Comparison of advanced discretization techniques for image-based modelling of heterogeneous porous rocks

  • Karim Ehab Moustafa Kamel
  • Jean-Baptiste Colliat
  • Pierre GerardEmail author
  • Thierry J. Massart
Research Paper


This contribution presents an assessment of computational techniques enabling automated simulations of complex porous rocks microstructures based on 3D imaging techniques. A subset of a CT-scanned sandstone sample is used to compare the results obtained by two advanced discretization frameworks. Raw scan results are processed by a level-set-based segmentation technique to produce smooth geometries prone to finite element discretizations. A recently developed technique is outlined for conforming mesh generation for complex porous geometries described implicitly by functions. This allows generating high-quality tetrahedral meshes with selective refinement. Next to this, a technique that uses a kinematic enrichment by incompatible modes to represent the heterogeneous geometry is explained. Both techniques use the same implicit geometry as main input for the simulations. Mechanical simulations are conducted on a subset of a scanned sample of a sandstone under triaxial loading conditions for isotropic compressive loading and for loading conditions involving a stress deviator. The results are compared and discussed based on local stress distributions and on a Mohr–Coulomb criterion with tensile cut-off. The results show that both discretization strategies yield complementary tools and allow envisioning automated simulations based on raw CT scan data for porous rocks exhibiting complex pore space morphologies.


Automated meshing CT scan Embedded discontinuities Finite elements Image-based modelling Porous rocks Rock mechanics 



The first author acknowledges the support of FRIA under Grant No. 29340757.


