A multiscale multi-permeability poroplasticity model linked by recursive homogenizations and deep learning.

*(English)*Zbl 1440.74130Summary: Many geological materials, such as shale, mudstone, carbonate rock, limestone and rock salt are multi-porosity porous media in which pores of different scales may co-exist in the host matrix. When fractures propagate in these multi-porosity materials, these pores may enlarge and coalesce and therefore change the magnitude and the principal directions of the effective permeability tensors. The pore-fluid inside the cracks and the pores of host matrix may interact and exchange fluid mass, but the difference in hydraulic properties of these pores often means that a single homogenized effective permeability tensor field is insufficient to characterize the evolving hydraulic properties of these materials at smaller time scale. Furthermore, the complexity of the hydro-mechanical coupling process and the induced mechanical and hydraulic anisotropy originated from the micro-fracture and plasticity at grain scale also makes it difficult to propose, implement and validate separated macroscopic constitutive laws for numerical simulations. This article presents a hybrid data-driven method designed to capture the multiscale hydro-mechanical coupling effect of porous media with pores of various different sizes. At each scale, data-driven models generated from supervised machine learning are hybridized with classical constitutive laws in a directed graph that represents the numerical models. By using sub-scale simulations to generate database to train material models, an offline homogenization procedure is used to replace the up-scaling procedure to generate cohesive laws for localized physical discontinuities at both grain and specimen scales. Through a proper homogenization procedure that preserves spatial length scales, the proposed method enables field-scale simulations to gather insights from meso-scale and grain-scale micro-structural attributes. This method is proven to be much more computationally efficient than the classical DEM-FEM or FEM\(^2\) approach while at the same time more robust and flexible than the classical surrogate modeling approach. Due to the usage of bridging-scale technique, the proposed model may provide multiple opportunities to incorporate different types of simulations and experimental data across different length scales for machine learning. Numerical issues will also be discussed.

##### MSC:

74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |

74S05 | Finite element methods applied to problems in solid mechanics |

74L10 | Soil and rock mechanics |

##### Keywords:

dual-porosity; data-driven modeling; directed graph; embedded discontinuity; recurrent neural network; multiscale method
PDF
BibTeX
Cite

\textit{K. Wang} and \textit{W. Sun}, Comput. Methods Appl. Mech. Eng. 334, 337--380 (2018; Zbl 1440.74130)

Full Text:
DOI

##### References:

[1] | Bai, Mao; Elsworth, Derek; Roegiers, Jean-Claude, Multiporosity/multipermeability approach to the simulation of naturally fractured reservoirs, Water Resour. Res., 29, 6, 1621-1634 (1993) |

[2] | Zimmerman, Robert W.; Chen, Gang; Hadgu, Teklu; Bodvarsson, Gudmundur S., A numerical dual-porosity model with semianalytical treatment of fracture/matrix flow, Water Resour. Res., 29, 7, 2127-2137 (1993) |

[3] | Kuhlman, Kristopher L.; Heath, Jason E.; Gardner, W. Payton; Robinson, David G., Multiporosity flow of gases in fractured shale formations, J. Coal Geol., 109, 110, 101-146 (2013) |

[4] | Lewis, Roland W.; Ghafouri, Hamid R., A novel finite element double porosity model for multiphase flow through deformable fractured porous media, Int. J. Numer. Anal. Methods Geomech., 21, 11, 789-816 (1997) |

[5] | Callari, C.; Federico, F., FEM validation of a double porosity elastic model for consolidation of structurally complex clayey soils, Int. J. Numer. Anal. Methods Geomech., 24, 4, 367-402 (2000) |

[6] | Choo, Jinhyun; White, Joshua A.; Borja, Ronaldo I., Hydromechanical modeling of unsaturated flow in double porosity media, Int. J. Geomech., D4016002 (2016) |

[7] | Borja, Ronaldo I.; Choo, Jinhyun, Cam-Clay plasticity, Part VIII: A constitutive framework for porous materials with evolving internal structure., Comput. Methods Appl. Mech. Engrg., 309, 653-679 (2016) |

