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A stochastic multiscale algorithm for modeling complex granular materials

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Abstract

Modeling of granular materials in which the grains have irregular shapes and surface is a long-standing problem that has been studied for decades. Almost all the current models either represent the grains as particles with geometrically-regular shapes or attempt to infer some low-order statistical properties of the materials in order to describe granular media. We use an approach to modeling of granular materials that utilizes a two- or three-dimensional image of the material’s morphology. It reconstructs realizations of the image based on a Markov process, and uses a multiscale approach and graph-theoretical concepts to refine the realizations and make them free of artefacts. The method is applied to several complex 2D and 3D examples of granular materials. Various morphological properties of the models are computed and are compared with those of the original images; very good agreement is found for all the cases. Furthermore, the computational cost of the method is very low and, therefore, the method can generate large-size models for complex granular materials.

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References

  1. Desrues, J., Viggiani, G.: Strain localization in sand: an overview of the experimental results obtained in Grenoble using stereophotogrammetry. Int. J. Numer. Anal. Methods Geomech. 28, 279–321 (2004). https://doi.org/10.1002/nag.338

    Article  Google Scholar 

  2. Hall, S.A., Rnert, M.B., Desrues, J., Pannier, Y., Lenoir, N., Viggiani, B., Suelle, G.P.: Discrete and continuum analysis of localised deformation in sand using X-ray CT and volumetric digital image correlation. Géotechnique 60, 315–322 (2010). https://doi.org/10.1680/geot.2010.60.5.315

    Article  Google Scholar 

  3. Cavarretta, I., Coop, M., O’Sullivan, C.: The influence of particle characteristics on the behaviour of coarse grained soils. Géotechnique 60, 413–423 (2010). https://doi.org/10.1680/geot.2010.60.6.413

    Article  Google Scholar 

  4. Aste, T., Saadatfar, M., Senden, T.: Geometrical structure of disordered sphere packings. Phys. Rev. E 71, 061302 (2005). https://doi.org/10.1103/PhysRevE.71.061302

    Article  ADS  Google Scholar 

  5. Jerves, A.X., Kawamoto, R.Y., Andrade, J.E.: Effects of grain morphology on critical state: a computational analysis. Acta Geotech. 11, 493–503 (2016). https://doi.org/10.1007/s11440-015-0422-8

    Article  Google Scholar 

  6. Tahmasebi, P.: Packing of discrete and irregular particles. Comput. Geotech. (2018). https://doi.org/10.1016/j.compgeo.2018.03.011

    Google Scholar 

  7. Cundall, P.A., Strack, O.D.L.: Development of constitutive laws for soil using the distinct element method. SAE Prepr. 1, 289–298 (1979)

    Google Scholar 

  8. Thomas, P.A., Bray, J.D.: Capturing nonspherical shape of granular media with disk clusters. J. Geotech. Geoenviron. Eng. 125, 169–178 (1999). https://doi.org/10.1061/(ASCE)1090-0241(1999)125:3(169)

    Article  Google Scholar 

  9. Salot, C., Gotteland, P., Villard, P.: Influence of relative density on granular materials behavior: DEM simulations of triaxial tests. Granul. Matter 11, 221–236 (2009). https://doi.org/10.1007/s10035-009-0138-2

    Article  MATH  Google Scholar 

  10. Stahl, M., Konietzky, H.: Discrete element simulation of ballast and gravel under special consideration of grain-shape, grain-size and relative density. Granul. Matter 13, 417–428 (2011). https://doi.org/10.1007/s10035-010-0239-y

    Article  Google Scholar 

  11. Katagiri, J., Matsushima, T., Yamada, Y.: Simple shear simulation of 3D irregularly-shaped particles by image-based DEM. Granul. Matter 12, 491–497 (2010). https://doi.org/10.1007/s10035-010-0207-6

    Article  MATH  Google Scholar 

  12. Lu, M., McDowell, G.R.: The importance of modelling ballast particle shape in the discrete element method. Granul. Matter 9, 69–80 (2006). https://doi.org/10.1007/s10035-006-0021-3

