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Formation of gas hydrates in a porous medium during an injection of cold gas
International Journal of Heat and Mass Transfer 84 (2015) 1030–1039
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Formation of gas hydrates in a porous medium during an injection of cold gas V.Sh. Shagapov a, N.G. Musakaev b,⇑, M.K. Khasanov c a
Institute of Mechanics of Ufa Branch RAS, Ufa, Russia Tyumen Branch of Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Tyumen State University, Tyumen, Russia c Sterlitamak Branch of Bashkir State University, Sterlitamak, Russia b
a r t i c l e
i n f o
Article history: Received 17 March 2014 Received in revised form 16 December 2014 Accepted 22 January 2015
Keywords: Gas hydrates Hydrate formation Self-similar solution Porous medium
a b s t r a c t This research paper examines various features of the formation of gas hydrates in a porous medium during an injection of cold gas (the injected gas temperature is lower than the initial temperature of the porous medium). Self-similar solutions of this problem were built in a rectilinear-parallel approximation. These solutions explain the distribution of the main parameters in a porous medium during the formation of gas hydrates on a frontal surface as well as in an extended zone. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Technological processes in the oil and gas industry often include the formation of gas hydrates in a reservoir and inside both the underground and the surface oilfield equipment [1–3]. In most cases, the gas hydrate formation has a negative impact, due to partial or complete plugging of the tube flow area. However, the gas hydrate formation can also have a positive impact [4–6]; for example, it can be used to increase the capacity of the underground reservoirs to store hydrocarbon gas. The point being that in a hydrate state the same gas mass can be stored under considerably lower pressure and in less volume [1], thereby eliminating the need to build compressor stations and high pressure vessels when storing gas in a hydrate state. The most efficient way would be to build gas depots in areas of permafrost [7]. In this case, placing the underground vessels near the surface will ensure that building these hydrate depots is economically efficient and can be used to balance the variations of gas consumption in small industrial centers on a daily and seasonal basis [8,9]. Also, gas hydrates can be used for the disposal of greenhouse gases in underground depots (in a porous medium) [7,10]. To intensify the process of hydrate formation, it is necessary to create conditions which ensure high areas of contact between the ⇑ Corresponding author. E-mail addresses: [email protected] (V.Sh. Shagapov), [email protected] (N.G. Musakaev), [email protected] (M.K. Khasanov). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.01.105 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
gas and the liquid. Such favorable conditions are present in a porous medium, where the contact area is great. The equilibrium temperature for gas hydrates depends on the pressure. Therefore, under the thermal and depression effects of a porous medium, gas hydrate formation and decomposition are possible on an frontal surface as well as in a voluminous unit. This paper [11–13] is devoted to researching the gas hydrate formation process in a porous medium. This paper differs from the one aforementioned by additionally building analytic solutions of the problem in a rectilinear-parallel approximation and by making an analysis of these solutions. Furthermore, we have obtained the desired conditions by segregating modes of the gas hydrate formation on a frontal surface and in an extended zone. The influence of the pressure and temperature of the injected gas and the initial parameters of a porous medium on the hydrate formation process was examined. 2. Mathematical model 2.1. Main assumptions and equations Assume that a reservoir, initially filled with gas (methane) and water, fills half-space x > 0. Before the injection of gas, pressure p0 and temperature T0 in the reservoir correspond to thermodynamic conditions of the existence of gas and water in a free state:
t ¼ 0; x > 0 :
T ¼ T 0 ; p ¼ p0 :
ð1Þ
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Assume that through the left boundary of the reservoir (x = 0) the gas (methane) is being injected. Pressure pe and temperature Te of the injected gas correspond to the conditions of gas hydrate formation and are supported at this boundary at constant level (Fig. 1):
t > 0; x ¼ 0 :
T ¼ T e ; p ¼ pe : ðpe > ps ðT e ÞÞ:
ð2Þ
We adopt several assumptions enumerated here. A hydrate is a two-component system with a mass concentration of gas G. The mass concentration of the methane in a hydrate does not strongly depend on pressure and temperature (G = 0.11–0.13). Therefore, we will neglect the relationship of the mass concentration of the gas in a hydrate with the pressure and temperature. Suppose that porosity m is constant; the rock skeleton, the gas hydrate and water are all incompressible and immovable. In this paper we research the processes, the duration of which is considerably longer than the typical time of temperature equilibrium
s d2 =vðTÞ (d is the typical pore dimensions, vðTÞ ¼ k=qc is the thermal diffusivity of a porous medium). Therefore, we assume that the temperatures of a porous medium and of a saturant (gas, hydrate or water) are similar (one temperature process). With consideration of the accepted assumptions, the system of main equations can be written in the following form [11–14]:
@ qg mSg þ Gqh mSh þ div qg mSg~ tg ¼ 0; @t @ k ðq mSl þ ð1 GÞqh mSh Þ ¼ 0; mSg~ tg ¼ g grad p; @t l lg @ @ ðqcTÞ þ qg cg mSg~ tg grad T ¼ divðk grad TÞ þ ðqh Lh mSh Þ; @t @t X Sj ¼ 1; kg ¼ k0 S3g ; p ¼ qg Rg T; j¼g; l; h
qc ¼ ð1 mÞqsk csk þ m
X
qj Sj cj ; k ¼ ð1 mÞksk þ m
j¼g; l; h
X
Sj k j :
j¼g; l; h
ð3Þ Hereafter, subscripts sk, h, w and g are related to the parameters of the rock skeleton, hydrate, water and gas, respectively; qj (j = sk, h, w and g) is the true specific density of the jth phase, Sj (j = g, l, h) is the pore saturation of the jth phase, k0 is the absolute permeability of the reservoir, tg and lg are the velocity and dynamic viscosity of the gas phase, p is the pressure, T is the temperature, Lh is the specific heat of hydrate formation, cj (j = g, l, h) is the specific heat capacity of the jth phase, kj (j = sk, g, l, h) is the coefficient of the thermal conductivity of the jth phase, qc and k are the specific volumetric heat capacity and the coefficient of the thermal conductivity of the system.
pe
T s ðpÞ ¼ T 0 þ T lnðp=ps0 Þ;
ð4Þ
where T0 is the initial temperature of the system, ps0 is the equilibrium pressure corresponding to the initial temperature, T⁄ is the empirical parameter having temperature dimension and depending on the composition of the natural gas. During hydrate formation in a porous medium, in general, three typical zones can appear. In the zone located close to the surface, where gas is injected into a porous medium, pores are filled with gas and hydrate. In the zone farther away from the surface, the pores are filled with gas and water. In the transition zone, gas hydrate is being formed, that is why pores in this zone are filled with gas, water and hydrate. At the boundaries between these zones, jumps in phase saturation as well as jumps in mass and heat flows are observed. At these boundaries, the relations which follow from conditions of mass and heat balance are realized [11,14]:
½mðSh qh ð1 GÞ þ Sl ql ÞxðiÞ ¼ 0; h i m Sg qg ðtg xðiÞ Þ Sh qh G xðiÞ ¼ 0;
Here [w] is the jump of the parameter w on the boundary x(i) (i = n, d); x(n) is the boundary between the near zone and the transition zone, x(d) is the boundary between the transition zone and the far zone; the dot over x(i) denotes velocity of the movement of the boundary. Temperature and pressure at these boundaries are considered to be continuous.
GAS + WATER
p0 T0 x
0
x(n)
h i ½k grad T ¼ mLh qh Sh xðiÞ : ð5Þ
GAS + WATER + HYDRATE
GAS + HYDRATE
Te
The fourth equation of the system (3) omits summands related to the barothermal effect, as the change of temperature due to this effect is negligible in comparison with temperature changes due to hydrate formation. We assume that the specific heat of the hydrate formation is independent of pressure [11]. 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 water saturation of the formation 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. It should be noted that the system of Eq. (3) is not complete. For the zones in which the phase transition (hydrate formation) does not occur, this system must be supplemented by the condition of constant hydrate saturation. For the zone where the hydrate is formed, the system of Eq. (3) must be supplemented by the condition of phase equilibrium:
x(d)
Fig. 1. Pattern of gas injection into a porous medium with hydrate formation.
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2.2. Formation of gas hydrate
For intensity of hydrate formation jh, related to the unit of the surface area of the hydrate layer, we can take the relation [16]:
It should be noted that in the accepted equilibrium scheme (on the frontal boundaries as well as in the volumetric zones) the intensity of hydrate formation is limited by the removal of the latent heat of such formation.According to modern concepts [15], the formation of hydrate particles is in general accompanied by unbalanced processes. These processes are affected by the diffusion of hydrate formation gas ‘‘water–hydrate’’ through layers of water and hydrate to the surface of the contact. If we consider that the typical thickness of such layers of the size of pore channels is pffiffiffiffiffiffiffiffiffiffi k=m, then for the typical time tD of relaxation of diffusion at k , where D – difnon-equilibrium we get the assessed value tD ffi mD fusion coefficient. These times tD are very insignificant in comparison with the time periods of interest in the considered problem [11]. From the second equation of the system (3) we get ql ðSl Sl0 Þ þð1 GÞqh Sh ¼ 0, Sl0 is the initial water saturation. From here, P considering the relation j¼g; l; h Sj ¼ 1, we get:
Sl ¼ Sl0
qh q ð1 GÞSh ; Sg ¼ 1 Sl0 1 h ð1 GÞ Sh : ql ql
Further, assuming Sl = 0, for the near zone using these relations we can write:
Sh ¼
ql Sl0 ; Sg ¼ 1 Sh : qh ð1 GÞ
ð6Þ
Thus, in the far and near zones, there is a filtration of gas in a porous medium with constant values of ‘‘live’’ porosity and permeability, but these values are not equal for different zones. In this case, the initial water saturation must satisfy the condition Sl0 6 qh ð1 GÞ=ql , because Sh 6 1=ql . Let’s review the process of hydrate formation in a porous medium, saturated in the initial state with water and gas, assuming that the velocity of the hydrate formation is limited by the kinetics of this process. Assume that the gas is slightly soluble in water, and the relative mass content of water vapors in the gas phase is negligible. In this case, hydrate formation will take place on the boundary of contact ‘‘gas–water’’ in the form of a hard hydrate film, separating these two phases [8]. Equilibrium relative mass concentration of gas in the hydrate on the boundary with gas is higher than on the boundary with water. Therefore, the diffusion flow of gas to the surface of the water is appearing in the layer of hydrate [15]. As a result, the process of hydrate formation is taking place on the boundary of contact of the water and the hydrate. To determine the intensity of hydrate formation we use the following scheme. Pore volume is a set of cylindrical channels with the typical radius a. Water is fully wetting the rock, i.e. the wall of the pore channels is covered by a water film with an average thickness dl(t). Hydrate deposit builds up on the surface of the film with the typical thickness dh(t). For simplicity we restrict ourselves to the case when dl þ dh a. Based on the aforementioned scheme for water saturation and hydrate saturation we can write:
Sl ¼
a2 ða dl Þ2 2dl ; a2 a
Sh ¼
2 ða dl Þ2 ða dl Þ2 dh a2
2dh : a ð7Þ
During description of the velocity of the growth of the layer of gas hydrates, we assume that the radius of the curvature of the pore surface a is much thinner than the thickness of the water hydrate layer. Therefore, the process of the growth of the gas hydrate layer can be viewed within the confines of a onedimensional pattern.
