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Application of the Stefan Problem to the Modelling
the Decomposition of a Gas Hydrate Pipeline Plug
Bentum, Emmanuel
Bentum, E. (2017). Application of the Stefan Problem to the Modelling the Decomposition of a
Gas Hydrate Pipeline Plug (Unpublished master's thesis). University of Calgary, Calgary, AB.
doi:10.11575/PRISM/24968
http://hdl.handle.net/11023/3809
master thesis
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UNIVERSITY OF CALGARY
Application of the Stefan Problem to the Modelling the Decomposition of a Gas Hydrate
Pipeline Plug
by
Emmanuel Christian Bentum
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
GRADUATE PROGRAM IN CHEMICAL AND PETROLEUM ENGINEERING
CALGARY, ALBERTA
MAY, 2017
© Emmanuel Christian Bentum 2017
i
ABSTRACT
This study deals with the application of the Stefan problem to modelling dissociation of hydrate
plug in which the hydrates were formed from a gas mixture. In the previous attempt to simulate
the decomposition of a hydrate pipeline plug, the hydrates have always been assumed to be pure
methane, will lead to erroneous prediction for the rate of decomposition of a hydrate plug
because the presence of even a small amount of ethane and or propane could drastically alter the
three-phase equlibrium conditions for a gas hydrate formation.
In the current study, the Stefan problem for heat conduction at a moving boundary is written in
radial coordinates for the case of double sided-depressurization of a pipeline hydrate plug. The
plug is assumed to have formed in the presence of various mixtures of methane and ethane, some
of which formed structure I hydrates and some which formed structure II hydrates. The effect of
the gas mixture composition, on the rate of hydrate plug decomposition is included by
incorporating Giraldo and Clarke’s model, for the rate of decomposition of gas hydrates formed
from a gas mixture, into one of the boundary conditions. In formulating the equations, it was
assumed that the depressurization is always occurring at a pressure whose corresponding
equilibrium temperature is greater than 273.15K, thus, there was no need to include an ice phase.
The resulting partial differntial equation is highly non-linear and was solved by using the method
of lines. At the time of writing,there were no publication available in which the method of lines
had been applied to stefan problem in radial coordinates. A base case model was run, in which
only heat conduction was considered.
ii
The base case scenario was able to satisfactorily model the experimental data from Peters et al.
[1] with an Absolute Average deiviation (AAD) of 5%. The kinetic model was subsequently
applied to the base case scenario and and it was found that the results were almost identical to
those obtained without the kinetic term. From this, it was concluded that heat transfer controls
the decomposition of the methane hydrate plug at the base case conditions.
Subsequently, the model was used to simulate the decomposition of hydrate plugs formed from
the mixtures of methane and ethane, some of which formed were sI hydrates and some formed
were sII hydrates. Without the addition of the kinetic term there would be no means for including
the composition of the gas mixture, in the heat transfer equations.
A sensitivity analysis on the kinetic model was conducted. The geometric parameter which is
related to the surface area was investigated and it was found out that by changing the ratio from 1
to 4 times varied very little indicating that the parameter was not very sensitive to the kinetic
model. It was also observed that at pressures of 7.4MPa the rate of dissociation was heat
controlled. However, when the pressure was lowered to 3.4MPa intrinsic kinetics became more
predominant indicating a sharp difference between the heat transfer model and the kinetic model.
At a temperature of 273.15K the model showed that the dissociation rate was only heat transfer
controlled at a pressure of 7.8MPa. The Absolute Average Deviation (AAD) was less than 1%.
However as the temperature was increased to 275.15K and eventually to 277.15K there was a
sharp deviation from the kinetic model to the heat transfer model.
iii
ACKNOWLEDGEMENTS
The author would like to express his honest gratitude to his supervisor Dr. Matthew A.
Clarke for his patience, continuous support, encouragement and supervision of this thesis.
The author also expresses his appreciation to the members of the examining committee
for their valuable comments.
The author also thanks Dr. Amitabha Majumdar for his assistance .The author also wants
to mention his research group members: Marlon Mendoza and Fahd Alquatani for the excellent
support.
Dedication
This thesis is dedicated to my family.
iv
LIST OF FIGURES
Figure 1: structure I hydrate cavities Reproduced from Sloan and Koh [10] ................................. 3
Figure 2: structure II hydrate cavities Reproduced from Sloan and Koh [10] ................................ 3
Figure 3: (a) unit cell of structure I (b) unit cell of structure II Reproduced from Sloan and Koh
[10] .................................................................................................................................................. 4
Figure 4: Unit cell of structure H Reproduced from Sloan and Koh [10] ...................................... 5
Figure 5: Phase diagram for a water-gas-hydrate-gas system Reproduced from Giovanni and
Hester [4] ........................................................................................................................................ 8
Figure 6: Natural gas gravity chart Reproduced from Sloan and Koh [10] .................................... 9
Figure 7: Plug formation through aggregation in an oil-dominated system. Reproduced
from Sloan and Koh [10, p. 653] ................................................................................................ 12
Figure 8: Removal of hydrates by pigging method (Reproduced from Giovanni and Hester) [8] 13
Figure 9: Hydrate plug formed from inside a pipeline in Brazil (photo by Petrobas reproduced
from Koh et al.) [22] ..................................................................................................................... 13
Figure 10 Pipeline rupture owing to excessive pressure buildup generated by hydrate dissociation
(Reproduced from Giovanni and Hester [8] ................................................................................. 14
Figure 11: Hydrate plug dissociation by one sided depressurization. (Reproduced from Giovanni
and Hester [8] ................................................................................................................................ 14
Figure 12: A phase diagram showing the general hydrate dissociation methods (Reproduced from
Sloan and Koh. [10, p. 585] .......................................................................................................... 15
Figure 13: Radial dissociation of hydrate plug Reproduced from Peters et al. [1] ....................... 34
Figure 14: Hydrate plug dissociation schematic Reproduced from Hong et al. [68, p. 1851] ..... 35
v
Figure 15: Discretization scheme with fictitious enthalpy values, Reproduced from Chun et al
[71] ................................................................................................................................................ 42
Figure 16: Schematic Linear enthalpy distribution for one-dimensional grid system, Reproduced
from Chun et al [71] ...................................................................................................................... 42
Figure 17: Discretization Scheme at the interface, Reproduced from Lacoa et al. [66] ............... 45
Figure 18: Plot of temperature profiles at different radial positions ............................................. 51
Figure 19: Plot of Temperature profiles of both pure methane (C1) and mixture of
methane/ethane (C1/C2) hydrate composition at different radial positions ................................. 52
Figure 20:3 D Plot of Temperature Profile of methane/ethane mixture at different radial positions
....................................................................................................................................................... 53
Figure 21: 3 D Plot of Temperature Profile of pure methane at different radial positions ........... 54
Figure 22: Plot of Temperature Profiles with pure methane and mixture at different radial
positions using kinetic model........................................................................................................ 55
Figure 23: 3D plot of pure methane at different radial position using the kinetic model ............. 56
Figure 24: Plot of structure I hydrate dissociation rate compared with the experimental data [77]
....................................................................................................................................................... 61
Figure 25: Plot of dissociation rate heat model for structure I with experimental data of Peters et
al. [77] ........................................................................................................................................... 62
Figure 26: Plot of heat transfer model (HT) compared with kinetic model at different pressures.
at a geometric ratio of one ............................................................................................................ 64
Figure 27: Plot of heat transfer (HT) model with present model (Kinetic) at different
temperatures .................................................................................................................................. 66
Figure 28: Plot of hydrate kinetics using different geometric parameters .................................... 67
vi
LIST OF TABLES
Table 1: Geometry of cages [10] .................................................................................................... 2
Table 2: Jeffrey’s List of series of seven Hydrate Crystal structures [15] ...................................... 7
Table 3: Table showing the number of iterations for the method of lines using different ODE
solvers ........................................................................................................................................... 50
Table 4: Table of calculated stoichiometric values, W of hydrate mixtures (CH4+C2H6) ........... 59
Table 5: Simulation Parameters of the hydrate plug ..................................................................... 59
Table 6: Table of activation and intrinsic kinetics constants for methane and ethane hydrates
source [42] .................................................................................................................................... 60
Table 7: Simulation Parameters source [27] ................................................................................. 70
Table 8: Hydrate dissociation models, Reproduced from Sloan and Koh [10] ............................ 71
vii
TABLE OF CONTENTS
ABSTRACT ......................................................................................................................... i ACKNOWLEDGEMENTS ............................................................................................... iii
LIST OF FIGURES ........................................................................................................... iv LIST OF TABLES ............................................................................................................. vi TABLE OF CONTENTS .................................................................................................. vii LIST OF SYMBOLS ......................................................................................................... ix CHAPTER 1: INTRODUCTION ........................................................................................1
1.1 Historical Background .......................................................................................................... 1
1.2 Structures of Gas Hydrates ................................................................................................... 2
Structure I................................................................................................................................ 3
Structure II .............................................................................................................................. 4
Structure H .............................................................................................................................. 5
1.3 Phase Equilibrium for gas hydrates ...................................................................................... 8
1.4 Determination of equilibrium conditions of gas hydrates..................................................... 9
1.5 Hydrate as a potential of source of energy.......................................................................... 10
1.6 Formation and removal of hydrate plugs ............................................................................ 11
1.6.1 Hydrate forming conditions ......................................................................................... 11
1.6.2 Remediation and removal of Hydrate plug .................................................................. 13
1.7 REVIEW OF HYDRATE KINETIC MODELS .........................................................17 1.7.1 Review of hydrate dissociation models ....................................................................... 18
1.7.2 Review of hydrate plug decomposition kinetics .......................................................... 21
1.7 3 The effect of Kinetics on the dissociation rate of the hydrate plug ............................. 23
1.7.4 Review of solution techniques for moving boundary heat transfer problems. ............ 24
viii
1.7.5 Application of Method of lines in solving partial differential equation ...................... 29
1.7.6 Comparison between finite difference method and method of lines ........................... 31
1.8 Scope of study ..................................................................................................................... 32
CHAPTER 2 MODELLING OF HYDRATE PLUG DISSOCIATION RATE ...............34 2.1 Hydrate dissociation by double-sided depressurization ...................................................... 34
2.2 Derivation of the moving boundary equation with the Kinetic term .................................. 38
2.3 Derivation of moving boundary of the heat transfer model using method of lines in radial
coordinates ................................................................................................................................ 41
2.4 Derivation of the heat equation by method of lines in the cylindrical coordinates ............ 43
CHAPTER 3-RESULTS AND DISCUSSION .................................................................50 3.1: Simulation Results for Temperature Profile in the Pipeline .............................................. 50
3.2 Decomposition of Hydrate plug kinetics ............................................................................ 57
3.3 Sensitivity analysis of the Model ........................................................................................ 64
CHAPTER 4: CONCLUSIONS AND RECOMMENDATIONS .....................................68
4.1 Conclusions ......................................................................................................................... 68
4.2 Recommendations ............................................................................................................... 69
APPENDIX ........................................................................................................................70 BIBLIOGRAPHY ..............................................................................................................72
ix
LIST OF SYMBOLS
Ap surface area of hydrate, m2
C Langmuir constant, 1/Pa
Cbo initial concentration, mole/dm
3
Ceq equilibrium concentration of dissolved gas in the presence of hydrate, mole/dm3
Cint concentration, mole/dm3
CpI heat capacity of ice phase, J/ (Kg. K)
Cpw heat capacity of water phase, J/ (Kg. K)
dn change in moles
dt change in time, s
e exponent
Ea Activation energy, KJ/mol
f fugacity, MPa
feq equilibrium fugacity, MPa
G linear dissociation rate, m/s
K* kinetic constant, Overall rate constant around a hydrate particle, mol/(m2.Mpa.s)
Kb mass transfer coefficient from liquid bulk to the surface of the particle, m/s
KI thermal conductivity of ice, watts per meter Kelvin, W/(m.K)
Kw thermal conductivity of water, watts per meter Kelvin, W/(m.K)
Ms molecular weight g/mol
n(t) number of moles remaining in the hydrate and water at the start of dissociation, moles
Nw Number of water moles
x
P Pressure, MPa
R Universal gas constant, 8.314KJ/ (mole. K)
R radial distance in pipe, cm
ro radial position at pipewall ,cm
Ry(t) Global rate of reactions, mole/(m3.s)
s1 water-hydrate interface position layer, cm
Td dissociation Temperature, K
Teq equilibrium Temperature, K
Tm melting Temperature, K
Tw Temperature water in pipe, K
V volume of mass reaction, m3
W stoichiometric amount of moles with respect to methane
W(r) cell potential function, J
xb gas mole fraction in the liquid bulk in equilibrium with hydrates phase
xint gas mole fraction in the liquid bulk in equilibrium with gas phase at the interface
yi vapour phase mole fraction of species ‘i’
yi,j Fractional occupancy of cavity species ‘ i’ in cavity j
Z compressibility factor
Ψ geometric ratio of hydrate plug
фs Surface volume of hydrate plug, cm3
фv Volume of hydrate plug, cm3
∆r radial grid size, cm
xi
Greek Letters
heat of dissociation of hydrate, J/(Kg. K)
heat of dissociation of ice, J/(Kg. K)
∆R radial change in position, cm
∆t time step, s
Pi
Density, kg/m3
Porosity
Thermal diffusivity of water, m2/s
Ψ Sphericity
νi number of cavities per type i
. Partial fugacity coefficient,
xii
Superscripts
H hydrate
MT empty lattice
o pure liquid water at reference conditions
Subscripts
eq equilibrium condition
s system condition
i index
j index
o reference conditions
w water
m molecule
b bulk solution
p particle
g gas
1
CHAPTER 1: INTRODUCTION
1.1 Historical Background
Gas hydrates are non-stoichiometric crystalline compounds which are formed as result of
association of water molecules and lower molecular weight mostly non polar gases under low
temperature and elevated pressures. The water forms three dimensional network structures with
spaces that can be occupied by these low molecular gases like methane, ethane and carbon
dioxide. The forces holding the hydrates together are the weak van der Waal forces. In 1778 Sir
Joseph Priestley [2] first formed hydrates from ice and sulphur during the winter season.
