REPORTS IN METEOROLOGY AND OCEANOGRAPHY

REPORTS IN METEOROLOGY AND OCEANOGRAPHY
REPORTS IN METEOROLOGY AND OCEANOGRAPHY
UNIVERSITY OF BERGEN
1 - 2008
Hydro- and thermodynamics related to
CO2-fluxes through the sea floor
By
Sönke Maus
Geophysical Institute
Allegaten 70
5007 Bergen
Norway
«REPORTS IN METEOROLOGY AND OCEANOGRAPHY»
utgis av Geofysisk Institutt ved Universitetet I Bergen.
Formålet med rapportserien er å publisere arbeider av personer som er tilknyttet
avdelingen.
Redaksjonsutvalg:
Peter M. Haugan, Frank Cleveland, Arvid Skartveit og Endre Skaar.
Redaksjonens adresse er : «Reports in Meteorology and Oceanography»,
Geophysical Institute.
Allégaten 70
N-5007 Bergen, Norway
RAPPORT NR: 1 - 2008
ISSN 1502-5519
ISBN 82-8116-013-6
Preface
The CLIMIT programme under the Norwegian Research Council, awarded a contract to
the University of Bergen in April 2007 for a review study on possible marine impacts
from storing CO2 under the seabed. The work has been conducted by the following
partners: Christian Michelsen Research (CMR), NIVA, Geophysical Institute (GFI, UiB),
Department of Mathematics (MI, UiB), and Bergen Center for Computational Science
(BCCS, Unifob).
The study “Geological storage of CO2 - The marine component”
was divided into four work packages with the following tasks:
• WP 1: Project coordination, including arranging a workshop (BCCS)
• WP 2: Impact on sediments, seawater and marine biota (NIVA)
• WP 3: The benthic boundary layer dynamics (UiB, GFI)
• WP 4: Monitoring and detection of seeps and CO2 in seawater (CMR)
The results from the work packages 1, 2 and 4 can be found as:
• WR 1: Presentations from a workshop: http://www.cmu.uib.no/subseastorage
• WP 2: NIVA report nr. 5478-2007 “Sub-seabed storage of CO2. Impact on sediments,
seawater and marine biota from leaks”
• WP 4: “Monitoring and detection of seeps and CO2 in seawater” as report CMR-07F10808-RA-1.
The present report is the contribution WP 3 by the Geophysical Institute (GFI, UiB. The
latter was originally planned as a review of oceanic bottom boundary dynamics. Due to
the manifold in ongoing rsearch on physical problems related to the leakage of CO2,
including the thermodynamics of pure CO2 and mixed seawater-CO2 systems, droplet
plume dynamics, ocean-seabed interaction, as well as stability and permeability aspects
of the seabed itself, the original topic has been extended. The title of the present report
“Hydro- and thermodynamics related to CO2-fluxes through the seafloor” reflects
this extension.
Bergen, January 2008
Sönke Maus
Geological storage of CO2
The marine component
CLIMIT Report
Hydro- and thermodynamics related to
CO2-fluxes near the sea floor
by Sönke Maus
Geophysical Institute, University of Bergen
Allegaten 70, 5007 Bergen, Norway
Bergen, January 2008
Contents
1 Background: CO2 storage
1.1 Climate change mitigation . . . . . . . . . . . . . . . . . . . .
1.2 Carbon capture and storage (CCS) . . . . . . . . . . . . . . .
1.2.1 Sequestration in the ocean: Dissolution is no solution .
1.2.2 Geological storage versus other approaches . . . . . . .
1.2.3 Safety of geological storage . . . . . . . . . . . . . . .
1.2.4 Sustainability . . . . . . . . . . . . . . . . . . . . . . .
1.3 Present Report . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Phase equilibria and thermodynamics of
2.1 Density and equation of state . . . . . .
2.2 Solubility . . . . . . . . . . . . . . . . .
2.3 Diffusion and dissolution . . . . . . . . .
2.4 Hydrate formation and stability . . . . .
2.5 Oceanic hydrate stability regimes . . . .
3 Phase transitions and metastability
3.1 Droplets with hydrate shells . . . .
3.2 Hydrates in porous media . . . . .
3.2.1 Gibbs-Thomson effect . . .
3.2.2 Metastability: Undercooling
CO2 and seawater
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and Superheating
4 Droplet dynamics and thermodynamics
4.1 Low Reynolds number flow . . . . . . .
4.2 High Reynolds numbers . . . . . . . . .
4.3 Turbulent mass diffusion and dissolution
4.4 Oceanic experiments and simulations . .
4.5 Future challenges . . . . . . . . . . . . .
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5 Oceanic bottom boundary layer
5.1 Classical scaling laws . . . . . . . . . . . . . . .
5.1.1 Ekman Layer . . . . . . . . . . . . . . .
5.1.2 Outer boundary of log-layer . . . . . . .
5.1.3 Viscous sublayer . . . . . . . . . . . . .
5.1.4 Diffusive sublayer . . . . . . . . . . . . .
5.2 Improved models . . . . . . . . . . . . . . . . .
5.2.1 Ekman layer and stratification . . . . .
5.3 Ocean-seabed interactions . . . . . . . . . . . .
5.3.1 Flux closure and diffusive sublayer δdif f
5.3.2 Internal wave-seabed interaction . . . .
5.3.3 Benthic storms . . . . . . . . . . . . . .
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6 Mesoscale simulations and observations
6.1 Deep sea lake . . . . . . . . . . . . . . . . . . . . . . .
6.2 Droplet plume modelling . . . . . . . . . . . . . . . . .
6.3 Observations . . . . . . . . . . . . . . . . . . . . . . .
6.4 Longterm perspectives and climate model simulations
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7 Fluxes through the seabed
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7.1 Permeability and Percolation . . . . . . . . . . . . . . . . . . . . 41
8 Summary and outlook
44
1
1
Background: CO2 storage
The past decades have resulted in the consent among scientists that the release
of greenhouse gases by human beings has begun to change climate on earth and will increasingly do so in the future (IPCC, 2007a). More than 60 % of
the anthropogenic greenhouse effect is related to carbondioxide release due to
burning of fossil fuels. The global emmissions of C02 show, both due to the
increasing energy usage in developing countries, and the continuous increase in
industrial countries, a rise in atmospheric C02 that even exceeds the worst scenarios from the IPCC (Figure 1). The severity of the problem may be envisaged
by quoting the lifetime of anthropogenic atmospheric C02 in the biosphere in
the popular form suggested by Archer (2005): 300 years, plus 25 % that lasts
forever.
1.1
Climate change mitigation
To avoid the serious risks of rapid climate change it is necessary to prosecute
actively all available mitigation options. The sustainable longterm goal, a stabilisation of atmospheric C02 concentration by energy saving and renewable
production, requires severe transformations of the present global energy infrastructure and societies (Parson and Keith, 1998). A realistic evaluation of the
present global situation points to the likelihood, that the expectable transformation will be too slow to mitigate climate change problems, if not methods
of geoengineering are considered. One of the first suggested geoengineering approaches was injection and dissolution of liquid C02 in the ocean (Marchetti,
59-68). Recently also the possibility to influence the radiation balance of the
upper atmosphere has been considered (MacCracken, 2006). The at present,
in terms of technology, safety and scientific clrity, most realistic approaches of
climate change mitigation by geoengineering appear to be related to the field
of carbon capture and storage (CCS) (IPCC, 2007b). These employ carbon
capture in an oil-, gas- or coal-fired power plant, followed by (i) mineral carbonation, (ii) direct injection into the ocean or (iii) storage in geological formations,
terrestrial and below the ocean floor (Figure 3). Assessment models of climate
change mitigation indicate that CCS is capable to provide a fraction of 20 to
30% of global emission reductions during the next century (Figure 2).
1.2
Carbon capture and storage (CCS)
The present scientific state of the art on Carbone Capture and Storage (CCS)
has been summarised by the Intergovernmental Panel of Climate Change IPCC
(2007b) and others (Göttlicher, 2006; Fischedick et al., 2007). On the one hand,
≈ 85% of CO2 emitted from a power plant may be captured by modern techniques. On the other hand, capture implies an increase in the energy requirement
by 15 to 30 %, and an increase in the capital coasts for energy production by
40 to 80 %. For transport and storage another coast penalty of each 5 − 15 %
has to be added, which raises the overall coast penalty of CCS to 50 to 110 %.
As coasts are, in the present world, intimately related to energy consumption
2
Figure 1: Evolution of C02 emissions for different groups of countries. From
Raupach et al. (2007).
one expects that CCS will increase the brutto emission of CO2 by 20 − 40%. In
connection with the capturing potential of ≈ 85% this means that effective CO2
emissions may be reduced to ≈ 20% of the values without CCS. This potential
may apply to ≈ 60% of the present fossil fuel burn, excluding emissions from
dispersed sources like buildings and vehicles, and it thus leads to the reduction
potential indicated in Figure 2.
Due to the ≈ 20 − 40% increase in energy consumption it is clear that the
storage of CCS must be safe and sustainable. Otherwise it will increase the
problems that it aims to solve. In this sense one must distinguish between time
scales of oceanic and geological storage, to be outlined below.
1.2.1
Sequestration in the ocean: Dissolution is no solution
The possibility of CO2 sequestration by direct injection into the ocean was
first considered by Marchetti (59-68). The present state-of-the-art simulations
indicate that, in case of shallow injection between 800 and 1500m ocean depth,
most of the CO2 captured in the ocean will return to the atmosphere within a few
centuries (IPCC, 2007b). Due to the higher brutto emission of CO2 by capture
the ocean storage option thus rather delays and increases the problem to future
generations. In case of injection in the deep ocean there is, due to liquid CO2
becoming heavier than seawater, a longer residence of 500-1000 years expected,
yet the risk of considerable ecological damage of oceanic benthic life is much
higher (IPCC, 2007b; Johnston et al., 1999). In this connection it is notable
3
Figure 2: Projections of primary energy use (a and b) due to two assessment
models (IPCC, 2007b). The reduction potential of atmospheric emissions indicates the role of CCS in a global mitigation portfolio.
that oceanic pH will already change seriously during the next centuries, due to
present and the expectable future emissions to the atmosphere (Figure 4). An
enhancement of this signal by direct injection of CO2 into the ocean is likely
very problematic (Caldeira and Wickett, 2003; IPCC, 2007b). These aspects
make a direct CO2 release into the ocean a rather unsustainable approach to
climate change mitigation.
1.2.2
Geological storage versus other approaches
The overall storage capacity of CO2 in geological formations is likely 1.5 to 2
×103 GtCO2 , with saline formations and oil/gas fields contributing roughly 2/3
and 1/3 and coal seams less than 1% (IPCC, 2007b). Provided that geological
storage is save, it is useful to compare these numbers to costs of other mitigation
options, and it is important to be aware of the limits of these numbers.
