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ANNUAL
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Thermodynamics and Mass
Transport in Multicomponent,
Multiphase H2O Systems
of Planetary Interest
Xinli Lu and Susan W. Kieffer
Department of Geology, University of Illinois, Urbana, Illinois 61801;
email: [email protected], [email protected]
Annu. Rev. Earth Planet. Sci. 2009. 37:449–77
Key Words
First published online as a Review in Advance on
February 10, 2009
geothermal systems, cryogenic systems, thermodynamics, fluid dynamics,
clathrates, Mars, Enceladus, sound speed
The Annual Review of Earth and Planetary Sciences is
online at earth.annualreviews.org
This article’s doi:
10.1146/annurev.earth.031208.100109
c 2009 by Annual Reviews.
Copyright !
All rights reserved
0084-6597/09/0530-0449$20.00
Abstract
Heat and mass transport in low-temperature, low-pressure H2 O systems
are important processes on Earth, and on a number of planets and moons
in the Solar System. In most occurrences, these systems will contain other
components, the so-called noncondensible gases, such as CO2 , CO, SO2 ,
CH4 , and N2 . The presence of the noncondensible components can greatly
alter the thermodynamic properties of the phases and their flow properties
as they move in and on the planets. We review various forms of phase diagrams that give information about pressure-temperature-volume-entropyenthalpy-composition conditions in these complex systems. Fluid dynamic
models must be coupled to the thermodynamics to accurately describe flow
in gas-driven liquid and solid systems. The concepts are illustrated in detail by considering flow and flow instabilities such as geysering in modern
geothermal systems on Earth, paleofluid systems on Mars, and cryogenic
ice-gas systems on Mars and Enceladus. We emphasize that consideration
of single-component end-member systems often leads to conclusions that
exclude many qualitatively and quantitatively important phenomena.
449
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INTRODUCTION: PLANETARY CONTEXT
Annu. Rev. Earth Planet. Sci. 2009.37:449-477. Downloaded from arjournals.annualreviews.org
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A fundamental observation on terrestrial H2 O geothermal systems, and even on cryogenic H2 O
systems, is that H2 O is rarely the only component in the system. CO2 , CO, N2 , NH3 , CH4 ,
H2 S, SO2 , C2 H2 (acetylene), C3 H8 (propane), C2 H6 (ethane), and other gases appear in varying
concentrations both in liquid and solid H2 O systems. The systems are thus more typically multicomponent than single component. Freezing, melting, vaporization, condensation, sublimation,
and deposition occur in these systems, so they are additionally multiphase. Any dynamic flow
processes in such systems are inherently more complex than in single-component systems (Wallis
1969a,b,c,d). We illustrate these complexities here by focusing on flow from reservoirs in singleand multicomponent aqueous systems with both liquid and solid reservoirs.
Conditions conducive to multicomponent, multiphase flow occur on, and in, a wide variety of
Solar System bodies, including Earth (Figure 1a,b), Mars, Jupiter, Neptune, Saturn, and comets,
as well as numerous icy satellites, such as Enceladus (Figure 1c). We present thermodynamic
data and methods for analyzing such systems at low temperatures and illustrate with a few examples how combining the thermodynamics with fluid dynamics models leads to complex fluid flow
phenomena that differ from the single-component counterparts. The eruption of a pure H2 O
geyser (or one with only minute traces of other gases) is driven by processes related to the boiling conditions of H2 O, e.g., Old Faithful geyser (Kieffer 1984, Ingebritsen & Rojstaszer 1993),
although it is possible that even Old Faithful has significant noncondensible gases (Hutchinson
et al. 1997). Eruptions of CO2 -rich geysers are driven by conditions of gas exsolution rather
than boiling of H2 O (Lu 2004; Lu et al. 2005, 2006). Serious errors can be made by assuming
that a partial pressure due to an H2 O component is equal to the total pressure of a mixture.
In engineering, it is well known that drilled wells into gas-rich liquid are much more difficult
to manage and contain than those that contain only H2 O liquid or solid ice. Ignoring gases in
a
b
c
Figure 1
(a) Old Faithful geyser, Yellowstone National Park. Photo by Susan W. Kieffer. (b) A discharging geothermal
well from a CO2 -rich source area, Karawau, New Zealand. Note supersonic plume at top of wellhead. Photo
by Susan W. Kieffer. (c) Multiple jets in the plume emerging from Enceladus, the frigid satellite of Saturn.
Photo from the NASA Cassini Mission.
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analysis of fracture flow can lead to erroneous conclusions about the nature of the flow or reservoir
properties.
Our initial perspective from Earth is from studies of low-temperature geothermal systems
(White 1967; Rinehart 1980a,b; Henley et al. 1984; Economides & Ungemach 1987). We include
the even lower-temperature clathrate systems (Kvenvolden 1993). Clathrates are ices that contain
foreign gas molecules in cages of H2 O molecules. Both geothermal fluids and clathrates have
significant amounts of non-H2 O components—dissolved gases ranging from approximately 0.01%
to several tens of percent (Giggenbach 1980, Henley & Ellis 1983, Sloan 1998). We especially
focus on those systems that are below the boiling point of pure water in which dissolved gases
play a major role in the fluid dynamics (Lu et al. 2005, 2006). Similar dynamics were involved
in the explosive eruptions of Lake Nyos, Cameroon (Zhang & Kling 2006). In the frozen form,
the formation of methane clathrates in glaciers (Miller 1961) and on the ocean floor (Sloan 1998,
Max et al. 2006), and their degassing upon either heating or release from high pressure, involves
multicomponent, multiphase processes.
Pressure (P), temperature (T), and composition (x) relations of these systems can be found in the
fluid inclusion literature (Bodnar 2003). However, quantitative analysis of dynamics in geothermal
and planetary systems requires knowledge of other valuables, such as specific entropy (s), enthalpy
(h), and volume (v), as well as different choices of independent and dependent variables, and these
are the focus of this review.
For planetary applications, we have focused on two bodies that are the current topics of intense
exploration: Mars and Enceladus, a satellite of Saturn. On Mars, CO2 and H2 O in solid, gas,
and possibly liquid and clathrate forms are the dominant volatile components in the crust and
at the icy poles (Miller & Smythe 1970, Milton 1974, Longhi 2006). Methane (CH4 ) has been
observed in the atmosphere (Formisano et al. 2004) and is of interest because it could indicate
either life or active volcanic processes. The behavior of subsurface ices versus latitude and season
is of great interest for Mars, and the recent arguments for “seeps” of a fluid (Malin & Edgett 2000,
Hartmann 2001) versus granular flow (Shinbrot et al. 2004) have made the studies of volatiles of
current interest. It is also possible that there is a liquid water–CO2 fluid at shallow depths, or a
liquid water–CH4 fluid (Lyons et al. 2005).
The discovery of a large plume emanating from the frigid south pole of Enceladus (Figure 1c)
during the Cassini spacecraft mission to Saturn has raised the fundamental question, “What kind
of reservoir produces these plumes?” Porco et al. (2006) used a single-component H2 O model
(“Cold Faithful”) to suggest that the reservoir is liquid water. The proposal that liquid water could
be as close as 7 m to the surface and related models for a deeper “ocean” (Nimmo et al. 2007)
have generated much speculation that Enceladus could host near-surface life and thus might be
an astrobiological target in near-term Solar System exploration.
Kieffer et al. (2006) pointed out that the Cold Faithful model cannot account for the presence of at least two of the three noncondensible gases in the observed abundances (N2 and
CH4 )1 ; the third major gas, CO2 , could be in solution in liquid water, but only at depths greater
than 20 km. They proposed instead a multicomponent model (dubbed “Frigid Faithful”) in
which cold decomposing clathrates host all of these gases (as well as the other minor organic
1
The substance observed by the INMS instrument on the first Cassini encounter with Enceladus in 2006 has a molecular
weight of 28. Initially it was attributed to N2 (Waite et al. 2006). During later encounters in 2008, J.H. Waite (NASA/JPL/
SwRI news conference of 3/26/08, available at http://saturn.jpl.nasa.gov; unpublished as of the time of preparation of this
review) proposed that it was CO, which also has a molecular weight of 28. Either molecule forms clathrates with roughly
similar properties and thus the conclusions here are unaffected by this uncertainty.
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constituents seen in the plume). This system forms a self-sealing advection machine that operates
at a much lower temperature than required for liquid water (Gioia et al. 2007). Initially, the difference between the Cold Faithful and Frigid Faithful scenarios appears to be temperature, but
more fundamentally, the difference is the behavior of single- versus multicomponent systems.
Although ammonia (NH3 ) has not been detected on Enceladus, there has been much speculation that it has existed in the interior in the past, and possibly at present (Matson et al. 2007).
Clathrate and ammonia hydrates have long been suspected to play key roles in the evolution and
present state of Titan (Lunine & Stevenson 1985, 1987). We do not discuss this complex system
in this review.
Analysis of multicomponent, multiphase systems is complicated, and systems are often simplified in composition to a single component. In this review, we discuss the thermodynamic
phase diagrams that apply to a number of multicomponent, multiphase systems using various
coordinate systems that are appropriate to the particular problems of fluid dynamics considered, with applications to Earth, Mars, and Enceladus. We focus on depressurizing processes as
a fluid moves from a high-pressure reservoir to a lower-pressure environment, and we show how
differing conclusions are reached from analysis of single- versus multicomponent systems. Our
approach brings in some analyses that are common in engineering practice and explains these
in the context of geologic and planetary problems. In many ways, this review complements, and
does not overlap, that of Zhang & Kling (2006) by considering the complex thermodynamics of
multiphase multicomponent systems, combining the thermodynamics with fluid dynamics models of flow in cracks, and extending the work to cryogenic conditions and to other planets and
moons.
