cracking pressure.
The static downhole pressure in the tubing will be the sum of the wellhead
pressure and the hydrostatic pressure. The pressure wave generated at the
wellhead will be superimposed on the static pressure. To create a low pressure
inside the tubing downhole, the wellhead pressure needs to be equally large.
At the wellhead, the pressure wave generator generates a pressure wave with a
positive part and a negative part. The negative part of the pressure wave can
never lower the static pressure below zero absolute pressure. It follows that
the negative part of a pressure wave can never be larger than the wellhead
pressure.
Assume a water well 100 m deep with a wellhead pressure of 10 bara. The
downhole pressure (at the inside tubing inlet) will be 20 bara. Assume a
pressure wave is generated at the wellhead with amplitude of 20 bar such
that the negative part of the wave is 10 bar. When this negative part of
the pressure wave arrives at a non-return inlet valve downhole, the valve is
closed and the pressure will reflect without phase change. In other words, the
negative pressure will reflect as negative pressure, hence further decreasing the
pressure.
As the reflected negative pressure reflects onto itself to decrease the pressure
further, there comes a point where the inside tubing pressure is less than the
outside wellbore pressure. When that point is reached, the non-return valve
will open and fluid will flow into the production tubing. While the non-return
valve is open, the negative pressure wave will no longer reflect as negative,
but it will reflect positively because of phase change. Thereafter there comes
a point when the low pressure inside the tubing is not sufficient to keep the
non-return valve open. The flow stops and the non-return valve remains closed
209

Appendix B
until the next negative pressure wave arrives downhole.
For the assumed 100 m deep water well above, the pressure wave has the
potential to decrease the downhole pressure by maximum 20 bar; that is,
twice the negative part of the pressure wave. Assuming that the wellbore
pressure outside the production tubing is 6 bara, a non-return inlet valve with
1 bar cracking pressure will open when 75% of the negative pressure wave
has reflected. The pressure difference between the outside wellbore and the
inside production tubing will be 1 bar (pressure inside production tubing will
be 5 bara). It is this pressure difference that makes fluid flow from outside to
inside. The valve stays open until the negative pressure wave swings back to
three-quarters (75%) of its value.
The pressure with depth in the assumed 100 m deep water well is shown in
Figure 5. Superimposed on the pressure with depth is the pressure wave reflected
(travelling from downhole to wellhead) from the non-return inlet valve.
The vertical lift in the assumed well is 40 m. For the assumed wellhead pressure
and non-return inlet valve cracking pressure, the theoretical maximum
vertical lift is 90 m for the case presented. However, for such vertical lift the
pumping rate will be zero. For vertical lift less than 90 m but greater than 40
m, the pumping rate will be less than for 40 m. This because the non-return
inlet valve will be open for a shorter time (during the full cycle of the negative
pressure wave) the greater the vertical lift.
Pump Performance
It is of interest to estimate the performance of impulse pumping in terms
of pressure and flow rate. First, a relationship between the stroke length of
the actuator and the pressure wave amplitude will be established. Second, a
relationship between the cracking pressure of the non-return inlet valve and
the flow rate through the valve will be suggested.
The isothermal compressibility, K, of fluids is given by the equation
K = −1 dV
V dp
(B.4)
where V is volume and p is pressure. In impulse pumping the system volume
V is the fluid volume in the pipe between the generator and the non-return
inlet valve, depending on the pipe length L and diameter d. And the change in
fluid volume dV is that resulting from the compression exerted on the system
by the pressure wave generator, depending on the stroke length x and effective
diameter. Assuming that the actuator and pipe have the same diameter, it
210

Appendix B
can be shown (using the compressibility equation) that the amplitude of a
pressure wave generated can be expressed by the relationship:
∆pImpulse = x 1
L K
(B.5)
The isothermal compressibility of water at standard conditions is reported as
4.65 10 −10 Pa −1 (Yaws 1999). For a pipe length of 100 m and stroke length
of 5 mm, the pressure amplitude was calculated 10.75 bar. We observe that
the maximum pressure reported by Sagov (2004) was about 10 bara.
In the present paper, the water hammer equation is assumed to control the
fluid velocity through the non-return inlet valve. The equation, also called the
Joukowski equation, can be expressed as:
∆pHammer = ρa∆u (B.6)
where ∆u represents the change in fluid velocity from zero to the velocity
through the non-return inlet valve. If the water hammer pressure drop is
assumed equal to the cracking pressure of the non-return inlet valve, the effective
fluid velocity can be calculated. Using speed of sound 1400 m/s and
water density of 1000 kg/m3 and a cracking pressure of 1 bar, the flow velocity
becomes 0.0714 ms −1 .
Fluid will flow into the production tubing only while the non-return valve is
open. The valve will open periodically. This period can be expressed as a
fraction of the negative pressure wave. In the example above the non-return
valve opened when the pressure reach 75% of the negative pressure wave.
Approximating a sinus wave by a triangle the non-return valve will be opened
25% of the full wave cycle. Therefore, the volumetric flow rate, q, through the
non-return valve can be expressed by the relationship:
q = αuA (B.7)
where, α is the cycle fraction and A the flow area. The flow rate depends on
the diameter of the production tubing. For example, for 1 inch ID the flow
rate will be 0.55 litres per minute. Similarly, for 2 inch and 4 inch ID the
volumetric flow rate will be 2.2 litres per minute and 4.4 litres per minute,
respectively. The flow rates are ideal because pressure losses due to valve
mechanics, fluid acceleration/deceleration and wall friction in the production
tubing have not been included.
211

Appendix B
Conclusions
Numerical simulations were used to illustrate the nature of pressure wave
propagation in impulse pumping systems. In particular how pressure waves
reflect at open and closed boundaries.
Analytical considerations presented illustrate how the wellhead pressure determines
the vertical lift capability of liquid filled wells. The higher the wellhead
pressure, the greater the vertical lift and flow rate achievable.
One particular example was presented where liquid water was lifted 40 m. The
wellhead pressure assumed was 10 bara, while no lifting would be achieved with
a wellhead pressure of 6 bara.
The stroke length of the pressure wave generator determines the amplitude of
the pressure wave superimposed on the static pressure.
The flow rate in impulse pumping can be related to the water hammer equation.
For the example presented, assuming 4 inch production tubing, the flow
rate was 4.4 litres per minute.
Acknowledgement
We thank Clavis Impulse Technology AS for providing a Ph.D. scholarship to
Benjamin Pierre. However, the ideas and results presented in the paper are
solely the responsibility of the authors.
References
LeVeque R.J. (2002): Finite Volume Methods for Hyperbolic Problems, Cambrigde
University Press.
Pain H.J. (2005): The Physics of Vibrations and Waves, Wiley.
Sagov M. (2004): Method and Apparatus for Transporting Fluid in a Conduit,
Norwegian Patent No. 20045382 (in Norwegian).
Yaws C.L. (1999): Chemical Properties Handbook, Mac-Graw Hill.
212

Nomenclature
a Acoustic Velocity [ms−1 ]
A Flow Area [m2 ]
K Fluid Compressibility [Pa−1 ]
L Pipe Length [m]
p Pressure [Pa]
q Volumetric Flow Rate [l min−1 ]
u Fluid Velocity [ms−1 ]
V Fluid Volume [m3 ]
x Stroke Length [m]
z Acoustic Impedance [kg s−1 m−2 ]
z1 Acoustic Impedance in Medium 1 [kg s−1 m−2 ]
z2 Acoustic Impedance in Medium 2 [kg s−1 m−2 ]
∆pHammer Water Hammer Amplitude [Pa]
∆pImpulse Half of the Pressure Wave Amplitude [Pa]
α Cycle Fraction
ρ Fluid Density [kg m−3 ]
Figures
1
h
3
2
L
4
Appendix B
Figure B.1: Impulse Pumping Apparatus, Horizontal Configuration
1: Impulse Generator
2: Outlet Non-Return Valve
3: Outlet Tank
4: Inlet Non-Return Valve
5: Inlet Tank
213
5

