- Email: [email protected]
A research into thermal field in fluid-saturated porous media
Powder Technology 148 (2004) 72 – 77 www.elsevier.com/locate/powtec
A research into thermal field in fluid-saturated porous media Rim A. Valiullin, Ramil F. Sharafutdinov*, Airat Sh. Ramazanov Department of Physics, Bashkir State University, ul. Frunze 32, Ufa, Bashkortostan 450074, Russia
Abstract Until now, the theory, methodology of investigations, and interpretation of thermometry data have been most completely developed for single-phase (oil, water, or gas) flows in formations. However, multiphase (oil+gas, oil+water, and oil+water+gas) flows in formations are more common in practice. This is primarily typical for fields featuring a high value of gas factor and saturation pressure, as well as for cases of formation tests at low values of bottom-hole pressure. Analysis of actual thermograms under these conditions has shown that the earlierdeveloped techniques for the cases of single-phase flows in the formation and the well cannot be applied here. This paper presents research data on the influence of the adiabatic and Joule–Thomson effects and the heat of fluid degassing on temperature field in porous medium. D 2004 Elsevier B.V. All rights reserved. Keywords: Thermal field; Fluid-saturated; Porous media
1. Introduction Recent years have witnessed an extensive application of thermal methods for investigating wells when monitoring the development of oil and gas fields [1,17]. Temperature changes in the formation are brought about by manifestations of the adiabatic and Joule–Thomson effects, as well as by the heat of fluid degassing when the pressure in the well is reduced below the saturation of oil with gas. Initial studies in the area of temperature field theory taking into account thermodynamic effects in nonisothermal filtration of fluids in porous media were carried out in 1940 by Lapuk [2]. He considered a steady-state filtration of liquid and gas in horizontal porous layers as a throttle (isenthalpic) process (i.e., he showed that change of the fluid temperature was caused by display of the Joule–Thomson effect). Temperature field investigations taking into account Joule–Thomson effects and effects of adiabatic expansion are presented in the works of Chekayuk [1]. For the first time, Zolotarev and Nikolaevsky [4] and Kurbanov and Rozenberg [5] have developed the application of nonequilibrium thermodynamics for filtration. * Corresponding author. E-mail address: [email protected] (R.F. Sharafutdinov). 0032-5910/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2004.09.023
In Tesluk and Palii [6], the system of differential equations describing the movement of a liquid is offered under nonisothermal conditions of filtration, derived on the basis of irreversible processes thermodynamics. From the specified systems, a special case turned out—a known system of equations for filtration of aerated oil, developed by Muscat [7]. Approximately at the same time, the works of Bredberg et al. [21] and Miller [20] appeared, where a question of transition from a liquid phase into a gaseous (and on the contrary) one with easy components under nonisothermal conditions is considered. In the work of Tesluk and Palii, the system of equations for nonisothermal filtration of a multiphase, multicomponent system in an elastic deformable collector at phase and chemical transformations results. From these equations, at certain simplifications, the equation of energy close to the equation of Chekayuk turns out, but, which, in addition, takes into account the energy of phase and chemical transformations [8]. Atkison and Ramey [9] presented several models of heat transfer in fractured and nonfractured porous media. Problems of nonisothermal one-dimensional filtration are considered in the works of Alishaev et al. [10], Entov and Vinogradov [11], Rubinstein [12], and Bondarev [13].
R.A. Valiullin et al. / Powder Technology 148 (2004) 72–77
Research into problems of two-dimensional nonisothermal filtration was started later when more effective methods, numerical solutions of systems, and nonlinear equations with applications of finite difference method were developed, and powerful computers had appeared. On the given problem, the works of Alishaev et al., Zolotarev and Nikolaevsky, and others have been executed and published. They considered problems of filtration in various formulations. The theory of nonisothermal filtration including hydromechanics of viscous fluids, theory of phase transformation (relative permeabilities), and heat exchange were developed in works of outstanding domestic and foreign researchers [14–19]. To calculate the formation of the temperature field with due account taken of the above effects, one has to know the thermodynamic coefficient for actual formation fluids. With this aim in view, an experimental installation was developed based on the PVT bomb to determine the coefficients of the Joule–Thomson and the adiabatic effects, as well as to obtain the specific heat of oil degassing. The thermodynamic coefficients thus obtained are used for investigating a nonstationary temperature field in the porous medium for various conditions of pressure gradient and the content of gas dissolved in the fluid.
2. Experimental determination of thermodynamic coefficients 2.1. The Joule–Thomson coefficient The value of the Joule–Thomson coefficient e is affected by the physical and chemical properties of the percolating fluid, and it is different for fluids found in different fields. Therefore, to determine the value of the Joule–Thomson coefficients for formation fluids of specific oil fields, an experimental installation was developed whose diagram is presented in Fig. 1. It consists of the main units as follows: the choking element I, the temperature regime measuring circuit II, and the system to create a pressure gradient III. The choking element is a controllable (allowing to change the aperture width) needle valve, whose chamber (A) is connected to a standard pressure gauge 1, whereas chamber (B) is maintained at atmospheric pressure. To reduce heat exchange of the choking fluid with the body, the heat-insulating ebonite sleeve 3 was squeezed into the chamber (B). Temperature transducers 2 are implanted into the chambers. Semiconductor diodes incorporated in a direct shift bridge circuit are used as temperature sensors. The temperature characteristic of the measuring circuit is linear in a wide temperature range. This measuring circuit enables one to measure both temperature difference in chambers and absolute temperatures in each of the choker chambers. The system to create the pressure gradient is the commercially available UIPK1M installation for determining rock permeability, with this
73
Fig. 1. Diagram of the experimental installation to determine the Jopule– Thomson coefficient.
installation comprising the principal parts as follows: two electric motor-driven presses (4), the piston separator (5), tanks for the investigated fluid (6) and the working fluid (7), and the standard pressure gauge (1). Due to the fine adjustment of the presses, the amount of fluid that passes through the choke can be precisely determined, whereas a wide range within which the speed of the presses can be controlled enables one to maintain flow rates of up to 0.67106 m3/day. It should be underlined here that the flow is strictly constant at constant speed. The UIPK-1M can create pressure Be gradients Be of up to 30 MPa. If we assume that BT and BP T are small in relation to e P with one measurement taken, then measuring the pressure gradient (DP) and the temperature gradient (DT) in the choker chambers with the fluid choking established, one can determine the Joule–Thomson coefficient according to the formula e=DT/DP. The Joule–Thomson coefficient can be calculated using Þ the expression ecalculated ¼ ð1aT CP q , where a is thermal expansion coefficient of fluid, C p is the specific thermal capacity of the fluid, and q is the density of the fluid. The analysis demonstrates that the Joule–Thomson coefficient is practically independent of pressure, whereas Be 4 MPa1 . Under such conditions, measBT P ¼ 6 10 uring the temperature change that follows the final pressure gradient under constant enthalpy is sufficient for a direct evaluation of e. Several values (experimental and calculated) of the Joule–Thomson coefficients are presented in Table 1. 2.2. Adiabatic coefficient The value of the temperature change under the adiabatic process is proportional to the pressure change DP in the system: DT=gDP.