jh ¼ qh
@dh : @t
ð8Þ
On the other hand this value is limited by the ‘‘stoichiometric’’ condition with the value of gas diffusion flow (on the boundary ‘‘water–hydrate’’) jg:
jh ¼ jg =kgh ;
ð9Þ
where kgh is the mass gas concentration in the gas hydrate on the fluid boundary. The value jg can be found using Fick’s law, considering the distribution of gas concentration in the gas hydrate as linear:
jg ¼ Dqh
Ggh Ghl : dh
ð10Þ
Values Ggh and Ghl are known functions of pressure and temperature and can be found with the diagram of phase equilibrium of the system ‘‘water–gas’’. Herewith, their difference DG ¼ Ggh Ghl is much less than they are themselves. On the basis of Eqs. (7)–(10) we get:
Sh ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8D t DG ; a2 G
ðSl > 0Þ;
where t is the duration of the process. On the basis of the last correlation, assuming Sh = 0, Sl = Sl0 with t = 0 and Sh = She, Sl = 0 with t = sh (sh – time of hydrate development), we get ql Sl0 ¼ ð1 GÞqh She . From here, with consideration 2 Sl0 ql Ga2 . of (10), we get that sh ¼ 8D DG ð1GÞq h
For values of the parameters Sl0 = 0.2, a = 105 m, D = 1012 m2/c, G = 0.12, and DG = 103, we get the assessed value for sh = 10 s. As is well known, for hydrate formation it is necessary that the temperature of the system ‘‘porous medium–saturant’’ was below the equilibrium temperature of the hydrate formation. But it should be taken into account that the process of hydrate formation is accompanied by a heat release. Therefore, hydrate formation can take place in a diffusion mode only in those cases when the typical size of the system is so small that the heat exchange with the external environment is sufficiently fast. This size of the porous sample can be defined by the following expression:
l¼
qffiffiffiffiffiffiffiffiffiffiffiffiffi
sh vðTÞ 0:03 m:
If the typical size of a porous sample is less than the value of l, then the velocity of the hydrate formation will be limited by diffusion, i.e. by the kinetics of the process. Since in the examined problem the size of the system ‘‘porous medium– saturant’’ is much greater than the value l, the process of hydrate formation will be limited by heat transmission in the porous medium. 3. Problem with a frontal boundary of phase transitions 3.1. Problem statement Let’s consider the case when hydrate formation occurs entirely on a frontal boundary. The transition zone is missing such that in the porous medium there are only two zones. In the first (near) zone, water was fully changed into gas hydrate condition, therefore, in the pores there are only gas and gas hydrate. In the second (far) zone, the porous medium is filled with gas and water. In this case, the gas mass-conservation equation, Darcy’s law and heat flow equation can be presented in the form:
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@ mSgðiÞ tgðiÞ qgðiÞ ¼ 0; @x @t kðiÞ @pðiÞ @T ðiÞ @T ðiÞ @ @T ðiÞ ; ; qc þ mcg SgðiÞ tgðiÞ qgðiÞ ¼ k mSgðiÞ tgðiÞ ¼ @x lg @x @t @x @x
mSgðiÞ
@ qgðiÞ
þ
ð11Þ where pðiÞ ¼ qgðiÞ Rg T ðiÞ , k(i) is the permeability coefficient for the gas; parameters of the first and second zones are provided with subscripts in parentheses i = 1, 2. On the surface x = x(s) separating the near and far zones, a jump þ of hydrate saturation occurs from S h ¼ Shð1Þ to Sh ¼ 0. Subsequently, on this surface, on the grounds of conditions of the balance of mass and heat (5), we get:
mShð1Þ qh ð1 GÞxðsÞ ¼ mSl0 ql xðsÞ ; mSgð2Þ qgðsÞ tgð2Þ xðsÞ þ mShð1Þ qh GxðsÞ ¼ mSgð1Þ qgðsÞ tgð1Þ xðsÞ ; k
@T ð1Þ @T ð2Þ k ¼ mShð1Þ qh Lh xðsÞ ; @x @x
ð12Þ
where Sh(1) is the hydrate saturation of the first zone, xðsÞ is the velocity of the movement of the boundary of phase transition. Subscript s in parentheses refers to parameters on the boundary of the hydrate formation. In this problem, the phase-transition front velocity is limited by the rate of removal of the heat released during the hydrate formation process, i.e., by the heat transfer in the reservoir. This is determined by the fact that heat exchange in extended natural reservoirs takes place very slowly (mainly at the expense of heat conductivity). For example, for the zone of length L = 1 m, the typical heat exchange time is given by L2/v(T) 106 s. For time values such as those much larger than the sh, the condition of temperature equilibrium in the zone of phase transition is met. Subsequently, the pressure and temperature on the boundary between the zones are linked by condition of phase equilibrium (4). For the frontal pattern of gas hydrate formation, conditions (1) and (2), considering the emergence of two zones, will change into:
t ¼ 0; x > 0 :
T ð2Þ ¼ T 0 ; pð2Þ ¼ p0 ;
t > 0; x ¼ 0 :
T ð1Þ ¼ T ð2Þ ¼ T ðsÞ :
If typical variations of temperature DT in the zone of filtration are not large [17] (for example, at DT << T0)), then in the equation of reservoir-pressure conductivity, which results from Eq. (11), the summand is negligible due to the variation of the temperature. Therefore, the system of Eq. (11) after changing can be written in the following form:
ð13Þ
The system of equations for finding coordinates of the boundary x(s) and values of parameters on it can be presented, considering the Eqs. (11) and (12), in the following form:
! @pð2Þ @pð1Þ qh G qh ð1 GÞ kð2Þ kð1Þ ¼ mShð1Þ lg þ 1 xðsÞ ; @x @x qgðsÞ ql k
@T ð1Þ @T ð2Þ k ¼ mShð1Þ qh Lh xðsÞ : @x @x
! 2 2 n dpðiÞ d dpðiÞ ; ¼ gðiÞ 2 dn dn dn
2 n dT ðiÞ PeðiÞ dpðiÞ dT ðiÞ d dT ðiÞ ; ¼ 2 þ 2 dn dn dn 2p0 dn dn ð15Þ
vðPÞ
where gðiÞ ¼ vðiÞ ðTÞ ,
k p q p c k ðPÞ vðiÞ ¼ mSðiÞgðiÞ 0l , PeðiÞ ¼ g0 k0l g ðiÞ , kðiÞ ¼ k0 S3gðiÞ (i = 1, 2). g g
After integration of the Eq. (15), for distribution of pressure and temperature in each zone the following relations:
Rn 2 p2e p2ðsÞ n ðsÞ exp 4gn dn ð1Þ p2ð1Þ ¼ p2ðsÞ þ 0 < n < nðsÞ ; R nðsÞ n2 exp 4g dn 0 ð1Þ 2 R nðsÞ Pe T e T ðsÞ n exp n4 2pð1Þ2 p2ð1Þ dn 2 0 0 < n < nðsÞ ; T ð1Þ ¼ T ðsÞ þ R nðsÞ Peð1Þ 2 n exp 4 2p2 pð1Þ dn 0 0
ð16Þ R 2 1 p2ðsÞ p20 n exp 4gn dn ð2Þ p2ð2Þ ¼ p20 þ ðnðsÞ < n < 1Þ; R1 n2 exp dn nðsÞ 4gð2Þ 2 R1 Pe T ðsÞ T 0 n exp n4 2pð2Þ2 p2ð2Þ dn 2 0 T ð2Þ ¼ T 0 þ ðnðsÞ < n < 1Þ: R1 Peð2Þ 2 n exp p dn nðsÞ 4 2p2 ð2Þ 0
ð17Þ Eq. (14) for the introduction of the self-similar variable n can be presented in the following form:
dT ð1Þ dT ð2Þ mSh qh Lh ¼ nðsÞ ; dn dn 2qc kð2Þ
On the boundary x = x(s) we involve the conditions of pressure and temperature continuity:
@pðiÞ @pðiÞ kðiÞ @ ; pðiÞ ¼ @t mSgðiÞ lg @x @x cg kðiÞ qgðiÞ @pðiÞ @T ðiÞ @T ðiÞ k @ @T ðiÞ þ ¼ : @t qc @x @x qclg @x @x
pffiffiffiffiffiffiffiffiffiffi Let us introduce the self-similar variable: n ¼ x= vðTÞ t. For that variable, the system of Eq. (11), with application of Leibenzon linearization, can be presented in the form of the system of ordinary differential equations of the 2nd order:
2
T ð1Þ ¼ T e ; pð1Þ ¼ pe :
pð1Þ ¼ pð2Þ ¼ pðsÞ ;
3.2. Self-similar solutions
dpð2Þ dn
2
kð1Þ
dpð1Þ
!
ql G ql ¼ mSl0 lg þ 1 nðsÞ vðTÞ pðsÞ : ð1 GÞqgðsÞ ð1 GÞqh dn ð18Þ
From these equations, after substituting the relations (16) and (17) by some numeric method, we can find the coordinate of the boundary of phase transitions n(s) and the values of parameters on it. 3.3. Calculation results Fig. 2 presents the distribution of temperature and pressure during injection of methane at the temperature Te = 278 K into a reservoir with the initial pressure p0 = 4 MPa and the initial water saturation Sl0 = 0.2. For other parameters defining the system, the following values were accepted: m = 0.1, G = 0.12, T0 = 280 K, T⁄ = 10 K, ps0 = 5.5 MPa, k0 = 1014 m2, Rg = 520 J/(K kg), qsk = 2000 kg/m3, qh = 900 kg/m3, ql = 1000 kg/m3, csk = 1000 J/ (K kg), ch = 2500 J/(K kg), cl = 4200 J/(K kg), cg = 1560 J/(K kg), ks = 2 W/(m K), kh = 2.11 W/(m K), kl = 0.58 W/(m K), lg = 105 kg/(m s), Lh = 5 105 J/kg [2,9,18]. From the layout of the curves on Fig. 2a, we can make the conclusion that during the injection of methane under the pressure pe = 5 MPa, the temperature of the reservoir before the front of hydrate formation is lower than the equilibrium temperature, and after the front this temperature is higher than that at
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Fig. 2. The reservoir temperature, equilibrium temperature and pressure at pe = 5 (a) and 6 MPa (b), Te = 278 K depending on the self-similar variable n. Dashed line is the equilibrium temperature which corresponds to the presented pressure distribution.
equilibrium. Hence, in this case, the solution for the model with a frontal surface of hydrate formation is consistent. If gas is injected at the same temperature Te = 278 K, but under the pressure pe = 6 MPa, the temperature of reservoir after the front of hydrate formation drops below than the equilibrium temperature (dashed line) which corresponds to the supercooling of water in this zone (Fig. 2b). Hence, in this case, we cannot find physically consistent solution using the model with a frontal surface of hydrate formation. In the result of the analysis of Fig. 2 we can make the conclusion that there is some injection pressure of gas pcr, which we call the critical injection pressure of gas, and if the injection pressure of gas is higher than pcr, then it is necessary to consider a volumetric zone of hydrate formation.The volumetric zone appears in the cases when on the boundary of phase transitions (n = n(s)) the following inequation is met
dT ð2Þ dn
<
dT ð2Þs , dn
where T(2)s is the tempera-
ture of hydrate formation corresponding to the pressure p(2). The derivative from the right part of the aforementioned inequation, with consideration of the Eq. (4), can be changed to:
dT ð2Þs dn
¼ 2pT 2
ðsÞ
dp2ð2Þ dn
The calculation experiments, based on the last inequation, to determine influence of different parameters on the critical injection pressure of gas pcr were held. Fig. 3 presents relation pcr (T0) during injection into reservoir of gas at the temperature Te = 278 K. The initial reservoir pressure p0 = 3 MPa. This figure shows that during injection of cold gas (Te < T0) the value pcr is increasing when the initial reservoir temperature is increasing, at that the critical pressure pcr is closely equal to the equilibrium pressure ps0 when the reservoir permeability is high (curve 1). During injection of warm gas (Te > T0) into reservoir with high permeability the pressure pcr is equal to the equilibrium pressure ps(Te) which corresponds to the temperature of the injected gas, i.e. the pressure pcr is equal to the minimum pressure of gas injection under which the emergence of a zone saturated with hydrates is possible. Hence, during injection of warm gas into reservoir with high permeability the mode with a frontal mode of hydrate formation is impossible.
.
Using the boundary conditions (18) we get the following condition for the emergence of an extended zone:
! 2 dpð1Þ dT ð1Þ Shð1Þ DT T nðsÞ < kð1Þ þ KShð1Þ nðsÞ ; dn dn 2 2kð2Þ p2ðsÞ where DT ¼ mqqhcLh ; K ¼ mlg vðTÞ pðsÞ
qh G qgðsÞ
þ qh ð1GÞ 1 . q l
Considering the relations in the near zone (16) this condition will look like:
2 n Pe ðT ðsÞ T e Þ exp 4ðsÞ 2pð1Þ2 p2ðsÞ DT Shð1Þ 0 2 nðsÞ R nðsÞ Peð1Þ 2 n 2 exp 4 2p2 pð1Þ dn 0 0 1 0 n2 kð1Þ p2ðsÞ p2e exp 4gðsÞ C ð1Þ T B C: BKShð1Þ nðsÞ þ < A 2 2 @ R nðsÞ nðsÞ 2kð2Þ pðsÞ dn exp 0 4g ð1Þ
Fig. 3. The critical gas injection pressure pcr depending on the initial reservoir temperature at different values of absolute reservoir permeability k0. Lines 1 and 2 correspond to k0 = 1015 and 1017 m2. Dashed line is the equilibrium pressure ps0 which corresponds to the initial reservoir temperature; p0 = 3 MPa.
V.Sh. Shagapov et al. / International Journal of Heat and Mass Transfer 84 (2015) 1030–1039
Fig. 4. The critical gas injection pressure pcr depending on the absolute reservoir permeability k0 at different values of initial reservoir temperature T0. Lines 1 and 2 correspond to T0 = 280 and 285 K. Dashed lines, parallel to abscissa axis, being horizontal asymptotes of solid curves, show equilibrium pressures ps0 which correspond to two initial reservoir temperatures; p0 = 3 MPa.