Appendix B
Reflection Coefficient
0
−0.1
−0.2
−0.3
−0.4
−0.5
−0.6
−0.7
−0.8
−0.9
−1
0 0.1 0.2 0.3 0.4 0.5
z / z
2 1
0.6 0.7 0.8 0.9 1
Figure B.2: Representative Values of the Reflection Coefficient
10
0
−10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Distance [m]
10
p [bar] a)
0
−10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Distance [m]
10
p [bar] b)
0
−10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Distance [m]
p [bar] c)
Figure B.3: Pressure Wave (a) After Passing a Non-return Outlet
Valve, (b) After Reflecting From a Closed Boundary, (c)
After Reflecting on an Open Boundary
214

Depth [m]
a)
0
10
20
30
40
50
60
70
80
90
100
0 10 20 30
Pressure [bara]
b)
0
10
20
30
40
50
60
70
80
90
100
0 10 20 30
Pressure [bara]
Appendix B
Figure B.4: Truncated Pressure Wave Travelling (a) Downward, (b)
After Reflecting From a Non-Return Valve
Depth [m]
0
10
20
30
40
50
60
70
80
90
100
0 5 10 15 20 25 30
Pressure [bara]
Figure B.5: Truncated Pressure Wave Travelling After Reflecting
From a Non-Return Valve
215

216

Appendix C
Modelling of Pressure Wave
Propagation in Pipelines:
Steady-State and Transient
Friction
Presented at MekIT’09, Fifth National Conference on Computational Mechanics,
Trondheim, 26-27 May 2009. Paper reformatted for the present thesis.
Abstract
The physics of the water hammer effect are used to model rapid pressure transients
in wells and pipelines and more generally in fluid transport systems.
Traditionally, the Method of Characteristics (MOC) is used to solve numerically
the governing partial differential equations. In addition to the water
hammer effect, effects due to line packing, pressure attenuation and transient
friction need to be included in comprehensive modelling. Recent works have
indicated that the errors inherent in the use of the MOC tend to mimic the
errors in the formulation of the transient friction factor. Numerical simulations
have been carried out using several transient friction factor formulations
in the hyperbolic system of equations describing rapid pressure transients. The
results argue for the necessity to develop new formulations that accurately represent
the physics of rapid pressure transient propagation in fluid transport
systems. The MOC needs also to be replaced by higher resolutions numerical
methods such as a second order finite volume method.
217

Appendix C
Introduction
Fluid transport systems, for instance transport and distribution systems of
oil and gas, are subject to rapid pressure transients. Rapid pressure transients
arise when a quick-acting valve is actuated, or when a pump is suddenly
stopped. Rapid pressure transient in the form of water hammer can lead
to pipe rupture (Pierre and Gudmunsson, 2008). Therefore, rapid pressure
transients need to be accurately predicted.
Rapid pressure transients in the form of pressure wave can be used in fluid
transport systems. In impulse pumping (Pierre and Gudmundsson, 2009),
pressure waves are used to pump liquids from bottom-hole to wellhead. In
pressure pulse technology (Gudmundsson and Celius, 1999), pressure wave
propagation properties are used to find the mass flow rate, gas-oil ratio and
deposits on pipe wall. Predicting the attenuation of a pressure wave due to
wall friction is therefore important to evaluate how far a pressure wave will
travel and therefore, assess the technologies limitations.
Numerical methods have long been used to predict rapid pressure transients.
The method of characteristics (MOC) presents an engineer-friendly method
for solving a simplified one dimensional unsteady flow of compressible fluid
system of equations. Recent model improvements for the governing system
of equations were presented Brunone et al (1991) and Ramos et al. (2004),
where the source term includes an unsteady friction factor in addition to the
steady-state Darcy-Weisbach friction factor. The proposed models of transient
friction factors were numerically studied by the method of characteristics.
However, a recent study by Szymkiewicz and Mitosek (2007) showed that
the improvements generated by the addition of transient friction factors were
negligible. Szymkiewicz and Mitosek showed also that the agreement between
experimental results and numerical simulations were not reproducible if not
using the method of characteristics for solving the governing equations. In
particular, the damping and smoothing of a pressure wave in fluid transport
pipeline are inaccurately predicted.
In this paper, a second order finite volume method will be used for solving
the commonly used governing equations. The finite volume method will be
compared with the standard method of characteristics in order to justify its
use. Two transient friction factor models will then be included in the selected
finite volume numerical scheme and the results will be compared with the
steady friction modelling. The adequacy of the two models will be evaluated
and the results will be compared to Szymkiewicz and Mitosek observations.
218

Pressure Wave in Pipelines
Appendix C
Oil and gas pipelines are used to transport hydrocarbons over long distances.
Transport pipelines are generally a few inches to tens of inches in diameter,
and cover hundreds of kilometers. For instance, the Europipe gas transport
pipeline between Draupner and Dornum measures 40 inches in diameter and
660 kilometres in length. Flow prediction in transport pipelines is therefore
best assessed using one-dimensional models.
Oil and gas transport systems include quick acting valves. The actuation of a
quick acting valve generates a pressure surge, also called water hammer. The
pressure surge generated propagates along the transmission line as a pressure
wave. The water hammer pressure amplitude is calculated using the
Joukowsky equation (Gudmundsson and Celius, 1999):
∆p = ρa∆u (C.1)
where, ρ, is fluid density, a, speed of sound in the fluid-pipe system, ∆u,
change in fluid velocity, and, ∆p, water hammer pressure amplitude.
The pressure surge generated by the actuation of a quick acting valve propagates
along the transmission line at the speed of sound. The speed of sound
depends on the fluid properties; that is the fluid density, ρ, and the fluid compressibility,
K. The speed of sound depends also on the pipe material; that is
the Young’s modulus, E, and on the pipe geometry; that is the diameter, D,
and the wall thickness, e. The speed of sound is calculated using Korteweg’s
formula (Tijsseling and Anderson, 2007):
1
a = � �
ρ K + D
�
Ee
(C.2)
The water hammer and pressure wave propagation, in fluid transport pipelines,
are modelled using the mass balance equation, and the momentum equation,
for 1D unsteady and compressible fluids in flexible pipelines. The mass balance
and momentum equations form the following system of equations:
∂u
∂t
+ u∂u
∂x
1 ∂p
ρ ∂x
1
ρa2 Dp ∂u
+ = 0
Dt ∂x
1 ∂ (β − 1)ρAu
+
ρA
2
+
∂x
τwπD
+ gsin(α) + = 0
ρA
219
(C.3)

Appendix C
where, A, is pipe cross-sectional area, g, gravitational acceleration, α, pipeline
inclination, and, τw, wall shear stress. The pressure wave propagation in fluid
transport generates an unsteady flow of a compressible fluid. Typical fluid
velocities range from 10 m/s to 20 m/s for gas transport and from 2 m/s to 4
m/s for oil transport. Typical speeds of sound in gas and oil are of order 300
m/s and 1200 m/s, respectively. The flow Mach number, defined by the ratio
of the fluid velocity and the acoustic velocity, is therefore much less than 1,
both in oil and gas pipelines.
The relative magnitude of the various terms in (C.4) is investigated for a full
wave scale; that is four times the travel time of a pressure wave propagating
through the fluid transport pipeline. The low Mach number approximation
is also used in the magnitude analysis. In addition, changes in density in
unsteady compressible flows are of the order of magnitude of the Mach number.
The simplifications applied on (C.4) and assuming a horizontal pipeline read:
∂p
+ ρa2∂u = 0
∂t ∂x
∂u
∂t
∂p
+ = s
∂x
(C.4)
where, s, represents a source term accounting for the energy losses. The
above system of equations is the one used in the industry for prediction of
pressure transients. The method of characteristics is commonly used to solve
numerically the above system of equations (Wylie et al., 1993). A typical
experimental result of the water hammer effect in a pipeline is shown in Figure
C.1. The maximum pressure value decays and the pressure trace smooths with
time.
Source Term
The source term, s, in the system of equations (C.5), accounts for the energy
losses in the fluid transport system. External forces on the fluid are assumed
negligible. Heat transfer between the fluid and its surrounding is assumed
negligible. The pipe visco-elastic behaviour is not considered in the present
study, neither in the speed of sound formulation (C.2) nor in the mass conservation
equation. The source term, s, is then reduced to the frictional pressure
loss of the flow along the pipeline and is expressed by:
s = −f
ρu | u | (C.5)
2D
220