74
R.A. Valiullin et al. / Powder Technology 148 (2004) 72–77
Table 1 Values of the Joule–Thomson coefficients
Table 2 Values of adiabatic expansion coefficient for fluids
Fluid under study
e calculated (K/MPa)
e experimental (K/MPa)
Fluid under study
g calculated (K/MPa)
g experimental (K/MPa)
Fresh water Mineralized water Transformer oil Arlan crude (Bashkiriya) Fyodorovskaya crude (Western Siberia)
0.220 – 0.46 – –
0.216 0.225 0.410 0.415 0.337
Fresh water Mineralized water (q=1.18 g/cm3) Transformer oil Arlan crude (Bashkiriya) Tuimazy crude (Bashkiriya) Fyodorovskaya crude (Western Siberia)
0.015 – 0.097 – – –
0.016 0.030 0.098 0.134 0.059 0.137
In practice, the adiabatic processes comprise the socalled fast processes during which no significant heat exchange with the environment takes place, as well as processes in large-volume thermodynamic systems (in the atmosphere, for example). The proportionality coefficient g bears the name of the adiabatic one, and it can be calculated using the expression g ¼ CaTP q : The sign of g coincides with that of a, usually aN0 (ab0 only for water within the 0–4 8C temperature range). The value and the pattern of the system temperature change due to the adiabatic effect depend primarily on the velocity and the value of the pressure change in the system, and are determined by the fluid’s physical and chemical properties. The determination of the adiabatic coefficient for the fluid involves measuring temperature, depending on the pressure change subject to constant entropy. The latter means that a pressure change should take place within such a period of time as to enable one to ignore any heat exchange with the environment. To investigate the adiabatic effect, the authors employed the heat-insulated piston-type separator (5) of the UIPK-1M installation (Fig. 1). The chamber (C), which is filled with the fluid under study, is provided with a tube with the temperature sensor (12). The STZ-19 thermoresistor with the time constant of not more than 1 s is used as a sensor. The pressure generated by the installation is fed through the valve (9) and piston (8) into the chamber (C). With valves (11) and (13) closed, the opening of valve (10) immediately releases pressure in the chamber (D) and, correspondingly, in the chamber (C). By measuring pressure and temperature changes before and after the pressure release and bearing in mind that temperature in the chambers equalizes with the ambient temperature within the period of 900–1200 s whereas the duration of temperature measurement is 3–5 s, the adiabatic coefficient can be determined. The installation and the measurement technique thus developed enabled the authors to experimentally determine the value of adiabatic coefficients with an error not exceeding 2%. Table 2 presents experimental and calculated values of the coefficient for certain fluids. As opposed to the Joule–Thomson effect, the adiabatic effect both in liquids and in gases acts in one direction (i.e., their temperature increases when they are compressed and decreases when they expand).
2.3. Phase transition heat When the pressure (P) drops below the saturation pressure (Psat), there takes place in the porous medium a degassing of the fluid or a release of the gas dissolved in the fluid. With this taking place, energy is absorbed, which brings about a change in the system temperature. This process is characterized by latent phase transition heat or degassing heat (L; J/kg). When gas is dissolved, the corresponding amount of energy is produced. Estimates are known of phase transition heat when formation crude is degassed; these estimates range between 100 and 300 kJ/kg [2,3]. To estimate the specific heat of fluid degassing, the authors created an experimental installation whose principal diagram is shown in Fig. 2. The principal element is the thermally insulated vessel I (PVT bomb). The vessel is filled with a liquid and a gas. Pressure inside the vessel is built up by the manual press (2) and is registered by the pressure gauge (4). Opening valve (6), one can reduce the pressure inside the cylinder down to P1bP 0, where P0 is the pressure inside the vessel before the valve is opened, whereas P1 is the one after the valve is opened. When the pressure drops below the saturation pressure, gas starts to emanate from the liquid (i.e., the degassing process commences). The volume of gas that has been released is measured using the gas meter (10). Temperature changes in
Fig. 2. Principal diagram of the installation. (1) Thermally insulated container (PVT bomb); (2) piston separator; (3) pressure gauge; (4) thermostatic container; (5) temperature sensor; (6) gas canister; (7) ice-filled reservoir.
R.A. Valiullin et al. / Powder Technology 148 (2004) 72–77
the vessel were measured using two copper–constantan thermocouples. One thermocouple (9) was located in the vessel (1), whereas the other one was located in the thermostatic vessel (14). Temperature was measured using the microvoltmeter (15). Temperature variations were logged by an automatic recorder. The technique for evaluating specific heat of fluid degassing is as follows. Let us assume that there is some liquid in the vessel of volume V with a gas dissolved in this liquid. The volume is completely filled with the gassed fluid. The pressure in the system is reduced by the value of DP. In doing so, a temperature change takes place by the value of DT brought about by the adiabatic effect and that of degassing. Let us assume that the mass of the gas thus released is small, so we shall ignore the change in the fluid mass in the unit of volume. We shall apply here the linear law of Henry of isothermal solubility of gas in liquid. The process of adiabatic expansion of the gas–liquid mixture can be divided into two stages: the first one involves a reduction of pressure from the initial to the saturation one, whereas the second one comprises a reduction of pressure below the saturation one. Let us investigate both of these stages in a greater detail. We shall start with the first one (i.e., when PNPsat). The liquid oversaturated with gas is under the pressure P1, which is greater than the saturation pressure Psat. The temperature change that results from the adiabatic expansion of the gas and liquid mixture can be presented using the adiabatic coefficient g l for the liquid DT=g lDP. When the pressure in the mixture is equal to the saturation pressure (i.e., P=Psat), the mixture temperature will change by the value of T=g l(P1Psat). Let us now consider the second stage (i.e., at PbPsat). At some moment of time t, the liquid and the gas are at temperature T(t). Within the time DT, at the moment of time t+DT, the pressure will change by DP. The liquid temperature will change by g lDP, whereas the gas temperature will change by g gDP due to the adiabatic process. Besides, the issuing gas will change its temperature by the value of L/C g, where L is the specific latent heat of the phase transition, and C g is the specific heat capacity of the gas. After simple transformations, we shall obtain L=C lm lDT/m g, where m i=q iVi; Cl is the specific heat capacity of the liquid; V l is the volume of the liquid; DT is the change of the mixture temperature; and V g is the volume of gas released under normal conditions.
