Yu. A. Poveshchenko¹, I. R. Yanglyaev^{2, 3}, V. O. Podryga¹, O. S. Yazovtseva^{2, 3}, I. M. Gubaydullin^{2, 4}

¹Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences (Moscow, Russian Federation)

²Ufa State Petroleum Technological University (Ufa, Russian Federation)

³National Research Mordovia State University (Saransk, Russian Federation)

⁴Institute of Petrochemistry and Catalysis – a separate structural subdivision of the Ufa Federal Research Centre of the Russian Academy of Sciences (Ufa, Russian Federation)

Abstract. The work studies the order of approximation of the symbolic tridiagonal matrix algorithm proposed for solving piezoconductivity problems in oil-bearing fractured-porous formations. The symbolic tridiagonal matrix algorithm is a modification of the classical tridiagonal matrix algorithm and consists in finding the values of the sought grid function in a polynomial form. This also entails a polynomial representation of one tridiagonal matrix coefficient and of the residual. The domain decomposition is organized as follows: over the entire oil-bearing formation a macrogrid with a large spatial step is introduced, and in one of its cells a microgrid with a finer spatial step is constructed. The algorithm allows obtaining an exact solution on the microgrid to refine the calculations independently of the function values at the macronodes. The algorithm is verified on problems with a known analytical solution, and the second order of accuracy of the algorithm is obtained in the formulation with nested grids, which corresponds to the accuracy of the stated difference scheme. The algorithm is applied to study the piezoconductivity problem in a fractured-porous reservoir, and pressure distribution patterns in the fracture system are obtained.

Key Words: mathematical modeling, symbolic tridiagonal matrix algorithm, fractured-porous reservoir, piezoconductivity equation

For citation: Yu. A. Poveshchenko, I. R. Yanglyaev, V. O. Podryga, O. S. Yazovtseva, I. M. Gubaydullin. On the Order of Approximation of the Symbolic Tridiagonal Matrix Algorithm. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 28:3(2026), 225–239. DOI: https://doi.org/10.15507/2079-6900.28.202603.225-239

Information about the authors:

Yury A. Poveshchenko, D.Sc. (Phys. and Math.), Leading Researcher, Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences (4 Miusskaya Sq., Moscow 125047, Russia), ORCID: https://orcid.org/0000-0001-9211-9057, hecon@mail.ru

Ilnur R. Yanglyaev, Analyst, Ufa State Petroleum Technological University (1 Kosmonavtov St., Ufa 450064, Russia), Laboratory Research Assistant, National Research Mordovia State University (68/1 Bolshevikskaya St., Saransk 430005, Russia), ORCID: https://orcid.org/0009-0008-4659-965X, ilnur.yanglyaev@mail.ru

Viktoriia O. Podryga, D.Sc. (Phys. and Math.), Leading Researcher, Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences (4 Miusskaya Sq., Moscow 125047, Russia), ORCID: https://orcid.org/0000-0001-7874-6978, pvictoria@list.ru

Olga S. Yazovtseva, Ph.D. (Phys. and Math.), Senior Researcher, Ufa State Petroleum Technological University (1 Kosmonavtov St., Ufa 450064, Russia), Associate Professor, National Research Mordovia State University (68/1 Bolshevikskaya St., Saransk 430005, Russia), ORCID: https://orcid.org/0000-0001-8075-4491, yazovtsevaos@yandex.ru

Irek M. Gubaydullin, D.Sc. (Phys. and Math.), Professor, Ufa State Petroleum Technological University (1 Kosmonavtov St., Ufa 450064, Russia), Head of the Laboratory of Mathematical Chemistry, Institute of Petrochemistry and Catalysis – a separate structural subdivision of the Ufa Federal Research Centre of the Russian Academy of Sciences (141 Oktyabrya Ave., Ufa 450075, Russia), ORCID: https://orcid.org/0000-0001-9787-0325, irekmars@mail.ru