Solution of Two-Dimensional Problems of Heat Conduction with Convection

About the author:

Rodion M. Ganopolskij, Cand. Sci. (Phys.-Math.), Head of the Department of Modeling of Physical Processes and Systems, School of Natural Sciences, Scientific Supervisor of the High-Performance Computing Center, Tyumen State University, Tyumen, Russia
r.m.ganopolskij@utmn.ru, https://orcid.org/my-orcid?orcid=0000-0002-7682-9830

Abstract:

Easily accessible deposits of low-viscosity oil are gradually being depleted. As a result, oil companies are forced to move towards developing more complex resources, including high-viscosity oils that were previously considered unprofitable. Thermal methods are the most promising for increasing extraction of oil. It is necessary for mathematical description to take into account several physical processes. First of all, this is the heat transfer process, as well as the convection process. There may be several reasons for convection. For example, filtration or rising of a hot liquid. At least a two-dimensional heat equation with convection is required for simulation several types of physical processes. The convection coefficient is different for each dimension. Numerical methods are widely used, but the question of their convergence and accuracy remains. An analytical method for solving the heat equation is the Fourier method. To use this algorithm, it is necessary to achieve zero boundary conditions. At first we find a stationary solution. Then subtract it from the original equation. In the work, the solution of the heat conduction equation without convection is first found. Then the same steps are performed for the full equation. The obtained solutions are analyzed. Two-dimensional images of the temperature distribution are constructed. The dimensionless parameters of the system are studied. Further complications of the problem are proposed, this is bringing it closer to real processes.

References:

Shpurov, I. V. (2014). The state of reproduction of hydrocarbon reserves in the Russian Federation and the tasks of the new classification of reserves // Subsoil Use XXI century, 4. pp. 58–67.

Antoniadi, D. G., Garushev, A. R., & Ishikhanov, V. G. (2000). A handbook on thermal methods of oil production. Sovetskaya Kuban.

Ibatullin, R. R., Takhautdinov Sh. F., Ibragimov N. G., Khisamov R. S. (2011). Development of methods for increasing the degree of oil recovery from hard-to-recover reserves. - Kazan: Publishing House “FEN” of the Academy of Sciences of the Republic of Tatarstan. pp. 31–34.

Khisamov, R. S., Gatiyatullin N. S., Makarevich V. N., Iskritskaya N. I., Bogoslovsky S. A. (2009). Features of development of heavy high-viscosity oils and natural bitumens of the East European platform. St. Petersburg: VNIGRI. 192 p.

Sergeev, R. V. (1981). Thermal methods of influencing the bottomhole formation zone of heavy and high-viscosity oil fields. Oilfield Business. 16.

Bourget, J., Surio, P., & Kombarnu, M. (1989). Thermal methods of enhanced oil recovery. Nedra. [In Russian]

Lykov, A. V. (1978) Teplomassoobmen [Heat and mass transfer]. Moscow, Energiya Publ. 480 p. [In Russian]

Isachenko, V. P., Osipova V. A., Sukomel A. S. (1981) Teploperedacha [Heat transfer]. 4th ed., revised and expanded. Moscow, Energoizdat Publ. 415 p. [In Russian]

Dulnev, G. N. (2012) Theory of heat and mass transfer. St. Petersburg: NRU ITMO. 195 p. [In Russian]

Chernyshov, V. E., Pivovarova I. I. (2020) Numerical solution of the heat equation on the example of calculating the amount of heat loss when hot water is injected into the well // STUDENT OF THE YEAR 2020. pp. 8–13.

Zhumaev, Zh., Tosheva M. M. (2022) Simulation of stationary heat conduction with free convection in a limited volume // UNIVERSUM. pp 34–37.

Gilmanov, A. Ya., Shevelev A. P. (2021) Modeling of steam-cyclic impact on oil reservoirs taking into account convective flows // Experimental methods for studying reservoir systems: problems and solutions. p. 82.

Kanevskaya, R. D. (2002). Mathematical modeling of hydrodynamic processes of development of hydrocarbon reserves. Institute of Computer Research. [In Russian]

Gilmanov, A. Ya., Shevelev A. P. (2022) Calculation of temperature distribution in a reservoir at the initiation stage of steam-assisted gravity drainage / Bulletin of the Tomsk Polytechnic University. Geo Аssets Engineering. V. 333(5) p. 108–115.

Skoblikov, R. M., Gil’manov A. Ya., Shevelev A. P. (2023). Methodology for express assessment of the main technological parameters of steam cyclic processing // Solving applied problems of oil and gas production based on the classical works of A. P. Telkov and A. N. Laperdin: Proceedings of the National Scientific and Technical Conference, Tyumen, April 21, 2023 / Responsible. editor S.I. Grachev. - Tyumen: Tyumen Industrial University. pp. 179–182. EDN UORTAV.

Galanin, M.P., Proshunin N.N., Rodin A.S., Sorokin D.L. (2016). Solution of three-dimensional non-stationary heat equation by finite element method taking into account phase transitions, Preprints of Keldysh Institute of Applied Mathematics. 66. 27 p. DOI 10.20948/prepr-2016-66

Gilmanov,  A.  Ya., Kovalchuk,  T.  N., Skoblikov,  R.  M., Fedorov,  A.  O., Khodzhiev, Ye. N., & Shevelev, A. P. (2023). Analysis of the influence of the thermophysical parameters of the reservoir and fluid on the process of cyclic steam stimulation. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, 9(3), 6–27. DOI 10.21684/2411-7978-2023-9-3-6-27

Bashkirtseva, I. A., Ryazanova T. V., Ryashko L. B. (2017). Computer modeling of nonlinear dynamics: Continuous models: tutorial.

Ganopolskij, R. M. (2023). Analytical solution of heat equation taking into account convection with isothermal boundary conditions. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, 9(3), 66–82. DOI 10.21684/2411-7978-2023-9-3-66-82

Ganopolsky, R. M. 2024. Solution of one-dimensional problems of heat conduction with convection using the Poisson integral. Tyumen State University Herald. Physical and Mathematical Modeling.. Oil, Gas, Energy. 10(1). pp. 41–54.

Tikhonov, A. N., Samarsky A. A. (2004) Equation of mathematical physics. Moscow // Higher school.

Petrovsky I. G. (2009) Lectures on the theory of ordinary differential equations. Moscow: FIZMATLIT. 208 p.

Cannon, John Rozier (1984) The One–Dimensional Heat Equation, Addison-Wesley publishing company.

Krainov, A. Yu., Moiseeva K. M. (2017) Convective heat transfer and heat transfer: textbook. allowance. Tomsk: STT. 80 p.

Krainov, A. Yu., Minkov L. L. (2016) Numerical methods for solving problems of heat and mass transfer: textbook. allowance. Tomsk: STT. 92 p.

Krainov, A. Yu., Ryzhykh Yu. N., Timokhin A. M. (2009) Numerical methods in heat transfer problems. Educational and methodical manual. Tomsk: TSU.

Samarsky, A. A., Vabishchevich P. N. (1999) Numerical methods for solving convection–diffusion problems. Moscow: Editorial URSS. 248 p.