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THERMODYNAMIC PROCESSES AND APPROPRIATE PHASE
DIAGRAMS FOR ANALYSIS
When analyzing a complex, multicomponent, multiphase process, the first and most basic step is
to understand the first and second laws of thermodynamics and then to map the behavior required
by the first and second laws onto relevant phase diagrams to get an estimate of the behavior of
the system. However, these two steps must be combined with calculation of another fundamental
parameter, the sound speed, to constrain the behavior of a system. Sound speed is the velocity at
which small disturbances in pressure or density are propagated through a material. The relation
of flow velocities to the sound velocity determines the flow regime.
If flow velocities are significantly lower than the sound speed, flow is subsonic and, therefore,
nearly incompressible. This assumption underlies much of traditional geophysical fluid dynamics
in understanding oceans, atmospheres, and Earth’s interior. If flow velocities are greater than the
sound speed, flow is supersonic, and compressibility and nonlinear effects, such as shock waves,
must be considered in analyses. Physical phenomena and analytic approaches are very different in
the two different flow regimes. We demonstrate that these effects are important in decompression
of geothermal and volcanic systems on Earth and other planets. As we demonstrate here, the
sound speed in multiphase systems can be very low and this allows supersonic flow regimes not
encountered in single-component regimes to be possible.
When the fluid velocity equals the sound speed, the phenomenon of “choking” plays a major,
and controlling, role in high-velocity fluid dynamics because choked conditions control the mass
flux from depressurizing systems. Low sound speed means that the flow would choke at relatively low velocities and thus low mass fluxes. Choking phenomena are found in many terrestrial
geothermal exploration and exploitation situations, and they control mass flux and, therefore,
power production and economic potential.
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Thermodynamic Processes
Standard thermodynamic variables used in many analyses are state variables such as pressure,
temperature, density, entropy, and enthalpy. Fluid dynamic analyses require additionally some
first partial derivatives of these variables, such as the sound speed. These derivatives have some
fundamentally different properties in multiphase and multicomponent systems.
Annu. Rev. Earth Planet. Sci. 2009.37:449-477. Downloaded from arjournals.annualreviews.org
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First law, second law, and sound speed. For an open system with fluids flowing in and out, the
general form of the first law is as follows (Huang 1988a):
!
"
!
"
V2
V2
Q̇syst = Ėsyst + Ẇs + ṁout h +
− ṁin h +
(1)
+ gz
+ gz ,
2
2
out
in
where Q̇syst is the rate of the heat gained by the system, Ėsyst is the rate of change of stored energy
within the system, Ẇs is rate of work (power) done by the system (except flow work), ṁ is mass
flow rate of the fluid, h is specific enthalpy, V is velocity of the fluid, and z is the elevation. The
subscripts in and out represent the thermodynamic state of the fluid flowing in and out of the
system, respectively. Below in this review, we use subscripts such as this, or 1, 2, 3, etc., to indicate
the values of variables along flow paths without further definition where obvious. Enthalpy h is
h = u + Pv = u +
P
,
ρ
(2)
where u is specific internal energy, and ρ is density.
For a steady-state steady flow with no power done by the system, ṁout = ṁin = ṁ, Ėsyst = 0,
and Ẇs = 0. Considering a thermodynamic process from state 1 to 2 (Adkins 1983), Equation 1
becomes
$
1# 2
q = (h2 − h1 ) +
V − V21 + g(z2 − z1 ),
(3)
2 2
where q is the heat gained by the system per unit mass. In the cases we discuss here, the depressurizing process is very fast, and the heat transfer rate to or from the surrounding of the system is
much smaller compared with other terms in Equation 3, so q = 0 can be assumed for the energy
balance. In many decompression-driven flows on Earth and other planetary bodies, the potential
energy term g(z2 − z1 ) is also negligible in Equation 3. When heat transfer q and initial velocity
V1 are also negligible, the exit velocity V2 in Equation 3 can be expressed as
%
(4)
V2 = 2 (h1 − h2 ).
The general form of the second law of thermodynamics for an open system (Huang 1988b) can
be expressed mathematically as
& '
(
( Q̇k
d̄S
dS (
+
ṁs −
ṁs −
=
,
(5)
dt
dt
Tk
out
in
irr
where ( d̄S
) is the rate of entropy production (the so-called path function) owing to internal
dt irr
)
)
irreversibility; dS
is the rate of the entropy change within the open system; out ṁs and in ṁs
dt
are total entropy contents of the outgoing and incoming streams, respectively; and Q̇k is the heat
transfer rate at temperature Tk (k is the index indicating the different heat sources and their
temperatures).
The second law of thermodynamics requires ( d̄S
) > 0 for any irreversible process (i.e., for
dt irr
) = 0 only for reversible ideal processes. Irreversibility can be caused by
all real processes). ( d̄S
dt irr
heat transfer driven by a temperature difference, free-expansion owing to a pressure difference
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(not quasi-static), and dissipation (Thompson 1972) in a viscous flow (viscous work going into
deformation of a fluid particle). In steady-state conditions, dS
= 0.
dt
If heat transfer is negligible or the process is adiabatic, Equation 5 becomes
& '
(
(
d̄S
=
ṁs −
ṁs ≥ 0.
(6a)
dt
out
in
irr
Note that, in Equation 6a, “>” corresponds to an irreversible process and “ = ” is for a reversible
process. For one stream of fluid from state 1 to 2, Equation 6 can be further expressed as
s2 − s1 ≥ 0.
(6b)
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Although not usually thought of as a fundamental thermodynamic parameter in the context of
P-v-T-s phase relations, the sound speed is an essential parameter for analysis of moving materials
in planetary systems where pressure gradients can be great, and it is directly related to the static
variables discussed above. Sound speed is a partial derivative of two fundamental static variables,
pressure and density, and is related to the bulk modulus, K (which is a measure of incompressibility,
and K = 1/β, where β is the compressibility):
! "
K
∂P
c2 =
=
.
(7)
ρ
∂ρ s
This derivative is taken for constant entropy conditions because sound wave propagation involves
infinitesimally small and rapid amplitude changes assumed to be adiabatic and reversible, i.e., constant entropy. Although defined in terms of the static variables K and ρ, the sound speed is a dynamic
variable and plays an important role in the simple wave equation. Some finite amplitude processes,
such as rapid decompression of a pressurized reservoir, are also well approximated by the adiabatic
and reversible isentropic condition (Kieffer & Delany 1979, pp. 1612–15). We show in sections
below how the sound speed varies in multicomponent, multiphase systems, and emphasize that it
is a fundamental parameter that must be considered in modeling flow in these complex systems.
Single-Component H2 O System: Phase Diagrams
Temperature-entropy diagrams. Although pressure-temperature (P-T) diagrams are commonly used in Earth and planetary sciences, this representation of the phase relations does not
show details of the two-phase region. Therefore, in many common flows where a flow undergoes a phase change, the P-T diagram is limited. The temperature-entropy (T-s) diagram is
much more useful in analyzing the thermodynamic processes if the system is undergoing phase
changes. These diagrams are commonly used in engineering and geothermal applications (thermodynamic analysis of power cycles, steam turbine thermal process design, flow in geothermal
wells, and geothermal steam separation processes), and the phase diagrams (or their equivalent
Mollier charts) are widely available for some substances. However, T-s diagrams typically do not
include the low-temperature range and have had only limited application in planetary sciences.
Although Kieffer (1982) generated T-s diagrams for SO2 , S, and CO2 for very-low-temperature
conditions, we could not find one for H2 O at low temperatures. We present one for temperatures
down to 145 K in Figure 2.
Why are T-s diagrams useful? Because, as discussed above, many flow processes—to first
order—are adiabatic and reversible, i.e., isentropic. An isentropic process is a vertical line on the
T-s diagram, and thus many properties can be inferred directly from such a representation. The
properties of each phase are represented and the properties of the mixture can be calculated from
the proportions of the phases.
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a
b
2 x 106
0.005
105
611 10
0.05
350
1 Pa
0.5
Temperature (°C)
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300
10
8.328 x 10-5
L
150
206 m3/kg
L+V
400
S
x = 0.2
–300 0
kJ/kg
–1
1
0.4
400
3
1
S+V
800 1200 1600
5
7
9
250
5.687 x 107
0.8
0.6
200
2000
1'
11
Entropy (kJ/kg–K)
13
10–2
D
Liquid
C
Solid A
10–4
Vapor
10–6
10–8
10–10
B
100
300
500
Temperature (K)
700
300
V
S +L
0
350
26146
0.2 0.4 0.6 0.8
50
–3
550
450
100
–100
650
100
500
200
–50
102
600
1.468 x 10-3
250
700
Pressure (bar)
107
108
400
Temperature (K)
450
15
17
19
150
21
Vapor mass fraction
Specific volume
Pressure
Enthalpy
Figure 2
(a, b) Diagrams of water system. Generated with Engineering Equation Solver, EES (Klein 2007). Thermodynamic properties from
Harr et al. (1984). Correlations are valid up to a pressure of 10,000 bar. Properties of ice T < 273.15 K and pressures above the
saturation vapor pressure of ice are based on Hyland & Wexler (1983) for temperatures from 173.15 K to 473.15 K. Data below
−100◦ C (with vapor mass fraction shown by dashed curves) are extrapolated. In (b), A is the triple point, C is the critical point. AC is the
vapor-pressure curve and AB is the sublimation-pressure curve. The melting-point temperature for ice Ih decreases with increasing
pressure, AD. For pressure increases from triple point to 300 bar, the melting point slightly decreases approximately 2 K; however,
when the pressure continues to increase to 2000 bar, the melting point decreases by approximately 18 K (Li et al. 2005, Luscher et al.
