ISSN 10283358, Doklady Physics, 2014, Vol. 59, No. 10, pp. 463–466. © Pleiades Publishing, Ltd., 2014. Original Russian Text © G.G. Tsypkin, 2014, published in Doklady Akademii Nauk, 2014, Vol. 458, No. 4, pp. 422–425.
MECHANICS
A Mathematical Model of Carbon Dioxide Flooding with Hydrate Formation G. G. Tsypkin Presented by Academician A.G. Kulikovskii January 21, 2014 Received January 30, 2014
Abstract—The injection of carbon dioxide into a reservoir that contains methane and water in a free state is investigated. A mathematical model of this process is proposed that suggests the formation of the CO2 hydrate on the surface of the phase transition separating regions of methane and carbon dioxide. The conditions on the interface are derived, and an asymptotic solution of the problem is found. Critical diagrams are obtained that define parameter ranges in which there is full or partial transition of gaseous carbon dioxide to a hydrate state. DOI: 10.1134/S1028335814080035
1. In recent years, much attention has been paid to the utilization of carbon dioxide in underground reser voirs to reduce its concentration in the atmosphere and to prevent global climate change [1]. The most effective and safest method for storing of CO2 can be implemented in a lowtemperature reservoir, where the gas is in a gas hydrate form. In this case, significant amounts of gas are stored at relatively low pressures. In [2], it was proposed to combine the utilization of car bon dioxide with the stimulation of the reservoir that contains methane hydrate in order to release addi tional amounts of methane. The proposed technique is based on replacing methane molecules that comprise the hydrate with carbon dioxide molecules. Various techniques of initiating the conversion of methane hydrate into carbon dioxide hydrate have been dis cussed [3–5]. Another urgent problem associated with the forma tion of hydrates is to build models describing the for mation of deposits of gas hydrates in global ocean sed iments and natural gas fields. In [6, 7] the mechanisms of the formation of methane hydrates in ocean sedi ments were studied. It was shown that the hydrate can be formed from the gas dissolved in seawater in the process of upward filtering in mud volcanoes [6] and as a result of the generation of methane by μorganisms in the region of thermodynamic stability of the hydrate. Mechanisms of hydrate formation in natural gas fields are scantily explored. Therefore, the investi
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia email:
[email protected]
gation of carbon dioxide injection with a transition to the hydrated state may be of paramount importance. In this paper, we propose a mathematical model for the formation of carbon dioxide hydrate during CO2 injection in the gas phase in a depleted natural gas field that contains methane and water in the free state. In this case, a methane hydrate displacement front is cre ated in which carbon dioxide is formed. A system of boundary conditions at the interface is established, and an asymptotic solution of the problem is obtained. Critical diagrams of injection modes of carbon dioxide that describe the conditions of full or partial hydrate formation are given. 2. Consider a lowtemperature depleted reservoir with porosity φ, permeability k, a saturated heteroge neous mixture of methane and water with tempera ture T0, and pressure P0; the reservoir water saturation Sw does not exceed the percolation threshold. Such a sit uation arises in the development of gas fields in the Arctic regions and permafrost areas. Initially, water can exist in gas fields as a methane hydrate component. Then the point (T0, P0) in the Clausius–Clapeyron plane lies above the curve 2 (Fig. 1), and in the process of gas production, the pressure decreases and the hydrate decomposes. As a result, water and methane transit to a free state with energy absorption and decrease in temperature. If at this time the thermodynamic conditions correspond to a point lying between curves 2 and 3, then the CO2 hydrate formation requires only carbon dioxide molecules. As a result of the injection of CO2 in such reservoir, two areas are formed saturated with a mixture of methane and water (area 1) and a mixture of carbon dioxide and
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TSYPKIN
of gases and are established similarly to hydrate disso ciation conditions [8]
P, atm 50 45
1
40 35
Hydrate 30 CH4 2 25
Hydrate CО2 3
20 15 10
274
276
278
280
282
284 T, K
Fig. 1. Regions of existence of hydrates of methane and carbon dioxide in the phase plane. Below curve 1 carbon dioxide is in the gaseous state. Curves 2 and 3 are dissocia tion curves of hydrate of CH4 and CO2, respectively. Sus tainable thermodynamic state regions of the hydrate are located above the curves.
its hydrate (area 2) separated by the hydrate formation surface. Heat and mass transfer processes provided the equilibrium transition of gas and water in the hydrate state are described by the conservation laws of mass and energy, Darcy’s law for gases, equations of state, and thermodynamic relations. The hydrate, as well as water, is assumed fixed because of the smallness of Sw. Then the system of basic equations for both areas has the form φ ∂ ( 1 – S j )ρ j + divρ i v i = 0, ∂t
P = ρ i R i T,
(2)
ρ eff ⎞ kf c ( S h ) S h ⎛ ( gradP ) n2 , – 1 + 1 V n = – ⎝ ρ* ⎠ φμ c c
(3)
φS h ρ h q h V n = – λ 1 ( gradT ) n1 + λ 2 ( gradT ) n2 .