Appendix C
where, f, is Darcy-Weisbach friction factor. The friction factor can be accurately
estimated by Haaland’s explicit formula (Haaland, 1983):
1
√ =
f −1.8
n log
��6.9�n � � �
1.1n
k
+
(C.6)
Re 3.75D
where, Re, defines flow Reynolds number and, n, a constant equal to 3 for a
gas and 1 for a liquid.
The aforementioned system of equations for predicting pressure transient propagation
is generally solved numerically by the methods of characteristics. The
Darcy-Weisbach frictional pressure loss is the only energy loss mechanism
taken into account. The computed solution does not predict accurately pressure
wave behaviour in fluid transport pipelines (Szymkiewicz and Mitosek,
2007). The pressure wave decay is underestimated, and the pressure wave is
not smoothed correctly; therefore the computed pressure traces do not correspond
to experimental data. Several models have therefore been proposed to
increase the accuracy of computer models.
The models proposed in the literature rely on a decomposition of the frictional
pressure drop in the source term, s. The frictional pressure drop is split in
two terms. The first term is the steady frictional pressure drop calculated
using the standard Darcy-Weisbach friction factor, fq. The second term is
a transient friction term, fu. The transient friction factor is time dependent
and is related to the flow parameters. The transient friction factor increases
locally the frictional pressure drop at the wave front, where the flow gradient
is important. The selected models for the present work are expressing the
transient friction factor value.
Two proposed models of transient friction factors are herein studied. The first
transient friction factor model studied was proposed by Brunone et al. (1991).
The formulation proposed by Brunone includes the local and instantaneous
convective acceleration of the fluid. The second transient friction factor model
studied was proposed by Ramos et al. (2004). The formulation proposed by
Ramos is an improvement of Brunone’s formulation aiming at increasing the
accuracy of the solution. Brunone’s transient friction factor formulation is
expressed by:
fu = k0D
u | u |
� ∂u
∂t
�
− a∂u
∂x
(C.7)
where, k0, is an empirical constant related to the Reynolds number. k0 is of
order 0.02-0.03. Ramos’ transient friction factor formulation is expressed by:
fu = D
�
� ��
∂u u �
k1 − k2a �
∂u�
�
u | u | ∂t | u | �∂x
�
(C.8)
221

Appendix C
where, k1, is a constant and of order 0.003, and, k2, a constant of order
0.04. Both transient friction factor formulations include the local and the
instantaneous convective accelerations which were neglected in (4) because of
the low Mach Number and the magnitude analysis. It is noted that the system
of equations including a transient friction factor has been studied and solved
numerically by their respective authors using the method of characteristics.
Modelling Equations
The system of equations (C.5) is a hyperbolic system of two first-order partial
differential equations. The transient friction factors (C.7) and (C.8) are
characterised by first-order partial differential equations. The one-dimensional
compressible flow continuity and mass equations (C.5) for horizontal conduits
may be written in their conservative form:
∂Q
∂t
+ ∂F (Q)
∂x
= S (C.9)
consisting of the vector variable, Q, the flux vector, F, and the source term
vector, S:
� � � � � �
p
ρa2u 0
Q = F(Q) = S =
(C.10)
u
p s
The well-established method of characteristics is based on finite difference
approximations of the partial derivations. The method of characteristics is
therefore first order. Szymkiewicz and Mitosek (2007) used a third order
finite element method to model the governing system (9). We chose to apply
a second order finite volume method in order to compare the results based on
the involved physical phenomena and not mathematical considerations.
The modelling and numerical simulation in the present work is based on a
finite volume method for hyperbolic systems included in CLAWPACK (LeVeque,
2002). CLAWPACK is a software package of high-resolution, multidimensional
wave propagation algorithms, for time dependent hyperbolic problems.
CLAWPACK includes Riemann solvers of first and second order in time
and space. Flux limiters are then applied to the resulting waves, increasing
the accuracy of the solution.
The Lax-Wendroff numerical scheme with Van-Leer flux limiter was selected
for solving the system of equations herein studied. The homogeneous system
is first solved using the Lax-Wendroff numerical scheme. The Van-Leer flux
limiter is applied directly on the flux vector F. Then, the source term approximated
by finite differences is applied to the solution. The Lax-Wendroff
222

Appendix C
flux limited method applied to a homogeneous system takes the form (Laney,
1998):
if Cr > 0
Q n+1
i
= Qn i − Cr(Q n i − Q n i−1)−
Cr
(1 − Cr)
2
if Cr < 0
Q n+1
i
�
Φ(θ n i+1/2 )(Qn i+1 − Qn i ) − Φ(θn i−1/2 )(Qn i − Qn i−1 )
= Qn i − Cr(Q n i+1 − Q n i )−
Cr
(1 + Cr)
2
�
Φ(θ n i+1/2 )(Qn i+1 − Qn i ) − Φ(θn i−1/2 )(Qn i − Qn i−1 )
�
�
(C.11)
where, Cr, represents Courant number, Φ(θ), flux limiter function, subscript
i, computational node and, superscript n, computational time step. The calculated
Courant number is relative to each solution wave and its direction of
propagation; that is the Courant number is positive when the wave propagates
in the positive direction +x, and negative otherwise. The Van-Leer flux
limiter function takes the form (Laney, 1998):
Φ(θ) =
θ + |θ|
1 + |θ|
where the variable θ is defined by:
if Cr > 0
θ n i−1/2 = Qn i−1 − Qn i−2
Q n i − Qn i−1
if Cr < 0
θ n i−1/2 = Qn i+1 − Qn i
Q n i − Qn i−1
(C.12)
(C.13)
The Lax-Wendroff numerical scheme is explicit, centered, second order accurate
in both time and space, linearly stable provided that the Courant number
is less than one. The transient friction factor equations (7) and (8) are first
order partial differential equations and are solved using finite differences.
223

Appendix C
Modelling Results
A test case is used to compare the two transient friction factors studied in this
work. The test case is a water pipeline, 1 inch in diameter, 100 meter in length.
The speed of sound is 1400 m/s. The initial fluid velocity is 2 m/s. Upstream;
the transport pipeline is connected to a reservoir at constant pressure, 50 bara.
Downstream, the pipeline is connected to a quick acting valve. A schematic
representation of the transport system is shown in Figure C.2. The upstream
reservoir is shown on the left-hand side, and the quick-acting valve is shown
on the right-hand side. The water transport system is meshed using 10,000
grid points. The Courant number is set to 0.99.
The flow is fully established, then the quick-acting valve is actuated at time
t=0.01s. The quick acting of the check valve is ideal. In other word, the
actuation of the valve assumes no energy loss. The quick acting of the valve
generates a pressure wave that propagates along the transmission line as a
pressure wave. Following the Joukowsky equation (C.1), the pressure surge
generated is 28 bar in amplitude. The pressure surge is superimposed on top
of the in-situ static pressure. The Reynolds number is 50800. The flow is
therefore fully turbulent before the valve closure.
The selected finite volume method is first compared to the standard method of
characteristics. The comparison is performed by simulating the pressure wave
successive transmissions and reflections without the Darcy-Weisbach steadystate
friction factor or a transient friction factor; that is, the pressure wave
propagates without energy loss. The result is shown in Figure C.3. Both
models predict the same pressure wave amplitude according to (C.1). However,
the MOC result decays with time as shown in Figure C.4, while the result
simulated by the finite volume method remains periodic without noticeable
decay over the simulated period of time. The MOC shows discrepancy with
the expected result while the selected numerical scheme decay can be neglected
for the period of time simulated.
The method of characteristics introduces numerical friction into the solution.
That is the damping is greater than what the physical model describes. Therefore,
the predicted results do not correspond entirely to the physics of the problem
and discrepancy are observed between experimental data and computed
solutions. Such observations suit the modelling of the system of equations
when including a source term or not. Therefore, the transient friction factor is
not modelled in the present work with the method of characteristics. Instead,
the present work focuses on the effect of the transient friction factor on the
solution computed with the selected finite volume method.
The pressure wave transmission and reflection are modelled successively using
224

