Proceedings of the 7th International Conference on Gas Hydrates (ICGH 2011),
Edinburgh, Scotland, United Kingdom, July 17-21, 2011.
METHANE SOLUBILITY IN WATER UNDER HYDRATE EQUILIBRIUM
CONDITIONS: SINGLE PORE AND PORE NETWORK STUDIES
Ioannis N. Tsimpanogiannis
National Center for Scientific Research “Demokritos”
Aghia Paraskevi, Athens, 15310
GREECE
Peter C. Lichtner
Los Alamos National Laboratory
Los Alamos, NM 87545
UNITED STATES OF AMERICA
ABSTRACT
In this work we perform a series of mass-balance-type calculations, in order to estimate the
minimum volume of liquid water required to dissolve, completely, a single methane-gas bubble,
located inside different types of domains that are under hydrate equilibrium pressure/temperature
conditions. We examine the case of bubble dissolution in the bulk, along with the cases of bubble
dissolution within simple networks of pores with the same pore/throat size. In our calculations
we consider experimental values for the equilibrium solubilities of methane in water, under
hydrate forming conditions, as well as, values obtained from predictive tools that are based on
thermodynamic models. As a result of the relatively low solubility of methane in water, large
volumes of water are required for complete dissolution of a methane bubble.
Keywords: gas hydrates, dissolution, methane solubility, hydrate equilibrium, porous medium.
NOMENCLATURE
L length
MW molecular weight
n number of moles
P pressure
R gas constant
r radius
T temperature
V volume
x mole fraction
z compressibility factor
Greek
 proportionality constant
 interfacial tension
 coordination number
 proportionality constant
 proportionality constant

Corresponding author: E-mail: [email protected]
 density
constant
Ψ geometrical factor
Ω proportionality constant
Superscripts
eq equilibrium
PM porous medium
Subscripts
b bubble
c capillary
g gas
H hydrate
l liquid
p pore
t throat
w water
INTRODUCTION
Upward gas migration due to buoyancy forces and
immiscible fluid displacement inside a porous
medium are important processes occurring in the
subsurface. Consider the schematic depicted in
Figure 1a. Two gas-phase clusters that occupy
several pores migrate upward, as a result of
buoyancy forces, displacing during the migration
process the liquid phase that is present in the
porous medium. The mole fraction of the gas
component in the liquid phase is initially (at time
t=0) equal to x g . For a given system (temperature,
pressure, and salinity) corresponds a value x geq
such that the system is at thermodynamic
equilibrium. As long as, x g  x geq (i.e. the liquid
phase becomes undersaturated), the gas cluster
continues to dissolve in the liquid phase until the
point that x g  x geq (i.e. the liquid phase is
saturated). On the other hand, if x g  x geq (i.e. the
liquid phase is supersaturated), then the gas phase
can grow, as a result of the gas component that
comes out of the liquid solution. This process is
known as gas-bubble growth resulting from
“solution-gas-drive” [1-2] to the petroleum
industry.
Under appropriate conditions of pressure,
temperature and salinity the migrating gas phase
can form solid hydrates inside a porous medium.
Of significant interest is the case of methane
migration in the subsurface, and the possible
formation of methane hydrate. Oceanic or
permafrost, naturally-occurring, methane hydrates
are important because they are considered a
possible source of methane for energy uses [3], a
possible environmental threat (due to sudden
release in the atmosphere of methane which is a
“green-house” gas) [4], a geologic hazard (creation
of a tsunami resulting from ocean-floor collapse as
a result of sudden hydrate dissociation), and a
drilling hazard for offshore oil/gas platforms [4].
During methane hydrate formation inside
oceanic sediments the gas phase can be of either
biogenic origin [5], produced locally by
microorganisms inside the Hydrate Stability Zone
(HSZ), or of thermogenic origin, produced deep in
the subsurface and subsequently migrating into the
HSZ [6]. In either case, part of the methane gas
dissolves into the liquid water phase to saturate it
for the prevailing pressure and temperature
conditions, part is consumed to form hydrate, and
part could remain as a free gas phase (provided
(a)
(b)
Figure 1. Schematics: (a) Gas-bubble
upward migration inside a porous medium
driven by buoyancy forces. (b) Spherical gas
bubble in a bulk phase.
adequate amounts of gas are present). If during the
hydrate formation process inside a porous domain,
a subdomain of the HSZ exists, such that the liquid
water present there is undersaturated, then part of
the formed hydrate can dissociate [7-11]. The
hydrate dissociation continues up to the point that
all the liquid becomes saturated with methane and
the three phases (liquid/gas/hydrate) are at
thermodynamic equilibrium. Therefore, methane
solubility in the liquid phase is an essential
parameter to be determined (either experimentally
or computationally), if one needs to calculate
correctly the amount of hydrate formed from a
given amount of gas phase present inside a porous
medium. Methane solubility in water at hydrate
formation conditions has been studied extensively
using theoretical [12-17] and experimental
methods [18-22]. Here we reported only some
representative works in the area.
The objective of the paper is to calculate the
minimum volume of liquid required to dissolve
completely, a single methane-gas bubble, located
inside different types of domains that are under
hydrate
equilibrium
pressure/temperature
conditions.
This paper is organized as follows: Initially, we
present the mathematical formulation for the case
of a gas bubble in a bulk phase, namely in the
absence of interference with any solid material.
Next, the case of a gas bubble inside a regular pore
network is described. Then, we present and discuss
the results for both aforementioned cases, and
finally, we end with the conclusions.
MATHEMATICAL FORMULATION: BULK
PHASE
Consider a spherical gas (CH4) bubble with a
radius rb and volume Vb placed inside an aqueous
(H2O) liquid phase, located in a container of
arbitrary shape that has a volume Vbox. Such a
system is shown in Figure 1b for the particular
case of a square container. The pressure Pg in the
gas bubble is given by the ideal gas law, corrected
with the compressibility factor z, in order to
account for the non-idealities of the gas phase at
higher pressures.
PgVb  zn g RT
(1)
where R is the gas constant, T is the temperature
and ng the number of gas moles. In the current
study we used the component-specific Equation of
State developed by Duan et al. [24] to calculate the
compressibility factor of methane.
The pressure inside the gas phase of the
spherical bubble, Pg, is also related to the capillary
pressure, Pc, and the pressure in the surrounding
liquid phase, Pl, as follows:
Pg  Pc  Pl 
2
 Pl
rb
(2)
where  is the interfacial tension, which is a
function of temperature and pressure. In the
current study  is calculated using the
methodology reported in [25]. Emphasis is placed
in values of pressure along the methane hydrate
equilibrium curve PHeq  f THeq . Therefore, we
 
