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Theoretical research of the gas hydrate deposits development using the injection of carbon dioxide
International Journal of Heat and Mass Transfer 107 (2017) 347–357
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Theoretical research of the gas hydrate deposits development using the injection of carbon dioxide V.Sh. Shagapov a, M.K. Khasanov b,⇑, N.G. Musakaev c, Ngoc Hai Duong d a
Institute of Mechanics and Engineering of Kazan Science Center RAS, Kazan, Russia Sterlitamak Branch of Bashkir State University, Sterlitamak, Russia c Tyumen Branch of Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Tyumen State University, Tyumen, Russia d Graduate University of Science and Technology VAST, Hanoi, Viet Nam b
a r t i c l e
i n f o
Article history: Received 15 June 2016 Received in revised form 7 November 2016 Accepted 10 November 2016
Keywords: Gas hydrates Porous medium Methane Carbon dioxide Replacement Decomposition
a b s t r a c t In this paper the mathematical model of the carbon dioxide injection into a natural reservoir initially saturated with methane and its hydrate was constructed and also the research, using this model, was carried out. Self-similar solutions of this problem in the one-dimensional formulation were built. These solutions explain the distribution of the main parameters in a reservoir. It is shown that there are two possible regimes: the first – the recovery of methane from hydrate can occur without decomposition of CH4 hydrate to gas and water, the second – with decomposition. In the first regime, the replacement of methane in hydrate with carbon dioxide is occurs. In the second regime the decomposition of CH4 hydrate to gas and water with the subsequent formation of hydrate from water and carbon dioxide is occurs. The critical diagrams of the each regime existence were built and analyzed. The influence of the injection pressure of carbon dioxide and the reservoir permeability to the speed of methane recovery from hydrate for the each regime was researched. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Currently the natural deposits of gas hydrates due to significant resources and concentrated gas state are the serious alternative to conventional resources of natural gas [1–4]. Currently the main methods of the extraction of hydrocarbon gas (mostly methane) from a hydrate reservoir are the pressure drop and the heating of hydrate saturated rocks [1–4]. However, in some cases, these methods are inefficient because of large expenditure of energy. For example, when the pressure drop method of the gas hydrate deposits development, it is necessary that the reservoir pressure was below than the equilibrium decomposition pressure of hydrate [1–4]. This is achieved also due to the pumping-out of water from wells and this requires a large expenditure of energy. In addition, when the wells exploitation, the temperature decreasing due to the absorption of latent heat of hydrate decomposition, as well as adiabatic cooling and throttle effect. So the water which becomes free after the hydrate decomposition can freeze and plug the equipment [4].
⇑ Corresponding author. E-mail addresses: [email protected] (V.Sh. Shagapov), [email protected] (M.K. Khasanov), [email protected] (N.G. Musakaev), [email protected] (N.H. Duong). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.11.034 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
When the thermal method of the gas hydrate deposits development the reservoir temperature increasing which necessary for the gas hydrate decomposition is achieved mainly by hot water circulation in a closed loop in a well or by injection into the reservoir of hot fluid (water or steam) [4]. However, this method also requires a large expenditure of energy for the hydrates decomposition and for the heating of the reservoir to the temperature that causes decomposition. In connection with the foregoing, the actual problem is to research new methods of influence on gas hydrate reservoirs, which minimize expenditure of energy for the gas extraction. Promising in this respect is the technology of replacement, which consists in the displacement of methane from hydrates by filling their by other gas [5–11]. By several researches the most promising gas for the replacement process is carbon dioxide. Also its pumping into the hydrate saturated reservoir would at the same time to solve the problem of disposal of CO2, in connection with its negative role in the development of the greenhouse effect. This method is based on the fact that the hydrate of carbon dioxide is more stable than methane hydrate. Therefore the molecules of carbon dioxide can to replace the molecules of methane in the methane hydrate. Experimental researches proved that the replacement process CH4-CO2 in the hydrate of methane does not require a supply of external energy, because the latent heat per unit volume of
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the CO2 hydrate formation (from water and carbon dioxide) more than the latent heat of the CH4 hydrate decomposition (to water and methane). In addition, a significant feature of the replacement process CH4-CO2 in the methane hydrate is that this reaction occurs at pressures and temperatures corresponding to the conditions of stable existence of the methane hydrate [5,6] and not accompanied by the release of free water [5–8]. This eliminates the necessity to heat a reservoir to the temperatures causing the decomposition of methane hydrate to gas and water. These facts make this technology promising owing to the low energy costs. Technological ideas must be confirmed by the calculations based on valid theoretical models. Therefore the construction of the mathematical model of the dioxide carbon injection into the gashydrate reservoir is the actual problem. Experimental researches of the replacement process of methane in hydrate with carbon dioxide are described, for example, in [5– 11]. In these works, the researches were conducted in samples of small size and, as a rule, in the thermostatic and barostatic conditions. The replacement process in these researches is limited by the process kinetics due to the small sizes of the samples and the maintenance of constant temperature and pressure conditions. In the cases of long natural reservoirs at constant injection of carbon dioxide into these reservoirs the process of replacing methane with carbon dioxide in the hydrate will be limited by the mass transfer in a porous medium. Therefore, these researches do not give a complete picture of the processes that take place into natural reservoirs. This significantly complicates the comparison of the experimental data with the results of mathematical modeling of the processes for long reservoirs. The mathematical models of the gas hydrate formation in extended porous mediums during the gas injection are formulated, in particular, in works [12–14], in which the process of the gas injection into the reservoir initially saturated with the same gas is researched. The mathematical model for the carbon dioxide injection into a reservoir, which contains methane and water in a free state, is presented in [15]. The mathematical model of the carbon dioxide injection into the natural reservoir saturated with methane and its hydrate is presented in this work. Note that some authors proposed to pump CO2 into porous medium in the liquid state [6,8,9,11]. However, there are significant differences at the mathematical descriptions of the injection process of carbon dioxide in the gaseous or liquid state. It is known that the equilibrium temperature of the hydrate decomposition depends on pressure. Therefore, the possibility of the realization of the different regimes is determined by the overlay of the temperature and pressure fields. There are different equations of state when the different aggregate state of carbon dioxide. Also, difference in the coefficients of viscosity more than an order of magnitude. All this leads to substantial difference of pressure fields for the cases of the carbon dioxide injection in the liquid or gaseous state. This has a significant impact to the possibility of the realization of the different regimes. Let us explain this in more detail. In the case of the liquid CO2 injection the main hydraulic resistance has only the first reservoir zone, which is saturated with the liquid. In the case of the gaseous carbon dioxide injection the hydraulic resistance is uniformly by the all filtering area, because this area is saturated with gas. This is due, firstly, with the different equations of state for the liquid and gas, and, secondly, with a very significant difference in viscosity for the liquid and gas. Therefore, under the same conditions in the case of the CO2 injection in the gaseous state the pressure drop in the near area is significantly lower than in the case of liquid CO2. In this connection, in the case of the gaseous CO2 injection the dependence of pressure before the front (i.e., in the far area) on the injection pressure is much more significant than in the case of the liquid CO2 injection. The realiza-
tion of the different regimes is defined by the equilibrium temperature in the far area, i.e. in fact by the pressure in the far area. When pumping liquid CO2 a greater impact has the temperature of the injected CO2 but not the injection pressure. In the case of the gaseous CO2 injection a greater impact to the features of the process (in terms of the control of the process regimes) has the injection pressure. Therefore, in this work the effect of the pressures (the initial reservoir pressure and the injected fluid pressure) to the process features is researched. Also the concept of the critical pressure of the injected CO2 was introduced and the value of this pressure was examined. In addition, an essential feature of the heat and mass transfer processes with phase transitions is the release or absorption of the latent heat of phase transitions. When the injection of gaseous CO2 the CO2ACH4 replacement process in methane hydrate is exothermic. This process is endothermic when the injection of liquid CO2. This circumstance is taken into account in this work. Therefore, when the gaseous CO2 injection the regime with the decomposition of CH4 hydrate to gas and water can be realized even when the injected gas temperature is equal to the initial reservoir temperature. At the liquid CO2 injection this regime can be realized only when the liquid CO2 temperature is greater than the initial reservoir temperature. Therefore, the replacement regime while injecting liquid CO2 more strongly depends on the temperature of the injected carbon dioxide than when the gaseous CO2 injection. 2. The regime with the replacement of methane with carbon dioxide in hydrate 2.1. Problem statement and basic equations Assume that the horizontal porous reservoir, initially saturated with methane hydrate (initial saturation m) and methane, fills halfspace x > 0. The top and bottom of the reservoir are impermeable. Initial pressure p0 and temperature T0 correspond to the thermodynamic conditions of the existence of mixture of methane and its hydrate:
t ¼ 0;
x>0:
T ¼ T0;
p ¼ p0 ;
Sh ¼ m :
Assume that through the left boundary (x = 0) gaseous carbon dioxide is being injected. The pressure of injection pw and temperature of this gas Tw correspond to the conditions of the existence of heterogeneous mixture of carbon dioxide and its hydrate and are supported at this boundary at constant level:
t > 0;
x¼0:
T ¼ Tw;
p ¼ pw :
Conditions of the existence of hydrates of carbon dioxide and methane are presented on the phase diagram (Fig. 1) [16]. In this chart the curves glh determine three-phase equilibrium ‘‘gas-wat er-hydrate”, and the curve lg – two-phase equilibrium ‘‘liquidvapor”. The subscript 1 corresponds to CO2 in gaseous or liquid state or its hydrate, and the subscript 2 – methane and its hydrate. In this problem the initial values of pressure and temperature of the reservoir are above the curve g2lh2 (in the field of the existence of heterogeneous mixture of methane and its hydrate). The pressure and temperature of the injected carbon dioxide are above the curve g1lh1 (in the field of the existence of carbon dioxide hydrate), but below the curve l1g1 (in the field of the existence of gaseous carbon dioxide). The values of temperature and pressure on the curve of threephase equilibrium ‘‘gas-water-hydrate” are quite well described by the following equation [17]:
p ¼ ps0 exp
T T0 ; T
ð2:1Þ
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The reservoir pressure conductivity factor is equal:
kp
vðpÞ ¼
l/
;
where p is the pressure, l is the average viscosity of gas phase. Then for the estimation of the time t2 can be obtained:
t2
l
l/ : kp
2
If t1 << t2 then the kinetics of process can be neglected. Then:
k/ l l/ : << D1 kp 2
Whence it follows that:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi k << l lD1 =p:
Fig. 1. Phase diagrams of the systems ‘‘CO2AH20” and ‘‘CH4AH20”.
where ps0 is the equilibrium pressure corresponding to the initial temperature T0, T⁄ is the is the empirical parameter, which depends on the type of gas hydrate. Flows in natural porous reservoirs can be considered laminar. The injection process is much faster compared to the diffusion of gases, i.e. the rate of mass transfer due to filtration significantly exceeds the rate of mass transfer due to diffusion. In the considered range of pressures and temperatures, the viscosity of carbon dioxide is about one and a half times higher than the viscosity of methane. Given these factors, we can neglect the mixing of the gases, and the front of displacement of methane with carbon dioxide can be considered stable. Therefore we can assume that, at the injection of CO2, two characteristic zones are formed. In the first (near) zone the pores are saturated with carbon dioxide and its hydrate, and in the second (far) zone the pores are saturated with methane and its hydrate. Thus, given these assumptions, the replacement of methane in hydrate with carbon dioxide is occurs on the movable front surface separating these two zones. In this mathematic model, the nonequilibrium processes connected with the diffusion of the gas which forms hydrate through the hydrate film are not taken into account. Let’s estimate the typical relaxation times of the diffusion nonequilibrium under the assumption that the kinetic mechanism of hydrate formation is related with diffusion. The characteristic relaxation time t1 of the diffusion nonequilibrium can be estimated as: 2
t 1 d =D1 ; where D1 is the diffusion coefficient in a hydrate, d is the typical thickness of the hydrate film (approximately equal to the pores size). The characteristic pores size can be estimated as:
Assume that the reservoir length is quite large, for example l = 100 m. Then for the values of parameters p = 1 MPa, l = 105 Pas, D1 = 1013 m2/s the kinetics can be neglected if the permeability satisfies the condition:
k << 1010 m2 : This condition is satisfied for the most natural reservoirs. Therefore, the characteristic time of the kinetics of the process will be much shorter than the characteristic time of the process of displacement (for the specified length of a reservoir). Consequently in long natural reservoirs the kinetics of the process, associated with diffusion in hydrated films at the level of individual pores, can be neglected. In [18] it is shown that the filtration mass transfer is much greater than the diffusion mass transfer at the displacement of miscible fluids at the natural porous systems. Therefore we will neglect the diffusion mixing of gases in considered problem. Let us assume the following assumptions to describe the mass transfer processes during CO2 injection into the porous reservoir. The process is one-temperature, i.e. the temperatures of a porous medium and of a saturant are similar. Hydrates of CO2 and CH4 are the two-component systems with the mass concentrations of carbon dioxide and methane Gc and Gm, respectively. The porous medium skeleton and gas hydrate are incompressible, porosity is constant, methane and carbon dioxide are the calorically perfect gases. The system of main equations, which describes processes of filtration and heat-transfer in porous medium, consist of conservation laws of masses and energy, Darcy’s law and gas law. This system in the rectilinear-parallel case using foregoing assumptions in the each zone is following [13–15,19]:
@ @ ðq /Si Þ þ ðqi /Si ti Þ ¼ 0; ði ¼ c; mÞ @t i @x @ @T @ @T ðqCTÞ þ qi ci /Si ti ¼ k ; @t @x @x @x ki @p
d k/;
/Si ti ¼
where k is the reservoir permeability, / is the reservoir porosity. Then for the estimation of the time t1 can be obtained:
qi ¼ p=Rgi T:
2
t 1 k/=D1 : The characteristic time t2 of the displacement front attainment of the reservoir right edge can be estimated as:
t 2 l =vðpÞ ; 2
where v(p) is the pressure conductivity factor, l is the reservoir length.
li @x
ð2:2Þ
;
Here bottom indexes i = c, m correspond to the parameters of carbon dioxide and methane; p is the pressure; T is the temperature; / is the porosity; qi, ti, ki, Ci, Rgi and li are the true density, velocity, permeability, specific mass heat capacity, gas constant and the dynamic viscosity of the i-th phase, respectively; Si is the i-th phase saturation in pores; qC and k are the specific heat capacity per unit volume and coefficient of heat conductivity of the system.