[8] | Liu, Yang; Sun, WaiChing; Fish, Jacob, Determining material parameters for critical state plasticity models based on multilevel extended digital database, J. Appl. Mech., 83, 1, 011003 (2016) |

[9] | Wang, Kun; Sun, WaiChing; Salager, Simon; Na, SeonHong; Khaddour, Ghonwa, Identifying material parameters for a micro-polar plasticity model via X-ray micro-computed tomographic (Ct) images: Lessons learned from the curve-fitting exercises, Int. J. Multiscale Comput. Eng., 14, 4 (2016) |

[10] | Friedman, Jerome H., On bias, variance, 0I—loss, and the curse-of-dimensionality, Data Min. Knowl. Discov., 1, 1, 55-77 (1997) |

[11] | Schmidt, U.; Mergheim, J.; Steinmann, P., Identification of elastoplastic microscopic material parameters within a homogenization scheme, Internat. J. Numer. Methods Engrg. (2015) |

[12] | Ghaboussi, J.; Garrett Jr., J. H.; Wu, Xiping, Knowledge-based modeling of material behavior with neural networks, J. Eng. Mech., 117, 1, 132-153 (1991) |

[13] | Graf, Wolfgang; Freitag, Steffen; Kaliske, Michael; Sickert, J.-U., Recurrent neural networks for uncertain time-dependent structural behavior, Comput. Aid. Civ. Infrastruct. Eng., 25, 5, 322-323 (2010) |

[14] | Furukawa, Tomonari; Yagawa, Genki, Implicit constitutive modelling for viscoplasticity using neural networks, Internat. J. Numer. Methods Engrg., 43, 2, 195-219 (1998) |

[15] | Kirchdoerfer, Trenton; Ortiz, Michael, Data-driven computational mechanics, Comput. Methods Appl. Mech. Engrg., 304, 81-101 (2016) |

[16] | Lefik, M.; Boso, D. P.; Schrefler, B. A., Artificial neural networks in numerical modelling of composites, Comput. Methods Appl. Mech. Engrg., 198, 21, 1785-1804 (2009) |

[17] | Foster, C. D.; Regueiro, R. A.; Fossum, A. F.; Borja, R. I., Implicit numerical integration of a three-invariant, isotropic/kinematic hardening cap plasticity model for geomaterials, Comput. Methods Appl. Mech. Engrg., 194, 50, 5109-5138 (2005) |

[18] | Sun, WaiChing; Chen, Qiushi; Ostien, Jakob T., Modeling the hydro-mechanical responses of strip and circular punch loadings on water-saturated collapsible geomaterials, Acta Geotech. (2013) |

[19] | Zhu, Wenlu; Wong, Teng-fong, The transition from brittle faulting to cataclastic flow: Permeability evolution, J. Geophys. Res. Solid Earth, 102, B2, 3027-3041 (1997) |

[20] | Nuth, Mathieu; Laloui, Lyesse, Advances in modelling hysteretic water retention curve in deformable soils, Comput. Geotechn., 35, 6, 835-844 (2008) |

[21] | Sun, WaiChing; Andrade, José E.; Rudnicki, John W.; Eichhubl, Peter, Connecting microstructural attributes and permeability from 3D tomographic images of in situ shear-enhanced compaction bands using multiscale computations, Geophys. Res. Lett., 38, 10 (2011) |

[22] | Guo, Ning; Zhao, Jidong, The signature of shear-induced anisotropy in granular media, Comput. Geotechn., 47, 1-15 (2013) |

[23] | Kuhn, Matthew R.; Sun, WaiChing; Wang, Qi, Stress-induced anisotropy in granular materials: fabric, stiffness, and permeability, Acta Geotech., 10, 4, 399-419 (2015) |

[24] | Sun, WaiChing; Cai, Zhijun; Choo, Jinhyun, Mixed Arlequin method for multiscale poromechanics problems, Internat. J. Numer. Methods Engrg. (2016) |

[25] | Na, SeonHong; Sun, WaiChing, Computational thermo-hydro-mechanics for multiphase freezing and thawing porous media in the finite deformation range, Comput. Methods Appl. Mech. Engrg., 318, 667-700 (2017) |