take the pressure in the container as Pl  PHeq for
the calculations presented in this study.
Figure 2. Capillary pressure as a function of
temperature for various gas bubbles with
radii in the range 10-9–10-3 m. The dasheddotted line shows the hydrate equilibrium
curve for methane hydrate [26].
Figure 2 shows the capillary pressure as a
function of temperature for various methane
bubbles with radii in the range 10-9–10-3 m. For
reasons of comparison the hydrate equilibrium
curve for pure methane hydrate is also shown in
the same figure. Note that for methane bubbles
with diameter less than rb  106 m, the capillary
pressure can contribute significantly to the gas
pressure, Pg, while for rb  106 m the contribution
is minor.
As is shown in the schematic of Figure 1b the
mole fraction of the gas component in the liquid
phase, at the boundary of the container (i.e. far
away from the bubble) is equal to x g  0 initially
(at t=0). In principal, however, it can have a higher
value than zero, initially, as a result of preexisting
dissolved-gas phase in the liquid. Inside the liquid
phase in the vicinity of the bubble, the mole
fraction of the gas component x g is equal to the
equilibrium value x geq (for the given P, T, salinity).
The mole fraction in the liquid phase is defined
as x g 
ng
ng  nw
, where nw denotes the moles of
the liquid water phase. Since, x geq  x g , there is a
driving force in the liquid phase for countercurrent diffusion of water and methane molecules
to occur between the regions near the bubble and
far away from it. Consequently, the mole fraction
of methane near the bubble decreases and becomes
x g  x geq as time evolves, which then creates an
additional driving force for further gas dissolution
in the liquid phase. Gas dissolution continues as
long as x g  x geq , and stops when xg  xgeq . The
transient dissolution process ends with one of the
following two scenarios: (i) with complete
dissolution of the gas phase if xg  xgeq , and (ii)
with undissolved gas phase still present in the
container if x g  x geq .
The solution of the transient problem is an
interesting research topic, however, it is beyond
the scope of this study. Our concern is the state of
the system when it has equilibrated and no further
changes can be observed. In particular, we are
interested for the case when at the end of the
process there is complete dissolution of the gas
bubble. We ask the question: “What is the
minimum volume of the liquid phase, Vw, that is
required to completely dissolve a single methanegas bubble of a given radius at pressures and
temperatures along the hydrate equilibrium
curve?” Therefore, we need a volume of water
such that at the end of the process:
ng
 x geq .
ng  nw
Equivalently, in order for complete dissolution of
a single methane bubble to occur, the number of
methane moles inside the bubble should be equal
to:
 x eq 
g
n g  n w 