However, it was in 1810 that Davy [3] discovered that hydrates could be formed by reacting
chlorine with water. Faraday in 1823 [4] came out with the formula for chlorine hydrates. Then,
in 1888 it was discovered that hydrates could be formed with the gaseous hydrocarbons like
methane, ethane and propane. [5]
During 1930 Hammerschmidt [6] found out that the blockages in oil and gas pipelines were due
to the presence of hydrates. It soon become known that hydrate pose serious threat to petroleum
productions and explorations. At these operating conditions of high pressure and low
temperature the hydrates formed could block other flowlines like, subsea lines, risers, blow-out
preventer (BOP) and chokelines.
The hydrate plug formed could lead to massive economic losses, destruction to life and property
in the oil and gas industry if best practices and safety measures are not put in place. In 1967 [7],
a huge deposit of naturally occurring hydrates were found under the permafrost regions of
2
Siberia. Later on more methane hydrate were discovered in the Alaska, Mackenzie delta in
Canada [8]. Though it has been found to be a nuisance to the oil industry it been seen to be
potential store of energy [8]. The potential amount of natural gas stored in situ gas hydrate in the
world has been estimated to be twice the amount of the world’s conventional gas reserves [9].
Ever since its discovery increased research has been conducted to better understand how to
control formation of hydrates and more recently it kinetics.
1.2 Structures of Gas Hydrates
There are three main types of hydrates structures namely Structure I, Structure II and Structure
H. The determination of the structures sI and sII were done using X-ray diffraction techniques
[10] Table 1 shown below gives the structural formula of the types of hydrates. The structural
formulae are illustrated in the Figures 2, 3 and 4.
Table 1: Geometry of cages [10]
Hydrate crystal
structure
I II H
Cavity Small Large Small Large Small Medium Large
Description 5
12 5
126
2 5
12 5
12 5
12 4
35
66
3 5
126
8
Number of cavities 2 6 16 8 3 2 1
Average cavity
radius (Å)
3.95 4.33 3.91 4.73 3.94 4.04 5.79
No. of water
molecules/cavity
20 24 20 28 20 20 36
3
Figure 1: structure I hydrate cavities Reproduced from Sloan and Koh [10]
Structure I
The unit cell of structure I is pentagonal dodecahedra which are packed together with
tetrakaidecahedra to form twelve pentagonal and two hexagonal faces. The sI is made up of 46
water molecules, two small 512
cavities and two large cavities 512
62. The structure I can be
occupied with small sized guest molecules which is less than 3 Å in molecular radius such as
methane, ethane, carbon dioxide, and hydrogen sulfide [10]
Figure 2: structure II hydrate cavities Reproduced from Sloan and Koh [10]
4
Structure II
The unit cell of structure II is also pentagonal dodecahedra which are packed together with
hexakaidecahedra to form twelve pentagonal and four hexagonal faces. Structure II is made up of
146 water molecules, sixteen small 512
cavities and eight large cavities 512
64. This structure can
be occupied by both small and larger sized molecules; for instance, propane and isobutene may
be entrapped in the larger cavities.
Figure 3: (a) unit cell of structure I (b) unit cell of structure II Reproduced from Sloan and Koh [10]
5
Structure H
Structure H unit cell is made up of 34 water molecules, three small 512
cavities, two medium size
435
66
3 cavities and one large size 5
126
8 cavity. The small guest molecules usually are caged in
small and medium cavities whereas molecules larger than 7.4Å, such as 2-methylbutane, 2, 2-
methylbutane, neohexane and cyclo-heptane, enter the larger cavity.
Figure 4: Unit cell of structure H Reproduced from Sloan and Koh [10]
The three main cavities in sH gas hydrates are the pentagonal dodecahedron 512
,
tetrakaidecahedra 512
62, hexakaidecahedron 5
126
4, irregular dodecahedron 4
35
66
3 and
icosahedrons (512
68). The 14 sided cavity tetrakaidecahedron is also referred as 5
126
2 since it has
12 pentagonal and 2 hexagonal faces.
The 16-hedron (hexakaidecahedrai cavity) are represented by the formula 512
64 because in
addition to 12 pentagonal faces it contains 4 hexagonal faces. The irregular dodecahedron cavity
(435
66
3) consists of three square faces and six pentagonal faces together with three hexagonal
faces. The biggest icosahedron cavity (512
68) has 12 pentagonal faces together with a girdle of 6-
6
hexagonal faces and a hexagonal face each at the cavity crown and foot. Several other forms of
hydrates have been discovered by some researchers. In the analysis of the simple and combined
cavities of Dyandin et al. [11], it was proposed that in addition to the cavities present in sI, sII
and sH were the following structures 512
63,4
45
4, 4
35
96
27
3and 4
66
8. The cavities expand in size in
comparison to ice and it is stabilized by the repulsive presence of the hydrate formers (guest
molecules) or the neighbouring molecules.
Tabushi et al. [12] also found out that the 15-hedron (512
63) was absent in clathrate except
bromine owing to an unfavourable strain in relation to the other cavities in structures I and II.
Rogers and Yevi [13] indicated that guest repulsion is more pronounced than attraction which
leads to cavity expansion. The mean polyhedral volume 12, 14 and 16 -hedral cavities were
found to vary with temperature, guest size and shape as observed by Chakumakos et al. [14].
However, Jeffery [15] as shown in Table 2 found out that 12, 14 and 16 -hedral cavities are not
stable in pure water structure. Sorensen and Walrafen [16] found out that liquid water may be
structured as cavities.
7
Table 2: Jeffrey’s List of series of seven Hydrate Crystal structures [15]
1 11 111 IV V VI VII
12-Hedra
512
12-Hedra
512
12-Hedra
512
12-Hedra
512
12-Hedra
512
8-Hedra
445
4
14-Hedra
512
62
16-Hedra
512
64
14-Hedra
512
62
15Hedra
(512
63)
14Hedra
(512
62)
512
63
16-Hedra
(512
64)
17-Hedra
(435
96
27
3)
14-Hedra
(466
8)
Cubic(s1)
Cubic(s11) Tetragonal Hexagonal Hexagonal Cubic Cubic
Pm3n Fd3m P42/mnm P6/mmm P63/mmc 143d Im3m
a=12Ả a=17.3 Ả c 12.4 Ả
a 25.5 Ả
c Ả
a
c ,
a 12 Ả
a=18.8(2) A=7.7 Ả
6x.2Y.
46.H 20
8X.16Y.
136.H 20
20X.10Y.
172.H20
8X.6Y.
80.H20
4X.8Y.
68.H2 0
16X156H20 2X.12H20
Gas
hydrates
Gas
hydrates
Bromine
hydrate
None
known
None
known
Me3CNH2
hydrate
HPF6
hydrate
X indicates the guest molecules in 14-hedra or larger voids, Y refers to those in 12-hedra
8
1.3 Phase Equilibrium for gas hydrates
The phase equilibria of gas hydrates provide the most significant set of properties that determine
the boundary for which hydrate exists. Figure 5 illustrates the key features of a phase diagram
when hydrates form from pure hydrocarbons.
Lw-H-G is the liquid water-gas hydrate-gas equilibrium line; I-H-G is the ice-gas hydrate-gas
equilibrium line and therefore at F if at constant temperature a hydrate plugs dissociation occurs
by moving F through equilibrium point E and finally to point D through depressurization.
Figure 5: Phase diagram for a water-gas-hydrate-gas system Reproduced
from Giovanni and Hester [4]
9
1.4 Determination of equilibrium conditions of gas hydrates
The equilibrium conditions of gas hydrates can be determined by the as gravity method. Using
the criteria for phase equilibrium the following conditions must be met.
1. Temperature and pressure of all the phase in the equilibria must be equal
2. Chemical potential of each component in each phase must be equal
3. Gibbs free energy is minimum
The gas gravity method was introduced by Katz [17] and it is a simple graphical technique in
which the gravity of the gas is required for determination of the equilibrium conditions. Gas
gravity is defined as the ratio of the molecular weight of the gas over the density of air. After
obtaining either temperature or pressure and the gas gravity, the other variable (either pressure or
temperature) can be read directly from the graph in Figure 6
Figure 6: Natural gas gravity chart Reproduced from Sloan and Koh [10]
10
1.5 Hydrate as a potential of source of energy
Gas hydrates can be potential source of energy. They are present in huge quantities in all the
continental shelves and in permafrost areas. The technology required for the utilization of this
energy is currently under research. Understanding the geology and properties of the reservoir is
very important in the successful utilization of this resource. Current estimates by the National
Research Council [18] and also by Association of America Petroleum engineers and Geologist
AAPG [19] estimates that the in-place volume of gas in gas hydrates for Gulf of Mexico to be
about 600TCM (6x1014
m3) and at North Slope of Alaska with a mean estimate of 2.4TCM
(2.4x1012
m3).