[26] | Miehe, Christian, Computational micro-to-macro transitions for discretized micro-structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy, Comput. Methods Appl. Mech. Engrg., 192, 5, 559-591 (2003) |

[27] | Nitka, Michał; Combe, Gaël; Dascalu, Cristian; Desrues, Jacques, Two-scale modeling of granular materials: a DEM-FEM approach, Granular Matter, 13, 3, 277-281 (2011) |

[28] | Sun, WaiChing; Kuhn, Matthew R.; Rudnicki, John W., A multiscale DEM-LBM analysis on permeability evolutions inside a dilatant shear band, Acta Geotech., 1-16 (2013) |

[29] | Guo, Ning; Zhao, Jidong; Sun, WaiChing, Multiscale analysis of shear failure of thick-walled hollow cylinder in dry sand, Géotechn. Lett., 6, 1 (2016) |

[30] | Wellmann, Christian; Wriggers, Peter, A two-scale model of granular materials, Comput. Methods Appl. Mech. Engrg., 205, 46-58 (2012) |

[31] | Sun, WaiChing; Mota, Alejandro, A multiscale overlapped coupling formulation for large-deformation strain localization, Comput. Mech., 1-18 (2014) |

[32] | Feyel, Frédéric; Chaboche, Jean-Louis, FE 2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials, Comput. Methods Appl. Mech. Engrg., 183, 3, 309-330 (2000) |

[33] | Miehe, Christian; Bayreuther, C. G., On multiscale FE analyses of heterogeneous structures: from homogenization to multigrid solvers, Internat. J. Numer. Methods Engrg., 71, 10, 1135-1180 (2007) |

[34] | Geers, Marc G. D.; Kouznetsova, Varvara G.; Brekelmans, W. A.M., Multi-scale computational homogenization: Trends and challenges, J. Comput. Appl. Math., 234, 7, 2175-2182 (2010) |

[35] | Zhou, X. W.; Moody, N. R.; Jones, R. E.; Zimmerman, J. A.; Reedy, E. D., Molecular-dynamics-based cohesive zone law for brittle interfacial fracture under mixed loading conditions: Effects of elastic constant mismatch, Acta Mater., 57, 16, 4671-4686 (2009) |

[36] | Elsworth, Derek; Bai, Mao, Flow-deformation response of dual-porosity media, J. Geotechn. Eng., 118, 1, 107-124 (1992) |

[37] | Ji, Yuntao; Hall, Stephen A.; Baud, Patrick; Wong, Teng-fong, Characterization of pore structure and strain localization in Majella limestone by X-ray computed tomography and digital image correlation, Geophys. J. Int., 200, 2, 701-719 (2015) |

[38] | Kuhlman, Kristopher L.; Malama, Bwalya; Heath, Jason E., Multiporosity flow in fractured low-permeability rocks, Water Resour. Res., 51, 2, 848-860 (2015) |

[39] | Köhne, J. Maximilian; Mohanty, Binayak P.; Šimŭnek, Jirka, Inverse dual-permeability modeling of preferential water flow in a soil column and implications for field-scale solute transport, Vadose Zone J., 5, 1, 59-76 (2006) |

[40] | Auriault, Jean-Louis; Boutin, Claude; Geindreau, Christian, Homogenization of Coupled Phenomena in Heterogenous Media, Vol. 149 (2010), John Wiley & Sons |

[41] | Lewandowska, Jolanta; Auriault, J.-L., Extension of Biot theory to the problem of saturated microporous elastic media with isolated cracks or/and vugs, Int. J. Numer. Anal. Methods Geomech., 37, 16, 2611-2628 (2013) |

[42] | Sun, WaiChing; Andrade, José E.; Rudnicki, John W., Multiscale method for characterization of porous microstructures and their impact on macroscopic effective permeability, Internat. J. Numer. Methods Engrg., 88, 12, 1260-1279 (2011) |

[43] | Liu, Yang; Sun, WaiChing; Yuan, Zifeng; Fish, Jacob, A nonlocal multiscale discrete-continuum model for predicting mechanical behavior of granular materials, Internat. J. Numer. Methods Engrg. (2015) |