(4)
Here V is the velocity of a moving boundary of the for mation of carbon dioxide hydrate, ρeff the effective density of the CO2 hydrate, ρh the hydrate density, and qh the heat of hydrate formation. Index n and the aster isk denote the normal components and the values on the front. Relations at the interface were obtained under the assumption that there was no mixing of gases, because the injection process is quite fast comparatively with the gas diffusion, and the flow in permeable rocks is laminar. Note also that the displacement front of methane with carbon dioxide is stable because of the greater viscosity of carbon dioxide. 3. The analysis of the injection of carbon dioxide with the hydrate formation can be carried out for the case of onedimensional unsteady flow. Let the reser voir occupies the half space x > 0 at the initial time. The initial water saturation, pressure, and temperature are constant t = 0: S = S w ,
λ i = φ ( 1 – S j )λ g + φS j λ j + ( 1 – φ )λ s , ( ρC ) i = φ ( 1 – S j )ρ i C i + φS j ρ j C j + ( 1 – φ )ρ s C s . Here S is the saturation, v the filtration rate, μ the vis cosity, ρ the density, λ the thermal conductivity, φ the porosity, C the specific heat, f the relative permeability, and R the gas constant. Index i takes values m and c accordingly to the areas which contain methane and carbon dioxide, and index j takes values w and h, which correspond to water and the carbon dioxide hydrate. The conditions on the surface of the hydrate for mation are the laws of conservation of energy and mass
T = T0 .
Then at constant pressure and constant tempera 0 ture T of injection at the point x = 0 that models the injection well, the problem admits a selfsimilar solu tion of the following form: T = T ( ζ ),
x , ζ = 2 a1 t (1)
P = P0 , P0
P = P ( ζ ),
∂T ( ρC ) i + ρ i C p v i ⋅ gradT = λ i ΔT, ∂t kf ( S ) v i = – jgradP, μi
kf m ( S w ) ( 1 – S w )V n = – ( gradP ) n1 , φμ m
V(t) =
a1 γ, t
(5)
λm a 1 = . ( ρC ) m
Hydrate saturation distribution Sh in the area 2 behind the front is constant, as follows from the continuity equation for the hydrate. Since the consideration is limited by the injection of only gaseous carbon dioxide, the point (T 0, P 0) in the Clausius–Clapeyron plane (Fig. 1) should be located below the line separating the liquid and gas phases of CO2 (curve 1). Furthermore, this point is located above the curve of the dissociation of the car bon dioxide hydrate (curve 3) in order to admit the existence of the hydrate at the final state. The upper limits for the pressure makes it possible to simplify sig nificantly nonlinear equations of system (1) because in the considered area the perturbations of the pressure DOKLADY PHYSICS
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A MATHEMATICAL MODEL OF CARBON DIOXIDE FLOODING
and temperature are much smaller than the absolute values and the linearization procedure can be applied. As a result, by eliminating velocity and density, we obtain a system of linear equations for the pressure and temperature, which describes gas flow in both areas
T' 0.025
∂T = a i ΔT, ∂t
0.015
∂P = κ i ΔP, ∂t kP κ 1 = 0 , φμ m
0
kP κ 2 = , φμ c
i = 1, 2,
λc a 2 = . ( ρC ) c
(6)
Similarity solution (5) of system (6) is expressed in terms of integrals of probability and contains the required parameters: the selfsimilar velocity γ, pres sure P∗, and temperature T∗ at the interface. By sub stituting these solutions in system of boundary condi tions (2)–(4), we obtain a system of transcendental equations that is solved numerically for typical values of the parameters. Figure 2 shows typical temperature profiles corre sponding to a slow (curve 1) and rapid injection of car bon dioxide (curve 2). If the front of the CO2 hydrate moves slowly, then it is influenced by the injection temperature, in this case, by cooling the area in the vicinity of the front. If the injection pressure or the res ervoir permeability grows, the velocity of the interface increases, and the injection temperature does not affect the front parameters. In this case, the interface temperature is determined mainly by the initial state of the system and the velocity of the front movement. In the case of high movement velocity, the heat penetration ahead of the front is reduced, and all the energy is consumed to raise the temperature on the front and thus to warm the area behind the front. It should be noted that the temperature of the injected gas will not affect the motion of the front in general when the convective energy transfer is taken into account, since the velocity of the temperature front is orders of magnitude lower than the displacement sur face velocity for cases of interest from a practical standpoint. This conclusion is also valid for gases of greater density and liquids with the heat capacity higher than the specific heat of gases [9]. The rise of the temperature T∗ on the front is an important factor determining regime of the flow. If the temperature rises significantly, it may exceed the equi librium temperature of carbon dioxide hydrate. It means that the point (T∗, P∗) on the phase plane (Fig. 1) will lie below the curve 3, which contradicts the assumptions. Thus, the criterion for the existence of the solution of the formulated problem is the condi tion P∗ > F(T∗), where P = F(T) is the equation of the equilibrium curve 3. Obviously, the increase in the ini tial water saturation leads to the growth of the amount DOKLADY PHYSICS
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0.020
0.010 1
0.005
2
0 −0.005
0
1
2
3
4
5 ζ
Fig. 2. Distributions of the dimensionless temperature T–T T ' = 0 for slow (curve 1) and rapid (curve 2) injec T0 tions of carbon dioxide to form a hydrate; φ = 0.1, k = 0.5 × 10–16 m2, Sw = 0.2, T0 = T 0 = 275 K, P0 = 2.9 MPa. P 0 = 3.1 MPa (1), 3.5 MPa (2).
