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.
Similar content being viewed by others
References
Levin, V.A.: Repeatedly superimposed large elastic deformation. Int. J. Fract. 79, R11–R15 (1996). https://doi.org/10.1007/BF00017717
Levin, V.A., Morozov, E.M.: Nonlocal criteria for determining a prefracture zone in the process of defect growth for finite strains. Dokl. Phys. 52, 391–393 (2007). https://doi.org/10.1134/S1028335807070129
Levin, V.A., Zingerman, K.M.: Interaction and microfracturing pattern for successive origination (introduction) of pores in elastic bodies: finite deformation. J. Appl. Mech. Trans. ASME. 65, 431–435 (1998). https://doi.org/10.1115/1.2789072
Levin, V.A., Zingerman, K.M., Vershinin, A.V.: Non-stationary plane problem of the successive origination of stress concentrators in a loaded body. Finite deformations and their superposition. J. Commun. Numer. Methods Eng. (2007)
Levin, V., Zingerman, K., Vershinin, A.: Geomechanical modelling of fracture propagation under finite strain. Prefracture zones. Seism. Technol. 11(4), 1–11 (2014)
Zingerman, K.M., Levin, V.A.: Redistribution of finite elastic strains after the formation of inclusions. Approximate analytical solution. J. Appl. Math. Mech. 73, 710–721 (2009). https://doi.org/10.1016/j.jappmathmech.2010.01.011
Duretz, T., Souche, A., de Borst, R., Le Pourhiet, L.: The benefits of using a consistent tangent operator for viscoelastoplastic computations in geodynamics. Geochem. Geophys. Geosyst. (2018). https://doi.org/10.1029/2018GC007877
Luo, J., Ramazani, A., Sundararaghavan, V.: Simulation of micro-scale shear bands using peridynamics with an adaptive dynamic relaxation method. Int. J. Solids Struct. 130–131, 36–48 (2018). https://doi.org/10.1016/j.ijsolstr.2017.10.019
Coussy, O.: Mechanics of Porous Continua. Wiley, New York (1995)
Coussy, O.: Poromechanics. Wiley, New York (2004)
Yarushina, V.M., Podladchikov, Y.Y.: (De)compaction of porous viscoelastoplastic media: model formulation. J. Geophys. Res. Solid Earth 120, 4146–4170 (2015)
Levin, V.A., Lokhin, V.V., Zingerman, K.M.: Effective elastic properties of porous materials with randomly dispersed pores: finite deformation. J. Appl. Mech. 67, 667–670 (2000). https://doi.org/10.1115/1.1286287
Vershinin, A.V., Levin, V.A., Zingerman, K.M., Sboychakov, A.M., Yakovlev, M.Y.: Software for estimation of second order effective material properties of porous samples with geometrical and physical nonlinearity accounted for. Adv. Eng. Softw. 86, 80–84 (2015). https://doi.org/10.1016/j.advengsoft.2015.04.007
Levin, V.A., Podladchikov, Y.Y., Zingerman, K.M.: An exact solution to the Lame problem for a hollow sphere for new types of nonlinear elastic materials in the case of large deformations. Eur. J. Mech. A/Solids 90, 104345 (2021). https://doi.org/10.1016/j.euromechsol.2021.104345
Myasnikov, A., Vershinin, A., Sboychakov, A.: A generalization of geomechanical model for naturally fractured reservoirs. In: Proceedings of the SPE Russian Petroleum Technology Conference and Exhibition, 24-26 October 2016, Moscow, Russia, vol. 2, pp. 1050–1092. Moscow (2016)
Sharma, M., Wang, H. A.: Fully 3-D, Multi-Phase, Poro-Elasto-Plastic Model for Sand Production // SPE 181566-MS. (2016)
Benallal, Ahmed, Botta, Alexandre S., Venturini, Wilson S.: Consolidation of elastic-plastic saturated porous media by the boundary element method. Comput. Methods Appl. Mech. Eng. 197(51–52), 4626–4644 (2008)