• The world-wide storage capacity of CCS implies that the annual capture
in Figure 2 may be realistic for 50 to 150 years beyond the year 2100. This
is a large potential, but it clearly points to the bridging character of CCS.
• The costs for capture, transport and storage will most likely be in the range
30 to 70 Euro/tCO2 IPCC (2007b); Fischedick et al. (2007). This compares to an approximately two times larger range of 70 to 150 Euro/tCO2
for capture and re-mineralisation. Notably, emission reductions of 30%
4
Figure 3: Options of C02 capture and storage. From IPCC (2007b).
on the basis of intelligent technology and energy efficiency, also included
in the scenarios from Figure 2, will cost less than 25 Euro/tCO2 , while
a certain amount of savings will be available for free. Costs for sequestration by forestation, considering a similar 20-30 % reduction of present
emissions, have been estimated as 30-40 Euro/tCO2 (Stavins, 1999).
Hence, there is sufficient storage capacity for CCS to take the role of a bridging
function - on the order of a century - as a climate change mitigation option.
Costs are comparable to other approaches like, for example, re-forestation. Savings and intelligent renewable energy use must take over in the long-term.
At present the global annual injection into geological reservoirs is around 30
MtCO2 , mostly in west Texas, to recover oil (Enhanced Oil Recovery). Other
noteworthy pilot studies are the Weyburn project (Canada), which combines
EOR with monitoring, modelling and research, and the annual injection of the
order of 1 MtCO2 in the Sleipner formation (Norway). The climate mitigation
target proposed by the IPCC thus requires an increase of the present day storage
rates by a factor of 500.
1.2.3
Safety of geological storage
The basic mechanisms to be expected for injection of CO2 in (subsea) saline
formations are the following (IPCC, 2007b): As liquid CO2 and water are immiscible, the less dense and less viscous liquid CO2 will move upwards in form
of plumes. Viscous fingering makes upward movement more easy. When the
pore fluid becomes denser by dissolution of CO2 , convection cells may form and
create a downward transport and effective mixing of CO2 in the formation. Two
5
Figure 4: From Caldeira and Wickett (2003): a) Simulated evolution of CO2
emissions, atmospheric pCO2 and oceanic ∆pH compared to its preindustrial
value in a probable emission scenario; b) Natural variability and timescales
(A,B,C) versus expectable anthropogenic signal (D) in ∆pH.
other processes are expected to be relevant: (i) Some CO2 will also be trapped
by capillary forces or dead-ends in the pore space, (ii) CO2 dissolved in water
will react with the silicate minerals to form carbonate minerals, thereby chemically trapping the CO2 . Adsorption as a third mechanism is relevant in coal
seams, yet these contribute only by 1 % to the overall storage capacity, which
is mostly due to saline formations.
Convective transport is likely an important trigger for dissolution of CO2
in a saline formation. Simulations of systems with slowly flowing water indicate that during tens of years 30 % of the CO2 will dissolve in the formation
fluid, with complete dissolution over centuries. Without fluid flow dissolution
is governed by diffusion and local concentration gradients and will take much
longer time, 102 to 104 years. Mineralisation is an even slower process for which
time scales of > 106 years can be estimated. Also mineralisation may be enhanced by convection. When saturation levels are reached, convection may lead
to outgassing and more complex situations of multiphase flow.
The main aspects of the safety evaluation of geological formations in question
for CCS, as evaluated by the IPCC IPCC (2007b) may be summarised as:
• It is considered likely that 99 % of injected CO2 will be retained for 1000
years (at carefully injected sites).
• Trapping below a confining layer (cap-rock) may be enhanced by (i) longterm dissolution in the in situ formation fluids, (ii) adsorption onto organic matter in coal and shale, (iii) trapping by reaction with minerals to
produce carbonate minerals.
6
• Potential risks to humans and ecosystems arise from (i) leaking injection
wells, (ii) leakage across faults and (iii) ineffective confining layers.
However, there are a number of processes that are not well understood and
therefore present uncertainty factors in the evaluation of longterm safety of
storage:
• Warming of saturated pore fluid implies degassing of CO2 and possible
upward migration of gas bubbles. Such a mechanism may be triggered
once convection sets in.
• Reaction of CO2 with rocks and formation water may effect the porosity and permeability of the formation. Observations are not available and
modelling studies are sparse. Xu et al. (2003) simulated that precipitation
of carbonates may slightly decrease the average porosity and permeability
in a reservoir on time scales of 1000-10,000 years. However, such simulations do not account for a possible heterogeneity of reactions and flow. The
formation of critical flow paths and localised pore structure changes may,
in contrast to homogeneous chemical model predictions, rather enhance
the permeability.
• Changes in the chemical, hydrological and mechanical properties may create stresses in the seabed and thus trigger crack formation.
The mentioned unknowns relate to a lack in coupled models of geochemical
modelling on the pore-scale and simulations of macroscopic fluid flow. Direct
validation of these processes on the field scale is difficult, and the present uncertainty evaluations rely mainly on the comparison of different numerical models
(IPCC, 2007b).
1.2.4
Sustainability
The term sustainability should be taken literally by recognising that CCS can
only overtake a bridging function during the transformation of energy infrastructures and societies towards renewability: This transition time is, due to
storage capacities, limited to one to two centuries. Any way of thinking that
proposes, due to the possibility of CO2 capture and storage, the continuation
of present-day burn of fossil fuels must be questioned. An example of such a
way of thinking is the term negative storage coasts, introduced by the IPCC in
connection with Enhanced Oil Recovery by injection of CO2 (EOR). The latter is considered as an option that avoids CO2 emissions, although the process
essentially discharges oil to be converted into CO2 emissions. This logical contradiction clearly points to the necessity to solve future problems by means of
rational and ethical, and not simply economical considerations.
7
1.3
Present Report
For a proper evaluation of CCS as a climate change mitigation option model and
observational studies are required. These may be grouped into investigations of
(i) geological formation safety and (ii) environmental problems that arise when
leakage from a storage reservoir occurs. From a geological point of view (i), one
has to combine geochemical modelling efforts with monitoring and careful selection of sites for which criteria have been summarised (IPCC, 2007b). According
to present day monitoring and modelling one may, if these criteria are followed,
expect a retention of more then 99 % of the injected CO2 for 1000 years (IPCC,
2007b). However, these numbers must still be viewed as provisoric and needing
more validation studies. While there is considerable work ongoing with respect
to more complex simulations and monitoring of geological formation storage,
there is a lack in studies of the effects of leakage on the ecosystems near the
seafloor (Gale, 2004; IPCC, 2007b).
The present report focuses mainly on the situation that leakage takes place,
considering the physical processes that are relevant when a CO2 source is present
at the sea floor. It is concerned with hydrodynamic and thermodynamic aspects in the oceanic Bottom Boundary Layer (BBL) and in the bottom-near
sediments. Investigated topics are (i) thermodynamics of hydrates and their
stability, (ii) the hydrodynamics and dissolution of rising droplets, (iii) turbulence and mixing in the oceanic bottom boundary layer, (iv) interaction between
sediments, ocean currents and waves. Last but not least, are some stability aspects of porous media, eventually also relevant for seabed fluid flow and seepage
through deeper layers, pointed out. Within the frame of the present work it is
hardly possible to present a detailed discussion or overview of the mentioned
topics. Instead it is focused on specific problems that have been recently addressed in investigations related to CO2 release and storage. It is attempted to
review these investigations critically while summarising the main hydrodynamic
or thermodynamic background. For more detailed discussions of the physical
problems encountered the reader is referred to selected basic textbooks. It is
hoped that the present overview helps to get a basic idea of the thermo- and
hydrodynamical problems in connection with fluxes near the seafloor. Most of
them are not only relevant in terms of climate change mitigation, yet important
for a general understanding of the earth’s climate system.
8
2
Phase equilibria and thermodynamics of CO2 and
seawater
To understand and model two-phase flow and dissolution in mixtures of seawater
brine and CO2 one must know the phase equilibria and PVTx-properties for the
separated and combined systems. For seawater these are well established in the
relevant pressure and temperature range (Fofonoff and Millard, 1983; Feistel,
2003). For high-salinity brines and wide temperature and pressure ranges, empirical algorithms show a wide spread and must be viewed with caution (Adams
and Bachu, 2002). However, theoretically well-founded extensions may be found
on the basis of Pitzer’s semi-empirical approach, as demonstrated for aqueous
NaCl (Pitzer et al., 1984; Archer, 1992). Also for pure CO2 an equation of state
has been validated for a sufficiently wide P-T range (Ely et al., 1989; Pitzer
and Sterner, 1995). For the binary CO2 -H2 0 and ternary CO2 -H2 0-NaCl systems, data and models have been reviewed by Hu et al. (2005). These authors
conclude that presently available models and data sets are not consistent and
that this poses some limits on the detailed modelling of CO2 sequestration in
reservoirs. These restrictions are not that severe for the limited PVTx-regime
in the ocean, however.
CO2 may occur in seawater in the pure gaseous and liquid phases and in
dissolved form. It may also occur as a clathrate hydrate, where a cubic solid
lattice of water molecules encloses gas molecules of CO2 (v. Stackelberg, 1949;
Miller, 1961; Sloan, 1998; Buffett, 2000). Hydrates look like ice and have a
similar density, yet the gas molecules are trapped in water cages at two orders
of magnitude larger concentration than in the gas form. The stability of the
hydrate, in dependence on pressure, temperature and solute content, is of major
interest to understand dissolution processes in seawater and marine sediments.
2.1
Density and equation of state
Figure 5 shows the density regimes of liquid C02 , seawater and CO2 -saturated
seawater relevant in the ocean. The transition pressure between the gaseous
and liquid phases is normally reached at 400-500 m depth (Figure 6). Due to
its higher compressibility liquid C02 becomes, for typical oceanic temperatures,
denser than seawater below ≈ 3000 depth. CO2 -saturated seawater is considerably denser than seawater. This density difference exceeds the typical density
differences that drive the ocean-circulation by an order of magnitude.
The seawater density increase due to dissolution of CO2 may be approximated by a linear concentration dependence (Bradshaw, 1973), and such an
equation of state has been used in oceanic simulations (Fer and Haugan, 2003;
Haugan and Alendal, 2005). For the temperature range 3 to 12 ℃, and a
pressure up to 12 MPa (1200 m ocean depth), this assumption appears to be
justified (Song et al., 2005). As mentioned above, models and observations are
less conclusive over the full PVTx range that needs to be considered for storage
in subsea formations, and a sufficiently accurate equation of state has still to be
established for these regimes (Hu et al., 2005).
9
Figure 5: Density of liquid C02 ,
seawater and CO2 -saturated seawater
(for 5 ℃) versus pressure. Arrows indicate the typical oceanic transition
levels to the gas (upper) and hydrate
(lower) phases. From Fer and Haugan
(2003).
2.2
Figure 6: Phase equilibrium depthT diagram of C02 along with a typical upper-ocean temperature gradient
(solid curve).
