geophystechГеофизические технологииRussian Journal of Geophysical Technologies2619-1563IPGG SB RAS10.18303/2619-1563-2024-1-19geophystech-344Research ArticleСтатьиИспользование итерационного решателя с разделением полей при решении квазистатического уравнения БиоUsing an iterative field-split solver for the quasistatic Biot equationhttps://orcid.org/0000-0001-9984-8367СоловьевС. А.SolovyevS. А.

СОЛОВЬЕВ Сергей Александрович – кандидат физико-математических наук, научный сотрудник,

630090, Новосибирск, просп. Акад. Коптюга, 4.

4, Koptyug Ave., Novosibirsk, 630090.

SolovevSA@ipgg.sbras.ruКостинВ. И.KostinV. I.

КОСТИН Виктор Иванович – кандидат физико-математических наук, старший научный сотрудник лаборатории вычислительной физики горных пород,

630090, Новосибирск, просп. Акад. Коптюга, 3.

3, Koptyug Ave., Novosibirsk, 630090.

https://orcid.org/0000-0003-3544-4878ЛисицаВ. В.LisitsaV. V.

ЛИСИЦА Вадим Викторович – доктор физико-математических наук, заведующий лабораторией вычислительной физики горных пород,

630090, Новосибирск, просп. Акад. Коптюга, 3. 

3, Koptyug Ave., Novosibirsk, 630090.

https://orcid.org/0000-0002-6373-3370НовиковМ. А.NovikovM. A.

НОВИКОВ Михаил Александрович – младший научный сотрудник лаборатории,

630090, Новосибирск, просп. Акад. Коптюга, 4.

4, Koptyug Ave., Novosibirsk, 630090.

Институт математики им. С.Л. Соболева СО РАНРоссияSobolev Institute of Mathematics SB RASRussian FederationИнститут нефтегазовой геологии и геофизики им. А.А. Трофимука СО РАНРоссияTrofimuk Institute of Petroleum Geology and Geophysics, SB RASRussian Federation20241607202401Спецвыпуск1928Copyright © Соловьев С.А., Костин В.И., Лисица В.В., Новиков М.А., 20242024Соловьев С.А., Костин В.И., Лисица В.В., Новиков М.А.Solovyev S.А., Kostin V.I., Lisitsa V.V., Novikov M.A.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.rjgt.ru/jour/article/view/344

С использованием уравнений Био в частотной области в квазистатической постановке моделируется низкочастотное нагружение образца трещиновато-пористой горной породы. Уравнения Био аппроксимируются конечно-разностной схемой на сдвинутых сетках. Для решения полученной системы линейных алгебраических уравнений (СЛАУ) с числом неизвестных более 106 используется итерационный метод (стабилизированный алгоритм бисопряженных градиентов) с предобуславливателем на основе расщепления полей для разделения уравнений и переменных на две группы: описывающих деформацию твердого тела и переноса жидкости. Численные эксперименты демонстрируют быструю сходимость итерационного процесса и его преимущество по сравнению с прямым методом решения СЛАУ на больших задачах.

Quasistatic Biot equations in the frequency domain are applied to model low-frequency loading of the fractured porous rock sample. We approximate Biot equations by a finite-difference scheme on the staggered grid, defining different field components at different grid points. To solve the resulting system of linear algebraic equations (SLAE) with more than 106 unknown values, we use an iterative method (Biconjugate gradient stabilized method, BCGStab) with a preconditioner based on field-split approach to divide equations into two groups. The first group describes solid deformation, and the second group describes fluid transport. Numerical experiments demonstrate fast convergence of the iterative process for applied method and its advantages in the case of large computational grid in comparison with direct method.