2004, Saitta & Datchi 2003, Longhi 2001, Chou et al. 1998). This is because water expands on freezing, causing the density of ice to be
less than liquid water, which is different from most substances for which the solid density is greater than that of the liquid (Moore
1962). In (a), the data are extrapolated down to 145 K. Isobars, constant specific volume lines, isenthalpic lines, and x lines (mass
fraction of saturated water vapor) in a two-phase region are all presented. The liquid-vapor (L + V) two-phase region corresponds to
the vapor-pressure curve AC in (b). The solid-vapor (S + V) two-phase region corresponds to the sublimation-pressure curve AB in
(b). The curve AD in (b) corresponds to many nearly horizontal lines in the solid-liquid region (S + L) in (a), with temperature very
close to 273.16 K for pressures less than 300 bar. These lines are almost identical and look like a horizontal line in (a). Process 1-1# is
mentioned in the text.
Two typical isentropic decompression processes in the two-phase region are shown in
Figure 3a (the high-temperature part of Figure 2). Process 1–1# is decompression from a saturated
liquid; process 2–2# is decompression from a saturated vapor. If the reservoir condition (either 1
or 2) is known, then the mass fraction of steam (or liquid) can be determined by the lever rule, the
enthalpy can be calculated, and an estimate of flow velocity can be calculated from Equation 4, assuming that the flow has not choked. Conversely, if the exit velocity and the vapor mass fraction (x1# )
are known, state 1# can be determined and hence the corresponding reservoir condition (state 1)
can be inferred. Similar arguments apply to any initial state within the two-phase region instead
of on the phase boundary.
For an adiabatic process, if the changes of kinetic energy and potential energy are negligible
compared with the change of the enthalpy, the process is isenthalpic. In Figure 3a, the process
3–3# is isenthalpic, a flow path typical of that in a geothermal well. One of the critical questions
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a
b
P1
1
20
300
3
10
3'
4
1'
2.3
0.4
3.4
1 bar
V
0.6
0
4.5
5.6
6.7
P4
P2, T2
P3, T3
2
3
P1
V
P4, T4
4
P2
P3
L+V
L
P3
P4
0.8
Entropy (kJ/kg–K)
P1, T1
P2
2' 2"
L+V
x = 0.2
1.2
5
3"
L
100
6
2
1
200
80
Critical point
Critical point
100
0
–1.0 0.1
7.8
8.9 10.0
Entropy
c
353
L
S+L
3
0
10
1
258 K
–40
–80
–120
–2.5
L+V
2
313
612
273
10 0.2
233
2'
243 K
1'
206
230 K
3'
x = 0.05
x = 0.01 26146
–300 –250
–350
–400
S+V
–450
7
5.687 x 10
m3/kg
3"
S
–500
kJ/kg
–2
–1.5
x = 0.15
1 Pa
x = 0.1
–200
–1
–0.5
Entropy (kJ/kg–K)
81
–150
–100
1"
0
2"
–50
0.5
0
193
153
1
Temperature (K)
40
Temperature (°C)
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Temperature (°C)
400
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Temperature
ANRV374-EA37-19
Vapor mass fraction
Specific volume
Pressure
Enthalpy
Figure 3
Different depressurizing processes in the two-phase regions. (a) Process 1–1# is an isentropic depressurizing process from reservoir state
1 (saturated liquid, P = 20 bar and x = 0) to and exit state 1# (liquid-vapor two-phase region, P = 1 bar). Process 2–2# shows an
isentropic depressurizing process from saturated vapor (state 2, P = 20 bar and x = 1) to the two-phase region (state 2# , P = 1 bar).
From the lever rule, or Equation 8 in the text, states 1# and 2# can be estimated to have 19% and 83% vapor content, respectively.
(b) Schematic diagram showing different depressurizing processes in liquid region. (c) Depressurizing processes in a low-pressure and
low-temperature region. Data below −100 ◦ C (with vapor mass fraction shown by dashed curves) are extrapolated. L, liquid; V, vapor; S,
solid.
in reservoir engineering is, “Can the enthalpy of the reservoir be inferred from measurements of
the properties at the well head?”
Given the enthalpies of the saturated liquid and vapor of the outflow, hf and hg , and the mass
fraction of vapor in the outflow, x3# , and the assumption that the process is isenthalpic, then the
reservoir enthalpy, h3 = h3# , can be inferred from the lever rule:
x 3# =
456
Lu
·
Kieffer
h3 − h f
.
hg − h f
(8)
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In the liquid-alone region, these processes can only be shown at an exaggerated scale
(Figure 3b). 0–1 is an isobaric process, 0–2 is isothermal, 0–3 is isenthalpic, and 0–4 is isentropic. On or near the surface of planets and moons where the surface pressure and temperature
are very low, the depressurizing process will involve the low-temperature ice-vapor part of the
phase diagram (Figure 3c, the enlarged low temperature region of Figure 2).
We can illustrate the usefulness of this representation for examining the plumes erupting from
Enceladus. The plumes were reported to have a mass ratio of solid/vapor of 0.42, corresponding
to a fraction vapor of x = 0.7 (70%)2 (Porco et al. 2006). What kind of reservoir and process
produces these plumes? We illustrate the processes suggested for the single-component reservoir
here, and then contrast these results with a multicomponent analysis below.
In a single-component H2 O system, two processes are available to produce the mixture in the
plume: sublimation of ice with subsequent recondensation of some of the vapor, and boiling of
liquid. Consider first the recondensation scenario path 1–1# in Figure 2. The initial state chosen
for our example is the sublimated vapor on the phase boundary (state 1 in Figure 2) corresponding
to 225 K. We chose the final state to correspond to the relatively low temperature of 145 K. From
the lever rule, a plume in equilibrium at this final state has a composition of that reported, x ≈ 0.7.
Porco et al. (2006, p. 1398) were considering the masses of ice and vapor to be comparable, which
would correspond to x ≈ 0.5. It can be seen from Figure 2 that this is a more difficult composition
to obtain from sublimation under plausible Enceladus conditions. If the kinetic limitations are such
that the final temperature of reaction would be approximately 180–190 K, it is difficult to obtain
x ≈ 0.5 − 0.7, because the final state is more vapor rich x ≈ 0.8 − 0.9. Note that this corresponds
to the recalculated conditions mentioned in Footnote 2.
Instead, the behavior of a reservoir of boiling water at 273 K was considered as the most
plausible candidate for producing the plume. Three potential reservoirs that correspond to the
Cold Faithful model are illustrated in Figure 3c: (a) a liquid water reservoir just at triple point
conditions (process 1–1# -1## ) corresponding to initial conditions proposed by Porco et al. (2006),
(b) a liquid water reservoir 20◦ above the triple point (process 2–2# -2## ), and (c) a 50%–50% mixed
ice-liquid slurry at conditions very close to triple point (3-3# -3## ). The final state is assumed to be
145 K. From this T-s diagram, it is easily seen that these conditions only produce approximately
10% vapor; a higher final temperature of 180–190 would produce even less vapor. As noted by
Porco et al., a mixture forming from a liquid reservoir is mostly ice. If all ice was entrained, the
plume should contain more ice than observed, and more than 90% of the ice must be left behind.
The two different conclusions (boiling water versus recondensation of sublimated vapor) are
at the root of the current controversy over the subsurface conditions at Enceladus’ south pole. We
discuss below how the presence of noncondensible gases also affects this conclusion.
T-v phase diagrams to illustrate nonequilibrium effects in condensing vapor flow. Nonequilibrium thermodynamic processes occur in condensing flow because supersaturation of the vapor
is required for the nucleation of condensed phases. These effects become pronounced even at
temperatures characteristic of terrestrial geothermal regions (e.g., ∼200◦ C) and may be especially
prominent at the low temperatures of interest for planetary surface conditions (e.g., see Kieffer
1982 for a discussion of this in the context of SO2 volcanism on Io; Ingersoll et al. 1985, 1989 for
sublimation dynamics on Io; and Byrne & Ingersoll 2003 for a discussion of sublimation of Mars
polar caps).
2
We picked the example to address the mass fraction reported by Porco et al. (2006). Kieffer et al. (2009) subsequently
recalculated the mass fraction and show that it is significantly higher than this value (x > 0.8).
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673
Critical point
350
Saturation line
Two-phase equilibrium expansion (T0 = 473 K)
Two-phase equilibrium expansion (T0 = 190 K)
Metastable vapor expansion (T0 = 473 K)
Metastable vapor expansion (T0 = 190 K)
Schematic real expansion process
after supersaturation
300
Temperature (°C)
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250
A (473 K, 15.5 bar)
200
V
150
L L+V
100
D
50
E
473
423
273
G
223
I (190 K, 3.5 x 10-7 bar)
S
S+V
173
V
–150
–200
–4
523
323
–50
–100
573
373
F (375 K, 1.013 bar)
273 K
0
623
Temperature (K)
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C
–2
0
B
J
2
4
6
8
10
12
14
16
18
20
22
24
26
28
73
log specific volume (m3/kg)
Figure 4
Temperature-volume semilog plot for H2 O. L, liquid; V, vapor; S, solid. See text for discussion of lettered points.
Although the topic of nonequilibrium flows with condensation could be the subject of a separate
review article because so many new computational and experimental techniques have shed light
on this problem, the basic concepts can be illustrated with yet a different phase diagram: the
temperature-volume (T-v) diagram introduced by Zel’dovich & Raizer (1966) and applied to SO2
condensation in the context of Io by Kieffer (1982). This diagram is useful because expansions are
well described by the specific volume of the system, and kinetic effects can be specified in terms
of temperature or temperature differences.
We illustrate the phenomena for the specific case of decompression of H2 O from a typical
geothermal reservoir at 200◦ C, 15.5 bars (Figure 4). Decompression from icy conditions follows
as a subset of this case (discussed below starting at 190 K), and we then apply the analysis to
nonequilibrium condensing flow in fractures on Enceladus.