From Brewer et al.
(2002).
Solubility
For the PTSx-range of interest in the ocean and seabed, solubility of CO2 decreases with temperature and composition and decreases with pressure (Figures
7 and 7). Models and observations of the solubility of CO2 in water and aqueous
solutions have been reviewed by Duan and Sun (2003). These authors applied
Pitzer’s ion interaction approach to model CO2 solubility in seawater for temperature of 273 K and pressures down to 2000 bar and have recently improved
the computations and accuracy Duan et al. (2006).
2.3
Diffusion and dissolution
Dissolution depends on the concentration gradient at an interface via Fick’s law
Fs = Ds (
dC
)int ,
dz
(1)
where Ds is the molecular diffusivity of a dissolved species with concentration
C. For the applications relevant in the present study the interfacial gradient is normally controlled by convection and a function of the saturation concentration Csat (from solubility), a concentration C∞ far from the interface,
and the convecto-diffusive mass transport V∗ . It is frequently parametrised as
Ds (dC/dz)int = V∗ (Csat − C∞ ). V∗ depends on the hydrodynamics near the interface and scales as some function of the molecular viscosity ν and the Schmidt
number Sc = ν/Ds .
Some diffusion coefficients for CO2 and other gases and ions are listed in
Figure 9. Note that the viscosity of water decreases for the range 0 to 20
10
Figure 7: Solubility of CO2 in pure
water. From Duan and Sun (2003).
Figure 8: Solubility of CO2 in seawater. From Duan and Sun (2003).
℃ by a factor 1.8 (IAPWS, 2003; Zaytsev and Aseyev, 1992). These values are
consistent with classical thermodynamics and the Stokes-Einstein relation which
implies Ds ∼ θ/µ, where θ the absolute (Kelvin) temperature and µ the dynamic
viscosity νρ. As µ decreases with temperature and Ds increases with θ and 1/µ,
the Schmidt number depends considerably on temperature. This has often been
ignored in studies to be mentioned below and needs to be properly accounted
for in models of interface dissolution enhanced by convective transport.
2.4
Hydrate formation and stability
At pressures higher than the gas-liquid-hydrate equilibrium (at 433 m) the phase
equilibrium between liquid CO2 and gas hydrate shows a weak pressure dependence (Figure 6). Thermodynamic predictions of this stability curve for gas
hydrate based on Pitzer’s method have been presented for seawater by Duan
and Sun (2006). According to this model the equilibrium temperature increases
with depth from 8.43 ℃(433 m) to 10 ℃(1874 m) and 11 ℃(3171 m). In oceanic
intermediate and deep regimes CO2 may thus, under appropriate saturation conditions, be present in form of hydrates.
The density and composition of hydrates depends on pressure (Handa, 1990;
Sloan, 1998). Observations at deep ocean conditions of 30 MPa indicate an
expected range of 1.09 to 1.11 g/cm3 (Aya et al., 1997). The hydrate density 1.12
g/cm3 indicated in Figure 5 is based on a structural X-ray diffraction study at a
comparable high pressure (Udachin et al., 2001). While the thermodynamics of
hydrate formation is reasonably understood, the role of kinetics and interfacial
processes in determining the exact composition, microstructure and density of
hydrates is still a challenge for future research (Sloan, 1998, 2004).
2.5
Oceanic hydrate stability regimes
The general picture of hydrate stability in the deep ocean is indicated in figure 10. In the ocean temperatures decreases towards the bottom, and under
most conditions one enters the hydrate stability regime below 300-400 m depth.
Near-bottom temperatures of 0 to 5 ℃ imply, with a geothermal gradient of
≈ 0.05K/m, a regime of 100-200m below sea bottom, where CO2 would be
11
Figure 9: Molecular diffusion coefficients of several gases, ions and
molecules. From Jørgensen (2001).
Figure 10: Principal regimes of hydrate stability in the ocean. From
Buffett (2000).
present in form of hydrate. Only at deeper levels it is expected in its stable liquid form. For somewhat different pressure and temperature ranges the problem
also applies to methane hydrate stability, where the transition has been realised
as a possible driver of climate change (Kvenvolden, 1993). Some further remarks
on this problem are given in section 3.2 below. Conditions that prevail in warm
shallow seas are thermodynamically different: if temperatures are low enough
for hydrates to form, the transition will be from the gaseous to the hydrate
phase (Figure 6).
The hydrate stability conditions are important in terms of phase transitions
and permeability of the seabed. In the ocean hydrate will under most cases only
be metastable, as its stability requires saturation. As will be discussed in the
following section 3, the details of this metastability are rather challenging in the
dissolution problem of rising CO2 droplets covered by a hydrate shell.
12
3
3.1
Phase transitions and metastability
Droplets with hydrate shells
Observations of the rapid formation of hydrates in containers at large ocean
depths have been published by Brewer et al. (1999). It has also been documented that CO2 and methane droplets released in the oceanic hydrate stability
regime become covered with a thin hydrate shell, while their buoyancy is still
governed by the density of liquid CO2 (Brewer et al., 2002). Sugaya and Mori
(1996) proposed that the detailed morphology of such a hydrate layer depends
strongly on the degree of supersaturation, and thus the hydrodynamic conditions adjacent to the interface. In laboratory experiments the latter authors
estimated a hydrate layer thickness of ≈ 10µm based on optical observations.
To explain such a shell, Mori and Mochizuki (1997) have suggested a perforated
plate model from which a hydrate thickness in dynamic equilibrium between
dissolution and dissociation can be predicted. The thickness δsh may be written
as
rc φγw,co2
δsh =
F (Csat , C∞ , n),
(2)
4τ 2 V∗ µ
with capillary radius rc , porosity φ, tortuosity τ , liquid CO2 -water interfacial
tension γw,co2 , kinematic viscosity µ and a function F that depends on solubility
Csat , ambient dissolved concentration C∞ and hydration number n. The main
point is the dependence on unknown structural parameters like the porosity,
pore radius and tortuosity, as well the mass transfer rate V∗ at the hydratewater interface. Due to the lack in mass transfer information in the laboratory
experiments made by (Sugaya and Mori, 1996), it was not possible to constrain
the model parameters (Mori and Mochizuki, 1997). A number of alternative
models and explanations have been discussed by Mori (1998). He concluded that
the perforated plate model was one of the most realistic physical explanations
for the stability of a thin hydrate shell.
An approach to estimate the plate thickness by a mechanistic approach,
based on correlation with observed droplet rise and dimensions (Gabitto and
Tsouris, 2006) has been criticised by Mori and Murakami (2007). It is worth a
note that in the perforated plate model the hydrate shell has no direct insulating
influence on the dissolution rate (Mori, 1998; Mori and Murakami, 2007), yet the
reduction comes from a change in solubility in the presence of hydrate (Zhang,
2005). The thickness may, however, be of interest when considering the elasticity
and deformation of the film. In a different context, considering a lake of CO2
covered by hydrate, Fer and Haugan (2003) have speculated on the possibility
of tensile fracture leading to the instability of such shell. Replacing the liquidliquid surface tension in a hydrodynamical problem by the tensile strength of
the hydrate layer, these authors calculated the most instable wave length and
critical interface velocities for the layer break-up.
Under large enough supercoolings hydrates form with a dendritic structure
as known for ice, which has been shown by optical observation methods (Ohmura
et al., 2004; Katsuki et al., 2007), yet there are at present no observations that
could validate the porous structure of hydrate shells forming in the oceanic
13
Figure 11: FE-SEM image of CH4 -gas
hydrate grown at 264 K and 60 bar.
From Staykova et al. (2003).
Figure 12: Submicron structure of
CH4 -gas hydrate grown at 264 K and
60 bar. From Staykova et al. (2003).
environment with submicron resolution, as assume by (Mori and Mochizuki,
1997). However, the PT-regime where hydrate forms from the ice phase is
accessible in the laboratory. Microstructure observations performed to date and
reveal a number of results that may be relevant for the hydrate shell problem
Kuhs et al. (2000); Staykova et al. (2003); Klapproth et al. (2003); Stern et al.
(2004); Genov et al. (2004), as they agree remarkably with hydrates found in
nature (Kuhs et al., 2004; Stern et al., 2004). In particular it was found by
Kuhs et al. (2000) that CO2 hydrate grown from the ice phase is a structure
with porosity of 10-20% and pores with diamters of 20 to 100 nm. Methane
hydrate had almost an order of magnitude wider pores. Using these numbers
in the model from (Mori and Mochizuki, 1997), one would obtain a hydrate
film thickness of ≈ 1 mm for a typical rising or flow velocity of ≈ 0.1 m/s,
if interfacial fluxes are based on the hydrodynamical scalings used by Fer and
Haugan (2003). Notably, this thickness is two orders of magnitude larger than
one obtains on the basis of a low porosity of 0.1%, the value assumed in studies
by Mori and Mochizuki (1997) and Fer and Haugan (2003).
A shell thickness of 1 mm is inconsistent with observations. It would imply
that droplets in the field study by Brewer et al. (2002) should not have been rising as they did. To yield a value of the order of 10 µm, as estimated by Sugaya
and Mori (1996), one would have to assume that only a small fraction of the
pores is interconnected. Indeed, observations made by Kuhs et al. (2004) seem
to indicate this constraint. These authors report that the pores are predominantly closed and thus only accessible at the surface. Such observations suggest
another idea. It is that not the pore structure and the mass transfer control
the film thickness, yet that the film thickness is controlled by growth kinetics
and diffusive transport at its surface. The connectivity of the pores might then
adjust to these conditions instead of primarily constraining them. Dynamical
length scales of the order of some ten micrometers have been reported in two
studies: Stern et al. (2005) reported that during re-texturing many hydrate
grains developed into hollow shells, typically 5 to 20 microns thick, while Stern
et al. (2004) mentioned a hydrate rind of 5 to 30 microns as an initial scale for
hydrates that formed from melting ice.
14
Recently, rejecting the perforated-plate model, it has been proposed that
the mass transfer from the hydrate shell is balanced by a diffusive flux of CO2
through the hydrate layer, driven by a hopping mechanism (Radhakrishnan
et al., 2003). This approach, for which the authors adopted the shell diffusivity
of CO2 predicted in molecular dynamics simulations by (Demurov et al., 2002),
appears to give the correct order of magnitude of the shell thickness for some
experiments. However, the data available so far are very sparse and cannot be
viewed as a validation. The length scale of several tens of micrometer that the
model predicts also appear in a number of other observations on porous hydrates
mentioned above. This points to the importance of modelling and imaging of
microstructure formation of hydrates, to date an open field of research. Recently
it has been reported that a CO2 -hydrate lake may exist at depths where liquid
CO2 is lighter than seawater, if it is covered by a pavement layer (Nealson,
2006). In this case the hydrate layer was rather thick, of the order of 10 cm, yet
contained bacteria and other substances from the sediment. Interpretation of
this regime, and other natural samples, requires a proper chemical and structural
analysis on the submicrometer scale.