3. An investigation of temperature fields brought about by thermodynamic effects in porous media 3.1. Single-phase nonstationary movement of liquid The temperature change in a porous medium when a liquid is choked is determined by convective transfer and
75
the thermodynamic effects that accompany percolation— these effects being the choking effect, the adiabatic expansion effect, and that of phase transitions. The mathematic formulation of the problem used to calculate the temperature field with no account taken of the fluid degassing in a uniform porous medium with a singlephase percolation has the form: BT BT BP BP þ uðr; t Þ ¼ euðr; t Þ þ gn Bt Br Br Bt
ð1Þ
T ðr; 0Þ ¼ 0
ð2Þ
where u(r,t) is the velocity of convective heat transfer [m/s]; uðr; t Þ ¼ cvðr; t Þ ¼ c lk BP ˜ =C l /C p is the ratio of Br , where n volume heat capacity of the fluid and that of the porous medium saturated with this fluid; e,g are the Joule– Thomson and the adiabatic expansion coefficients of the fluid [K/Pa]; j=mc is the parameter that takes into account instantaneous heat exchange between the fluid and the skeleton of the porous medium; and P(r,t) is the pressure at the distance r from the well axis at the moment of time t, which is obtained by solving the corresponding boundary value problem of piezo-conductance. A general solution of Eqs. (1) and (2) along the characteristics r t (t,r 1) can be written as: T ðrt ; t Þ ¼ e½pðr1 ; 0Þ pðrt ; tÞ þ ðe þ gjÞ Z t Bpðts ; sÞ ds Bs 0
ð3Þ
Obviously, the first term is unambiguously determined by the initial and the current values of pressure, whereas the second one points at the dependence of temperature change on the nature of pressure change versus time along the fluid path. The calculations show that upon a lapse of some time after choking, the main contribution to the value of temperature change is made by the choking effect, and the following formula can be used for estimating calculations (for tVt 0): eDP qt Rk ln 1 þ c 2 ; R ¼ T ðrc ; tÞc ð4Þ 2lnR prc rc where R k is the radius to the border of the porous medium; r c is the well radius, q is the specific flow, and DP is the pressure gradient. Fig. 3 demonstrates the results of calculating, using formula (4), the temperature change brought about by the choking effect when the crude and water flow under the influence of the constant depression DP=5.0 MPa were: R=100, e o=0.5 K/MPa, e w=0.2 K/MPa, co=0.8, cw=1.2, q o=1 m3/day m, and q w=100 m3/day m. It is evident from the figure that initially water is heated more than oil. Thereafter, one can observe a bnormalQ change of oil and water temperature (ToNTw). In this sense, one can talk about an inversion in time of the value of the choking-induced
76
R.A. Valiullin et al. / Powder Technology 148 (2004) 72–77
Fig. 3. Dependence of temperature on time.
temperature anomaly within the intervals of oil and water inflow. This is primarily explained by different mobilities of oil and water: the more mobile water, within the same period of time, covers a larger portion of the depression funnel and, in spite of the lower value of the Joule– Thomson coefficient, it is heated more than oil. 3.2. Multiphase nonstationary movement accompanied by liquid degassing A two-phase movement of oil and gas is observed when the pressure is reduced below the one of saturating oil with gas. The mathematical model for calculating the temperature field in a nonstationary pressure field with due account taken of the Joule–Thomson effect, the adiabatic effect, and the heat of fluid degassing in a porous medium is written as: m
Bq0i cik Si Bt 1 B kKi BP rq0i cik ¼ 0; þ r Br li Br
i ¼ 1; 2; 3; k ¼ 1; 2 ð5Þ
# " 3 X B 0 0 qi Ci Si T þ mq1 S1 L ð1 mÞq0 C0 T þ m Bt i¼1 " # 3 X 1 B 0 0 þ r qi mSi vi Ci T þ mq1 S1 v1 L r Br i¼1 þ
3 X i¼0
q0i Ci ei mSi vi
3 X BP BP ¼ 0: q0i Ci Si gi Br Bt i¼1
Fig. 4 shows the dependencies for the values of thermodynamic parameters of phases as follows: C 0=800 J/(kg K), C 1=3000 J/(kg K), C 2=4000 J/(kg K), C 3=2000 J/ (kg K); e 1=4 K/MPa, e 2=0.2 K/MPa, e 3=0.4 K/MPa; g 1=20 K/MPa, g 2=0.015 K/MPa, g 3=0.13 K/MPa. The value of phase transfer heat during degassing oil is taken as L=100 kJ/kg. The viscosities of the gaseous, gassed aqueous, and oil phases are, respectively: l 1=0.01 mPa s, l 2=0.2 mPa s, and l 3=0.4 mPa s. Let us consider the peculiarities of forming the temperature field in the porous medium in a nonstationary pressure field. Curves 1–3 correspond to the following values of the established ratio of the dissolved gas volume to the volume of the fluid: 50, 100, and 150 m3/m3. It is seen from Fig. 4 that at the instantaneous pressure drop at the initial moments of time, one can observe a reduction in temperature resulting from the manifestation of the adiabatic effect and the heat of fluid degassing. As time passes, the formation of the heat field is effected by the choking heating of the fluid and gas cooling. It is obvious that as the portion of gas in the flow increases, one can observe a more intensive cooling of the fluid, which is connected with degassing gas choking heat. For the sake of comparing the influence of degassing heat on the temperature field, curve 4 in Fig. 4 ignores the heat of fluid degassing. Thus, when the pressure is instantaneously reduced below the one of saturation of oil with gas, at the initial moments, the main contribution to forming the temperature field is made by the adiabatic cooling of the fluid and the heat of fluid degassing. Thereafter, the contribution of the choking effect increases. The extent of the temperature drop is affected by the value of fluid degassing heat, the Joule–Thomson coefficient, and that of the adiabatic effect, by gas solubility and the pressure gradient. The combined influence of the cooling and heating effects results in the formation of a temperature minimum on the temperature curve. As this takes, depending on the amount of gas dissolved in the fluid and on the pressure gradient, the stationary distribution of temperature may
ð6Þ
Different i indices correspond to different phases, namely: 0—rock, 1—gas, 2—water, 3—oil; S i and v i are saturation and motion speed of the ith phase; Cik is concentration of the kth component in the ith phase; q i is density of the ith phase; K is absolute permeability; k i is phase permeability; l i is viscosity of the ith phase; m is porosity; P is pressure; T is temperature; C i is heat capacity; e i is the Joule–Thomson coefficient; g i is the adiabatic coefficient; and L is degassing heat.