[44] | Wang, Kun; Sun, WaiChing, Anisotropy of a tensorial bishop coefficient for wetted granular materials, J. Eng. Mech., B4015004 (2015) |

[45] | Wang, Kun; Sun, WaiChing, A semi-implicit discrete-continuum coupling method for porous media based on the effective stress principle at finite strain, Comput. Methods Appl. Mech. Engrg., 304, 546-583 (2016) |

[47] | Choo, Jinhyun; Borja, Ronaldo I., Stabilized mixed finite elements for deformable porous media with double porosity, Comput. Methods Appl. Mech. Engrg., 293, 131-154 (2015) |

[48] | Borja, Ronaldo I., A finite element model for strain localization analysis of strongly discontinuous fields based on standard galerkin approximation, Comput. Methods Appl. Mech. Engrg., 190, 11, 1529-1549 (2000) |

[49] | Callari, C.; Armero, Francisco, Finite element methods for the analysis of strong discontinuities in coupled poro-plastic media, Comput. Methods Appl. Mech. Engrg., 191, 39, 4371-4400 (2002) |

[50] | Mosler, J.; Meschke, G., 3D modelling of strong discontinuities in elastoplastic solids: fixed and rotating localization formulations, Internat. J. Numer. Methods Engrg., 57, 11, 1553-1576 (2003) |

[51] | Callari, C.; Armero, F.; Abati, A., Strong discontinuities in partially saturated poroplastic solids, Comput. Methods Appl. Mech. Engrg., 199, 23, 1513-1535 (2010) |

[52] | White, Joshua A.; Borja, Ronaldo I., Stabilized low-order finite elements for coupled solid-deformation/fluid-diffusion and their application to fault zone transients, Comput. Methods Appl. Mech. Engrg., 197, 49, 4353-4366 (2008) |

[53] | Sun, WaiChing; Ostien, Jakob T.; Salinger, Andrew G., A stabilized assumed deformation gradient finite element formulation for strongly coupled poromechanical simulations at finite strain, Int. J. Numer. Anal. Methods Geomech., 37, 16, 2755-2788 (2013) |

[54] | Sun, WaiChing, A stabilized finite element formulation for monolithic thermo-hydro-mechanical simulations at finite strain, Internat. J. Numer. Methods Engrg., 103, 11, 798-839 (2015) |

[55] | Krischok, Andreas; Linder, Christian, On the enhancement of low-order mixed finite element methods for the large deformation analysis of diffusion in solids, Internat. J. Numer. Methods Engrg., 106, 4, 278-297 (2016) |

[56] | Frankenreiter, Ilona; Rosato, Daniele; Miehe, Christian, Hybrid micro-macro-modeling of evolving anisotropies and length scales in finite plasticity of polycrystals, PAMM, 11, 1, 515-518 (2011) |

[57] | Fish, Jacob, Practical Multiscaling (2013), John Wiley & Sons |

[58] | Keshavarz, Shahriyar; Ghosh, Somnath, Multi-scale crystal plasticity finite element model approach to modeling nickel-based superalloys, Acta Mater., 61, 17, 6549-6561 (2013) |

[59] | Fish, Jacob; Wu, Wei, A nonintrusive stochastic multiscale solver, Internat. J. Numer. Methods Engrg., 88, 9, 862-879 (2011) |

[60] | Shahir, Hadi; Pak, Ali; Taiebat, Mahdi; Jeremić, Boris, Evaluation of variation of permeability in liquefiable soil under earthquake loading, Comput. Geotechn., 40, 74-88 (2012) |

[61] | Tawhai, Merryn; Bischoff, Jeff; Einstein, Daniel; Erdemir, Ahmet; Guess, Trent; Reinbolt, Jeff, Multiscale modeling in computational biomechanics, IEEE Eng. Med. Biol. Mag., 28, 3 (2009) |

[62] | Jain, Jayesh R.; Ghosh, Somnath, Damage evolution in composites with a homogenization-based continuum damage mechanics model, Int. J. Damage Mech., 18, 6, 533-568 (2009) |

[63] | Pawlowski, Roger P.; Phipps, Eric T.; Salinger, Andrew G., Automating embedded analysis capabilities and managing software complexity in multiphysics simulation, Part I: Template-based generic programming, Sci. Program., 20, 2, 197-219 (2012) |

