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ISSN 0015-4628, Fluid Dynamics, 2016, Vol. 51, No. 5, pp. 672–679. ©Pleiades Publishing, Ltd., 2016.
Original Russian Text ©G.G. Tsypkin, 2016, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2016, Vol. 51, No. 5,
pp. 99–107.
Formation of Hydrate in Injection of Liquid Carbon
Dioxide into a Reservoir Saturated
with Methane and Water
G. G. Tsypkin
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences,
pr. Vernadskogo 101, Moscow, 119526 Russia
e-mail: tsypkin@ipmnet.ru
Received March 15, 2016
Abstract—Injection of liquid carbon dioxide into a depleted natural gas field is investigated. A mathe-
matical model of the process which takes into account forming CO2hydrate and methane displacement
is suggested. An asymptotic solution of the problem is found in the one-dimensional approximation.
It is shown that three injection regimes can exist depending on the parameters. In the case of weak
injection, liquid carbon dioxide boils up with formation of carbon-dioxide gas. The intense regime is
characterized by formation of CO2hydrate or a mixture of CO2and CH4hydrates. Critical diagrams of
the process which determine the parameter ranges of the corresponding regimes are plotted.
Keywords: gas field, methane, liquid carbon dioxide, injection, hydrate
DOI: 10.1134/S0015462816050112
In recent twenty years considerable attention has been given to minimization of atmospheric emissions
and utilization of greenhouse gases leading to global climate change [1]. One of the most abundant green-
house gases is carbon dioxide. As the prospective technology of CO2utilization, its injection into deeply
embedded permeable rocks, for example, coal seams [2] or depleted natural gas fields [3], was proposed.
In [4, 5] it was suggested to use carbon dioxide injection to intensify methane recovery from the fields
containing the gas in the hydrate state.
The problem of leakage due to the high pressure in storing arises in underground utilization of CO2.
One of the possible solutions of this problem consists in conversion of carbon dioxide into the hydrate state
which can be implemented at relatively low temperatures. Then the storing of the same volumes can be
implemented at a low pressure. These conditions are present in arctic regions and permafrost areas.
In [6, 7] a mathematical model of carbon dioxide injection into a depleted low-temperature hydrocarbon
field containing methane and water in the free state was proposed. Water in the free state interacts with the
injected carbon dioxide gas and carbon dioxide hydrate is formed. In this case heat is released and increase in
the temperature can be significant. This leads to an undesirable phenomenon, namely, incomplete conversion
of gas into hydrate. Consideration is restricted to the low injection pressures (≤3.5 MPa) since at the higher
pressures carbon dioxide exists in the liquid state.
In the present paper the mathematical model is generalized to include the case of injection of liquid
carbon dioxide. The problem is investigated in the self-similar formulation in the one-dimensional approxi-
mation. It is shown that, as compared with injection of the gas phase, no significant reservoir heating takes
place in the case of injection of liquid carbon dioxide. At the high injection pressures formation of carbon
dioxide hydrate on the front can be also accompanied by formation of methane. At the moderate injection
rates only CO2hydrate is formed, while at the low rates the process is accompanied by boiling of liquid
carbon dioxide with formation of the gas phase. Critical diagrams which illustrate the parameter ranges
corresponding to three possible regimes of injection of the liquid phase are given.
672
FORMATION OF HYDRATE IN INJECTION OF LIQUID CARBON DIOXIDE 673
Fig. 1. Domains of existence of methane hydrate and carbon dioxide in the phase plane. Curve 1is the curve of thermo-
dynamic equilibrium of the liquid and gaseous phases of carbon dioxide and curves 2and 3are the curves of dissociation
of CH4and CO2hydrates, respectively. The domains of the stable thermodynamic states of hydrates are located above the
curves.