An equilibrium expansion process is one end-member of possibilities in which the mixture composition is always the equilibrium composition at given conditions (curve A-F-B of
Figure 4). Condensation of droplets must occur rapidly enough for equilibrium to be maintained.
The expansion path lies so close to the saturation curve boundary that it is nearly indistinguishable,
even in semilog coordinates.
If condensation does not occur (e.g., if there is insufficient supersaturation of the vapor, lack
of nucleation sites, or insufficient time to form nucleation centers), then the fluid remains in the
vapor phase and expands isentropically. The fluid can be treated as an ideal gas. For a given initial
condition (T0 , P0 , v0 at point A) and a given final temperature T, the specific volume v and pressure
P along the isentropic metastable process can be determined using the perfect gas law equations
for v(T) and P(T), giving the metastable vapor isentrope, path A-D-C.
There are significant differences between equilibrium and metastable expansion. For example,
the pressure at 375 K is six times higher for a metastable expansion than for an equilibrium one, and
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the corresponding volumes are different by a factor of 6. A real expansion process is usually a path
between these two limiting processes, shown schematically by the path A-D-E-G in Figure 4,
where we have arbitrarily illustrated a metastable vapor isentrope to 100◦ C supersaturation.
The enthalpy drop for the metastable expansion from A to D is 234 kJ kg−1 , less than the
297 kJ kg−1 enthalpy drop of the equilibrium expansion from A to F. Conduit velocities, which
are directly proportional to the enthalpy change, are correspondingly less (684 m s−1 versus
771 m s−1 ).
These effects become more extreme as the temperature difference between initial and final state
increases, as shown by the spread between curves A-D-C and A-F-B in Figure 4. The specific
volume difference between the two processes at 348 K is approximately 1 order of magnitude,
whereas it becomes 5 orders of magnitude at 223 K. This volume difference is mainly due to the
difference in system pressures at a given temperature.
At the triple point, ice is the stable phase in equilibrium with vapor and water. Vapor expansion
at a temperature below its freezing point can be different from the behavior in the liquid-vapor
two-phase region. For the temperature range analogous to that on the south pole of Enceladus,
curves I-J (blue curve) and I-B (blue dots) in Figure 4 show the metastable vapor expansion
and the two-phase equilibrium expansion, respectively, from the initial state of saturated vapor at
190 K. At a condition with a very low temperature and pressure, the specific volume becomes very
large and the spacing between molecules is so rarified that the vapor may not find solid nuclei to
grow on.
On Enceladus, nonequilibrium conditions may be dominant because of the very low temperature. To illustrate the effects of nonequilibrium conditions on condensation in low-temperature
flows, consider the formation of a micron-sized ice particle. Assuming the mass flux onto the
surface of an ice particle is ρv vrms (2π)−1/2 (Ingersoll 1989), the radius growth rate of a spherical
particle in a condensing flow is
!
"
dr
ρv
(9)
=
vrms (2π)−1/2 ,
dt
ρice
where the root mean square molecular velocity vrms = (3RT/M )1/2 , R is the universal gas
constant = 8.314 J mol-K−1 , T is temperature in Kelvin, M is molar mass of vapor in kg mol−1 ,
and ρv and ρice are the vapor and ice densities in kg m−3 , respectively.
In a low-temperature vapor that is flowing fast and has a relatively short flow path, such as
water vapor escaping from fractures on Enceladus, it is more likely that flow would follow a
nonequilibrium process (I-J in Figure 4) than an equilibrium process. If an expanding vapor
follows a nonequilibrium path, such as I-J, the vapor density at any given temperature is greater
than that at equilibrium condition (about 1 order of magnitude at 175 K). Growth rates will
therefore be higher, possibly by an order of magnitude, and conclusions based on growth rates
in single-component systems may tend to underestimate possible growth rates. However, the
presence of noncondensible gases in conduits, as advocated by Kieffer et al. (2006), will also
influence growth rates, so many factors must be considered.
Sound speeds in multiphase systems: s-ρ phase diagrams. Finally, we consider a fourth phase
diagram that illustrates yet another thermodynamic property: the sound speed. The sound speed
of end-member phases can be quite high: 1435 m s−1 for liquid water and 470 m s−1 for steam.
However, the sound speed of a mixed liquid-gas fluid can be as low as a meter or a few meters per
second (Figure 5). The speed of sound in a multiphase system depends on whether heat and mass
transfer are maintained in equilibrium as the small pressure pulses associated with the sound wave
pass through the fluid. If equilibrium is not maintained, the sound speed drops by approximately
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a
b
100
(L)
10–4
10–4
10–3
η
10–2
100
10 –2
10–2
1 bar
25
(V)
(L+V)
10–3
100 bars (311° C)
re
2Φ
200 bars (366° C)
100
100
300
250
200
175
150
125
100
75
50
ssu
Pre
Sound speed (m s–1)
1000
Mass frac
tion
10–1
10 –1
Water-steam phases
not in thermodynamic
equilibrium
25
1
10–5
10–1
Density (g cm–3)
10
Critical
point
50
Sound speed (m s–1)
100
200
50 100 ba bars (3
bar
6
10
s (2 rs (311 6° C)
bar
6 4°
° C)
s
C
)
(
5b
ars 180° C
(15
2° C )
)
1b
ar (
100
° C)
125 10 0 75
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1000
300
250
1Φ
10 –3
10,000
50 bars (264° C)
10
1
10–5
10 bars (180° C)
5 bars (152° C)
1 bar (100
10–4
° C)
10–3
Water-steam phases
in thermodynamic
equilibrium
10–2
10–1
η (mass fraction, steam)
H20
100
10–4
1
5
10
Entropy (Joule g–1 K–1)
Figure 5
(a) Sound speed of water-steam mixtures not in thermodynamic equilibrium (top) and in thermodynamic
equilibrium (bottom). Reproduced from Kieffer 1977. (b) Entropy-density phase diagram for H2 O, with
contours of pressure and mass fraction. Reproduced from Kieffer and Delany 1979.
one order of magnitude (Figure 5a, top), whereas if it is maintained, the sound speed drops by
nearly two orders of magnitude (Figure 5a, bottom). It can be as low as 1 m s−1 (magnitude varies
with pressure).
The sound speed depends on the mass fraction of vapor in the liquid-vapor system. The fluid
can be either a liquid with gas bubbles (boiling liquid) or a vapor with liquid droplets (aerosol or
mist). What causes this phenomenon? The definition of sound speed has two terms (Equation 7): a
direct proportionality to the compressibility, K, and an inverse proportionality to the density, ρ. In
a boiling two-phase mixture, the density is nearly that of the liquid phase, but the compressibility
is that of the gas phase—much lower than that of the liquid phase. These two effects dramatically
diminish the sound speed (left side of Figures 5a). However, as the vapor mass fraction increases
toward the aerosol state, the mixture density decreases and the effect is reversed (right side of
Figure 5a).
Referring again to Equation 7, note that the sound speed (squared) is (δP/δρ)s , which suggests
that changes of sound speed between liquid, gas, and (gas + liquid) phases (both boiling liquid and
aerosol phases) can be visually seen on a plot of entropy versus density with contours of constant
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pressure (Figure 5b) (Kieffer & Delany 1979). This plot is analogous to a normal topographic
map where contours of elevation (z) are plotted against two directions, x and y: (δz/δy)x . Thus,
when a s-ρ graph is read as a topo map and P is considered equivalent to z, the topography of a
s-ρ graph visually illustrates the large variations in sound speed that characterize the single and
multiphase regions (Figure 5b). When traversed in a vertical direction at constant entropy, the
spacing between isobars on such a plot reflects the magnitude of the sound speed—large spacing
indicates small gradients reflecting low sound speeds, and tight spacing indicates large gradients
and high sound speeds.
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Multicomponent, Multiphase Liquid-Gas Systems
Unfortunately, it is difficult to make similar representations of these phase diagrams in multicomponent systems because solubility of the gas into the liquid or solid adds additional complexity.
We use the commonly occurring systems, including H2 O-CO2 and H2 O-CH4 , to illustrate how
Henry’s law and solubility curves are used in the study of multiphase and multicomponent flows.
If gas dissolves in a liquid in equilibrium, the molar concentration of a solute gas in a solution,
Xaq , is proportional to the partial pressure of that gas above the solution, Pg :
Pg = KH Xaq .
(10)
KH is the Henry’s law constant on the molar concentration scale, which can be determined by
experiments, Xaq :
ng,aq
,
(11)
Xaq =
ng,aq + nl
where ng,aq is the number of moles of the gas in aqueous phase and nl is the number of moles of
liquid water.
Henry’s law is accurate if concentrations or partial pressures are relatively low. Deviations from
Henry’s law become noticeable when concentrations and partial pressures are high, but first-order
effects can be considered even in this case by using Henry’s law. Detailed descriptions of the
Henry’s coefficient are given in Lu (2004).
H2 O-CO2 and H2 O-CH4 systems. The H2 O-CO2 system is a typical aqueous system found
in many geothermal systems on Earth, and it could be present on Mars (Lyons et al. 2005, Okubo
& McEwen 2007, Than 2007) or other planetary bodies, such as Enceladus as discussed below.
Several experimental studies on the solubility of CO2 in water were carried out by previous
researchers (Ellis & Golding 1963, Malinin 1974; a review of work is in Lu 2004). Using Malinin’s
correlation, Lu (2004) analyzed the thermodynamics of the H2 O-CO2 system and generated
solubility curves versus temperatures and total system pressures (Figure 6a,b). As is well known,
and can be seen from these figures, at temperatures less than 160◦ C, the CO2 solubility increases
with decreasing temperature, but at temperatures above 160◦ C, the CO2 solubility increases with
increasing temperature.