Besides the mentioned observations of microstructure of hydrates growing
from the ice phase (and thus accessible to a number of observational methods),
first basic thermodynamic phase field simulations of hydrate growth have been
performed (Svandal et al., 2006), indicating future possibilities to understand
different hydrate morphologies. Also the tensile strength of hydrate films has
been measured for different pressure and temperature regimes (Yamane et al.,
2000). Indirect derivations of and effective tensile strength of the hydrate layer
in a deep sea experiment have been proposed based on hydrodynamic stability
theory (Hove and Haugan, 2005). The strength was found to be of the order of
the liquid CO2 and hydrate-water interfacial energies. A strong abnormal behaviour of the strength close to the dissociation temperature has been reported
by (Yamane et al., 2000) and also Tewes and Boury (2004) observed a dynamical
response of liquid CO2 -H2 O interface energy during increasing pressure.
In conclusion, there is much progress going on to elucidate the physical processes related to hydrate formation in the environment. However, a proper description in applied sciences, to produce a better understanding of the diffusive,
kinetic processes and possibly hydrodynamical processes during the formation
of porous gas hydrates, still requires a lot of fundamental work. This work must
be highly multidisciplinary, linking microscopic and macroscopic observations
and models.
3.2
Hydrates in porous media
The understanding of the hydrate stability regime, implied by the hydrothermal
and geothermal gradients near the sea floor (Figure 10) has given rise to a
discussion of its role in terms of longterm climate changes, driven by catastrophic
release of the strong greenhouse has methane (Kvenvolden, 1993; Wefer et al.,
1994; Dickens et al., 1995; Nisbet, 2002; Zhang, 2003). The mechanisms are clear
from Figure 10: The instability of hydrates in the sediments can be reached by
both warming of the ocean or, a mechanism that will be dominant in shallow
15
Figure 13: A conventional model for the sealing of pores by hydrates in the
stability zone close to the seafloor. From Clennell et al. (1999).
seas, a decrease in sea level (and bottom pressure).
A conventional view of the role of hydrates in marine sediments is illustrated
in 13. The hydrates may be thought to fill the pores of sediments, making them
impermeable to liquid and gas fluxes from the lower layers (Kvenvolden, 1993;
Clennell et al., 1999; Henry et al., 1999; Østergård et al., 2002). Hydrate stability
may thus also be viewed as a relevant condition in connection with the safety
of CO2 storage.
The idea brings about another question. If CO2 approaches the seafloor
from below, will it be transformed into hydrate, thereby stopping its leakage by
self-sealing? Two mechanisms relevant to answer this question will be discussed
as follows.
3.2.1
Gibbs-Thomson effect
Freezing of a non wetting liquid in small capillaries is controlled by the GibbsThomson relation. It gives the freezing point depression ∆T in dependence on
the capillary radius R as
2Γ
∆TΓ =
,
(3)
R
where Γ = Tm γ/Lv is the Gibbs-Thomson parameter, γ being the solid-liquid
surface free energy, Lv the volumetric latent heat of fusion, Tm the melting
point in Kelvin. For pure water/ice Γ ≈ 2.7 × 10−8 m K which implies a
freezing point depression of 1.08 K for a pore of 100 nm diameter. Relation 3
has been classically used to determine the interfacial free energy of the ice-water
16
system, e.g., Hillig (1998). Uchida et al. (2002) derived 29 mJ/m2 for ice and
determined values of 17 mJ/m2 and 13 mJ/m2 for CH4 and CO2 , respectively.
Anderson et al. (2003a) proposed that during melting experiments one should
use equation 3 with the factor of 2 missing on the right hand side. The latter
authors then obtained very similar values of 32 ± 2 mJ/m2 , 32 ± 3 mJ/m2 and
30 ± 3 mJ/m2 for ice, CH4 and CO2 , respectively. They further argued that the
data for ice reported by Uchida et al. (2002) might have been in error due to
the presence of a broad pore size distribution. A difference between γ obtained
during melting and freezing in pores has been reported earlier (Ishikiriyama
et al., 1995). Considering the available data it seems most plausible to assume
a smaller Γ ≈ 1.8 × 10−8 m K in equation 3, when melting is considered. In
general, it can be concluded that Γ is very similar for ice and gas hydrates.
An example of interpretation of field data in terms of equation 3 is shown
in Figure 14 It shows cumulative pore size distributions derived by mercury
injection into samples from Blake Ridge, taken near the basis of the hydrate
stability zone. The ’percolation threshold’ given by the inflection point may be
interpreted as the lower limit of pore throat radii that determines the permeability. This approach is based on the critical path analysis (CPA) of heterogeneous media, which assumes that flow is primarily taking place through the
larger channels (Thompson et al., 1987). The ’percolation threshold’ marks the
fraction of the overall porosity where the pore space becomes interconnected,
notably taking a value of φc ≈ 10 % (in terms of absolute porosity) for these
samples. The interpretation of Figure 14 on the basis of the above mentioned Γ
is that both methane and CO2 , permeating through the base into the hydrate
stability zone, would not experience a stability temperature depression of more
then 0.6 K. Such a limit has been pointed out by Henry et al. (1999) by using
ice-water interfacial energy values.
Modelling and interpretation of hydrate stability in connection with natural pore size distributions, considering effects of dissolved salts, have been performed (Østergård et al., 2002; Anderson et al., 2003b) with a recent application
to marine sediments (Sun and Duan, 2007). A laboratory study of the morphology of hydrate growth in a porous medium in dependence on the supercooling
between 3.4 and 14 K, showing dendritic growth at lowest supercoolings, was
performed by Katsuki et al. (2007).
3.2.2
Metastability: Undercooling and Superheating
A second aspect of hydrate stability in porous media is related to nucleation.
For the ice-water system it is well known that freezing above the homogeneous
nucleation temperature of ≈ −40 ℃requires the presence of nuclei, e.g., Hobbs
(1974); Pruppacher and Klett (1997). High supercooling of water can be realised
in emulsions, as the separation into many small volumes limits the probability
of activation of nuclei. A similar principal morphology is given in porous media
and has been investigated by Zatsepina and Buffett (2001) for CO2 -hydrate.
These authors interpret resistance measurements during cooling and warming
of water solutions with dissolved CO2 in a porous medium of grain size 0.4-0.6
mm. Nucleation of both vapour and hydrate during crossing of the three-phase17
Figure 14: Cumulative pore size distribution derived by mercury injection. The
’percolation’ threshold given by the inflection point may be interpreted as the
lower limit of pore throat radii that determines the permeability. From Henry
et al. (1999).
stability temperature (Figure 16) was associated with plots as shown in Figure
15. It was found that superheating and supercooling by 3-4 K was possible,
before the phase-transition took place.
Henry et al. (1999) pointed out that an observed shift in the hydrate stability
base temperature by 1-2 K cannot be explained by pore size effects via the GibbsThompson relation. In a statistical nucleation model, parametrised on the basis
of their experiments, Zatsepina and Buffett (2003) predicted that nucleation of
hydrate in pores of 100 nm radius would, even for supercooling of 5 K, take
109 years. These observations indicate the possible delay of nucleation when
gas-bearing water rises into the hydrate stability zone.
18
Figure 15: Resistance measurements
during cooling and warming of CO2
in a porous medium of grain size 0.40.6 mm, P= 2.6 MPa and T3 (P ) = 6.1
℃, indicating superheating and supercooling prior to nucleation. From Zatsepina and Buffett (2001).
4
Figure 16: Phase diagram of CO2 -H2 0
mixture at constant pressure 3 MPa,
corresponding to the experiment in
figure 15. From Zatsepina and Buffett (2001)
Droplet dynamics and thermodynamics
When a droplet or gas bubble of CO2 moves through undersaturated seawater it
will dissolve. At the same time a rising (sinking) gas bubble expands (contracts),
an effect that is opposite to the volume change by dissolution. The velocity V
of perfectly spherical noninteracting droplets due to buoyancy may be written
in the form
V =
8 grρ0
3 Cd
1/2
(4)
where ρ0 = (1 − ρCO2 /ρsw ) is the buoyancy of CO2 in seawater, r the radius
of a droplet and Cd the drag coefficient. This equation is obtained by equating
buoyancy force (4/3πr3 (ρsw − ρCO2 )g) with a quadratic drag force due to the
effective cross-sectional area, (πr2 Cd V 2 ρsw /2). It is valid for spheres, and in
a more general form one must use the cross-sectional area normal to the flow
in connection with the volume of the body. Equation 4 is in principle the
definition of an effective drag coefficient Cd (Davies and Taylor, 1950; Levich,
1962; Batchelor, 1967), which either has to be found from theory or observation.
The velocity V , in the steady state when forces balance, is often termed the
terminal velocity.
19
4.1
Low Reynolds number flow
At low Reynolds numbers, for Re = 2RV /ν < 1 (and thus for very tiny droplets),
the classical Stokes-flow relation for a solid sphere (Levich, 1962; Batchelor,
1967) gives
2 gr2 ρ0
24
V =C
,
Cd =
(5)
9 ν
Re
and a dependence of V ∼ r2 . In the derivation of the Stokes equation inertia
forces are neglected and buoyancy is balanced by friction. The shape factor C
is 1 for a sphere and decreases with specific surface area, taking for example
the value of 0.921 for a cube (Clift et al., 1978; Happel and Brenner, 1986).
When the droplet or bubble has no infinite viscosity, a more realistic equation
is the Rybczynski-Hadamard formula (Levich, 1962; Batchelor, 1967), which for
a sphere (setting C = 1) may be written as
V =
3 + 3kν 2 gr2 ρ0
.
2 + 3kν 9 ν
(6)
Here kν is the ratio of the dynamic viscosity ratio of the droplet to that of the
fluid. For a solid particle (kν = ∞) it passes into the Stokes equation 5. For a
gas (kν << 1) it gives a 3/2 larger velocity V and a lower drag Cd = 16/Re than
the Stokes flow. The difference relates to the difference in the surface mobility of
a solid and a low-viscosity droplet (Levich, 1962; Moore, 1963; Batchelor, 1967).
The observations that gas bubbles sometimes behave according to Stokes law
has been explained in terms of surface-active substances that let the bubble
behave like a solid (Levich, 1962; Batchelor, 1967). As liquid CO2 has a more
than an order of magnitude lower viscosity than seawater, equation 6 should be
applicable, with a prefactor close to 3/2. The low Reynolds number Stokes limit
has been discussed in terms of settling velocities of (solid) sediments (Lerman,
1979).
4.2
High Reynolds numbers
To include inertial effects one may consider sufficiently large gas bubbles that rise
at high enough Re, to let boundary-layer ideas become applicable, but still small
enough Re to keep their spherical shape. Assuming a free boundary condition
one may then obtain the solution Cd = 48/Re (Levich, 1962; Batchelor, 1967;
Clift et al., 1978) which is two times the Stokes drag. Implementing boundary
theory leads to the solution (Moore, 1963; Batchelor, 1967)
Cd =
48
2.21
(1 −
),
Re
Re1/2
(7)
which has later been derived including higher order terms (Kang and Leal,
1988). The terminal velocity is thus half the Stokes velocity from equation 5.