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In the second equation of the system (2.2) the components related to barothermal effect are discarded, because in the problem the relatively small range of changes in pressure and temperature is considered. As only gas is being filtered, the main input into the values qC and k is introduced by the parameters of the rock skeleton of a porous medium. In fact, the assessed values show that if the initial hydrate saturation of the reservoir is equal to 0.5 and its porosity is equal to 0.1, then during the full transition of water into hydrate the specific heat capacity of the system will reduce by approximately 4%, and the thermal conductivity coefficient of the system will increase by 5%. That is why the values qC and k can be considered constant in the whole reservoir. The dependence of the coefficient of permeability ki for the ith gas phase from the gas saturation Si and absolute permeability k0 is defined by the formula of Kozeny [20]:
ki ¼ k0 S3i
ði ¼ c; mÞ:
ð2:3Þ
the piezoconductivity equation the term of equation responsible for the variability of temperature is negligible. With this in mind, on the basis of the system (2.2), the equations of piezoconductivity and thermal diffusivity can be written as:
@p2ðjÞ @t
where
carbon dioxide (i = c) and methane (i = m). xðnÞ is the velocity of the movement of phase transitions boundary. Hereinafter bottom index n corresponds to the parameters at the boundary between the zones. Taking into account the system of Eq. (2.4) and Darcy’s law, the conditions of mass and heat balance at the boundary between the zones can be written as follows:
kc @pð1Þ qhc Shc Gc þ Sc xðnÞ ; ¼/ lc @x qc km @pð2Þ
lm @x
ð2:5Þ
qhm x ; S G þS qm hm m m ðnÞ
¼/
kc pð1Þ k pð2Þ ðpÞ ðpÞ q0c C c kc q0m C m km vð1Þ ¼ l /ð1S , vð2Þ ¼ l m/ð1 mÞ, Peð1Þ ¼ 2klc pð1Þ , Peð2Þ ¼ 2klm pð2Þ , hc Þ c m
dn
2vðjÞ d dp2ðjÞ vðTÞ dn dn ðpÞ
2
dpðjÞ
¼
! ðj ¼ 1; 2Þ;
ð2:7Þ
2 dpðjÞ dT ðjÞ dT ðiÞ 2d dT ðjÞ ¼ þ 2PeðjÞ : dn dn dn dn dn
pressure conductivity factor
vðpÞ ðjÞ include the unknown function
p(j). To obtain approximate analytical solution we usedA the Leibenson linearization [21]. For this purpose the variable pressure p(j) in the parameter
vðpÞ ðjÞ assumed constant and equal to the initial
reservoir pressure p0. Note that this will not make a significant error in the solution, because in the work there is the case of small pressure gradients (Dp p0). Similarly the pressure p(j) at the parameter Pe(j) assumed constant and equal to the initial reservoir pressure p0. After the piezoconductivity and temperature conductivity equations integration (2.7), the solution for the pressure and temperature distributions in each zone can be written as:
p2ð1Þ
Rn 2 ðp2w p2ðnÞ Þ n ðnÞ exp 4gn dn ð1Þ ¼ p2ðnÞ þ ; R nðnÞ 2 exp 4gn dn 0
0 < n < nðnÞ ;
ð2:8Þ
ð1Þ
/ Shc qhc ð1 Gc Þ xðnÞ ¼ / Shm qhm ð1 Gm Þ xðnÞ ; @T ð1Þ @T ð2Þ k k ¼ /ðShc qhc Lhc Shm qhm Lhm Þ xðnÞ : @x @x
Here Lhi is the heat of formation of the hydrates of carbon dioxide (i = c) and methane (i = m); p(j) and T(j) are the pressure and temperature in the j-th zone, bottom indexes in parentheses j = 1, 2 correspond to the parameters of the first and second zones, respectively. The temperature and pressure at the boundary between the zones can be considered continuous:
pnð1Þ ¼ pnð2Þ ¼ pðnÞ ;
ðj ¼ 1; 2Þ;
Here the piezoconductivity equation is nonlinear, because of
where qhi and Shi are the density and saturation of the hydrates of
ð2:6Þ
pffiffiffiffiffiffiffiffiffiffi Let us introduce the self-similar variable: n ¼ x= vðTÞ t. For this variable the equations of piezoconductivity and temperature conductivity (2.6) can be written in the form:
n
ðj ¼ 1; 2Þ;
2.2. Self-similar solution
/ Sc qc ðtc x ðnÞ Þ ¼ / Shc qhc Gc x ðnÞ ;
!
vðTÞ ¼ qkC.
n
/ Sm qm ðtm x ðnÞ Þ ¼ / Shm qhm Gm xðnÞ ;
2 @ @pðjÞ @x @x
@p2ðjÞ @T ðjÞ @T ðjÞ @ @T ðjÞ þ vðTÞ PeðjÞ ¼ vðTÞ @x @x @t @x @x
As stated above in this problem the mixing of gases can be neglected and the front of displacement of methane with carbon dioxide can be considered stable. Then, with a glance of the replacement of CH4 in the methane hydrate with CO2, the conditions of mass balance of carbon dioxide and methane at the boundary between the zones are following:
ð2:4Þ
¼v
ðpÞ ðjÞ
T nð1Þ ¼ T nð2Þ ¼ T ðnÞ :
The methane hydrate saturation in the second zone Shm ¼ m. Then on the basis of the third equation of the system (2.5), representing the balance of mass of the bound in the hydrate water, it is possible to find the value of hydrate saturation in the first zone:
Shc ¼ qhm ð1 Gm Þm=qhc ð1 Gc Þ: In the examined problem the temperature difference DT = Tw T0 in the filtration area is small (DT T0). Therefore in
T ð1Þ
p2ð2Þ
2 Rn ðT w T ðnÞ Þ n ðnÞ exp n4 Peð1Þ p2ð1Þ dn 2 ¼ T ðnÞ þ ; R nðnÞ exp n4 Peð1Þ p2ð1Þ dn 0 R1 2 ðp2ðnÞ p20 Þ n exp 4gn dn ð2Þ ¼ p20 þ ; R1 n2 dn exp nðnÞ 4g
0 < n < nðnÞ ;
nðnÞ < n < 1;
ð2:9Þ
ð2Þ
T ð2Þ
2 R1 ðT ðnÞ T 0 Þ n exp n4 Peð2Þ p2ð2Þ dn 2 ¼ T0 þ ; R1 exp n4 Peð2Þ p2ð2Þ dn nðnÞ
nðnÞ < n < 1;
vðpÞ
ðjÞ where gðjÞ ¼ vðTÞ (j = 1, 2).On the basis of the conditions (2.5) and using the obtained solutions (2.8) and (2.9) we obtain the equations for determination of the coordinate of the displacement front and the values of the parameters p(n) and T(n) on this front:
2 ðp2w p2ðnÞ Þ exp 4gn ð1Þ ¼ Að1Þ pðnÞ nðnÞ ; R nðnÞ 2 exp 4gn dn 0 ð1Þ
ð2:10Þ
V.Sh. Shagapov et al. / International Journal of Heat and Mass Transfer 107 (2017) 347–357
2 n ðp2ðnÞ p20 Þ exp 4gðnÞ ð2Þ ¼ Að2Þ pðnÞ nðnÞ ; R1 n2 exp 4g dn nðnÞ
ð2:11Þ
ð2Þ
2 n ðT ðnÞ T w Þ exp ðnÞ Peð1Þ p2ðnÞ 4 2 R nðnÞ exp n4 Peð1Þ p2ð1Þ dn 0 2 n 2 ðT 0 T ðnÞ Þ exp ðnÞ Pe p ð2Þ ðnÞ 4 2 R1 ¼ BnðnÞ ; n 2 exp Pe p dn ð2Þ ð2Þ nðnÞ 4 where B ¼ /ðqhc Lhc S2hcqCqhm Lhm mÞ, /vðTÞ lm qhm Gm Shm þ1m . km q
Að1Þ ¼ /vkc lc ðTÞ
qhc Gc Shc q0c
ð2:12Þ þ 1 Shc ,
Að2Þ ¼
0m
The system of equations was solved as follows. Expressing from the Eq. (2.11) the value p(n) as a function of n(n) (by solving a quadratic equation) and substituting it into the Eq. (2.10), we obtain a transcendental equation with one unknown n(n). This equation is solved by the bisection method. Then, from the Eqs. (2.10) and (2.12) we determine the values of pressure p(n) and temperature T(n) at the boundary between the zones n(n). As previously mentioned, in this paper it is assumed that the diffusion and kinetic mechanisms do not limit the process of hydrate formation. Let us estimate the conditions at which the transition zone can be neglected and the thickness of which is limited by the diffusion of the gas in hydrate. In the pores of this zone there are particles of the hydrate methane and carbon dioxide hydrate. The thickness of the transition zone can be estimated by following condition:
Dx ¼ nðnÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi vðTÞ ðt þ t1 Þ vðTÞ t ; 2
where t 1 d =D1 is the is the characteristic times of the kinetics of the process, due to the diffusion of the gas in hydrate. pffiffiffiffiffiffiffiffiffiffi The length of the near zone is xðnÞ ¼ nðnÞ vðTÞ t. The thickness of the transition zone can be neglected if Dx << xðnÞ . Then we obtain following condition:
nðnÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi vðTÞ ðt þ t1 Þ vðTÞ t << nðnÞ vðTÞ t:
Then t >> t 1 =3. In particular, for the characteristic values of the sizes of porous channels d = 105 m, and the diffusion coefficient in a hydrate D1 = 1013 m2/s we can obtain that t1 103 s. I.e. the self-similar scheme with the clear front of phase transition can be applied if the characteristic times of the problem of the order of the day. Note that this estimate was obtained for the hydrate with solid structure. If the hydrate has a porous structure, then the characteristic time of the kinetics of the process and consequently the thickness of the transition zone will be even smaller. Then the process will come to self-similar regime with the clear front of phase transitions much earlier. 2.3. Calculation results When constructing the mathematical model it was assumed that in the first zone there is CO2 hydrate and in the second – CH4 hydrate. Therefore, the constructed model adequately describes the process if the local pressure in the first and second zones is higher than the local pressure of formation of the hydrates of carbon dioxide and methane, respectively. This pressure is determined from the formula (2.1) by using the temperature distri-
351
bution, which was found in the process of solving (the condition of thermodynamic consistency). The initial pressure and temperature of the reservoir correspond on the phase diagram to the zone of stable existence of methane hydrate (above the curve of phase equilibrium g2lh2 of the mixture ‘‘methane - water - methane hydrate”). Then, when the carbon dioxide injection with the pressure and temperature also located in this area the condition of thermodynamic consistency is obviously satisfied. Therefore, in this case, the mathematical model with the frontal boundary of replacement, in compliance with the conditions for permeability (paragraph 2.2), is adequately describes the process. If the pressure and temperature of the injected carbon dioxide are in the zone of the existence of methane and water (below the equilibrium curve g2lh2 of mixture ‘‘methane – water – methane hydrate”), then it is possible that the conditions at some part of the second zone correspond to the conditions of the CH4 hydrate decomposition to gas and water. Let us study the conditions at which the mathematical model with the frontal boundary of replacement is adequately describes the process for the case of the injection of carbon dioxide at the pressures and temperatures, at which there are conditions of methane hydrate decomposition to the gas and water. Fig. 2 shows the distributions of temperature and pressure for the different values of the injection pressure of carbon dioxide pw with the temperature, which satisfy to the condition of decomposition of CH4 hydrate to gas and water:
T w > T 0 þ T ln
pw ps0
ð2:13Þ
For the parameters characterizing the system, the following values are taken: / = 0.1, m = 0.2, p0 = 3 MPa, T0 = 274 K, Tw = 277 K, Gc = 0.28, Gm = 0,13, k0 = 1016 m2, k = 2 W/(mK), qC = 2.5106 J/ (Km3), qhc = 1100 kg/m3, qhm = 900 kg/m3, Cc = 800 J/(Kkg), Cm = 1560 J/(Kkg), Rgc = 189 J/(Kkg), Rgm = 520 J/(Kkg), Lhm = 4.5105 J/kg, Lhc = 4.1105 J/kg, lc = 1.4105 Pas, lm = 105 Pas, Tc⁄ = 7.6 K, Tm⁄ = 10 K, ps0c = 1.29 MPa, ps0m = 2.87 MPa [22,23,24]. Fig. 2 shows that at the high injection pressure of carbon dioxide (case a) the reservoir pressure (solid line) in the first zone is above the equilibrium pressure of the carbon dioxide hydrate formation (dashed line 1) and in the second zone is above the equilibrium pressure of the methane hydrate formation (dashed line 2). Therefore, in this case, the solution with the one boundary of phase transitions provides adequate mathematical description of the process. At the lower value of the injection pressure of carbon dioxide (case b) the reservoir pressure ahead of the front of phase transitions (i.e. in the second zone) at certain region falls below then the equilibrium pressure of the methane hydrate decomposition. This corresponds to the overheating of the heterogeneous mixture of methane and its hydrate at this region. Consequently, it is necessary to introduce the second boundary of phase transitions at which the decomposition of CH4 hydrate to methane and water is occurs. Accordingly, in this case, it is necessary to add intermediate zone, which partially saturated with free water. Let us study the critical conditions, which determine the exercise of the one or another regime. As already noted, the condition (2.13) is the necessary condition for the existence of the possibility of the CH4 hydrate decomposition to gas and water. The sufficient condition for the existence of the solution with the two boundaries of phase transitions is determined by the inequality:
pðnÞ < pðsÞ ; where p (n) is the pressure on the phase transitions boundary, p(s) is the equilibrium pressure of the methane hydrate decomposition,
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Fig. 2. The reservoir pressure and temperature and the equilibrium temperature depending on the self-similar coordinate n. pw = 3.5 (a) and 3.2 MGa (b). The dashed line 1 shows the equilibrium pressure of carbon dioxide hydrate formation and the dashed line 2 – the equilibrium pressure of methane hydrate decomposition.