пороупругостьмодель Биоконечные разностипрямые методы решения СЛАУитерационные методы решения СЛАУпредобуславливатель с разделением полейporoelasticityBiot equationfinite differencesdirect methods for SLAEiterative methodsfield-split preconditionerРабота выполнена в рамках проекта ФНИ FWZZ-2022-0022 и при поддержке Российского научного фонда, грант № 19-77-20004-П. The study was carried out as part of government assignment to the Russian Academy of Sciences in basic research, Project FWZZ-2022-0022, and supported by the Russian Science Foundation, Project No. 19-77-20004-P.ReferencesНовиков М.А., Лисица В.В., Козяев А.А. Численное моделирование волновых процессов в трещиноватопористых флюидозаполненных средах // Вычислительные методы и программирование. 2018. Т. 19, № 2. С. 130–149.Alekseev A.S., Mikhailenko B.G. The solution of dynamic problems of elastic wave propagation in inhomogeneous media by a combination of partial separation of variables and finite-difference methods // Journal of Geophysics. 1980. Vol. 48 (1). P. 161–172.Соловьев С.А., Новиков М.А., Лисица В.В. Численное решение уравнений Био анизотропной пороупругой флюидонасыщенной среды в квазистатической постановке для численного решения задачи апскейлинга // Вычислительные методы и программирование. 2023. Т. 24, № 1. С. 67–88. doi: 10.26089/NumMet.v24r106.Amestoy P.R., Duff I.S., L’Excellent J.-Y., Koster J. A fully asynchronous multifrontal solver using distributed dynamic scheduling // SIAM Journal on Matrix Analysis and Applications. 2001. Vol. 23 (1). P. 15–41. doi: 10.1137/S089547989935819.Alekseev A.S., Mikhailenko B.G. The solution of dynamic problems of elastic wave propagation in inhomogeneous media by a combination of partial separation of variables and finite-difference methods // Journal of Geophysics. 1980. Vol. 48 (1). P. 161–172.Biot M.A. Theory of propagation of elastic waves in fluid-saturated porous solid. I. Low-frequency range // Journal of the Acoustical Society of America. 1956a. Vol. 28 (2). P. 168–178. doi: 10.1121/1.1908239.Amestoy P.R., Duff I.S., L’Excellent J.-Y., Koster J. A fully asynchronous multifrontal solver using distributed dynamic scheduling // SIAM Journal on Matrix Analysis and Applications. 2001. Vol. 23 (1). P. 15–41. doi: 10.1137/S089547989935819.Biot M.A. Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range // Journal of the Acoustical Society of America. 1956b. Vol. 28 (2). P. 179–191. doi: 10.1121/1.1908241.Biot M.A. Theory of propagation of elastic waves in fluid-saturated porous solid. I. Low-frequency range // Journal of the Acoustical Society of America. 1956a. Vol. 28 (2). P. 168–178. doi: 10.1121/1.1908239.Bollhöfer M., Schenk O., Janalik R., Hamm S., Gullapalli K. State-of-the-art sparse direct solvers // Grama A., Sameh A. (Eds.) Parallel Algorithms in Computational Science and Engineering. Modeling and Simulation in Science, Engineering and Technology. Baser: Birkhäuser Cham, 2020. P. 3–33. doi: 10.48550/arXiv.1907.05309.Biot M.A. Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range // Journal of the Acoustical Society of America. 1956b. Vol. 28 (2). P. 179–191. doi: 10.1121/1.1908241.Calandrini S., Aulisa E., Ke G. A field-split preconditioning technique for fluid-structure interaction problems with applications in biomechanics // International Journal for Numerical Methods in Biomedical Engineering. 2020. Vol. 36 (3). P. e3301. doi: 10.1002/cnm.3301.Bollhöfer M., Schenk O., Janalik R., Hamm S., Gullapalli K. State-of-the-art sparse direct solvers // Grama A., Sameh A. (Eds.) Parallel Algorithms in Computational Science and Engineering. Modeling and Simulation in Science, Engineering and Technology. Baser: Birkhäuser Cham, 2020. P. 3–33. doi: 10.48550/arXiv.1907.05309.Masson Y.J., Pride S.R., Nihei K.T. Finite difference modeling of Biot's poroelastic equations at seismic frequencies // Journal of Geophysical Research: Solid Earth. 2006. Vol. 111 (B10305). P. 1–13. doi: 10.1029/2006JB004366.Calandrini S., Aulisa E., Ke G. A field-split preconditioning technique for fluid-structure interaction problems with applications in biomechanics // International Journal for Numerical Methods in Biomedical Engineering. 2020. Vol. 36 (3). P. e3301. doi: 10.1002/cnm.3301.Novikov M.A., Lisitsa V.V., Kozyaev A.A. Numerical modeling of wave processes in fractured porous fluidsaturated media // Numerical Methods and Programming. 