In 2009 China made a huge discovery of onshore deposit at Qinghai province and Tibet plateau
at a depth of 130-300m below the permafrost [8]. Economic and commercial production of
hydrates is being considered by Japan and other nations like USA, South Korea and Canada in
the coming decades [8]. Japan drilled an offshore methane hydrate production test well in 2013.
This research will convert natural gas to gas hydrates for easier transportability and avoid the
prohibitive cost of pipeline transport.
11
1.6 Formation and removal of hydrate plugs
Hydrate plugs are may form when there is a change in flow geometry (e.g. bend, or pipeline dip
along the sea floor). It may also occur at a nucleation site (e.g. sand, weld slag etc). Hydrate
plugs may also occur during transient operations for instance during an emergency shut-down
due to the failure of inhibitor injection failure or dehydrator failure. It may also occur following
restart after shutdown.
Another possibility of formation occurs when it is made to flow through a valve which may
result in further cooling. Structural imperfections, weld spot and pipeline fitting like elbow, tee
and valve are suitable sites for nucleation of hydrates. Also high velocity flow which occurs in
narrow orifices and valves causes mixing, which may enhance the formation of hydrates. When
natural gas flow through choke valves, hydrates may form is due to joule Thomson effect [20] .
1.6.1 Hydrate forming conditions
Formation of hydrate normally happens when conditions of elevated pressure and low
temperature are present in undersea pipelines. Other causes of hydrate formation are, natural gas
below its water dew point in the presence of water, turbulence and high velocity in the flowlines
as reported by Ikoku [21] . Again, the presence of H2S and CO2 which has a higher solubility
than hydrocarbons [21] could also promote hydrate formation.
12
Figure 7: Plug formation through aggregation in an oil-dominated system. Reproduced from Sloan and Koh [10, p. 653]
The Figure 7 above shows the process of agglomeration leading to the plug formation. The
process of plug formation begins at the nucleation site, then hydrate particles begins to grow with
time. Then the hydrate formed begins to aggregate leading to the plugging of the pipeline. There
has been steady progress in allowing hydrates to be formed but inhibiting the aggregation
process to avoid the formation of the hydrate plug. Hydrates are in transportable form and could
flow in flowlines as reported by Sloan and Koh [10]
13
1.6.2 Remediation and removal of Hydrate plug
When a hydrate plug is formed in a pipeline it needs to be remediated almost immediately to
mitigate economic losses. When hydrate forms there is down-time in production leading to
revenue shortfalls, and extra cost for remediation.
Figure 8: Removal of hydrates by pigging method (Reproduced from Giovanni and Hester) [8]
Figure 9: Hydrate plug formed from inside a pipeline in Brazil (photo by Petrobas reproduced from Koh et al.) [22]
If the hydrate formation is slow and is able to flow, pigging is the most suitable method
employed which can be used to clean and clear out the pipeline. The other methods of
remediation include injecting chemical inhibitors, heating, mechanical and depressurization
source [8].
14
One or a combination of methods can be employed. Each of the methods carries its own potential
dangers. When using thermal method care must be taken to avoid excessive pressure build-ups
and possible rupture and explosion as shown in Figure 10. The depressurization technique is the
most widely used in field operations where absolute hydrate plugs in pipelines are detected. One
sided depressurization is quite dangerous since the plug may be dislodged and move swiftly as a
projectile and cause equipment damage and injury to personnel as illustrated in Figure.10.
Figure 10 Pipeline rupture owing to excessive pressure buildup generated by hydrate dissociation (Reproduced from
Giovanni and Hester [8]
Figure 11: Hydrate plug dissociation by one sided depressurization. (Reproduced from Giovanni and Hester [8]
Lysne in 1995 thesis [23] reported three incidences in which hydrate projectiles exploded from
pipelines at elbows causing the death of three oil workers and loss of over US$7 million in
capital cost.
15
Figure 12: A phase diagram showing the general hydrate dissociation methods (Reproduced from Sloan and Koh. [10, p.
585]
As shown above in Figure 12 depressurization results in the heat being transferred to the hydrate
plug to decompose it. The heat is absorbed to decompose the hydrate at a constant interface
temperature. For the broken lines the inhibitor injection is represented by shifting the hydrate
formation curve by injection of 10 (w/v) % methanols in the free water phase.
In practice two sided- depressurization is the most recommended method over the single-sided
method. Double-sided depressurization method has become the preferred choice for removing
hydrate plugs in undersea natural gas and condensate pipelines as observed by Yousif and
Donayevsky [24].
16
Hydrate dissociation by depressurization is a heat and mass transfer dependent process. This
process involves a gradual reduction of pressure below the hydrate equilibrium pressure at which
the hydrates will be unstable as temperature rises at the hydrate interface. This method does not
necessarily require knowledge of its location, size or composition before it is implemented.
However, caution is required not to depressurize the pipeline quickly without heat transfer
(adiabatic) as Joule-Thomson cooling may rather worsen the problem by causing ice formation.
Again if the depressurization is done very slowly (isothermal depressurization) is also not the
most suitable option.
The most ideal approach is to use intermediate depressurization which ensures that the hydrate
interface temperature is always much lower than the surrounding temperature which will allow
the heat move from the surroundings to dissociate the hydrate from the pipe boundary inwardly.
In addition to finding suitable and economical methods of remediating hydrate plug, an offshore
pipeline will need suitable techniques of dealing with hydrate plugs formed in pipeline which are
submerged in very deep and cold waters especially if the plug is a few kilometers away from
shore. Formation of such a plug may potentially occur in unplanned shutdown probably due to
equipment failures. Hence thorough understanding of the mechanisms and processes involved is
key in developing techniques in controlling hydrates plugs.
17
1.7 REVIEW OF HYDRATE KINETIC MODELS
Clathrate hydrates have gained a very huge attention during these recent past decades owing to
its different potential application in its transportation, storage of natural gas and carbon
sequestration in the ocean. These applications require the development of effective hydrate
reactors thus requiring a deeper understanding of hydrate dissociation kinetics. Unlike hydrate
thermodynamics hydrate kinetics are still not fully understood.
In this section review of the literature on hydrate kinetics is presented which lays emphasis on
modeling. The models presented touches on the techniques of hydrate decomposition with its
kinetics investigated. The various dissociation models developed by most research groups
indicate that a greater number of them are based on heat transfer. Again some of the heat transfer
models have been coupled with kinetics in other to investigate the intrinsic kinetics at different
temperatures and pressures. Furthermore it has been observed that based on the comparisons
with experimental data most of the models are heat controlled rather than intrinsic kinetic
controlled [22].
Gupta et al. [25] in 2006 investigated the dissociation of methane hydrate by the use of nuclear
magnetic resonance and found out that intrinsic kinetics did not play a significant role in the
overall process. Hydrate dissociation is mainly controlled by heat transfer with kinetics playing
an insignificant role in majority of the cases.
\
18
1.7.1 Review of hydrate dissociation models
Kelkar et al. [26] used a model based on rectilinear coordinates that suggested that at an
optimum pressure the most rapid dissociation of the hydrates or solid phase occurs. However,
rapid depressurization caused ice to form due to joule Thomson cooling which delayed the
dissociation rate though ice had a higher thermal diffusivity which transmitted higher heat flux.
Thus, for low pressure depressurization the ice melts more slowly than the hydrate.
Jamaluddin et al. [27] investigated the decomposition of hydrate plugs and used a mathematical
model which coupled intrinsic kinetics with mass and heat transfer model to compare
experiments carried out under the condition of shutdown in pipeline in the laboratory at
temperature of 274K and pressures of 4MPa, 5MPa, and 7MPa.The model was used to compare
experimental values which indicated that if there was planned shut down under 48hours plugging
may not occur based on the effective diffusivities of methane gas through the hydrate. The
accurate estimation of the effective diffusivities is vital to the model prediction.
Vazquez-Roman [28] used a one -dimensional flow model to calculate pressure and temperature
profiles of gas production system based on average velocity between inlet and outlet conditions
of the pipe. The flow model was coupled with a heat transfer model to account for the
surrounding effects. The diameter, length and overall heat transfer coefficient of the hydrate plug
were then fine-tuned in the model to match the experimental data of the pressure and temperature
profiles which significantly improve the accuracy of its predictions.
19
Bollavaram et al. [29] investigated hydrate dissociation mechanism by method depressurization
for both (two-sided and single-sided). The model was based on Peters et al. [1] model that used
the two sided hydrate dissociation as radial moving boundary model. The model was able to
estimate the hydrate dissociation time and the total time for the complete melting of the hydrate
plug. The one-sided depressurization model was also investigated .This single sided model was
able to predict the time required to re-start the flow in a pipeline. This time also was dependent
on the downstream pressure, length, porosity, and permeability of the plug.
A safety model was developed to estimate a safe and optimum pressure for one sided
depressurization. The one-sided model was validated with laboratory and Tommeliten field plugs
and the predicted gas evolution curves matched the data [30] within 10% absolute error. The
model predicted that the Tommeliten field plugs were re-started when the annulus spacing was
8% of pipeline radius. The safety model also compared well with the simulations of Xiao et al.
[31] but predicted higher comparable plug velocities.
Nguyen et al. [32] developed a numerical model to predict the dissociation time of hydrate plugs
in oil subsea pipelines. The experimental hydrate plugs were dissociated by the method of
symmetric depressurization the model was in very good agreement with their experimental data.
The dissociation of the hydrate plug was dependent on the following variables diameter, porosity
and the dissociation temperature (Td).
20
Osokogwu and Ajienka [33] presented model that was based on Fourier heat law. The model
considered a radial dissociation from two-sided approach on a fixed boundary which applied
depressurization technique in a pipeline. Results from the model indicated at initial dissociation
temperature of 285.7K and a dissociation time was 13hours.
Vlasov [34] developed a model based on theory of chemical kinetics for the dissociation of gas
hydrate. The driving force for dissociation was highlighted in the model and the rate constant of
the hydrate dissociation was found to be dependent on the temperature. It was found out that the
temperature reliance of hydrate dissociation was confirmed with available experimental data in
the case where the interface involved liquid water. The case involving ice interface was yet to be
determined.
Uddin et al. [35] has proposed a new model based on deep investigation of a previous laboratory
scale study of methane hydrate decomposition and some observations from molecular dynamics
study. The model has not been tested with the appropriate data yet.
Lekvan et al. [36] investigated methane hydrate dissociation rate as a function of temperature.
Using a model that was based on the pseudo-elementary processes, it was able to predict the
kinetics using temperature gradients. The temperature range used for the study was narrow and
therefore was limited in its application in hydrates in subsea pipelines.
21
Chen et al. [37] used previous experimental data that were carried out in the laboratory for gases
in pipelines. A hydrodynamics model with an integrated model was used to simulate gas
dissociation. The model based on Englezos et al. [38] was used to estimate rate of evolution of
gas bubbles from the hydrates as it moved upward and also estimate the range of values of D
(mass transfer coefficient in the dispersion film) for which it was sensitive.