The treatment of the inner boundary layer of the droplet is more difficult and
several approaches have been made to refine equation 7 in terms of viscosity
and density ratios (Clift et al., 1978). A detailed discussion of many aspects of
the problem was given by Harper (1972).
20
Figure 17: Drag coefficient of bubbles in dependence on the Reynolds number.
From Joseph (2003) after Batchelor (1967).
Equation 7 is valid for undeformed spherical gas bubbles at high Re, while
at low Re equation 6 becomes valid, yielding a three times smaller limiting
Cd = 16/Re. The low Re-limit is in agreement with observations (Clift et al.,
1978; Bhaga and Weber, 1981). However, at Re above ≈ 100 it starts to fail.
Up to Re ≈ 300 − 500 the flow is typically characterised by a quasi-constant
drag, with a sharp increase at larger Re (Batchelor, 1967; Clift et al., 1978;
Maxworthy et al., 1996), see Figure 17. This can be understood in terms of the
deformation from a spherical shape and compares to the onset of wake instability
in the case of rigid spheres, taking place at 130 < Re < 400 (Clift et al., 1978).
However, solid spheres experience a much smaller variation in Cd at high Re
and a standard drag curve compiled from many data sources indicates a value
0.4 < Cd 0.5 for a wide range of 500 < Re < 105 . A value of Cd = 0.445 has
been mentioned by Clift et al. (1978) as most reasonable approximation.
Furthermore it is important to note, that experiments have shown that rigid
discs and spheroids have a larger Cd by a factor 2 to 3 when aspect ratios become
large (Clift et al., 1978).
The case of liquid and gas bubbles, which begin to deform at high Reynolds
numbers, is more complex. The dependence of the the drag coefficient on shape
and flow conditions is, in addition to Re, often described in terms of two other
non-dimensional numbers that include the surface tension σ: The Morton number M o = gν 4 ρ3sw /σ 3 and the Eötvös number Eo = g(2r)2 (∆ρ)/σ (Moore, 1959;
Harper, 1972; Clift et al., 1978). Small deformations are, for example, given in
a liquid of small M o when bubbles rise at moderate Re, because surface tension
21
keeps the circular shape. For low Morton number systems (< 10−8 ) like water,
the shape of bubbles changes with increasing Re from spherical to increasingly
oblate, then fluctuates, until they take the form of an umbrella, steady at its
front even if the rear fluctuates (Moore, 1959; Clift et al., 1978). Much progress
in the understanding of these umbrellas or spherical caps has been made by
Davies and Taylor (1950) in their famous study. On the one hand these authors
showed that the velocity of spherical caps can be described by a stagnation-point
condition
V =
4 0
gρ r0
9
1/2
,
(8)
where r0 is the radius of curvature at the tip of the spherical cap. On the other
hand they also showed that the drag coefficient based on Cd = gV/(1/2V 2 F ),
2 transverse to the flow, and bubble
with maximum cross-sectional area F = πr⊥
volume V, was close to unity for the spherical cap bubbles. Davies and Taylor
(1950) further pointed out their experiments strongly supported the geometrical
similarity of spherical cap bubbles and a constant Cd . Their experiments have
been later confirmed and interpreted by many authors in terms of the relation
Cd =
8 gρ0 re
8
≈
3 V2
3
(9)
for the drag coefficient of spherical cap bubbles at large Re, which is now based
on the equivalent radius re = (3V/4π)1/3 (Moore, 1959; Harper, 1972; Clift
et al., 1978; Bhaga and Weber, 1981; Joseph, 2003). The limiting velocity is
thus
V ≈ gρ0 re
1/2
.
(10)
once the regime of spherical cap bubbles has been reached.
The described regimes are summarised in Figure 18. It is seen that, in dependence on bubble rigidity and Re, a wide number of drag coefficients is possible.
For gas bubbles in water Figure 19 illustrates the intermediate regime between
spherical and spherical-cap drag, where the rise velocity is relatively constant
over a wide range of Reynolds numbers. Algorithms for particular regimes of
fluid properties and shapes may be found in the standard work by Clift et al.
(1978). A number of more recent algorithms was discussed by Kalbaliyev and
Ceylan (2007), who proposed a preferred set of equations for the prediction of
Cd for the solid sphere and gas bubble regimes.
The behaviour in Figures 18 and 19 may be approximated by parameterising
the shape of particles in terms of Re, M o and Eo. A number of such approaches
have been made, but in particular at high Re, when secondary motion exist, the
problem is not well understood (Clift et al., 1978) and the complexity is indicated
by the role of contamination in Figure 18. In this sense any suggested algorithm
that is not based on coupled flow and shape simulations, like the recent ones
by Kalbaliyev and Ceylan (2007) or Bozzano and Dente (2001), must be viewed
with caution.
22
Figure 18: Comparison of standard drag curve (solid spheres), Cd for bubbles
in pure solutions and when influence by surface effects due to contamination.
From Clift et al. (1978).
4.3
Turbulent mass diffusion and dissolution
The complexity and lack of universal solutions for the flow field indicates, that
the problem of dissolution from a droplet or crystal interface is not an easy task.
The success of simplifying drag-laws is limited. A proper analysis involves more
detailed boundary layer and interface flux modelling. In general, the prediction
of the solute flux Fs from a dissolving interface under conditions of convection
is described by
dC
Fs = Ds (
)int = V∗ (Csat − C∞ ).
(11)
dz
For specific geometries it is convenient to write the solute flux velocity as V∗ =
ShDs /δ, in terms of a Sherwood number Sh and a thickness scale δ. The
general problem is the prediction of the interfacial solute flux velocity scale V∗
by boundary layer modelling. Boundary layer theories for horizontal surfaces
and confined geometries, e.g. Gebhart et al. (1988); Schlichting (2004), need to
be properly modified for flow around objects. However, for regular objects like
the sphere and not too large Reynolds numbers, the problem has been treated
by many authors. Again, Levich (1962) has provided some theoretical groundwork and Clift et al. (1978) have discussed empirical relations and hydrodynamic
scalings for different shapes and Reynolds number regimes, rigid objects and gas
bubbles.
A second relevant mechanism in the problem is compositional free convection due to dissolving solute of a droplet. Convection introduces different length
(and transfer velocity) scales than the flow due to the terminal velocity. Dissolu23
Figure 19: Terminal rise velocity for gas bubbles in water. From Clift et al.
(1978).
tion driven by convection alone is reasonably understood for horizontal surfaces
(Thomas and Armistead, 1968; Selman and Tobias, 1978; Kerr, 1994) in terms
of classical theory of hydrodynamic instability (Chandrasekhar, 1961; Turner,
1973). Theory and experiment also exists for other geometries Gershuni and
Zhukovitskii (1976); Selman and Tobias (1978); Gebhart et al. (1988). Also
for free convection from the surface of spheres and arbitrary-shaped particles
theories and scaling laws have been developed (Clift et al., 1978).
The problem becomes complex when both free convection (from the crystal
or bubble interface) and forced convection (due to the crystal’s terminal velocity) are present and interact, influencing both the terminal velocity and the
dissolution. However, in the limiting cases of large and small Re, excluding situations where Re ≈ 1, predictions and observations agree reasonably. For low Re,
assuming Stokes flow, Kerr (1995) has found, in experiments with salt crystals,
reasonable agreement with predictive equations from Clift et al. (1978). For Re
up to 350 Zhang and Xu (2003) also found reasonable agreement with theoretical scalings, by combining equations for drag, terminal velocity and solute flux
given by Clift et al. (1978) for solid particles. The approach is expected to be
valid to large Reynolds numbers Re < 105 and appears in reasonable agreement
with the survival time of methane hydrate (Zhang and Xu, 2003) and observed
diameters and dissolution rates of CO2 droplets surrounded by a hydrate shell
(Zhang, 2005).
Due to the above discussion the application of dissolving bubbles, as suggested by Zhang and Xu (2003) and Zhang (2005), should be further critically
24
evaluated in terms of the following aspects: (i) while Cd is affected by shape
changes, the mass transfer and Sh appears to be more independent of the latter
(Clift et al., 1978) and it is critical to incorporate this into the analysis; (ii)
the approach assumes a uniform equivalent boundary layer over the surface of
a droplet, while the real concentration field will be much more complex with a
compressed boundary layer in front of it; (iii) the dissolution from the surface
of a droplet covered by a hydrate shell will depends on the porous structure and
renewal of the shell and thus on processes that are not well understood yet.
4.4
Oceanic experiments and simulations
Some authors have applied equation 4 with a constant drag coefficient and
a constant dissolution velocity to illustrate the levels to which gas or liquid
droplets may rise a until they are dissolved (Holder et al., 1995; Clark et al.,
2000).
The first dataset to analyse this problem was provided by Brewer et al.
(2002) and also discussed on the basis of equation 4.1 Droplets were released
in the deep ocean and followed by an ROV to provide data on rise velocity and
droplet radius. They found that the rise of a droplet could be closely described
by equation 4 when using a constant Cd = 1 with a constant dissolution rate.
The droplets were covered by a hydrate shell but irregular in form. From the
above discussion it is recalled that Cd ≈ 0.45 would be expected for a spherical
droplet, but a factor of 2 to 3 larger Cd is realistic for anisotropic shapes (Clift
et al., 1978). In this sense the value Cd = const is consistent with theory.
However, a proper analysis would have to account for the detailed geometry
of the followed feature, which actually was a droplet pair. As mentioned in
the previous paragraph, Zhang (2005) provided a consistent simulation of the
droplet rise rate and dissolution. The dissolution rate that he obtained by
application of a simple forced convection algorithm (Clift et al., 1978) almost
exactly matched the observations. However, the formula for the drag coefficient
that he used was one for a spherical rigid droplet, which gives 0.47 < Cd < 0.62
for the range 1000 > Re > 400 typical for the experiment. His algorithm
therefore should have slightly overestimated the rise velocity (a depth-time curve
was not shown). In another study Gangstø et al. (2005) made predictions of
the rise on the basis of the scalings from Bozzano and Dente (2001). The
latter model however, as discussed above, should in principle only apply to gas
bubbles, for which it parametrises the drag change with deformation at high
Re. The assumed functional dependence of Cd on Re is therefore not physically
justified, which may explain the poorer performance of the model by Gangstø
et al. (2005) at low Re.
As shown by Gangstø et al. (2005), Brewer et al. (2002) only were successful
in predicting the depth-time curve of droplets by assuming a constant drag and
dissolution rate, while the constant drag predictions of the rise velocity alone
did not give the observed behaviour. The better performance of the predictions
by Gangstø et al. (2005) may indicate that despite a hydrate shell the droplets
1
The authors use the incorrect expression ρ0 = (ρsw /ρCO2 − 1) instead of ρ0 = (1 −
ρCO2 /ρsw ) and quote equation 4 as the ’Stokes velocity’, which is neither correct.