1 x eq 
g 

(3)
where the number of moles of water are given by
nw 
 wVw
MW w
(4)
and w and MWw denote the density and molecular
weight of water respectively. For all the
calculations presented in this study, liquid
densities for water are obtained from Wagner and
Pruss [27].
In order to solve the problem we make the
following assumption: Vw  Vb , where  is the
proportionality constant that needs to be calculated
in order to answer the aforementioned question.
By combining Equations (1-4), and introducing the
previous assumption we can obtain an equation for
 as follows:
2
 PHeq
rb

 x eq  
zRTHeq  g eq  w
 1  x  MW
g 
w

(5)
Once  is calculated for a particular radius of a
gas bubble through Eq. (5), we can compute Vw as
Vw  Vb . During the methane dissolution
process in water we assumed that there is no
change in the temperature T of the system, which
remains equal to THeq .
MATHEMATICAL FORMULATION: PORE
NETWORK
In the previous section we developed an
expression for the minimum volume required to
dissolve completely a gas bubble for the case
when the gas bubble is located in a large container
(i.e. in the bulk). Namely, this result holds in the
absence of any interactions between the bubble
and the solid walls of the container. Of interest
also is the case when the gas bubble is under
confinement (i.e. located within a porous
medium). In the current section we develop an
expression for the minimum volume required to
dissolve completely a gas bubble inside a regular
pore network.
Initially, we describe the construction of the
pore network. Consider the schematic shown in
Figure 3a. At the center of a cubic solid block,
with side-length LUPB, we place a pore space (void)
that is made up of a spherical pore body with
radius rp and a number  of pore throats with
radius rt.  is known as the coordination number.
The coordination number could have values in the
approximate range 1-6. Alternatively, other
geometrical shapes can be adopted for pore bodies
(e.g. cubic, cylindrical, etc.). However, we limit
the discussion in this study, to spherical pore
bodies only.
The pore bodies provide the main storage
capacity and the pore throats the secondary storage
capacity. In addition, pore throats provide the
connections between neighboring pore bodies.
Pore throats also provide the resistance to flow, as
a result of the capillary threshold that needs to be
exceeded, in order for the throats to be penetrated.
solution to our problem. However, this assumption
can be relaxed requiring more complicated
numerical calculations to solve the problem. In
real porous media, both pore bodies and pore
throats can have sizes that follow appropriate size
distributions.
The total volume of the pore space inside each
UPB should take into account the contribution of
the volume of the pore body, Vp, and the volume of
each throat, Vti, among all the  pore throats that
are connected to the pore. The total volume of the
pore space is given as follows:

PM
VUPB
 V p  Vti  V p  Vti
(a)
(6)
i1
The total volume of each UPB (including both
solid and void spaces), VUPB, is equal to:
VUPB  LUPB  . We define the porosity of the
3
PM
system such that:   VUPB
VUPB . We can make
the assumption that V p  Vti where  is given by:
 2 2 

   
  2 
(b)
Figure 3. Schematics: (a) The reconstruction
process for the creation of a porous medium.
(b) A 2-D projection of a regular pore
network.
The combination of solid and void spaces
described above consists a Unit Porous Block
(UPB). This porous unit can be repeated in the
three dimensions to create a 3-D pore network.
Figure 3b depicts a 2-D cut of such a pore
network, where the various characteristic lengths
of interest to this study are clearly shown. In this
study we have assumed that rp  rt , with   1,
and LUPB  rp , with   2 . As a result, the length
Lt
of
a
pore
throat
is
given
by:
L
   2 
Lt   UPB  rp  rp 
 . As a result of the
 2
  2 
way the pore network is constructed, all pore
bodies have the same radius, and therefore the
same pore volume. The same observation also
applies to all pore throats. Following such an
approach enables us in obtaining an analytical
(7)
where  is a geometrical factor depending on the
geometry of the pore throat used. In particular for
the cases of pore throats examined in this study,
and assuming a spherical pore body, we have:
4
3

 

3


for cylindrical throats 



for rectangular throats

(8)
By substituting into Eq. (6) we find that the
total volume of the void (pore) space inside each
UPB is given by:
 
PM
VUPB
 V p 1    V p 
 
(9)
where
  1


(10)
As
PM
UPB
V
  1 the contribution to the volume
originates only from the contribution of the
values from CSMGem are used in Eq. (5) in order
to calculate  .
pore body. When   1 the pore throats begin to
PM
contribute to the volume VUPB
. In order to solve
the problem, for the volume required to dissolve a
gas bubble occupying completely a single pore
body with volume Vp, we make the assumption
PM
that Vw  VP  VUPB
, where  is the
parameter that was calculated in the previous
section. Therefore, the new proportionality
constant  that needs to be calculated is given by:



(11)
Note that in the limiting case when   2 ,
essentially when there are no pore throats in the
system, which is equivalent to the case of the gas
bubble in the bulk fluid, then   1 and    ,
as one should expect. Essentially,  denotes the
number of UPB’s required to dissolve a given gas
bubble.
RESULTS AND DISCUSSION
Thermodynamic issues
In this study we are interested in the methane
solubility in water under pressure and temperature
conditions along the methane hydrate equilibrium
curve. Figure 4a shows the P-T hydrate
equilibrium curve as calculated: (i) by the
correlation of Moridis [26], and (ii) by using the
commercial code CSMGem [23]. Also shown in
Figure 4a are the P, T conditions where
measurements for the methane solubility in the
liquid phase, x g , under hydrate formation
conditions are available [18-22]. Figure 4b shows
the methane solubility in the liquid phase, along
the hydrate equilibrium curve, as calculated by
using the CSMGem model. Figure 4b shows also
available experimental methane solubilities under
hydrate equilibrium conditions. In most cases, the
experiments were performed at pressures that are
higher than the PHeq  f THeq (see also Figure 4a).
Very good agreement is obtained between the
available experimental values along the hydrate
equilibrium curve and those calculated from
CSMGem, as clearly shown in Table 1 for some
representative cases. Therefore, the solubility
 
(a)
(b)
Figure 4. (a) The two lines denote the hydrate
equilibrium curve for methane hydrate
calculated by different methods (solid line:
CSMGem [23], and dashed line: Moridis
[26]). The experimental data show the P, T
conditions where measurements are available
for the methane solubility in the liquid
phase, x g ,
under
hydrate
formation
conditions. (b) Methane solubility in the
liquid phase vs. temperature, along the
hydrate equilibrium curve, using the
CSMGem model and comparison with
experimental data. (Experimental data used.
A: Servio and Englezos, [20]; B: Kim et al.,
[21]; C: Seo et al., [22]; D: Yang et al., [19];
E: Bergeron and Servio, [18]).
THeq
PHeq
(K)
276.25
279.65
282.05
(bar)
35.
50.
65.
xgeq (exp) Ref
xgeq (calc)
%
dev
radii in the range 10-9–10-3 m. We observe that for
bubble radii with rb  105 m, the values of
0.001240
0.001600
0.001850
0.001240
0.001520
0.001753
0.0
5.0
5.2
 essentially collapse on a single curve, showing
no effect of  vs. T. On the other hand, for bubble
radii with rb  105 m, the values of  increase
20
20
20
Table 1. Comparison between experimental
and calculated values for the mole fraction
of methane in water under hydrate
equilibrium conditions.
significantly as rb decreases (for a given P, T along
the methane hydrate equilibrium curve). This
should be expected since the contribution of Pc to
the total gas pressure Pg becomes important as rb
decreases (see also Figure 2).
Bulk Phase
Figure 5 shows the calculated values for  as
a function of temperature and for various bubble
(a)
(a)
(b)
(b)
Figure 5. Calculated values of  vs. T for
various bubble radii in the range: (a) 10-9–10-5
m. (b) 10-6–10-3 m.
Figure 6. Calculated values for  for
rectangular pore throats as a function of
and and two different pore network
coordination numbers: (a) =6. (b) =3.
Pore Network
Figure 6 shows the calculated values
for  corresponding to the case of rectangular pore
throats. The values of  are plotted as a function
of and and two different pore network
coordination numbers (i.e. =6, =3). As can be
seen the values of  are in the range 1-7
approximately. Similar results are obtained for the
case of cylindrical pore throats with the values of
 being in the range 1-5.5 approximately.
Figure 7 shows the calculated values for  as a
function of temperature for various radii of gas
bubbles, located inside a single spherical pore
body. Three cases are considered for  . In
particular curves are plotted for  (=1, 4, 7).
Higher values of  imply increased contribution
to the volume of the UPB from pore throats and
therefore, less UPB’s are required to dissolve the
same size of a gas bubble.
CONCLUSIONS
In this work we have performed analytical
calculations to estimate the minimum volume of
liquid water required to dissolve, completely, a
single methane gas bubble, located inside different
types of domains that are under hydrate
equilibrium
pressure/temperature
conditions.
Initially we considered the case of a single gas
bubble dissolving in the bulk. In addition, we
examined the case of gas bubble dissolution inside
a regular pore network.
ACKNOWLEDGEMENTS
Partial funding by the European Commission
DG Research (contract SES6-2006-518271/
NESSHY) is gratefully acknowledged by the first
author (I. N. T.).
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