Rehder et al. [39] measured the dissociation rates of methane and carbon dioxide hydrates in
seawater underfloor. The hydrate dissociation was due to the differences between the
concentration of the guest molecule in the hydrate interface and the bulk solution. Thus a
solubility-controlled boundary layer model was able to satisfactorily predict the dissociation
data. The dissociation rate of carbon dioxide was higher than methane due to its higher solubility
in water.
1.7.2 Review of hydrate plug decomposition kinetics
In 1987, Kim et al. [40] studied the kinetics of methane hydrate dissociation. He found that the
dissociation rate was directly proportional to the surface area of the hydrate particle and the
fugacity difference of methane at equilibrium pressure and the decomposition pressure. An
experimental determination of the hydrate particle diameter led to the development of the
intrinsic model for the kinetics of hydrate decomposition.
22
Yousif et al. [24] simulated methane hydrate dissociation using a three-phase ID model .The
model matched very well the experimental data of gas and water produced from depressurization
and the movement of dissociation front .
Clarke and Bishnoi [41] developed a mathematical model to estimate the intrinsic rate constant
of decomposition. The model accounted for the particle size distribution in the hydrate phase as
the intrinsic rate constant was determined. Data was obtained from experiments performed using
temperatures of 274.15 to 281.15K and at pressures between 5 and 11bar was used to estimate
the intrinsic rate constant. The intrinsic rate constant was estimated to be 2.56.x103
mol/m2Pas.
Giraldo and Clarke [42] presented the experimental data from the kinetics of decomposition of
hydrates from pure ethane and methane and its mixtures 25%-75% methane at temperature and
pressure range of 274-278K and 6.39 and 14.88bar respectively. The model predicted accurately
well the experiment data for sI hydrates. For gas mixtures that formed sII hydrates
(72%CH4/28%C2H6 and 75%CH4/25%C2H6) the rate constants for both methane and ethane in
sII was regressed. The intrinsic rate constant of ethane decomposition was assumed to be the
same for sI and sII. The intrinsic rate constant regressed for methane in sI was estimated to be
8.06x103
mol/m2Pas
23
1.7 3 The effect of Kinetics on the dissociation rate of the hydrate plug
When kinetics is negligible or intrinsic kinetics is extremely fast (i.e. heat transfer only), the
interface fugacity becomes nearly equal to the equilibrium fugacity and therefore heat transfer
equation is employed to determine the dissociation rate of the hydrate plug. The hydrate plug is
assumed to attain extremely fast the interface temperature, Td. Even though the gas phase has a
much higher resistance than the liquid phase, it is generally assumed that all the heat absorbed by
the dissociating hydrate at the hydrate-water interface is conducted only through the water from
the pipewall.
In order to investigate if kinetics is influencing the dissociation rate (i.e. kinetics model) would
be applied for the simulation study. The driving force for the dissociation becomes the difference
in fugacity of methane/ethane mixture at the interface and the equilibrium fugacity of the system,
as reported by Englezos et al. [38] . The temperature distribution in the water phase which is
dependent on the temperature gradient between the pipewall and the interface and the radial
distance which could be determined by equations (17) and (18).
The hydrate plug is assumed to be a solid which occupy the region 0<R<s(t) which divides the
two phases(i.e. hydrae-water). The pipewall temperature is assumed to be at a constant
temperature, To and Tm which is the temperature at the interface is also assumed to be constant.
The surface of the hydrate plug is initially regarded as rough. As the hydrate plug dissociates
with time it surface becomes smoother with time at the latter stages.
24
Therefore the value of the dimensionless surface ratio , approaches the value of unity. The
intrinsic kinetic rate of hydrate dissociation is proportional to the surface area. The dimensionless
parameter (surface roughness factor), which indicates the real surface area ratio to the geometric
one is difficult to determine. Due to the difficulty in determining the geometric parameter Ψ,
however, the value of unity was assumed for it in the modeling by Englezos et al. [38] in their
experimental determination of the kinetic rate constant.
1.7.4 Review of solution techniques for moving boundary heat transfer problems.
When hydrate plugs begins to dissociate it involves heat transfer at the moving boundary.
Several mathematical and numerical techniques have been employed in solving this type of
problem which is usually referred as Stefan [43] type moving boundary involving phase changes
such as from solid from liquid.
Verma [44] et al. used the fixed grid method, which was based on explicit finite difference
approach to solve the moving boundary problem. This method used an alternative equation to
describe the grid containing the interface with the energy balance of the grid point next to the
interface. This equation reduced the mass balance error that generally affected the numerical
solution of moving boundary problems.
In addition to the increased accuracy of its prediction due to the elimination of the mass balance
error, the method had the advantages of a fully implicit scheme. The results given by that
approach was identical to the one developed by Murray and Landis [45] who employed fictitious
temperature which contained the fusion front. They used variable grid method by discretizing
either the time or space domain into equal sized increments, while the other is allowed to vary in
25
grid size. The time increment is maintained constant and the liquid and solid regions are at any
instant in time divided into a fixed number of equally spaced intervals. However, the size of the
intervals varied as the proportion of solid/liquid changes, this type of subdivision presented some
difficulties for the initial start up of the numerical computation particularly when the liquid or
solid is very small in dimension.
Kutluay [46] used variable space grid and boundary immobilization method which was based on
the explicit finite difference to determine temperature history and interface position. The variable
space grid which always ensures that the moving boundary is always on the Nth
grid is achieved
by the varying the grid size with time. However, the boundary immobilization method which was
first proposed by Laudau [47] is much more convenient is implemented by transforming the
moving boundary into a fixed boundary. This approached significantly simplifies the numerical
method and by applying the finite difference method a solution is obtain with high accuracy and
convergence is easily attained.
Chernousko [48] independently studied the isotherm migration method with Dix and Cizek [49].
In this method the dependent variable (Temperature) can be interchanged with the space
variable ( R) so that the solution is evaluated as R(T, t) instead of the more common T(R,t).This
method provides fixed boundaries in solving the model problem. Thus the method is most often
used to obtain numerical solutions especially subject to time dependent boundary conditions.
Esen and Kutluay [50] [went further to apply a Neumann boundary condition to obtain numerical
solutions of the Stefan problem.
26
Myers and Mitchell [51] presented the combined integral method (CIM) in which the Stefan
problem was solved by the introduction of a heat penetration depth. This was followed by an
approximating function in a polynomial form that described the temperature range .Then the heat
equation was integrated over that heat range to obtain heat balance integral. Then the resultant
single ordinary differential equation was solved analytically to determine the interface position.
Reutskiy [52] introduced a meshless technique (which did not need any domain or boundary
discretization) for solving one-dimensional problems for the moving boundary. The key idea of
this method was the use of modified particular solutions which satisfy the homogenous boundary
conditions. This technique utilizes truncated Fourier series as approximate fundamental solutions
The Fourier representation allows to write the solution on each time-layer, the right hand side
with the particular solution in the same form of truncated series resulting into two unknowns. It
was based on the application of delta-shaped functions and the method of approximate
fundamental solutions. The results show remarkable very high accuracy in tracking the interface
position. Again it was further produced higher accuracy results for cylindrical geometries and 2-
Dimensional Stefan problems where comparison was very good with analytical solutions by
Chein-Shan Liu [53].
Case and Tausch [54] used Green’s method to transform the partial differential equation into a
system of integral equations with unknown fluxes on the interface. This is discretized with
Nystrom method using the Stefan condition to track interface position. The results were quite
accurate.
27
Mitchel and Vynnycky [55] used numerical algorithm for one-dimensional time dependent
Stefan problem by Keller box finite difference scheme. The significance of the work was the use
of variable transformation that was built into the numerical algorithm to resolve the boundary
condition discontinuity arising from the onset of phase change. The method allowed the delay
time until dissociation/melting begins to be determined. Its results gave a second order accuracy
in both time and space.
.
Papac et al. [56] applied a numerical method using a hybrid finite-difference and finite-volume
framework which also encompassed the level-set finite difference discretization. The goal of this
approach (level-set finite difference) is to compute and predict the moving interface under
velocity field. This velocity depended on the time, position and the geometry of the interface.
This method is very simple and adaptable in its implementation.
Tadi [57] used a fixed grid local method by coordinate transformation which could be applied to
obtain exact solutions within a small local region. The method is used to cover the whole domain
resulting into implicit scheme with first order accurate in time. Ping et al. [58] presented Stefan
problem with exact solution method to solve the one-dimensional elliptic problem with Dirichlet
boundary condition. Results generated were quite accurate.
Voller [59] employed a sharp and diffuse interface model of fractional Stefan problems with
implicit time stepping numerical solution for the diffuse interface fractional Stefan model. The
results were accurate and the method was able to capture sharp physical changes
28
Hetmaniok et al. [60] employed the alternating phase truncation method to solve two
dimensional inverse Stefan problem based on the knowledge of selected locations of the region
of interest to determine the heat transfer coefficient. Each phase is treated alternately which
simplifies into algebraic problem. It is thus easy to implement and avoids numerical instability.
Song et el. [61] used isogeometric approach to solve the Stefan problem using algebraic distance
estimations and point projection algorithms developed under the condition of Gibbs-Thomson
conditions. Its method was much more effective than Newton-Raphson iterations.
Krasnova and Levashov [62] used numerical approach to solve one-dimensional two-phase
Stefan problem based on ultrafast processes in solids (which lasted less than the time of electron-
phonon relaxation) paving way for the heat equations to be solved separately for ions and
electrons at constant volume. The numerical method derived is used to determine ionic and
electronic temperatures in aluminum alloy subject to a femtosecond (10-15
s). The ionic
temperature showed a jump at the Stefan condition whereas the electronic temperature did not.
However the kinetics of the melting process had an effect on the model prediction.
Myers and Font [63] solved the one-phase Stefan problem by transformation of the phase change
temperature to a variable using, Cartesian, cylindrical and spherical coordinates. The resultant
equations were simple to implement very adaptable. Mitchell [64] applied the method of
combined integral method (CIM) which is characterized by delayed onset of phase change based
on Robin condition. The method was stable and versatile.
29
Layeni and Johnson [65] used a differential-difference equation reformulation exact closed-form
solution to a class of Stefan problems. The transformation reduces into analytical sets of
equations which is easily solved. The accuracy is high and very easy to implement. In this work
method of lines is employed by discretizing the partial differential equation in the space
dimension and leaving the time domain continuous as adopted by Campos and Lacoa [66] .
1.7.5 Application of Method of lines in solving partial differential equation
The method of lines is a well-developed numerical technique sometimes referred as a semi-
numerical method which was originally employed in the analysis of transmission lines wave
guide structures and scattering problems. It was initially developed by mathematicians and
applied to solve boundary value problems [67] .
It is a special numerical technique in which the space variable is discretized in one or two
dimensions while the time variable is allowed to be continuous or maintained in its analytical
form in a given differential equation. Thus this technique has both the merit of the finite
difference method and analytical method which does not produce spurious mode nor oscillatory
solutions and has no problem relating to relative convergence. Even though method of lines was
initially used to solve hyperbolic wave equation it could be applied to solve parabolic heat
equation. In this simulation study the method of lines would be applied to solve the partial
differential equation from Fourier’s heat equation.