Appendix C
four different expressions of the source term. The first expression of the source
term does not include any frictional pressure loss; that is, the system is ideal
without energy loss. The second expression of the source term includes only
Darcy-Weisbach steady friction factor. The third expression of the source
term adds Brunone’s transient friction factor to the steady friction factor.
The fourth expression of the source term adds Ramos’ transient friction factor
to the steady friction factor.
The pressure trace at the downstream valve is computed in the four cases.
The pressure traces are shown in Figure C.5 and Figure C.6. The pressure
wave propagates on top of the in-situ static pressure. The water hammer
pressure amplitude is accurately predicted according to (1). The wave propagation
speed is respected. The modelling results are in good agreement with
the physics of the water hammer phenomenon. However, the steady friction
model solution and the two transient friction models solution are almost superimposed.
It means that the proposed models by Brunone et al. and Ramos
et al. do not improve significantly the predicted solution. The result is similar
to observations of Szymkiewicz and Mitosek (2007).
A detail of the pressure trace at the valve is shown in Figure C.7. The solutions
using only the steady friction factor and using the steady friction factor and
Brunone’s transient friction factor are superimposed. Therefore, Brunone’s
proposed model does not improve the prediction. The solution using the steady
friction factor with Ramos transient friction factor formulation is slightly lower
than the two aforementioned solutions. Especially, the wave front pressure is
lower. However, the difference between the Ramos et al. ’s proposed model
and Brunone et al. ’s proposed model is negligible. Consequently, none of the
proposed model used in the present work is evaluated as being satisfactory.
Discussion
The test case has been modelled in four different ways. None of the solution
exhibits the same characteristics as the typical water hammer pressure trace
(Figure C.1). The transient friction factors included in the source term modelling
did not improve significantly the accuracy of the computed solution. The
result from the present work confirms Szymkiewicz and Mitosek observations
(2007).
Historically, the first source term model using transient effects was developed
by Zielke (1966). In Zielke’s model, the source term included the past history
of the fluid velocity. The transient friction models proposed by Brunone et al.
(1991) and Ramos et al. (2004) use the past history of the flow by taking into
account the convective acceleration of the fluid. All transient friction mod-
225

Appendix C
els developed were tested using the method of characteristics. The method
of characteristics does not ensure mass conservation, only momentum conservation.
The method of characteristics is also diffusive and dissipative; that
is the pressure wave decay included in the mathematical model is increased
by an artificial decay due to the numerical method. The inherent properties
of the method of characteristics are consequently inappropriate for accurate
unsteady flow predictions.
Szymkiewicz and Mitosek used a modified finite element method with adjustable
accuracy, up to the third order in space and time (Szymkiewicz and
Mitosek, 2005) for modelling the mass conservation and momentum equations.
Szymkiewicz and Mitosek results are in good agreement with the results
herein presented; that is the transient friction factor does not improve
the prediction. They demonstrated that the mathematical functions developed
for modelling the transient friction factor does not change the hyperbolic nature
of the system of equations. Szymkiewicz and Mitosek suggested that
an advection-diffusion transport can explains the smoothing. They concluded
that the current models are not suitable for predicting the damping and the
smoothing of a pressure wave.
The present work uses a finite volume method, second order accurate in both
space and time. The presented results are not compared with experimental
data, however, the conclusions are similar to (Szymkiewicz and Mitosek,
2007); that is the current transient friction models do not improve the accuracy
of a pressure wave behaviour. While the pressure wave is damped, it
is not smoothed. Besides, it is observed that the transient friction formulation
is in contradiction with the low Mach number approximation used for
establishing the mass and momentum conservation equations. Such approximation
minimizes the effect of the convective acceleration. Nevertheless, the
transient friction models are centred on the local and instantaneous convective
acceleration terms.
The transient friction models are inadequate for explaining both the damping
and the smoothing of the pressure wave along the pipeline. Besides work on
the mathematics of the source term and the system of equations to be solved,
additional fundamental work should be performed on the pressure wave propagation
phenomena. Ramos et al. (2004) added the visco-elastic behaviour
of the pipe material to the mass conservation equation. The proposed model
was again tested using the well-established method of characteristics.
Better modelling of pressure wave propagation is still to be established. Further
modelling of pressure wave propagation will have to include new physical
statements not included in the current models; that is the modelling with the
standard assumptions of low Mach number and simplifications after order of
226

Appendix C
magnitude analysis, as used in the present work. Moreover, further models
will have to be tested using several numerical methods to demonstrate their
independence with respect to the simulation tool.
Conclusions
A second order finite volume has been applied successfully to model one dimensional
compressible mass conservation and momentum equations. Two
different expressions have been applied to model the source term; that is
steady-state friction and transient friction.
The present work confirms that the transient friction factor in the literature
do not improve the pressure wave predictions currently used. Of particular
interest, it is shown that the transient friction factor is not responsible for the
smoothing and damping of a pressure wave propagating in a fluid transport
pipeline.
Further study of pressure wave propagation should focus on the fundamentals.
Assumptions currently used to establish the governing system of equations
will have to be re-evaluated. New models will have to be tested with diverse
numerical methods to ensure their reliability.
Acknowledgment
The authors want to thank Clavis Impulse Technology AS for providing a
Ph.D scholarship to Benjamin Pierre.
References
Brunone B., Golia U. and Greco M. (1991): Some Remarks on the Momentum
Equation for Fast Transients, Int. Meeting on Hydraulic Transients with
Column Separation, volume 9, pages 140-148.
Gudmundsson J.S. and Celius H.K. (1999): Gas-Liquid Metering Using Pressure-
Pulse Technology, SPE Annual Technical Conference and Exhibition, Houston,
SPE 56584.
Haaland S. (1983): Simple and Explicit Formulas for the Friction Factor in
Turbulent Pipe Flow, Journal of Fluids Engineering, volume 105(1).
Laney C. (1998): Computational Gasdynamics, Cambridge University Press.
227

Appendix C
LeVeque R.J. (2002): Finite Volume Methods for Hyperbolic Problems, Cambridge
University Press.
Pierre B. and Gudmundsson J.S. (2008): Water Hammer Simulation of Pressure
in the Failed Flexible Hose for Off-Loading Oil at Statfjord A, Department
of Petroleum Engineering and Applied Geophysics, NTNU, 7491 Trondheim,
Technical Report.
Pierre B. and Gudmundsson J.S. (2009): Pumping of Fluids Using Pressure
Impulses, SPE EUROPEC, Amsterdam, SPE 120896.
Ramos H., Covas D., Borga A. and Loureiro D. (2004): Surge Damping Analysis
in Pipe Systems - Modelling and Experiments, Journal of Hydraulic Research,
Volume 42(4).
Szymkiewicz R. and Mitosek M. (2005): Analysis of Unsteady Pipe Flow Using
the Modified Finite Element Method, Communications in Numerical Methods
in Engineering, volume 21(4).
Szymkiewicz R. and Mitosek M. (2007): Numerical Aspects of Improvement of
the Unsteady Pipe Flow Equations, International Journal for Numerical Methods
in Fluids, volume 55(11).
Tijsseling A. and Anderson A. (2007): Johannes Von Kries and the History
of Water Hammer, Journal of Hydraulic Engineering, volume 133(1).
Wylie E.B., Streeter V.L. and Suo L. (1993): Fluid Transients in Systems,
Prentice Hall.
Zielke W. (1966): Frequency Dependent Friction in Transient Pipe Flow, Industry
Program of the College of Engineering, University of Michigan.
228