    Article  Google Scholar 

  13. Jensen, R.P., Edil, T.B., Bosscher, P.J., Plesha, M.E., Kahla, N.B.: Effect of particle shape on interface behavior of DEM-simulated granular materials. Int. J. Geomech. 1, 1–19 (2001). https://doi.org/10.1061/(ASCE)1532-3641(2001)1:1(1)

    Article  Google Scholar 

  14. Pournin, L., Weber, M., Tsukahara, M., Ferrez, J.-A., Ramaioli, M., Liebling, T.M.: Three-dimensional distinct element simulation of spherocylinder crystallization. Granul. Matter 7, 119–126 (2005). https://doi.org/10.1007/s10035-004-0188-4

    Article  MATH  Google Scholar 

  15. Andrade, J.E., Lim, K.-W., Avila, C.F., Vlahinić, I.: Granular element method for computational particle mechanics. Comput. Methods Appl. Mech. Eng. 241, 262–274 (2012). https://doi.org/10.1016/j.cma.2012.06.012

    Article  ADS  MATH  Google Scholar 

  16. Ng, T.-T.: Particle shape effect on macro- and micro-behaviors of monodisperse ellipsoids. Int. J. Numer. Anal. Methods Geomech. 33, 511–527 (2009). https://doi.org/10.1002/nag.732

    Article  MATH  Google Scholar 

  17. Azéma, E., Radjai, F., Saussine, G.: Quasistatic rheology, force transmission and fabric properties of a packing of irregular polyhedral particles. Mech. Mater. 41, 729–741 (2009). https://doi.org/10.1016/j.mechmat.2009.01.021

    Article  Google Scholar 

  18. Peña, A.A., García-Rojo, R., Herrmann, H.J.: Influence of particle shape on sheared dense granular media. Granul. Matter 9, 279–291 (2007). https://doi.org/10.1007/s10035-007-0038-2

    Article  MATH  Google Scholar 

  19. Mollon, G., Zhao, J.: Fourier–Voronoi-based generation of realistic samples for discrete modelling of granular materials. Granul. Matter 14, 621–638 (2012). https://doi.org/10.1007/s10035-012-0356-x

    Article  Google Scholar 

  20. Tillemans, H.-J., Herrmann, H.J.: Simulating deformations of granular solids under shear. Phys. A Stat. Mech. Appl. 217, 261–288 (1995). https://doi.org/10.1016/0378-4371(95)00111-J

    Article  Google Scholar 

  21. Galindo-Torres, S.A., Pedroso, D.M.: Molecular dynamics simulations of complex-shaped particles using Voronoi-based spheropolyhedra. Phys. Rev. E 81, 061303 (2010). https://doi.org/10.1103/PhysRevE.81.061303

    Article  ADS  Google Scholar 

  22. Fu, P., Dafalias, Y.F.: Fabric evolution within shear bands of granular materials and its relation to critical state theory. Int. J. Numer. Anal. Methods Geomech. 35, 1918–1948 (2011). https://doi.org/10.1002/nag.988

    Article  Google Scholar 

  23. Azéma, E., Radjaï, F., Peyroux, R., Saussine, G.: Force transmission in a packing of pentagonal particles. Phys. Rev. E 76, 011301 (2007). https://doi.org/10.1103/PhysRevE.76.011301

    Article  ADS  Google Scholar 

  24. Houlsby, G.T.: Potential particles: a method for modelling non-circular particles in DEM. Comput. Geotech. 36, 953–959 (2009). https://doi.org/10.1016/j.compgeo.2009.03.001

    Article  Google Scholar 

  25. Biswal, B., Manwart, C., Hilfer, R., Bakke, S., Øren, P.E.: Quantitative analysis of experimental and synthetic microstructures for sedimentary rock. Phys. A Stat. Mech. Appl. 273, 452–475 (1999). https://doi.org/10.1016/S0378-4371(99)00248-4

    Article  Google Scholar 

  26. Biswal, B., Øren, P.-E., Held, R.J., Bakke, S., Hilfer, R.: Stochastic multiscale model for carbonate rocks. Phys. Rev. E 75, 061303 (2007). https://doi.org/10.1103/PhysRevE.75.061303