Another parameter, which affects to the pressure pcr, is the reservoir permeability. The higher permeability, the closer the critical pressure to the corresponding equilibrium pressure (Fig. 4). The pressure pcr is increasing when the permeability is decreasing. The numerical calculations show that the value of pressure pcr goes to infinity when the value of absolute permeability is close to values of the order 1019 m2. Hence, when the reservoir permeability is very low then the mode with a frontal surface of hydrate formation is realized at any gas injection pressure. In reservoirs with low permeability the pressure pcr is considerably decreasing when the initial reservoir pressure is increasing (Fig. 5). In reservoirs with high permeability the pressure pcr is almost independent of reservoir pressure and pcr is close to the equilibrium pressure ps0. Thus, the closer the initial system condition to the condition of phase equilibrium, the lesser pressure on the reservoir boundary is needed for the emergence of a volumetric zone of gas hydrate formation. Figs. 4 and 5 show that the mode with a frontal surface of hydrate formation is typical for a porous medium with low permeability, low pressure, and high initial temperature, i.e. for reservoirs in which the initial condition of gas and water is far from the conditions of gas hydrate formation.
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In the result of the calculations it was determined that, depending both on the injected gas parameters and on the parameters determining the initial system condition, there are possible two radically different solutions. For the gas injection pressure pe there is some maximum value, and if this value is exceeded, then by using a frontal model of hydrate formation it is impossible to find physically consistent solution. For the permeabilities k0 = 1012 1015 m2 (typical for real reservoirs) this maximum pressure with high accuracy coincides with the equilibrium pressure ps0 corresponding to initial temperature.This result can also be gotten from the following physical considerations. It follows from Fig. 2 that, with the considered values of permeability, the pressure on the front of phase transition is approximately coincides with the gas injection pressure. If pe > ps0 then the temperature on the front is higher than the initial temperature (temperature at infinity). Therefore, in the second zone, the reservoir temperature is gradually decreases. At this the coefficient of piezoconductivity v(p) is considerably higher than the coefficient of thermal diffusivity v(T). For example, for parameters corresponding to Fig. 2:
vðPÞ ¼ mSk pg l0 g ¼ 0:05 m2 =s, vðTÞ ¼ k=qc ¼ 105 m2 =s.
Therefore, the reservoir temperature in the second zone decreases faster than the pressure and the equilibrium temperature of hydrate formation which depends on this pressure. And, if we use the frontal model, this leads to the supercooling of water after the front. Hence, the model with a frontal surface of hydrate formation can describe the process without physical inconsistency only in the cases when the gas injection pressure is not exceeding the equilibrium pressure which corresponds to the initial temperature. To describe the hydrate formation process during injection of gas under higher pressure we need to build a model with a volumetric zone of hydrate formation. 4. Gas hydrate formation in a volumetric zone 4.1. Main equations In this case besides the near (first) and far (third) zones there will appear the transition zone (second) where process of hydrate formation is taking place. And accordingly two boundaries appear. The first boundary x = x(n) appears between the near and transition zones. On this boundary the hydrate formation process stops. The second boundary x = x(d) appears between the far and transition zones. On this boundary the hydrate formation process starts. Processes of heat and mass transfer in the first and third zones can be described by the system of Eq. (12). In the transition zone (subscript i = 2) main equations shall be written with consideration of gas absorption and heat generation during hydrate formation:
@ @Shð2Þ @ qgð2Þ Sgð2Þ þ ; mSgð2Þ tgð2Þ qgð2Þ ¼ mGqh @t @x @t @Slð2Þ @Shð2Þ kð2Þ @pð2Þ ¼ mð1 GÞqh ; mSgð2Þ tgð2Þ ¼ ; mql @t @t lg @x @Shð2Þ @T @T @T ð2Þ @ þ mqh Lh qc ð2Þ þ mcg Sgð2Þ tgð2Þ qgð2Þ ð2Þ ¼ k : @x @t @x @x @t m
ð19Þ
Values of the temperature and pressure in the second zone are not limited by the condition of phase equilibrium (4). The value of water saturation in the transition zone can be found by integrating the second equation of the system (19):
ql ðSlð2Þ Sl0 Þ þ ð1 GÞqh Shð2Þ ¼ 0: Fig. 5. The critical gas injection pressure pcr depending on the absolute reservoir permeability k0 at different values of initial reservoir pressure p0. Lines 1 and 2 correspond to T0 = 3 and 5 MPa. Dashed line, parallel to abscissa axis, shows equilibrium pressure ps0 which corresponds to the initial reservoir temperature T0 = 280 K.
From here:
Slð2Þ ¼ Sl0
qh ð1 GÞShð2Þ : ql
ð20Þ
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The process of hydrate formation is possible simultaneously in the transition zone and on the first boundary. We assume that on the surface x = x(n) separating the near and transition zones (on the first boundary), a jump of hydrate saturation occurs from þ S h ¼ Shð1Þ to Sh ¼ ShðnÞ . This jump is caused by the transition on this boundary of part of water into gas hydrate. The value Sh(1) can be defined from (6) and hydrate saturation Sh(n) is needed to be found.For the gas saturation in the zone of phase transitions, conP sidering j¼g; l; h Sj ¼ 1, we can write the following equation:
Sgð2Þ
q ¼ 1 Sl0 1 h ð1 GÞ Shð2Þ :
ql
Let us evaluate the maximum change in the gas saturation Sg in the transition zone. With full transition of water, which initially saturates a porous medium, into gas hydrate the value Sg changes from Sg ¼ Sg0 ¼ 1 Sl0 to Sg ¼ Sgð1Þ ¼ 1 Shð1Þ ¼ 1 ql Sl0 =qh ð1 GÞ. Hence, with consideration of immobility of water and hydrate, for the maximum change in the gas saturation we can write:
DSg ¼ Sg0 Sgð1Þ ¼ Sl0 ð1 þ ql =qh ð1 GÞÞ: This equation implies that if the initial water saturation of porous medium (Sl0) less than 0.3, then the following condition is met: DSg << Sg0. For example, for the system ‘‘methane–water’’ at Sl0 = 0.2 we get: Sg0 = 0.8, a DSg = 0.05. Therefore, in this case we can neglect the change of gas saturation in the transition zone and we will consider that the value of gas saturation in this zone is constant and equal to the value of gas saturation in the far zone:
! qh G qh ð1 GÞ kð2Þ @pð2Þ kð1Þ @pð1Þ ¼ m Shð1Þ ShðnÞ þ 1 xðnÞ ; lg @x lg @x qgðnÞ ql ! @T ð1Þ @T ð2Þ @T ð2Þ T @pð2Þ ; k k ¼ m Shð1Þ ShðnÞ qh Lh xðnÞ ¼ @x @x @x pðnÞ @x ð24Þ
@pð3Þ @pð2Þ mlg ShðdÞ qh G xðdÞ ¼ ; @x @x kð3Þ qgðdÞ k
! @T ð2Þ T @pð2Þ ¼ : @x pðdÞ @x
@T ð2Þ @T ð3Þ k ¼ mShðdÞ qh Lh xðdÞ @x @x
ð25Þ
We will consider that on the second boundary x = x(d) the hydrate saturation Sh(d) is continuous and equal to zero:
ShðdÞ ¼ SþhðdÞ ¼ 0:
ð26Þ
This condition provides thermodynamic consistency of the solution in the third zone. In fact, let us assume that on the second boundary a jump of hydrate saturation occurs from S h ¼ ShðdÞ > 0 to Sþ h ¼ 0: Then, based on the Eq. (25), on the boundary x = x(d) we get:
@pð2Þ @pð3Þ < ; @x @x
@T ð2Þ @T ð3Þ > : @x @x
Besides, on this boundary we can write:
Sgð2Þ ¼ Sgð3Þ ¼ 1 Sl0 : The value of permeability for gas in the transition zone also is constant and equal to its value in the far zone:
kð2Þ ¼ kð3Þ ¼ k0 ð1 Sl0 Þ3 : On the first boundary (x = x(n)) the conditions of the balance of mass and heat can be presented in the following form:
m Shð1Þ ShðnÞ qh ð1 GÞxðnÞ ¼ mSlðnÞ ql xðnÞ ; q mSgð2Þ tgð2Þ xðnÞ þ m Shð1Þ ShðnÞ G h xðnÞ ¼ mSgð1Þ tgð1Þ xðnÞ ;
qgðnÞ
k
We can also rewrite the Eqs. (21) and (22):
@T ð1Þ @T ð2Þ k ¼ m Shð1Þ ShðnÞ qh Lh xðnÞ ; @x @x
@T ð2Þ T @pð2Þ ¼ ; @x pðdÞ @x
@T ð3Þs T @pð3Þ ¼ ; @x pðdÞ @x
where T(3)s is the equilibrium temperature of hydrate formation which corresponds to the pressure p(3). After substitution we get the following inequalities:
@T ð3Þ @T ð3Þ @T ð3Þs T @pð2Þ @T ð3Þs < < or < : @x pðdÞ @x @x @x @x The last inequation demonstrates that in the third zone appears area where water is in supercooled state. Thus, the solution (26) is the only possible thermodynamically consistent solution for the hydrate saturation on the second boundary.