30
The justification of using method of lines is its high computational efficiency. The presence of its
semi-analytical part makes it simple to make compact algorithm while at the same instant
yielding more accurate results. Again less computational effort is required as compared to the
finite difference numerical methods. Since the discretized space variable is analyzed separately
from the continuous time variable it is easy to achieve numerical stability and convergence for
wide range of problems.
Programming effort is also significantly reduced by employing the well-established and
dependable Ordinary differential Equation (ODE) solvers Computational time is decreased since
only a small amount of discretization lines are required in the computation avoiding the need to
solve large system of equations. ODE solvers like ODE45, ODE15s and ODE23 are preferred
because of their ability to handle stiff Equations.
In order to apply the Method of lines the following procedure is required
i. Partitioning the solution into two layers
ii. Discretization of the differential heat equation in one coordinate direction (space
variable)
iii. Transformation of the Partial differential equation to systems of ordinary
differential equation
iv. Applying the required boundary conditions
v. Then finally solving the systems of equations with the suitable ODE solver.
31
1.7.6 Comparison between finite difference method and method of lines
Even though finite difference could be used to solve the partial differential equation it has some
weakness as compared with the robust method of lines. The finite difference method has two
main approaches that is the finite explicit and the implicit methods. For the finite explicit the
major weakness is the constraint with the time step which should always be less than a half for
stability to be achieved.
Another problem is the error introduced in the time domain as a result of the finite difference
approximation which unlike the method of lines is left continuous and therefore has no error
introduced from the time domain. The finite explicit also requires a very large number of
iterations thereby increasing significantly the calculation time.
Regarding the implicit method which has no stability issue it still encounters errors as a result of
the finite difference approximation at both domains time and space. If the time steps are too large
it may become less accurate unlike the method of lines which retains its accuracy since there are
no time steps involved in its implementation Again the solution requires more computational
effort and storage in order to perform the solutions since the nodes are solved simultaneously
32
1.8 Scope of study
In previous work done by Kelkar et al. [26] to model the hydrate dissociation rate, a
mathematical model using Cartesian coordinates was employed that was based on Fourier’s heat
transfer equation. The resultant systems of equations were solved analytically. Peters et al. [1]
followed up on the previous work using cylindrical coordinates which was also based on Fourier
heat equation. The resultant systems of equations were solved using finite difference
approximation on both the time and space domain. He finally compared his model predictions
with the experiment he conducted using ice to produce the methane hydrate and subsequently
dissociated the plug by depressurization. The model prediction was within 5% of the
experimental data.
In this work, hydrate plug which is a problem in the oil and gas pipelines is simulated to find the
regimes where intrinsic kinetics and heat transfer is predominating at various range of
temperatures and pressures. The use of mathematical model which employs the numerical
method of lines was applied in the study of hydrate dissociation in this thesis. The mathematical
model based on heat conduction only by Fourier’s heat equation in cylindrical coordinates is
used. The heat model was used to determine the temperature profile with time. It was also used
to estimate the location of the interface with time. Then the kinetic model was used to investigate
regimes where the dissociation rate was heat controlled and also when it was intrinsic kinetic
controlled.
33
The Chapter 1 gives insight into hydrates in general and its occurrences and its thermodynamics
and kinetics. It concludes with a literature review of kinetics models by different authors and
methods. Then the computational procedure using numerical method of lines is applied to
discretize Fourier’s heat equation in cylindrical coordinates. The heat equation model developed
would be use to predict the dissociation rates for both structure I and structure II hydrates from
experimental data. Sensitivity analysis would be conducted to determine regimes where
dissociation rates are heat transfer controlled or intrinsic kinetics controlled. Chapter 2 deals with
application of method of lines and its correct implementation in deriving the heat equation in
cylindrical coordinates and also the kinetic equation for cylindrical coordinates. The Chapter 3
presents results of the numerical simulation and finally Chapter 4 discusses the conclusions and
recommendations.
34
CHAPTER 2 MODELLING OF HYDRATE PLUG DISSOCIATION RATE
2.1 Hydrate dissociation by double-sided depressurization
The dissociation of hydrate plug could be achieved by several methods. Among the methods,
includes, electrical heating, direct heating methods, chemical treatment and depressurization. In
this simulation study depressurization was used. Depressurization method involves a gradual
reduction of pressure in the pipewall without an external heat supply in order to destabilize the
thermodynamic equilibrium condition of the gas hydrate plug in the pipe .When the pressure is
reduced, sensible heat is supplied from the pipewall surroundings to both the hydrate interface
and vapour phase.
The dissociation of the hydrate plugs start as the heat move radially towards the hydrate plug
whereas the temperature at the pipewall is constant. The evolved gas is assumed to be
immediately removed from the surface of the dissociating plug and hence has negligible effect
on dissociation rate. The porosity of the hydrate is assumed to be uniform throughout the hydrate
plug. The dissociation temperature is also assumed uniform. The radial dissociation is the
dominant process rather than the axial dissociation as shown in Figure 13
Figure 13: Radial dissociation of hydrate plug Reproduced from Peters et al. [1]
35
Figure 14: Hydrate plug dissociation schematic Reproduced from Hong et al. [68, p. 1851]
Figure 14 illustrates a hydrate inner core which is surrounded by water layer next to the pipewall.
The temperature profiles are determined according to the Fourier’s law of heat conduction in
radial coordinates.
For water-hydrate phase only
…………………………………………………………………...........1
The boundary conditions in the system are given below;
36
At the pipewall
Tw=To r=ro, t>0……………………………………………………………………….…...2
At the interface
Tw=Td r=s1, t>0..............................................................................................................…...3
The Stefan condition
-
r=s1 t>0…………….….………………………..................4
The Stefan condition in equation 4 is obtain by the conservation of energy across the inter phase.
The interface velocity is determined by the Stefan [43] condition which is the thermal
conductivity, kw divided by the density of dissociating medium multiplied by the latent heat.
Therefore for a hydrate plug the dissociation rate,
is obtained by multiplying the temperature
gradient
by the Stefan condition [43] which will be
where ρH and γH are the density
of heat of dissociation of hydrate respectively. The hydrate porosity is given by θ which is
always less than one but greater than zero.
The boundary condition (2) indicates a constant temperature at the pipewall. The boundary
condition (3) also indicates the constant temperature at the water hydrate interface. Then the
boundary condition (4) also known as the Stefan condition shows that the heat conducted
through the water layer is equivalent to the heat required to dissociate hydrate plug.
There is no analytical solution to this system partial differential equation therefore it may be
solved by a numerical method.
Assuming the dissociation of a hydrate plug in a cylindrical region 0≤R≤ro is initially at a
constant pipewall temperature To, that is above the hydrate formation temperature at a given
pressure, P. When the pressure of the hydrate is reduced gradually until the temperature rises
37
above the hydrate equilibrium temperature, dissociation of the hydrate may start. However, if the
equilibrium temperature is below the ice point, ice will form around the hydrate as it dissociates
thus there would be three phases in the system (water-ice-hydrate). On the other hand if the
hydrate dissociation temperature is above ice point then only two phases would be present that is
water and hydrate.
When the pipeline is depressurized the hydrate dissociation temperature (Td) becomes less than
the pipewall temperature To. Thus, heat moves from the external pipe and travels radially into the
system to dissociate the hydrate plug. The hydrate plug begin to reduce in size as it detaches
itself radially from the pipe wall. The modeling of hydrate plugs was based on heat transfer by
conduction only.
The double-sided depressurization of the hydrate plug in the pipewall was considered as the
method for dissociation. The hydrate plug was assumed to undergo dissociation towards the
center of the pipe. As the hydrate plug decomposed, it became surrounded by stationary water
phase which transmitted heat to the dissociating front of the hydrate plug. Since the dissociation
temperatures, Td (273.25K) was greater than the ice point the hydrate system had only one
moving boundary.
38
2.2 Derivation of the moving boundary equation with the Kinetic term
In order to investigate the kinetics of the hydrate plug dissociation, the heat transfer model was
coupled with the kinetics. The coupling of intrinsic hydrate dissociation kinetics with heat
transfer rate is done in order to investigate the influence of kinetics in the overall rate of
dissociation. The approach was to vary both the system pressures and temperature independently
between a certain ranges in the simulation to observe those effects.
Clarke and Bishnoi [69] presented the total hydrate decomposition as the sum total rate of
evolution of each hydrate forming compound as follows;
……………………………………………. 5
NC is total number of hydrate-forming gases and Kd,j the intrinsic rate constant of the gas
hydrate dissociation rate for component j. Giraldo et al. [70]simplified the equation eliminating
the need for the intrinsic rate constant of the other hydrate forming gases apart from methane.
This was accomplished by relating the hydrate stoichiometric coefficients of the hydrate forming
compounds making the methane as the reference component as shown in equations below;
1M1 2M2 3M3…….. NC WH20……………………………………..........................6
where w1M1 w2M2, w3M3, and wNCw represent the stoichiometric coefficients and molar
masses of M1, M2, M3 and NCw respectively.
……………………………………………………………….......................... 7
39
Plugging equation 7 into 5 yield
........................................................................................................................................... 8
Sum of the relative stoichiometric ratios, W is given as;
Where W=
.......................................................................................................9
Hence the resulting dissociation equation is given as
………………………………………………………………...10
………………………………………………….. 11
Therefore the molar hydrate dissociation rate is given as;
……………………………………………………………........ …12
Where W (sum total of the relative stoichiometric ratios) is given by;
W=
……………………………………………………………………………… 13
And Kd the intrinsic dissociation constant
………………………………………………………………………......... 14
From the mass balance around the hydrate surface the molar rate of hydrate dissociation to the
plug thickness is given as;
…………………………………………………………………………...… .15
40
where Ageo is the geometric area of the hydrate plug and s is the interface location of the hydrate
plug,
The dimensionless surface roughness factor is given as;
Where
……………………………………………………………………........ ………16
Hence the kinetic model methane hydrate dissociation is used from Kim et al [40]
…………………………………………………………............17
When the heat transfer rate is combined with the kinetic equation we would be able to derive the
moving boundary system with the kinetic term.
Again applying the Stefan condition in equation 4
………………………………………………………….......18
Plugging (17) into (18) yields the equation 19 below
……………………………………19
Discretizing equation 19 gives the equation below
-Kw(
-
…………20
41
2.3 Derivation of moving boundary of the heat transfer model using method of
lines in radial coordinates
The partial differential equation (1) and its boundary conditions is solved numerically by the
method of lines. This numerical technique discretizes the space coordinates by finite difference
method while it leaves the time derivative continuous. This hybrid numerical methodology
transforms the partial differential equation to a system of ordinary differential equations which is
then solved using an appropriate ODE solver algorithm. The Fourier heat conduction equation in
cylindrical coordinates may be expressed by the following;
For moderate temperature range at constant pressure, dh (enthalpy change) can be replaced by
(temperature change.), dT.
dh=cpdT………………………………………………………………………………………... .21
……………………………………………………………………………... 22
The Stefan condition for a constant temperature at the hydrate – water interphase is given as;
……………………………………………………………………………………23
where L is latent heat of fusion of the solid phase and is the thermal diffusitiviyy of the
dissociated phase.