Fig. 4. Dependence of temperature on time with due account taken of fluid degassing. Volume content of gas: (1) 50 m3/m3; (2) 100 m3/m3; (3) 150 m3/m3; and (4) without taking into account degassing heat, 150 m3/m3.
R.A. Valiullin et al. / Powder Technology 148 (2004) 72–77
2.
77
of the choking effect, on the other hand, result, in due course of time, to the formation of a temperature minimum. Using the peculiarities of the temperature field in the multiphase flow environment, simultaneously taking into account fluid degassing, enables one to address the tasks of identifying oil and water influxes from the formation and expands the potentialities of temperature well logging.
References Fig. 5. Results of investigating well no. 1378.
feature both positive (curves 1 and 2) and negative (curve 3) temperature anomalies. Below are examples of interpreting temperature investigations of wells in the multiphase flow environment. Fig. 5 presents the results of investigating well no. 1378 (Bashkiria). The average gas factor at this field is 47.7 m3/ metric tons. Two formations are perforated in this well within the intervals 2374.8–2376.8 and 2380–2381.2 m. The investigations were carried out in the process of compressor-assisted well stimulation. The compressor pump column is lowered to the depth of 2275 m. The following thermograms have been recorded: (1) a background thermogram, recorded before the compressor operation; (2) during compression; (3) immediately after compression and pressure release in the annular space; and (4 and 5) 2 and 8 h after the compression. Besides, diagrams of density measurements (6) are provided. Analyzing thermogram (2), one can see that, with the compressor operating, the lower perforated range is marked by slight heating due to choking of the liquid phase. After the pressure release in the annular space (lowering of the bottom-hole pressure), thermogram (3) recorded a temperature drop (cooling anomaly) opposite both of the two perforated subformations. The shape of the temperature curve above the operating ranges of the formation and the corresponding changes of temperature opposite the formations are characteristic of the predominant inflow from the gas phase formations. curve 4 corresponds to the case of the gas and oil mixture influx, whereas thermogram (5) corresponds to the inflow of oil from the lower formation. The upper formation is marked as an operating one only if gas flows in. When oil flows in, this formation is not identified, evidently, due to a low flow rate. It follows from Ref. [6] that the formations yield oil and that degassing takes place in the hole. The well operation figures show the sucker rod pump produce 2.4 m3/day oil. The gas factor for the well is 47.9 m3/metric ton.
4. Conclusions 1.
Oil cooling at the expense of degassing heat and gas choking, on the one hand, and its heating at the expense
[1] E.B. Chekayuk, Thermodynamics of Oil Formation, M. Nedra, 1965. [2] B.B. Lapuk, Thermodynamic Processes of Gassed Oil Moving in Porous Media, in: Azer. neftyanoye khozyaistvo, vol. 2, 1940. [3] G.F. Trebin, F.Yu. Kapyrin, O.G. Limanski, in: Estimation of Temperature Depression in the Well-Bore Zone of Producing Wells._I`. Collection of Papers of VNII, vol. 64, 1978, pp. 16 – 22. [4] P.P. Zolotarev, V.N. Nikolaevsky, Thermodynamical analysis of nonsteady-state processes in water and oil saturated flexible porous media, Year-Book, VNII, M. Nedra C, 1966, pp. 49 – 61. [5] A.K. Kurbanov, M.D. Rozenberg, Non-isothermal filtration of multiphase fluids, Year-Book: VNII. The Theory and Practice of Oil Recovery, M. Nedra, 1968. [6] E.V. Tesluk, A.O. Palii, Non-isothermal processes in the theory and practice of oil recovery, The Ways of the Development of Oil Recovery Industry in Kazakhstan, M. VNIIOENG. C, 1967, pp. 192 – 208. [7] M. Muscat, Flow of Homogeneous Fluids Through Porous Media, 1937 New York. [8] I.A. Charni, Underground Hydro-Gas Mechanics, M. Gostoptehizdat, 1963, 310c. [9] P.G. Atkison, H.J. Ramey, Problems of heat transfer in porous media. SPE 6792. [10] M.G. Alishaev, M.D., Rozenberg, E.V. Tesluk, Non-isothermal displacement of paraffin oil in heterogenetic collector with taking into account the kinetics of wax deposition. [11] V.M. Entov, I.M. Vinogradov, Effects of Dissipative Phenomena on Non-isothermal Displacement of Oil by a Solution of an Active Additive: Numerical Experiment. Institute for Problems in Mechanics RAS, Moscow, Russia. [12] L.I. Rubinstein, The total heat losses in injection of a hot fluid into a stratum, Neft Gas 2 (9) (1959). [13] E.A. Bondarev, V.E. Nikolaev, K.K. Argunova, Computing experiment in non-isothermal gas filtration, in: Proceedings of International Conference RDAMM, vol. 6, Pt. 2, 2001. [14] R.A. Valiullin, A.Sh. Ramazanov, R.F. Sharafutdinov, Barothermal effect under three-phase percolation with phase transitions, in: Izv. RAN, MZhG, vol. 6, 1994, pp. 113 – 117. [15] J.K. Roberts, Heat and Thermodynamics, Blackie and Son, London, 1942, pp. 267 – 271. [16] D.L. Katz, Thermodynamic analysis of frictional heat effects in pipeline flow, Oil Gas J. (1972 (March 6)) 87 – 90. [17] R.C. Smith, R.J. Steffensen, Interpretation of temperature profiles in water-injection wells, J. Pet. Technol. 27 (1975 (June)) 777 – 784. [18] M. Kaviany, Principles of Heat Transfer in Porous Media, SpringerVerlag, 1991. [19] Donald A. Nield, A. Bejan, Convection in Porous Media, SpringerVerlag, 1998. [20] C.D. Miller. //SPEJ.-1975.-v.21.-p.474–479. [21] J. Bredberg, S.-H. Peng, L. Davidson, On the wall boundary condition for computing heat transfer with k-omega models, in: J.H. Kim (Ed.), Proceedings of the ASME Heat Transfer Division - 2000, vol. 5, The American Society of Mechanical Engineers, 2000, pp. 243 – 250.