ð21Þ
where xðnÞ is the velocity of the movement of the first boundary. On the surface x = x(d), which separates the transition and far zones (on the second boundary), equation of the balance of mass and heat can be written in the form:
mShðdÞ qh ð1 GÞxðdÞ ¼ m Sl0 Slð2Þ ql xðdÞ ; q mSgð3Þ tgð3Þ xðdÞ þ mShðdÞ G h xðdÞ ¼ mSgð2Þ tgð2Þ xðdÞ ;
qgðdÞ
k
ð22Þ
@T ð2Þ @T ð3Þ k ¼ mShðdÞ qh Lh xðdÞ ; @x @x
where xðdÞ is the velocity of the movement of the second boundary. We will consider that on the both boundaries the pressure and temperature are continuous. Based on the equation of gas state and the system (19) the equations of piezoconductivity and thermal diffusivity in the transition zone can be presented in the following form:
@pð2Þ @pð2Þ qh Gpð2Þ @Shð2Þ kð2Þ @ ¼ ; pð2Þ @t mSgð2Þ lg @x @x qg0 ð1 Sl0 Þ @t cg kð2Þ qgð2Þ @pð2Þ @T ð2Þ mqh Lh @Shð2Þ @T ð2Þ k @ @T ð2Þ þ ¼ þ : @t qc @x @x qclg @x @x qc @t
4.2. Self-similar solutions Let us introduce the self-similar variable: n. Eq. (23), with application of Leibenzon linearization, can be presented in the form:
! 2 2 dShð2Þ n dpð2Þ d dpð2Þ ¼ gð2Þ ; þ /h p2ð2Þ n 2 dn dn dn dn 2 Peð2Þ dpð2Þ dT ð2Þ n dT ð2Þ d dT ð2Þ n dShð2Þ þ ¼ DT ; 2 dn dn dn 2 dn 2p20 dn dn vðPÞ
kð2Þ p0 ðPÞ hG vð2Þ ¼ mð1S ; /h ¼ q qð1S ; DT ¼ mqqhcLh : l0 Þlg l0 Þ g0
ð2Þ where gð2Þ ¼ vðTÞ ;
Let us express the value tem (27) and
dT ð2Þ dn
¼
T 2p2ð2Þ
dp2ð2Þ dn
dShð2Þ dn
from the first equation of the sys-
from the condition of phase equilibrium
and then let us substitute these derivatives in the second equation of the system (27): 2
2
/ T þ DT dpð2Þ Peð2Þ dpð2Þ h n ¼ 2/h T dn dn 2p20 ð23Þ
ð27Þ
!2
! 2 /h T þ gð2Þ DT d dpð2Þ : þ dn dn /h T ð28Þ
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Let us transform Eq. (28), accepting the ratio of the first summand in the right part of this equation to the second one in the following form:
B¼
Peð2Þ Dp2ð2Þ /h T 2p20 ð/h T
þ gð2Þ DTÞ
:
where Dp2ð2Þ is the typical change of p2ð2Þ in the second zone. For the value B we can write:
B¼
Gcg T Dp2ð2Þ 2p20 Lh
! 2 2 n dpð2Þ d dpð2Þ ~ ð2Þ ¼g ; 2 dn dn dn
~ ð2Þ ¼ where g
/h T þgð2Þ DT /h T þDT
ð29Þ
:
After integration of the Eq. (29), we get distribution of pressure in the transition zone:
p2ð2Þ
¼
p2ðdÞ
þ
p2ðnÞ
p2ðdÞ
R nðdÞ n
R nðdÞ nðnÞ
2 exp 4g~nð2Þ dn : 2 exp 4g~nð2Þ dn
ð30Þ
Distribution of temperature T(2) which corresponds to the distribution of pressure p(2) is linked to it by the condition of phase equilibrium (4). Based on the first equation of the system (27) and using Eq. (30), we can write the following equation for distribution of hydrate saturation in the transition zone:
Shð2Þ ¼
qh G qgðnÞ
þ qh ð1GÞ 1 : q l
This system, after substituting previously obtained distributions of pressure and temperature, can be solved numerically. For this paper we used iteration method for solving this system. Obtained solutions for pressure, temperature and other parameters were used for analysis of the influence of different factors to the process of hydrate formation in a porous medium. 4.3. Calculation results
:
Analysis shows that, in the most of cases of practical interest, there are B << 1: Then the Eq. (28) can be presented in the form:
where DT ¼ mqqhcLh ; K ¼ mlg vðTÞ pðnÞ
T ðgð2Þ 1Þ 2/h T þ 2gð2Þ DT
ln
p2ð2Þ p2ðdÞ
ð31Þ
:
One of the parameters, which can be varied at the gas injection into a reservoir, is the pressure of gas injection pe. Depending on the pe three different situations can occur. Fig. 6 presents distribution of pressure, temperature and hydrate saturation at the different values of injection pressure. We accept values p0 = 3 MPa, Sl0 = 0.2 for the parameters which determine the initial condition of a porous medium, the rest of the parameters are the same as written in Section 3.3. The first curve corresponds to the case of gas injection without hydrate formation, the second curve – the mode with a frontal surface of hydrate formation, the third and the fourth curves – the mode with a volumetric zone of hydrate formation. Fig. 6 shows that, in the case of hydrate formation in a volumetric zone, the temperature in the transition zone rises above the initial reservoir temperature. Hence, in this case, injection of cold gas into a reservoir leads to heating of this reservoir. At that, if the injection pressure is increasing then the temperature is increasing. Besides, in accordance with Fig. 6, if the injection pressure is increasing then the length of transition zone is increasing. As well as, the quantity of formed hydrate, related exactly to the transition zone (excluding hydrate which formed on the first boundary
From this equation we can write the following equation for the value of Sh on the first boundary:
ShðnÞ ¼
T ðgð2Þ 1Þ 2/h T þ 2gð2Þ DT
ln
p2ðnÞ p2ðdÞ
ð32Þ
:
The value of water saturation on the first boundary can be defined by Eq. (20). Equations for the thermal diffusivity and for the piezoconductivity with the self-similar variable for the near (i = 1) and far (i = 3) zones can be presented in the form (15). Therefore, for distribution of pressure and temperature in the near zone we get relations which coincide with (16). For the far zone we can write:
R 2 n1 exp 4gn dn ð3Þ ; p2ð3Þ ¼ p20 þ p2ðdÞ p20 R 1 n2 exp dn nðdÞ 4gð3Þ 2 R1 Pe exp n4 2pð3Þ2 p2ð3Þ dn n 0 2 ;: T ð3Þ ¼ T 0 þ ðT ðdÞ T 0 Þ R 1 Peð3Þ 2 n exp p dn nðdÞ 4 2p2 ð3Þ 0
ðPÞ ð3Þ ðTÞ
v
q p c k kð3Þ p0 ðPÞ gð3Þ ¼ v ; vð3Þ ¼ mð1S ; Peð3Þ ¼ g0 k0l g ð3Þ ; l0 Þlg g
where
k(3)
is
the
permeability coefficient for the gas in the far zone. Positions of boundaries n = n(n) and n = n(d) and the values of parameters on these boundaries can be found from the following system of equations, which was obtained based on the Eqs. (21) and (22) with consideration of conditions (26):
kð2Þ dp2ð2Þ dn
dp2ð2Þ dn
¼
kð1Þ
dp2ð3Þ dn
;
dp2ð1Þ dn
dT ð2Þ dn
dT dT ¼ K Shð1Þ ShðnÞ nðnÞ ; dnð1Þ dnð2Þ ¼ D2T Shð1Þ ShðnÞ nðnÞ :
¼
dT ð3Þ ; dn
Fig. 6. The pressure, temperature and hydrate saturation depending on the selfsimilar variable n at different values of gas injection pressure pe. Lines 1, 2, 3 and 4 correspond to pe = 4, 5, 6 and 7 MPa; Te = 278 K.
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Fig. 7. The length of the volumetric zone of hydrate formation Dn = n(d) n(n) depending on the gas injection pressure pe at different values of absolute reservoir permeability k0. Lines 1, 2 and 3 correspond to k0 = 1014, 5 1015 and 1015 m2; p0 = 3 MPa, T0 = 280 K, Te = 278 K.
n = n(n)), is increasing. Thus, with increasing the pe the pattern of the process more and more differs from the pattern with a frontal surface of hydrate formation. The volumetric zone appears when the pe exceeds the some value which is close to the equilibrium pressure which corresponds to the initial reservoir temperature (Fig. 7). In our calculations the equilibrium pressure is 5.5 MPa. Further, with increasing the pe the length of volumetric zone of hydrate formation (Dn = n(d) n(n)) is increasing and the higher the reservoir permeability the faster this length increasing. Fig. 8 shows that with decreasing the reservoir permeability the length of volumetric zone of phase transitions is decreasing as well as the quantity of hydrate, which formed in the transition zone, is decreasing (comparatively with hydrate formation on the first boundary n = n(n)). Therefore, in the low permeability reservoirs the features of the process behavior become more typical for the mode with a frontal surface of hydrate formation. In this case the piezoconductivity of a reservoir is directly proportional to its absolute permeability. Generically, we can distinguish three intervals for the pe which corresponds: (1) Hydrate formation is not occurs; (2) The mode with a frontal surface of hydrate formation; (3) The mode with a volumetric zone of hydrate formation. The length of this zone is increasing with increasing the pressure of injected gas pe and with increasing the permeability of reservoir k0.
Fig. 8. The length of the volumetric zone of hydrate formation (a) and the hydrate saturation on the first boundary (from the side of the second zone) (b) depending on the absolute reservoir permeability k0 at different values of gas injection pressure pe. Lines 1 and 2 correspond to pe = 7 and 8 MPa; p0 = 3 MPa, T0 = 280 K, Te = 278 K.
~0 at which a Let’s define the value of absolute permeability k volumetric zone of hydrate formation changes into a frontal surface. For this we equate the coefficients of piezoconductivity and ~ð2Þ ¼ k ~0 ð1 S Þ3 : thermal diffusivity (g ¼ 1), considering that k ð2Þ
l0
~0 ð1 S Þ p mklg k k l0 0 ~0 ¼ ¼ ) k : mð1 Sl0 Þlg q p0 ð1 Sl0 Þ2 q 3
Analysis shows that the features of hydrate formation in a volumetric zone (big length of the transition zone and big quantity of hydrate formed in this zone) are strongly pronounced when the reservoir piezoconductivity
vðPÞ ð2Þ is much higher than its thermal
(T)
diffusivity v , i.e. when gð2Þ 1: At some values of permeability the
(T) vðPÞ (gð2Þ 1). In this case the values Dn ð2Þ is close to the v
and Sh(n) are vanish, i.e. a volumetric zone of hydrate formation changes into a frontal surface. It can be demonstrated that the transition zone changes into the frontal boundary of phase transitions if the following condition is ðPÞ
met: vð2Þ ¼ vðTÞ (gð2Þ ¼ 1). In fact, if this condition is met then from Eqs. (31) and (32) for the values of hydrate saturation in the transition zone we get: Sh(2) = 0 and Sh(n) = 0. Consequently, in this case the transition zone does not exist and all hydrate is formed on a frontal surface.
Fig. 9. Location of the self-similar coordinate of the boundary n(n), separating the near and transition zones, depending on the gas injection pressure pe at different values of absolute reservoir permeability k0. Curves labeling and other parameters are the same as in Fig. 8.
V.Sh. Shagapov et al. / International Journal of Heat and Mass Transfer 84 (2015) 1030–1039
With increasing the pressure of gas injection pe the length of the first zone, saturated with gas and hydrate, is increasing (Fig. 9). This is determined by the fact that hydrate formation is accompanied by gas absorption and heat generation. And as the pressure of cold gas injection is increasing the rate of gas inflow and heat withdrawal (because of convection) in the zone of hydrate formation is increasing. Increase of the length of the near zone with the increasing of permeability of reservoir is triggered by the fact that in the high permeability reservoirs heat-mass-exchange is more intensive. 5. Conclusions Analytical solutions of the self-similar problem of gas hydrate formation during injection of gas (with the temperature less than the initial temperature of reservoir) into a porous medium, which is in the initial state saturated with gas and water, were built. It is shown that hydrate formation can occur on a frontal surface as well as in a volumetric zone. We have obtained conditions which separate these modes of gas hydrate formation. It was determined that the mode with a frontal surface of hydrate formation is typical for a porous medium with low permeability, low pressure, and high initial temperature. In the result of the analysis of the aforementioned self-similar solutions, it was demonstrated that during the injection of a cold gas into a porous medium (with pressure above the equilibrium pressure corresponding to the initial reservoir temperature), the coefficient of piezoconductivity higher than the coefficient of thermal diffusivity is required for the emergence of a volumetric zone of gas hydrate formation. Conflict of interest None declared. Acknowledgment This work was supported of the Russian Foundation for Basic Research (Grant 14-01-31089).