which is determined by the Eq. (2.1). The pressure p(s) corresponds to the temperature T(n), which is determined by the Eq. (2.12). Based on this inequality the computational experiments in a wide range of parameters for determining the value of the injected gas pressure, above which there are no the decomposition of CH4 hydrate to gas and water pcr (call it the critical pressure), were performed. Fig. 3 shows the dependence of the critical pressure on the absolute permeability of the reservoir. It can be seen that the critical pressure is decreasing when the permeability is increasing and the faster the lower the initial reservoir pressure p0. Furthermore, figure shows that when the permeability is high then the value pcr is almost equal to the initial reservoir pressure p0. Thus, the regime with the one boundary of phase transitions is realized into the high permeability porous mediums, as well as in the cases when the initial reservoir pressure is low. Fig. 4 shows the critical injection pressure pcr depending on the initial hydrate saturation m at injection CO2 into the reservoir with absolute permeability k0 = 1015 m2. It is seen that with increasing the initial hydrate saturation the critical pressure is increasing and the faster the lower the porosity. Thus, the regime with the one phase transition boundary is typical for the reservoirs with the low porosity and low initial hydrate saturation. Figs. 5 and 6 show the dependences of the self-similar coordinate of the phase transitions boundary on the permeability and initial reservoir hydrate saturation. According to Fig. 5 with increasing the permeability the velocity of the movement of the
Fig. 3. The critical injection pressure pcr depending on the absolute reservoir permeability k0 at different values of initial reservoir pressure p0. Lines 1 and 2 correspond to p0 = 3 and 3.1 MPa.
phase transitions boundary is increasing and the faster the lower initial reservoir pressure p0. This is due to the fact that with increasing the permeability and pressure drop the filtration rate is increasing.
V.Sh. Shagapov et al. / International Journal of Heat and Mass Transfer 107 (2017) 347–357
Fig. 4. The critical injection pressure pcr depending on the initial hydrate saturation m at different values of reservoir porosity /. Lines 1 and 2 correspond to / = 0.1 and 0.2.
Fig. 6 shows that with increasing the hydrate saturation the velocity of the phase transitions boundary is decreasing and the faster the higher the porosity. This is due to the fact that with decreasing the phase permeability of gas and with increasing the porosity the piezoconductivity coefficient and, respectively, the intensity of the mass transfer in porous medium is decreasing. Thus with increasing the permeability k0, decreasing the initial reservoir pressure p0, porosity / and hydrate saturation m the rate of displacement front becomes higher than the rate of the temperature front. This reduces the risk of the methane hydrate overheating in the second zone. It should be noted, that the rate of the temperature front caused by convection, is always much less than the rate of the pressure front. This is due to the smallness of the specific volumetric heat capacity of gas qg C g compared with the specific volumetric heat capacity of the porous medium skeleton qC. Also this is due to the almost instantaneous heat exchange between the gas and the porous medium skeleton.
qg C g << qC:
353
Fig. 6. The self-similar coordinate of boundary n(n) depending on the initial hydrate saturation m at different values of reservoir porosity /. Lines 1 and 2 correspond to / = 0.1 b 0.2.
Also note that the rate of heat transfer, due to thermal conductivity, becomes comparable to the rate of mass transfer at the low values of permeability or low pressure drop. Therefore, the injected gas temperature, at the cases of the low medium permeability or low pressure drop, has a significant impact (by conduction) to the temperature in the second zone of the reservoir, making it possible, under certain conditions, the overheating of the hydrate in this zone, within the frontal model. In other words, the regime with the one boundary of phase transitions adequately describes the process even when the pressure and temperature of the carbon dioxide injection correspond to the conditions of the decomposition of methane hydrate (to gas and water) in the case when the phase transitions boundary (the law of motion which is limited by mass transfer in a porous medium) extends considerably farther than the heating zone of porous medium. In this case, the second zone does not have time to warm to the temperature values which correspond to the methane hydrate decomposition to gas and water even when the hot gas injection. 3. The regime with the decomposition of gas hydrate to gas and water 3.1. Problem statement and basic equations
Fig. 5. The self-similar coordinate of boundary n(n) depending on the absolute reservoir permeability k0 at different values of initial reservoir pressure p0. Lines 1 and 2 correspond to p0 = 3 and 3.1 MPa.
Let us consider the case of the carbon dioxide injection with the temperature, at which the regime with the two boundaries of phase transitions is realized (T > Tcr). This regime is characterized by the CH4 hydrate decomposition to gas and water and by the CO2 hydrate formation from water and carbon dioxide. In this case, three characteristic zones are formed. The pores of the first (near) zone are saturated with carbon dioxide and its hydrate, the porous medium of the second (intermediate) zone is saturated with methane and water, and the pores of the third (far) zone are saturated with methane and its hydrate. In this way, the methane hydrate decomposition is occurs at the movable frontal surface separating the second and third zones (the far boundary of phase transitions). And the carbon dioxide hydrate formation is occurs using free water and carbon dioxide at the movable frontal surface separating the first and second zones (the near boundary of phase transitions).
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Let us consider the case when the initial hydrate saturation value is low (not exceeding 0.2). In this case, water can be considered immovable owing to the low value of water saturation in the second zone. The conditions of the mass and energy balances on the boundary between the first and second zones can be written as:
kc @pð1Þ qhc Shc Gc þ Sc xðnÞ ; ¼/ lc @x qc
kmð2Þ @pð2Þ ¼ / Smð2Þ xðnÞ ; lm @x
kmð2Þ @pð2Þ kmð3Þ @pð3Þ qhm mG þ S S ¼/ lm @x lm @x qmðdÞ m mð3Þ mð2Þ
!
xðdÞ ;
ð3:2Þ
@T ð3Þ @T ð2Þ k ¼ / mqhm Lhm xðdÞ : @x @x
Here qm(j), Sm(j), p(j) and T(j) are the density, saturation, pressure and temperature of methane in the second (j = 2) and third zones
(j = 3), ql and Sl are the water density and saturation, xðdÞ the movement speed of the far boundary of phase transitions. The temperature and pressure at this boundary are continuous and associated by the condition of phase equilibrium:
T ðdÞ ¼ T 0 þ T lnðpðdÞ =ps0 Þ: Using the second Eq. (3.2) for the value of water saturation in the second zone can be obtained:
Slð2Þ ¼ mqhm ð1 Gm Þ=ql : The equations for piezoconductivity and thermal diffusivity can be written similar to the equations (2.5). The coefficients of these equations in the second and third zones can be presented in the form:
kmðjÞ p0
;
PeðjÞ ¼
q0m C m kmðjÞ ; kmðjÞ ¼ k0 S3mðjÞ ; ðj ¼ 2; 3Þ: 2k lm p0
3.2. Self-similar solution For the self-similar variable n the solution for the pressure and temperature distributions in the first zone coincides with (2.8). The equations for piezoconductivity and thermal diffusivity in the second and third zones can be written similar to the Eq. (2.7) when j = 2, 3. After integration the piezoconductivity and thermal diffusivity equations the solutions for the pressure and temperature distributions in the second and third zones can be written in the form:
p2ð2Þ
2 R1 ðT ðdÞ T 0 Þ n exp n4 Peð3Þ p2ð3Þ dn 2 ¼ T0 þ ; R1 exp n4 Peð3Þ p2ð3Þ dn n
nðnÞ < n < 1;
ðjÞ where gðjÞ ¼ vðTÞ (j = 2, 3). On the basis of the conditions (3.1) b (3.2), with consideration the solutions (3.3), (3.4) and (2.8), we get the equations for determination of coordinates of the phase-transitions boundaries n(n), n(d) and the values of parameters p(n), T(n), p(d) and T(d) at these boundaries:
2 ðp2w p2ðnÞ Þ exp 4gn ð1Þ ¼ Að1Þ pðnÞ nðnÞ ; R nðnÞ 2 exp 4gn dn 0
ð3:5Þ
ð1Þ
/ Slð2Þ ql xðdÞ ¼ /m qhm ð1 Gm ÞxðdÞ ;
lm / SmðjÞ
ð3:4Þ
vðpÞ
@T ð1Þ @T ð2Þ k ¼ / Shc qhc Lhc xðnÞ : @x @x
vðpÞ ðjÞ ¼
nðdÞ < n < 1;
ð3Þ
ðdÞ
The conditions of the mass and energy balances on the boundary between the second and third zones can be written as:
k
R1 2 ðp2ðdÞ p20 Þ n exp 4gn dn ð3Þ ¼ p20 þ ; R1 n2 exp dn n 4g ðdÞ
T ð3Þ
p2ð3Þ
ð3:1Þ
/ Shc qhc ð1 Gc Þ xðnÞ ¼ / Slð2Þ ql xðnÞ ; k
2 Rn ðT ðnÞ T d Þ n ðdÞ exp n4 Peð2Þ p2ð2Þ dn 2 T ð2Þ ¼ T ðdÞ þ ; nðnÞ < n < nðdÞ ; R nðdÞ n 2 nðnÞ exp 4 Peð2Þ pð2Þ dn
Rn 2 ðp2ðnÞ p2ðdÞ Þ n ðdÞ exp 4gn dn ð2Þ ¼ p2ðdÞ þ ; R nðdÞ n2 dn nðnÞ exp 4g
nðnÞ < n < nðdÞ ;
ð2Þ
ð3:3Þ
2 n ðp2ðnÞ p2ðdÞ Þ exp 4gðnÞ ð2Þ ¼ Að2Þ pðnÞ nðnÞ ; R nðdÞ n2 dn nðnÞ exp 4g
ð3:6Þ
ð2Þ
2 2 n n ðp2ðdÞ p20 Þ exp 4gðdÞ p2ðnÞ p2ðdÞ exp 4gðdÞ ð3Þ ð2Þ kmð3Þ R 1 kmð2Þ R nðdÞ n2 n2 exp 4g dn dn nðnÞ exp 4g nðdÞ ð3Þ
ð2Þ
¼ Að3Þ pðdÞ nðdÞ ;
ð3:7Þ
2 n ðT ðnÞ T w Þ exp ðnÞ Peð1Þ p2ðnÞ 4 2 R nðnÞ exp n4 Peð1Þ p2ð1Þ dn 0 2 n 2 ðT 0 T ðnÞ Þ exp ðnÞ Pe p ð2Þ ðnÞ 4 2 Rn ¼ Bð1Þ nðnÞ ; ðdÞ n 2 nðnÞ exp 4 Peð2Þ pð2Þ dn 2 n ðT 0 T ðdÞ Þ exp ðdÞ Peð3Þ p2ðdÞ 4 2 R1 exp n4 Peð3Þ p2ð3Þ dn nðdÞ 2 n ðT ðdÞ T ðnÞ Þ exp ðdÞ Peð2Þ p2ðdÞ 4 2 ¼ Bð2Þ nðdÞ ; R nðdÞ n 2 nðnÞ exp 4 Peð2Þ pð2Þ dn T ðdÞ ¼ T 0 þ T lnðpðdÞ =ps0 Þ;
ð3:8Þ
ð3:9Þ
ð3:10Þ
ðTÞ m Þm , ¼ / vkmð2Þlm 1 qhm ð1G q
where Að1Þ ¼ /vkc lc qhcqGc Shc þ 1 Shc , Að2Þ 0c l Gm qhm ð1Gm Þ / qhc Lhc Shc / qhm Lhm m þ 1 , B ¼ , B ¼ . Að3Þ ¼ / vðTÞ lm m qhm ð1Þ ð2Þ q q 2q C 2q C ðTÞ
0m
l
The system of the boundary Eqs. (3.5)–(3.10) is solved as follows. Firstly, we set zeroth approximation of the unknown quantities at the first (near) phase transitions boundary n(n). At this paper this approximation is the parameters at the left boundary of a reservoir. Secondly, from the Eq. (3.7) we obtain the expression for the pressure p(d) (the root of quadratic equation). Thirdly, substituting this expression in the (3.10), we obtain the expression for the temperature T(d). Fourthly, substituting the temperature expression in the
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Eq. (3.9), we obtain the transcendental equation in the one unknown n(d). Fifthly, we get the value n(d) by solving the transcendental equation by bisection method. Sixthly, we get the values p(d) and T(d). Seventh, from the Eq. (3.6) we obtain the expression for the pressure p(n) (the root of quadratic equation). Eighth, substituting the expression for p(n) in the (3.5), we obtain the transcendental equation in the one unknown n(n). Ninthly, we get the new approximate value of the first boundary n(n). In-tenths, we get the new approximate values T(n) (Eq. (3.8)) and p(n). As a result of cyclic repetition of described iterative procedure we obtain the sequence of approximate values which converges to the desired values of the parameters at the phase-transition boundaries. 3.3. Calculation results Fig. 7. presents the reservoir temperature and pressure at the injected dioxide carbon temperature Tw = 281 K. The values of other system parameters are equal to the values for Fig. 2. Fig. 7 shows that the thermodynamic conditions in the second zone correspond to the conditions of existence of a heterogeneous mixture
Fig. 7. The reservoir temperature and pressure depending on the self-similar coordinate n. Dashed line show the equilibrium pressure of methane hydrate decomposition.