  1. 1.
    Akono AT, Miguel Reis P, Ulm FJ (2011) Scratching as a fracture process: from butter to steel. Phys Rev Lett 106:204302. Google Scholar
  2. 2.
    Andó E, Cailletaud R, Roubin E, Stamati O, Wiebicke M, Couture CB, Matsushima T, Okubadejo O, Bertoni F, Sun Y, Colliat JB (2019) Spam: software for practical analysis of materials a python toolkit built on Numpy and Scipy. (In preparation) Google Scholar
  3. 3.
    Bernard O, Friboulet D, Thevenaz P, Unser M (2009) Variational b-spline level-set: a linear filtering approach for fast deformable model evolution. IEEE Trans Image Process 18(6):1179–1191MathSciNetzbMATHGoogle Scholar
  4. 4.
    Besuelle P, Desrues J, Raynaud S (2000) Experimental characterisation of the localisation phenomenon inside a vosges sandstone in a triaxial cell. Int J Rock Mech Min Sci 37(8):1223–1237Google Scholar
  5. 5.
    Beucher S (2001) Geodesic reconstruction, saddle zones & hierarchical segmentation. Image Anal Stereol 20(3):137–141zbMATHGoogle Scholar
  6. 6.
    Bottasso CL, Detomi D, Serra R (2005) The ball-vertex method: a new simple spring analogy method for unstructured dynamic meshes. Comput Methods Appl Mech Eng 194(39):4244–4264zbMATHGoogle Scholar
  7. 7.
    Cai M (2010) Practical estimates of tensile strength and Hoek–Brown strength parameter \(m_{\text{ i }}\) of brittle rocks. Rock Mech Rock Eng 43(2):167–184Google Scholar
  8. 8.
    Caselles V, Kimmel R, Sapiro G (1997) Geodesic active contours. Int J Comput Vision 22(1):61–79zbMATHGoogle Scholar
  9. 9.
    Chan TF, Vese LA (2001) Active contours without edges. Trans Image Proc 10(2):266–277zbMATHGoogle Scholar
  10. 10.
    Chen S, Yue ZQ, Tham LG (2006) Digital image based approach for three-dimensional mechanical analysis of heterogeneous rocks. Rock Mech Rock Eng 40(2):145Google Scholar
  11. 11.
    Ehab Moustafa Kamel K, Sonon B, Massart TJ (2019) An integrated approach for the conformal discretization of complex inclusion-based microstructures. Comput Mech. Google Scholar
  12. 12.
    Feng XT, Chen S, Zhou H (2004) Real-time computerized tomography (CT) experiments on sandstone damage evolution during triaxial compression with chemical corrosion. Int J Rock Mech Min Sci 41(2):181–192Google Scholar
  13. 13.
    Frey P, George P (2000) Mesh generation: application to finite elements. Hermes Science, OxfordzbMATHGoogle Scholar
  14. 14.
    Furrer R, Sain SR (2010) Spam: a sparse matrix R package with emphasis on MCMC methods for Gaussian Markov random fields. J Stat Softw 36(10):1–25Google Scholar
  15. 15.
    Gerber F, Furrer R (2015) Pitfalls in the implementation of Bayesian hierarchical modeling of areal count data: an illustration using BYM and Leroux models. J Stat Softw Code Snippets 63(1):1–32Google Scholar
  16. 16.
    Gerber F, Moesinger K, Furrer R (2017) Extending R packages to support 64-bit compiled code: an illustration with spam64 and GIMMS NDVI3g data. Comput Geosci 104:109–119. Google Scholar
  17. 17.
    Geuzaine C, Remacle JF (2009) Gmsh: A 3-d finite element mesh generator with built-in pre- and post-processing facilities. Int J Numer Methods Eng 79(11):1309–1331MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hashemi MA, Khaddour G, François B, Massart TJ, Salager S (2014) A tomographic imagery segmentation methodology for three-phase geomaterials based on simultaneous region growing. Acta Geotech 9(5):831–846Google Scholar
  19. 19.
    Hashemi MA, Massart TJ, Salager S, Herrier G, François B (2015) Pore scale characterization of lime-treated sandbentonite mixtures. Appl Clay Sci 111:50–60Google Scholar
  20. 20.
    Hu C, Agostini F, Skoczylas F, Jeannin L, Potier L (2018) Poromechanical properties of a sandstone under different stress states. Rock Mech Rock Eng 51:3699–3717Google Scholar
  21. 21.
    Ibrahimbegovic A, Wilson E (1991) Modified method of incompatible modes. Commun Appl Numer Methods 7(3):187–194zbMATHGoogle Scholar
  22. 22.
    Kim KY, Zhuang L, Yang H, Kim H, Min KB (2016) Strength anisotropy of Berea sandstone: results of X-ray computed tomography, compression tests, and discrete modeling. Rock Mech Rock Eng 49(4):1201–1210Google Scholar
  23. 23.
    Legrain G, Cartraud P, Perreard I, Moës N (2011) An X-FEM and level set computational approach for image-based modelling: application to homogenization. Int J Numer Methods Eng 86(7):915–934zbMATHGoogle Scholar
  24. 24.
    Li G, Liang ZZ, Tang CA (2015) Morphologic interpretation of rock failure mechanisms under uniaxial compression based on 3d multiscale high-resolution numerical modeling. Rock Mech Rock Eng 48(6):2235–2262Google Scholar
  25. 25.
    Li CS, Zhang D, Du SS, Shi B (2016) Computed tomography based numerical simulation for triaxial test of soilrock mixture. Comput Geotech 73:179–188Google Scholar
  26. 26.
    Li J, Konietzky H, Frühwirt T (2017) Voronoi-based dem simulation approach for sandstone considering grain structure and pore size. Rock Mech Rock Eng 50(10):2749–2761Google Scholar
  27. 27.
    Lin TJ, Guan ZQ, Chang JH, Lo SH (2014) Vertex-ball spring smoothing: an efficient method for unstructured dynamic hybrid meshes. Comput Struct 136:24–33Google Scholar
  28. 28.
    Lo D (2015) Finite element mesh generation. Taylor & Francis, LondonzbMATHGoogle Scholar
  29. 29.
    Lorensen WE, Cline HE (1987) Marching cubes: a high resolution 3d surface construction algorithm. SIGGRAPH Comput Graph 21(4):163–169Google Scholar
  30. 30.
    Macri M, De S (2008) An octree partition of unity method (octpum) with enrichments for multiscale modeling of heterogeneous media. Comput Struct 86(7):780–795Google Scholar
  31. 31.
    Mahabadi O, Randall N, Zong Z, Grasselli G (2012) A novel approach for micro-scale characterization and modeling of geomaterials incorporating actual material heterogeneity. Geophys Res Lett 39:1303. Google Scholar
  32. 32.
