On the stability of a difference scheme for the three-phase non-isothermal flow problem
Keywords:
three-phase non-isothermal flow, finite difference method, stability, non-linear term, priori estimateAbstract
The article presents a study of stability of a finite difference scheme in terms of initial values and right-hand sides of the equations for the problem of three-phase non-isothermal flow in homogeneous isotropic porous media without capillary, gravitational forces and phase transitions. It is assumed that oil is homogeneous non-evaporable fluid, and phases are in local thermal equilibrium which means that fluids saturating the porous media and the rock have the same temperature in any elementary volume. The model describing this process consists of the mass conservation equation, equation of motion in the form of linear Darcy’s law, equation of state, and phase balance equation. In the present work, so-called “global” formulation of the problem is used which is based on the introduction of a change of variables for pressure, called "global pressure". Using this approach, the original model equations reduce to a system of five partial differential equations with respect to pressure, temperature, velocity, and two saturations. The stability analysis of the scheme is carried out using the method of a priori estimates. A priori estimate for the solution of the difference problem is obtained with limitations on the value of the time step and the norm of the temperature derivative.
References
[2] Bokserman A. A., Yakuba S. I. Chislennoe issledovanie protsessa vytesneniya nefti parom // Izvestiya AN SSSR. – 1987. – No 4. – S. 78-84. (in Russian)
[3] Abdramanova M. B. Chislennoe modelirovanie processa vytesneniya nefti parom // Vestnik KazGU. Seriya matematika, mehanika, informatika. – Almaty, 1998. – No 10. – S. 3-10. (in Russian)
[4] Akhmed-Zaki D. ZH. Ob odnoi zadache dvuhfaznoi filtratsii smesi v poristoi srede s uchetom teplovogo vozdeistviya // Nauchnye trudy NIPI «Neftegaz». – 2010. – No 3. – S. 29-33. (in Russian)
[5] Abirov A. K., Mukhambetzhanov S. T. Modelirovanie zadach fazovyh perehodov pri neizotermicheskoi filtratsii i kachestvennye svoistva resheniya // Vestnik KazGU. Seriya matematika, mehanika, informatika. – 1996. – No 5. – S. 3-11. (in Russian)
[6] Bocharov O. B., Telegin I. G. O nekotoryh osobennostyah neizotermicheskoi fil’tracii nesmeshivayushhihsya zhidkostei // Teplofizika i aeromehanika. – Novosibirsk, 2002. – No 3. – S. 459-466. (in Russian)
[7] Temirbekov N. M., Baigereyev D. R. Modeling of three-phase non-isothermal flow in porous media using the approach of reduced pressure // Mathematical Modeling of Technological Processes: 8th International Conference, CITech-2015, Almaty, Kazakhstan, September 24-28, 2015, Proceedings / edited by N. Danaev, Yu. Shokin, D. Akhmed-Zaki. – Almaty, 2015. — P. 166-176.
[8] Antontsev S. N., Monakhov V. N. Kraevye zadachi dlya nekotoryh vyrozhdayushhihsya uravnenii mehaniki sploshnoi sredy. – Novosibirsk: Novosibirskii gosudarstvennyi universitet, 1977. – 48 s. (in Russian)
[9] Chavent G., Jaffre J. Mathematical models and finite elements for reservoir simulation. – Elsevier, 1986. – 375 p.
[10] Samarsky A. A., Andreyev V. B. Raznostnye metody dlya ellipticheskih uravnenii. – Moskva: Nauka, 1976. – 352 s. (in Russian)