A research into thermal field in fluid-saturated porous media Rim A. Valiullin, Ramil F. Sharafutdinov*, Airat Sh. Ramazanov Department of Physics, Bashkir State University, ul. Frunze 32, Ufa, Bashkortostan 450074, Russia
Abstract Until now, the theory, methodology of investigations, and interpretation of thermometry data have been most completely developed for single-phase (oil, water, or gas) flows in formations. However, multiphase (oil+gas, oil+water, and oil+water+gas) flows in formations are more common in practice. This is primarily typical for fields featuring a high value of gas factor and saturation pressure, as well as for cases of formation tests at low values of bottom-hole pressure. Analysis of actual thermograms under these conditions has shown that the earlierdeveloped techniques for the cases of single-phase flows in the formation and the well cannot be applied here. This paper presents research data on the influence of the adiabatic and Joule–Thomson effects and the heat of fluid degassing on temperature field in porous medium. D 2004 Elsevier B.V. All rights reserved. Keywords: Thermal field; Fluid-saturated; Porous media
1. Introduction Recent years have witnessed an extensive application of thermal methods for investigating wells when monitoring the development of oil and gas fields [1,17]. Temperature changes in the formation are brought about by manifestations of the adiabatic and Joule–Thomson effects, as well as by the heat of fluid degassing when the pressure in the well is reduced below the saturation of oil with gas. Initial studies in the area of temperature field theory taking into account thermodynamic effects in nonisothermal filtration of fluids in porous media were carried out in 1940 by Lapuk [2]. He considered a steady-state filtration of liquid and gas in horizontal porous layers as a throttle (isenthalpic) process (i.e., he showed that change of the fluid temperature was caused by display of the Joule–Thomson effect). Temperature field investigations taking into account Joule–Thomson effects and effects of adiabatic expansion are presented in the works of Chekayuk [1]. For the first time, Zolotarev and Nikolaevsky [4] and Kurbanov and Rozenberg [5] have developed the application of nonequilibrium thermodynamics for filtration. * Corresponding author. E-mail address: [email protected] (R.F. Sharafutdinov). 0032-5910/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2004.09.023
In Tesluk and Palii [6], the system of differential equations describing the movement of a liquid is offered under nonisothermal conditions of filtration, derived on the basis of irreversible processes thermodynamics. From the specified systems, a special case turned out—a known system of equations for filtration of aerated oil, developed by Muscat [7]. Approximately at the same time, the works of Bredberg et al. [21] and Miller [20] appeared, where a question of transition from a liquid phase into a gaseous (and on the contrary) one with easy components under nonisothermal conditions is considered. In the work of Tesluk and Palii, the system of equations for nonisothermal filtration of a multiphase, multicomponent system in an elastic deformable collector at phase and chemical transformations results. From these equations, at certain simplifications, the equation of energy close to the equation of Chekayuk turns out, but, which, in addition, takes into account the energy of phase and chemical transformations [8]. Atkison and Ramey [9] presented several models of heat transfer in fractured and nonfractured porous media. Problems of nonisothermal one-dimensional filtration are considered in the works of Alishaev et al. [10], Entov and Vinogradov [11], Rubinstein [12], and Bondarev [13].
R.A. Valiullin et al. / Powder Technology 148 (2004) 72–77
Research into problems of two-dimensional nonisothermal filtration was started later when more effective methods, numerical solutions of systems, and nonlinear equations with applications of finite difference method were developed, and powerful computers had appeared. On the given problem, the works of Alishaev et al., Zolotarev and Nikolaevsky, and others have been executed and published. They considered problems of filtration in various formulations. The theory of nonisothermal filtration including hydromechanics of viscous fluids, theory of phase transformation (relative permeabilities), and heat exchange were developed in works of outstanding domestic and foreign researchers [14–19]. To calculate the formation of the temperature field with due account taken of the above effects, one has to know the thermodynamic coefficient for actual formation fluids. With this aim in view, an experimental installation was developed based on the PVT bomb to determine the coefficients of the Joule–Thomson and the adiabatic effects, as well as to obtain the specific heat of oil degassing. The thermodynamic coefficients thus obtained are used for investigating a nonstationary temperature field in the porous medium for various conditions of pressure gradient and the content of gas dissolved in the fluid.
2. Experimental determination of thermodynamic coefficients 2.1. The Joule–Thomson coefficient The value of the Joule–Thomson coefficient e is affected by the physical and chemical properties of the percolating fluid, and it is different for fluids found in different fields. Therefore, to determine the value of the Joule–Thomson coefficients for formation fluids of specific oil fields, an experimental installation was developed whose diagram is presented in Fig. 1. It consists of the main units as follows: the choking element I, the temperature regime measuring circuit II, and the system to create a pressure gradient III. The choking element is a controllable (allowing to change the aperture width) needle valve, whose chamber (A) is connected to a standard pressure gauge 1, whereas chamber (B) is maintained at atmospheric pressure. To reduce heat exchange of the choking fluid with the body, the heat-insulating ebonite sleeve 3 was squeezed into the chamber (B). Temperature transducers 2 are implanted into the chambers. Semiconductor diodes incorporated in a direct shift bridge circuit are used as temperature sensors. The temperature characteristic of the measuring circuit is linear in a wide temperature range. This measuring circuit enables one to measure both temperature difference in chambers and absolute temperatures in each of the choker chambers. The system to create the pressure gradient is the commercially available UIPK1M installation for determining rock permeability, with this
73
Fig. 1. Diagram of the experimental installation to determine the Jopule– Thomson coefficient.
installation comprising the principal parts as follows: two electric motor-driven presses (4), the piston separator (5), tanks for the investigated fluid (6) and the working fluid (7), and the standard pressure gauge (1). Due to the fine adjustment of the presses, the amount of fluid that passes through the choke can be precisely determined, whereas a wide range within which the speed of the presses can be controlled enables one to maintain flow rates of up to 0.67106 m3/day. It should be underlined here that the flow is strictly constant at constant speed. The UIPK-1M can create pressure Be gradients Be of up to 30 MPa. If we assume that BT and BP T are small in relation to e P with one measurement taken, then measuring the pressure gradient (DP) and the temperature gradient (DT) in the choker chambers with the fluid choking established, one can determine the Joule–Thomson coefficient according to the formula e=DT/DP. The Joule–Thomson coefficient can be calculated using Þ the expression ecalculated ¼ ð1aT CP q , where a is thermal expansion coefficient of fluid, C p is the specific thermal capacity of the fluid, and q is the density of the fluid. The analysis demonstrates that the Joule–Thomson coefficient is practically independent of pressure, whereas Be 4 MPa1 . Under such conditions, measBT P ¼ 6 10 uring the temperature change that follows the final pressure gradient under constant enthalpy is sufficient for a direct evaluation of e. Several values (experimental and calculated) of the Joule–Thomson coefficients are presented in Table 1. 2.2. Adiabatic coefficient The value of the temperature change under the adiabatic process is proportional to the pressure change DP in the system: DT=gDP.