de Buhan, P., Chateau, X., Dormieux, L.: The constitutive equations of finite strain poroelasticity in the light of a micro-macro approach. Eur. J. Mech. A/Solids 17(6), 909–921 (1998). https://doi.org/10.1016/S0997-7538(98)90501-0. (hal-01983081)
Callari, C., Armero, F.: Analysis and numerical simulation of strong discontinuities in finite strain poroplasticity. Comput. Methods Appl. Mech. Eng. 193(27–29), 2941–2986 (2004). https://doi.org/10.1016/j.cma.2004.02.002
Dormieux, L., Maghous, S.: Poroelasticity and Poroplasticity At Large Strains. Oil & Gas Science and Technology—Revue d’IFP Energies nouvelles. Institut Français du Pétrole 54(6), 773–784 (1999). https://doi.org/10.2516/ogst:1999065.hal-02075845
Nedjar, Boumediene: On finite strain poroplasticity with reversible and irreversible porosity laws. Formulation and computational aspects. Mech. Mater. 68, 237–252 (2014). https://doi.org/10.1016/j.mechmat.2013.08.011
dell’Isola, F., Hutter, K.: Variations of porosity in a sheared pressurized layer of saturated soil induced by vertical drainage of water. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 455(1988), 2841–2860 (1999)
Zienkiewicz, O., Taylor, R.: The Finite Element Method for Solid and Structural Mechanics, 7th edn. Elsevier, Amsterdam (2014)
Gagneux, G., Millet, O.: Modeling capillary hysteresis in unsatured porous media. Math. Mech. Complex Syst. 4(1), 67–77 (2016)
Placidi, L., Barchiesi, E., Misra, A.: A strain gradient variational approach to damage: a comparison with damage gradient models and numerical results. Math. Mech. Complex Syst. 6(2), 77–100 (2018)
Madeo, A., dell’Isola, F., Darve, F.: A continuum model for deformable, second gradient porous media partially saturated with compressible fluids. J. Mech. Phys. Solids 61(11), 2196–2211 (2013)
Giorgio, I., De Angelo, M., Turco, E., Misra, A.: A Biot-Cosserat two-dimensional elastic nonlinear model for a micromorphic medium. Contin. Mech. Thermodyn. 32(5), 1357–1369 (2020)
Sciarra, Giulio, Dell’Isola, Francesco, Coussy, Olivier: Second gradient poromechanics. Int. J. Solids Struct. 44(20), 6607–6629 (2007)
Eremeyev, V.A., Cazzani, A., dell’Isola, F.: On nonlinear dilatational strain gradient elasticity. Contin. Mech. Thermodyn. 33, 1429–1463 (2021). https://doi.org/10.1007/s00161-021-00993-6
Marin, M., Öchsner, A., Othman, M.I.A.: On the evolution of solutions of mixed problems in thermoelasticity of porous bodies with dipolar structure. Contin. Mech. Thermodyn. 34, 491–506 (2022). https://doi.org/10.1007/s00161-021-01066-4
Biot, M.A.: Theory of propagation of elastic waves in a fluid-saturated porous solid. J. Acoust. Soc. Am. 28, 168–191 (1956)
Charara, M., Vershinin, A., Sabitov, D., Pekar, G.: SEM wave propagation in complex media with tetrahedral to hexahedral mesh. In: 73rd European Association of Geoscientists and Engineers Conference and Exhibition pp 41–45 (2011)
Konovalov, D., Vershinin, A., Zingerman, K. , Levin, V.: The implementation of spectral element method in a CAE system for the solution of elasticity problems on hybrid curvilinear meshes. Modell. Simul. Eng. (2017)
Vershinin, A., Levin, V., Kukushkin, A., Konovalov, D.: 2019 Application of variable order spectral element method on non-conformal unstructured meshes for an engineering analysis of assemblies with geometric inaccuracies. In: Proceedings of the International SPDM Conference NAFEMS World Congress. Quebec City, Canada
Charara, M., Vershinin, A., Deger, E., Sabitov, D., Pekar, G. (2011) 3D spectral element method simulation of sonic logging in anisotropic viscoelastic media SEG Expanded Abstracts 30, pp. 432–437