25
behave like liquid or gas. This appears unlikely because, even if the hydrate shell
is flexible, it may not promote shear to the inner of the droplets. It seems more
likely that an inaccurate drag law is shaded by an inaccurate dissolution model,
or that shape changes were accidentally taking place in a way that approaches
this law. The many open questions indicate the need for more observations,
proper evaluation of the geometry of droplets and, last not least, an evaluation
of the role of the movement of the ROV, that may have stabilised the rise
velocities. Such artificial effects, would be largest when the droplets are near
the boundary of the ROV which, as pointed out by (Gangstø et al., 2005), might
indeed have happened. The question was also raised by Mori and Murakami
(2007).
The drag parametrisation used by Zhang (2005) has been criticised by Alendal et al. (2006) as giving 50 % too high rising velocities. It is, however,
more realistic than a liquid-or gas bubble approach from Gangstø et al. (2005),
and would, with an increase of the drag by a factor of 2-3 due to ellipticity of
the droplets, perform reasonably well. To what degree the dissolution approach
(Zhang and Xu, 2003; Zhang, 2005) needs to be modified due to assymetric
boundary layer and droplet shape needs to be investigated. The model is a
physically consistent treatment of surface renewal diffusion coupling that reproduces the observations well.
Another question that has been addressed is the reduction of dissolution
rates by the presence of the hydrate shell (Aya et al., 1997; Ogasawara et al.,
2001; Zhang, 2005; Gangstø et al., 2005). The explanation in terms of a different
solubility in the presence of hydrate gives a quantitative correct reduction by a
factor of ≈ 2 (Zhang, 2005) for the dataset from Brewer et al. (2002). However,
to what degree the dissolution may be affected by shape and surface mobility
changes, needs also to be investigated. It seems plausible that a larger effective
drag on a perturbed droplet will also enhance its dissolution. More recent data
on rise velocities (Chen et al., 2003) indicate that also in the presence of a
hydrate shell the shape of droplet changes to elliptical forms, with increasing
drag and a terminal velocity plateau as seen in Figure 19. The droplets in the
experiment discussed in the latter study were, however, strongly interacting.
More proper evaluations of all effects on the drag are needed.
4.5
Future challenges
The above discussion indicates the future need of small-scale modelling of the
flow, drag and dissolution of particles of different shapes. Some aspects are
important for correct interpretation of existing and future studies. The application of a liquid droplet algorithm to predict Cd (Gangstø et al., 2005), as it was
also applied in a larger-scale study (Alendal and Drange, 2001), is not justified
for rigid droplets with a hydrate shell. It gives a too large drag. On the other
hand, also a non-spherical hydrate-shelled drop will experience some increase
in the drag, and this increase and the typical shell geometry need to be established. If present, liquid CO2 and seawater through which the latter rise have a
viscosity ratio of the order of 10, and may thus reflect a case between gas and
rigid bubbles. In any case, dissolution needs then to be modelled on the droplet
26
scale with high-resolution boundary layers. For non-dissolving particles a better understanding can be provided by numerical models. Both front-tracking
approaches (Hua and Lou, 2007) and lattice Boltzmann simulations (Inamuro
et al., 2004; Frank et al., 2006) have recently be applied to the problem, with
realistic results on shapes.
There is a further need for the study of coagulation and droplet interaction.
Boundary effects are known to become relevant when d/D, the ratio of droplet
diameter to free flow path cross-sectional diameter, becomes larger than ≈ 0.1
(Clift et al., 1978). With larger d/D the effective drag increases while the rise
velocity decreases. Similar results have been obtained by analytic approaches
to bubble interactions (Zhang and Fan, 2003) and by numerical modelling of
a swarm of bubbles (Krishna et al., 1999). Coalescence of bubbles and the
accompanying shape changes may also be investigated by lattice Boltzmann
simulations (Inamuro et al., 2004).
Finally, there should be a large potential to learn from other research fields
that long have been treating analogous problems. Convective heat and mass
transfer have been treated in the field of crystal growth (Tiller, 1991; Xu, 2004)
and solutions found there may also serve as guides to further progress. In cloud
physics many of the problems like droplet deformation, terminal velocity, droplet
coagulation and diffusional growth of particles have been treated theoretically
and experimentally (Pruppacher and Klett, 1997).
27
Figure 20: Oceanic boundary layers with approximate scaling laws.
Boudreau and Jørgensen (2001).
5
From
Oceanic bottom boundary layer
One may think of situations where the flux through the bottom is not due
to rising CO2 liquid bubbles, either because it is of diffusive nature from a
hydrated seafloor, or if the liquid CO2 is denser than seawater. In this case it
will be important to model the bottom boundary layer of the ocean. Solute flux
depend on the coupling of the interface flux with the turbulence in the boundary
layer.
The overall thickness of the oceanic boundary is conventionally defined as
the depth where the current has approached the value in the interior of the
ocean. Changes within the boundary layer are due to friction in terms of the
flow field, but also due to stratification. The frictional boundary layer near the
ocean bottom may be characterised by different physical regimes indicated in
Figure 20.
5.1
Classical scaling laws
A boundary layer drag or friction velocity U∗ is conventionally defined as
U∗2 =
τ
2
= Cw U∞
ρ
28
(12)
where Cw is the quadratic drag coefficient and U∞ the velocity far away from
the interface. Under most oceanic conditions one finds 0.001 < Cw < 0.0025
and U∞ /U∗ ≈ 20 − 30 (Armi and Millard, 1976; Weatherly et al., 1980; Thorpe,
2005). The following thickness estimates assume U∞ /U∗ = 25 and illustrate
the case U∞ = 5 cm/s (U∗ = 0.2 cm/s), which is a realistic figure for the deep
ocean.
5.1.1
Ekman Layer
The largest boundary layer scale is conventionally termed the Ekman Layer and
often estimated as
κU∗
δE =
,
(13)
f
where κ ≈ 0.4 is van-Karman’s constant and f the Coriolis parameter. With
U∞ = 5 cm/s and f ≈ 10−4 s−1 the Ekman Layer is δE ≈ 8 m.
5.1.2
Outer boundary of log-layer
The outer boundary layer or log-layer has its name due to the logarithmic velocity profile given by the law of the wall. It is also termed the constant stress
layer. One has commonly for rough flow
U (z) =
U∗ z
ln ,
κ z0
z0 ≈ zsedi /30
(14)
where z is the distance from the bottom and zsedi is the roughness of the sediment.
For smooth flow
U (z) =
U∗ z
ln ,
κ z0
z0 ≈ 0.1
ν
≈ δν /100
U∗
(15)
the effective roughness z0 is determined by the viscosity ν (≈ 0.016 cm2 /s for
deep ocean conditions), or the thickness δν of the laminar viscous sublayer, see
below. Flow is termed smooth if zsedi > 3ν/U∗ . For U∞ /U∗ = 25 the log layer
has a scale δlog ≈ e10 × z0 or 1.8 meter for smooth flow and U∗ = 0.2 cm/s.
5.1.3
Viscous sublayer
In the viscous sublayer flow is laminar and unaffected by eddies or bottom
roughness and the velocity shear is constant. It’s scaling
δν ≈ 10
ν
U∗
gives δν ≈ 0.8 cm for smooth flow and U∗ = 0.2 cm/s.
29
(16)
5.1.4
Diffusive sublayer
The fluxes from the bottom are are strongly dependent on the diffusive sublayer
commonly scaled as
δdif f ≈ δν Sc−1/3 ,
Sc = ν/Ds (Schmidt
number),
(17)
where for heat the Schmidt number Sc should be replaced by the Prandtl number. For salt and deep ocean values ν/Ds ≈ 2.0 × 103 and δdif f ≈ 0.6 mm in
case of U∗ = 0.2 cm/s.
The solute flux in the diffusive sublayer is frequently parametrised in the
form
(Csat − C∞ )
Fs ≈ Ds
≈ 0.1U∗ Sc−2/3 (Csat − C∞ ).
(18)
δdif f
Equation 18 contains the concentration Csat at the interface and C∞ far away
from it where U∗ is defined.
5.2
Improved models
The boundary layer scales mentioned so far are simplified models that do not
consider the complexity of turbulent mixing near the seafloor. The simplest
example to illustrate the complexity is to imagine a bottom current driven by
its own density. The structure in its boundary layer, and its interaction with
the seabed, can be expected to be very different if compared to a boundary
layer where a homogeneous or barotropic ocean velocity drops to zero when
approaching the sea floor. For proper computation of sea floor solute fluxes it
is apparently critical to compute C∞ and U∗ by a turbulence model. Such a
model needs to consider several mechanisms that are important near the seafloor,
like (i) flow-sediment interaction, (ii) tides, (iii) internal waves, (iv) biological
activity.
5.2.1
Ekman layer and stratification
Observations show that equation 13 frequently tends to underestimate the bottom boundary layer thickness (Armi and Millard, 1976; Armi, 1978; Weatherly
and Martin, 1978). In particular a large variability was found in areas of variable topography (Figure 21). In some cases also a too shallow boundary layer
was predicted. Based on dimensional grounds, Weatherly and Martin (1978)
proposed the the improved formula
U∗
δE = A
f
1
,
1 + N 2 /f 2
(19)
where N 2 = ρg dρ
dz represents the stratification and A ≈ 1.3. For N f the
equation passes into δE = AU∗ /(f N )1/2 derived by Pollard et al. (1973) for
deepening wind-mixed layers. Equation 19 accounts both for stratification effects and the generally larger observed δE , but it still underestimated the latter
in several cases (Weatherly et al., 1980). The latter authors thus agreed with
30
Figure 21: Variability in deep sea bottom boundary layers. From Armi and
Millard (1976).
Armi (1977) who had pointed out the importance of internal waves as a mixing
agent in the deep ocean.
Mixed layer models with more appropriate vertical resolution and turbulence
closure schemes have been used in later studies, e.g. (Martin, 1985; Galperin
et al., 1988; Kantha and Clayson, 1994; Thorpe, 2005). A semi-empirical treatment of the mixing agency of internal waves was presented by Kantha and
Clayson (1994). Diffusion of CO2 from a lake has been simulated both by the
standard bulk mixed layer equations given above (Fer and Haugan, 2003) and
a more advanced turbulence closure scheme including internal wave parametrisation (Haugan and Alendal, 2005). Some results will be mentioned below.
5.3
Ocean-seabed interactions
To find the solute fluxes between the ocean and the seabed the vertical structure
in the log-layer close to the sea bottom is particularly important. As an example,
Figure 22 from Sanford and Lien (1999) indicates the presence of two log-loglayers close to the bottom, of which one has been attributed by the authors to
the presence of form drag, with ripples of the order of 0.3 m height at the sea
bottom. An alternative interpretation of this observation has been suggested
by Perlin et al. (2005, 2007). The latter authors proposed a simplified approach
to calculate the logarithmic velocity profile in the presence of stratification,
provided that the mixed layer depth is δE is known.