74
R.A. Valiullin et al. / Powder Technology 148 (2004) 72–77
Table 1 Values of the Joule–Thomson coefficients
Table 2 Values of adiabatic expansion coefficient for fluids
Fluid under study
e calculated (K/MPa)
e experimental (K/MPa)
Fluid under study
g calculated (K/MPa)
g experimental (K/MPa)
Fresh water Mineralized water Transformer oil Arlan crude (Bashkiriya) Fyodorovskaya crude (Western Siberia)
0.220 – 0.46 – –
0.216 0.225 0.410 0.415 0.337
Fresh water Mineralized water (q=1.18 g/cm3) Transformer oil Arlan crude (Bashkiriya) Tuimazy crude (Bashkiriya) Fyodorovskaya crude (Western Siberia)
0.015 – 0.097 – – –
0.016 0.030 0.098 0.134 0.059 0.137
In practice, the adiabatic processes comprise the socalled fast processes during which no significant heat exchange with the environment takes place, as well as processes in large-volume thermodynamic systems (in the atmosphere, for example). The proportionality coefficient g bears the name of the adiabatic one, and it can be calculated using the expression g ¼ CaTP q : The sign of g coincides with that of a, usually aN0 (ab0 only for water within the 0–4 8C temperature range). The value and the pattern of the system temperature change due to the adiabatic effect depend primarily on the velocity and the value of the pressure change in the system, and are determined by the fluid’s physical and chemical properties. The determination of the adiabatic coefficient for the fluid involves measuring temperature, depending on the pressure change subject to constant entropy. The latter means that a pressure change should take place within such a period of time as to enable one to ignore any heat exchange with the environment. To investigate the adiabatic effect, the authors employed the heat-insulated piston-type separator (5) of the UIPK-1M installation (Fig. 1). The chamber (C), which is filled with the fluid under study, is provided with a tube with the temperature sensor (12). The STZ-19 thermoresistor with the time constant of not more than 1 s is used as a sensor. The pressure generated by the installation is fed through the valve (9) and piston (8) into the chamber (C). With valves (11) and (13) closed, the opening of valve (10) immediately releases pressure in the chamber (D) and, correspondingly, in the chamber (C). By measuring pressure and temperature changes before and after the pressure release and bearing in mind that temperature in the chambers equalizes with the ambient temperature within the period of 900–1200 s whereas the duration of temperature measurement is 3–5 s, the adiabatic coefficient can be determined. The installation and the measurement technique thus developed enabled the authors to experimentally determine the value of adiabatic coefficients with an error not exceeding 2%. Table 2 presents experimental and calculated values of the coefficient for certain fluids. As opposed to the Joule–Thomson effect, the adiabatic effect both in liquids and in gases acts in one direction (i.e., their temperature increases when they are compressed and decreases when they expand).
2.3. Phase transition heat When the pressure (P) drops below the saturation pressure (Psat), there takes place in the porous medium a degassing of the fluid or a release of the gas dissolved in the fluid. With this taking place, energy is absorbed, which brings about a change in the system temperature. This process is characterized by latent phase transition heat or degassing heat (L; J/kg). When gas is dissolved, the corresponding amount of energy is produced. Estimates are known of phase transition heat when formation crude is degassed; these estimates range between 100 and 300 kJ/kg [2,3]. To estimate the specific heat of fluid degassing, the authors created an experimental installation whose principal diagram is shown in Fig. 2. The principal element is the thermally insulated vessel I (PVT bomb). The vessel is filled with a liquid and a gas. Pressure inside the vessel is built up by the manual press (2) and is registered by the pressure gauge (4). Opening valve (6), one can reduce the pressure inside the cylinder down to P1bP 0, where P0 is the pressure inside the vessel before the valve is opened, whereas P1 is the one after the valve is opened. When the pressure drops below the saturation pressure, gas starts to emanate from the liquid (i.e., the degassing process commences). The volume of gas that has been released is measured using the gas meter (10). Temperature changes in
Fig. 2. Principal diagram of the installation. (1) Thermally insulated container (PVT bomb); (2) piston separator; (3) pressure gauge; (4) thermostatic container; (5) temperature sensor; (6) gas canister; (7) ice-filled reservoir.
R.A. Valiullin et al. / Powder Technology 148 (2004) 72–77
the vessel were measured using two copper–constantan thermocouples. One thermocouple (9) was located in the vessel (1), whereas the other one was located in the thermostatic vessel (14). Temperature was measured using the microvoltmeter (15). Temperature variations were logged by an automatic recorder. The technique for evaluating specific heat of fluid degassing is as follows. Let us assume that there is some liquid in the vessel of volume V with a gas dissolved in this liquid. The volume is completely filled with the gassed fluid. The pressure in the system is reduced by the value of DP. In doing so, a temperature change takes place by the value of DT brought about by the adiabatic effect and that of degassing. Let us assume that the mass of the gas thus released is small, so we shall ignore the change in the fluid mass in the unit of volume. We shall apply here the linear law of Henry of isothermal solubility of gas in liquid. The process of adiabatic expansion of the gas–liquid mixture can be divided into two stages: the first one involves a reduction of pressure from the initial to the saturation one, whereas the second one comprises a reduction of pressure below the saturation one. Let us investigate both of these stages in a greater detail. We shall start with the first one (i.e., when PNPsat). The liquid oversaturated with gas is under the pressure P1, which is greater than the saturation pressure Psat. The temperature change that results from the adiabatic expansion of the gas and liquid mixture can be presented using the adiabatic coefficient g l for the liquid DT=g lDP. When the pressure in the mixture is equal to the saturation pressure (i.e., P=Psat), the mixture temperature will change by the value of T=g l(P1Psat). Let us now consider the second stage (i.e., at PbPsat). At some moment of time t, the liquid and the gas are at temperature T(t). Within the time DT, at the moment of time t+DT, the pressure will change by DP. The liquid temperature will change by g lDP, whereas the gas temperature will change by g gDP due to the adiabatic process. Besides, the issuing gas will change its temperature by the value of L/C g, where L is the specific latent heat of the phase transition, and C g is the specific heat capacity of the gas. After simple transformations, we shall obtain L=C lm lDT/m g, where m i=q iVi; Cl is the specific heat capacity of the liquid; V l is the volume of the liquid; DT is the change of the mixture temperature; and V g is the volume of gas released under normal conditions.