Nomenclature
a Acoustic Velocity [ms −1 ]
A Flow Area [m 2 ]
Cr Courant Number
D Diameter [m]
f Friction Factor
g Gravitational Acceleration [ms −2 ]
K Fluid Compressibility [Pa −1 ]
L Pipe Length [m]
p Pressure [Pa]
Re Reynolds Number
u Fluid Velocity [ms −1 ]
α Angle [rad]
ρ Fluid Density [kg m −3 ]
Figures
Pressure
2 4 6 8 10 12
Figure C.1: Typical Water Hammer Pressure Trace
229
Appendix C
Time [2L/a]

Appendix C
L = 100m
Figure C.2: Schematic Representation of Test Case System
Figure C.3: Comparison Between the Method of Characteristics
(dotted line) and the Selected Finite Volume Method
(solid line)
Ideal Pressure Wave Propagation; that is Without Energy Loss.
230
D

Figure C.4: Detail of Figure (C.3)
Solid Line: Finite Volume Method
Dotted Line: Method of Characteristics
231
Appendix C

Appendix C
Figure C.5: Pressure Trace at Non-Return Valve
Dotted Line: No Source Term
Dashed Line: Source Term with Steady-State Friction Factor Only
Plus Sign: Source Term with Steady-State Friction and Brunone’s Transient
Friction factor
232

Appendix C
Figure C.6: Pressure Trace at the Valve
Dotted Line: No Source Term
Dashed Line: Source Term with Steady-State Friction Factor Only
Cross Sign: Source Term with Steady-State Friction and Ramos Transient
Friction factor
233

Appendix C
Figure C.7: Detail of the Pressure Trace at the Valve
Dotted Line: No Source Term
Dashed Line: Source Term with Steady-State Friction Factor Only
Plus Sign: Source Term with Steady-State Friction and Brunone’s Transient
Friction factor
Cross Sign: Source Term with Steady-State Friction and Ramos Transient
Friction factor
234

Appendix D
Simulation of Rapid Pressure
Transients in
Statfjord A Off-Loading
Pipeline
Paper to be submitted.
Abstract
The second largest oil spillage in Norwegian petroleum history occurred on
December 12, 2007, when an off-loading pipeline ruptured. Results of numerical
simulations of pressure transients in the off-loading system are presented.
Of particular interest is the influence of acoustic properties of pipeline sections
and equipment on the pressure transient propagation. The damaged Statfjord A
off-loading system geometry and characteristics as well as the acoustic properties
are described. The paper describes numerical simulations of pressure transients
in the off-loading pipeline. The paper shows particularly how changes
in acoustic properties generate pressure wave reflections that superimpose and
create rapid and local pressure peaks.
Introduction
Hydrocarbon transport pipelines are subject to rapid pressure transients in the
form of water-hammer events. Rapid pressure transients can arise when quickacting
valves are suddenly activated or after failure of a pump. Consequences
235

Appendix D
of rapid pressure transients in hydrocarbon transport pipelines range from
operational damages such as leakages (Moura et al., 2004; Wang et al., 2004),
to pipe rupture (Misiunas et al.; 2005, NPD, 2008). Therefore, rapid pressure
transients need to be accurately predicted in hydrocarbon transport pipelines.
Fluid transport systems consist typically of many individual pipe sections and
pieces of equipment. Valves and pumps are among the pieces of equipment
integrated into a fluid transport system. Calculations and modelling of rapid
pressure transient events need to account for all the sections and pieces of
equipment included in the system.
The Statfjord A off-loading system is a 2,808 m long oil transport system,
made of 38 sections. The off-loading system connects a storage tank located
on a production platform, to a single path coupler located on a tanker. The
Statfjord A off-loading system ruptured on December 12, 2007, due to a rapid
pressure transient event generated by sudden actuation of the single path
coupler (Pierre and Gudmundsson, 2008).
Rupture of the Statfjord A off-loading system caused 4,400 m 3 of oil being
pumped into the sea; thus generating the second largest spillage in the history
of oil activities in Norway. Predictions of rapid pressure transients need to be
accurate in order to prevent future accidents.
Predictions of rapid pressure transients with and without considering all the
sections produce different results. Of particular interest is the influence of the
many sections geometrical and material properties on the maximum pressure
reached.
Statfjord Off-Loading System
Statfjord is an oil field in the Tampen area in the northern part of the North
Sea (NPD, 2008). Statfjord is one of the oldest producing field on the Norwegian
continental shelf. The Statfjord reservoirs lie between 2,500 and 3,000
m deep. The sea depth in the area is about 150 m. A total of approximately
47,000 barrels of oil were produced per day in 2008 from the Statfjord field.
The field has been developed with three fully integrated facilities: Statfjord
A, Statfjord B and Statfjord C. Statfjord A was the first facility in place and
is centrally positioned in the field.
The produced oil is stored in cells (tanks) at each facility. Storage cells are
located inside the platform structure at the bottom of the sea. The storage
cell on Statfjord A holds 206,000 standard cubic metres; that is 1.3 million
barrels of oil. Oil from the storage cell is loaded to tankers approximately
every two days. Tankers connect to the off-loading system at a safe distance
236

Appendix D
Figure D.1: Illustration of the Off-Loading System between the Riser
Base and the Single Path Coupler
away from the platform. The off-loading system is the whole system from the
storage cell to the off-loading point and comprises two loading pumps located
near the platform (PSA, 2008).
The Statfjord A off-loading system is made of several sections as partly shown
in Figure D.1 (Sangesland, 2008). The first section, not shown in Figure D.1,
is a 2,516 m long steel pipe connecting the storage cells on the production
platform, to a riser. A 80 m long flexible hose then connects the base of
the riser to a swivel. A sequence of flexible hoses inter-connected by steel
components links the swivel to a single path coupler located on a tanker. The
whole system is 2,808 m in length.
The composition of the Statfjord A processed oil determines such properties
as density, viscosity and compressibility. An exact composition of the relevant
oil is not available in the literature. Therefore, an early composition of the
Statfjord A reservoir oil was used to find the properties of the processed oil.
The processed oil composition from Statfjord is given in Table D.1 (Johnsen,
2008). The reservoir hydrocarbon has been flashed down to 1 bara, 5 ◦ C before
being pressurized to 30 bara.
Hydrocarbon properties have been obtained using the Peng Robinson fluid
package included in HYSYS. The fluid flowing in the transport pipeline be-
237