42
Figure 15: Discretization scheme with fictitious enthalpy values, Reproduced from Chun et al [71]
Figure 16: Schematic Linear enthalpy distribution for one-dimensional grid system, Reproduced from Chun et al [71]
43
2.4 Derivation of the heat equation by method of lines in the cylindrical
coordinates
In order to use the radial equation for the Fourier heat equation in equation (22) and the Stefan
condition in equation (23), Chun et al [71]approach are employed as shown in Figures 15 and 16
The linear enthalpy distribution for the one-dimensional radial equation in equation (22)
is discretized as shown in equation (24),with as fictitious nodal enthalpy
……………………………………………………....24
The finite difference approximation of equation (6) along the spacial coordinates is given below;
= -
………………………………………………………………….…………25
Solving for the fictitious nodes;
Since
= -
…………………………………………………………………………….26
setting hm to zero,
……………………………………………………………………………27
Plugging 10 into 7
…………………..................28
44
Simplifying equation (28)
………………………………………………..29
From equation 21 the enthalpy change dh can be represented by temperature change, dT the
equation 29 thus can be expressed in terms of temperature as below;
……………………………………………30
Then the discrete form in equation 30 can be expressed in the derivative form in the equation
below
…………………………………………………… 31
The equation (31) would be required to solve in cylindrical coordinates as was similarly
implemented by Campo and Ulises [72] in Cartesian coordinates by the application method of
lines in predicting the phase boundary velocity (dissociation rate) and temperature history.
45
1 2 3 i-1 i i+1 N-1 N
interface
∆r δ∆r (1-δ)∆r
Figure 17: Discretization Scheme at the interface, Reproduced from Lacoa et al. [66]
Problem Description
Using the discretization method obtained from [71, 73].The spatial domain is divided into N-1
uniform grid cells. The interface is assumed to be located between the nodes i and i+1 as
illustrated in Figure 17.The present model will be implementing the method of lines in
cylindrical coordinates. The spatial domain is divided into N-1 constant step size. Since the
interface is found between the lines i and i+1 as illustrated in Figure17, the line temperatures, Tk,
are determined from the systems of first order differential equations. Assuming constant thermal
properties, the mathematical formulation of the Stefan problem start with a one dimensional heat
46
conduction equation. The hydrate begins to dissociate from the pipewall r=0 at a constant
pipewall temperature at 277.15K.
The hydrate-water interface moves in a positive R-direction towards the center of the pipeline.
Hence at any time t, the temperature distribution and the radial position of r1 (t) could be
determined.
For the initial condition
T(r, t=0) = Ti……………………………………………………………………………………32
With the boundary conditions
T(r=Rw,t)=277.15K…………………………………………………………………………….33
Where the temperature at the pipewall is To and the temperature at the interface are given as
follows;
T(r=s(t),t)=273.25K…………………………………………………………………………….34
After applying the first boundary condition r=Rw, dissociation of the hydrate begins at this
location. The water hydrate interface moves inwards into the center of the plug in a positive R-
direction; r>0 to reach the moving boundary condition at R(t)
The governing heat conduction equation cannot be solved at the phase change interface and also
R(t) is not known a priori .Thus to overcome this problem the Stefan equation is introduced
which brings a closure to the set of equations. The Stefan condition indicates that the front
velocity (i.e. the dissociation rate by conduction) is proportional to the jump in heat flux across
the front. Generally, pure substances change phases isothermally and there is latent heat
associated with phase change (dissociation of hydrate). The latent heat of a given phase is the
47
heat liberated or absorbed when a unit mass undergoes that phase completely and isothermally.
The latent heat has dimensions of (energy/mass) which is given in units of joule/gram.
The Stefan condition is a physical constraint which comes from the conservation of energy at the
water-hydrate interface. Thus the local interface velocity (i.e. dissociation rate) depends upon the
flux discontinuity. The non-linear condition from equation 4 with initial condition at S(0)=0.The
density ,the thermal diffusivity of the water phase and heat of dissociation, of the hydrate is
provided in Table 8.
At r=0; the first derivative radial group becomes undefined in equation (1), therefore in order to
overcome this difficulty L’Hȏpital rule is applied by taking limit of the ratio of the derivatives as
r→0
…………………………………………………………………………………... 35
Thus at r=0 the equation below is obtained
……………………………………………………………………...36
For r>0 for K=2...i-1
………………………………………………………...37
While at i (i.e. interface), it is given by (from equation 31)
…………………………………………..…..38
Where the index i indicate
i=max (Tm K=1…N-1…………………………………………………………… …...39
48
is the dimensionless distance from line i to the interface from Figure 16
………………………………………………………………………………...40
Where 0 < …………………………………………………………………………….41
……at r=s (t)………………………………………………………………..42
There is singularity that arises from the hydrate-water interface as it gets extremely close to the
grid line. According to Verma et al [44] the temperature gradient at the interface at r=s(t) could
be evaluated .using. Taylor’s series expansion and taking only the first two terms .Therefore the
temperature gradient can be expressed as
……………………………………………………………..43
Hence plugging equation 24 into 23 yields
……………………………………………...44
Now in order to determine the temperature profile and the interface velocity (i.e. dissociation
rate), equations (37, 38 and 44) are applied respectively.
Therefore the systems of ODEs in equations (36), (37), (38) and (44) are solved using ODE15s
solver. Generally ODE assumes smoothness in the solution procedure but since there is a
discontinuity at the interface which would subsequently lead to structural changes. MATLAB
2016T and higher is equipped with event location which is able to identify the point of this
discontinuity to enable the integration to stop momentarily so it could be re-initialized for the
integration to start again.
49
The equilibrium fugacity is found using the Van der Waals Platteuw [74] theory. The fugacities
(fugacity at the interface) and (equilibrium fugacity) were determined by application of the
Trebble-Bishnoi Equation of state [75]. The activation energy, E and Universal gas constant, R
was obtained from Giraldo et al. [69] and geometric ratio, , was assumed to be unity for base
case. For the methane hydrate the stoichiometric factor, W was taken as unity. The
stoichiometric factors , W of the hydrate mixture gas mixtures with varying amount of mole
fractions of the methane and ethane were determined and presented in Table 4.
The porosity was assumed to be 0.5 which was a reasonable estimation since Lysne [23] showed
that porosities of plugs were ranged from 13% to 83%. For hydrate mixtures the calculated W
values was greater than unity. The parameter surface ratio is determined from Mullin et al [76].
They described the parameter as the ratio of the surface of the sphere having the same volume
as the particle to the apparent estimated surface area of the particle.
50
CHAPTER 3-RESULTS AND DISCUSSION
3.1: Simulation Results for Temperature Profile in the Pipeline
The major three driving forces hydrate dissociation mechanism are namely, heat transfer, mass
transfer and intrinsic kinetics. The Fourier heat equation for radial coordinates was applied to
model the process. The discretized equation 36, 37 generated the temperature profiles in the
radial coordinates.
The temperature profile illustrates a general increase in the water phase temperature at
decreasing radial positions from the pipewall. When the hydrate dissociate at the interface, it
absorbs heat that is transmitted from the pipewall though the water phase or water layer around
the hydrate phase. Temperature gradient was established between the pipewall and the hydrate-
water interface at various radial positions. The temperature profile in the water phase depends on
the rate of heat consumed at the interface and the thermal properties of both the water and the
dissociating hydrate phase.
Table 3: Table showing the number of iterations for the method of lines using different ODE solvers
ODE15s 118
ODE 23s 604
ODE45 7477
The ODE 15s was used in our simulations it had the lowest iterations to achieve convergence.
52
Figure 19: Plot of Temperature profiles of both pure methane (C1) and mixture of methane/ethane (C1/C2) hydrate
composition at different radial positions
53
Figure 20:3 D Plot of Temperature Profile of methane/ethane mixture at different radial positions
55
Figure 22: Plot of Temperature Profiles with pure methane and mixture at different radial positions using kinetic model
57
3.2 Decomposition of Hydrate plug kinetics
. The model presented here is to describe the depressurization mechanism in the pipe by coupling
the intrinsic kinetics of the hydrate dissociation with the heat transfer. The model tracks the
movement of the hydrate interface which involves only one moving boundary problem that is
water and hydrate only. The model also reveals the range of the pressures where the heat
transfers resistances is dominating as well as the range where the intrinsic kinetics of the hydrate
dissociation is predominant or kinetics controlled.
The model is initially applied to a finite one-dimensional cylindrical pipeline as presented in
Figure 17. The heat model equation 38 is then matched with the experimental data. The heat
transfer model is then finally extended to predict the hydrate dissociation rate and matched with
the data from Peters and Sloan [77] on the hydrate dissociation rate for both the hydrate plugs for
structures I and II. The heat transfer model developed in this work was validated by matching the
simulated results with the data from Peters and Sloan [77].
The hydrate plug porosities were assumed to be 0.5.The model predicted the position of the
dissociating front of the hydrate plug as a function of time for each of the experimental data. The
latent heat of dissociation for structure I is 460.24kJ/kg, while that of structure II hydrates is
640.15kJ/kg [1] . There was good in agreement between the model prediction for structure II and
the data with an AAD of less than 5%.
58
The possible causes of the deviation from the data could be attributed to the experimental error,
equipment error or set-up and environmental conditions. The dissociation rate for structure I was
faster than structure II. The faster dissociation rate of structure I was probably due to its lower
latent heat of dissociation. The radial dissociation of the hydrate was assumed to be in the center
of the pipeline during the dissociation while the remaining space was occupied by the water.
However, buoyancy effects as well as the presence of hydrocarbon phase on the fluid thermal
diffusivity of the aqueous phase might have influenced the heat transfer rate.
The surrounding fluid composition and the liquid volume fraction for an industrial hydrate plug
will depend on the flowline geometry and plug location. The porosity is not known a priori on
the industrial scale a value of 0.5 was assumed because it was the average value which is a fair
representation of most hydrate plugs. The experimental data that was used to validate the heat
model had the initial equilibrium pressure of 9.8MPa with 7.8MPa as the initial system pressure
at a temperature of 273.25K.
Further simulations using 5.4MPa and 3.8MPa at the same temperature 273.25K was done to
find out which pressure regimes the dissociations rates were kinetically controlled. The equation
37 was applied to predict the dissociation rate in a heat transfer limited system. The validation of
the present model was used to fit the experimental data of structures I and II of gas hydrate plug
dissociation data of Peters and Sloan. It was found that the percentage average absolute deviation
(% AAD) of the predicted hydrate dissociation rate for structure1 were less than 4% from the
experimental data [77] as illustrated in Figure 24.