2018. Vol. 19 (2). P. 130–149.Masson Y.J., Pride S.R., Nihei K.T. Finite difference modeling of Biot's poroelastic equations at seismic frequencies // Journal of Geophysical Research: Solid Earth. 2006. Vol. 111 (B10305). P. 1–13. doi: 10.1029/2006JB004366.Novikov M.A., Lisitsa V.V., Bazaikin Ya.V. Wave propagation in fractured-porous media with different percolation length of fracture systems // Lobachevskii Journal of Mathematics. 2020. Vol. 41 (8). P. 1533–1544. doi: 10.1134/S1995080220080144.Novikov M.A., Lisitsa V.V., Bazaikin Ya.V. Wave propagation in fractured-porous media with different percolation length of fracture systems // Lobachevskii Journal of Mathematics. 2020. Vol. 41 (8). P. 1533–1544. doi: 10.1134/S1995080220080144.Novikov M., Lisitsa V., Khachkova T., Reshetova D., Vishnevsky D. Numerical algorithm of seismic wave propagation and seismic attenuation estimation in anisotropic fractured porous fluid-saturated media // 21st International Conference on Computational Science and Its Applications, ICCSA 2021. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). 2021. Vol. 12949 LNCS. P. 434–448. doi: 10.1007/978-3-030-86653-2_32.Novikov M., Lisitsa V., Khachkova T., Reshetova D., Vishnevsky D. Numerical algorithm of seismic wave propagation and seismic attenuation estimation in anisotropic fractured porous fluid-saturated media // 21st International Conference on Computational Science and Its Applications, ICCSA 2021. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). 2021. Vol. 12949 LNCS. P. 434–448. doi: 10.1007/978-3-030-86653-2_32.Li X.S. An overview of SuperLU: Algorithms, implementation, and user interface // ACM Transactions on Mathematical Software. 2005. Vol. 31 (3). P. 302–325. doi: 10.1145/1089014.1089017.Li X.S. An overview of SuperLU: Algorithms, implementation, and user interface // ACM Transactions on Mathematical Software. 2005. Vol. 31 (3). P. 302–325. doi: 10.1145/1089014.1089017.Quintal B., Steeb H., Frehner M., Schmalholz S.M. Quasi-static finite element modeling of seismic attenuation and dispersion due to wave-induced fluid flow in poroelastic media // Journal of Geophysical Research. 2011. Vol. 116 (B1). P. 1–17. doi: 10.1029/2010JB007475.Quintal B., Steeb H., Frehner M., Schmalholz S.M. Quasi-static finite element modeling of seismic attenuation and dispersion due to wave-induced fluid flow in poroelastic media // Journal of Geophysical Research. 2011. Vol. 116 (B1). P. 1–17. doi: 10.1029/2010JB007475.Reshetova G., Romenski E. Diffuse interface approach to modeling wavefields in a saturated porous medium // Applied Mathematics and Computation. 2021. Vol. 398. P. 12598. doi:10.1016/j.amc.2021.125978.Reshetova G., Romenski E. Diffuse interface approach to modeling wavefields in a saturated porous medium // Applied Mathematics and Computation. 2021. Vol. 398. P. 12598. doi: 10.1016/j.amc.2021.125978.Saad Y. Iterative Methods for Sparse Linear Systems. New York: PWS Publishing, 1996. 447 p.Saad Y. Iterative Methods for Sparse Linear Systems. New York: PWS Publishing, 1996. 447 p.Solovyev S., Novikov V., Lisitsa V. Numerical solution of anisotropic Biot equations in quasi-static state // 22nd International Conference on Computational Science and Its Applications, ICCSA 2022. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). 2022. Vol. 13378 LNCS. P. 310–327. doi: 10.1007/978-3-031-10562-3_23.Solovyev S., Novikov V., Lisitsa V. Numerical solution of anisotropic Biot equations in quasi-static state // 22nd International Conference on Computational Science and Its Applications, ICCSA 2022. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). 2022. Vol. 13378 LNCS. P. 310–327. doi: 10.1007/978-3-031-10562-3_23.Solovyev S.A., Novikov M.A., Lisitsa V.V. Numerical solution of anisotropic Biot equations of poroelastic fluidsaturated media in quasi-static state for numerical upscaling // Numerical Methods and Programming. 2023. Vol. 24 (1). P. 67–88. doi: 10.26089/NumMet.v24r106.Stuben K. A review of algebraic multigrid // Journal of Computational and Applied Mathematics. 2001. Vol. 128 (1–2). P. 281–309. doi: 10.1016/S0377-0427(00)00516-1.Stuben K. A review of algebraic multigrid // Journal of Computational and Applied Mathematics. 2001. Vol. 128 (1–2). P. 281–309. doi: 10.1016/S0377-0427(00)00516-1.

The authors declare that there are no conflicts of interest present.