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of methane and water, which accords with the assumption of the absence of methane hydrate in the second zone, which was made in the problem statement. The pressure conductivity factor for real reservoirs is several orders more than the thermal diffusivity. Therefore, the pressure from the far frontal boundary to the near frontal boundary always increases slowly than the equilibrium temperature and the pressure associated with it. This eliminates thermodynamic contradiction, when the reservoir pressure in the second zone is higher than the equilibrium pressure. Accordingly, methane hydrate in the second zone is absent. Fig. 8 presents that the coordinate of the near boundary of phase transitions is increasing when the injected gas pressure is increasing and the initial reservoir pressure is decreasing. This is because the rate of methane displacement with carbon dioxide is limited by mass transport in a porous medium, the intensity of which is increasing when the pressure drop (Dp = pw – p0) is increasing. Also, Fig. 8 shows that the rate of the far boundary of phase transitions decreases with increasing the initial reservoir pressure and little affected by the gas injection pressure. This is due to the fact that the equilibrium temperature of the decomposition of methane hydrate is increasing when the injected gas pressure or the initial reservoir pressure are increasing, that slows the process of the dissociation of methane hydrate to gas and water. On the other hand with an increase in the pressure drop the formation intensity of carbon dioxide hydrate is increasing and thus the heat, which emitted at the near boundary, is increasing. It promotes the growth of the decomposition rate of CH4 hydrate to gas and water when the injected gas pressure is increasing, and this rate is decreasing when the initial reservoir pressure is increasing. Thus, when the injected gas pressure is increasing, these two factors cancel each other, so the intensity of methane hydrate decomposition to gas and the water very little depends on the pressure of injected gas. And with an increase of the initial pressure these factors contribute to the slowdown of the CH4 hydrate dissociation to gas and water. Increasing the injected gas pressure in the range of parameters corresponding to the mode with two phase transition boundaries is not effective way to increase the intensity of the CH4 hydrate decomposition to gas and water. Also, Fig. 8 shows that the rate of near boundary tends to zero when the pressure drop value tends to zero. This case corresponds to the heating of a reservoir without the carbon dioxide injection and hence without the formation of carbon dioxide hydrate. At this regime (Dp = 0), the rate of far boundary in the case a (pw = p0 = 3 MPa) almost one and a half times more than in the case b (pw = p0 = 3.5 MPa). This is because the pressure and respectively the equilibrium temperature of the hydrate decomposition in the first case are lower than in the second case. I.e. the length of region, where the temperature causes decomposition of methane hydrate, in the first case is more than in the second case. In the considered problem one of the main factors determining the intensity of mass transfer is the gas relative permeability, which, according to the equation (2.3), is increasing when the absolute reservoir permeability is increasing and is decreasing when the reservoir hydrate saturation is increasing. Fig. 9 shows that with increasing the gas permeability (i.e., with increasing the absolute permeability and decreasing the initial reservoir hydrate saturation) the coordinate values of both boundaries of phase transitions are increasing. This is due to increasing the rate of the displacement front of methane by carbon dioxide and this is due to corresponding increasing the intensity of the carbon dioxide hydrate formation with increasing the gas permeability. Since the process of the carbon dioxide hydrate formation is accompanied by the release of heat, the motion speed of the far boundary (although only slightly) also is increasing when the permeability is increasing. Furthermore, Fig. 9 shows that at high values of gas
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Fig. 8. The self-similar coordinates of the phase-transition boundaries n(n) (line 1) b n(d) (line 2) depending on the injection pressure pw (a) and initial pressure p0 (b). Tw = 280 K, p0 = 3 MPa (a), pw = 3.5 MPa (b). Dashed line corresponds to the coordinate of the far boundary, which was calculated without consideration of the CO2 hydrate formation heat that is released on the near boundary (Lhc = 0).
Fig. 9. The self-similar coordinates of the phase-transition boundaries n(n) (line 1) b n(d) (line 2) depending on the absolute reservoir permeability k0 (a) at m = 0.2 and depending on the initial hydrate saturation m (b) at k0 = 1016 m2.
permeability the intermediate area degenerates into a frontal surface (i.e. the infinitely small thickness area) that corresponds to the regime when the replacement of methane in hydrate with carbon dioxide without release free water is occurs (Section 2). This is entirely consistent with the results of Section 2 and this is due to the following. The movement speed of the near boundary is limited by the mass transport and the propagation speed of the temperature front is mainly limited by the thermal conductivity. The movement speed of the near boundary becomes more than the propagation speed of the temperature front when the gas permeability is increasing due to the absolute reservoir permeability is increasing and the initial reservoir hydrate saturation is decreasing. This prevents heating of the area saturated with methane hydrate to temperatures causing its decomposition. Sources of heat, which is necessary for the methane hydrate decomposition to gas and water, are the latent heat of CO2 hydrate formation and the heat flux from the left border. So, the dissociation of methane hydrate to gas and water can occur even without the injection of carbon dioxide. However, it should be noted that the heat, which generated during the formation of CO2 hydrate (approximately 400 kJ per 1 kg of hydrate), can increase the rate of the methane hydrate decomposition. Fig. 10 shows the calcula-
Fig. 10. The parameter N depending on the injected gas temperature Tw at different values of the injected gas pressure pw. Lines 1, 2 and 3 correspond to pw = 3.1, 3.2 and 3.3 MPa.
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tions, which allow assessing the contribution of latent heat of the CO2 hydrate formation in the CH4 hydrate decomposition rate. Here N is the ratio of values of the coordinates of the boundaries between the second and third zones, calculated taking into account the generated heat (Lhc = 4.1 105 J/kg) and without it (Lhc = 0) at the boundary between the first and second zones. This figure shows that the contribution of the heat, which generated during the carbon dioxide hydrate formation, is particularly noticeable when: 1) increasing the gas injection pressure, 2) low temperature of the injected gas. The rate of the CO2 hydrate formation boundary is increasing when the carbon dioxide injection pressure is increasing (Fig. 8). Therefore, the intensity of the heat generation at this boundary is increasing. The intensity of the heat flux from the left boundary is decreasing when the injected gas temperature is decreasing. Therefore, the relative contribution of the generated heat at the boundary of the CO2 hydrate formation becomes more prominent. 4. Conclusions The mathematical model of the warm carbon dioxide injection into a natural reservoir saturated with methane and its hydrate was constructed. It was determined that the methane recovery from hydrate at such influence can occur in two regimes. At the first regime the displacement of methane with carbon dioxide in the CH4 hydrate is occurs. This regime is typical for porous mediums with high permeability and when the injected gas pressure is high as well as for porous mediums with low porosity, low hydrate saturation and low initial pressure. In the second regime the decomposition of CH4 hydrate to gas and water with the subsequent formation of the CO2 hydrate from water and carbon dioxide is occurs. At this regime, the heat, which released at the formation of carbon dioxide hydrate, has a significant impact on the decomposition rate of methane hydrate to gas and water. Acknowledgement This work was supported by Grant of the RFBR 17-51-540001 and VAST.HTQT/17-18 ‘‘Wave and filtration flow with phase transitions in porous structures”. References [1] Y.F. Makogon, Hydrates of Hydrocarbons, PennWell Publishing Company, Tulsa, OK, USA, 1997. [2] Y.F. Makogon, Natural gas hydrates – A promising source of energy, J. Nat. Gas Sci. Eng. 2 (1) (2010) 49–59.