3. An investigation of temperature fields brought about by thermodynamic effects in porous media 3.1. Single-phase nonstationary movement of liquid The temperature change in a porous medium when a liquid is choked is determined by convective transfer and
75
the thermodynamic effects that accompany percolation— these effects being the choking effect, the adiabatic expansion effect, and that of phase transitions. The mathematic formulation of the problem used to calculate the temperature field with no account taken of the fluid degassing in a uniform porous medium with a singlephase percolation has the form: BT BT BP BP þ uðr; t Þ ¼ euðr; t Þ þ gn Bt Br Br Bt
ð1Þ
T ðr; 0Þ ¼ 0
ð2Þ
where u(r,t) is the velocity of convective heat transfer [m/s]; uðr; t Þ ¼ cvðr; t Þ ¼ c lk BP ˜ =C l /C p is the ratio of Br , where n volume heat capacity of the fluid and that of the porous medium saturated with this fluid; e,g are the Joule– Thomson and the adiabatic expansion coefficients of the fluid [K/Pa]; j=mc is the parameter that takes into account instantaneous heat exchange between the fluid and the skeleton of the porous medium; and P(r,t) is the pressure at the distance r from the well axis at the moment of time t, which is obtained by solving the corresponding boundary value problem of piezo-conductance. A general solution of Eqs. (1) and (2) along the characteristics r t (t,r 1) can be written as: T ðrt ; t Þ ¼ e½pðr1 ; 0Þ pðrt ; tÞ þ ðe þ gjÞ Z t Bpðts ; sÞ ds Bs 0
ð3Þ
Obviously, the first term is unambiguously determined by the initial and the current values of pressure, whereas the second one points at the dependence of temperature change on the nature of pressure change versus time along the fluid path. The calculations show that upon a lapse of some time after choking, the main contribution to the value of temperature change is made by the choking effect, and the following formula can be used for estimating calculations (for tVt 0): eDP qt Rk ln 1 þ c 2 ; R ¼ T ðrc ; tÞc ð4Þ 2lnR prc rc where R k is the radius to the border of the porous medium; r c is the well radius, q is the specific flow, and DP is the pressure gradient. Fig. 3 demonstrates the results of calculating, using formula (4), the temperature change brought about by the choking effect when the crude and water flow under the influence of the constant depression DP=5.0 MPa were: R=100, e o=0.5 K/MPa, e w=0.2 K/MPa, co=0.8, cw=1.2, q o=1 m3/day m, and q w=100 m3/day m. It is evident from the figure that initially water is heated more than oil. Thereafter, one can observe a bnormalQ change of oil and water temperature (ToNTw). In this sense, one can talk about an inversion in time of the value of the choking-induced
76
R.A. Valiullin et al. / Powder Technology 148 (2004) 72–77
Fig. 3. Dependence of temperature on time.
temperature anomaly within the intervals of oil and water inflow. This is primarily explained by different mobilities of oil and water: the more mobile water, within the same period of time, covers a larger portion of the depression funnel and, in spite of the lower value of the Joule– Thomson coefficient, it is heated more than oil. 3.2. Multiphase nonstationary movement accompanied by liquid degassing A two-phase movement of oil and gas is observed when the pressure is reduced below the one of saturating oil with gas. The mathematical model for calculating the temperature field in a nonstationary pressure field with due account taken of the Joule–Thomson effect, the adiabatic effect, and the heat of fluid degassing in a porous medium is written as: m
Bq0i cik Si Bt 1 B kKi BP rq0i cik ¼ 0; þ r Br li Br
i ¼ 1; 2; 3; k ¼ 1; 2 ð5Þ
# " 3 X B 0 0 qi Ci Si T þ mq1 S1 L ð1 mÞq0 C0 T þ m Bt i¼1 " # 3 X 1 B 0 0 þ r qi mSi vi Ci T þ mq1 S1 v1 L r Br i¼1 þ
3 X i¼0
q0i Ci ei mSi vi
3 X BP BP ¼ 0: q0i Ci Si gi Br Bt i¼1
Fig. 4 shows the dependencies for the values of thermodynamic parameters of phases as follows: C 0=800 J/(kg K), C 1=3000 J/(kg K), C 2=4000 J/(kg K), C 3=2000 J/ (kg K); e 1=4 K/MPa, e 2=0.2 K/MPa, e 3=0.4 K/MPa; g 1=20 K/MPa, g 2=0.015 K/MPa, g 3=0.13 K/MPa. The value of phase transfer heat during degassing oil is taken as L=100 kJ/kg. The viscosities of the gaseous, gassed aqueous, and oil phases are, respectively: l 1=0.01 mPa s, l 2=0.2 mPa s, and l 3=0.4 mPa s. Let us consider the peculiarities of forming the temperature field in the porous medium in a nonstationary pressure field. Curves 1–3 correspond to the following values of the established ratio of the dissolved gas volume to the volume of the fluid: 50, 100, and 150 m3/m3. It is seen from Fig. 4 that at the instantaneous pressure drop at the initial moments of time, one can observe a reduction in temperature resulting from the manifestation of the adiabatic effect and the heat of fluid degassing. As time passes, the formation of the heat field is effected by the choking heating of the fluid and gas cooling. It is obvious that as the portion of gas in the flow increases, one can observe a more intensive cooling of the fluid, which is connected with degassing gas choking heat. For the sake of comparing the influence of degassing heat on the temperature field, curve 4 in Fig. 4 ignores the heat of fluid degassing. Thus, when the pressure is instantaneously reduced below the one of saturation of oil with gas, at the initial moments, the main contribution to forming the temperature field is made by the adiabatic cooling of the fluid and the heat of fluid degassing. Thereafter, the contribution of the choking effect increases. The extent of the temperature drop is affected by the value of fluid degassing heat, the Joule–Thomson coefficient, and that of the adiabatic effect, by gas solubility and the pressure gradient. The combined influence of the cooling and heating effects results in the formation of a temperature minimum on the temperature curve. As this takes, depending on the amount of gas dissolved in the fluid and on the pressure gradient, the stationary distribution of temperature may
ð6Þ
Different i indices correspond to different phases, namely: 0—rock, 1—gas, 2—water, 3—oil; S i and v i are saturation and motion speed of the ith phase; Cik is concentration of the kth component in the ith phase; q i is density of the ith phase; K is absolute permeability; k i is phase permeability; l i is viscosity of the ith phase; m is porosity; P is pressure; T is temperature; C i is heat capacity; e i is the Joule–Thomson coefficient; g i is the adiabatic coefficient; and L is degassing heat.