    Article  ADS  Google Scholar 

  27. Bryant, S., Blunt, M.: Prediction of relative permeability in simple porous media. Phys. Rev. A 46, 2004–2011 (1992). https://doi.org/10.1103/PhysRevA.46.2004

    Article  ADS  Google Scholar 

  28. Coelho, D., Thovert, J.-F., Adler, P.M.: Geometrical and transport properties of random packings of spheres and aspherical particles. Phys. Rev. E 55, 1959–1978 (1997). https://doi.org/10.1103/PhysRevE.55.1959

    Article  ADS  Google Scholar 

  29. Øren, P.E., Bakke, S.: Process based reconstruction of sandstones and prediction of transport properties. Transp. Porous Media 46, 311–343 (2002). https://doi.org/10.1023/A:1015031122338

    Article  Google Scholar 

  30. Rechenmacher, A., Finno, R.: Digital image correlation to evaluate shear banding in dilative sands. Geotech. Test. J. 27, 10864 (2004). https://doi.org/10.1520/GTJ11263J

    Article  Google Scholar 

  31. Tahmasebi, P., Sahimi, M., Kohanpur, A.H.A.H., Valocchi, A.: Pore-scale simulation of flow of CO2 and brine in reconstructed and actual 3D rock cores. J. Pet. Sci. Eng. (2016). https://doi.org/10.1016/j.petrol.2016.12.031

    Google Scholar 

  32. Alshibli, K.A., Hasan, A.: Spatial variation of void ratio and shear band thickness in sand using X-ray computed tomography. Géotechnique 58, 249–257 (2008). https://doi.org/10.1680/geot.2008.58.4.249

    Article  Google Scholar 

  33. Andò, E., Hall, S.A., Viggiani, G., Desrues, J., Bésuelle, P.: Grain-scale experimental investigation of localised deformation in sand: a discrete particle tracking approach. Acta Geotech. 7, 1–13 (2012). https://doi.org/10.1007/s11440-011-0151-6

    Article  Google Scholar 

  34. Senatore, C., Wulfmeier, M., Vlahinić, I., Andrade, J., Iagnemma, K.: Design and implementation of a particle image velocimetry method for analysis of running gear-soil interaction. J. Terramech. 50, 311–326 (2013). https://doi.org/10.1016/j.jterra.2013.09.004

    Article  Google Scholar 

  35. Garboczi, E.J.: Three-dimensional mathematical analysis of particle shape using X-ray tomography and spherical harmonics: application to aggregates used in concrete. Cem. Concr. Res. 32, 1621–1638 (2002). https://doi.org/10.1016/S0008-8846(02)00836-0

    Article  Google Scholar 

  36. Bowman, E.T., Soga, K., Drummond, W.: Particle shape characterisation using Fourier descriptor analysis. Géotechnique 51, 545–554 (2001). https://doi.org/10.1680/geot.2001.51.6.545

    Article  Google Scholar 

  37. Andrade, J.E., Tu, X.: Multiscale framework for behavior prediction in granular media. Mech. Mater. 41, 652–669 (2009). https://doi.org/10.1016/j.mechmat.2008.12.005

    Article  Google Scholar 

  38. Mollon, G., Zhao, J.: Generating realistic 3D sand particles using Fourier descriptors. Granul. Matter. 15, 95–108 (2013). https://doi.org/10.1007/s10035-012-0380-x

    Article  Google Scholar 

  39. Lim, K.-W., Andrade, J.E.: Granular element method for three-dimensional discrete element calculations. Int. J. Numer. Anal. Methods Geomech. 38, 167–188 (2014). https://doi.org/10.1002/nag.2203

    Article  Google Scholar 

  40. Yeong, C.L.Y., Torquato, S.: Reconstructing random media. Phys. Rev. E 57, 495–506 (1998). https://doi.org/10.1103/PhysRevE.57.495

    Article  ADS  MathSciNet  Google Scholar 

  41. Yeong, C.L.Y., Torquato, S.: Reconstructing random media. II. Three-dimensional media from two-dimensional cuts. Phys. Rev. E 58, 224–233 (1998). https://doi.org/10.1103/PhysRevE.58.224