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[3] Y.F. Makogon, S.A. Holditch, T.Y. Makogon, Natural gas-hydrates – A potential energy source for the 21st Century, J. Petrol. Sci. Eng. 56 (2007) 14–31. [4] Energy from Gas Hydrates: Assessing the Opportunities and Challenges for Canada, Report of the Expert Panel on Gas Hydrates, Council of Canadian Academies, Ottawa, ON Canada, 2008. [5] J.W. Jung, J.C. Santamarina, CH4-CO2 replacement in hydrate-bearing sediments: a pore-scale study, Geochem. Geophys. Geosyst. 11 (2010) Q0AA13, http://dx.doi.org/10.1029/2010GC003339. [6] Qing Yuan, Chang-Yu Sun, Bei Liu, Xue Wang, Zheng-Wei Ma, Qing-Lan Ma, Lan-Ying Yang, Guang-Jin Chen, Qing-Ping Li, Shi Li, Ke Zhang, Methane recovery from natural gas hydrate in porous sediment using pressurized liquid CO2, Energy Convers. Manage. 67 (2013) 257–264. [7] A. Nago, A. Nieto, Natural gas production from methane hydrate deposits using CO2 clathrate sequestration: state-of-art review and new technical approaches, J. Geol. Res. 2011 (2011), http://dx.doi.org/10.1155/2011/ 239397, Article ID 239397. [8] D.N. Espinoza, J.C. Santamarina, P-wave monitoring of hydrate-bearing sand during CH4–CO2 replacement, Int. J. Greenhouse Gas Control 5 (4) (2011) 1032–1038. [9] M. Ota, K. Morohashi, Y. Abe, M. Watanabe, R.L. Smith Jr., H. Inomata, Replacement of CH4 in the hydrate by use of liquid CO2, Energy Conversation Manage. 46 (11–12) (2005) 1680–1691. [10] K. Ohgaki, K. Takano, H. Sangawa, H. Sangawa, T. Matsubara, S. Nakano, Methane exploitation by carbon dioxide from gas hydrates – phase equilibria for C02–CH4 mixed hydrate system, J. Chem. Eng. Jpn. 29 (3) (1996) 478–483. [11] S. Hirohama, Y. Shimoyama, A. Wakabayashi, S. Tatsuta, N. Nishida, Conversion of CH4-hydrate to CO2-hydrate in liquid CO2, J. Chem. Eng. Jpn. 29 (6) (1996) 1014–1020. [12] V.Sh. Shagapov, M.K. Khasanov, N.G. Musakaev, Formation of a gas hydrate due to injection of a cold gas into a porous reservoir partly saturated by water, J. Appl. Mech. Tech. Phys. 49 (3) (2008) 462–472. [13] M.K. Khasanov, Investigation of regimes of gas hydrate formation in a porous medium, partially saturated with ice, Thermophys. Aeromech. 22 (2) (2015) 255–265. [14] V.Sh. Shagapov, N.G. Musakaev, M.K. Khasanov, Formation of gas hydrates in a porous medium during an injection of cold gas, Int. J. Heat Mass Transfer 84 (2015) 1030–1039. [15] G.G. Tsypkin, Formation of carbon dioxide hydrate at the injection of carbon dioxide into a depleted hydrocarbon field, Fluid Dyn. 49 (6) (2014) 789–795. [16] V.A. Istomin, V.S. Yakushev, Gas Hydrates under Natural Conditions (in Russian), Nedra, Moscow, 1992. [17] S.Sh. Byk, Y.F. Makogon, V.I. Fomina, Gas Hydrates (in Russian), Khimiya, Moscow, 1980. [18] R.E. Collins, Flow of Fluids Through Porous Materials, Reinhold Publishing Corp, New York, 1961. [19] R.I. Nigmatulin, Dynamics of Multiphase Media, Hemisphere Publ. Corp, New York, 1991. [20] G.I. Barenblatt, V.M. Entov, V.M. Ryzshik, Theory of Fluid Flows Through Natural Rocks, Kluwer Academic Publishers, Dordrecht, Boston, 1990. [21] K.S. Basniev, I.N. Kochina, V.M. Maksimov, Underground Hydrodynamics (in Russian), Nedra, Moscow, 1993. [22] N.G. Musakaev, M.K. Khasanov, Theoretical research of the carbon dioxide injection process into the rock saturated with ice, J. Phys.: Conf. Ser. 754 (2016) 032015. [23] G.K. Anderson, Enthalpy of dissociation and hydration number of carbon dioxide hydrate from the Clapeyron equation, J. Chem. Thermodyn. 35 (2003) 1171–1183. [24] I.K. Gimaltdinov, S.R. Kildibaeva, About the theory of initial stage of oil accumulation in a dome-separator, Thermophys. Aeromech. 22 (3) (2015) 387–392.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Theoretical research of the gas hydrate deposits development using the injection of carbon dioxide V.Sh. Shagapov a, M.K. Khasanov b,⇑, N.G. Musakaev c, Ngoc Hai Duong d a
Institute of Mechanics and Engineering of Kazan Science Center RAS, Kazan, Russia Sterlitamak Branch of Bashkir State University, Sterlitamak, Russia c Tyumen Branch of Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Tyumen State University, Tyumen, Russia d Graduate University of Science and Technology VAST, Hanoi, Viet Nam b
a r t i c l e
i n f o
Article history: Received 15 June 2016 Received in revised form 7 November 2016 Accepted 10 November 2016
Keywords: Gas hydrates Porous medium Methane Carbon dioxide Replacement Decomposition
a b s t r a c t In this paper the mathematical model of the carbon dioxide injection into a natural reservoir initially saturated with methane and its hydrate was constructed and also the research, using this model, was carried out. Self-similar solutions of this problem in the one-dimensional formulation were built. These solutions explain the distribution of the main parameters in a reservoir. It is shown that there are two possible regimes: the first – the recovery of methane from hydrate can occur without decomposition of CH4 hydrate to gas and water, the second – with decomposition. In the first regime, the replacement of methane in hydrate with carbon dioxide is occurs. In the second regime the decomposition of CH4 hydrate to gas and water with the subsequent formation of hydrate from water and carbon dioxide is occurs. The critical diagrams of the each regime existence were built and analyzed. The influence of the injection pressure of carbon dioxide and the reservoir permeability to the speed of methane recovery from hydrate for the each regime was researched. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Currently the natural deposits of gas hydrates due to significant resources and concentrated gas state are the serious alternative to conventional resources of natural gas [1–4]. Currently the main methods of the extraction of hydrocarbon gas (mostly methane) from a hydrate reservoir are the pressure drop and the heating of hydrate saturated rocks [1–4]. However, in some cases, these methods are inefficient because of large expenditure of energy. For example, when the pressure drop method of the gas hydrate deposits development, it is necessary that the reservoir pressure was below than the equilibrium decomposition pressure of hydrate [1–4]. This is achieved also due to the pumping-out of water from wells and this requires a large expenditure of energy. In addition, when the wells exploitation, the temperature decreasing due to the absorption of latent heat of hydrate decomposition, as well as adiabatic cooling and throttle effect. So the water which becomes free after the hydrate decomposition can freeze and plug the equipment [4].
⇑ Corresponding author. E-mail addresses: [email protected] (V.Sh. Shagapov), [email protected] (M.K. Khasanov), [email protected] (N.G. Musakaev), [email protected] (N.H. Duong). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.11.034 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
When the thermal method of the gas hydrate deposits development the reservoir temperature increasing which necessary for the gas hydrate decomposition is achieved mainly by hot water circulation in a closed loop in a well or by injection into the reservoir of hot fluid (water or steam) [4]. However, this method also requires a large expenditure of energy for the hydrates decomposition and for the heating of the reservoir to the temperature that causes decomposition. In connection with the foregoing, the actual problem is to research new methods of influence on gas hydrate reservoirs, which minimize expenditure of energy for the gas extraction. Promising in this respect is the technology of replacement, which consists in the displacement of methane from hydrates by filling their by other gas [5–11]. By several researches the most promising gas for the replacement process is carbon dioxide. Also its pumping into the hydrate saturated reservoir would at the same time to solve the problem of disposal of CO2, in connection with its negative role in the development of the greenhouse effect. This method is based on the fact that the hydrate of carbon dioxide is more stable than methane hydrate. Therefore the molecules of carbon dioxide can to replace the molecules of methane in the methane hydrate. Experimental researches proved that the replacement process CH4-CO2 in the hydrate of methane does not require a supply of external energy, because the latent heat per unit volume of
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the CO2 hydrate formation (from water and carbon dioxide) more than the latent heat of the CH4 hydrate decomposition (to water and methane). In addition, a significant feature of the replacement process CH4-CO2 in the methane hydrate is that this reaction occurs at pressures and temperatures corresponding to the conditions of stable existence of the methane hydrate [5,6] and not accompanied by the release of free water [5–8]. This eliminates the necessity to heat a reservoir to the temperatures causing the decomposition of methane hydrate to gas and water. These facts make this technology promising owing to the low energy costs. Technological ideas must be confirmed by the calculations based on valid theoretical models. Therefore the construction of the mathematical model of the dioxide carbon injection into the gashydrate reservoir is the actual problem. Experimental researches of the replacement process of methane in hydrate with carbon dioxide are described, for example, in [5– 11]. In these works, the researches were conducted in samples of small size and, as a rule, in the thermostatic and barostatic conditions. The replacement process in these researches is limited by the process kinetics due to the small sizes of the samples and the maintenance of constant temperature and pressure conditions. In the cases of long natural reservoirs at constant injection of carbon dioxide into these reservoirs the process of replacing methane with carbon dioxide in the hydrate will be limited by the mass transfer in a porous medium. Therefore, these researches do not give a complete picture of the processes that take place into natural reservoirs. This significantly complicates the comparison of the experimental data with the results of mathematical modeling of the processes for long reservoirs. The mathematical models of the gas hydrate formation in extended porous mediums during the gas injection are formulated, in particular, in works [12–14], in which the process of the gas injection into the reservoir initially saturated with the same gas is researched. The mathematical model for the carbon dioxide injection into a reservoir, which contains methane and water in a free state, is presented in [15]. The mathematical model of the carbon dioxide injection into the natural reservoir saturated with methane and its hydrate is presented in this work. Note that some authors proposed to pump CO2 into porous medium in the liquid state [6,8,9,11]. However, there are significant differences at the mathematical descriptions of the injection process of carbon dioxide in the gaseous or liquid state. It is known that the equilibrium temperature of the hydrate decomposition depends on pressure. Therefore, the possibility of the realization of the different regimes is determined by the overlay of the temperature and pressure fields. There are different equations of state when the different aggregate state of carbon dioxide. Also, difference in the coefficients of viscosity more than an order of magnitude. All this leads to substantial difference of pressure fields for the cases of the carbon dioxide injection in the liquid or gaseous state. This has a significant impact to the possibility of the realization of the different regimes. Let us explain this in more detail. In the case of the liquid CO2 injection the main hydraulic resistance has only the first reservoir zone, which is saturated with the liquid. In the case of the gaseous carbon dioxide injection the hydraulic resistance is uniformly by the all filtering area, because this area is saturated with gas. This is due, firstly, with the different equations of state for the liquid and gas, and, secondly, with a very significant difference in viscosity for the liquid and gas. Therefore, under the same conditions in the case of the CO2 injection in the gaseous state the pressure drop in the near area is significantly lower than in the case of liquid CO2. In this connection, in the case of the gaseous CO2 injection the dependence of pressure before the front (i.e., in the far area) on the injection pressure is much more significant than in the case of the liquid CO2 injection. The realiza-
tion of the different regimes is defined by the equilibrium temperature in the far area, i.e. in fact by the pressure in the far area. When pumping liquid CO2 a greater impact has the temperature of the injected CO2 but not the injection pressure. In the case of the gaseous CO2 injection a greater impact to the features of the process (in terms of the control of the process regimes) has the injection pressure. Therefore, in this work the effect of the pressures (the initial reservoir pressure and the injected fluid pressure) to the process features is researched. Also the concept of the critical pressure of the injected CO2 was introduced and the value of this pressure was examined. In addition, an essential feature of the heat and mass transfer processes with phase transitions is the release or absorption of the latent heat of phase transitions. When the injection of gaseous CO2 the CO2ACH4 replacement process in methane hydrate is exothermic. This process is endothermic when the injection of liquid CO2. This circumstance is taken into account in this work. Therefore, when the gaseous CO2 injection the regime with the decomposition of CH4 hydrate to gas and water can be realized even when the injected gas temperature is equal to the initial reservoir temperature. At the liquid CO2 injection this regime can be realized only when the liquid CO2 temperature is greater than the initial reservoir temperature. Therefore, the replacement regime while injecting liquid CO2 more strongly depends on the temperature of the injected carbon dioxide than when the gaseous CO2 injection. 2. The regime with the replacement of methane with carbon dioxide in hydrate 2.1. Problem statement and basic equations Assume that the horizontal porous reservoir, initially saturated with methane hydrate (initial saturation m) and methane, fills halfspace x > 0. The top and bottom of the reservoir are impermeable. Initial pressure p0 and temperature T0 correspond to the thermodynamic conditions of the existence of mixture of methane and its hydrate:
t ¼ 0;
x>0:
T ¼ T0;
p ¼ p0 ;
Sh ¼ m :
Assume that through the left boundary (x = 0) gaseous carbon dioxide is being injected. The pressure of injection pw and temperature of this gas Tw correspond to the conditions of the existence of heterogeneous mixture of carbon dioxide and its hydrate and are supported at this boundary at constant level:
t > 0;
x¼0:
T ¼ Tw;
p ¼ pw :
Conditions of the existence of hydrates of carbon dioxide and methane are presented on the phase diagram (Fig. 1) [16]. In this chart the curves glh determine three-phase equilibrium ‘‘gas-wat er-hydrate”, and the curve lg – two-phase equilibrium ‘‘liquidvapor”. The subscript 1 corresponds to CO2 in gaseous or liquid state or its hydrate, and the subscript 2 – methane and its hydrate. In this problem the initial values of pressure and temperature of the reservoir are above the curve g2lh2 (in the field of the existence of heterogeneous mixture of methane and its hydrate). The pressure and temperature of the injected carbon dioxide are above the curve g1lh1 (in the field of the existence of carbon dioxide hydrate), but below the curve l1g1 (in the field of the existence of gaseous carbon dioxide). The values of temperature and pressure on the curve of threephase equilibrium ‘‘gas-water-hydrate” are quite well described by the following equation [17]:
p ¼ ps0 exp
T T0 ; T
ð2:1Þ
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The reservoir pressure conductivity factor is equal:
kp
vðpÞ ¼
l/
;
where p is the pressure, l is the average viscosity of gas phase. Then for the estimation of the time t2 can be obtained:
t2
l
l/ : kp
2
If t1 << t2 then the kinetics of process can be neglected. Then:
k/ l l/ : << D1 kp 2
Whence it follows that:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi k << l lD1 =p:
Fig. 1. Phase diagrams of the systems ‘‘CO2AH20” and ‘‘CH4AH20”.