Any model with emphasis on the solute fluxes near the bottom must focus on
31
Figure 22: Appearance of two log-layers near the seabed. From Sanford and
Lien (1999).
eddy viscosity parameterisations for the particular type of boundary layer. Possible deviations from a log-layer may be related to roughness, interaction with
the sediment, the influence of internal waves. Alternative analytical solutions
of eddy viscosity schemes have been summarised by Sideman and Pinczewski
(1975). Considering the role of suspended sediments, Dade et al. (2001) have
compared some model results with the equation 19 from Weatherly and Martin
(1978). In Figure 23 the Burger number S = gρ0 W C/(U∗2 f ), where C is sediment concentration and W settling velocity, gives a similar dependence in the
reduction of δE .
Furthermore may stratification and suspension of sediment in the boundary
layer interact in a complex manner with the flow, as shown in Figure 24. It is
notable that the settling velocity in this case is computed in a similar way as in
section 4 for the rising bubbles.
5.3.1
Flux closure and diffusive sublayer δdif f
It is expected that the solute flux from the seafloor not only depends on the
parametrisation of the eddy transport close to the interface, yet also on the suspension modes from figure 24. An example of observations of such a diffusive
boundary layer is shown in Figure 25. Figure 26 compares two fine-scale boundary layer models in terms of eddy diffusivities. It is seen that, while the models
agree on the scale of the viscous boundary layer δν , they differ on the scale of
the diffusive boundary layer. The difference corresponds almost to a factor two
in the parametrised interface fluxes, emphasising the problem of boundary layer
type.
32
Figure 23: Decrease in boundary layer depth by suspended sediments. From
Dade et al. (2001).
Also from an observational viewpoint there is room for model refinement.
The values that have been obtained for the mass transfer velocity, V∗ = c∗ U∗ Sc−2/3 ,
often differ from c∗ = 0.10 approximated in equation 18. Comparing different
sources for Sc = 1000 the prefactor in equation 18 varies between c∗ = 0.04 and
c∗ = 0.08, with most values between 0.06 and 0.07 (Boudreau, 2001). It is thus
smaller than the frequently assumed value 0.1 Boudreau (2001). Many different
schemes and correlations leading to slightly modified parametric dependencies
were reviewed by Sideman and Pinczewski (1975). Hence, also the diffusive flux
parametrisation at the ocean bottom should be properly validated. The likelihood that a reasonably accurate parameterisation can be found is indicated
in Figure 27, which shows the dependence of boundary layer thickness δdif f on
ocean depth. The increase with depth is expected, as one expects slower flow
in the deep ocean. The variability is moderate.
In this context also a more detailed look on the fine-scale bottom topography
and interaction with the sediment is worth a look. A fine-scale simulation (Figure 28) indicates that bottom irregularities may trigger a convective circulation
from as deep as 10 cm within the sediments. The influence of this circulation
on matter distribution (figure 28) and diffusive fluxes has been discussed in connection with bio-geochemical processes important on the small scale (Huettel
et al., 1998). While obstacles as shown in Figure 28 do not destroy the diffusive
sublayer, they have a pronounced effect on the mass transfer. The change of
the latter may be expressed in the form
V∗,rough
U∗ zb b c
) Sc .
= a(
V∗,smooth
ν
33
(20)
Figure 24: Regimes of stratification in dependence on particle size, friction
velocity, settling velocity and z/L, where L is the Monin-Obukhov length. From
Hill and McCave (2001).
for which a number of studies were compared by Dade et al. (2001). These
parameterisations indicate that for U∗ = 0.2 cm/s the mass transfer from a
bottom with roughness 0.1 to 1 cm is likely to be enhanced by a factor of 2 to
4, when compared to a flat surface. Turbulence in flows over rough walls have
recently reviewed by Jimenez (2004).
In connection with the convective circulation as deep as 10 cm in sediments
(Figure 28) it is of interest that Boudreau (1998) has reported a mean mixed
depth of 9.8 ± 4.5 cm for sediments. The direct impact of the slow flow on
overall diffusive fluxes is likely to be small (if not bio-geo-chemical reactions are
considered). However, such an information would be important in understanding the fluxes from the seafloor by consistent modeling of resuspension rates,
sublayer stratification and small scale turbulence near the bottom.
5.3.2
Internal wave-seabed interaction
While the average mean mixed depth of sediments appears to be ≈ 10 cm
(Boudreau, 1998), during storm events fluid flow penetration depths of 2-4 meters have been observed (Moore and Wilson, 2005). Comparable magnitudes of
seabed instability have been predicted by models of wave-seabed interaction in
34
Figure 25: High resolution diffusive
boundary layer observations above
the seafloor and within the sediments.
From Jørgensen (2001).
Figure 26: Difference in the diffusive
sublayer thickness due to two different
model approaches. From Boudreau
(2001).
shallow waters (Jeng, 1997). Recently also a theoretical framework of interaction between internal waves and the seabed has been published Chen and Hsu
(2005). The latter authors predict that soil displacement on the seabed may
reach 40 and 15 m in the horizontal and vertical directions. These scales are
similar to the dimension of pockmarks on the seafloor (Judd and Hovland, 2007),
pointing to the relevance of internal waves in terms of sediment resuspension.
5.3.3
Benthic storms
Benthic storms with velocities of ≈ 20 cm/s have been reported for different
ocean basins, e.g. Polzin et al. (1996); Ledwell et al. (2000); Woodgate and
Fahrbach (1999). One might suppose that, in an integral sense, they occur too
seldom to influence the slow dissolution from the ocean bottom. However, their
role in breaking up a stable stratification could be highly relevant, a process
worth of investigation. As extreme events they might shape the seabed and
also lead to an effective spreading of leakage over larger distances. The role of
benthic storms to eventually breakup the hydrate layer that might form on a
CO2 lake in the deep layer was discussed by Fer and Haugan (2003) and Hove
and Haugan (2005).
35
Figure 27: Dependence of diffusive boundary layer thickness on ocean depth.
From Jørgensen (2001).
6
6.1
Mesoscale simulations and observations
Deep sea lake
Fer and Haugan (2003) have performed two-dimensional simulations of dissolution and advection from a 3000 m deep CO2 lake. They performed runs with
and without a hydrate layer and reported a 2.7 time higher dissolution rate
in the absence of hydrate. However, these results do not appear to be linked
to the use of different solubilities in the absence and presence of the hydrate
phase, as discussed by Zhang (2005) on the basis of thermodynamic arguments.
It appears to be linked to a difference in turbulent closure an flux parametrisation for the two different phases. For example, for the low friction velocity run
(U∞ = 5 cm/s), the account of stratification only led to a difference of a factor
1.5. This again points to the need to consider the flux parameterisations near
the interface, and their uncertainty, in more detail.
The results obtained by from Fer and Haugan (2003) for the hydrate layer
run, based on a solute flux given by equation 18 and a near-bottom velocity
of 5 cm/s, are shown in Figure 30. It is seen that the CO2 is confined to a
bottom layer of the order of 10 m and spreads over 15 km in the course of three
days, which is the expected advection scale. Stratification effects, resulting from
the densification of seawater when dissolving CO2 , implemented on the basis of
equation 19, have also been discussed in this simplified model and are actually
included in figure 30. They have the potential to decrease the dissolution rate
by a factor of 3 to 5, confining CO2 to a much thinner boundary layer Fer and
Haugan (2003).
36
Figure 28: Simulation of flow triggering in the sediment by an obstacle.
From Huettel et al. (1998).
Figure 29: Fine-scale distribution of
matter in sediments due to flow. From
Huettel et al. (1998).
Similar simulations have been performed with an improved turbulence closure scheme Haugan and Alendal (2005). The latter authors in particular outlined the effect of internal waves which was substantial in increasing the height
of the bottom boundary layer, in particular at low velocities. Results are illustrated in Figure 31 for runs without stratification (NIW), including stratification
(SIW) and for different velocities 5, 10 and 20 cm/s.
The simulations presented so far Fer and Haugan (2003); Haugan and Alendal (2005) are two-dimensional, yet parametrisations are based on validated
ocean models that take this idealisation into account. The simulations show
the expected influence of stratification on the mixing efficiency of CO2 that dissolves from the ocean bottom and demonstrates that it will be confined to a
O(10) m thick bottom boundary layer. Long term simulations are not available.
However, based on the generally slow eddy diffusion above the Ekman layer
one expects a wide spreading of such a confined signal. For example, a rough
estimate of additional layer growth and spread of CO2 by slow eddy diffusion is
∆H ≈ (Kv L/U∞ )1/2 and yields 32 m for L=500 km and Kv = 10−4 m2 /s.
6.2
Droplet plume modelling
Alendal and Drange (2001) simulated the release of droplets of CO2 from a
source at oceanic mid depth, where CO2 is lighter than seawater. While the
bubbles are rising they dissolve and make the ambient water more dense, which
then starts sinking. The principal behaviour of such a system is to some degree
resembled by Figure 24. There is a critical diameter above which droplets continue rising and below which they are simply advected by the mean flow. In
the simulations by Alendal and Drange (2001) this diameter was 0.5 mm. The
vertical extension of the evolving plume thus increases with increasing initial
droplet size. For droplets of the order of 1 − 2 cm diameter it was found to be
typically 30-50 m, when evaluated some 100 m from the source.
Simulations by Alendal and Drange (2001) assumed (i) liquid, (ii) noninteracting bubbles (iii) without a hydrate skin. In reality (i) underestimates
37
Figure 30: Simulations of height and advection of CO2 (and pH changes) dissolving from a hydrate-covered lake over the course of 3 days. From Fer and
Haugan (2003).
the spreading because the terminal velocity should be based on solid particles
(although, as discussed in section 4, irregular droplet shapes would in turn
increase the drag). The neglect of interaction (ii) also overestimates the rise
velocity. Furthermore, the use of CO2 solubility should be corrected for the
presence of hydrate (Zhang, 2005), which would decrease dissolution and thus
increase the height to which bubbles rise. The results from Alendal and Drange
(2001) also indicate the behaviour of droplet plumes, when released from the
bottom of shallower seas. In such a scenario concentration of CO2 will also be
restricted to a shallow regime close to the source. In shallow seas, however,
mixing by winds is much more relevant and the mixing can be expected to be
more efficient on length-scales of hundreds of kilometers. Then, of course, CO2
returns to the atmosphere.
Recent observations of rise velocities (Chen et al., 2003) indicate that also
in the presence of a hydrate shell droplets are elliptical, with increasing drag
and a terminal velocity plateau as seen in Figure 19. The droplets in the experiment discussed by Chen et al. (2003) were, however, strongly interacting.