of the CO2 hydrate behind the front and also increases the production of heat on the front. In the plane (P0, P 0) (Fig. 3), critical diagrams of the process are presented. Since the injection pressure should be higher than the initial pressure, all the solu tions that have a physical meaning correspond to the points lying above the dashed curve. Critical curves 1– 4 are obtained for different values of the reservoir per meability, and areas located between the dashed curve and critical curves correspond to the injection regimes in which the solution satisfies the criterion of exist ence. If the permeability is reduced, the front velocity decreases, and a significant amount of energy is con sumed for heating of the area ahead of the front. Therefore, the existence region increases. Figure 3b illustrates the behavior of critical curves with decreasing initial water saturation. In this case, energy production is reduced, and the region of the solution existence increases. The lower parts of the critical curves correspond to the modes when the injection pressure is close to the initial pressure, the front moves slowly, and the temperature rise on the front is limited by the influence of the injection well. At high injection pressures, the pressure P∗ on the front is increased significantly, which prevents the transition of the point (T∗, P∗) (Fig. 1) to the region below the curve 3, where the carbon dioxide hydrate does not exist.
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TSYPKIN P 0, MPa 3.6
(a)
4
3 2
3.4
1
3.2 3.0 2.8 (b)
3.6 3.4
2
3
4
1
wise, the transition in the hydrated state is carried out partially. In previously studied models of the formation and decomposition of gas hydrates, the phase transition has always occurred under the condition that the point (T∗, P∗), corresponding to the interface belongs to the saturation curve. In the proposed model, the condi tions on the surface of the carbon dioxide hydrate for mation do not correspond to the phase equilibrium curve (curve 3, Fig. 1), and in general the point (T∗, P∗) lies above the curve and is in the region of the sustain able state of the CO2 hydrate. This is explained by the fact that the initial state lies in the thermodynamic region of existence of the carbon dioxide hydrate, which is not formed only because of the lack of CO2 molecules. When these molecules are introduced, the gas immediately reacts with water to form a hydrate. ACKNOWLEDGMENTS
3.2
This work was supported by the Russian Foundation for Basic Research, project no. 120192603KOa.
3.0
REFERENCES 2.8 2.7
2.8
2.9
3.0
3.1
3.3 3.2 P0, MPa
Fig. 3. Critical curves separating modes of full and partial formation of the carbon dioxide hydrate: φ = 0.1, T0 = T 0 = 276 K, (a) Sw = 0.2, (b) Sw = 0.18; k = 10–14 (1), 10–15 (2), 0.2 × 10–15 (3), and 10–16 m2 (4).
If pressures are such that the point (P0, P 0) (Fig. 3) lies to the left of the critical curves, it corresponds to an excess of heat from the hydrate formation. Formally, the hydrate on the front is superheated. In this case, part of the water is not included into the hydrate but it remains in the free state, thereby reducing heat gener ation. Thus, the hydrate full formation mode is imple mented at low water content in the free state. Other
1. C. M. Oldenburg, R. Pruess, and S. M. Benson, Energy Fuels 15, 293 (2001). 2. K. Ohgaki, K. Takano, H. Sangawa, et al., J. Chem. Eng. Jpn. 29, 478 (1996). 3. N. Goel, J. Pet. Sci. Eng. 51, 169 (2006). 4. M. D. White, S. K. Wurstner, and B. P. McGrail, Mar. Pet. Geol. 28, 546 (2011). 5. G. G. Tsypkin, A. W. Woods, and C. N. Richardson, Preprint, No. 1035, IPMech RAS (Institute for Prob lems in Mechanics, Russian Academy of Sciences, 2013). 6. A. V. Egorov and A. N. Rozhkov, Fluid Dyn., No. 5, 103 (2010). 7. A. V. Egorov and G. G. Tsypkin, Fluid Dyn., No. 5, 71 (2011). 8. G. G. Tsypkin, Ann. N. Y. Acad. Sci. 912, 428 (2000). 9. G. G. Tsypkin and C. Calore, Int. J. Heat Mass Transfer 20, 3195 (2007).
Translated by O. Pismenov
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