59
Table 4: Table of calculated stoichiometric values, W of hydrate mixtures (CH4+C2H6)
Components W Type of Hydrate
structure
Pure methane 1.00 Structure I
65%CH4+35%C2H6 2.66 Structure II
85%CH4+15%C2H6 2.85 Structure II
95%CH4+5%C2H6 2.99 Structure II
Table 5: Simulation Parameters of the hydrate plug
Radius of hydrate plug, ro 50cm
Dissociation temperature, Td 273.18K
Pipewall Temperature, To 277.15K
Porosity, θ 0.5 unitless
Geometric ratio for base case, Ψ 1.0 unitless
Stoichiometric coefficient for gas mixture 2.66 unitless
60
Table 6: Table of activation and intrinsic kinetics constants for methane and ethane hydrates
source [42]
Species
∆E(kJ/mol)
Kdo (mol m
-2 Pa
-1s
-1)
CH4 in s1 81.0 3.60*104
CH4 in s11 77.3 8.06*103
C2H6 in s1 and s11 104.0 2.56*108
61
Figure 24: Plot of structure I hydrate dissociation rate compared with the experimental data [77]
0
10000
20000
30000
40000
50000
60000
0 1000 2000 3000 4000 5000 6000
Co
mm
ula
tive
vo
l/L
Time/s
comparison of sI data with heat transfer model Model Data
62
Figure 25: Plot of dissociation rate heat model for structure I with experimental data of Peters et al. [77]
The accuracy of the prediction of the hydrate dissociation rate depends on parameters which are
needed by the heat transfer model based on the assumptions associated with the model. There is
an uncertainty of the value of the porosity in the model. A value of 0.5 has been assumed .The
porosity value depends on the original characteristics of both the hydrocarbon and the source of
geological formation of the hydrocarbon. In the present work both structures I and II hydrates
used the same assumed porosity value of 0.5. The Figures 24 and 25 shows that the model agrees
fairly well with the experimental data for both structures I and II.
0
10000
20000
30000
40000
50000
60000
0 1000 2000 3000 4000 5000 6000
cum
mu
lati
ve v
ol /
L
Time/s
comparison of sII data with heat transfer model Model Data
63
At the initial stage there is a little under prediction of the model values with the experimental
data probably due to the intrinsic kinetics but as the dissociation progresses to the latter stages
there is a significant improvement of the model prediction which eventually gives a very good
match with the data. This satisfactorily good agreement between the model and data suggest that
the assumptions stated in our model were quite valid
64
3.3 Sensitivity analysis of the Model
The model was simulated to investigate pressure regimes where intrinsic kinetics became more
predominant. Therefore a the pressures were varied in the simulation to identify which pressure
regimes indicated where intrinsic kinetics were pre-dominanting.and also where it has little or no
influence. The following results show the trend analysis of the temperature, pressure and the
geometric ratios.
Figure 26: Plot of heat transfer model (HT) compared with kinetic model at different pressures. at a geometric ratio of
one
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 7 8 9 10
no
of
mo
les
of
gas
evo
lve
d/n
Time/hours
Plot of no of moles against time at different Pressures(T=273.15K,Ψ=1)
Ht 3.8Mpa
HT 5.8MPa
Model 3.8MPa
HT 7.8MPa
Model 5.8 MPa
Model 7.8 Mpa
65
The sensitivity analysis illustrates that by changing the pressure from 7.8MPa to 3.8MPa the
dissociation rate moves from heat controlled to a regime where intrinsic kinetics begins to
predominate the process. As can be seen from Figure 26 the percentage deviation between the
heat transfer and the kinetic model is less than one percentage. However reducing the pressure to
5.8MPa and finally to 3.8MPa shows a sharp deviation from the heat transfer model. The
significant deviation indicates that the intrinsic kinetics becomes more prominent when the
pressure is decreased to 3.8MPa and 5.8MPa respectively.
66
Figure 27: Plot of heat transfer (HT) model with present model (Kinetic) at different temperatures
The second parameter that was investigated was the temperature. As can be seen in Figure 27 the
temperature at 273.15K showed that the dissociation rate was only heat transfer controlled at the
pressure of 7.8MPa .The AAD was less than 1% .However as the temperature was increased to
275.15K and eventually to 277.15K there was significant deviation from the model to the heat
transfer model. As the temperatures were raised intrinsic kinetics became more significant, this
could be attributed to the rate constant which is dependent on temperature. Therefore the
increased temperature raises the value of the rate constant thus the overall rate of dissociation is
ultimately increased.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 2 4 6 8 10
no
of
mo
les
gas
evo
lve
d/n
Time/hours
Plot of no of moles against time(P=7.8Mpa,Ψ=1)
273.15K HT
275.15K HT
277.15K HT
273.15K model
275.15K model
277.15K model
67
Figure 28: Plot of hydrate kinetics using different geometric parameters
The last parameter that was investigated was the geometric factor. As can be seen from Figure 28
the model was not very sensitive to the geometric factor. There was not a significant difference
between the different geometric factors indicating that the factor was very negligible.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 7 8 9 10
no
of
mo
leso
f g
as e
volv
ed
/n
Time/hours
Plot of different different geometric values versus time for the present model
Ψ=1---Ψ =2---Ψ=4
68
CHAPTER 4: CONCLUSIONS AND RECOMMENDATIONS
4.1 Conclusions
A moving boundary heat conduction model was formulated to describe the decomposition of a
pipeline hydrate plug. The model includes, in one of the boundary conditions, a term that can
describe the rate of gas hydrate decomposition for hydrates that have been formed in the
presence of a gas mixture. At the time of writing no other modeling attempt had included a term
to explicitly account for the composition of the gas phase, in the heat equation. The dissociation
of a hydrate plug was studied using a very powerful method of lines to discretize the Fourier heat
equation in radial coordinates. The simulation results showed that the hydrate dissociation raw
data matched well the heat transfer model with a deviation of 4% and 5% for structure I and
structure II hydrates plug respectively. It was then coupled with the intrinsic kinetics with the
heat transfer rates.
The simulation results indicated that the rate of hydrate dissociation was sensitive to several
factors which include system pressure, temperature and roughness of the surface of the hydrate.
It was found that by changing the pressure from 7.8MPa to 3.8MPa we could move from a heat
controlled regime to a regime where both heat transfer and intrinsic kinetics have profound effect
on the global rate of dissociation. Also changing the temperature from 273.15K to 277.15K the
system moved from heat controlled to kinetic controlled. The numerical method of lines was the
technique that was used to discretize the heat equations. It was then coupled with the kinetics to
develop the Kinetic model.
69
4.2 Recommendations
From the simulation results of this study, the following suggestions and recommendations must
be considered for future studies:
1) Experiments should be performed to track the decomposition of a hydrate plug in which
the hydrate was formed in the presence of a gas mixture.
2) The model could also be investigated for three phase behavoiur where ice could be
present with the hydrate in the pipe.
3) The model derived in this study could be used as a starting point of a gas hydrate plug
that is subject to a single sided depressurization.
70
APPENDIX
Table 7: Simulation Parameters source [27]
Activation Energy, ∆E 81.0 kJ/mol
Universal gas constant, R 8.3146 JK-1
mol-1
Porosity, θ 0.5
Intrinsic rate constant, Ko(CH4) 3.60*104 mol m
-2 Pa
-1 s
-1
Geometric ratio of hydrate plug, 1.0
Thermal conductivity of water ,kw 1.31*10-3
cm2/s
Thermal conductivity of ice, KI 1.31*10-2
cm2/s
Thermal diffusivity of ice ,αI 1.0007*10-3
m2/s
Thermal diffusivity of water ,αw 1.486*10-7
m2/s
Density of hydrate, 914 kg/m3
Dissociation temperature, Td 0.3oC
Density of ice, kI 917kg/m3
71
Table 8: Hydrate dissociation models, Reproduced from Sloan and Koh [10]
Model Heat transfer Fluid flow Kinetics Solution
Method Conduction Convection Gas Water
Holder and
Angert(1982)
X X Numerical
Burshears et
al(1986)
X X X Numerical
Jamuladin et al
(1989)
X . X Numerical
Selim and
Sloan(1989)
.......X .....X ......X Analytical
Yousif and
Sloan(1991)
...... X X ......X Numerical
Makogon(1997) X X X ......X Analytical
Tsypkin(2000) .......X ......X .....X ......X Analytical
Masuda et al
(2002)
X X .....X ......X .....X Numerical
Moridi et
al(2002)
......X .....X X .......X X Numerical
Pooladi-
Darvish et al
(2003)
X .....X X .......X X
72
BIBLIOGRAPHY
[1] D. Peters, S. M. Selim and E. D. Sloan, "Hydrate Dissociation in Pipelines by Two-Sided
Depressurization,Center for Hydrate Research, Colorado School of Mines, Golden,
Colorado 80401, USA," 1999.
[2] J. Priestley, "Experimentsand Observations on Different kinds of Air and Other Branches of
Natural Philosophy connected with the subject(in three volumes)," Birmingham, 1790, p.
359.
[3] H. Davy, Phil.Trans.Royal .Society London 101,1, 1811.
[4] M. Faraday, "On Hydrate of Chlorine," Quart. Journal Sci. Lit. Arts., vol. 15, pp. 71-74,
1823.
[5] P. Vlillard, Sur quelques nouveaux hydrates de gaz. Compt Rend 106:1602-1603, 1888.
[6] E. G. Hammerschmidt, Formation of Gas Hydrates in Natural Gas Transmission
Lines,Ind.Eng.Chem., Vol.26,(8), 851-855, 1934.
[7] Y. F. Makogon, Hydrates of Natural gases .Nedra, Moscow, 1974.
[8] C. Giavanni and K. Hester, Gas Hydrate Immense Energy Potential and Environmental
changes, New York: Springer, 2011.
[9] T. Collect, A. Johnson, G. Knapp and R. Boswell, Natural gas hydrates ,energy resource
potential and associated geological hazards. AAPG memoir 89, Tulsa, 2009.
[10] E. D. Sloan and C. A. Koh, Clathrates Hydrates of Natural Gases, New York: CRC Press-
Taylor and Francis Group, 2008.
[11] Y. A. Dyandin and K. A. Udachin, "Clathrate Compounds," Molecular Inclusion
Phenomena and Cyclodextrins, vol. 3, 1984.
73
[12] I. Tabushi, Y. Kiyosuke and K. Yamamura, Bull. Chem. Soc. of Japan, 54, 2260, 1981.
[13] R. Rogers and G. Yevi, "in Proc. Second International Conference on Gas Hydrates,"
Toulouse, 1996.
[14] B. C. Chakoumakos, C. J. Rawn, A. J. Rondinone, L. A. Stern, S. Circone and S. H. Kirby,
Can. J. Phys., 81, 183, 2003.
[15] G. A. Jeffrey, "'Hydrate Inclusion Compounds'," J. Inclusion Phenoma, vol. 1, pp. 211-222,
1984.
[16] C. Sorensen, "“Clathrate Structures and the Anomalies of Supercooled Water”," in
Proc.12th Int. Conference on Properties of water and steam, Orlando, September,1994.
[17] D. L. Katz, "Prediction of conditions for Hydrate Formation in Natural gases," Trans. AIME,
vol. 160, pp. 140-149, 1945.
[18] N. R. Council, Realizing the energy potential of methane hydrate for United States,
Washington DC: National Academic Press, 2010.
[19] T. Collett, A. Johnson, G. Knapp and R. Boswell, Natural gas hydrates: energy resource
potential and asssociated geological hazrads, AAPG Memoir 89, Tulsa, 2009.
[20] B. Guo, W. C. Lyons and A. Ghalambor, 'Petroleum Production Engineering-A computer
Assisted Approach', Gulf Professional Publishing, 2007.
[21] C. U. Ikoku, 'Natural gas Production Engineering', Malabar, Florida: John Wily and Son Inc,
1984.