    Article  ADS  MathSciNet  Google Scholar 

  42. Jiao, Y., Stillinger, F.H., Torquato, S.: A superior descriptor of random textures and its predictive capacity. Proc. Natl. Acad. Sci. USA 106, 17634–17639 (2009). https://doi.org/10.1073/pnas.0905919106

    Article  ADS  Google Scholar 

  43. Zachary, C.E., Torquato, S.: Improved reconstructions of random media using dilation and erosion processes. Phys. Rev. E 84, 056102 (2011). https://doi.org/10.1103/PhysRevE.84.056102

    Article  ADS  Google Scholar 

  44. Guo, E.-Y., Chawla, N., Jing, T., Torquato, S., Jiao, Y.: Accurate modeling and reconstruction of three-dimensional percolating filamentary microstructures from two-dimensional micrographs via dilation–erosion method. Mater. Charact. 89, 33–42 (2014). https://doi.org/10.1016/j.matchar.2013.12.011

    Article  Google Scholar 

  45. Jiao, Y., Stillinger, F.H., Torquato, S.: Modeling heterogeneous materials via two-point correlation functions. II. Algorithmic details and applications. Phys. Rev. E 77, 031135 (2008). https://doi.org/10.1103/PhysRevE.77.031135

    Article  ADS  MathSciNet  Google Scholar 

  46. Ashmawy, A.K., Sukumaran, B., Hoang, V.V.: Evaluating the influence of particle shape on liquefaction behavior using discrete element modeling. In: The Thirteenth International Offshore and Polar Engineering Conference, International Society of Offshore and Polar Engineers, pp. 2003–2005 (2003)

  47. Dubois, F., Jean, M.: The non smooth contact dynamic method: recent LMGC90 software developments and application. In: Wriggers, P., Nackenhorst, U. (eds.) Analysis and Simulation of Contact Problems, pp. 375–378. Springer, Berlin/Heidelberg (2006)

    Chapter  Google Scholar 

  48. Krumbein, W.C.: Measurement and geological significance of shape and roundness of sedimentary particles. J. Sediment. Res. (1941). https://doi.org/10.1306/D42690F3-2B26-11D7-8648000102C1865D

    Google Scholar 

  49. Wadell, H.: Volume, shape, and roundness of quartz particles. J. Geol. 43, 250–280 (1935)

    Article  ADS  Google Scholar 

  50. Cho, G.-C., Dodds, J., Santamarina, J.C.: Particle shape effects on packing density, stiffness, and strength: natural and crushed sands. J. Geotech. Geoenviron. Eng. 132, 591–602 (2006). https://doi.org/10.1061/(ASCE)1090-0241(2006)132:5(591)

    Article  Google Scholar 

  51. Saadatfar, M., Garcia-Moreno, F., Hutzler, S., Sheppard, A.P., Knackstedt, M.A., Banhart, J., Weaire, D.: Imaging of metallic foams using X-ray micro-CT. Colloids Surf. A Physicochem. Eng. Asp. 344, 107–112 (2009). https://doi.org/10.1016/j.colsurfa.2009.01.008

    Article  Google Scholar 

  52. Karimpouli, S., Tahmasebi, P.: Conditional reconstruction: an alternative strategy in digital rock physics. Geophysics 81, D465–D477 (2016). https://doi.org/10.1190/geo2015-0260.1

    Article  Google Scholar 

  53. Askari, R., Hejazi, S.H., Sahimi, M.: Effect of deformation on the thermal conductivity of granular porous media with rough grain surface. Geophys. Res. Lett. 44, 8285–8293 (2017). https://doi.org/10.1002/2017GL074651

    Article  ADS  Google Scholar 

  54. Chen, D., Torquato, S.: Designing disordered hyperuniform two-phase materials with novel physical properties. Acta Mater. (2017). https://doi.org/10.1016/J.ACTAMAT.2017.09.053

    Google Scholar 

  55. Torquato, S.: Random Heterogeneous Materials. Springer, New York (2002)

    Book  MATH  Google Scholar 

  56. Tahmasebi, P.: Accurate modeling and evaluation of microstructures in complex materials. Phys. Rev. E. (2018). https://doi.org/10.1103/PhysRevE.97.023307

