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Poroelastoplastic modeling of a borehole stability under small and finite strains using isoparametric spectral element method

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Abstract

Simulation of the localization and development of plastic shear bands in fluid-saturated rocks is considered using a nonlinear poroelastoplastic model generalizing Biot’s model for a two-phase fluid-saturated porous medium under small and finite strains. A Drucker-Prager yield criterion and a non-associated plastic flow rule are applied to describe an accumulation and localization of plastic strains in a rock. Additionally, a nonlinear dependence of the model parameters (elastic moduli, Biot’s modulus, permeability, etc.) on porosity is considered as well as a dynamic variation of porosity due to the volumetric deformation of the pore space. An isoparametric spectral element method is used to discretize a geometric model and PDEs on curvilinear unstructured meshes of high order in space. A distinctive feature of the developed algorithm for numerical solving the system of nonlinear PDEs of poroelastoplasticity is the use of the dynamic relaxation method, which provides a quasi-stationary solution using an explicit time integration scheme and an optimal choice of the damping parameter. The suggested algorithm allows efficient implementation on a massively parallel high-performance computing system using CUDA technology. The spectral element mesh is naturally mapped onto the CUDA Grid representing GPU’s multiprocessors, and accordingly, each spectral element is mapped onto a streaming block, within which element’s internal nodes are processed by the corresponding threads of the block. Numerical results of solving a series of model problems of the development of plastic shear bands nearby a borehole drilled in a porous fluid-saturated rock are presented. The dynamic variations of porosity and permeability because of the accumulation of plastic deformations are analyzed.

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Acknowledgements

The research for this article was performed in Schmidt Institute of Physics of the Earth of the Russian Academy of Sciences and was financially supported by Russian Science Foundation (project No. 19-77-10062) in the part related to the development of mathematical models and numerical modeling and was performed in Lomonosov Moscow State University and was supported by the grant of the President of the Russian Federation for young scientists—doctors of sciences MD-208.2021.1.1 in the part related to the development of numerical algorithms for problem solving and their parallel implementation on massively parallel high-performance computing systems. Author is grateful and thank the full professor of Lomonosov Moscow State University Vladimir Anatolievich Levin for his valuable advices on the problem statement and for the numerous consultations while its solving.

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  1. Lomonosov Moscow State University, Institute of Physics of the Earth of the Russian Academy of Sciences, Moscow, Russian Federation

    Anatoly Vershinin

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Correspondence to Anatoly Vershinin.

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Communicated by Francesco dell’Isola.

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Vershinin, A. Poroelastoplastic modeling of a borehole stability under small and finite strains using isoparametric spectral element method. Continuum Mech. Thermodyn. 35, 1245–1262 (2023). https://doi.org/10.1007/s00161-022-01117-4

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