[22] C. A. Koh, E. D. Sloan, A. K. Sum and D. T. Wu, "Fundamentals and Applications of Gas
Hydrates," Annual Review Chem.Biomol.Eng, vol. 2, pp. 237-257, 2011.
[23] D. Lysne, ""Hydrate plug dissociation by pressure reduction" Dr. ing. thesis," Norwegian
74
Institute of Technology(NTH), Trondheim, 1995.
[24] M. H. Yousif and V. A. Dunayevsky, 'Hydrate Plug decomposition:Measurements and
modelling', SPE 30641, 1995.
[25] A. Gupta, "Methane hydrate dissociation Measurements and Modelling: The role of Heat
transfer and Reaction Kinetics,PhD Thesis ,Colorado School of Mines," Golden, 2007.
[26] S. K. Kelkar, M. S. Selim and E. D. Sloan, "Hydrate dissociation rates in pipelines,Center
for Hydrate Research, Colorado School of Mines, Golden, CO 80401, USA," Fluid Phase
Equilibria 150–151, p. 371–382, 1998.
[27] A. K. Jamaluddin, N. Kalogerakis and P. R. Bishnoi, "Hydrate plugging problems in
undersea natural gas pipelines under shutdown conditions," Journal of Petroleum Science
and Engineering,, no. 5, pp. 323-335, 1991.
[28] R. Vazquez-Romain, "A Model Based on Average Velocity for Gas Production Pipes
Simulation," chem. Engng Vol. 22,, pp. pp. 307-314,, 1998.
[29] P. Bollavaram and E. D. Sloan, "Hydrate Plug Dissociation by Method of Depressurization,"
in Offshore Technology Conference, Houston, 2003.
[30] T. Austvik, "Tommeliten gamma field experiments," in 7th International Conference on
Multiphase Floow, Cannes, 1996.
[31] J. Xiao, G. Shoup, G. Hatton and V. Kruka, "Predicting hydrate plug movement during
subsea flowline depressurization operations,," in in:Proceedings of the Offshore Technology
Conference, OTC 8728,, 1998.
[32] F.-D. Nguyen, D. Nguyen and J. M. Herri, "Formation and Dissociation of hydrates plugs in
water in Oil Emulsion," in 4th International Conference on Gas Hydrates, Yokohama, 2002.
75
[33] J. Ajienka and U. Osokogwu, ""Modelling of hydrate Dissociation in subsea natural Gas',"
in 34th Annual SPE International Conference and Exhibition held in Calabar,31 July -7
August 2010, Calabar, 2010.
[34] V. Vlasov, "Formation and dissociation of gas hydrates in terms of chemical kinetics,"
Akademiai Kiado, no. 110, pp. 5-13, 2013.
[35] M. Uddin, F. Wright, Dallimore and D. Coombe, "Gas Hydrate dissociations in Mallik
hydrate bearing zones A,B and C by depressurization," Journal of Natural gas science and
engineering, vol. 21, pp. 40-63, 2014.
[36] K. Lekvan and P. Ruoff, "A reaction kinetic mechanism for methane hydrate formation in
liquid water," Journal of the American Chemical Sociiety, vol. 115, pp. 8565-8569, 1993.
[37] F. Chen, "Estimating hydrate formation and decomposition of gases released in deepwater
ocean plume," Journal of Marine systems,, no. 30, pp. 21-32, 2001.
[38] P. Englezos, N. E. Kalogerakis, P. D. Dholabhai and P. R. Bishnoi, "Kinetics of formation of
methane and ethane gas hydrates,Chemical Engineering Science 42;," p. 2647–2658..
[39] G. Rehder, S. Kirby, B. Durham, L. Stern, E. T. Peltzer, J. Pinkston and P. Brewer,
"Geochimica Cosmochimica Acta," no. 68, p. 285, 2004.
[40] H. C. Kim, P. R. Bishnoi, R. A. Heidermann and S. H. Rizvi, "Kinetics of Methane Hydrate
Decomposition," Chemical Engineering Science, no. 42, pp. 1645-1653, 1987.
[41] M. A. Clarke and P. R. Bishnoi, "Determination of intrinsic rate of ethane gas hydrate
decomposition," Chemical engineering Science, no. 55, pp. 4869-4883, 2000.
[42] C. Giraldo and M. A. Clarke, "'Measuring and Modelling the rate of decomposition of gas
hydrates formed from mixtures of methane and ethane'," Chemical Engineering Science, no.
76
56, pp. 4715-4724, 2001.
[43] J. Stefan, "Uver die Theorie der Eisbildung,Insbesondere Uver die Eisbildung im
Polarmeere," Annalen der Physik und Chemie,Vol. 42,, vol. 42, pp. 269-286, 1891.
[44] A. K. Verma, S. Chandra and B. K. Dhindaw, "An alternative fixed grid method for solution
of the classical one-phase stefan problem," Applied Mathematics and computation, no. 158,
pp. 573-584, 2003.
[45] W. D. Murray and F. Landis, "Numerical and machine solution of transient heat conduction
involving phase change," Journal Heat Transfer 81, pp. 106-112, 1959.
[46] S. Kutluay, "Numerical schemes for one-dimensional Stefan-like problems with a forcing
term," Applied Mathematics and Computation, no. 168, pp. 1159-1168, 2005.
[47] H. G. Landau, "Heat Conduction in a melting Solid," Quart. Journal Appl. Math, no. 8, pp.
81-94, 1950.
[48] F. L. Chermousko, "Solution of non-linear problems in medium with changes," Int. Chem.
Engng, no. 10, pp. 42-48, 1970.
[49] R. C. Dix and J. Cizek, "The isotherm migration method of transient heat conduction
anaysis,U. Gigull and E. Hahre. Eds,Heat transfer," in 4th Znternation. Heat transfer conf. ,
Paris, 1970.
[50] A. Esen and S. Kutluay, "An isotherm migration formulation for one-phase stefan problem
with time dependent Neumann condition," Applied Mathematics and computation, no. 150,
pp. 59-67, 2004.
[51] T. G. Myers and S. L. Mitchel, "Application of combined integral method to stefan
problems," Applied Mathematical modelling, no. 35, pp. 4281-4294, 2011.
77
[52] S. Y. Reutskiy, "A meshless method for one-dimension Stefan problems," Applied
Mathematics and computation, no. 217, pp. 9689-9701, 2011.
[53] L. Chein-Shan, "Solving two typical inverse stefan problems by using the Lie-group
shooting method," Internal Journal of Heat and Mass Transfer, no. 54, pp. 1941-1949,
2011.
[54] J. Tausch and E. Case, "An integral equation method for spherical Stefan problems," Applied
Mathematics and Computation, vol. 218, pp. 11451-11460, 2012.
[55] S. L. Mitchell and M. Vynncky, "An accurate finite-difference method for ablation-type
stefan problems," Journal of Computational and Applied Mathematics, vol. 236, pp. 4181-
4192, 2012.
[56] J. Papac and R. C. G. F. Helgadottir Asdis, "A level set approach for diffusion and stefan-
type problems with Robin boundaty conditions on quadtree/octree adapttive Cartesian
grids," Journal of Computational Physics , vol. 233, pp. 241-261, 2013.
[57] M. Tadi., "A fixed-grid local method for 1-D Stefan problems," A fixed-grid local method
for 1-D stefan problems, vol. 219, pp. 2331-2341, 2012.
[58] Y. Ping, J. Liandrat, W. Shen and Z. Chen, "A multiresolution and smooth fictitions domain
method for one-dimensional elliptical and Stefan problems," Mathematical and Computer
Modelling, vol. 58, pp. 1727-1737, 2013.
[59] V. R. Voller, "Fractional Stefan problems," International Journal of Heat and Mass
Transfer, vol. 74, pp. 260-277, 2014.
[60] D. Hetmanoik, A. Slota and E. Zieionka, "Using the swarm intelligence algorithmns in
solution of the two-dimensional inverse Stefan problem," Computer and Mathematics with
78
Applications, vol. 69, pp. 347-361, 2015.
[61] T. Song, K. Upreti and G. Subbarayan, "A sharp interface isogeometric solution to the
Stefan problem," Computer methods in Applied mechanical engineering, vol. 284, pp. 556-
582, 2015.
[62] P. A. Krasnova and P. R. Levashov, "Two-phase isochoric Stefan problem for ultrafast
process," International Journal of heat and Mass Transfer, vol. 83, pp. 311-316, 2015.
[63] T. G. Myers and F. Font, "On the one-phase reduction of Stefan problem with a variable
phase change temperature," International Communications in Heat and Mass Transfer, vol.
61, pp. 37-41, 2015.
[64] S. Mitchell, "Applying the combined integral method to two-phase Stefan problems with
delayed onset of phase change," Journal of computational and Applied Mathematics, vol.
281, pp. 58-73, 2015.
[65] O. P. Layeni and J. V. Johnson, "Exact closed-form solution of some Stefan problems in
thermally heterogeneous cylinders," Mechanics Research Commmunications, vol. 71, pp.
32-37, 2016.
[66] U. Lacoa and A. Campos, "Adaptation of Method of lines (MOL) to the MATLAB code for
the Analysis of Stefan problem," WSEAS Transactions on Heat and Mass Transfer, vol. 9,
2014.
[67] A. Zafarullah, " 'Application of the method of lines to parabolic diffrential equations with
error estimates'," J.ACM, vol. 17, no. 2, pp. 294-302, 1970.
[68] D. N. Hong, F. Gruy and J.-M. Herri, "Experimental data and approximation estimation for
dissociation time of hydrate plugs," Chemical Engineering Science, no. 61, pp. 1846-1853,
79
2005.
[69] C. Giraldo and M. A. Clarke, "Stoichiometric Approach toward Modelling the
Decomposition Kinetics of Gas Hydrates Formed from Mixed Gases," American Chemical
Society (Energy and Fuels), pp. 1204-1211, 2013.
[70] C. Giraldo, B. Miani and P. R. Bishnoi, "A simplified approach to modelling the rate of
formation of gas hydrates formed from mixtures of gases.," Energy Fuels, vol. 27, pp. 1204-
1211, 2013.
[71] C. Chun and S. Park, "A fixed-grid finite-diiference method for phase- change problems.,"
Numerical Heat Transfer,Part, vol. 38, pp. 59-73, 2000.
[72] U. Lacoa and A. Campo, "Adaptation of Method of Lines(MOL) to the MATLAB Code for
the Analysis of the stefan Problem," de Mecanica Computacional, vol. Vol XXVII, pp.
1573-1579, 10-13 November 2008.
[73] B. Furenes and B. Lie, "Using event location in fionite-diffrence methods for phase
problems.," Numerical Heat Transfer , Part B, vol. 50, pp. 143-155, 2006.
[74] J. Van der Waal and J. Platteuw, "'Clathrate solutions'," Advances in Chemical Physics, pp.
1-57, 1959.
[75] M. A. Trebble and P. R. Bishnoi, "Extension of Trebble-Bishnoi Equation of State to fluid
mixtures," Fluid Phase Equilibrium, no. 40, pp. 1-21, 1988.
[76] J. Mullin, "'Crystallization in Butterworth-Heinemann'," London, 1997.
[77] D. Peters, M. S. Selim and E. D. Sloan, "Hydrate Plug Dissociation," AlChE, vol. 52, no. 12,
p. 4022, 2006.