Appendix D
Table D.1: Statfjord Oil Molar Composition at 5 ◦ C, 30 bara
Component N2 CO2 C1 C2 C3 iC4
% mol 0.04 0.19 9.32 6.25 10.27 1.81
Component nC4 iC5 nC5 C6 C7 C8
% mol 6.25 2.09 3.32 5.33 7.44 6.87
Component C9 C10 C11 C12
% mol 5.61 4.43 3.57 27.19
tween the storage cells (tanks) and the tanker is approximately at 30 bara
pressure and at 5 ◦ C temperature. For such conditions, a 705 kg m −3 density
and a 0.635 mPas viscosity are obtained. A 1.42 10 −9 Pa −1 isothermal
compressibility, KΓ, has been approximated using:
KΓ = 1
� �
∂ρ
ρ ∂p
Γ
(D.1)
where, K, defines compressibility, ρ, fluid density, p, pressure, and subscript
Γ, isothermal conditions. The partial derivation of density with pressure in
Eq. D.1 has been approximated by first order finite difference.
The complete Statfjord A off-loading system is made of several sections of
different sizes, lengths and materials. Components of the Statfjord A offloading
system are either steel sections or flexible hose sections. Steel has a
Young’s modulus of 200 10 9 Pa. A flexible hose is essentially a rubber hose
reinforced with steel elements. Rubber has a Young’s modulus of 0.1 10 9 Pa
(Cheremisinoff, 1996). However, wired steel elements increase the resistance to
stress (Horn and Kuipers, 1988). A 1.8 10 9 Pa Young’s modulus was selected
for the reinforced rubber material.
Every change in geometry and/or material creates an interface between two
media of different acoustic impedances where pressure waves can transmit
and reflect. Acoustic impedance is defined by the product of fluid density
and acoustic velocity. Isothermal acoustic velocities can be experimentally
measured or calculated using (Wylie et al., 1993):
1
a = � �
ρ KΓ + d
�
Y e
(D.2)
where, a, represents isothermal acoustic velocity in the fluid-filled pipeline,
d, pipeline diameter, e, pipeline wall thickness, and, Y , pipeline material’s
Young modulus. A constant ratio e/d along the pipeline of 0.1 was selected
238

Appendix D
for calculations of acoustic velocities. Calculated isothermal acoustic velocities
in steel and flexible elements were then 1000 ms −1 and 450 ms −1 , respectively.
Isothermal acoustic impedances can now be expressed by (Pain, 1993):
Z = ρa (D.3)
where, Z, represents isothermal acoustic impedance. Calculated isothermal
acoustic impedances in steel and flexible elements were then 705,000 and
317,250 in steel and flexible elements, respectively.
Pressure Transients
Failure of the Statford off-loading system was induced by sudden actuation of
the single path coupler located on the tanker. Sudden flow stoppage generates
a pressure wave whose amplitude can be calculated by the Joukowsky’s
equation (Wylie et al., 1993):
∆p = ρa∆u (D.4)
where, ∆p, represents pressure wave amplitude, and, ∆u, reduction in fluid
velocity.
The fluid velocity reduction ∆u depends on the type of valve (Thorley, 1987).
The ratio of fluid velocity reduction and initial fluid velocity for a valve closure
is shown with time in Figure D.2 for 6 types of valves (Wood and Jones, 1973).
First the fluid velocity decreases slowly until 80 % of the initial velocity value,
u0. Then, the fluid velocity decreases rapidly until full closure of the valve.
The valve closure function defines the shape of the front of the pressure wave.
The pressure wave front is shaped mainly during the last 20 % of the valve
closure process, where 80 % of the fluid velocity is lost. During the last 20 % of
the valve closure, the fluid velocity decreases faster for a needle valve than for
a circular gate-valve, as observed in Figure D.2. The resulting pressure wave
front will then be sharper if induced by a needle valve. Practically, most of
the pressure surge is created during the last 20 % of the valve closure action.
The fluid velocity reduction in Figure D.2 is theoretical and based on flow
area calculation (Wood and Jones, 1973). The above result is obtained for
a uniform valve closure. However the same result is obtained for a linearly
accelerated valve closure, Figure D.3. In case of an accelerated valve closure,
the 80 % fluid velocity reduction takes place in a shorter time than a uniform
valve closure; therefore creating a sharper pressure wave front.
239

Appendix D
u(t) / u 0 [%]
100
80
60
40
20
Needle
Gate − Circular
Globe
Gate − Square
Ball
Butterfly
0
0 0.1 0.2 0.3 0.4 0.5
τ
0.6 0.7 0.8 0.9 1
Figure D.2: Fluid Velocity Response for 6 Different Types of Valves.
Uniform Valve Closure (Wood and Jones, 1973)
τ: non-dimensional closure time
u(t) / u 0 [%]
100
80
60
40
20
Needle
Gate − Circular
Globe
Gate − Square
Ball
Butterfly
0
0 0.1 0.2 0.3 0.4 0.5
τ
0.6 0.7 0.8 0.9 1
Figure D.3: Fluid Velocity Response for 6 Different Types of Valves.
Accelerated Valve Closure (Wood and Jones, 1973)
τ: non-dimensional closure time
240

Appendix D
Calculations of pressure transients must be performed assuming a worst case
scenario. A pressure transient worst case scenario can be obtained assuming
instantaneous and ideal valve actuation. Another way to express this is that
the sharper the pressure wave front, the worse the consequences.
Pressure transients propagate in fluid-filled pipelines in the form of pressure
waves. Pressure wave propagation can be modelled based on the onedimensional
mass and momentum conservation equations. After order of magnitude
analysis, mass and momentum conservation equations can be expressed
by (Wylie et al., 1993):
∂p
∂t + ∂ � ρa2u �
= 0
∂x
(D.5)
∂u
∂t
∂p
+ = s
∂x
where, p, characterizes pressure, u, fluid velocity, t, time, x, axial direction,
and, s, source term. The source term accounts for pressure wave attenuation
mechanisms. Only steady state friction is herein considered due to the lack
of physical meaning and inaccuracies of unsteady friction models (Pierre and
Gudmundsson, 2009).
Propagation of a pressure wave in a fluid-filled pipeline is defined as an adiabatic
process. However, propagation of pressure waves in liquids can be
approximated by an isothermal process due to the rapidness of the event.
Acoustic velocities used in calculations are then isothermal acoustic velocities
and can be calculated using Eq. D.2.
In conservative forms, the hyperbolic system Eq. D.6 can be expressed by
∂Q
∂t
+ ∂F(Q)
∂x
= S (D.6)
consisting of the vector variable, Q, the flux vector, F, and the source term
vector, S:
Q =
� p
u
�
F(Q) =
� ρa 2 u
p
�
S =
� 0
s
�
(D.7)
Propagation of pressure transients in the Statfjord A off-loading system was
modelled and numerically simulated using a finite volume method (Pierre and
Gudmundsson, 2009). The Lax-Wendroff numerical scheme with Superbee
flux limiter function were selected. The numerical scheme is therefore second
241

Appendix D
order accurate in both space and time, with increased resolution. The Lax-
Wendroff numerical scheme applied to a homogeneous system can be expressed
by (Laney, 1998):
Ω n+1
i
ˆΦ n
i+ 1
2
a n
i+ 1
2
= Ωn i − ∆t
�
ˆΦ n
i+ 1 −
2
ˆ Φ n
i− 1
�
2
∆x
= 1 � n
Φ(Ω
2
i+1) + Φ(Ω n i ) � − ∆t
2∆x an
i+ 1
� n
Φ(Ωi+1) − Φ(Ω
2
n i ) �
⎧
⎪⎨ Φ(Ω
=
⎪⎩
n i+1 ) + Φ(Ωni )
Ωn i+1 − Ωn for Ω
i
n i �= Ωn i+1
Φ(Ω n i ) for Ωn i = Ωn i+1
(D.8)
The Lax-Wendroff numerical scheme coupled to a flux limiter function is expressed
by (Laney, 1998)
Ω n+1
i
= Ωni − ∆t
∆x ˆ �
φ(θ) ˆF n
i+ 1 −
2
ˆ F n
i− 1
�
2
The Superbee flux limiter function takes the form (Laney, 1998):
(D.9)
ˆφ(θ) = max (0, min(1, 2θ), min(2, θ)) (D.10)
where the variable θ is defined by:
Cr > 0
θ n i−1/2 = Qn i−1 − Qn i−2
Q n i − Qn i−1
Cr < 0
θ n i−1/2 = Qn i+1 − Qn i
Q n i − Qn i−1
(D.11)
The Lax-Wendroff numerical scheme is explicit, centred, second order accurate
in both time and space, linearly stable provided that the Courant number is
less than one.
Cases
Three cases were modelled to assess the importance of the sequence of offloading
system components on rapid pressure transient events. The first case
describes the off-loading system without considering steel components at each
242