    Massart TJ, Selvadurai A (2012) Stress-induced permeability evolution in a quasi-brittle geomaterial. J Geophys Res 117:B07207Google Scholar
  33. 33.
    Massart TJ, Selvadurai A (2014) Computational modelling of crack-induced permeability evolution in granite with dilatant cracks. Int J Rock Mech Min Sci 70:593–604Google Scholar
  34. 34.
    Moes N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131–150zbMATHGoogle Scholar
  35. 35.
    Moes N, Cloirec M, Cartraud P, Remacle JF (2003) A computational approach to handle complex microstructure geometries. Comput Methods Appl Mech Eng 192(28–30):3163–3177zbMATHGoogle Scholar
  36. 36.
    Osher S, Fedkiw R (2003) Signed distance functions. Springer, New York, pp 17–22Google Scholar
  37. 37.
    Otsu N (1979) A threshold selection method from gray level histograms. IEEE Trans Syst Man Cybern 9:62–66Google Scholar
  38. 38.
    Paiva A, Lopes H, Lewiner T, Figueiredo LHD (2006) Robust adaptive meshes for implicit surfaces. In: 2006 19th Brazilian symposium on computer graphics and image processing, pp 205–212Google Scholar
  39. 39.
    Perras MA, Diederichs MS (2014) A review of the tensile strength of rock: concepts and testing. Geotech Geol Eng 32(2):525–546Google Scholar
  40. 40.
    Persson PO (2005) Mesh generation for implicit geometries. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, aAI0807802Google Scholar
  41. 41.
    Persson PO, Strang G (2004) A simple mesh generator in matlab. SIAM Rev 46:2004MathSciNetzbMATHGoogle Scholar
  42. 42.
    Rabbani A, Jamshidi S (2014) Specific surface and porosity relationship for sandstones for prediction of permeability. Int J Rock Mech Min Sci 71:25–32Google Scholar
  43. 43.
    Randall NX, Vandamme M, Ulm FJ (2009) Nanoindentation analysis as a two-dimensional tool for mapping the mechanical properties of complex surfaces. J Mater Res 24(3):679690. Google Scholar
  44. 44.
    Raynaud S, Vasseur G, Soliva R (2012) In vivo CT X-Ray observations of porosity evolution during triaxial deformation of a calcarenite. Int J Rock Mech Min Sci 56:161–170Google Scholar
  45. 45.
    Roubin E, Colliat JB, Benkemoun N (2015) Meso-scale modeling of concrete: a morphological description based on excursion sets of random fields. Comput Mater Sci 102:183–195Google Scholar
  46. 46.
    Selvadurai APS, Glowacki A (2007) Permeability hysterisis of limestone during isotropic compression. Groundwater 46(1):113–119Google Scholar
  47. 47.
    Sethian J (1999) Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, CambridgezbMATHGoogle Scholar
  48. 48.
    Shao J, Hoxha D, Bart M, Homand F, Duveau G, Souley M, Hoteit N (1999) Modelling of induced anisotropic damage in granites. Int J Rock Mech Min Sci 36(8):1001–1012Google Scholar
  49. 49.
    Shewchuk JR (2002) Constrained delaunay tetrahedralizations and provably good boundary recovery. In: Proceedings of the 11th international meshing roundtable, IMR 2002, Ithaca, New York, USA, September 15–18, pp 193–204Google Scholar
  50. 50.
    Shi Y, Karl WC (2008) A real-time algorithm for the approximation of level-set-based curve evolution. IEEE Trans Image Process 17(5):645–656MathSciNetGoogle Scholar
  51. 51.
    Si H (2010) Constrained delaunay tetrahedral mesh generation and refinement. Finite Elem Anal Des 46(1–2):33–46MathSciNetGoogle Scholar
  52. 52.
    Si H (2015) TetGen, a delaunay-based quality tetrahedral mesh generator. ACM Trans Math Softw 41(2):11:1–11:36MathSciNetzbMATHGoogle Scholar
  53. 53.
    Simo J, Rifai M (1990) A class of mixed assumed strain methods and the method of incompatible modes. Int J Numer Methods Eng 29(8):1595–1638MathSciNetzbMATHGoogle Scholar
  54. 54.
    Simo J, Oliver J, Armero F (1993) An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids. Comput Mech 12(5):277–296MathSciNetzbMATHGoogle Scholar
  55. 55.
    Sonon B, François B, Massart TJ (2012) A unified level set based methodology for fast generation of complex microstructural multi-phase RVEs. Comput Methods Appl Mech Eng 223:103–122Google Scholar
  56. 56.
    Sufian A, Russell AR (2013) Microstructural pore changes and energy dissipation in Gosford sandstone during pre-failure loading using X-Ray CT. Int J Rock Mech Min Sci 57:119–131Google Scholar
  57. 57.
    Sun W, Wong T (2018) Prediction of permeability and formation factor of sandstone with hybrid lattice Boltzmann/finite element simulation on microtomographic images. Int J Rock Mech Min Sci 106:269–277Google Scholar
  58. 58.
    Washizu K (1982) Variational methods in elasticity and plasticity. Pergamon Press, New YorkzbMATHGoogle Scholar
  59. 59.
    White JA, Borja RI, Fredrich JT (2006) Calculating the effective permeability of sandstone with multiscale lattice Boltzmann/finite element simulations. Acta Geotech 1(4):195–209Google Scholar
  60. 60.
    Wilson E (1974) The static condensation algorithm. Int J Numer Methods Eng 8(1):198–203Google Scholar
  61. 61.
    Yu Q, Yang S, Ranjith PG, Zhu W, Yang T (2016) Numerical modeling of jointed rock under compressive loading using X-Ray computerized tomography. Rock Mech Rock Eng 49(3):877–891Google Scholar
  62. 62.
    Zhou XP, Xiao N (2017) A novel 3d geometrical reconstruction model for porous rocks. Eng Geol 228:371–384Google Scholar
  63. 63.
    Zhou XP, Xiao N (2018) 3d numerical reconstruction of porous sandstone using improved simulated annealing algorithms. Rock Mech Rock Eng 51(7):2135–2151Google Scholar
  64. 64.
    Zhou XP, Xiao N (2018b) A hierarchical-fractal approach for the rock reconstruction and numerical analysis. Int J Rock Mech Min Sci 109:68–83Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.BATir, Building, Architecture & Town PlanningUniversité Libre de Bruxelles (ULB)BrusselsBelgium
  2. 2.Laboratoire de mécanique, multiéchelle, multiphysique, CNRS, Centrale LilleUniversité de LilleLilleFrance

Personalised recommendations