[64] | Salinger, Andrew; Bartlett, Roscoe; Bradley, Andrew; Chen, Qiushi; Demeshko, Irina; Gao, Xujiao; Hanson, Glen; Mota, Alejandro; Muller, Richard; Nielsen, Erik, Albany: Using component-based design to develop a flexible, generic multiphysics analysis code, Int. J. Multiscale Comput. Eng. (2016) |

[65] | Biot, Maurice A., General theory of three-dimensional consolidation, J. Appl. Phys., 12, 2, 155-164 (1941) |

[66] | Terzaghi, Karl, Theory of Consolidation (1943), Wiley Online Library |

[67] | Coussy, Olivier, Poromechanics (2004), John Wiley & Sons |

[68] | Lefik, M.; Schrefler, B. A., Artificial neural network as an incremental non-linear constitutive model for a finite element code, Comput. Methods Appl. Mech. Engrg., 192, 28, 3265-3283 (2003) |

[69] | Guo, Ning; Zhao, Jidong, A coupled FEM/DEM approach for hierarchical multiscale modelling of granular media, Internat. J. Numer. Methods Engrg., 99, 11, 789-818 (2014) |

[70] | Yvonnet, Julien; He, Q.-C., The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains, J. Comput. Phys., 223, 1, 341-368 (2007) |

[71] | Zahr, Matthew J.; Avery, Philip; Farhat, Charbel, A multilevel projection-based model order reduction framework for nonlinear dynamic multiscale problems in structural and solid mechanics, Internat. J. Numer. Methods Engrg. (2017) |

[72] | Miehe, C.; Dettmar, J.; Zäh, D., Homogenization and two-scale simulations of granular materials for different microstructural constraints, Internat. J. Numer. Methods Engrg., 83, 8-9, 1206-1236 (2010) |

[73] | Du, X.; Ostoja-Starzewski, M., On the size of representative volume element for darcy law in random media, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci, 462, 2074, 2949-2963 (2006) |

[74] | Ostoja-Starzewski, Martin; Du, X.; Khisaeva, Z. F.; Li, W., Comparisons of the size of the representative volume element in elastic, plastic, thermoelastic, and permeable random microstructures, Int. J. Multiscale Comput. Eng., 5, 2 (2007) |

[75] | Bryant, Steven L.; King, Peter R.; Mellor, David W., Network model evaluation of permeability and spatial correlation in a real random sphere packing, Transp. Porous Media, 11, 1, 53-70 (1993) |

[76] | Chareyre, Bruno; Cortis, Andrea; Catalano, Emanuele; Barthélemy, Eric, Pore-scale modeling of viscous flow and induced forces in dense sphere packings, Transp. Porous Media, 94, 2, 595-615 (2012) |

[77] | Edelsbrunner, Herbert; Shah, Nimish R., Incremental topological flipping works for regular triangulations, Algorithmica, 15, 3, 223-241 (1996) |

[78] | Piri, Mohammad; Blunt, Martin J., Three-dimensional mixed-wet random pore-scale network modeling of two-and three-phase flow in porous media. I. Model description, Phys. Rev. E, 71, 2, 026301 (2005) |

[79] | Hilpert, Markus; Glantz, Roland; Miller, Cass T., Calibration of a pore-network model by a pore-morphological analysis, Transp. Porous Media, 51, 3, 267-285 (2003) |

[80] | Johari, A.; Habibagahi, G.; Ghahramani, A., Prediction of soil-water characteristic curve using genetic programming, J. Geotechn. Geoenviron. Eng., 132, 5, 661-665 (2006) |

[81] | Lamorski, Krzysztof; Pachepsky, Yakov; Sławiński, C.; Walczak, R. T., Using support vector machines to develop pedotransfer functions for water retention of soils in Poland, Soil Sci. Am. J., 72, 5, 1243-1247 (2008) |

[82] | Gardner, Matt W.; Dorling, S. R., Artificial neural networks (the multilayer perceptron)—a review of applications in the atmospheric sciences, Atmos. Environ., 32, 14, 2627-2636 (1998) |