Appendix D
end of a flexible hose. The second case describes the off-loading system taking
into consideration steel components at each end of a flexible hose. The third
case considers flexible hoses twice their original length with steel components
at each end. The three cases represent a similar system with the same length.
Differences lay in the sequence of sections.
The three cases are illustrated in Figure D.8, and the corresponding lengths of
every element are reported in Tables D.2, D.3 and D.4. We observe that case 2
corresponds to the actual geometry of the Statfjord A off-loading system while
cases 1 and 3 are used for analysis. Purpose of cases 1 and 3 is to evaluate
the importance of the flexible sections lengths in the sequence of the whole
off-loading pipeline. Distances are measured from the storage cell. The 2,808
metres distance corresponds to the single path coupler.
In the three cases, the 2,808 metres long off-loading system is modelled using
22,463 grid cells; that is, with a 12.5 centimetres spatial resolution. The
pressure wave propagation is simulated over 3 seconds with 28,800 steps; that
is, with a 0.1 millisecond temporal resolution. The 3 seconds simulated time
corresponds approximately to the travel time of the pressure wave propagating
from the single path coupler to the storage cells. During the 3 seconds, loading
pumps effects are assumed negligible.
The emergency shut-down valve connected to the single path coupler was suddenly
activated on December 12, 2007, generating a rapid pressure transient.
Before sudden valve actuation, the oil flow rate in the off-loading system was
6,000 m 3 hr −1 . The corresponding fluid velocity in a 20 inches pipeline is 8.2
ms −1 . The flow is fully turbulent with a 4.62 10 6 Reynolds number. A 0.02
steady-state friction factor is included as a source term in (D.6). The pressure
evolution in time is computed at every grid cell of the model during 3 seconds.
Every time step is iterated such that the Courant number is 0.9.
The emergency shut-down valve was assumed to close instantaneously, therefore
creating a sharp pressure wave front. Such assumption corresponds to
the worst case of valve closure. Pressure surge predictions aimed at increasing
safety and prevent damages to the fluid transport pipeline. Therefore, worst
case modelling was to be used.
Results
The three cases were solved numerically with the Lax-Wendroff numerical
scheme, including the Superbee flux limiter. The three cases were modelled
with three sets of acoustic velocities. The first set of acoustic velocities in the
steel sections and flexible sections was a 1000 ms −1 and 450 ms −1 , respectively.
The second set was 1000 ms −1 and 350 ms −1 , respectively. Finally
243

Appendix D
Pressure [bara]
60
50
40
30
20
10
0
0 500 1000 1500
Distance [m]
2000
Case 1
Case 2
Case 3
2500
Figure D.4: Maximum Pressure Achieved in the Oil Off-Loading
System in 3 seconds with asteel = 1000 ms and
aflexible = 450 ms
the third set was 1100 ms −1 and 450 ms −1 , respectively.
The first set of acoustic velocities corresponds to the standard speeds of sound
described and calculated previously. The second and third set of acoustic
velocities were used for analysis. Purpose of acoustic velocities sets 2 and 3 is
to evaluate the importance of the acoustic velocities on the maximum pressure
reached. For all three sets of acoustic velocities, all three cases were modelled
and results are shown in Figures D.4, D.5, D.6 and D.7.
The pressure wave generated at the single path coupler propagates towards
the storage cells. Interfaces between pipeline elements with different acoustic
impedances induce multiple pressure wave reflections along the oil off-loading
system. Superpositions of pressure waves reflections generate pressure peaks.
Locations of pressure peaks depend on the off-loading system geometry and
on the acoustic velocities in every element.
Maximum pressures reached along the off-loading pipeline for the first set
of acoustic velocities, during the first three seconds after valve closure, are
shown in Figure D.4 for the three cases. The maximum pressure reached is
the greatest for case 2. The maximum pressure reached is the lowest for case
1. Another way to express the result is that the more numerous changes in
244

Cumulated Time [s]
x 10−5
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
Case 1
Case 2
Case 3
0
0 500 1000 1500
Distance [m]
2000 2500
Figure D.5: Cumulated Time of the Pressure Surge being over 50
barg
Appendix D
acoustic impedances along the transport pipeline, the greater the maximum
pressure reached.
The maximum pressure reached increases with the number of changes in acoustic
impedance. Interfaces between media of different acoustic impedances generate
pressure wave reflections that travel separately of the original pressure
wave. The superposition of two or more pressure waves generate pressure
peaks that are responsible for high maximum pressure reached along the offloading
system for case 2.
Calculated pressures are gauge pressures. Another way to express this is that
the calculated pressures are on top of the in-situ static pressure. The maximum
calculated pressure reached in the oil off-loading system for case 2 is 58
barg and takes place approximately 80 m from the single path coupler. The
maximum calculated pressures are always less than 60 barg for all three cases.
Pressure wave superpositions are mainly responsible for the high maximum
pressure values. It means that maximum pressure values are due to pressure
peaks short in time. Repetitive pressure peaks induce fatigue in pipeline elements
and decrease the resistance to bursting. The cumulated time of pressure
surge being over 50 barg is shown in Figure D.5. The pressure surge is greater
than 50 barg during 1.62 s at 80 m from the single path coupler.
245

Appendix D
Pressure [bara]
60
50
40
30
20
10
0
0 500 1000 1500
Distance [m]
2000
Case 1
Case 2
Case 3
2500
Figure D.6: Maximum Pressure Achieved in the Oil Off-Loading
System in 3 s with asteel = 1000 ms and
aflexible = 350 ms
Flexible hoses and steel elements in the Statfjord A off-loading system have
a 95 bara resistance to bursting (PSA, 2008); that is almost 40 barg over the
calculated maximum pressure of 58 barg at 80 m from the single path coupler.
However, the calculated maximum pressure do not consider the in-situ static
pressure. Besides, the repetitive pressure peaks mentioned previously decrease
the resistance to bursting of the structure.
The calculated maximum pressure was located 80 m from the single path coupler.
However, the predicted location and amplitude depend on the acoustic
velocity in the flexible hose elements of the off-loading system. The same three
cases were then modelled with the second and third sets of acoustic velocities.
Case 2 was the worst case for both sets of acoustic velocities as shown in
Figures D.6 and D.7. For the second set of acoustic velocities, the maximum
pressure reached was 55 barg and was located 10 m from the single path
coupler. For the third set of acoustic velocities, the maximum pressure reached
is 65 barg and is located 170 m from the single path coupler. For both cases,
the maximum pressure reported was obtained in case 2.
For all three sets of acoustic velocities, case 2 was always the most severe case
and case 1 always the least severe case. The first set of acoustic velocities that
246

Pressure [bara]
70
60
50
40
30
20
10
Case 1
Case 2
Case 3
0
0 500 1000 1500
Distance [m]
2000 2500
Figure D.7: Maximum Pressure Achieved in the Oil Off-Loading
System in 3 seconds with asteel = 1100 ms and
aflexible = 450 ms
Appendix D
correspond to the calculated speed of sounds, generated the greater maximum
pressure. The Statfjord A off-loading system actually ruptured 37 m from the
single path coupler.
Differences between predicted maximum pressure values reached and values
reported in (PSA, 2008) are due to uncertainties. Density and compressibility
values were obtained assuming an hydrocarbon composition. Similarly, the
off-loading system geometry and material properties were assumed. Thus,
calculated acoustic impedances are the main uncertainties.
Acoustic impedances impact both the amplitude and location of the maximum
pressure reachable. Therefore, accurate descriptions of the hydrocarbon
composition and state, as well as an accurate description of the off-loading
geometry and material properties would help decreasing the uncertainties.
Conclusions
The sequence of pipe sections has been analyzed based on the existing Statfjord
A off-loading system. Rupture of the Statfjord A off-loading system
due to rapid pressure transient has been responsible for the second largest oil
spillage on the Norwegian continental shelf. Three different sequences have
247