Aqueous methane systems H2 O-CH4 may exist on Mars (Lyons et al. 2005). The solubility of
CH4 is much less than that of CO2 (Figure 6c). For example, at 50◦ C and 300 bar, the solubility of
CO2 is approximately 20 times that of CH4 . The lowest solubility region of CH4 is at approximately
100◦ C, with a wide, flat curve range, whereas CO2 has its lowest solubility at approximately 150◦ C,
with a narrower, flat curve range.
Based on recent spectroscopic detections of CH4 in the Martian atmosphere, Lyons et al. (2005)
have reexamined the role of C-O-H fluids in the Martian crust. They concluded that there may be
reservoirs of H2 O-CH4 fluids at depths less than 9.5 km in the crust, with H2 O-CO2 -rich fluids
www.annualreviews.org • Multicomponent, Multiphase H2 O Systems
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a
b
A
1000 bar
800 bar
600 bar
400 bar
200 bar
50 bar
0.30
0.20
B
0.10
C
0
0
50
100
150
200
10,000
250
Temperature (°C)
300
350
8000
10 bar (total pressure)
8
6000
6
4000
2000
Q
4
2
1
20
R
40
60
80
c
120,000
Solubility of CH4 (ppm)
A
1000 bar
900 bar
800 bar
700 bar
600 bar
500 bar
400 bar
300 bar
200 bar
100 bar
100,000
80,000
60,000
120
0.14
0.12
B
0.1
0.08
0.06
40,000
0.04
20,000
0
100
Temperature (°C)
Solubility of CH4 (mole fraction)
0.40
Solubility of CO2 (ppm)
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Solubility of CO2 (mole fraction)
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0.02
C
0
50
100
150
200
250
300
0
350
Temperature (°C)
Figure 6
(a) Solubility of CO2 at different pressure and temperature conditions. (b) Solubility of CO2 in a
low-temperature and low-pressure region. (c) Solubility of CH4 at different pressure and temperature
conditions. See text for discussion of lettered points.
at greater depths. This suggests two possible scenarios for fluid migration in the Martian crust:
(a) upward flow of H2 O-CO2 mixtures from a deeper reservoir, and (b) upward flow of H2 O-CH4
mixtures from shallow reservoirs. The existence of these reservoirs has strong implications for
planetary degassing because the volatiles exsolve as the fluids ascend (Lyons et al. 2005). When
H2 O-CH4 -rich fluids decompress and cool, they may encounter a two-phase region.
Slow ascent of the shallow fluids would allow degassing of a vapor-rich phase, providing
methane to the atmosphere (Figure 6c). The illustrative process line A-B-C in this figure corresponds to the initial and final temperatures and lithostatic pressures of Lyons et al. (2005), with
an initial concentration of CH4 of 0.1 mole fraction. Upon ascent from the reservoir, the flow
remains liquid with the gas in solution from A to B, ∼8.5 km, and 320◦ C, but degassing occurs
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at B as proposed by Lyons et al. (2005). From B to C, the flow is a liquid-gas two-phase mixture,
unless flow separation occurs, for example, the liquid stagnates and the gas rises as a separate
phase.
Upward flow from a H2 O-CO2 -rich reservoir at a depth greater than 9.5 km could also produce
two-phase flow as shown in Figure 6a, where the conditions for the process line A-B-C were again
based on Lyons et al. (2005). Degassing occurs at point B at shallower and cooler conditions,
∼6 km and 210◦ C, than for the H2 O-CH4 -rich fluids.
Sound speed in multicomponent liquid-gas systems. Sound speeds in liquid-gas systems such
as H2 O-CO2 or H2 O-CH4 are low because they have the increased compressibility of the gassy
liquid, but the sound speeds do not drop as low as for those in liquid-vapor systems because
there is no mass transfer between the two phases at the time scales of acoustic wave propagation. Therefore, to first order the H2 O-air system provides an illustration of the magnitude of
the multiphase effect. These systems mimic the nonequilibrium H2 O-steam system shown in
Figure 5a (top) (details in Kieffer 1977) and therefore we do not discuss this further here.
Multicomponent, Multiphase Solid-Gas Systems
At lower temperatures in multicomponent systems, the solid ice phase may contain dissolved gases.
In general, these systems are called clathrates, from Latin and Greek roots meaning lattice. When
the ice matrix is H2 O they are called gas hydrates; we use the terms interchangeably. With specific
gases they are called, for example, methane hydrate, carbon dioxide hydrate, etc., and if there are
multiple gases, simply mixed hydrate or mixed clathrate.
Forty seven natural-gas hydrate locations are known on Earth, in permafrost and in ocean
sediments (Sloan 1998), and they are potential natural gas resources. Gas hydrates can exist on
other planetary bodies where pressure, temperature, and other conditions are favorable for their
formation (Miller 1961, Kargel & Lunine 1998, Gautier & Hersant 2005, Hand et al. 2006). The
possible existence of CO2 clathrates on Mars has been proposed by many researchers (Miller &
Smythe 1970, Longhi 2006, Kargel et al. 2000), and it has been suggested that methane clathrates
may supply CH4 to the Martian atmosphere (Prieto-Ballesteros et al. 2006).
The pressure-temperature decomposition (Pd -Td ) curves for clathrate stability and decomposition depend on the nature of the gas and gas mixture. The phases involved are hydrate (H), ice
(I), and gas (G) (i.e., the decomposition reaction is H = G + I). The decomposition curves (Pd -Td
curves) of the N2 , CH4 , and CO2 gas hydrates, together with the data sources used to generate
the curves, are shown in Figure 7a.
Energy balance for the decomposition of a clathrate reservoir. In the section on T-s
diagrams, above, we considered the energy and mass balances for a single-component ice source
for the plume of Enceladus. Consider a parallel analysis for decomposition of a clathrate reservoir
to produce the plume. If a clathrate reservoir with temperature lower than 273 K is pierced by a
fracture (or well), the clathrates will decompose (hydrate → H2 O ice + gases) and material will
leave the system.
To analyze this, consider the energy balances (Figure 7b) with flow through a control volume,
CV. Denoting (ha,d − hr ) as &Hdis , the enthalpy of dissociation of the clathrate hydrate (Yamamuro
& Suga 1989, Kang et al. 2001), the energy balance of the system is
Q̇in + Q̇source − Q̇out = ṁout,g &Hdis + ṁout,v Lsub + ṁout
Pr
.
ρr
(12)
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a
102
101
b
N2 hydrate
Mixed hydrate
CH4 hydrate
CO2 hydrate
Plume
(H)
.
Qout
Vout
.
V2 + gz
mout h +
2
out
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Pd (bar)
100
(I + G)
10–1
Kuhs et al. (1997)
Sloan (1990)
Makogon & Sloan (1994);
also in Sloan (1998)
Sloan (1990)
Miller & Smythe (1970)
Longhi (2005)
Sloan (1990)
10–2
10–3
10–4
10–5
80
100 120 140 160 180 200 220 240 260 280
H2O/CO2
ice cap
z
.
Qsource
Clathrate reservoir
(hr , Tr , ur , ρr)
System boundary
(control volume)
ha,d
.
Qin
Td (K)
Figure 7
(a) Phase equilibrium (H = I + G) for N2 , CH4 , CO2 , and mixed gas clathrates at temperature below 273 K. The composition of the
mixed gas clathrate is that of the plume emanating from Enceladus’ south pole in the first Cassini encounter (Waite 2006).
(b) Schematic diagram of the derivation of energy balance of a clathrate reservoir with outgoing fluids due to decomposition.
The left side of Equation 12 is the net heat gained by the system, whereas the right side is the
heat that is required to dissociate the clathrate hydrate and to accelerate the outgoing fluids, but
includes a term for possible sublimation of ice as well. Assuming steady-state conditions, the left
side of Equation 12 is equal to the right side.
The rate of heat leaving the system (Q̇out ) from the south pole of Enceladus is estimated to
be 3 to 7 GW (Spencer et al. 2006). Based on the mole fractions of N2 , CH4 , and CO2 (Waite
et al. 2006) and the values of enthalpies of dissociation of pure N2 , CH4 , and CO2 hydrates (Kang
et al. 2001, Yamamuro & Suga 1989), the enthalpy of dissociation of the mixed hydrate &Hdis is
&Hdis = 890 kJ kg−1 of the gas mixture.
To estimate the magnitude of the terms on the right side, we use an approximate density
ρr = 1000 kg m−3 . For an order-of-magnitude reservoir pressure we use Pr = 5 bar (Kieffer et al.
2006). The molar ratio of the mixture of gases (N2 , CH4 , and CO2 ) to H2 O vapor is approximately
1:10 (Waite et al. 2006), which is equivalent to the mass ratio of 1:7 for the given molar fractions
detected. The total mass flow rate of H2 O in the plume is ṁout ' 100 kg s−1 (Hansen et al. 2006),
thus the mass flow rate of the gas mixture (N2 , CH4 , and CO2 ) is ṁout,g ∼ 15 kg s−1 . The first
two terms on the right side of Equation 12 are ∼1 × 10−2 GW and 2 × 10−4 GW, very small
compared with the observed energy flux.
Taking the mass rate of the sublimated H2 O vapor, ṁout,v , as 350 kg s−1 , and the latent heat for
the sublimation of ice Lsub = 2800 kJ kg−1 , we can estimate the third term as 0.84 GW. The three
terms are ∼1 GW, small compared with the radiated heat. This led Kieffer et al. (2006) to propose
that the plume represents a small leak on a massive advection system in which large amounts of
vapor are condensing on the walls releasing latent heat, which contributes an additional radiated
energy component.
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c=
!
γ RT
M
" 12
(13)
,
where γ is the isentropic exponent for the gas, and M is the molecular weight. For a perfect gas,
the sound speed is a direct function of temperature and γ , and inversely proportional to molecular
weight.