[83] | Cochocki, A.; Unbehauen, Rolf, Neural Networks for Optimization and Signal Processing (1993), John Wiley & Sons, Inc. |

[84] | Kalogirou, Soteris A., Artificial neural networks in renewable energy systems applications: a review, Renew. Sustain. Energy Rev., 5, 4, 373-401 (2001) |

[85] | Lisboa, Paulo J.; Taktak, Azzam F. G., The use of artificial neural networks in decision support in cancer: a systematic review, Neural Netw., 19, 4, 408-415 (2006) |

[86] | Zienkiewicz, Olgierd C.; Chan, A. H.C.; Pastor, M.; Schrefler, B. A.; Shiomi, T., Computational Geomechanics (1999), Wiley Chichester |

[87] | Jung, Sungmoon; Ghaboussi, Jamshid, Neural network constitutive model for rate-dependent materials, Comput. Struct., 84, 15, 955-963 (2006) |

[88] | Lukoševičius, Mantas; Jaeger, Herbert, Reservoir computing approaches to recurrent neural network training, Comput. Sci. Rev., 3, 3, 127-149 (2009) |

[89] | Zhu, Jian-Hua; Zaman, Musharraf M.; Anderson, Scott A., Modeling of soil behavior with a recurrent neural network, Can. Geotech. J., 35, 5, 858-872 (1998) |

[90] | Hochreiter, Sepp; Schmidhuber, Jürgen, Long short-term memory, Neural Comput., 9, 8, 1735-1780 (1997) |

[91] | Dafalias, Yannis F.; Manzari, Majid T., Simple plasticity sand model accounting for fabric change effects, J. Eng. Mech., 130, 6, 622-634 (2004) |

[92] | Fu, Pengcheng; Dafalias, Yannis F., Fabric evolution within shear bands of granular materials and its relation to critical state theory, Int. J. Numer. Anal. Methods Geomech., 35, 18, 1918-1948 (2011) |

[93] | Versino, Daniele; Tonda, Alberto; Bronkhorst, Curt A., Data driven modeling of plastic deformation, Comput. Methods Appl. Mech. Engrg., 318, 981-1004 (2017) |

[94] | Peng, S. S., Time-dependent aspects of rock behavior as measured by a servocontrolled hydraulic testing machine, Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 10, 3 (1973), 235IN21237-236IN22246 |

[96] | Borja, Ronaldo I.; Tamagnini, Claudio, Cam-clay plasticity Part III: Extension of the infinitesimal model to include finite strains, Comput. Methods Appl. Mech. Engrg., 155, 1-2, 73-95 (1998) |

[97] | Mota, Alejandro; Sun, WaiChing.; Ostien, Jakob T.; Foulk, James W.; Long, Kevlin N., Lie-group interpolation and variational recovery for internal variables, Comput. Mech., 1-19 (2013) |

[98] | Krysl, Petr; Endres, Lance, Explicit Newmark/Verlet algorithm for time integration of the rotational dynamics of rigid bodies, Internat. J. Numer. Methods Engrg., 62, 15, 2154-2177 (2005) |

[99] | Tuzel, Oncel; Porikli, Fatih; Meer, Peter, Learning on lie groups for invariant detection and tracking, (Computer Vision and Pattern Recognition, 2008. CVPR 2008. IEEE Conference on (2008), IEEE), 1-8 |

[103] | Pedregosa, Fabian; Varoquaux, Gaël; Gramfort, Alexandre; Michel, Vincent; Thirion, Bertrand; Grisel, Olivier; Blondel, Mathieu; Prettenhofer, Peter; Weiss, Ron; Dubourg, Vincent, Scikit-learn: Machine learning in Python, J. Mach. Learn. Res., 12, Oct, 2825-2830 (2011) |

[104] | Belytschko, Ted; Liu, Wing Kam; Moran, Brian; Elkhodary, Khalil, Nonlinear Finite Elements for Continua and Structures (2013), John wiley & sons |

[105] | Park, Kyoungsoo; Paulino, Glaucio H., Cohesive zone models: a critical review of traction-separation relationships across fracture surfaces, Appl. Mech. Rev., 64, 6 (2011), 060802 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.