10. Zhalnin R.V., Masyagin V.F., Peskova E.E., et al. Application of the Discontinuous Galerkin Method to the Study of the Dynamics of Temperature and Pressure Changes in a Formation with an Injection Well and a Hydraulic Fracture

UDK 519.85:622.276.42

DOI: 10.15507/2658-4123.031.202101.161-174

Application of the Discontinuous Galerkin Method to the Study of the Dynamics of Temperature and Pressure Changes in a Formation with an Injection Well and a Hydraulic Fracture

Ruslan V. Zhalnin
Leading Researcher, Head of the Chair of Applied Mathematics, Differential Equations and Theoretical Mechanics of Faculty of Mathematics and Information Technology, National Research Mordovia State University (68 Bolshevistskaya St., Saransk 430005, Russian Federation), Cand.Sc. (Phys.-Math.), Associate Professor, Researcher ID: Q-6945-2016, ORCID: https://orcid.org/0000-0002-1103-3321, This email address is being protected from spambots. You need JavaScript enabled to view it.

Victor F. Masyagin
Senior Researcher, Associate Professor of the Chair of Applied Mathematics, Differential Equations and Theoretical Mechanics of Faculty of Mathematics and Information Technology, National Research Mordovia State University (68 Bolshevistskaya St., Saransk 430005, Russian Federation), Cand.Sc. (Phys.-Math.), Researcher ID: C-2439-2013, ORCID: https://orcid.org/0000-0001-6738-8183, This email address is being protected from spambots. You need JavaScript enabled to view it.

Elizaveta E. Peskova
Researcher, Associate Professor of the Chair of Applied Mathematics, Differential Equations and Theoretical Mechanics of Faculty of Mathematics and Information Technology, National Research Mordovia State University (68 Bolshevistskaya St., Saransk 430005, Russian Federation), Cand.Sc. (Phys.-Math.), Researcher ID: U-7971-2019, ORCID: https://orcid.org/0000-0003-2618-1674, This email address is being protected from spambots. You need JavaScript enabled to view it.

Vladimir F. Tishkin
Head of the Department of Keldysh Institute of Applied Mathematics of Russian Academy of Sciences (4 Miusskaya Sq., Moscow 125047, Russian Federation), Corresponding Member of RAS, D.Sc. (Phys.-Math.), Professor, Researcher ID: R-5820-2016, ORCID: https://orcid.org/0000-0001-7295-7002, This email address is being protected from spambots. You need JavaScript enabled to view it.

Introduction. In this article, the problem of temperature distribution in an oil-bearing formation with a hydraulic fracture and a vertical injection well is numerically modeled.
Materials and Methods. To describe the process of temperature distribution in the formation under the action of the fluid injected into the formation, the Fourier-Kirchhoff equation of convective heat transfer is used. To solve this equation, the discontinuous Galerkin method on staggered unstructured grids is used. To describe the process of pressure change in the formation under the action of the injection well, an equation is used that is obtained based on the continuity equation and Darcy’s law. To solve it, the discontinuous Galerkin method on an unstructured triangular grid is used. To parallelize the numerical algorithm, the MPI library is used.
Results. The article presents a numerical algorithm and the results of modeling the dynamics of the temperature fields in an oil reservoir with a hydraulic fracture and a vertical injection well.
Discussion and Conclusion. A numerical algorithm based on the discontinuous Galerkin method for math modeling of the temperature and pressure fields in a oil-bearing formation with a hydraulic fracture and injection well was developed and implemented. The results obtained for the distribution of temperature and pressure in the fracture are adequate and in good agreement with the specified initial-boundary conditions. Further work in this direction involves modeling on tetrahedral unstructured meshes for a more accurate study of the ongoing processes.

Keywords: discontinuous Galerkin method, vertical injection well, hydraulic fracturing, convective heat transfer equation, continuity equation, Darcy’s law, unstructured grids, spaced grids, MPI

Funding: The study was supported by the Russian Foundation for Basic Research (Projects No. 18-41-130001, No. 18-31-00102) and the grant from the President of the Russian Federation for young Russian Candidates of Science (MK-2007.2018.1).

Conflict of interest: The authors declare no conflict of interest.

Acknowlegments: The authors express gratitude to the anonymous reviewers.

For citation: Zhalnin R.V., Masyagin V.F., Peskova E.E., et al. Application of the Discontinuous Galerkin Method to the Study of the Dynamics of Temperature and Pressure Changes in a Formation with an Injection Well and a Hydraulic Fracture. Inzhenerernyye tekhnologii i sistemy = Engineering Technologies and Systems. 2021; 31(1):161-174. DOI: https://doi.org/10.15507/2658-4123.031.202101.161-174

Contribution of the authors:
R. V. Zhalnin – discussion of the numerical algorithm for approximating convective terms in the heat transfer equation.
V. F. Masyagin – implementation of the main parallel numerical algorithm based on the Galerkin method with discontinuous basis functions, active participation in developing the mathematical model and in discussing the choice of initial and boundary conditions for the calculation problem, conducting numerical calculations.
E. E. Peskova – literature review of domestic and foreign sources, participation in the development of a mathematical model of the process under study.
V. F. Tishkin – formulation of the task and general management of the work.

All authors have read and approved the final manuscript.

Submitted 09.10.2020; approved after reviewing 10.12.2020;
accepted for publication 20.12.2020

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