1039
References [1] E.D. Sloan, C.A. Koh, Clathrate Hydrates of Natural Gases, third ed., CRC Press, Boca Raton, FL, USA, 2007. [2] Y.F. Makogon, Hydrates of Hydrocarbons, PennWell Publishing Company, Tulsa, OK, USA, 1997. [3] T. Ahmed, P.D. McKinney, Advanced Reservoir Engineering, Elsevier, Amsterdam, The Netherlands, 2005. [4] E.D. Sloan, C.A. Koh, Natural gas hydrates: recent advances and challenges in energy and environmental applications, AIChE J. 53 (7) (2007) 1636–1643. [5] I. Chatti, A. Delahaye, L. Fournaison, J.P. Petitet, Benefits and drawbacks of clathrate hydrates: a review of their areas of interest, Energy Convers. Manage. 46 (9–10) (2005) 1333–1343. [6] F.A. Kuznetsov, V.A. Istomin, T.V. Rodionova, Gas hydrates: history, modern state, prospects for investigations, Rossijskij Khimicheskij Zhurnal (Zhurnal Rossijskogo Khimicheskogo Obshchestva Im. D.I. Mendeleeva) 47 (3) (2003) 5– 19. [7] A.D. Duchkov, L.S. Sokolova, D.E. Ayunov, M.E. Permyakov, Assessment of potential of west siberian permafrost for the carbon dioxide storage, Earth’s Cryosphere 13 (4) (2009) 62–68. [8] P. Jadhawar, A.H. Mohammadi, J. Yang, B. Tohidi, Subsurface carbon dioxide storage through clathrate hydrate formation, Advances in the Geological Storage of Carbon Dioxide, Springer, The Netherlands, 2006, pp. 111–126. [9] S.Sh. Byk, Y.F. Makogon, V.I. Fomina, Gas Hydrates [in Russian], Khimiya, Moscow, 1980. [10] Y.F. Makogon, Hydrates of Natural Gas, PennWell Publishing Company, Tulsa, OK, USA, 1981. Translation by W.J. Cieslewicz. [11] 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. [12] O.R. Nurislamov, V.Sh. Shagapov, Gas injection into a moist porous medium with the formation of a gas hydrate, J. Appl. Math. Mech. 73 (5) (2009) 581– 591. [13] V.Sh. Shagapov, M.K. Khasanov, I.K. Gimaltdinov, M.V. Stolpovskii, Numerical modeling of formation of a gas hydrate in a finite-length porous bed purged by a gas, J. Appl. Mech. Tech. Phys. 52 (4) (2011) 599–607. [14] R.I. Nigmatulin, Dynamics of Multiphase Media, Hemisphere Publ. Corp., New York, 1991. [15] N.A. Gumerov, Self-similar growth of a gas hydrate layer separating gas and liquid, Fluid Dyn. 27 (5) (1992) 664–669. [16] E.A. Bondarev, L.N. Gabysheva, M.A. Kanibolotskii, Simulation study of the gashydrate formation process in piped gas flow, Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza (5) (1982) 105–112. [17] V.Sh. Shagapov, L.A. Nasyrova, The heating of porous medium partly filled with gas hydrate in the presence of impermeable boundaries, High Temp. 37 (5) (1999) 754–759. [18] E.D. Sloan, Natural Gas Clathrate Hydrates, Marcel Dekker, New York, 1998.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Formation of gas hydrates in a porous medium during an injection of cold gas V.Sh. Shagapov a, N.G. Musakaev b,⇑, M.K. Khasanov c a
Institute of Mechanics of Ufa Branch RAS, Ufa, Russia Tyumen Branch of Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Tyumen State University, Tyumen, Russia c Sterlitamak Branch of Bashkir State University, Sterlitamak, Russia b
a r t i c l e
i n f o
Article history: Received 17 March 2014 Received in revised form 16 December 2014 Accepted 22 January 2015
Keywords: Gas hydrates Hydrate formation Self-similar solution Porous medium
a b s t r a c t This research paper examines various features of the formation of gas hydrates in a porous medium during an injection of cold gas (the injected gas temperature is lower than the initial temperature of the porous medium). Self-similar solutions of this problem were built in a rectilinear-parallel approximation. These solutions explain the distribution of the main parameters in a porous medium during the formation of gas hydrates on a frontal surface as well as in an extended zone. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Technological processes in the oil and gas industry often include the formation of gas hydrates in a reservoir and inside both the underground and the surface oilfield equipment [1–3]. In most cases, the gas hydrate formation has a negative impact, due to partial or complete plugging of the tube flow area. However, the gas hydrate formation can also have a positive impact [4–6]; for example, it can be used to increase the capacity of the underground reservoirs to store hydrocarbon gas. The point being that in a hydrate state the same gas mass can be stored under considerably lower pressure and in less volume [1], thereby eliminating the need to build compressor stations and high pressure vessels when storing gas in a hydrate state. The most efficient way would be to build gas depots in areas of permafrost [7]. In this case, placing the underground vessels near the surface will ensure that building these hydrate depots is economically efficient and can be used to balance the variations of gas consumption in small industrial centers on a daily and seasonal basis [8,9]. Also, gas hydrates can be used for the disposal of greenhouse gases in underground depots (in a porous medium) [7,10]. To intensify the process of hydrate formation, it is necessary to create conditions which ensure high areas of contact between the ⇑ Corresponding author. E-mail addresses: [email protected] (V.Sh. Shagapov), [email protected] (N.G. Musakaev), [email protected] (M.K. Khasanov). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.01.105 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
gas and the liquid. Such favorable conditions are present in a porous medium, where the contact area is great. The equilibrium temperature for gas hydrates depends on the pressure. Therefore, under the thermal and depression effects of a porous medium, gas hydrate formation and decomposition are possible on an frontal surface as well as in a voluminous unit. This paper [11–13] is devoted to researching the gas hydrate formation process in a porous medium. This paper differs from the one aforementioned by additionally building analytic solutions of the problem in a rectilinear-parallel approximation and by making an analysis of these solutions. Furthermore, we have obtained the desired conditions by segregating modes of the gas hydrate formation on a frontal surface and in an extended zone. The influence of the pressure and temperature of the injected gas and the initial parameters of a porous medium on the hydrate formation process was examined. 2. Mathematical model 2.1. Main assumptions and equations Assume that a reservoir, initially filled with gas (methane) and water, fills half-space x > 0. Before the injection of gas, pressure p0 and temperature T0 in the reservoir correspond to thermodynamic conditions of the existence of gas and water in a free state:
t ¼ 0; x > 0 :
T ¼ T 0 ; p ¼ p0 :
ð1Þ
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Assume that through the left boundary of the reservoir (x = 0) the gas (methane) is being injected. Pressure pe and temperature Te of the injected gas correspond to the conditions of gas hydrate formation and are supported at this boundary at constant level (Fig. 1):
t > 0; x ¼ 0 :
T ¼ T e ; p ¼ pe : ðpe > ps ðT e ÞÞ:
ð2Þ
We adopt several assumptions enumerated here. A hydrate is a two-component system with a mass concentration of gas G. The mass concentration of the methane in a hydrate does not strongly depend on pressure and temperature (G = 0.11–0.13). Therefore, we will neglect the relationship of the mass concentration of the gas in a hydrate with the pressure and temperature. Suppose that porosity m is constant; the rock skeleton, the gas hydrate and water are all incompressible and immovable. In this paper we research the processes, the duration of which is considerably longer than the typical time of temperature equilibrium
s d2 =vðTÞ (d is the typical pore dimensions, vðTÞ ¼ k=qc is the thermal diffusivity of a porous medium). Therefore, we assume that the temperatures of a porous medium and of a saturant (gas, hydrate or water) are similar (one temperature process). With consideration of the accepted assumptions, the system of main equations can be written in the following form [11–14]:
@ qg mSg þ Gqh mSh þ div qg mSg~ tg ¼ 0; @t @ k ðq mSl þ ð1 GÞqh mSh Þ ¼ 0; mSg~ tg ¼ g grad p; @t l lg @ @ ðqcTÞ þ qg cg mSg~ tg grad T ¼ divðk grad TÞ þ ðqh Lh mSh Þ; @t @t X Sj ¼ 1; kg ¼ k0 S3g ; p ¼ qg Rg T; j¼g; l; h
qc ¼ ð1 mÞqsk csk þ m
X
qj Sj cj ; k ¼ ð1 mÞksk þ m
j¼g; l; h
X
Sj k j :
j¼g; l; h
ð3Þ Hereafter, subscripts sk, h, w and g are related to the parameters of the rock skeleton, hydrate, water and gas, respectively; qj (j = sk, h, w and g) is the true specific density of the jth phase, Sj (j = g, l, h) is the pore saturation of the jth phase, k0 is the absolute permeability of the reservoir, tg and lg are the velocity and dynamic viscosity of the gas phase, p is the pressure, T is the temperature, Lh is the specific heat of hydrate formation, cj (j = g, l, h) is the specific heat capacity of the jth phase, kj (j = sk, g, l, h) is the coefficient of the thermal conductivity of the jth phase, qc and k are the specific volumetric heat capacity and the coefficient of the thermal conductivity of the system.
pe
T s ðpÞ ¼ T 0 þ T lnðp=ps0 Þ;
ð4Þ
where T0 is the initial temperature of the system, ps0 is the equilibrium pressure corresponding to the initial temperature, T⁄ is the empirical parameter having temperature dimension and depending on the composition of the natural gas. During hydrate formation in a porous medium, in general, three typical zones can appear. In the zone located close to the surface, where gas is injected into a porous medium, pores are filled with gas and hydrate. In the zone farther away from the surface, the pores are filled with gas and water. In the transition zone, gas hydrate is being formed, that is why pores in this zone are filled with gas, water and hydrate. At the boundaries between these zones, jumps in phase saturation as well as jumps in mass and heat flows are observed. At these boundaries, the relations which follow from conditions of mass and heat balance are realized [11,14]:
½mðSh qh ð1 GÞ þ Sl ql ÞxðiÞ ¼ 0; h i m Sg qg ðtg xðiÞ Þ Sh qh G xðiÞ ¼ 0;
Here [w] is the jump of the parameter w on the boundary x(i) (i = n, d); x(n) is the boundary between the near zone and the transition zone, x(d) is the boundary between the transition zone and the far zone; the dot over x(i) denotes velocity of the movement of the boundary. Temperature and pressure at these boundaries are considered to be continuous.
GAS + WATER
p0 T0 x
0
x(n)
h i ½k grad T ¼ mLh qh Sh xðiÞ : ð5Þ
GAS + WATER + HYDRATE
GAS + HYDRATE
Te
The fourth equation of the system (3) omits summands related to the barothermal effect, as the change of temperature due to this effect is negligible in comparison with temperature changes due to hydrate formation. We assume that the specific heat of the hydrate formation is independent of pressure [11]. 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 water saturation of the formation 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. It should be noted that the system of Eq. (3) is not complete. For the zones in which the phase transition (hydrate formation) does not occur, this system must be supplemented by the condition of constant hydrate saturation. For the zone where the hydrate is formed, the system of Eq. (3) must be supplemented by the condition of phase equilibrium:
x(d)
Fig. 1. Pattern of gas injection into a porous medium with hydrate formation.
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2.2. Formation of gas hydrate
For intensity of hydrate formation jh, related to the unit of the surface area of the hydrate layer, we can take the relation [16]:
It should be noted that in the accepted equilibrium scheme (on the frontal boundaries as well as in the volumetric zones) the intensity of hydrate formation is limited by the removal of the latent heat of such formation.According to modern concepts [15], the formation of hydrate particles is in general accompanied by unbalanced processes. These processes are affected by the diffusion of hydrate formation gas ‘‘water–hydrate’’ through layers of water and hydrate to the surface of the contact. If we consider that the typical thickness of such layers of the size of pore channels is pffiffiffiffiffiffiffiffiffiffi k=m, then for the typical time tD of relaxation of diffusion at k , where D – difnon-equilibrium we get the assessed value tD ffi mD fusion coefficient. These times tD are very insignificant in comparison with the time periods of interest in the considered problem [11]. From the second equation of the system (3) we get ql ðSl Sl0 Þ þð1 GÞqh Sh ¼ 0, Sl0 is the initial water saturation. From here, P considering the relation j¼g; l; h Sj ¼ 1, we get:
Sl ¼ Sl0
qh q ð1 GÞSh ; Sg ¼ 1 Sl0 1 h ð1 GÞ Sh : ql ql
Further, assuming Sl = 0, for the near zone using these relations we can write:
Sh ¼
ql Sl0 ; Sg ¼ 1 Sh : qh ð1 GÞ
ð6Þ
Thus, in the far and near zones, there is a filtration of gas in a porous medium with constant values of ‘‘live’’ porosity and permeability, but these values are not equal for different zones. In this case, the initial water saturation must satisfy the condition Sl0 6 qh ð1 GÞ=ql , because Sh 6 1=ql . Let’s review the process of hydrate formation in a porous medium, saturated in the initial state with water and gas, assuming that the velocity of the hydrate formation is limited by the kinetics of this process. Assume that the gas is slightly soluble in water, and the relative mass content of water vapors in the gas phase is negligible. In this case, hydrate formation will take place on the boundary of contact ‘‘gas–water’’ in the form of a hard hydrate film, separating these two phases [8]. Equilibrium relative mass concentration of gas in the hydrate on the boundary with gas is higher than on the boundary with water. Therefore, the diffusion flow of gas to the surface of the water is appearing in the layer of hydrate [15]. As a result, the process of hydrate formation is taking place on the boundary of contact of the water and the hydrate. To determine the intensity of hydrate formation we use the following scheme. Pore volume is a set of cylindrical channels with the typical radius a. Water is fully wetting the rock, i.e. the wall of the pore channels is covered by a water film with an average thickness dl(t). Hydrate deposit builds up on the surface of the film with the typical thickness dh(t). For simplicity we restrict ourselves to the case when dl þ dh a. Based on the aforementioned scheme for water saturation and hydrate saturation we can write:
Sl ¼
a2 ða dl Þ2 2dl ; a2 a
Sh ¼
2 ða dl Þ2 ða dl Þ2 dh a2
2dh : a ð7Þ
During description of the velocity of the growth of the layer of gas hydrates, we assume that the radius of the curvature of the pore surface a is much thinner than the thickness of the water hydrate layer. Therefore, the process of the growth of the gas hydrate layer can be viewed within the confines of a onedimensional pattern.