where ps0 is the equilibrium pressure corresponding to the initial temperature T0, T⁄ is the is the empirical parameter, which depends on the type of gas hydrate. Flows in natural porous reservoirs can be considered laminar. The injection process is much faster compared to the diffusion of gases, i.e. the rate of mass transfer due to filtration significantly exceeds the rate of mass transfer due to diffusion. In the considered range of pressures and temperatures, the viscosity of carbon dioxide is about one and a half times higher than the viscosity of methane. Given these factors, we can neglect the mixing of the gases, and the front of displacement of methane with carbon dioxide can be considered stable. Therefore we can assume that, at the injection of CO2, two characteristic zones are formed. In the first (near) zone the pores are saturated with carbon dioxide and its hydrate, and in the second (far) zone the pores are saturated with methane and its hydrate. Thus, given these assumptions, the replacement of methane in hydrate with carbon dioxide is occurs on the movable front surface separating these two zones. In this mathematic model, the nonequilibrium processes connected with the diffusion of the gas which forms hydrate through the hydrate film are not taken into account. Let’s estimate the typical relaxation times of the diffusion nonequilibrium under the assumption that the kinetic mechanism of hydrate formation is related with diffusion. The characteristic relaxation time t1 of the diffusion nonequilibrium can be estimated as: 2
t 1 d =D1 ; where D1 is the diffusion coefficient in a hydrate, d is the typical thickness of the hydrate film (approximately equal to the pores size). The characteristic pores size can be estimated as:
Assume that the reservoir length is quite large, for example l = 100 m. Then for the values of parameters p = 1 MPa, l = 105 Pas, D1 = 1013 m2/s the kinetics can be neglected if the permeability satisfies the condition:
k << 1010 m2 : This condition is satisfied for the most natural reservoirs. Therefore, the characteristic time of the kinetics of the process will be much shorter than the characteristic time of the process of displacement (for the specified length of a reservoir). Consequently in long natural reservoirs the kinetics of the process, associated with diffusion in hydrated films at the level of individual pores, can be neglected. In [18] it is shown that the filtration mass transfer is much greater than the diffusion mass transfer at the displacement of miscible fluids at the natural porous systems. Therefore we will neglect the diffusion mixing of gases in considered problem. Let us assume the following assumptions to describe the mass transfer processes during CO2 injection into the porous reservoir. The process is one-temperature, i.e. the temperatures of a porous medium and of a saturant are similar. Hydrates of CO2 and CH4 are the two-component systems with the mass concentrations of carbon dioxide and methane Gc and Gm, respectively. The porous medium skeleton and gas hydrate are incompressible, porosity is constant, methane and carbon dioxide are the calorically perfect gases. The system of main equations, which describes processes of filtration and heat-transfer in porous medium, consist of conservation laws of masses and energy, Darcy’s law and gas law. This system in the rectilinear-parallel case using foregoing assumptions in the each zone is following [13–15,19]:
@ @ ðq /Si Þ þ ðqi /Si ti Þ ¼ 0; ði ¼ c; mÞ @t i @x @ @T @ @T ðqCTÞ þ qi ci /Si ti ¼ k ; @t @x @x @x ki @p
d k/;
/Si ti ¼
where k is the reservoir permeability, / is the reservoir porosity. Then for the estimation of the time t1 can be obtained:
qi ¼ p=Rgi T:
2
t 1 k/=D1 : The characteristic time t2 of the displacement front attainment of the reservoir right edge can be estimated as:
t 2 l =vðpÞ ; 2
where v(p) is the pressure conductivity factor, l is the reservoir length.
li @x
ð2:2Þ
;
Here bottom indexes i = c, m correspond to the parameters of carbon dioxide and methane; p is the pressure; T is the temperature; / is the porosity; qi, ti, ki, Ci, Rgi and li are the true density, velocity, permeability, specific mass heat capacity, gas constant and the dynamic viscosity of the i-th phase, respectively; Si is the i-th phase saturation in pores; qC and k are the specific heat capacity per unit volume and coefficient of heat conductivity of the system.
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In the second equation of the system (2.2) the components related to barothermal effect are discarded, because in the problem the relatively small range of changes in pressure and temperature is considered. As only gas is being filtered, the main input into the values qC and k is introduced by the parameters of the rock skeleton of a porous medium. In fact, the assessed values show that if the initial hydrate saturation of the reservoir is equal to 0.5 and its porosity is equal to 0.1, then during the full transition of water into hydrate the specific heat capacity of the system will reduce by approximately 4%, and the thermal conductivity coefficient of the system will increase by 5%. That is why the values qC and k can be considered constant in the whole reservoir. The dependence of the coefficient of permeability ki for the ith gas phase from the gas saturation Si and absolute permeability k0 is defined by the formula of Kozeny [20]:
ki ¼ k0 S3i
ði ¼ c; mÞ:
ð2:3Þ
the piezoconductivity equation the term of equation responsible for the variability of temperature is negligible. With this in mind, on the basis of the system (2.2), the equations of piezoconductivity and thermal diffusivity can be written as:
@p2ðjÞ @t
where
carbon dioxide (i = c) and methane (i = m). xðnÞ is the velocity of the movement of phase transitions boundary. Hereinafter bottom index n corresponds to the parameters at the boundary between the zones. Taking into account the system of Eq. (2.4) and Darcy’s law, the conditions of mass and heat balance at the boundary between the zones can be written as follows:
kc @pð1Þ qhc Shc Gc þ Sc xðnÞ ; ¼/ lc @x qc km @pð2Þ
lm @x
ð2:5Þ
qhm x ; S G þS qm hm m m ðnÞ
¼/
kc pð1Þ k pð2Þ ðpÞ ðpÞ q0c C c kc q0m C m km vð1Þ ¼ l /ð1S , vð2Þ ¼ l m/ð1 mÞ, Peð1Þ ¼ 2klc pð1Þ , Peð2Þ ¼ 2klm pð2Þ , hc Þ c m
dn
2vðjÞ d dp2ðjÞ vðTÞ dn dn ðpÞ
2
dpðjÞ
¼
! ðj ¼ 1; 2Þ;
ð2:7Þ
2 dpðjÞ dT ðjÞ dT ðiÞ 2d dT ðjÞ ¼ þ 2PeðjÞ : dn dn dn dn dn
pressure conductivity factor
vðpÞ ðjÞ include the unknown function
p(j). To obtain approximate analytical solution we usedA the Leibenson linearization [21]. For this purpose the variable pressure p(j) in the parameter
vðpÞ ðjÞ assumed constant and equal to the initial
reservoir pressure p0. Note that this will not make a significant error in the solution, because in the work there is the case of small pressure gradients (Dp p0). Similarly the pressure p(j) at the parameter Pe(j) assumed constant and equal to the initial reservoir pressure p0. After the piezoconductivity and temperature conductivity equations integration (2.7), the solution for the pressure and temperature distributions in each zone can be written as:
p2ð1Þ
Rn 2 ðp2w p2ðnÞ Þ n ðnÞ exp 4gn dn ð1Þ ¼ p2ðnÞ þ ; R nðnÞ 2 exp 4gn dn 0
0 < n < nðnÞ ;
ð2:8Þ
ð1Þ
/ Shc qhc ð1 Gc Þ xðnÞ ¼ / Shm qhm ð1 Gm Þ xðnÞ ; @T ð1Þ @T ð2Þ k k ¼ /ðShc qhc Lhc Shm qhm Lhm Þ xðnÞ : @x @x
Here Lhi is the heat of formation of the hydrates of carbon dioxide (i = c) and methane (i = m); p(j) and T(j) are the pressure and temperature in the j-th zone, bottom indexes in parentheses j = 1, 2 correspond to the parameters of the first and second zones, respectively. The temperature and pressure at the boundary between the zones can be considered continuous:
pnð1Þ ¼ pnð2Þ ¼ pðnÞ ;
ðj ¼ 1; 2Þ;
Here the piezoconductivity equation is nonlinear, because of
where qhi and Shi are the density and saturation of the hydrates of
ð2:6Þ
pffiffiffiffiffiffiffiffiffiffi Let us introduce the self-similar variable: n ¼ x= vðTÞ t. For this variable the equations of piezoconductivity and temperature conductivity (2.6) can be written in the form:
n
ðj ¼ 1; 2Þ;
2.2. Self-similar solution
/ Sc qc ðtc x ðnÞ Þ ¼ / Shc qhc Gc x ðnÞ ;
!
vðTÞ ¼ qkC.
n
/ Sm qm ðtm x ðnÞ Þ ¼ / Shm qhm Gm xðnÞ ;
2 @ @pðjÞ @x @x
@p2ðjÞ @T ðjÞ @T ðjÞ @ @T ðjÞ þ vðTÞ PeðjÞ ¼ vðTÞ @x @x @t @x @x
As stated above in this problem the mixing of gases can be neglected and the front of displacement of methane with carbon dioxide can be considered stable. Then, with a glance of the replacement of CH4 in the methane hydrate with CO2, the conditions of mass balance of carbon dioxide and methane at the boundary between the zones are following:
ð2:4Þ
¼v
ðpÞ ðjÞ
T nð1Þ ¼ T nð2Þ ¼ T ðnÞ :
The methane hydrate saturation in the second zone Shm ¼ m. Then on the basis of the third equation of the system (2.5), representing the balance of mass of the bound in the hydrate water, it is possible to find the value of hydrate saturation in the first zone:
Shc ¼ qhm ð1 Gm Þm=qhc ð1 Gc Þ: In the examined problem the temperature difference DT = Tw T0 in the filtration area is small (DT T0). Therefore in
T ð1Þ
p2ð2Þ
2 Rn ðT w T ðnÞ Þ n ðnÞ exp n4 Peð1Þ p2ð1Þ dn 2 ¼ T ðnÞ þ ; R nðnÞ exp n4 Peð1Þ p2ð1Þ dn 0 R1 2 ðp2ðnÞ p20 Þ n exp 4gn dn ð2Þ ¼ p20 þ ; R1 n2 dn exp nðnÞ 4g
0 < n < nðnÞ ;
nðnÞ < n < 1;
ð2:9Þ
ð2Þ
T ð2Þ
2 R1 ðT ðnÞ T 0 Þ n exp n4 Peð2Þ p2ð2Þ dn 2 ¼ T0 þ ; R1 exp n4 Peð2Þ p2ð2Þ dn nðnÞ
nðnÞ < n < 1;
vðpÞ
ðjÞ where gðjÞ ¼ vðTÞ (j = 1, 2).On the basis of the conditions (2.5) and using the obtained solutions (2.8) and (2.9) we obtain the equations for determination of the coordinate of the displacement front and the values of the parameters p(n) and T(n) on this front:
2 ðp2w p2ðnÞ Þ exp 4gn ð1Þ ¼ Að1Þ pðnÞ nðnÞ ; R nðnÞ 2 exp 4gn dn 0 ð1Þ
ð2:10Þ
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2 n ðp2ðnÞ p20 Þ exp 4gðnÞ ð2Þ ¼ Að2Þ pðnÞ nðnÞ ; R1 n2 exp 4g dn nðnÞ
ð2:11Þ
ð2Þ
2 n ðT ðnÞ T w Þ exp ðnÞ Peð1Þ p2ðnÞ 4 2 R nðnÞ exp n4 Peð1Þ p2ð1Þ dn 0 2 n 2 ðT 0 T ðnÞ Þ exp ðnÞ Pe p ð2Þ ðnÞ 4 2 R1 ¼ BnðnÞ ; n 2 exp Pe p dn ð2Þ ð2Þ nðnÞ 4 where B ¼ /ðqhc Lhc S2hcqCqhm Lhm mÞ, /vðTÞ lm qhm Gm Shm þ1m . km q
Að1Þ ¼ /vkc lc ðTÞ
qhc Gc Shc q0c
ð2:12Þ þ 1 Shc ,
Að2Þ ¼
0m
The system of equations was solved as follows. Expressing from the Eq. (2.11) the value p(n) as a function of n(n) (by solving a quadratic equation) and substituting it into the Eq. (2.10), we obtain a transcendental equation with one unknown n(n). This equation is solved by the bisection method. Then, from the Eqs. (2.10) and (2.12) we determine the values of pressure p(n) and temperature T(n) at the boundary between the zones n(n). As previously mentioned, in this paper it is assumed that the diffusion and kinetic mechanisms do not limit the process of hydrate formation. Let us estimate the conditions at which the transition zone can be neglected and the thickness of which is limited by the diffusion of the gas in hydrate. In the pores of this zone there are particles of the hydrate methane and carbon dioxide hydrate. The thickness of the transition zone can be estimated by following condition:
Dx ¼ nðnÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi vðTÞ ðt þ t1 Þ vðTÞ t ; 2
where t 1 d =D1 is the is the characteristic times of the kinetics of the process, due to the diffusion of the gas in hydrate. pffiffiffiffiffiffiffiffiffiffi The length of the near zone is xðnÞ ¼ nðnÞ vðTÞ t. The thickness of the transition zone can be neglected if Dx << xðnÞ . Then we obtain following condition:
nðnÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi vðTÞ ðt þ t1 Þ vðTÞ t << nðnÞ vðTÞ t:
Then t >> t 1 =3. In particular, for the characteristic values of the sizes of porous channels d = 105 m, and the diffusion coefficient in a hydrate D1 = 1013 m2/s we can obtain that t1 103 s. I.e. the self-similar scheme with the clear front of phase transition can be applied if the characteristic times of the problem of the order of the day. Note that this estimate was obtained for the hydrate with solid structure. If the hydrate has a porous structure, then the characteristic time of the kinetics of the process and consequently the thickness of the transition zone will be even smaller. Then the process will come to self-similar regime with the clear front of phase transitions much earlier. 2.3. Calculation results When constructing the mathematical model it was assumed that in the first zone there is CO2 hydrate and in the second – CH4 hydrate. Therefore, the constructed model adequately describes the process if the local pressure in the first and second zones is higher than the local pressure of formation of the hydrates of carbon dioxide and methane, respectively. This pressure is determined from the formula (2.1) by using the temperature distri-
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bution, which was found in the process of solving (the condition of thermodynamic consistency). The initial pressure and temperature of the reservoir correspond on the phase diagram to the zone of stable existence of methane hydrate (above the curve of phase equilibrium g2lh2 of the mixture ‘‘methane - water - methane hydrate”). Then, when the carbon dioxide injection with the pressure and temperature also located in this area the condition of thermodynamic consistency is obviously satisfied. Therefore, in this case, the mathematical model with the frontal boundary of replacement, in compliance with the conditions for permeability (paragraph 2.2), is adequately describes the process. If the pressure and temperature of the injected carbon dioxide are in the zone of the existence of methane and water (below the equilibrium curve g2lh2 of mixture ‘‘methane – water – methane hydrate”), then it is possible that the conditions at some part of the second zone correspond to the conditions of the CH4 hydrate decomposition to gas and water. Let us study the conditions at which the mathematical model with the frontal boundary of replacement is adequately describes the process for the case of the injection of carbon dioxide at the pressures and temperatures, at which there are conditions of methane hydrate decomposition to the gas and water. Fig. 2 shows the distributions of temperature and pressure for the different values of the injection pressure of carbon dioxide pw with the temperature, which satisfy to the condition of decomposition of CH4 hydrate to gas and water:
T w > T 0 þ T ln
pw ps0
ð2:13Þ
For the parameters characterizing the system, the following values are taken: / = 0.1, m = 0.2, p0 = 3 MPa, T0 = 274 K, Tw = 277 K, Gc = 0.28, Gm = 0,13, k0 = 1016 m2, k = 2 W/(mK), qC = 2.5106 J/ (Km3), qhc = 1100 kg/m3, qhm = 900 kg/m3, Cc = 800 J/(Kkg), Cm = 1560 J/(Kkg), Rgc = 189 J/(Kkg), Rgm = 520 J/(Kkg), Lhm = 4.5105 J/kg, Lhc = 4.1105 J/kg, lc = 1.4105 Pas, lm = 105 Pas, Tc⁄ = 7.6 K, Tm⁄ = 10 K, ps0c = 1.29 MPa, ps0m = 2.87 MPa [22,23,24]. Fig. 2 shows that at the high injection pressure of carbon dioxide (case a) the reservoir pressure (solid line) in the first zone is above the equilibrium pressure of the carbon dioxide hydrate formation (dashed line 1) and in the second zone is above the equilibrium pressure of the methane hydrate formation (dashed line 2). Therefore, in this case, the solution with the one boundary of phase transitions provides adequate mathematical description of the process. At the lower value of the injection pressure of carbon dioxide (case b) the reservoir pressure ahead of the front of phase transitions (i.e. in the second zone) at certain region falls below then the equilibrium pressure of the methane hydrate decomposition. This corresponds to the overheating of the heterogeneous mixture of methane and its hydrate at this region. Consequently, it is necessary to introduce the second boundary of phase transitions at which the decomposition of CH4 hydrate to methane and water is occurs. Accordingly, in this case, it is necessary to add intermediate zone, which partially saturated with free water. Let us study the critical conditions, which determine the exercise of the one or another regime. As already noted, the condition (2.13) is the necessary condition for the existence of the possibility of the CH4 hydrate decomposition to gas and water. The sufficient condition for the existence of the solution with the two boundaries of phase transitions is determined by the inequality:
pðnÞ < pðsÞ ; where p (n) is the pressure on the phase transitions boundary, p(s) is the equilibrium pressure of the methane hydrate decomposition,
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Fig. 2. The reservoir pressure and temperature and the equilibrium temperature depending on the self-similar coordinate n. pw = 3.5 (a) and 3.2 MGa (b). The dashed line 1 shows the equilibrium pressure of carbon dioxide hydrate formation and the dashed line 2 – the equilibrium pressure of methane hydrate decomposition.
which is determined by the Eq. (2.1). The pressure p(s) corresponds to the temperature T(n), which is determined by the Eq. (2.12). Based on this inequality the computational experiments in a wide range of parameters for determining the value of the injected gas pressure, above which there are no the decomposition of CH4 hydrate to gas and water pcr (call it the critical pressure), were performed. Fig. 3 shows the dependence of the critical pressure on the absolute permeability of the reservoir. It can be seen that the critical pressure is decreasing when the permeability is increasing and the faster the lower the initial reservoir pressure p0. Furthermore, figure shows that when the permeability is high then the value pcr is almost equal to the initial reservoir pressure p0. Thus, the regime with the one boundary of phase transitions is realized into the high permeability porous mediums, as well as in the cases when the initial reservoir pressure is low. Fig. 4 shows the critical injection pressure pcr depending on the initial hydrate saturation m at injection CO2 into the reservoir with absolute permeability k0 = 1015 m2. It is seen that with increasing the initial hydrate saturation the critical pressure is increasing and the faster the lower the porosity. Thus, the regime with the one phase transition boundary is typical for the reservoirs with the low porosity and low initial hydrate saturation. Figs. 5 and 6 show the dependences of the self-similar coordinate of the phase transitions boundary on the permeability and initial reservoir hydrate saturation. According to Fig. 5 with increasing the permeability the velocity of the movement of the
Fig. 3. The critical injection pressure pcr depending on the absolute reservoir permeability k0 at different values of initial reservoir pressure p0. Lines 1 and 2 correspond to p0 = 3 and 3.1 MPa.
phase transitions boundary is increasing and the faster the lower initial reservoir pressure p0. This is due to the fact that with increasing the permeability and pressure drop the filtration rate is increasing.
V.Sh. Shagapov et al. / International Journal of Heat and Mass Transfer 107 (2017) 347–357
Fig. 4. The critical injection pressure pcr depending on the initial hydrate saturation m at different values of reservoir porosity /. Lines 1 and 2 correspond to / = 0.1 and 0.2.
Fig. 6 shows that with increasing the hydrate saturation the velocity of the phase transitions boundary is decreasing and the faster the higher the porosity. This is due to the fact that with decreasing the phase permeability of gas and with increasing the porosity the piezoconductivity coefficient and, respectively, the intensity of the mass transfer in porous medium is decreasing. Thus with increasing the permeability k0, decreasing the initial reservoir pressure p0, porosity / and hydrate saturation m the rate of displacement front becomes higher than the rate of the temperature front. This reduces the risk of the methane hydrate overheating in the second zone. It should be noted, that the rate of the temperature front caused by convection, is always much less than the rate of the pressure front. This is due to the smallness of the specific volumetric heat capacity of gas qg C g compared with the specific volumetric heat capacity of the porous medium skeleton qC. Also this is due to the almost instantaneous heat exchange between the gas and the porous medium skeleton.
qg C g << qC:
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Fig. 6. The self-similar coordinate of boundary n(n) depending on the initial hydrate saturation m at different values of reservoir porosity /. Lines 1 and 2 correspond to / = 0.1 b 0.2.
Also note that the rate of heat transfer, due to thermal conductivity, becomes comparable to the rate of mass transfer at the low values of permeability or low pressure drop. Therefore, the injected gas temperature, at the cases of the low medium permeability or low pressure drop, has a significant impact (by conduction) to the temperature in the second zone of the reservoir, making it possible, under certain conditions, the overheating of the hydrate in this zone, within the frontal model. In other words, the regime with the one boundary of phase transitions adequately describes the process even when the pressure and temperature of the carbon dioxide injection correspond to the conditions of the decomposition of methane hydrate (to gas and water) in the case when the phase transitions boundary (the law of motion which is limited by mass transfer in a porous medium) extends considerably farther than the heating zone of porous medium. In this case, the second zone does not have time to warm to the temperature values which correspond to the methane hydrate decomposition to gas and water even when the hot gas injection. 3. The regime with the decomposition of gas hydrate to gas and water 3.1. Problem statement and basic equations
Fig. 5. The self-similar coordinate of boundary n(n) depending on the absolute reservoir permeability k0 at different values of initial reservoir pressure p0. Lines 1 and 2 correspond to p0 = 3 and 3.1 MPa.
Let us consider the case of the carbon dioxide injection with the temperature, at which the regime with the two boundaries of phase transitions is realized (T > Tcr). This regime is characterized by the CH4 hydrate decomposition to gas and water and by the CO2 hydrate formation from water and carbon dioxide. In this case, three characteristic zones are formed. The pores of the first (near) zone are saturated with carbon dioxide and its hydrate, the porous medium of the second (intermediate) zone is saturated with methane and water, and the pores of the third (far) zone are saturated with methane and its hydrate. In this way, the methane hydrate decomposition is occurs at the movable frontal surface separating the second and third zones (the far boundary of phase transitions). And the carbon dioxide hydrate formation is occurs using free water and carbon dioxide at the movable frontal surface separating the first and second zones (the near boundary of phase transitions).
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Let us consider the case when the initial hydrate saturation value is low (not exceeding 0.2). In this case, water can be considered immovable owing to the low value of water saturation in the second zone. The conditions of the mass and energy balances on the boundary between the first and second zones can be written as:
kc @pð1Þ qhc Shc Gc þ Sc xðnÞ ; ¼/ lc @x qc
kmð2Þ @pð2Þ ¼ / Smð2Þ xðnÞ ; lm @x
kmð2Þ @pð2Þ kmð3Þ @pð3Þ qhm mG þ S S ¼/ lm @x lm @x qmðdÞ m mð3Þ mð2Þ
!