To what degree the simulations by Chen et al. (2003) are realistic therefore
depends essentially on the the above mentioned question of droplet geometry
and interaction. The effective parameterisations that they used were similar to
those employed by Alendal and Drange (2001). The results were also comparable, with the difference that the Chen et al. (2003) performed simulations for
a bottom release: The main density increase was confined to a layer of a few
38
Figure 31: Simulations of distribution of CO2 and pH reduction in the bottom
boundary layer 7500m from the source after 3 days. From Haugan and Alendal
(2005).
tenths of meter above the bottom, and the thickness of this layer increased with
initial droplet diameter. Chen et al. (2005) have further compared simulations
for droplet release at mid depth.
Recently, Brewer et al. (2006) reported similar simulations, treating the
rising velocity on the basis of the liquid drop drag coefficient parameterisations
from Chen et al. (2005). Their application is in so far inconsistent, that they
use a drag parametrisation for a liquid droplet, while arguing that the observed
bubbles are assumed perfectly spherical, which they propose due to the rigid
hydrate shell.
Clearly, more accurate evaluations of shape and interaction effects on the
drag are needed. On the one hand, this may be done in sensitivity studies
of numerical simulations (Alendal and Drange, 2001; Chen et al., 2003, 2005).
On the other hand, it is likely that the sensitivity on droplet behaviour may
be obtained by simplified boundary layer models. An useful goal of future
studies could be to establish bounds and parameter regimes like in Figure 24
for interaction of bottom sediment buoyancy with the mean flow.
6.3
Observations
There are no observations available that might validate the modelling of dissolution from the sea bottom on the large scale (as investigated by Fer and
Haugan (2003) and Haugan and Alendal (2005)). The small-scale experiments
performed so far with small containers (Brewer et al., 2004; Hove and Haugan,
2005) not even allow the detection of a signal more than 1 m away from these
samples. Here the progress essentially depends on model studies.
Observations of single droplets tracked by an ROV (Brewer et al., 2002),
39
already discussed above, might also suffer from the influence of the ROV on the
droplet movement. Recently, experiments on the release of cloud of droplets
have been reported. These were monitored by acoustic means and could be
tracked by some 150 m (Brewer et al., 2006). Such observations seem to be
promising to analyse the rising behaviour of interacting droplets. However, the
volumes of CO2 that may be realised under such experiments appear too small
to create the critical increase in the ambient water density that results in the
double-plume formation as discussed by Alendal and Drange (2001) and Chen
et al. (2003).
Indirect validations of models may be performed by studying natural seep
areas of gas bubbles (Judd and Hovland, 2007). For example, an interesting area
is located in the Western Barents Sea where large methane plumes are observed
in a pockmark field (Lammers et al., 1995). The latter is insofar of interest as
horizontal currents and tidal excursions are relatively well known, and due to
seasonal variability in the stability characteristics of the bottom boundary in
this area.
6.4
Longterm perspectives and climate model simulations
In the long term the CO2 from the discussed potential source will spread with
ocean currents. Its return to the atmosphere will depend on the evolution of
CO2 content therein, and the bio-geo-chemical carbon cycle. A global simplified
model study has been performed by Khesgi and Archer (2004). Studies with
climate and global circulation models, that more thoroughly treat the release
and source functions of CO2 , focusing on the longterm changes and distribution
of near-bottom pH, do not exist yet.
40
7
Fluxes through the seabed
The present report closes with a remark on physical processes related to the long
term safety on geological sub-sea-storage, illustrated by Figure 32. Many studies
support that longterm climate changes were influenced by catastrophic release
of the strong greenhouse gas methane (Kvenvolden, 1993; Wefer et al., 1994;
Dickens et al., 1995; Nisbet, 2002; Zhang, 2003). One may therefore speculate
that changes in the ocean bottom temperature, as they are expected under the
global warming scenarios, have the potential to change certain seabed regimes,
in terms of their permeability and trapping behaviour, if methane hydrate is
liquefied by warming. Taking the propagation of an oceanic warming signal into
the seabed as Z ≈ 2(κt)1/2 , and using κ ≈ 10−6 m2 /s for the heat diffusivity,
one may obtain an estimate of 80-200 years for the propagation through the
typical hydrate stability zone of thickness 100-200 m. The response is slow, but
operates on the timescale of climate change (and its mitigation). Moreover may
gas hydrate destabilisation be triggered by other more rapid mechanisms related
to structural changes in the seabed, e.g., Suess et al. (1999).
Another important question is to what degree the seabed stability may be
modified when CO2 dissolves in the water of the formation where it is to be
stored. The following process chain may be imagined. 1.) Dissolution implies
an overall increase in the density. 2.) The density increase leads to convection
in the formation (this has been simulated, see IPCC (2007b)). 3.) Convection
then brings saturated brine to levels of higher temperature, making the solution
over-saturated. It needs to be considered to what degree these diffuso-convective
processes can lead to heterogeneity in the pore structure and might trigger
fracturing.
7.1
Permeability and Percolation
In any case it is important to consider the possible change in seabed permeability that either climate change or storage of a reactive compound as CO2 may
trigger, and be aware of the timescales. An important aspect of catastrophic
permeability changes may be illustrated as follows. Flow through porous media
at low Reynolds number is normally described by the classical seepage velocity
V law after Darcy
K dP
V =
,
(21)
µ dz
with dynamic viscosity µ and pressure gradient dP/dz. The permeability K is
a measure of the characteristic cross-sectional area of pores normal to flow. It
depends on tortuosity, connectivity and porosity φ. Empirically K ∼ φm with a
most frequent range 2 < m < 5 is found. Theoretically m = 2 and 3 for arrays
of cylinders and slots where the connectivity is not changing. A constant m ≈ 3
is also frequently assumed in reservoir modelling.
However, in natural porous media forming by some kind of random compaction the permeability is often controlled by a network of the largest pores.
Such a system may be described by percolation theory (Shante and Kirkpatrick,
1971; Thompson et al., 1987; Stauffer, 1991; Sahimi, 1993; Hunt, 2005b,a) which
41
Figure 32: Trapping of gas and liquids due to sealing in the gas hydrate stability
regime. From Kvenvolden (1993).
yields an effective K 0 via an equation of the form
K0
=∼ (φ − φc )q ,
K
(22)
where φc is the critical porosity at which the connectivity vanishes. Theoretically one finds thatφc ≈ 0.29 and q ≈ 1.9 for the three-dimensional isotropic
continuum. For anisotropic inclusions and systems of pores φc can be considerably smaller. Its value for most three-dimensional lattices where the bonds are
closed by a random process is ≈ 0.15 (Shante and Kirkpatrick, 1971; Sahimi,
1993).
Natural systems often operate near thresholds. In porous media the critical
behaviour near the threshold is often related to a certain porosity 2 . Then a very
small change in the porosity may lead to extreme changes in the permeability.
This is illustrated in Figures 33 and 34 and has been observed in many other
materials. The threshold may, however, have rather different values, depending
on the connectivity of different pores. The example near the base of the stability
illustrated in Figure 14 has, for example, an overall porosity of 0.5, but the
percolation apparently takes place through the largest pores that occupy ≈
0.2 × 0.5 = 0.1. A small change in the heterogenity and pore distribution of
this material, driven by either natural climate change, or CO2 -related chemical
2
Threshold behaviour may also be related to the onset of chemical reactions upon reaching
a certain thermodynamic state (Shante and Kirkpatrick, 1971; Stauffer, 1991)
42
Figure 33:
Permeability of hotpressed calcite, showing a sharp deviation from K ∼ φ3 above the percolation threshold of φc ≈ 0.04. From
Zhang et al. (1994).
Figure 34: Permeability of sandstone
with an increasing deviation from K ∼
φ3 near a crossover porosity of ≈ 0.08.
From Zhu et al. (1995).
processes within the formation, might shift the system to a state above the
percolation threshold.
Percolation theory allows the evaluation of the critical macroscopic behaviour of large systems on the basis of microscopic observations and localised
chemo-physical state transitions. Its importance to interpret geological systems
has recently been outlined (Sahimi, 1993; Selyakov and Kadet, 1996; Berkowitz
and Ewing, 1998; Hunt, 2005b). Future reservoir modelling has a clear perspective in this field of research. Percolation thresholds and nonlinear network
permeability of seabeds appear as an important topic in order to properly evaluate of longterm safety of geological carbon storage.
43
8
Summary and outlook
It is recalled from the introductory section 1, that Carbon Capture and Storage
(CCS) (i) requires a considerably larger overall energy production, and that (ii)
its role to mitigate climate change is limited to a bridging period of 100 to 200
years. Within this time frame a fully renewable energy infrastructure needs to
be established, if serious socio-economical and ecological problems for humans
on earth are to be avoided.
The goal of the present report was to provide an overview about the physical
processes that need to be understood to evaluate risks of CO2 leakage through
the sea floor, as a possible consequence of geological sub-sea storage during
this bridging period. The main conclusions of the present limited report are
summarised as follows:
• Section 2. Thermodynamics and phase equilibria of CO2 -seawater mixtures and of CO2 -hydrate are well established for the oceanic PVTxregime. There is, however, future need for proper extension of thermodynamic models and state equations to high pressure regimes that are
required for accurate reservoir modelling.
• Section 3. Microstructural details of porous media are important to properly model seepage through the seabed, in particular when considering
metastability and phase transitions in the hydrate-stability regime. The
yet unsolved problem of a hydrate shell around a CO2 droplet rising
through seawater might be adressed in a multidisciplinary way, for example by including structural information from hydrates forming from
the ice phase, in a much more easily accessible PT-regime.
• Section 4. Most applications to date have not properly distinguished between drag and dissolution of liquid and rigid CO2 -droplets rising through
seawater. The influence of droplet shape, interaction and coagulation, and
the details of local convective boundary layers also need to be properly
established. This also points to the importance of properly understanding
hydrate shell thermodynamics and morphology (section 3).
• Section 5. Modifications of classical boundary layer models in terms of
sediment suspension, stratification and internal waves are known to be
important to understand bottom boundary layer mixing in the ocean. To
understand CO2 fluxes from the seafloor also viscous and diffusive sublayer
models need to be critically reviewed. The topic of internal wave seabed
interactions has received little interest yet. It is of particular importance
in situations of strongly stratified CO2 -saturated bottom layers.
• Section 6. Long-term large-scale simulations of CO2 -leakage by global
models are lacking. To validate mesoscale simulations very large amounts
of CO2 would be required in release experiments. It appears to be most
realistic to look for information on ion fluxes from the sea floor, as available in the field of geochemistry and deep sea biology. Considering droplet
44
plume dynamics, the field of cloud physics stands as an established empiricaltheoretical framework, from which much can be learned to constrain modelling approaches.
• Section 7. Investigations of catastrophic fluxes through the seafloor should
be extended by modern theories of percolation and probability. This is a
challenge for future reservoir modelling.
45
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