Cuomo, M., Contrafatto, L., Greco, L.: A variational model based on isogeometric interpolation for the analysis of cracked bodies. Int. J. Eng. Sci. 80, 173–188 (2014)
Kirk, D., Hwu, W.-mei: CUDA textbook. Elsevier, Amsterdam (2010)
Johnson, D.L., Plona, T.J.: Probing porous media with first and second sound. II. Acoustic properties of water-saturated porous media. J. Appl. Phys. 76(1), 115–125 (1994)
Plyushchenkov, B.D., Turchaninov,V.I.: Construction principles of the efficient finite difference scheme for the refined Biot’s equations. In: Auriault et al., (ed), Poromechanics II (Swets & Zeitlinger, Lisse, ISBN 90 5809 394 8), pp. 757-764 (2002)
Pride, S.R.: Relationships between Seismic and Hydrological Properties. In: Rubin, Y., Hubbard, S.S. (eds.) Hydrogeophysics. Water Science and Technology Library, vol. 50. Springer, Dordrecht (2005)
Chapelle, D., Sainte-Marie, J., Gerbeau, J.-F., Vignon-Clementel, I.: A poroelastic model valid in large strains with applications to perfusion in cardiac modeling. Comput. Mech. 46(1), 91–101 (2010). https://doi.org/10.1007/s00466-009-0452-x.inria-00542672
Uzuoka, Ryosuke, Borja, Ronaldo: Dynamics of unsaturated poroelastic solids at finite strain. Int. J. Numer. Anal. Methods Geomech. 36, 1535–1573 (2012). https://doi.org/10.1002/nag.1061
Zingerman, K.M., Levin, V.A.: Some qualitative effects in the exact solutions of the Lamé problem for large deformations. J. Appl. Math. Mech. 76, 205–219 (2012). https://doi.org/10.1016/j.jappmathmech.2012.05.012
Levin, V.A., Zingerman, K.M., Vershinin, A.V., Yakovlev, M.: Numerical analysis of effective mechanical properties of rubber-cord composites under finite strains. Compos. Struct. 131, 25–36 (2015). https://doi.org/10.1016/j.compstruct.2015.04.037
Levin, V., Zingerman, K., Vershinin, A., Freiman, E., Yangirova, A.: Numerical analysis of the stress concentration near holes originating in previously loaded viscoelastic bodies at finite strains. Int. J. Solids Struct. 50(20–21), 3119–3135 (2013)
Komatitsch, D., Tromp, J.: Spectral-element simulations of global seismic wave propagation-I. Validation. Geophys. J. Int. 149, 390–412 (2002)
Komatitsch, D., Vilotte, J.-P.: The spectral element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures. Bull. Seismol. Soc. Am. 88(2), 368–392 (1998)
Schubert, B.: The spectral-element method for seismic wave propagation: theory, implementation and comparison to finite difference methods, Tech. rep., University of Munich, Germany, p. 165 (2003)
Bernardi, C., Maday, Y.: Polynomial interpolation results in Sobolev spaces. J. Comput. Appl. Math. 43(1–2), 53–80 (1992)
Karpenko, V., Vershinin, A., Levin, V., Zingerman, K.: Some results of mesh convergence estimation for the spectral element method of different orders in Fidesys industrial package. In: IOP Conference Series: Materials Science and Engineering vol. 158 (2016)
Dang, H.K., Meguid, M.A.: Evaluating the performance of an explicit dynamic relaxation technique in analyzing non-linear geotechnical engineering problems. Comput. Geotech. 37(1–2), 125–131 (2010). https://doi.org/10.1016/j.compgeo.2009.08.004
Styopin, N.E., Vershinin, A.V., Zingerman, K.M., Levin, V.A.: Comparative analysis of different variants of the Uzawa algorithm in problems of the theory of elasticity for incompressible materials. J. Adv. Res. 7, 703–707 (2016). https://doi.org/10.1016/j.jare.2016.08.001
https://cae-fidesys.com website of Fidesys LLC
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Francesco dell’Isola.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-022-01117-4