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## 6th International Conference on Gas Hydrates

### Prediction of Hydrate Plugs in Gas Wells in Permafrost Bondarev, Edward; Argunova, Kira; Rozhin, Igor 2008-07-31

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`Prediction of Hydrate Plugs in Gas Wells in Permafrost Edward Bondarev Leading Researcher Kira Argunova Senior Researcher Igor Rozhin Senior Researcher Institute of Oil and Gas Problems, Yakutsk, Russia  Abstract An approach to predictions of position and size of hydrate plugs inside gas wells has been proposed. It is based on the mathematical model of steady non-isothermal flow of real gas in tubes and an algorithm of calculation of equilibrium conditions of hydrate formation. The proposed approach includes the following steps. 1) Numerically solve the system of ordinary differential equations to find the distributions of pressure and temperature along a particular well. 2) Represent the results of calculations as connection between pressure and temperature. 3) Find the intersection of this function with the calculated or experimental equilibrium curve for a particular natural gas. 4) Find the depth of well from the results of numerical solution. Keywords: hydrate; gas well; optimal regime; gas extraction; computational experiment. growth in the cross section x  x 1 while at T 02  T e temperature drops. One can estimate the influence of gas flow rate on dissipative processes by modeling the real gas flow in the heat-insulated pipe. For all cases the increase in mass flow rate results in temperature drop at the fixed cross section. Therefore at T 01  T e the increase in mass flow rate leads to two contrary trends: 1) gas temperature grows due to the decrease in heat transfer while 2) it drops because of throttling. At low flow rates the first trend prevails while at higher ones – the second. Therefore if one plots the curve of temperature at the fixed cross section versus mass flow rate it will have maximum at some optimum value M * .  Introduction Analysis of factors determining reliability of gas supply to consumers located in the permafrost zone show that the first weak link in the technological chain is a well itself and the adjoining bottom-hole zone of the gas reservoir. Just here the intensive cooling of gas occurs due to throttle effect and heat exchange with the permafrost rocks surrounding the well. Due to the fact that many fields have sufficiently high reservoir pressures the danger of gas hydrate formation directly in the wells arises, which can cause either lowering of flow rate or their complete plugging. Moreover, risk of well plugging with gas hydrates can arise at their stoppage because of low temperature of surrounding rocks. To prevent hydrate formation in wells it is necessary to establish such regime of gas extraction at which its temperature will be higher than the equilibrium temperature of hydrate formation. Such regime can be provided for at the projecting stage of field development, constructive parameters of wells being chosen accordingly. Evidently, it is not always possible because of different engineering constraints. In this connection study of the possibility of controlling the gas temperature without changing the constructive parameters of wells is of a significant interest. Such possibility is based on the following peculiarities of gas well temperature conditions, which are influenced by two factors: external heat transfer and internal dissipative thermodynamic processes being the consequence of gas imperfection. Let us estimate the effect of mass flow rate on each of these factors n considering temperature at a fixed cross section of a well x  x 1 at two mass flow rates M  2   M 1 and two input temperatures T  T e and T 1 0  2 0  Formulation of the Problem To make the quantitative estimation of this effect let us use the mathematical model of stationary gas flow in pipes derived in the monograph (Bondarev et al. 1988) where it was shown that at the normal operation of gas velocity of gas flow is much lower than that of the sound. For example, at mass flow rate of 10 kg/s and at well depth of 3000 m it is approximately 5 m/s. This allows reducing the equation of flow and energy to the following: dp     g sin    dx  dT dx    dp dx    D  cpM   M 4   T e  2  2 .5   T   ,  (1) g  sin  ,  (2)  cp  where  – gas density; с p – gas specific heat capacity; g – gravity acceleration;  , D – cross section and diameter of a pipe; x – coordinate along the pipe axis; p – pressure;  – angle of the pipe slope counted from the fixed horizontal plane;  – coefficient of hydraulic resistance;   Te  (where T e is the ambient temperature). In case when gas is perfect the flow rate increase at T 01  T e causes temperature 1  2  NINTH INTERNATIONAL CONFERENCE ON PERMAFROST  T – gas temperature;  – total coefficient of heat transfer; M   – mass flow rate of a gas being a constant.  Density is connected with pressure and temperature by the equation of state    p  ,  (3)  zRT  Coefficient of throttling formula    RT  2  z  c p p T    is determined by the  ; coefficient of gas compressibility z  is the empirical function of pressure and temperature; R – gas constant. As stated above, one of the possibilities of gas temperature control in a well is based on its non-monotone dependence on mass flow rate. Let us note that similar dependence on the production rate will not have such peculiarity. This phenomenon is explained by the structure of the equation (2). Indeed when one transfers the second member from the left to the right part of the equation and notices the dependence of the pressure gradient on flow rate it becomes clear that the intensity of gas heat exchange with the environment (the first member in the right part of the equation (2)) is inversely proportional to mass flow rate while that of throttling is directly proportional to the square of this magnitude. Therefore, to determine the possibility of controlling the gas temperature it is necessary to find its dependence on M . Let us formulate the initial conditions for the system of equations (1)-(2) in the form: (4) p ( 0 )  p 0 , T ( 0 )  T0 . In case when the Bertolou equation of state is used, z  1 k  coefficient k  0 . 