1. FORMULATION OF THE PROBLEM
We will consider injection of liquid carbon dioxide into a homogeneous low-temperature reservoir with
the porosity
φ
, the permeability k, the temperature T0, and the pressure P0. The reservoir is saturated with a
heterogeneous mixture of methane and water in the free state. The conditions of existence of the components
are so that the point (T0,P0) in the Clausius–Clapeyron phase plane (Fig. 1) lies below the curve 2of methane
hydrate dissociation and the point (Tinj,Pinj) determined by the injection temperature and pressure lies above
the curve 1separating the liquid and gaseous states of carbon dioxide. We will consider small values of the
water saturation Swwhich are not greater than the flow threshold. This makes it possible to consider water
to be immobile.
Contacting with water, injected liquid carbon dioxide forms hydrate. As a result, zone 2saturated with
a mixture of carbon dioxide and CO2hydrate and separated from zone 1saturated with methane and water
by the hydrate formation surface are formed.
The governing equations which describe the transfer processes in both regions represent the mass and
energy conservation laws, the Darcy’s laws for methane in the gas phase and liquid carbon dioxide, the
equations of state, and thermodynamic relations. Under the assumption of immobility of hydrate and the
matrix of the porous medium the system of equations for both regions takes the form:
φ∂
∂
t(1−Sj)
ρ
i+div
ρ
ivi=0,(
ρ
C)i
∂
T
∂
t+
ρ
iCpvi⋅gradT=
λ
iΔT,
∂
Sj
∂
t=0,vi=−kf(Sj)
μ
igradP,P=
ρ
mRmT,
ρ
c=
ρ
c01+
α
(P−P0)−
β
(T−T0),
λ
1,2=
φ
(1−Sj)
λ
i+
φ
Sj
λ
j+(1−
φ
)
λ
s,
(
ρ
C)1,2=
φ
(1−Sj)
ρ
iCi+
φ
Sj
ρ
jCj+(1−
φ
)
ρ
sCs.
(1.1)
Here, Tis the temperature, Pis the pressure, Sis the saturation, vis the velocity of flow through the
porous medium,
μ
is the viscosity,
ρ
is the density,
λ
is the thermal conductivity, Cis the specific heat, fis
the relative phase permeability, and
α
and
β
are the compressibility of liquid carbon dioxide and the thermal
expansion coefficient, respectively. The subscripts “m”and“c” correspond to methane and liquid carbon
dioxide, the subscript “i” takes the values mand cin the zones containing methane and carbon dioxide, and
the subscript “ j” takes the values wand hwhich correspond to water and carbon dioxide hydrate.
The conditions on the hydrate formation surface represent the energy conservation laws and the methane
and carbon dioxide mass conservation laws
FLUID DYNAMICS Vol. 51 No. 5 2016
674 TSYPKIN
φ
ShQhVn=−
λ
1(gradT)n1+
λ
2(gradT)n2,(1.2)
1−SwVn=−kf
m(Sw)
φμ
mgradPn1,(1.3)
Sh
ρ
ef
c
ρ
∗
c−1+1Vn=−kf
c(Sh)
φμ
c(gradP)n2.(1.4)
Here, Vis the velocity of the movable boundary of forming carbon dioxide hydrate,
ρ
ef
cis the effective
density of CO2in hydrate, and Qhis the heat evolved in forming hydrate from liquid carbon dioxide and
water. The subscript “n” and asterisk denote the normal component and the values of quantities on the front.
For the sake of simplicity we will assume that the relative phase permeabilities are linear functions.
We note that in the general case the point (T∗,P∗)in the phase plane (Fig. 1) corresponding to the
temperatures and pressures on the CO2hydrate formation surface does not belong to curve 3but lies above
it. This is attributable to the fact that the thermodynamic conditions of existence of carbon dioxide hydrate
can be reached in the zone ahead of the front but hydrate cannot be formed due to the absence of liquid
carbon dioxide.
We can eliminate the densities and the velocity of components from the system of governing equations
(1.1). If the difference between the temperature of injected liquid carbon dioxide and the initial reservoir
temperature is much less than the absolute temperature, the energy equation can be linearized. We will as-
sume that the change in pressure in region 1 is small as compared with the initial pressure. Then, neglecting
the small terms, we obtain the following system of linear equations for the saturation, the pressure, and the
temperature in both regions:
∂
Sj
∂
t=0,
∂
P
∂
t=
κ
iΔP,
∂
T
∂
t=aiΔT(j=w,h,i=1,2)
κ
1=kP0
φμ
m,
κ
2=k
φαμ
c,ai=
λ
i
(
ρ
C)i.