jh ¼ qh
@dh : @t
ð8Þ
On the other hand this value is limited by the ‘‘stoichiometric’’ condition with the value of gas diffusion flow (on the boundary ‘‘water–hydrate’’) jg:
jh ¼ jg =kgh ;
ð9Þ
where kgh is the mass gas concentration in the gas hydrate on the fluid boundary. The value jg can be found using Fick’s law, considering the distribution of gas concentration in the gas hydrate as linear:
jg ¼ Dqh
Ggh Ghl : dh
ð10Þ
Values Ggh and Ghl are known functions of pressure and temperature and can be found with the diagram of phase equilibrium of the system ‘‘water–gas’’. Herewith, their difference DG ¼ Ggh Ghl is much less than they are themselves. On the basis of Eqs. (7)–(10) we get:
Sh ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8D t DG ; a2 G
ðSl > 0Þ;
where t is the duration of the process. On the basis of the last correlation, assuming Sh = 0, Sl = Sl0 with t = 0 and Sh = She, Sl = 0 with t = sh (sh – time of hydrate development), we get ql Sl0 ¼ ð1 GÞqh She . From here, with consideration 2 Sl0 ql Ga2 . of (10), we get that sh ¼ 8D DG ð1GÞq h
For values of the parameters Sl0 = 0.2, a = 105 m, D = 1012 m2/c, G = 0.12, and DG = 103, we get the assessed value for sh = 10 s. As is well known, for hydrate formation it is necessary that the temperature of the system ‘‘porous medium–saturant’’ was below the equilibrium temperature of the hydrate formation. But it should be taken into account that the process of hydrate formation is accompanied by a heat release. Therefore, hydrate formation can take place in a diffusion mode only in those cases when the typical size of the system is so small that the heat exchange with the external environment is sufficiently fast. This size of the porous sample can be defined by the following expression:
l¼
qffiffiffiffiffiffiffiffiffiffiffiffiffi
sh vðTÞ 0:03 m:
If the typical size of a porous sample is less than the value of l, then the velocity of the hydrate formation will be limited by diffusion, i.e. by the kinetics of the process. Since in the examined problem the size of the system ‘‘porous medium– saturant’’ is much greater than the value l, the process of hydrate formation will be limited by heat transmission in the porous medium. 3. Problem with a frontal boundary of phase transitions 3.1. Problem statement Let’s consider the case when hydrate formation occurs entirely on a frontal boundary. The transition zone is missing such that in the porous medium there are only two zones. In the first (near) zone, water was fully changed into gas hydrate condition, therefore, in the pores there are only gas and gas hydrate. In the second (far) zone, the porous medium is filled with gas and water. In this case, the gas mass-conservation equation, Darcy’s law and heat flow equation can be presented in the form:
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@ mSgðiÞ tgðiÞ qgðiÞ ¼ 0; @x @t kðiÞ @pðiÞ @T ðiÞ @T ðiÞ @ @T ðiÞ ; ; qc þ mcg SgðiÞ tgðiÞ qgðiÞ ¼ k mSgðiÞ tgðiÞ ¼ @x lg @x @t @x @x
mSgðiÞ
@ qgðiÞ
þ
ð11Þ where pðiÞ ¼ qgðiÞ Rg T ðiÞ , k(i) is the permeability coefficient for the gas; parameters of the first and second zones are provided with subscripts in parentheses i = 1, 2. On the surface x = x(s) separating the near and far zones, a jump þ of hydrate saturation occurs from S h ¼ Shð1Þ to Sh ¼ 0. Subsequently, on this surface, on the grounds of conditions of the balance of mass and heat (5), we get:
mShð1Þ qh ð1 GÞxðsÞ ¼ mSl0 ql xðsÞ ; mSgð2Þ qgðsÞ tgð2Þ xðsÞ þ mShð1Þ qh GxðsÞ ¼ mSgð1Þ qgðsÞ tgð1Þ xðsÞ ; k
@T ð1Þ @T ð2Þ k ¼ mShð1Þ qh Lh xðsÞ ; @x @x
ð12Þ
where Sh(1) is the hydrate saturation of the first zone, xðsÞ is the velocity of the movement of the boundary of phase transition. Subscript s in parentheses refers to parameters on the boundary of the hydrate formation. In this problem, the phase-transition front velocity is limited by the rate of removal of the heat released during the hydrate formation process, i.e., by the heat transfer in the reservoir. This is determined by the fact that heat exchange in extended natural reservoirs takes place very slowly (mainly at the expense of heat conductivity). For example, for the zone of length L = 1 m, the typical heat exchange time is given by L2/v(T) 106 s. For time values such as those much larger than the sh, the condition of temperature equilibrium in the zone of phase transition is met. Subsequently, the pressure and temperature on the boundary between the zones are linked by condition of phase equilibrium (4). For the frontal pattern of gas hydrate formation, conditions (1) and (2), considering the emergence of two zones, will change into:
t ¼ 0; x > 0 :
T ð2Þ ¼ T 0 ; pð2Þ ¼ p0 ;
t > 0; x ¼ 0 :
T ð1Þ ¼ T ð2Þ ¼ T ðsÞ :
If typical variations of temperature DT in the zone of filtration are not large [17] (for example, at DT << T0)), then in the equation of reservoir-pressure conductivity, which results from Eq. (11), the summand is negligible due to the variation of the temperature. Therefore, the system of Eq. (11) after changing can be written in the following form:
ð13Þ
The system of equations for finding coordinates of the boundary x(s) and values of parameters on it can be presented, considering the Eqs. (11) and (12), in the following form:
! @pð2Þ @pð1Þ qh G qh ð1 GÞ kð2Þ kð1Þ ¼ mShð1Þ lg þ 1 xðsÞ ; @x @x qgðsÞ ql k
@T ð1Þ @T ð2Þ k ¼ mShð1Þ qh Lh xðsÞ : @x @x
! 2 2 n dpðiÞ d dpðiÞ ; ¼ gðiÞ 2 dn dn dn
2 n dT ðiÞ PeðiÞ dpðiÞ dT ðiÞ d dT ðiÞ ; ¼ 2 þ 2 dn dn dn 2p0 dn dn ð15Þ
vðPÞ
where gðiÞ ¼ vðiÞ ðTÞ ,
k p q p c k ðPÞ vðiÞ ¼ mSðiÞgðiÞ 0l , PeðiÞ ¼ g0 k0l g ðiÞ , kðiÞ ¼ k0 S3gðiÞ (i = 1, 2). g g
After integration of the Eq. (15), for distribution of pressure and temperature in each zone the following relations:
Rn 2 p2e p2ðsÞ n ðsÞ exp 4gn dn ð1Þ p2ð1Þ ¼ p2ðsÞ þ 0 < n < nðsÞ ; R nðsÞ n2 exp 4g dn 0 ð1Þ 2 R nðsÞ Pe T e T ðsÞ n exp n4 2pð1Þ2 p2ð1Þ dn 2 0 0 < n < nðsÞ ; T ð1Þ ¼ T ðsÞ þ R nðsÞ Peð1Þ 2 n exp 4 2p2 pð1Þ dn 0 0
ð16Þ R 2 1 p2ðsÞ p20 n exp 4gn dn ð2Þ p2ð2Þ ¼ p20 þ ðnðsÞ < n < 1Þ; R1 n2 exp dn nðsÞ 4gð2Þ 2 R1 Pe T ðsÞ T 0 n exp n4 2pð2Þ2 p2ð2Þ dn 2 0 T ð2Þ ¼ T 0 þ ðnðsÞ < n < 1Þ: R1 Peð2Þ 2 n exp p dn nðsÞ 4 2p2 ð2Þ 0
ð17Þ Eq. (14) for the introduction of the self-similar variable n can be presented in the following form:
dT ð1Þ dT ð2Þ mSh qh Lh ¼ nðsÞ ; dn dn 2qc kð2Þ
On the boundary x = x(s) we involve the conditions of pressure and temperature continuity:
@pðiÞ @pðiÞ kðiÞ @ ; pðiÞ ¼ @t mSgðiÞ lg @x @x cg kðiÞ qgðiÞ @pðiÞ @T ðiÞ @T ðiÞ k @ @T ðiÞ þ ¼ : @t qc @x @x qclg @x @x
pffiffiffiffiffiffiffiffiffiffi Let us introduce the self-similar variable: n ¼ x= vðTÞ t. For that variable, the system of Eq. (11), with application of Leibenzon linearization, can be presented in the form of the system of ordinary differential equations of the 2nd order:
2
T ð1Þ ¼ T e ; pð1Þ ¼ pe :
pð1Þ ¼ pð2Þ ¼ pðsÞ ;
3.2. Self-similar solutions
dpð2Þ dn
2
kð1Þ
dpð1Þ
!
ql G ql ¼ mSl0 lg þ 1 nðsÞ vðTÞ pðsÞ : ð1 GÞqgðsÞ ð1 GÞqh dn ð18Þ
From these equations, after substituting the relations (16) and (17) by some numeric method, we can find the coordinate of the boundary of phase transitions n(s) and the values of parameters on it. 3.3. Calculation results Fig. 2 presents the distribution of temperature and pressure during injection of methane at the temperature Te = 278 K into a reservoir with the initial pressure p0 = 4 MPa and the initial water saturation Sl0 = 0.2. For other parameters defining the system, the following values were accepted: m = 0.1, G = 0.12, T0 = 280 K, T⁄ = 10 K, ps0 = 5.5 MPa, k0 = 1014 m2, Rg = 520 J/(K kg), qsk = 2000 kg/m3, qh = 900 kg/m3, ql = 1000 kg/m3, csk = 1000 J/ (K kg), ch = 2500 J/(K kg), cl = 4200 J/(K kg), cg = 1560 J/(K kg), ks = 2 W/(m K), kh = 2.11 W/(m K), kl = 0.58 W/(m K), lg = 105 kg/(m s), Lh = 5 105 J/kg [2,9,18]. From the layout of the curves on Fig. 2a, we can make the conclusion that during the injection of methane under the pressure pe = 5 MPa, the temperature of the reservoir before the front of hydrate formation is lower than the equilibrium temperature, and after the front this temperature is higher than that at
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Fig. 2. The reservoir temperature, equilibrium temperature and pressure at pe = 5 (a) and 6 MPa (b), Te = 278 K depending on the self-similar variable n. Dashed line is the equilibrium temperature which corresponds to the presented pressure distribution.
equilibrium. Hence, in this case, the solution for the model with a frontal surface of hydrate formation is consistent. If gas is injected at the same temperature Te = 278 K, but under the pressure pe = 6 MPa, the temperature of reservoir after the front of hydrate formation drops below than the equilibrium temperature (dashed line) which corresponds to the supercooling of water in this zone (Fig. 2b). Hence, in this case, we cannot find physically consistent solution using the model with a frontal surface of hydrate formation. In the result of the analysis of Fig. 2 we can make the conclusion that there is some injection pressure of gas pcr, which we call the critical injection pressure of gas, and if the injection pressure of gas is higher than pcr, then it is necessary to consider a volumetric zone of hydrate formation.The volumetric zone appears in the cases when on the boundary of phase transitions (n = n(s)) the following inequation is met
dT ð2Þ dn
<
dT ð2Þs , dn
where T(2)s is the tempera-
ture of hydrate formation corresponding to the pressure p(2). The derivative from the right part of the aforementioned inequation, with consideration of the Eq. (4), can be changed to:
dT ð2Þs dn
¼ 2pT 2
ðsÞ
dp2ð2Þ dn
The calculation experiments, based on the last inequation, to determine influence of different parameters on the critical injection pressure of gas pcr were held. Fig. 3 presents relation pcr (T0) during injection into reservoir of gas at the temperature Te = 278 K. The initial reservoir pressure p0 = 3 MPa. This figure shows that during injection of cold gas (Te < T0) the value pcr is increasing when the initial reservoir temperature is increasing, at that the critical pressure pcr is closely equal to the equilibrium pressure ps0 when the reservoir permeability is high (curve 1). During injection of warm gas (Te > T0) into reservoir with high permeability the pressure pcr is equal to the equilibrium pressure ps(Te) which corresponds to the temperature of the injected gas, i.e. the pressure pcr is equal to the minimum pressure of gas injection under which the emergence of a zone saturated with hydrates is possible. Hence, during injection of warm gas into reservoir with high permeability the mode with a frontal mode of hydrate formation is impossible.
.
Using the boundary conditions (18) we get the following condition for the emergence of an extended zone:
! 2 dpð1Þ dT ð1Þ Shð1Þ DT T nðsÞ < kð1Þ þ KShð1Þ nðsÞ ; dn dn 2 2kð2Þ p2ðsÞ where DT ¼ mqqhcLh ; K ¼ mlg vðTÞ pðsÞ
qh G qgðsÞ
þ qh ð1GÞ 1 . q l
Considering the relations in the near zone (16) this condition will look like:
2 n Pe ðT ðsÞ T e Þ exp 4ðsÞ 2pð1Þ2 p2ðsÞ DT Shð1Þ 0 2 nðsÞ R nðsÞ Peð1Þ 2 n 2 exp 4 2p2 pð1Þ dn 0 0 1 0 n2 kð1Þ p2ðsÞ p2e exp 4gðsÞ C ð1Þ T B C: BKShð1Þ nðsÞ þ < A 2 2 @ R nðsÞ nðsÞ 2kð2Þ pðsÞ dn exp 0 4g ð1Þ
Fig. 3. The critical gas injection pressure pcr depending on the initial reservoir temperature at different values of absolute reservoir permeability k0. Lines 1 and 2 correspond to k0 = 1015 and 1017 m2. Dashed line is the equilibrium pressure ps0 which corresponds to the initial reservoir temperature; p0 = 3 MPa.
V.Sh. Shagapov et al. / International Journal of Heat and Mass Transfer 84 (2015) 1030–1039
Fig. 4. The critical gas injection pressure pcr depending on the absolute reservoir permeability k0 at different values of initial reservoir temperature T0. Lines 1 and 2 correspond to T0 = 280 and 285 K. Dashed lines, parallel to abscissa axis, being horizontal asymptotes of solid curves, show equilibrium pressures ps0 which correspond to two initial reservoir temperatures; p0 = 3 MPa.
