S.O. Huseynzade

Èlektron. model. 2017, 39(3):37-46


The process of gas displacement by boundary water in the layer has been considered. The process is described by a nonlinear parabolic equation in a domain with a moving boundary. The problem of control of moving boundary is formulated as a boundary inverse problem, which consists in determining the mode of the operational gallery by a given law of motion of the moving boundary. Applying the method of straightening fronts based on transformation of independent variables, the domain of their determination is reduced to a rectangular form with fixed boundaries. A discrete analogue of the problem is proposed and computational algorithm is developed to solve the resulting system of linear algebraic equations.


as deposits, gas-water drive mode, the problem with moving boundary, the method of straightening of fronts, finite difference method.


1. Charny, I.A. (1963), Podzemnaya gidrogazodinamika [Underground hydro-gas dynamics], Gostoptekhizdat, Moscow, USSR.
2. Basniev, K.S., Dmitriev, N.M. and Rosenberg, G.D. (2005), Neftegazovaya gidromekhanika [Oil and gas hydromechanics], Institut kompyuternykh issledovaniy, Moscow-Izhevsk, Russia.
3. Kanevskaya, R.D. (2002), Matematicheskoe modelirovanie gidrodinamicheskikh protsesov razrabotki mestorozhdeniy uglevodorodov [Mathematical modeling of hydrodynamic processes of development of hydrocarbon fields], Institut kosmicheskikh issledovaniy, Moscow-Izhesk, Russia.
4. Rubinshtein, L.I. (1967), Problema Stefana [Stefan problem], Zvaigzne, Riga, Latvian SSR.
5. Crank, J. (1984), Free and moving boundary problems, Clarendon Press, Oxford, UK.
6. Alexiades, V. and Solomon, A.D. (1993), Mathematical modeling of melting and freezing processes, Hemisphere Publ. Co, Washington DC, USA.
7. Meirmanov, A.M. (1986), Zadacha Stefana [Stefan problem], Nauka, Novosibirsk, USSR.
8. Samarskiy, A.A. and Vabishchevich, P.N. (2009), Chislennyie metody resheniya obratnykh zadach matematicheskoi fiziki [Numerical methods for solution of inverse problems of mathematical physics], LKI, Moscow, Russia.
9. Javierre-Perez, E. (2003), Literature study: Numerical methods for solving Stefan problems, Report 03-16, Delft University of Technology, Delft, USA.
10. Caldwell, J. and Kwan, Y.Y. (2004), Numerical methods for one-dimensional Stefan problems, Commun. Numer. Meth. Engng., no. 20, pð. 535-545.
11. Gamzaev, Kh.M. (2015), “Numerical solution of problem of unsaturated filtration with a moving boundary”, Elektronnoe modelirovanie, Vol. 37, no. 1, pp. 15-24.

Full text: PDF (in Russian)