(1.5)
Using the small compressibility of the liquid phase in region 2, we can simplify the equation for the
pressure in the system (1.5) [8]. The term in the left-hand side of the equation is of the order of
δ
P/tx,
where
δ
Pis the variation of pressure and txis the characteristic time. Similarly, the right-hand side is of
the order of k
δ
P/
φμ
c
α
L2
P,whereLPis the characteristic dimension. Then the ratio of the left-hand to the
right-hand side is equal to
ε
=
φμ
c
α
L2
P/ktx. From relations on the interface surface (1.4) it follows that
L2
P/tx∼k
δ
P/
φμ
c. Combining two last expressions, we obtain
ε
=
αδ
P≪1. Hence it follows that we can
neglect the derivative with respect to time in the equation for the pressure and, as a result, we obtain
ΔP=0.
Thus, the estimates show that the front on which hydrate is forming moves slowly as compared with the
pressure redistribution rate in region 2 and the motion in this region is quasi-steady-state.
The formulation of the problem must be supplemented with restrictions of the thermodynamic nature
following from the component existence conditions. Firstly, the pressure and the temperature on the interface
calculated in the course of solving must correspond to the domain of existence of liquid carbon dioxide.
Otherwise, liquid carbon dioxide will evaporate and regions containing gaseous carbon dioxide will be
formed. This condition can be expressed by the relation P∗>Pliq(T∗),wherePliq(T)=−239.15 +1.0026T
is the approximate equation of the curve of carbon dioxide saturation (curve 1in Fig. 1) which separates the
regions of gaseous and liquid phases of CO2in the Clausius–Clapeyron phase plane.
Secondly, the point (T∗,P∗)corresponding to the quantities on the front must lie below the curve of
formation of methane hydrate on the phase plane (curve 2in Fig. 1) so that the pressure P∗on the interface
is not higher than the pressure Phm of methane hydrate formation, i.e., P∗<Phm (T∗). Otherwise, a mixture
of methane and carbon dioxide hydrates will be formed. The pressure of methane hydrate formation as a
function of the temperature can be determined from the relation Phm =exp(49.32 −9459/T).
FLUID DYNAMICS Vol. 51 No. 5 2016
FORMATION OF HYDRATE IN INJECTION OF LIQUID CARBON DIOXIDE 675
Finally, at the low injection rates the temperature of the components on the front can decrease to negative
values due to the low temperature of the injected carbon dioxide. This can lead to formation of ice in the
physical system.
In these cases the model suggested does not describe the physical process under consideration. Since
only the fairly high injection rates are of practical interest, in what follows we will not consider the last
restriction.
2. SELF-SIMILAR SOLUTION
The basic properties of the process of injection of liquid carbon dioxide into a reservoir with formation
of CO2hydrate can be illustrated with reference to the self-similar solution for the one-dimensional time-
dependent problem. We will assume that at the initial instant of time the reservoir containing methane and
water in the free state occupies half-space x>0 and the initial water saturation, pressure, and temperature
are constant
t=0: S=Sw,P=P0,T=T0.
At the point x=0, which simulates the injection well, the pressure Pinj and the temperature Tinj of injected
carbon dioxide are also constant. Then the problem admits the self-similar solution of the form:
P=P(
ζ
),T=T(
ζ
),V(t)=a1
t
γ
,
ζ
=x
2√a1t.(2.1)
From the first of the equations of the system (1.5) it follows that in the self-similar approximation the
hydrate saturation Shis constant in region 2 behind the front. From the H2O conservation law and the
condition of immobility of water and hydrate we can determine the hydrate saturation behind the front
Sh=Sw
ρ
w/
ρ
ef
w. Hence we can find Sh=Sw/0.784 when
ρ
ef
w=784 and
ρ
w=1000 kg/m3.
