Abstract

The presentation considers a generalization of classical Biot's equations to the poroelastoplastic medium in order to simulate a shear banding phenomena taking place around the borehole drilled in the prestressed solid under the artificial depression leading to the change in the pore pressure in the surrounding rock as well as redistribution of stresses and accumulated plastic strains. The mathematical problem formulation consists of a coupled system of dynamic poroelastoplastic equations in a solid skeleton and a saturating fluid for the small strains case (geometrically linear formulation): equilibrium equations, Darcy's law, and constitutive stress-strain and pore pressure relations. Different physically nonlinear relations are taken into account: dynamic porosity and permeability dependent on the pore pressure and volumetric strains, non-associative poroplasticity taken into account irreversible plastic strans and change in fluid volume content, dynamic poroelastic moduli (bulk, shear, Biot etc) dependent on the current porocity. A set of nonlinear PDEs is solved using a novel approach based on the fully explicit time integration scheme and the high order spectral element method (SEM) space discretization scheme. Two numerical algorithms are analyzed and compared with each other: the first one is based on the direct Lagrangian formalism leading to the full displacement integration and accumulation of the full plastic strains at each loading step, the second one is based on the Euler formalism leading to the seeking a numerical solution in terms of the relative displacements at each loading step (i.e. velocities) and accumulated stresses. Both approaches are implemented for solving quasi-static poroelastoplastic equations using pseudo-transient scheme with the inertia and dissipative terms allowing one to converge to the static solution at each loading "time" step. Obtained numerical results as well as the performance of both approaches are analyzed and compared with CAE Fidesys for the Drucker-Prager plasticity. CUDA technology is used to parallelize the implemented algorithm on the massively parallel Tesla V100 GPU. Different optimization strategies and details of parallelizing spectral element method using CUDA are discussed. In particular, an algorithm for the mapping of an unstructured spectral element mesh of an arbitrary order onto the grid of blocks of GPU multilevel thread hierarchy is presented. CUDA kernels' design, memory access patterns, synchronization issues are considered. Performance analysis is given for different SEM orders and mesh sizes as well as for the different floating-point precisions. The reported study was funded by Russian Science Foundation project -19-77-10062. References [1] M.A. Biot ?Theory of propagation of elastic waves in a fluid-saturated porous solid,? in J. Acoust. Soc. Am., 28, pp. 168-191 (1956). [2] Charara M, Vershinin A, Deger E, Sabitov D and Pekar G 2011 3D spectral element method simulation of sonic logging in anisotropic viscoelastic media SEG Expanded Abstracts 30 pp 432?437 [3] Duretz, T., Souche, A., de Borst, R., & Le Pourhiet, L. (2018). The benefits of using a consistent tangent operator for viscoelastoplastic computations in geodynamics. Geochemistry, Geophysics, Geosystems, 19. [4] O. Coussy, Poromechanics, John Wiley and Sons, 2004. [5] Komatitsch D and Vilotte J-P 1998 The spectral element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures Bulletin of Seismological Society of America 88(2) [6] Levin V, Zingerman K, Vershinin A, Freiman E and Yangirova A 2013 Numerical analysis of the stress concentration near holes originating in previously loaded viscoelastic bodies at finite strains International Journal of Solids and Structures 50(20-21) [7] Yarushina, V. M. & Podladchikov, Y. Y. (De)compaction of porous viscoelastoplastic media: Model formulation. J. Geophys. Res. Solid Earth 120, 4146?4170 (2015). References [8] CAE Fidesys website www.cae-fidesys.com