If the mass ratio of solids/gas is denoted by m, then the effect of the mass loading alone in
the pseudogas approximation is to change R/M from that of the gas to R/M(1+m) (Wallis 1969b,
equation 2.17), that is, mass loading mimics increasing molecular weight and decreases the sound
speed. If heat transfer is important, then γ , the ratio of heat capacities, is changed to
γ =
cp,gas + mcs
,
cv,gas + mcs
(14)
where cp,gas and cv,gas are the heat capacities at constant pressure and constant volume for the gas,
and cs is the averaged heat capacity for the solid phase. For significant mass loading, γ tends toward
unity. These effects are shown in Figure 8a, which compares the behavior of an H2 O-solid dusty
a
b
104
1200
1 bar
Critical point
400
900
700
Temperature (K)
1100
25
1Φ
600
100
SF6 (vapor)-solid mixture
750 K
600 K
400 K
101
0.1
1300
5
500
Temperature (°C)
c (m/s)
102
800
m=0
1200 K
600 K
400 K
103
Kimberlite
m=
2
1000
100
H2O (vapor)-solid mixture
4
m =m = 0.2
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Sound speeds in multicomponent, multiphase solid-gas systems. We conclude our review
of sound speeds in multicomponent, multiphase systems by summarizing the effects of particles
entrained in gases on the sound speed. We do not attempt to cover densely packed solid-gas
mixtures, such as granular materials (see Jaeger et al. 1996 for a review of the behavior of these
complex substances), or the more subtle effects of embedded particles or bubbles in solid H2 O.
Two effects are important in determining the sound speed of gas-solid mixtures: mass loading
of the gas by the particles and heat transfer from the particles to the gas. To first order, it is assumed
that the particles move with the gas, the so-called pseudogas approximation (Wallis 1969b). For a
perfect gas, Equation 7 gives the sound speed as
200
(L)
(V)
25
2Φ
500
1 bar
1
10
100
m (mass ratio solids : vapor)
0
0
(L + V)
2
4
6
Entropy (Joule g K )
–1
–1
8
300
10
Figure 8
(a) Dependence of sound speed on mass loading, temperature, and composition. (b) Temperature-entropy phase diagram for H2 O
comparing the thermodynamic history of the H2 O during isentropic ascent of the fluid alone (m = 0 path) with isentropic ascent of a
mixture containing weight ratio, m, of solids to vapor (reproduced from Kieffer 1984). L, liquid; V, vapor; S, solid.
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gas with that of a heavy-molecular-weight gas SF6 . SF6 could provide a good analog for dusty gas
behavior for m > 5.
Particles entrained in a gas also affect the thermodynamics of an ascending gas phase, as is
illustrated in Figure 8b (compare with Figure 3). If no heat is transferred from the particles to
the gas phase, the gas expands adiabatically and isentropically as discussed in Thermodynamic
Processes and Appropriate Phase Diagrams for Analysis, above (m = 0 path in Figure 8b). In
many cases, the initial conditions are such that adiabatic expansion would cause the fluid to expand
from the vapor or supercritical initial state into the two-phase field (e.g., m = 0, or m = 0.2 in
Figure 8b). However, if sufficient particles transfer heat to the vapor, the expansion is no longer
nearly adiabatic and the thermodynamic paths veer toward the vapor field. The amount of heat
available for transfer to the gas depends on the mass loading (m). For high values of mass loading
(m > 10), the expansion tends to be more isothermal than isentropic, and two-phase conditions
are avoided.
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Choked versus Unchoked Flow
When a fluid flows through a pressure gradient in pipes or fractures, the flow velocity might exceed
the sound speed. Normally, for pure liquids such situations do not arise in geologic settings because
the sound speeds are a kilometer to a few kilometers per second.
However, the situation is very different for liquid-gas or gas-solid systems. Fluids easily attain
the low velocities (meters to a few hundred meters per second) in a number of situations, particularly in geothermal and volcanic settings. In pipe, fracture, or nozzle flow, when the fluid velocity
reaches sonic velocity, the flow chokes and mass flux is limited. In such settings, full decompression
to the ambient pressure may occur downstream of the choke point, which is often underground
(e.g., in plumes). It has been suggested that Old Faithful geyser is choked (Kieffer 1989); Plinian
volcanic eruptions and steam blasts are choked (Buresti & Casarosa 1989, Mastin 1995); and that
choked flow played a role in triggering long-period seismic events at Redoubt Volcano, Alaska
(Morrissey & Chouet 1997), and in eruptions on Mars (Wilson & Head 2007). In such cases, flow
in the conduit only partially decompresses erupting material, and the remaining decompression
to ambient pressure occurs downstream of the choke point in an underexpanded (overpressured)
plume (Kieffer 1982).
Determination of choked conditions is crucial for understanding the relation of plumes to the
reservoirs and conduits that feed them. Transition from subsonic conduit flow to supersonic plume
flow is complicated and includes many nonlinear processes.
DYNAMICS AND NUMERICAL SIMULATIONS:
EXAMPLES OF GAS-LIQUID SYSTEMS
Thermodynamic analysis as reviewed above is very useful in defining relations between initial and final states and for estimating flow properties given certain thermodynamic paths. But
dynamical models are required when dealing with materials in motion for determining various properties (velocity, pressure, temperature, and density, etc.) of the material as functions
of space and time. The coupling of the thermodynamic equations of state with conservation
laws for motion results in complex equations that almost always require numerical solution.
Compared with the number of single-component models, there are relatively few multiphase,
multicomponent models, and they always contain many embedded assumptions. Here and in
the next section, we illustrate several specific examples for both gas-liquid flows and gas-solid
systems.
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Fluid flow in fractures may be steady or not. Steady fracture flow is indicated by the discharges
of liquid hot springs, or vapor/wet vapor fumaroles. However, two-phase flows may be unsteady
for at least two reasons. They may break down internally into various flow regimes (homogeneous,
bubbly, slug, annular, mist, etc.), which we do not consider here. They may also break down because
heat and mass discharges are not balanced, for example, the cyclic phenomenon of geysering, which
is observed both naturally and industrially (Rinehart 1980a,b; Jiang et al. 1995). Understanding
sources of cyclicity in fractures that feed plumes, and separating these effects from atmospheric
processes that may contribute to plume unsteadiness, is important for interpreting plumes and
their relation to the subsurface.
We use the words vapor and gas very specifically in this section: gas means a noncondensible constituent of the liquid, and vapor means the vapor phase, which may contain gas (e.g.,
CO2 ) in addition to the condensable phase (e.g., the H2 O). Gas-rich liquids can bubble3 in response to pressure or temperature changes at fluid temperatures well below the boiling point
of pure water. For example, in an H2 O-CO2 system (Figure 6b), at 1 bar pressure pure water boils at 100◦ C, but aqueous CO2 solutions will bubble (degas, fizz, flash) at only 70◦ C
(point R) if the CO2 concentration is greater than 500 ppm. At 8 bar, pure water boils at
170◦ C, but if the CO2 concentration is more than 4000 ppm, the solution bubbles at only
90◦ C at the same pressure (point Q). Unlike pure H2 O systems, the composition of the bubble is determined by gas solubility and the latent heat associated with partial pressure of H2 O is
negligible.
The exsolution of gases from gas-water aqueous systems can cause hazardous engineering
situations and the gas plays a crucial role in the low-temperature geysers. To analyze this complex
system and to describe the flow behavior, it is necessary to develop a multiphase, multicomponent
flow model, with mass transfer between the phases being considered.
Partial formulations of the flow equations have been provided for pure water systems with
phase change or water-air mixtures without mass transfer at the two-phase interface (e.g., Wallis
1969c). However, for systems with dissolved gases, a source term must be introduced into the
governing equations to account for the mass exchange between the phases and components. This
was provided recently for geysering flow (Lu 2004, 2006).
Conservation Equations
For one-dimensional, time-dependent flow in a vertical pipe, the mass conservation equations for
the liquid and gas phases are
∂
∂
[ρL (1 − α)] + [ρL (1 − α) vL ] = SLG
∂t
∂z
(15)
and
∂
∂
(16)
(ρ α) + (ρG α vG ) = −SLG ,
∂t G
∂z
where ρ, α, and v are density, void fraction, and velocity, respectively; t is time; and SLG
is the total mass exchange rate (including CO2 and water vapor) between liquid and gaseous
phases. Expressions for the mass exchange SLG can be found in Lu (2004) and Lu et al. (2006). The
3
We avoid using boiling to describe this phenomenon because boiling usually refers to a rapid vaporization of a pure liquid,
which typically occurs when a liquid is heated to its boiling point. We introduce the equivalent words bubbling, fizzing, or
flashing.
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momentum conservation equation is
∂P
=
∂z
!
∂P
∂z
"
G
+
!
∂P
∂z
"
A
+
!
∂P
∂z
"
,
(17)
F
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# $ # ∂P $
# $
where ∂P
, ∂z A , and ∂P
are the three components of pressure gradients of gravitation,
∂z G
∂z F
acceleration, and friction, respectively (Wallis 1969b).
These equations differ from the equations for boiling flow in a pipe discussed by Wallis with
regard to the mass exchange between the phases and components. Additional equations required
to close the model include (a) a drift flux model for coupling the velocities of the gas and liquid
phases for bubbly and slug flow regimes, such as that proposed by Zuber & Findlay (1965) or
Wallis (1969a,d); (b) a temperature profile along the flow path or energy equation for determining
the temperature profile; (c) a pressure distribution along the flow path at initial state; and (d )
equations for determining the thermophysical and fluid properties.
Model Implementation and Verification: Te Aroha Geyser, New Zealand
For verification of this formulation, Lu et al. (2004, 2006) carried out experimental and numerical
studies for flow in a vertical geothermal well at Te Aroha, on the north island of New Zealand.