07  Tc p 0 p cT0  p  b  1  2  , T  T   in the second variant (curve 2). In the case considered this difference makes approximately 12%. One must note that values of mass flow rates corresponding to the maximum temperature of gas at the wellhead practically do not depend on the temperature of rocks. The earlier investigations (Argunova & Bondarev 2005) show that depth and initial values of temperature and pressure have the basic effect on this parameter. The analysis of curves shows that the optimum conditions of gas extraction correspond to mass flow rate M *  5 kg/s and temperatures at the wellhead 275.2 K (the first variant) and 273 K (the second variant).  2  where  b6  Tc  2  ,  T0  , T c , p c – critical temperature and pressure  which depend on the natural gas composition.  Fig. 1. Dependence of dimensionless temperature at the wellhead on mass flow rate (bold numbers at the curves correspond to variants of calculation).  Results For the first variant calculations have been made at the following values of the parameters:   5 .82 W/(m2K); p 0  290  10  D  0 . 1 m; R  520 J/(kgK); p с  45 . 5  10  5  5  N/m2;  T0  323 K;  J/(kgK);  T с  191 K;  c p  2300  N/m2;    0 . 02 ;  Te  Te0  x ;  Te 0  323 K;   0 . 02 K/m; L  2900  variant  –  p 0  290  10  5  m. In the second 2  N/m ;  L  3000  m;  T e 0   x , 0  x  2593 , Te    271 . 15 K, 2593  x  3000  at the same values of the rest parameters. Thus, the first variant corresponds to the standard well, while the second one – to the well drilled in permafrost. Dependence of gas temperature at the wellhead on mass flow rate is presented in Fig. 1. As it has been expected gas temperature in the first variant (curve 1) is higher than that  Similar dependences for the pressure at the wellhead are presented in Fig. 2. The optimum conditions of gas extraction correspond to M  2 . 5 kg/s and pressures 216.3105 N/m2(the first variant) and 212.0105 N/m2(the second variant). Here non-monotone character of curves is explained by the fact that at low values of M the second member in the equation of motion (2) is very small and does not influence pressure distribution along the depth of the well. This distribution is determined only by the first member in this equation, which decreases with the temperature growth according to the equation of state (3). Due to the fact that at low flow rates the gas temperature increases with the well production rate (see Fig. 1) pressure in the wellhead will increase also until hydraulic resistance, described by the second member in the equation (1), becomes predominant. Thus, the above peculiarity of the wellhead pressure dependence on the well production rate reveals itself only at the non-isothermal regime of gas flow.  EDWARD BONDAREV ET AL. One must note particularly that the optimal regime of gas extraction providing the minimum loss of pressure corresponds to much less value of mass flow rate than in the case of the regime with the minimum heat loss.  Fig. 2. Dependence of dimensionless pressure in the wellhead on gas mass flow rate (bold numbers at the curves correspond to variants of calculation).  3  The presence of two different extrema indicates the possibility to state the problem of searching such gas extraction regime, which corresponds to the minimum loss of the full energy of gas flow. This energy is expended for resisting the gas friction against pipe walls and heat loss due to throttling and heat transfer to the environment. Comparative analysis of the results of these two variants allows drawing the following conclusion. The presence of the permafrost strata and slight increase in the well depth and bottom-hole pressure do not influence the location of temperature and pressure extrema. Yet pressure and temperature values at the wellhead decrease. To determine the danger of hydrate formation in the borehole one can use the following procedure suggested in (Argunova & Bondarev 2005). On the equilibrium curve of hydrate formation plotted according to the experimental data one lays on the dependence between pressure and temperature obtained by solving the problem (1) - (4). The coordinate x, above which gas will be cooled below the equilibrium temperature, is determined by the temperature corresponding to the intersection of these two curves. The example of using the above procedure is presented in Figs. 3, 5 (mass flow rate corresponds to the optimum temperature of a gas) and Figs. 4, 6 (mass flow rate corresponds to the optimum pressure of a gas). All calculations have been made for the second variant of the initial data.  Fig. 3. Intersection of the experimental equilibrium curve of hydrate formation (dots and dotted line) and calculated connection between gas pressure and temperature in the well for M = 5 kg/s.  4  NINTH INTERNATIONAL CONFERENCE ON PERMAFROST  Fig. 4. The same as in Fig. 3 for M = 2.5 kg/s.  Fig. 5. Dependence of temperature and pressure on the dimensionless depth of a well for M = 5 kg/s.  Fig. 6. The same as in Fig. 5 for M = 2.5 kg/s.  EDWARD BONDAREV ET AL. The data presented show that in those sections of wells, where depth changes from 0 to 1  0 .57   3000  1299 m (Fig. 5) and to 1  0 .56   3000  1330 m (Fig. 6), gas temperature will be lower than the equilibrium temperature of hydrate formation. It is seen that these depths are essentially larger than the permafrost thickness (the dotted vertical line in Figs. 5, 6). It should be noted in particular that for the variant where mass flow rate corresponds to the optimal gas pressure, the depth of the dangerous zone is larger than that in the variant where mass flow rate corresponds to the optimal gas temperature. The examples presented visually demonstrate possibilities of mathematical simulation and, particularly, computational experiment for solving the actual problems of improving the reliability of the gas production systems in the Northern regions.  References Argunova, K.K. & Bondarev, E.A. 2005. Control of gas well operation: possibilities of mathematical simulation. Science and Education, 1: 41-45. Bondarev, E.A., Vasiliev, V.I., Voevodin, A.F., et al. 1988. Thermohydrodynamics of the systems of gas production and transport. Novosibirsk: Nauka. Siberian Branch, 272 pp.  5  `

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