The temperature and pressure distributions can be determined from the expressions
γ
<
ζ
<∞:T(
ζ
)=T0+(T∗−T0)erf(
ζ
)
erf(
γ
),P(
ζ
)=P0+(P∗−P0)erfc(
ζ
a1/
κ
1)
erfc(
γ
a1/
κ
1),(2.2)
0<
ζ
<
γ
:T(
ζ
)=Tinj +(T∗−Tinj)erfc(
ζ
a1/a2)
erfc(
γ
a1/a2),P(
ζ
)=Pinj +P∗−Pinj
γζ
.(2.3)
Substituting the solutions (2.2) and (2.3) in the system of the boundary conditions (1.2)–(1.4), we obtain
the system of transcendental equations on the front on which hydrate is forming in dimensionless form:
√
πφ
ShQha1
λ
1T0
γ
1=T∗
T0−1exp(−
γ
2)
erfc(
γ
)+
λ
2
λ
1a1
a2T∗
T0−Tinj
T0exp(−
γ
2a1/a2)
erf(
γ
a1/a2),
γ
=
κ
1
π
a1P∗
P0−1exp(−
γ
2a1/
κ
1)
erfc(
γ
a1/
κ
1),(2.4)
Sh
ρ
ef
c
ρ
∗
c−1+1
γ
=−
κ
2a1
1−Sh
γ
P∗
P0−Pinj
P0,
κ
=kP0
φμ
c.
In the system of transcendental equations (2.4) the unknown parameters are the self-similar velocity
γ
,
the temperature T∗, and the pressure P∗on the surface on which hydrate is forming.
The system (2.4) was investigated numerically for the following values of the parameters: Qh=9.6×
106J/m3,
ρ
ef
c=316 kg/m3,
ρ
h=1100 kg/m3,
ρ
s=2×103kg/m3,
μ
m=1.12 ×10−5Pa s,
μ
c=10−4Pa s,
λ
w=0.58 W/(m K),
λ
s=2 W/(m K),
λ
h=2.11 W/(m K), Cw=4.2×103J/(kg K), Cs=103J/(kg K), and
Ch=2.5×103J/(kg K).
FLUID DYNAMICS Vol. 51 No. 5 2016
676 TSYPKIN
Fig. 2. Temperature distribution in injection of liquid carbon dioxide and formation of hydrate for
φ
=0.2, k=10−15 m2,
Sw=0.25, T0=278 K, Tinj =270 K, P0=3.3MPa,andPinj =7MPa.
3. CALCULATION RESULTS
When CO2is injected into a reservoir the pressure varies monotonically decreasing from the injection
well, while the temperature reaches a maximum on the interface. In Fig. 2 we have reproduced the char-
acteristic temperature distribution for high injection rates. The increase in the temperature on the front is
associated with heat emission due to forming hydrate; however, the heat emission is considerably smaller as
compared with the case of gaseous carbon dioxide injection [7] since the enthalpy of gas is greater than the
enthalpy of liquid carbon dioxide.
At the high injection rates (
γ
≫1) jump in the temperature from the initial to maximum value depends
on the thermophysical parameters and the amount of CO2hydrate formed, i.e., on the porosity and the
initial water saturation. In this case the temperature of injected carbon dioxide has no effect on the front
temperature. This statement is also valid in taking the convective energy transfer into account since, due to
the high heat capacity of the matrix of the porous medium, the velocity of the temperature front determined
by convective energy transfer is lower than the velocity of propagation of the front of the liquid phase by an
order of the magnitude [8].
As the injection pressure decreases, the front velocity also decreases and at
γ
≃1 a considerable part of
the evolved energy disperses ahead of the front due to heat conduction, thus decreasing the temperature in
the neighborhood of the front. In the case of weak injection (
γ
<1) the temperature of liquid carbon dioxide
injected into the reservoir becomes significant. In this case the convective energy transfer can be neglected
due to the low velocity of liquid carbon dioxide.
Formally, the problem has the solution for any values of the parameters; however, as mentioned above,
it is possible that the solution obtained does not satisfy the thermodynamic conditions of existence of the
components. In Fig. 3a we have reproduced the critical curves determining the domain of existence of the
thermodynamically noncontradictory solution in the case of low-permeability rocks.