The well is 70 m deep and 10 cm in diameter. The temperature at the bottom of the well is
approximately 90◦ C and approximately 70◦ C on the top. The fluid in the well is water with high
levels of dissolved CO2 (∼3000 ppm). The well can be shut down, but when it is fully opened to
the atmosphere, the flow is not steady, but cyclic, with a small geyser several meters high erupting
approximately every 12 min.
For the verification, the inlet boundary at the bottom was assumed to have a constant flux. The
outlet boundary condition at the wellhead is defined by atmospheric pressure, 1 bar at sea level on
Earth, i.e., the flow was not choked. The initial pressure distribution along the well is determined
from the momentum balance, assuming that pure water is flowing up the well.
The thermodynamic processes that occur as the fluid flows from the bottom of the well (point
Q) to the top (point R) are shown by the path Q-R in Figure 6b. As the fluid rises up the well,
the decompression causes the CO2 solution to become supersaturated and so the CO2 comes out
of solution. This degassing process forms a cyclic two-phase flow.
Close agreement was obtained between the analytical and measured results with regard
to the major variables, such as void fraction and pressure variations at different depths
(Figure 9a,b) of the well, cycle period, water level variations at the top part of the well, and
flash point (the start of degassing) variations at the lower part of the well (Lu et al. 2005, 2006).
Sensitivity studies showed the range of conditions under which the flow would and would not be
steady.
The study showed that the supersaturated CO2 solution, instead of superheated water, causes
bubble generation and intermittent flashing, which induces geysering in the well. The sensitivity
studies showed that a larger inflow rate does not make a positive contribution in generating
geysering flow. On the contrary, too large an inflow rate may cause the CO2 -driven geysering to
disappear and be replaced by a steady gas-liquid two-phase flow.
Geysering is a unique phenomenon of transient two-phase flow with intermittent discharge
characteristics that happen only when some key parameters have particular values that match one
another. In pure H2 O systems, boiling conditions determine the conditions; in multicomponent
systems, gas exsolution determines them.
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6.0
a
5.0
40 m (Earth)
4.0
30 m (Earth)
3.0
20 m (Earth)
50 m (Mars)
40 m (Mars)
30 m (Mars)
20 m (Mars)
2.0
1.0
0
500
1000
1500
2000
2500
b
0.50
50 m (Earth)
Void fraction
Pressure variations (bar)
There is strong evidence for fracture-controlled paleofluid flow on Mars (Okubo & McEwen
2007). How would Te Aroha behave if it were erupting on Mars, which has approximately onethird the gravitational field of Earth? Would it be geysering flow?
If the Martian atmospheric pressure was lower than ∼1 bar when the flow occurred, the proper
analog would be discharging terrestrial geothermal wells. The wellhead pressures of open flowing
geothermal wells are often tens of bars. In this case, the flow is choked at the exit plane and a
flared, noisy supersonic plume forms downstream of the exit plane (Armstead 1983, p. 123, plate 9).
The noise results from both edge noise and internal shock waves in the tulip-shaped plume. By
analogy, we can assume that flow from a fracture on Mars would be choked if the atmosphere were
at low pressure unless the flow itself erodes the conduit (Kieffer 1982).
To isolate the effect of gravity alone, we assume that in the past the atmosphere may have been
at pressures of approximately 1 bar. The key parameters determining flow rate are then gravity and
inflow rate. If the conduit is the same length as that of Te Aroha (70 m), the reservoir pressure is approximately one-third of that on Earth. After a transient initial phase of approximately 2000 s, the
geysering behavior stops and the flow becomes a continuous gassy-liquid discharge (Figure 9a,b,
0.40
20 m (Mars)
0.30
30 m (Mars)
0.20
50 m (Mars)
40 m (Mars)
20 m (Earth)
30 m (Earth)
40 m (Earth)
50 m (Earth)
0.10
0
3000
0
500
1000
Time (s)
6.5
c
0.3
2000
2500
3000
d
150 m
5.5
4.5
120 m
3.5
90 m
2.5
60 m
1.5
1500
Time (s)
0
1000
2000
3000
4000
Time (s)
5000
6000
Void fraction
Pressure variations (bar)
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An Example of How Gravity Conditions Influence the Behavior of CO2 -H2 O
Aqueous Flow: Earth versus Mars
60 m
0.2
90 m
0.1
120 m
150 m
7000
0
0
1000
2000
3000
4000
5000
6000
7000
Time (s)
Figure 9
Numerical simulation and comparison of terrestrial and Martian gravity conditions on the behavior of CO2 -H2 O aqueous flow:
(a) pressure variations and (b) void fractions. Numerical simulation of pressure variations and void fractions on Mars (scenario 2, see
text): (c) pressure variations and (d ) void fractions.
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curves marked with Mars). The void fraction is, however, greater for the Martian analog because of
the lower gravitation. On Earth, gravitational pressure suppresses bubble formation below a certain
level (the flash point, the start of degassing) that is changing during a cycle (Lu et al. 2005, 2006).
The reduced hydrostatic pressure on Mars results in (a) more CO2 released from the solution at
different depths in a upward flow, and (b) the same mass of gases in fluid occupying more volume
than on Earth and hence a bigger value of void fraction as shown in Figure 9b. As a result, the intermittent discharge (geysering) on Earth has turned into a continuous two-phase discharge on Mars.
However, if the geyser were extended to 210 m depth so that the reservoir has the same
hydrostatic pressure as Te Aroha, a different phenomenon is observed (Figure 9b,c). The geysering
phenomenon remains under Martian conditions, but the period changes to 37 min instead of
12 min. The hydrostatic pressure matches the given CO2 concentration and the inflow rate and
hence causes geysering to happen again. The hydrostatic pressures at 60 m, 90 m, 120 m, and
150 m on Mars are almost equal to those on Earth at 20 m, 30 m, 40 m, 50 m, respectively, i.e.,
the rate of change of hydrostatic pressure is approximately one-third of that on Earth. The time
for the whole changing process in each cycle (period) mainly depends on the rate of change of
hydrostatic pressure. The lower rate of change in pressure results in a longer cycle period. We
conclude that geysering flow is as likely to have occurred on Mars as on Earth.
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DYNAMICS AND NUMERICAL SIMULATIONS: EXAMPLES
OF GAS-ICE SYSTEMS (CLATHRATES)
Because sublimation of ice in the single-component H2 O system under terrestrial and planetary
conditions has been the subject of much literature (see, for example, Brown & Cruikshank 1997,
Brown et al. 1988, and references therein), we concentrate here on reviewing the dynamics of
multicomponent ice-gas systems, namely, clathrates. No comprehensive mathematical or numerical model has yet been formulated for simulating the clathrate reservoir degassing process under
the low-pressure and low-temperature conditions of other planets. Models have been developed
only in the past few years for terrestrial degassing of clathrates. They are typically 1D, or axisymmetric, models for simulating methane hydrate reservoir dissociation and predicting natural
gas production either by heating or depressurizing reservoir hydrates ( Ji et al. 2001, Goel et al.
2001, Moridis et al. 2004, Sun et al. 2005). All models are for temperatures greater than 273 K,
where the dissociation of the methane hydrate causes the hydrate to decompose into methane gas
and water. Effects that are considered in the models include reservoir temperature and pressure,
zone permeability, heat transfer into the decomposing zone, kinetics of decomposition, thermal
parameters of the materials, and flow regime.
The only reservoir on Earth for which there are data to validate such models is the Messo Yakhi
field in Siberia (Makogon 1997), and refinements of the available data in planetary settings are
not sufficiently abundant to allow verification for extrapolations. More importantly, the terrestrial
clathrates decompose from hydrate into gas plus liquid water, whereas the planetary clathrates are
so cold that decomposition would be into gas plus ice, for which there are no formulations. To
examine clathrate decomposition under cold, low-pressure conditions, we modified the model of
Ji et al. (2001) to apply to decomposition of the clathrate into ice and gas. We chose this model
because of its simplicity and because it is easy to use for a parametric study when extended to
low-temperature conditions, but we emphasize that many effects not considered in the model
need to be included in future studies.
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a
Impermeable rock
Zone 1
(Gas + Water)
Zone 2
(Hydrate + Gas)
Decomposition front
Flow
Td
Pd
γ
l (t)
x
0
60
Days
120
180
240
300
(Pr = 5 bar, Tr = 190 K, Pg = 2.5 bar)
10–2
Within 365 days (porosity = 0.2)
10–3
After 1 year (porosity = 0.2)
10–4
10–5
After 1 year (porosity = 0.3)
10–6
After 1 year (porosity = 0.1)
10–7
0
10
20
30
40
50
Years
60
70
80
c
360
90 100
10–3
Mass flux of gas mixture (kg/m2–s)
b
10–1
Mass flux of gas mixture (kg/m2–s)
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Well
10–4
Pr = 41 bar, Tr = 250 K, Pg = 20.5 bar
10–5
10–6
Pr = 1 bar, Tr = 160 K, Pg = 0.5 bar
10–7
Pr = 0.1 bar, Tr = 135 K, Pg = 0.05 bar
10–8
0
10
20
30
40
50
60
70
80
90 100
Years
Figure 10
(a) Schematics of one-dimensional hydrate reservoir degassing model on Earth (from Ji et al. 2001). (b) Calculated degassing fluxes
owing to decomposition of a hypothetical clathrate hydrate reservoir on Enceladus. Fluxes of the nominal reservoir (top axis in days
applies only to the top curve) and sensitivity analysis of porosity (dashed curves). (c) Sensitivity analysis for different reservoir pressure
and temperature conditions.