Fig. 4. Dependence of temperature on time with due account taken of fluid degassing. Volume content of gas: (1) 50 m3/m3; (2) 100 m3/m3; (3) 150 m3/m3; and (4) without taking into account degassing heat, 150 m3/m3.
R.A. Valiullin et al. / Powder Technology 148 (2004) 72–77
2.
77
of the choking effect, on the other hand, result, in due course of time, to the formation of a temperature minimum. Using the peculiarities of the temperature field in the multiphase flow environment, simultaneously taking into account fluid degassing, enables one to address the tasks of identifying oil and water influxes from the formation and expands the potentialities of temperature well logging.
References Fig. 5. Results of investigating well no. 1378.
feature both positive (curves 1 and 2) and negative (curve 3) temperature anomalies. Below are examples of interpreting temperature investigations of wells in the multiphase flow environment. Fig. 5 presents the results of investigating well no. 1378 (Bashkiria). The average gas factor at this field is 47.7 m3/ metric tons. Two formations are perforated in this well within the intervals 2374.8–2376.8 and 2380–2381.2 m. The investigations were carried out in the process of compressor-assisted well stimulation. The compressor pump column is lowered to the depth of 2275 m. The following thermograms have been recorded: (1) a background thermogram, recorded before the compressor operation; (2) during compression; (3) immediately after compression and pressure release in the annular space; and (4 and 5) 2 and 8 h after the compression. Besides, diagrams of density measurements (6) are provided. Analyzing thermogram (2), one can see that, with the compressor operating, the lower perforated range is marked by slight heating due to choking of the liquid phase. After the pressure release in the annular space (lowering of the bottom-hole pressure), thermogram (3) recorded a temperature drop (cooling anomaly) opposite both of the two perforated subformations. The shape of the temperature curve above the operating ranges of the formation and the corresponding changes of temperature opposite the formations are characteristic of the predominant inflow from the gas phase formations. curve 4 corresponds to the case of the gas and oil mixture influx, whereas thermogram (5) corresponds to the inflow of oil from the lower formation. The upper formation is marked as an operating one only if gas flows in. When oil flows in, this formation is not identified, evidently, due to a low flow rate. It follows from Ref. [6] that the formations yield oil and that degassing takes place in the hole. The well operation figures show the sucker rod pump produce 2.4 m3/day oil. The gas factor for the well is 47.9 m3/metric ton.
4. Conclusions 1.
Oil cooling at the expense of degassing heat and gas choking, on the one hand, and its heating at the expense
[1] E.B. Chekayuk, Thermodynamics of Oil Formation, M. Nedra, 1965. [2] B.B. Lapuk, Thermodynamic Processes of Gassed Oil Moving in Porous Media, in: Azer. neftyanoye khozyaistvo, vol. 2, 1940. [3] G.F. Trebin, F.Yu. Kapyrin, O.G. Limanski, in: Estimation of Temperature Depression in the Well-Bore Zone of Producing Wells._I`. Collection of Papers of VNII, vol. 64, 1978, pp. 16 – 22. [4] P.P. Zolotarev, V.N. Nikolaevsky, Thermodynamical analysis of nonsteady-state processes in water and oil saturated flexible porous media, Year-Book, VNII, M. Nedra C, 1966, pp. 49 – 61. [5] A.K. Kurbanov, M.D. Rozenberg, Non-isothermal filtration of multiphase fluids, Year-Book: VNII. The Theory and Practice of Oil Recovery, M. Nedra, 1968. [6] E.V. Tesluk, A.O. Palii, Non-isothermal processes in the theory and practice of oil recovery, The Ways of the Development of Oil Recovery Industry in Kazakhstan, M. VNIIOENG. C, 1967, pp. 192 – 208. [7] M. Muscat, Flow of Homogeneous Fluids Through Porous Media, 1937 New York. [8] I.A. Charni, Underground Hydro-Gas Mechanics, M. Gostoptehizdat, 1963, 310c. [9] P.G. Atkison, H.J. Ramey, Problems of heat transfer in porous media. SPE 6792. [10] M.G. Alishaev, M.D., Rozenberg, E.V. Tesluk, Non-isothermal displacement of paraffin oil in heterogenetic collector with taking into account the kinetics of wax deposition. [11] V.M. Entov, I.M. Vinogradov, Effects of Dissipative Phenomena on Non-isothermal Displacement of Oil by a Solution of an Active Additive: Numerical Experiment. Institute for Problems in Mechanics RAS, Moscow, Russia. [12] L.I. Rubinstein, The total heat losses in injection of a hot fluid into a stratum, Neft Gas 2 (9) (1959). [13] E.A. Bondarev, V.E. Nikolaev, K.K. Argunova, Computing experiment in non-isothermal gas filtration, in: Proceedings of International Conference RDAMM, vol. 6, Pt. 2, 2001. [14] R.A. Valiullin, A.Sh. Ramazanov, R.F. Sharafutdinov, Barothermal effect under three-phase percolation with phase transitions, in: Izv. RAN, MZhG, vol. 6, 1994, pp. 113 – 117. [15] J.K. Roberts, Heat and Thermodynamics, Blackie and Son, London, 1942, pp. 267 – 271. [16] D.L. Katz, Thermodynamic analysis of frictional heat effects in pipeline flow, Oil Gas J. (1972 (March 6)) 87 – 90. [17] R.C. Smith, R.J. Steffensen, Interpretation of temperature profiles in water-injection wells, J. Pet. Technol. 27 (1975 (June)) 777 – 784. [18] M. Kaviany, Principles of Heat Transfer in Porous Media, SpringerVerlag, 1991. [19] Donald A. Nield, A. Bejan, Convection in Porous Media, SpringerVerlag, 1998. [20] C.D. Miller. //SPEJ.-1975.-v.21.-p.474–479. [21] J. Bredberg, S.-H. Peng, L. Davidson, On the wall boundary condition for computing heat transfer with k-omega models, in: J.H. Kim (Ed.), Proceedings of the ASME Heat Transfer Division - 2000, vol. 5, The American Society of Mechanical Engineers, 2000, pp. 243 – 250.