    Google Scholar 

  57. Tahmasebi, P., Sahimi, M.: Enhancing multiple-point geostatistical modeling: 1. Graph theory and pattern adjustment. Water Resour. Res. 52, 2074–2098 (2016). https://doi.org/10.1002/2015WR017806

    Article  ADS  Google Scholar 

  58. Tahmasebi, P.: Structural adjustment for accurate conditioning in large-scale subsurface systems. Adv. Water Resour. 101, (2017). https://doi.org/10.1016/j.advwatres.2017.01.009

  59. Tahmasebi, P.: HYPPS: a hybrid geostatistical modeling algorithm for subsurface modeling. Water Resour. Res. 53, 5980–5997 (2017). https://doi.org/10.1002/2017WR021078

    Article  ADS  Google Scholar 

  60. Tahmasebi, P., Sahimi, M.: Reconstruction of three-dimensional porous media using a single thin section. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 85, 1–13 (2012). https://doi.org/10.1103/PhysRevE.85.066709

    Article  Google Scholar 

  61. Tahmasebi, P., Sahimi, M., Andrade, J.E.J.E.: Image-based modeling of granular porous media. Geophys. Res. Lett. (2017). https://doi.org/10.1002/2017GL073938

    Google Scholar 

  62. Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297 (1965). https://doi.org/10.2307/2003354

    Article  MathSciNet  MATH  Google Scholar 

  63. Oppenheim, A.V.: Discrete-Time Signal Processing. Pearson Education, India (1999)

    Google Scholar 

  64. Duhamel, P., Vetterli, M.: Fast Fourier transforms: a tutorial review and a state of the art. Sig. Process. 19, 259–299 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  65. Tahmasebi, P.: Nanoscale and multiresolution models for shale samples. Fuel 217, 218–225 (2018). https://doi.org/10.1016/j.fuel.2017.12.107

    Article  Google Scholar 

  66. Greig, P.B., Seheult, A.: Exact maximum a posteriori estimation for binary images. J. R. Stat. Soc. Ser. B 51, 271–279 (1989)

    Google Scholar 

  67. Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Trans. Pattern Anal. Mach. Intell. 23, 1222–1239 (2001). https://doi.org/10.1109/34.969114

    Article  Google Scholar 

  68. Montgomery, D.C., Peck, E.A., Vining, G.G.: Introduction to Linear Regression Analysis. Wiley, London (2012)

    MATH  Google Scholar 

  69. Goldberg, D.E., Holland, J.H.: Genetic algorithms and machine learning. Mach. Learn. 3, 95–99 (1988). https://doi.org/10.1023/A:1022602019183

    Article  Google Scholar 

  70. Tahmasebi, P., Sahimi, M.: Reconstruction of nonstationary disordered materials and media: Watershed transform and cross-correlation function. Phys. Rev. E 91, 032401 (2015). https://doi.org/10.1103/PhysRevE.91.032401

    Article  ADS  Google Scholar 

  71. Tahmasebi, P., Sahimi, M., Shirangi, M.G.: Rapid Learning-Based and Geologically Consistent History Matching. Transp. Porous Media. (2018). https://doi.org/10.1007/s11242-018-1005-6

  72. Tahmasebi, P., Kamrava, S.: A Multiscale Approach for Geologically and Flow Consistent Modeling. Transp. Porous Media. (2018). https://doi.org/10.1007/s11242-018-1062-x

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Acknowledgements

The first author would like to thank the financial support from the University of Wyoming for this research. Work at USC was supported by the Petroleum Research Fund, administered by the American Chemical Society.

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  1. Department of Petroleum and Energy Materials Engineering, University of Wyoming, Laramie, WY, 82071, USA

    Pejman Tahmasebi

  2. Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, CA, 90089-1211, USA

    Muhammad Sahimi

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Tahmasebi, P., Sahimi, M. A stochastic multiscale algorithm for modeling complex granular materials. Granular Matter 20, 45 (2018). https://doi.org/10.1007/s10035-018-0816-z

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