Appendix D
been modelled for assessing the importance of the sections sequence and the
lengths of the elements.
Acoustic interfaces are responsible for pressure peaks due to the superposition
of pressure wave reflections. The more sections and components in the sequence,
the greater the maximum pressure reached. The shorter the elements,
the greater the maximum pressure reached.
Flexible hoses low acoustic impedances decrease the maximum pressure reached;
while rigid sections great acoustic impedances increase the maximum pressure
reached. Another way to express these results is that the maximum pressure
achieved decreases with the stiffness of the sections making the off-loading
system sequence.
Quick-acting valve closure functions also influence the maximum pressure
reached. The greater the fluid velocity gradient during the last 20 % of the closure
process, the sharper the pressure wave front created. Practically, waterhammer
surges are generated mostly in the last 20 % of the valve closure
process.
Acknowledgement
The authors want to thank the Norwegian Research Council for providing financial
support to Ph.D. student Benjamin Pierre under contract 196054/S60.
References
Cheremisinoff N.P. (1996): Materials Selection Deskbook, Noyes Publications.
Ghidaoui M.S., Zhao M., McInnis D.A. and Axworthy D.H. (2005): A Review
of Water Hammer Theory and Practice, ASME Applied Mechanics Reviews,
vol. 58(1), pp 49-76.
Horn B.A. and Kuipers M. (1988): Strength and Stiffness of a Reinforced
Flexible Hose, Flow, Turbulence and Combustion, vol. 45(3), pp 251-281.
Johnsen E.E. (2008): Private Communication.
Laney C.B. (1998): Computational Gasdynamics, Cambridge University Press.
LeVeque R.J. (2002): Finite Volume Methods for Hyperbolic Problems, Cambridge
University Press.
248

Appendix D
Misiunas D., Vitkovsky J.P., Olsson G., Simpson A. and Lambert M.F. (2005):
Pipeline Break Detection Using Pressure Transient Monitoring, Journal of Water
Resources and Planning Management, vol. 131(4), pp 316-321.
Moura C.H.W., Sampaio D.S., de Lacerda I.M. and Selli M.F. (2004): Monitoring
Leakages on Oil Production Offloading at Open Seas Using Statistics Associated
With Mass Balance Methods, ASME Conference Proceedings 41766,
pp 2243-2247.
Norwegian Petroleum Directorate NPD (2008): Facts, The Norwegian Petroleum
Sector.
Pain H.J. (1993): The Physics of Vibrations and Waves, Jonn Wiley & Sons.
Petroleum Safety Authority PSA (2008): Oljeutslipp Statfjord OLS-A 12.12.2007.
Pierre B. and Gudmundsson J.S. (2008): Water Hammer Simulation of Pressure
in the Failed Flexible Hose for Off-Loading Oil at Statfjord A, Department
of Petroleum Engineering and Applied Geophysics, NTNU, 7491 Trondheim.
Pierre B. and Gudmundsson J.S. (2009): Modelling of Pressure Wave Propagation
in Pipelines: Steady-State and Transient Friction, Fifth National Conference
on Computational Mechanics, Trondheim 26-27 May.
Sangesland S. (2008): Private Communication.
Thorley A.R.D. (1987): Check Valve Behaviour Under Transient Flow Conditions.
A State of the Art Review , Fluid Transients in Fluid-Structure
Interaction (ASME).
Wang L., Li J., Peng K., Jin S. and Li Z. (2004): Petroleum Pipe Leakage
Detection and Location Embeded in S.C.A.D.A., ASME Conference Proceedings
41766, pp 2249-2253.
Wang X.J., Lambert M.F., Ligget J.A. and Vitkovsky J.P. (2002): Leak Detection
in Pipelines Using the Damping of Fluid Transients, Journal of Hydraulic
Engineering, vol. 128(7), pp 697-711.
Wood D.J. and Jones S.E. (1973): Water-Hammer Charts for Various Types
of Valves, Journal of Hydraulic Division, vol. 99(1), pp 167-178.
Wylie E.B., Streeter V.L and Suo L. (1993): Fluid Transients in Systems,
249

Appendix D
Prentice Hall.
Nomenclature
a Acoustic Velocity [ms −1 ]
Cr Courant Number
D Diameter [m]
e Pipe Wall Thickness [m]
E Material Young Modulus [Pa]
F Flux Vector
KT Isothermal Compressibility [Pa −1 ]
p Pressure [Pa]
Φ Flux Limiter
Q Vector Variable
ρ Density [kg m −3 ]
S Source Term
u Velocity [ms −1 ]
z Acoustic Impedance [kg m −2 s −1 ]
Appendix: Cases Geometry
Figure D.8: Illustration of the Three Cases Considered for the
Modelling
250

Table D.2: Geometry of Case 1
Appendix D
Element Length [m] Property Element Length [m] Property
0 0.75 Rigid 9 11.25 Flexible
1 10.00 Flexible 10 1.00 Rigid
2 11.25 Flexible 11 80.00 Flexible
3 11.25 Flexible 12 10.00 Flexible
4 11.25 Flexible 13 1.00 Rigid
5 2.00 Rigid 14 80.00 Flexible
6 11.25 Flexible 15 28.25 Flexible
7 11.25 Flexible 16 2516.00 Rigid
8 11.25 Flexible
Table D.3: Geometry of Case 2
Element Length [m] Property Element Length [m] Property
0 0.75 Rigid 20 0.75 Rigid
1 0.75 Rigid 21 9.75 Flexible
2 8.50 Flexible 22 0.75 Rigid
3 0.75 Rigid 23 0.75 Flexible
4 0.75 Rigid 24 9.75 Flexible
5 9.75 Flexible 25 0.75 Rigid
6 0.75 Rigid 26 1.00 Rigid
7 0.75 Rigid 27 0.75 Rigid
8 9.75 Flexible 28 78.50 Flexible
9 0.75 Rigid 29 0.75 Rigid
10 0.75 Rigid 30 10.00 Rigid
11 9.75 Flexible 31 1.00 Rigid
12 0.75 Rigid 32 0.75 Rigid
13 2.00 Rigid 33 78.50 Flexible
14 0.75 Rigid 34 0.75 Rigid
15 9.75 Flexible 35 0.75 Rigid
16 0.75 Rigid 36 26.75 Flexible
17 0.75 Rigid 37 0.75 Rigid
18 9.75 Flexible 38 2516.00 Rigid
19 0.75 Rigid
251

Appendix D
Table D.4: Geometry of Case 3
Element Length [m] Property Element Length [m] Property
0 0.75 Rigid 16 11.25 Flexible
1 0.75 Rigid 17 0.75 Rigid
2 10.00 Flexible 18 1.00 Rigid
3 9.75 Flexible 19 0.75 Rigid
4 0.75 Rigid 20 78.50 Flexible
5 0.75 Rigid 21 0.75 Rigid
6 11.25 Flexible 22 10.00 Rigid
7 9.75 Flexible 23 1.00 Rigid
8 0.75 Rigid 24 0.75 Rigid
9 2.00 Rigid 25 78.50 Flexible
10 0.75 Rigid 26 0.75 Rigid
11 9.75 Flexible 27 0.75 Rigid
12 11.25 Flexible 28 26.75 Flexible
13 0.75 Rigid 29 0.75 Rigid
14 0.75 Rigid 30 2516.00 Rigid
15 9.75 Flexible
252