The critical curve 1, which consists of two branches merging at the turning point, separates the injection
regimes with (domain I) and without (domain II) forming methane hydrate. On the critical curve the pres-
sure on the front is equal to the pressure of forming methane hydrate and the point (T∗,P∗)lies on the curve
of dissociation (curve 2in Fig. 1). The critical diagram shows that considerable increase in the injection
pressure at a fixed initial temperature leads to an increase in the pressure on the interface which is higher
than the pressure Pmof forming methane hydrate. Significant decrease in the injection pressure initiates
decrease in the velocity of front and cooling due to injected carbon dioxide initiates formation of methane
hydrate. In the first case methane hydrate is formed as a result of increase in the pressure (upper branch) and
in the second case as a result of decrease in the temperature (lower branch). Thus, the inner domain I cor-
responds to the thermodynamically noncontradictory solution, while the parameters of the outer domain II
correspond to the regimes with formation of CH4hydrate which cannot be described by the model proposed.
In utilization of CO2the transition of methane into the hydrate state is an undesirable phenomenon since in
FLUID DYNAMICS Vol. 51 No. 5 2016
FORMATION OF HYDRATE IN INJECTION OF LIQUID CARBON DIOXIDE 677
Fig. 3. Effect of permeability on the domain of existence of the solution (hatched domain) for
φ
=0.2, Sw=0.25,
Tinj =260 K, P0=3.3 MPa:(a) and (b) correspond to k=0.3×10−16 and 10−15 m2, respectively; curve 1is the criti-
cal curve which separates domain I of formation of CO2hydrate and domain II of formation of a mixture of CH4and CO2
hydrates; curve 2is the critical curve which bounds domain III of boiling of the liquid carbon dioxide.
this case a smaller amount of carbon hydrate gets fixed in hydrate, methane is incompletely displaced and
remains in the reservoir also in the hydrate state.
The critical curve 2is the curve of transition from the regime of injection of liquid carbon dioxide to the
regime of formation of the gas phase and also is two-valued. On this curve the pressure on the displace-
ment front is equal to the pressure of boiling of liquid carbon dioxide and the point (T∗,P∗)lies on curve 1
(Fig. 1). Here, at the fixed initial reservoir temperature boiling of the liquid CO2phase can be prevented by
two ways, namely, by increase in the pressure or by decrease in the temperature on the front. On the upper
branch of curve 2the liquid phase is stabilized due to increase in the injection pressure and, correspondingly,
the pressure on the front and on the lower branch due to decrease in the front temperature for slow injection.
The intermediate values of the injection pressure correspond to the regimes of boiling of the liquid CO2
phase and are located in the inner domain of curve 2(domain III). Thus, the thermodynamically noncontra-
dictory regimes of injection of liquid carbon dioxide occupy the hatched part of domain I. In domain III the
model proposed cannot be used and formation of gaseous carbon dioxide must be taken into account.
Within the framework of the model proposed it is impossible to give the quantitative description of the
role of the gaseous carbon dioxide interlayer developed and estimate its influence on utilization of CO2.
However, a comparison of the results of injection of gaseous and liquid carbon dioxide shows that in the last
case the mass of injected CO2is significantly greater. Therefore, it is natural to assume that boiling of liquid
carbon dioxide will lead to deceleration of the injection process.
With increase in the permeability (Fig. 3b) the critical curves are deformed and the critical curve 1is
displaced toward the domain of lower temperatures, while the upper branch of the critical curve 2rises.
This is attributable to the fact that in the high-permeability rocks the pressure on the front grows slower
due to the more intense methane outflow from the front zone and impedes formation of CH4hydrate. On
the other hand, the lower pressure on the front facilitates boiling of liquid carbon dioxide; therefore, the
parameter domains corresponding to formation of carbon dioxide is extended.