Conservation Equations
Consider natural gas production from methane hydrate due to depressurization of a porous
permeable clathrate zone by injection of a production well into the zone (Figure 10a) (see details
in Ji et al. 2001). The governing equations of mass, momentum, and energy, including heat transfer
and the effects of throttling (isenthalpic process), were linearized by Ji et al. (2001) to give a set
of self-similar solutions for temperature and pressure distributions in the reservoir. This results
in a system of coupled algebraic equations for the location of the decomposition front expressed by
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the distance of the decomposition front from the well, l(t), and the temperature (Td ) and pressure
(Pd ) at the front. The distance from the well l(t) is
√
l(t) = γ t,
(18)
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where γ is a constant in m2 /s.
For a given reservoir pressure and temperature and the pressure at the well, Td , Pd , and γ can
be determined through an iterative scheme by simultaneously satisfying three equations: (a) equilibrium pressure and temperature relation on the decomposition front (i.e., Td -Pd decomposition
curve), (b) linearized energy equation at the decomposition front, and (c) the combined equation of
momentum and mass balances at the decomposition front ( Ji et al. 2001). From Td , Pd , and γ , the
pressure and temperature distributions in the reservoir are calculated from the self-similar solutions of the linearized momentum and energy equations by satisfying the corresponding boundary
conditions.
For Enceladus, the question is, “Can this process produce the observed mass fluxes?” The
volumetric production rate Qv (degassing rate) and its time dependence are ( Ji et al. 2001)
Qv (t) =
2
2
1
k1 Pd − Pg 1
,
√
µg Pg erfα1 2 πχ1 t
(19)
which depends on the phase permeability of gas in zone 1 (dissociated hydrate zone) k1 , the
pressure of the gas in the fracture Pg is the pressure of the gas in the well, the viscosity of the gas,
µg . Two variables that appear here are mostly introduced for ease of calculation:
χ1 =
and
k1 Pg
*1 µg
(20a)
*
γ
,
(20b)
4χ1
one of which includes the porosity on the reservoir, *1 . Sensitivity of the natural gas production
from hydrate by depressurization to variations of reservoir parameters can be found in Ji et al.
(2001). The mass flux Qm (t) (degassing rate) of the gas mixture is then Qm (t) = Qv (t)/ρg .
α1 =
Model Implementation: Plumes on Enceladus
The general principles of clathrate degassing by depressurization for Enceladus were illustrated
by Kieffer et al. (2006) using an extension of the Ji et al. (2001) model. The model was extended
by Fortes (2007) and Halevy & Stewart (2008). The gas composition was modified to reflect the
mixed gases of the plume, using a simplified mixing law. Decomposition into liquid + gas assumed
by Ji et al. was changed to reflect decomposition into ice + gas. The degassing fluxes Qm were
calculated for two plausible reservoirs: one shallow and cool (5 km, 5 bar, 190 K) and one deeper
and warmer (40 km, 41 bar, 250 K) (Figure 10b,c). The pressure in the fracture, Pg , was assumed
to be half of the reservoir pressure based on the assumption that when gas flows in complicated
networks, pressure conditions are commonly determined by narrow constrictions where the flow
chokes to sonic conditions of pressure, temperature, and flow velocity (Mach number = 1) (Kieffer
1989). Sensitivity studies, and the assumed role of entrained ice particles, were considered.
The time-variable flux for these, shown in Figure 10b,c, is of the order of that measured in the
plume, 10−7 to 10−6 kg s−1 m−2 . The rapid time variability implied by the process (opening and
closing of small fractures) is similar to the time scale of variabilities observed by the instruments
on Cassini (<1 month). Much refinement of models needs to be done as data provide more
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constraints on interior processes on Enceladus (Kieffer & Jakosky 2008), but results from these
calculations suggest that degassing of clathrates on Enceladus can produce gases at roughly the rates
observed.
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SUMMARY POINTS
1. Mixed phases in a single-component system can have complicated properties, such as
sound speed in boiling liquids or aerosols. These properties complicate fluid motions.
The flow of H2 O as a pure liquid phase or a pure vapor phase alone in a conduit is much
simpler than the flow of boiling or gassy fluid, which can choke at sonic velocity, and can
develop flow instabilities such as geysering, under certain conditions of heat and mass
transport. Serious errors can be made by assuming that a partial pressure due to an H2 O
component is equal to the total pressure of a mixture.
2. Multiphase, multicomponent systems are inherently more complicated than multiphase,
single-component systems. Counterintuitive phenomena may occur, such as gases being
more soluble in H2 O ice (clathrate) than in H2 O liquid. Even phenomena that superficially appear to be similar can have significant differences when examined in detail.
Geysering, for example, in a cool CO2 -driven system is induced by a periodic bubble
generation that is caused not by the superheated water, as it is in single-component
systems, but by the supersaturated CO2 solution.
FUTURE ISSUES
1. The differences between multiphase, multicomponent systems and multiphase, singlecomponent systems need to be kept in mind in quantitative interpretations of multicomponent systems; for example, in efforts to infer reservoir characteristics on the planets
or moons from surface or plume observations. Resolution of the controversy about the
nature of the reservoir feeding Enceladus’ plume depends on accounting for all of the
gas components (CO2 , N2 , CH4 , plus trace simple organics) as well as the phases.
2. Creation of a thermodynamic database that would allow the exploration of the various
thermodynamic processes discussed in the first part of this paper and the various dynamic processes discussed in the latter part is essential for understanding the processes
in the colder planetary settings of Mars and of some moons. This includes the standard
thermodynamic properties and their derivatives, the properties of metastable H2 O at
temperatures below 0◦ C, the standard thermodynamic properties of clathrate phases, as
well as data for clathrates in cold saline environments.
3. Extension and development of degassing models to colder planetary conditions is much
needed, as are kinetic data.
4. Thinking and teaching about multicomponent systems is needed to educate future Earth
and planetary scientists about the similarities and dissimilarities of multicomponent and
single-component systems. The use of single-component approximation can lead to conclusions that are not only quantitatively wrong, but also qualitatively wrong. Degassing,
depressurizing, and flow instabilities occur for fundamentally different reasons in the two
systems.
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DISCLOSURE STATEMENT
The authors are not aware of any biases that might be perceived as affecting the objectivity of this
review.
ACKNOWLEDGMENTS
This research was supported by NASA NNX06AJ10G and NASA NNX08AP78G as well as
Charles R. Walgreen, Jr. Endowed Chair funds to SWK.
Annu. Rev. Earth Planet. Sci. 2009.37:449-477. Downloaded from arjournals.annualreviews.org
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Planetary Sciences
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Contents
Volume 37, 2009
Where Are You From? Why Are You Here? An African Perspective
on Global Warming
S. George Philander ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1
Stagnant Slab: A Review
Yoshio Fukao, Masayuki Obayashi, Tomoeki Nakakuki,
and the Deep Slab Project Group ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !19
Radiocarbon and Soil Carbon Dynamics
Susan Trumbore ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !47
Evolution of the Genus Homo
Ian Tattersall and Jeffrey H. Schwartz ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !67
Feedbacks, Timescales, and Seeing Red
Gerard Roe ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !93
Atmospheric Lifetime of Fossil Fuel Carbon Dioxide
David Archer, Michael Eby, Victor Brovkin, Andy Ridgwell, Long Cao,
Uwe Mikolajewicz, Ken Caldeira, Katsumi Matsumoto, Guy Munhoven,
Alvaro Montenegro, and Kathy Tokos ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 117
Evolution of Life Cycles in Early Amphibians
Rainer R. Schoch ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 135
The Fin to Limb Transition: New Data, Interpretations, and
Hypotheses from Paleontology and Developmental Biology
Jennifer A. Clack ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 163
Mammalian Response to Cenozoic Climatic Change
Jessica L. Blois and Elizabeth A. Hadly ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 181
Forensic Seismology and the Comprehensive Nuclear-Test-Ban Treaty
David Bowers and Neil D. Selby ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 209
How the Continents Deform: The Evidence from Tectonic Geodesy
Wayne Thatcher ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 237
The Tropics in Paleoclimate
John C.H. Chiang ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 263
vii
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Rivers, Lakes, Dunes, and Rain: Crustal Processes in Titan’s
Methane Cycle
Jonathan I. Lunine and Ralph D. Lorenz ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 299
Planetary Migration: What Does it Mean for Planet Formation?
John E. Chambers ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 321
The Tectonic Framework of the Sumatran Subduction Zone
Robert McCaffrey ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 345
Annu. Rev. Earth Planet. Sci. 2009.37:449-477. Downloaded from arjournals.annualreviews.org
by University of British Columbia Library on 12/29/09. For personal use only.
Microbial Transformations of Minerals and Metals: Recent Advances
in Geomicrobiology Derived from Synchrotron-Based X-Ray
Spectroscopy and X-Ray Microscopy
Alexis Templeton and Emily Knowles ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 367
The Channeled Scabland: A Retrospective
Victor R. Baker ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 393
Growth and Evolution of Asteroids
Erik Asphaug ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 413
Thermodynamics and Mass Transport in Multicomponent, Multiphase
H2 O Systems of Planetary Interest
Xinli Lu and Susan W. Kieffer ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 449
The Hadean Crust: Evidence from >4 Ga Zircons
T. Mark Harrison ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 479
Tracking Euxinia in the Ancient Ocean: A Multiproxy Perspective
and Proterozoic Case Study
Timothy W. Lyons, Ariel D. Anbar, Silke Severmann, Clint Scott,
and Benjamin C. Gill ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 507
The Polar Deposits of Mars
Shane Byrne ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 535
Shearing Melt Out of the Earth: An Experimentalist’s Perspective on
the Influence of Deformation on Melt Extraction
David L. Kohlstedt and Benjamin K. Holtzman ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 561
Indexes
Cumulative Index of Contributing Authors, Volumes 27–37 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 595
Cumulative Index of Chapter Titles, Volumes 27–37 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 599
Errata
An online log of corrections to Annual Review of Earth and Planetary Sciences articles
may be found at http://earth.annualreviews.org
viii
Contents