Another parameter, which affects to the pressure pcr, is the reservoir permeability. The higher permeability, the closer the critical pressure to the corresponding equilibrium pressure (Fig. 4). The pressure pcr is increasing when the permeability is decreasing. The numerical calculations show that the value of pressure pcr goes to infinity when the value of absolute permeability is close to values of the order 1019 m2. Hence, when the reservoir permeability is very low then the mode with a frontal surface of hydrate formation is realized at any gas injection pressure. In reservoirs with low permeability the pressure pcr is considerably decreasing when the initial reservoir pressure is increasing (Fig. 5). In reservoirs with high permeability the pressure pcr is almost independent of reservoir pressure and pcr is close to the equilibrium pressure ps0. Thus, the closer the initial system condition to the condition of phase equilibrium, the lesser pressure on the reservoir boundary is needed for the emergence of a volumetric zone of gas hydrate formation. Figs. 4 and 5 show that the mode with a frontal surface of hydrate formation is typical for a porous medium with low permeability, low pressure, and high initial temperature, i.e. for reservoirs in which the initial condition of gas and water is far from the conditions of gas hydrate formation.
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In the result of the calculations it was determined that, depending both on the injected gas parameters and on the parameters determining the initial system condition, there are possible two radically different solutions. For the gas injection pressure pe there is some maximum value, and if this value is exceeded, then by using a frontal model of hydrate formation it is impossible to find physically consistent solution. For the permeabilities k0 = 1012 1015 m2 (typical for real reservoirs) this maximum pressure with high accuracy coincides with the equilibrium pressure ps0 corresponding to initial temperature.This result can also be gotten from the following physical considerations. It follows from Fig. 2 that, with the considered values of permeability, the pressure on the front of phase transition is approximately coincides with the gas injection pressure. If pe > ps0 then the temperature on the front is higher than the initial temperature (temperature at infinity). Therefore, in the second zone, the reservoir temperature is gradually decreases. At this the coefficient of piezoconductivity v(p) is considerably higher than the coefficient of thermal diffusivity v(T). For example, for parameters corresponding to Fig. 2:
vðPÞ ¼ mSk pg l0 g ¼ 0:05 m2 =s, vðTÞ ¼ k=qc ¼ 105 m2 =s.
Therefore, the reservoir temperature in the second zone decreases faster than the pressure and the equilibrium temperature of hydrate formation which depends on this pressure. And, if we use the frontal model, this leads to the supercooling of water after the front. Hence, the model with a frontal surface of hydrate formation can describe the process without physical inconsistency only in the cases when the gas injection pressure is not exceeding the equilibrium pressure which corresponds to the initial temperature. To describe the hydrate formation process during injection of gas under higher pressure we need to build a model with a volumetric zone of hydrate formation. 4. Gas hydrate formation in a volumetric zone 4.1. Main equations In this case besides the near (first) and far (third) zones there will appear the transition zone (second) where process of hydrate formation is taking place. And accordingly two boundaries appear. The first boundary x = x(n) appears between the near and transition zones. On this boundary the hydrate formation process stops. The second boundary x = x(d) appears between the far and transition zones. On this boundary the hydrate formation process starts. Processes of heat and mass transfer in the first and third zones can be described by the system of Eq. (12). In the transition zone (subscript i = 2) main equations shall be written with consideration of gas absorption and heat generation during hydrate formation:
@ @Shð2Þ @ qgð2Þ Sgð2Þ þ ; mSgð2Þ tgð2Þ qgð2Þ ¼ mGqh @t @x @t @Slð2Þ @Shð2Þ kð2Þ @pð2Þ ¼ mð1 GÞqh ; mSgð2Þ tgð2Þ ¼ ; mql @t @t lg @x @Shð2Þ @T @T @T ð2Þ @ þ mqh Lh qc ð2Þ þ mcg Sgð2Þ tgð2Þ qgð2Þ ð2Þ ¼ k : @x @t @x @x @t m
ð19Þ
Values of the temperature and pressure in the second zone are not limited by the condition of phase equilibrium (4). The value of water saturation in the transition zone can be found by integrating the second equation of the system (19):
ql ðSlð2Þ Sl0 Þ þ ð1 GÞqh Shð2Þ ¼ 0: Fig. 5. The critical gas injection pressure pcr depending on the absolute reservoir permeability k0 at different values of initial reservoir pressure p0. Lines 1 and 2 correspond to T0 = 3 and 5 MPa. Dashed line, parallel to abscissa axis, shows equilibrium pressure ps0 which corresponds to the initial reservoir temperature T0 = 280 K.
From here:
Slð2Þ ¼ Sl0
qh ð1 GÞShð2Þ : ql
ð20Þ
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The process of hydrate formation is possible simultaneously in the transition zone and on the first boundary. We assume that on the surface x = x(n) separating the near and transition zones (on the first boundary), a jump of hydrate saturation occurs from þ S h ¼ Shð1Þ to Sh ¼ ShðnÞ . This jump is caused by the transition on this boundary of part of water into gas hydrate. The value Sh(1) can be defined from (6) and hydrate saturation Sh(n) is needed to be found.For the gas saturation in the zone of phase transitions, conP sidering j¼g; l; h Sj ¼ 1, we can write the following equation:
Sgð2Þ
q ¼ 1 Sl0 1 h ð1 GÞ Shð2Þ :
ql
Let us evaluate the maximum change in the gas saturation Sg in the transition zone. With full transition of water, which initially saturates a porous medium, into gas hydrate the value Sg changes from Sg ¼ Sg0 ¼ 1 Sl0 to Sg ¼ Sgð1Þ ¼ 1 Shð1Þ ¼ 1 ql Sl0 =qh ð1 GÞ. Hence, with consideration of immobility of water and hydrate, for the maximum change in the gas saturation we can write:
DSg ¼ Sg0 Sgð1Þ ¼ Sl0 ð1 þ ql =qh ð1 GÞÞ: This equation implies that if the initial water saturation of porous medium (Sl0) less than 0.3, then the following condition is met: DSg << Sg0. For example, for the system ‘‘methane–water’’ at Sl0 = 0.2 we get: Sg0 = 0.8, a DSg = 0.05. Therefore, in this case we can neglect the change of gas saturation in the transition zone and we will consider that the value of gas saturation in this zone is constant and equal to the value of gas saturation in the far zone:
! qh G qh ð1 GÞ kð2Þ @pð2Þ kð1Þ @pð1Þ ¼ m Shð1Þ ShðnÞ þ 1 xðnÞ ; lg @x lg @x qgðnÞ ql ! @T ð1Þ @T ð2Þ @T ð2Þ T @pð2Þ ; k k ¼ m Shð1Þ ShðnÞ qh Lh xðnÞ ¼ @x @x @x pðnÞ @x ð24Þ
@pð3Þ @pð2Þ mlg ShðdÞ qh G xðdÞ ¼ ; @x @x kð3Þ qgðdÞ k
! @T ð2Þ T @pð2Þ ¼ : @x pðdÞ @x
@T ð2Þ @T ð3Þ k ¼ mShðdÞ qh Lh xðdÞ @x @x
ð25Þ
We will consider that on the second boundary x = x(d) the hydrate saturation Sh(d) is continuous and equal to zero:
ShðdÞ ¼ SþhðdÞ ¼ 0:
ð26Þ
This condition provides thermodynamic consistency of the solution in the third zone. In fact, let us assume that on the second boundary a jump of hydrate saturation occurs from S h ¼ ShðdÞ > 0 to Sþ h ¼ 0: Then, based on the Eq. (25), on the boundary x = x(d) we get:
@pð2Þ @pð3Þ < ; @x @x
@T ð2Þ @T ð3Þ > : @x @x
Besides, on this boundary we can write:
Sgð2Þ ¼ Sgð3Þ ¼ 1 Sl0 : The value of permeability for gas in the transition zone also is constant and equal to its value in the far zone:
kð2Þ ¼ kð3Þ ¼ k0 ð1 Sl0 Þ3 : On the first boundary (x = x(n)) the conditions of the balance of mass and heat can be presented in the following form:
m Shð1Þ ShðnÞ qh ð1 GÞxðnÞ ¼ mSlðnÞ ql xðnÞ ; q mSgð2Þ tgð2Þ xðnÞ þ m Shð1Þ ShðnÞ G h xðnÞ ¼ mSgð1Þ tgð1Þ xðnÞ ;
qgðnÞ
k
We can also rewrite the Eqs. (21) and (22):
@T ð1Þ @T ð2Þ k ¼ m Shð1Þ ShðnÞ qh Lh xðnÞ ; @x @x
@T ð2Þ T @pð2Þ ¼ ; @x pðdÞ @x
@T ð3Þs T @pð3Þ ¼ ; @x pðdÞ @x
where T(3)s is the equilibrium temperature of hydrate formation which corresponds to the pressure p(3). After substitution we get the following inequalities:
@T ð3Þ @T ð3Þ @T ð3Þs T @pð2Þ @T ð3Þs < < or < : @x pðdÞ @x @x @x @x The last inequation demonstrates that in the third zone appears area where water is in supercooled state. Thus, the solution (26) is the only possible thermodynamically consistent solution for the hydrate saturation on the second boundary.