xðdÞ ;
ð3:2Þ
@T ð3Þ @T ð2Þ k ¼ / mqhm Lhm xðdÞ : @x @x
Here qm(j), Sm(j), p(j) and T(j) are the density, saturation, pressure and temperature of methane in the second (j = 2) and third zones
(j = 3), ql and Sl are the water density and saturation, xðdÞ the movement speed of the far boundary of phase transitions. The temperature and pressure at this boundary are continuous and associated by the condition of phase equilibrium:
T ðdÞ ¼ T 0 þ T lnðpðdÞ =ps0 Þ: Using the second Eq. (3.2) for the value of water saturation in the second zone can be obtained:
Slð2Þ ¼ mqhm ð1 Gm Þ=ql : The equations for piezoconductivity and thermal diffusivity can be written similar to the equations (2.5). The coefficients of these equations in the second and third zones can be presented in the form:
kmðjÞ p0
;
PeðjÞ ¼
q0m C m kmðjÞ ; kmðjÞ ¼ k0 S3mðjÞ ; ðj ¼ 2; 3Þ: 2k lm p0
3.2. Self-similar solution For the self-similar variable n the solution for the pressure and temperature distributions in the first zone coincides with (2.8). The equations for piezoconductivity and thermal diffusivity in the second and third zones can be written similar to the Eq. (2.7) when j = 2, 3. After integration the piezoconductivity and thermal diffusivity equations the solutions for the pressure and temperature distributions in the second and third zones can be written in the form:
p2ð2Þ
2 R1 ðT ðdÞ T 0 Þ n exp n4 Peð3Þ p2ð3Þ dn 2 ¼ T0 þ ; R1 exp n4 Peð3Þ p2ð3Þ dn n
nðnÞ < n < 1;
ðjÞ where gðjÞ ¼ vðTÞ (j = 2, 3). On the basis of the conditions (3.1) b (3.2), with consideration the solutions (3.3), (3.4) and (2.8), we get the equations for determination of coordinates of the phase-transitions boundaries n(n), n(d) and the values of parameters p(n), T(n), p(d) and T(d) at these boundaries:
2 ðp2w p2ðnÞ Þ exp 4gn ð1Þ ¼ Að1Þ pðnÞ nðnÞ ; R nðnÞ 2 exp 4gn dn 0
ð3:5Þ
ð1Þ
/ Slð2Þ ql xðdÞ ¼ /m qhm ð1 Gm ÞxðdÞ ;
lm / SmðjÞ
ð3:4Þ
vðpÞ
@T ð1Þ @T ð2Þ k ¼ / Shc qhc Lhc xðnÞ : @x @x
vðpÞ ðjÞ ¼
nðdÞ < n < 1;
ð3Þ
ðdÞ
The conditions of the mass and energy balances on the boundary between the second and third zones can be written as:
k
R1 2 ðp2ðdÞ p20 Þ n exp 4gn dn ð3Þ ¼ p20 þ ; R1 n2 exp dn n 4g ðdÞ
T ð3Þ
p2ð3Þ
ð3:1Þ
/ Shc qhc ð1 Gc Þ xðnÞ ¼ / Slð2Þ ql xðnÞ ; k
2 Rn ðT ðnÞ T d Þ n ðdÞ exp n4 Peð2Þ p2ð2Þ dn 2 T ð2Þ ¼ T ðdÞ þ ; nðnÞ < n < nðdÞ ; R nðdÞ n 2 nðnÞ exp 4 Peð2Þ pð2Þ dn
Rn 2 ðp2ðnÞ p2ðdÞ Þ n ðdÞ exp 4gn dn ð2Þ ¼ p2ðdÞ þ ; R nðdÞ n2 dn nðnÞ exp 4g
nðnÞ < n < nðdÞ ;
ð2Þ
ð3:3Þ
2 n ðp2ðnÞ p2ðdÞ Þ exp 4gðnÞ ð2Þ ¼ Að2Þ pðnÞ nðnÞ ; R nðdÞ n2 dn nðnÞ exp 4g
ð3:6Þ
ð2Þ
2 2 n n ðp2ðdÞ p20 Þ exp 4gðdÞ p2ðnÞ p2ðdÞ exp 4gðdÞ ð3Þ ð2Þ kmð3Þ R 1 kmð2Þ R nðdÞ n2 n2 exp 4g dn dn nðnÞ exp 4g nðdÞ ð3Þ
ð2Þ
¼ Að3Þ pðdÞ nðdÞ ;
ð3:7Þ
2 n ðT ðnÞ T w Þ exp ðnÞ Peð1Þ p2ðnÞ 4 2 R nðnÞ exp n4 Peð1Þ p2ð1Þ dn 0 2 n 2 ðT 0 T ðnÞ Þ exp ðnÞ Pe p ð2Þ ðnÞ 4 2 Rn ¼ Bð1Þ nðnÞ ; ðdÞ n 2 nðnÞ exp 4 Peð2Þ pð2Þ dn 2 n ðT 0 T ðdÞ Þ exp ðdÞ Peð3Þ p2ðdÞ 4 2 R1 exp n4 Peð3Þ p2ð3Þ dn nðdÞ 2 n ðT ðdÞ T ðnÞ Þ exp ðdÞ Peð2Þ p2ðdÞ 4 2 ¼ Bð2Þ nðdÞ ; R nðdÞ n 2 nðnÞ exp 4 Peð2Þ pð2Þ dn T ðdÞ ¼ T 0 þ T lnðpðdÞ =ps0 Þ;
ð3:8Þ
ð3:9Þ
ð3:10Þ
ðTÞ m Þm , ¼ / vkmð2Þlm 1 qhm ð1G q
where Að1Þ ¼ /vkc lc qhcqGc Shc þ 1 Shc , Að2Þ 0c l Gm qhm ð1Gm Þ / qhc Lhc Shc / qhm Lhm m þ 1 , B ¼ , B ¼ . Að3Þ ¼ / vðTÞ lm m qhm ð1Þ ð2Þ q q 2q C 2q C ðTÞ
0m
l
The system of the boundary Eqs. (3.5)–(3.10) is solved as follows. Firstly, we set zeroth approximation of the unknown quantities at the first (near) phase transitions boundary n(n). At this paper this approximation is the parameters at the left boundary of a reservoir. Secondly, from the Eq. (3.7) we obtain the expression for the pressure p(d) (the root of quadratic equation). Thirdly, substituting this expression in the (3.10), we obtain the expression for the temperature T(d). Fourthly, substituting the temperature expression in the
V.Sh. Shagapov et al. / International Journal of Heat and Mass Transfer 107 (2017) 347–357
Eq. (3.9), we obtain the transcendental equation in the one unknown n(d). Fifthly, we get the value n(d) by solving the transcendental equation by bisection method. Sixthly, we get the values p(d) and T(d). Seventh, from the Eq. (3.6) we obtain the expression for the pressure p(n) (the root of quadratic equation). Eighth, substituting the expression for p(n) in the (3.5), we obtain the transcendental equation in the one unknown n(n). Ninthly, we get the new approximate value of the first boundary n(n). In-tenths, we get the new approximate values T(n) (Eq. (3.8)) and p(n). As a result of cyclic repetition of described iterative procedure we obtain the sequence of approximate values which converges to the desired values of the parameters at the phase-transition boundaries. 3.3. Calculation results Fig. 7. presents the reservoir temperature and pressure at the injected dioxide carbon temperature Tw = 281 K. The values of other system parameters are equal to the values for Fig. 2. Fig. 7 shows that the thermodynamic conditions in the second zone correspond to the conditions of existence of a heterogeneous mixture
Fig. 7. The reservoir temperature and pressure depending on the self-similar coordinate n. Dashed line show the equilibrium pressure of methane hydrate decomposition.
355
of methane and water, which accords with the assumption of the absence of methane hydrate in the second zone, which was made in the problem statement. The pressure conductivity factor for real reservoirs is several orders more than the thermal diffusivity. Therefore, the pressure from the far frontal boundary to the near frontal boundary always increases slowly than the equilibrium temperature and the pressure associated with it. This eliminates thermodynamic contradiction, when the reservoir pressure in the second zone is higher than the equilibrium pressure. Accordingly, methane hydrate in the second zone is absent. Fig. 8 presents that the coordinate of the near boundary of phase transitions is increasing when the injected gas pressure is increasing and the initial reservoir pressure is decreasing. This is because the rate of methane displacement with carbon dioxide is limited by mass transport in a porous medium, the intensity of which is increasing when the pressure drop (Dp = pw – p0) is increasing. Also, Fig. 8 shows that the rate of the far boundary of phase transitions decreases with increasing the initial reservoir pressure and little affected by the gas injection pressure. This is due to the fact that the equilibrium temperature of the decomposition of methane hydrate is increasing when the injected gas pressure or the initial reservoir pressure are increasing, that slows the process of the dissociation of methane hydrate to gas and water. On the other hand with an increase in the pressure drop the formation intensity of carbon dioxide hydrate is increasing and thus the heat, which emitted at the near boundary, is increasing. It promotes the growth of the decomposition rate of CH4 hydrate to gas and water when the injected gas pressure is increasing, and this rate is decreasing when the initial reservoir pressure is increasing. Thus, when the injected gas pressure is increasing, these two factors cancel each other, so the intensity of methane hydrate decomposition to gas and the water very little depends on the pressure of injected gas. And with an increase of the initial pressure these factors contribute to the slowdown of the CH4 hydrate dissociation to gas and water. Increasing the injected gas pressure in the range of parameters corresponding to the mode with two phase transition boundaries is not effective way to increase the intensity of the CH4 hydrate decomposition to gas and water. Also, Fig. 8 shows that the rate of near boundary tends to zero when the pressure drop value tends to zero. This case corresponds to the heating of a reservoir without the carbon dioxide injection and hence without the formation of carbon dioxide hydrate. At this regime (Dp = 0), the rate of far boundary in the case a (pw = p0 = 3 MPa) almost one and a half times more than in the case b (pw = p0 = 3.5 MPa). This is because the pressure and respectively the equilibrium temperature of the hydrate decomposition in the first case are lower than in the second case. I.e. the length of region, where the temperature causes decomposition of methane hydrate, in the first case is more than in the second case. In the considered problem one of the main factors determining the intensity of mass transfer is the gas relative permeability, which, according to the equation (2.3), is increasing when the absolute reservoir permeability is increasing and is decreasing when the reservoir hydrate saturation is increasing. Fig. 9 shows that with increasing the gas permeability (i.e., with increasing the absolute permeability and decreasing the initial reservoir hydrate saturation) the coordinate values of both boundaries of phase transitions are increasing. This is due to increasing the rate of the displacement front of methane by carbon dioxide and this is due to corresponding increasing the intensity of the carbon dioxide hydrate formation with increasing the gas permeability. Since the process of the carbon dioxide hydrate formation is accompanied by the release of heat, the motion speed of the far boundary (although only slightly) also is increasing when the permeability is increasing. Furthermore, Fig. 9 shows that at high values of gas
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Fig. 8. The self-similar coordinates of the phase-transition boundaries n(n) (line 1) b n(d) (line 2) depending on the injection pressure pw (a) and initial pressure p0 (b). Tw = 280 K, p0 = 3 MPa (a), pw = 3.5 MPa (b). Dashed line corresponds to the coordinate of the far boundary, which was calculated without consideration of the CO2 hydrate formation heat that is released on the near boundary (Lhc = 0).
Fig. 9. The self-similar coordinates of the phase-transition boundaries n(n) (line 1) b n(d) (line 2) depending on the absolute reservoir permeability k0 (a) at m = 0.2 and depending on the initial hydrate saturation m (b) at k0 = 1016 m2.
permeability the intermediate area degenerates into a frontal surface (i.e. the infinitely small thickness area) that corresponds to the regime when the replacement of methane in hydrate with carbon dioxide without release free water is occurs (Section 2). This is entirely consistent with the results of Section 2 and this is due to the following. The movement speed of the near boundary is limited by the mass transport and the propagation speed of the temperature front is mainly limited by the thermal conductivity. The movement speed of the near boundary becomes more than the propagation speed of the temperature front when the gas permeability is increasing due to the absolute reservoir permeability is increasing and the initial reservoir hydrate saturation is decreasing. This prevents heating of the area saturated with methane hydrate to temperatures causing its decomposition. Sources of heat, which is necessary for the methane hydrate decomposition to gas and water, are the latent heat of CO2 hydrate formation and the heat flux from the left border. So, the dissociation of methane hydrate to gas and water can occur even without the injection of carbon dioxide. However, it should be noted that the heat, which generated during the formation of CO2 hydrate (approximately 400 kJ per 1 kg of hydrate), can increase the rate of the methane hydrate decomposition. Fig. 10 shows the calcula-
Fig. 10. The parameter N depending on the injected gas temperature Tw at different values of the injected gas pressure pw. Lines 1, 2 and 3 correspond to pw = 3.1, 3.2 and 3.3 MPa.
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tions, which allow assessing the contribution of latent heat of the CO2 hydrate formation in the CH4 hydrate decomposition rate. Here N is the ratio of values of the coordinates of the boundaries between the second and third zones, calculated taking into account the generated heat (Lhc = 4.1 105 J/kg) and without it (Lhc = 0) at the boundary between the first and second zones. This figure shows that the contribution of the heat, which generated during the carbon dioxide hydrate formation, is particularly noticeable when: 1) increasing the gas injection pressure, 2) low temperature of the injected gas. The rate of the CO2 hydrate formation boundary is increasing when the carbon dioxide injection pressure is increasing (Fig. 8). Therefore, the intensity of the heat generation at this boundary is increasing. The intensity of the heat flux from the left boundary is decreasing when the injected gas temperature is decreasing. Therefore, the relative contribution of the generated heat at the boundary of the CO2 hydrate formation becomes more prominent. 4. Conclusions The mathematical model of the warm carbon dioxide injection into a natural reservoir saturated with methane and its hydrate was constructed. It was determined that the methane recovery from hydrate at such influence can occur in two regimes. At the first regime the displacement of methane with carbon dioxide in the CH4 hydrate is occurs. This regime is typical for porous mediums with high permeability and when the injected gas pressure is high as well as for porous mediums with low porosity, low hydrate saturation and low initial pressure. In the second regime the decomposition of CH4 hydrate to gas and water with the subsequent formation of the CO2 hydrate from water and carbon dioxide is occurs. At this regime, the heat, which released at the formation of carbon dioxide hydrate, has a significant impact on the decomposition rate of methane hydrate to gas and water. Acknowledgement This work was supported by Grant of the RFBR 17-51-540001 and VAST.HTQT/17-18 ‘‘Wave and filtration flow with phase transitions in porous structures”. References [1] Y.F. Makogon, Hydrates of Hydrocarbons, PennWell Publishing Company, Tulsa, OK, USA, 1997. [2] Y.F. Makogon, Natural gas hydrates – A promising source of energy, J. Nat. Gas Sci. Eng. 2 (1) (2010) 49–59.
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