Numerical experiments show that in injection into high-permeability rocks the turning point of curve 2
goes in the low temperature domain and the upper branch is always located above the lower branch of
curve 1. Therefore, the domain of existence of the solution is located between the upper branches of curves 1
and 2and for finding this domain it is sufficient to determine only the location of the upper branches. Since
the case of high-permeability rocks is of greater interest, in what follows we will restrict our attention to
finding the upper branches of the critical curves.
Figure 4 illustrates the effect of the initial reservoir pressure on the domain of existence of the solution.
As the pressure decreases at the same initial temperature, it is necessary to have the higher injection pressure
FLUID DYNAMICS Vol. 51 No. 5 2016
678 TSYPKIN
Fig. 4. Fig. 5.
Fig. 4. Effect of the initial pressure on the domain of existence of the solution. Curves aand bcorrespond to P0=3.3and
3 MPa, respectively. The remaining parameters and the notation are the same as in Fig. 3b.
Fig. 5. Effect of the porosity on the domain of existence of the solution. Curves aand bcorrespond to
φ
=0.2, 0.3,
respectively. The remaining parameters and the notation are the same as in Fig. 3b.
Fig. 6. Effect of the water saturation on the domain of existence of the solution. Curves aand bcorrespond to Sw=0.25
and 0.23, respectively. The remaining parameters and the notation are the same as in Fig. 3b.
to prevent boiling of liquid carbon dioxide. Therefore, the critical curve 2(Fig. 4) is located considerably
higher and the domain of applicability of the model proposed converges. On the other hand, decrease in the
initial pressure leads to decrease in the pressure on the front. This prevents formation of methane hydrate
and the critical curve 1rises and the boundaries of applicability of the mathematical model are extended.
Increase in the rock porosity decreases the diffusion coefficient
κ
and, accordingly, the pressure redistri-
bution rate. In addition, the mass of hydrate formed per unit volume increases and, as a result, the amount of
the evolved heat also increases. The influence of the last effect is more significant. This prevents formation
of methane hydrate and displaces the point of intersection of the critical curves toward the low temperature
domain (Fig. 5). The influence of the porosity on the condition of boiling up of liquid carbon dioxide is less
significant; therefore, curve 2is displaced only slightly.
Decrease in the initial water saturation affects similarly (Fig. 6). On the one hand, the phase permeability
increases and, on the other hand, the amount of the heat evolved and the temperature on the front decrease.
Therefore, the conditions of formation of methane hydrate become favorable and the critical curve 1is
displaced to the right to the higher temperature domain and reduces significantly the parametric domain of
existence of the noncontradictory solution. However, decrease in the temperature prevents boiling of liquid
carbon dioxide and curve 2descents to the low pressure domain and the boundaries of applicability of the
mathematical model are extended.
FLUID DYNAMICS Vol. 51 No. 5 2016
FORMATION OF HYDRATE IN INJECTION OF LIQUID CARBON DIOXIDE 679
Summary. Injection of liquid carbon dioxide into a depleted natural gas field accompanied by forming
CO2hydrate is considered. It is assumed that initially the reservoir contains methane and water in the free
state. Carbon dioxide contacts with water on the front of displacement of residual methane. Under the corre-
sponding thermodynamic conditions this leads to forming carbon dioxide hydrate. A mathematical model of
the process is suggested and the one-dimensional problem is investigated in the self-similar approximation.
Calculations show that formation of CO2hydrate accompanied by heat evolution leads to increase in
the reservoir temperature. Depending on found values of the pressure and temperature on the front, three
different injection regimes determined by the thermodynamic state of the components can be implemented.
The relatively low pressure on the front leads to phase transition of liquid carbon dioxide into the gaseous
state. At the high pressures on the front carbon dioxide hydrate is formed together with methane hydrate.
At the moderate pressures liquid carbon dioxide does not boil up and only CO2hydrate is formed. The effect
of the reservoir parameters and the initial and boundary conditions on the injection regime is investigated.
The results are illustrated on the critical diagrams. The mathematical model proposed describes adequately
the process only in the last case and it is necessary to modify the model to describe the first two cases.
The work was carried out with support from the Russian Foundation for Basic Research (project
No. 16-01-00363).
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