ð21Þ
where xðnÞ is the velocity of the movement of the first boundary. On the surface x = x(d), which separates the transition and far zones (on the second boundary), equation of the balance of mass and heat can be written in the form:
mShðdÞ qh ð1 GÞxðdÞ ¼ m Sl0 Slð2Þ ql xðdÞ ; q mSgð3Þ tgð3Þ xðdÞ þ mShðdÞ G h xðdÞ ¼ mSgð2Þ tgð2Þ xðdÞ ;
qgðdÞ
k
ð22Þ
@T ð2Þ @T ð3Þ k ¼ mShðdÞ qh Lh xðdÞ ; @x @x
where xðdÞ is the velocity of the movement of the second boundary. We will consider that on the both boundaries the pressure and temperature are continuous. Based on the equation of gas state and the system (19) the equations of piezoconductivity and thermal diffusivity in the transition zone can be presented in the following form:
@pð2Þ @pð2Þ qh Gpð2Þ @Shð2Þ kð2Þ @ ¼ ; pð2Þ @t mSgð2Þ lg @x @x qg0 ð1 Sl0 Þ @t cg kð2Þ qgð2Þ @pð2Þ @T ð2Þ mqh Lh @Shð2Þ @T ð2Þ k @ @T ð2Þ þ ¼ þ : @t qc @x @x qclg @x @x qc @t
4.2. Self-similar solutions Let us introduce the self-similar variable: n. Eq. (23), with application of Leibenzon linearization, can be presented in the form:
! 2 2 dShð2Þ n dpð2Þ d dpð2Þ ¼ gð2Þ ; þ /h p2ð2Þ n 2 dn dn dn dn 2 Peð2Þ dpð2Þ dT ð2Þ n dT ð2Þ d dT ð2Þ n dShð2Þ þ ¼ DT ; 2 dn dn dn 2 dn 2p20 dn dn vðPÞ
kð2Þ p0 ðPÞ hG vð2Þ ¼ mð1S ; /h ¼ q qð1S ; DT ¼ mqqhcLh : l0 Þlg l0 Þ g0
ð2Þ where gð2Þ ¼ vðTÞ ;
Let us express the value tem (27) and
dT ð2Þ dn
¼
T 2p2ð2Þ
dp2ð2Þ dn
dShð2Þ dn
from the first equation of the sys-
from the condition of phase equilibrium
and then let us substitute these derivatives in the second equation of the system (27): 2
2
/ T þ DT dpð2Þ Peð2Þ dpð2Þ h n ¼ 2/h T dn dn 2p20 ð23Þ
ð27Þ
!2
! 2 /h T þ gð2Þ DT d dpð2Þ : þ dn dn /h T ð28Þ
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Let us transform Eq. (28), accepting the ratio of the first summand in the right part of this equation to the second one in the following form:
B¼
Peð2Þ Dp2ð2Þ /h T 2p20 ð/h T
þ gð2Þ DTÞ
:
where Dp2ð2Þ is the typical change of p2ð2Þ in the second zone. For the value B we can write:
B¼
Gcg T Dp2ð2Þ 2p20 Lh
! 2 2 n dpð2Þ d dpð2Þ ~ ð2Þ ¼g ; 2 dn dn dn
~ ð2Þ ¼ where g
/h T þgð2Þ DT /h T þDT
ð29Þ
:
After integration of the Eq. (29), we get distribution of pressure in the transition zone:
p2ð2Þ
¼
p2ðdÞ
þ
p2ðnÞ
p2ðdÞ
R nðdÞ n
R nðdÞ nðnÞ
2 exp 4g~nð2Þ dn : 2 exp 4g~nð2Þ dn
ð30Þ
Distribution of temperature T(2) which corresponds to the distribution of pressure p(2) is linked to it by the condition of phase equilibrium (4). Based on the first equation of the system (27) and using Eq. (30), we can write the following equation for distribution of hydrate saturation in the transition zone:
Shð2Þ ¼
qh G qgðnÞ
þ qh ð1GÞ 1 : q l
This system, after substituting previously obtained distributions of pressure and temperature, can be solved numerically. For this paper we used iteration method for solving this system. Obtained solutions for pressure, temperature and other parameters were used for analysis of the influence of different factors to the process of hydrate formation in a porous medium. 4.3. Calculation results
:
Analysis shows that, in the most of cases of practical interest, there are B << 1: Then the Eq. (28) can be presented in the form:
where DT ¼ mqqhcLh ; K ¼ mlg vðTÞ pðnÞ
T ðgð2Þ 1Þ 2/h T þ 2gð2Þ DT
ln
p2ð2Þ p2ðdÞ
ð31Þ
:
One of the parameters, which can be varied at the gas injection into a reservoir, is the pressure of gas injection pe. Depending on the pe three different situations can occur. Fig. 6 presents distribution of pressure, temperature and hydrate saturation at the different values of injection pressure. We accept values p0 = 3 MPa, Sl0 = 0.2 for the parameters which determine the initial condition of a porous medium, the rest of the parameters are the same as written in Section 3.3. The first curve corresponds to the case of gas injection without hydrate formation, the second curve – the mode with a frontal surface of hydrate formation, the third and the fourth curves – the mode with a volumetric zone of hydrate formation. Fig. 6 shows that, in the case of hydrate formation in a volumetric zone, the temperature in the transition zone rises above the initial reservoir temperature. Hence, in this case, injection of cold gas into a reservoir leads to heating of this reservoir. At that, if the injection pressure is increasing then the temperature is increasing. Besides, in accordance with Fig. 6, if the injection pressure is increasing then the length of transition zone is increasing. As well as, the quantity of formed hydrate, related exactly to the transition zone (excluding hydrate which formed on the first boundary
From this equation we can write the following equation for the value of Sh on the first boundary:
ShðnÞ ¼
T ðgð2Þ 1Þ 2/h T þ 2gð2Þ DT
ln
p2ðnÞ p2ðdÞ
ð32Þ
:
The value of water saturation on the first boundary can be defined by Eq. (20). Equations for the thermal diffusivity and for the piezoconductivity with the self-similar variable for the near (i = 1) and far (i = 3) zones can be presented in the form (15). Therefore, for distribution of pressure and temperature in the near zone we get relations which coincide with (16). For the far zone we can write:
R 2 n1 exp 4gn dn ð3Þ ; p2ð3Þ ¼ p20 þ p2ðdÞ p20 R 1 n2 exp dn nðdÞ 4gð3Þ 2 R1 Pe exp n4 2pð3Þ2 p2ð3Þ dn n 0 2 ;: T ð3Þ ¼ T 0 þ ðT ðdÞ T 0 Þ R 1 Peð3Þ 2 n exp p dn nðdÞ 4 2p2 ð3Þ 0
ðPÞ ð3Þ ðTÞ
v
q p c k kð3Þ p0 ðPÞ gð3Þ ¼ v ; vð3Þ ¼ mð1S ; Peð3Þ ¼ g0 k0l g ð3Þ ; l0 Þlg g
where
k(3)
is
the
permeability coefficient for the gas in the far zone. Positions of boundaries n = n(n) and n = n(d) and the values of parameters on these boundaries can be found from the following system of equations, which was obtained based on the Eqs. (21) and (22) with consideration of conditions (26):
kð2Þ dp2ð2Þ dn
dp2ð2Þ dn
¼
kð1Þ
dp2ð3Þ dn
;
dp2ð1Þ dn
dT ð2Þ dn
dT dT ¼ K Shð1Þ ShðnÞ nðnÞ ; dnð1Þ dnð2Þ ¼ D2T Shð1Þ ShðnÞ nðnÞ :
¼
dT ð3Þ ; dn
Fig. 6. The pressure, temperature and hydrate saturation depending on the selfsimilar variable n at different values of gas injection pressure pe. Lines 1, 2, 3 and 4 correspond to pe = 4, 5, 6 and 7 MPa; Te = 278 K.
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Fig. 7. The length of the volumetric zone of hydrate formation Dn = n(d) n(n) depending on the gas injection pressure pe at different values of absolute reservoir permeability k0. Lines 1, 2 and 3 correspond to k0 = 1014, 5 1015 and 1015 m2; p0 = 3 MPa, T0 = 280 K, Te = 278 K.
n = n(n)), is increasing. Thus, with increasing the pe the pattern of the process more and more differs from the pattern with a frontal surface of hydrate formation. The volumetric zone appears when the pe exceeds the some value which is close to the equilibrium pressure which corresponds to the initial reservoir temperature (Fig. 7). In our calculations the equilibrium pressure is 5.5 MPa. Further, with increasing the pe the length of volumetric zone of hydrate formation (Dn = n(d) n(n)) is increasing and the higher the reservoir permeability the faster this length increasing. Fig. 8 shows that with decreasing the reservoir permeability the length of volumetric zone of phase transitions is decreasing as well as the quantity of hydrate, which formed in the transition zone, is decreasing (comparatively with hydrate formation on the first boundary n = n(n)). Therefore, in the low permeability reservoirs the features of the process behavior become more typical for the mode with a frontal surface of hydrate formation. In this case the piezoconductivity of a reservoir is directly proportional to its absolute permeability. Generically, we can distinguish three intervals for the pe which corresponds: (1) Hydrate formation is not occurs; (2) The mode with a frontal surface of hydrate formation; (3) The mode with a volumetric zone of hydrate formation. The length of this zone is increasing with increasing the pressure of injected gas pe and with increasing the permeability of reservoir k0.
Fig. 8. The length of the volumetric zone of hydrate formation (a) and the hydrate saturation on the first boundary (from the side of the second zone) (b) depending on the absolute reservoir permeability k0 at different values of gas injection pressure pe. Lines 1 and 2 correspond to pe = 7 and 8 MPa; p0 = 3 MPa, T0 = 280 K, Te = 278 K.
~0 at which a Let’s define the value of absolute permeability k volumetric zone of hydrate formation changes into a frontal surface. For this we equate the coefficients of piezoconductivity and ~ð2Þ ¼ k ~0 ð1 S Þ3 : thermal diffusivity (g ¼ 1), considering that k ð2Þ
l0
~0 ð1 S Þ p mklg k k l0 0 ~0 ¼ ¼ ) k : mð1 Sl0 Þlg q p0 ð1 Sl0 Þ2 q 3
Analysis shows that the features of hydrate formation in a volumetric zone (big length of the transition zone and big quantity of hydrate formed in this zone) are strongly pronounced when the reservoir piezoconductivity
vðPÞ ð2Þ is much higher than its thermal
(T)
diffusivity v , i.e. when gð2Þ 1: At some values of permeability the
(T) vðPÞ (gð2Þ 1). In this case the values Dn ð2Þ is close to the v
and Sh(n) are vanish, i.e. a volumetric zone of hydrate formation changes into a frontal surface. It can be demonstrated that the transition zone changes into the frontal boundary of phase transitions if the following condition is ðPÞ
met: vð2Þ ¼ vðTÞ (gð2Þ ¼ 1). In fact, if this condition is met then from Eqs. (31) and (32) for the values of hydrate saturation in the transition zone we get: Sh(2) = 0 and Sh(n) = 0. Consequently, in this case the transition zone does not exist and all hydrate is formed on a frontal surface.
Fig. 9. Location of the self-similar coordinate of the boundary n(n), separating the near and transition zones, depending on the gas injection pressure pe at different values of absolute reservoir permeability k0. Curves labeling and other parameters are the same as in Fig. 8.
V.Sh. Shagapov et al. / International Journal of Heat and Mass Transfer 84 (2015) 1030–1039
With increasing the pressure of gas injection pe the length of the first zone, saturated with gas and hydrate, is increasing (Fig. 9). This is determined by the fact that hydrate formation is accompanied by gas absorption and heat generation. And as the pressure of cold gas injection is increasing the rate of gas inflow and heat withdrawal (because of convection) in the zone of hydrate formation is increasing. Increase of the length of the near zone with the increasing of permeability of reservoir is triggered by the fact that in the high permeability reservoirs heat-mass-exchange is more intensive. 5. Conclusions Analytical solutions of the self-similar problem of gas hydrate formation during injection of gas (with the temperature less than the initial temperature of reservoir) into a porous medium, which is in the initial state saturated with gas and water, were built. It is shown that hydrate formation can occur on a frontal surface as well as in a volumetric zone. We have obtained conditions which separate these modes of gas hydrate formation. It was determined that the mode with a frontal surface of hydrate formation is typical for a porous medium with low permeability, low pressure, and high initial temperature. In the result of the analysis of the aforementioned self-similar solutions, it was demonstrated that during the injection of a cold gas into a porous medium (with pressure above the equilibrium pressure corresponding to the initial reservoir temperature), the coefficient of piezoconductivity higher than the coefficient of thermal diffusivity is required for the emergence of a volumetric zone of gas hydrate formation. Conflict of interest None declared. Acknowledgment This work was supported of the Russian Foundation for Basic Research (Grant 14-01-31089).
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References [1] E.D. Sloan, C.A. Koh, Clathrate Hydrates of Natural Gases, third ed., CRC Press, Boca Raton, FL, USA, 2007. [2] Y.F. Makogon, Hydrates of Hydrocarbons, PennWell Publishing Company, Tulsa, OK, USA, 1997. [3] T. Ahmed, P.D. McKinney, Advanced Reservoir Engineering, Elsevier, Amsterdam, The Netherlands, 2005. [4] E.D. Sloan, C.A. Koh, Natural gas hydrates: recent advances and challenges in energy and environmental applications, AIChE J. 53 (7) (2007) 1636–1643. [5] I. Chatti, A. Delahaye, L. Fournaison, J.P. Petitet, Benefits and drawbacks of clathrate hydrates: a review of their areas of interest, Energy Convers. Manage. 46 (9–10) (2005) 1333–1343. [6] F.A. Kuznetsov, V.A. Istomin, T.V. Rodionova, Gas hydrates: history, modern state, prospects for investigations, Rossijskij Khimicheskij Zhurnal (Zhurnal Rossijskogo Khimicheskogo Obshchestva Im. D.I. Mendeleeva) 47 (3) (2003) 5– 19. [7] A.D. Duchkov, L.S. Sokolova, D.E. Ayunov, M.E. Permyakov, Assessment of potential of west siberian permafrost for the carbon dioxide storage, Earth’s Cryosphere 13 (4) (2009) 62–68. [8] P. Jadhawar, A.H. Mohammadi, J. Yang, B. Tohidi, Subsurface carbon dioxide storage through clathrate hydrate formation, Advances in the Geological Storage of Carbon Dioxide, Springer, The Netherlands, 2006, pp. 111–126. [9] S.Sh. Byk, Y.F. Makogon, V.I. Fomina, Gas Hydrates [in Russian], Khimiya, Moscow, 1980. [10] Y.F. Makogon, Hydrates of Natural Gas, PennWell Publishing Company, Tulsa, OK, USA, 1981. Translation by W.J. Cieslewicz. [11] 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. [12] O.R. Nurislamov, V.Sh. Shagapov, Gas injection into a moist porous medium with the formation of a gas hydrate, J. Appl. Math. Mech. 73 (5) (2009) 581– 591. [13] V.Sh. Shagapov, M.K. Khasanov, I.K. Gimaltdinov, M.V. Stolpovskii, Numerical modeling of formation of a gas hydrate in a finite-length porous bed purged by a gas, J. Appl. Mech. Tech. Phys. 52 (4) (2011) 599–607. [14] R.I. Nigmatulin, Dynamics of Multiphase Media, Hemisphere Publ. Corp., New York, 1991. [15] N.A. Gumerov, Self-similar growth of a gas hydrate layer separating gas and liquid, Fluid Dyn. 27 (5) (1992) 664–669. [16] E.A. Bondarev, L.N. Gabysheva, M.A. Kanibolotskii, Simulation study of the gashydrate formation process in piped gas flow, Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza (5) (1982) 105–112. [17] V.Sh. Shagapov, L.A. Nasyrova, The heating of porous medium partly filled with gas hydrate in the presence of impermeable boundaries, High Temp. 37 (5) (1999) 754–759. [18] E.D. Sloan, Natural Gas Clathrate Hydrates, Marcel Dekker, New York, 1998.