Rock Mechanics and Rock Engineering

, Volume 52, Issue 12, pp 5099–5121 | Cite as

Multiscale Poromechanical Modeling of Shales Incorporating Microcracks

  • Vasav Dubey
  • Mohammad Mashhadian
  • Sara AbediEmail author
  • Arash NoshadravanEmail author
Original Paper


A probabilistic multiscale model for poroelastic properties of organic-rich shale is extended to include the effect of microcracks. The approach is based on physics-based modeling of poroelastic properties and uncertainty propagation from nano- to macro-scale using homogenization schemes. The multiscale thought model used in this work improves on an existing thought model by adding another level to model embedded microcracks. From the scale of single clay particle to the scale of embedded microcracks, the model parameters are calibrated using experimental characterization at different length scales. To quantify the crack density and microcrack orientations, an inverse optimization approach is utilized which uses the UPV measurements at the macro-scale. Major model parameters representing compositional and mechanical properties at each length scale are modeled as a random variable. Probabilistic description of both scalar random variables and matrix-valued random variables is constructed using the maximum entropy principle (MaxEnt). Using such statistical description of model input parameters with Monte Carlo simulation, probabilistic descriptions of model outputs are obtained at each length scale. The results of stochastic upscaling are validated against available experimental values. Finally, a global sensitivity analysis is performed to identify the model input parameters which are most influential to the statistical fluctuation of upscaled poroelastic properties. The presented multiscale framework provides an enhanced understating regarding the effect of uncertainties associated with microcrack density and orientation, and other subscale features on homogenized poroelastic properties, thus further improves predictive capability for shale multiscale behavior.


Micro-poromechanics Organic-rich shale Probabilistic modeling Microcrack-induced damage 

List of symbols

\(\left[ \varSigma \right]\)

Stress matrix

\(\left[ E \right]\)

Strain matrix

\(\left[\kern-0.15em\left[ C \right]\kern-0.15em\right]\)

Fourth-order stiffness tensor

\(\left[ \alpha \right]\)

Second-order tensor of Biot pore pressure coefficients


Pore pressure




Solid Biot modulus

\(\left[\kern-0.15em\left[ A \right]\kern-0.15em\right]\)

Strain localization tensor


Representative volume element

\(\left[\kern-0.15em\left[ I \right]\kern-0.15em\right]\)

Fourth-order identity tensor

\(\left[\kern-0.15em\left[ P \right]\kern-0.15em\right]\)

Fourth-order Hill polarization tensor


Overall Biot modulus

\(\eta^{i} , f^{i}\)

Volume fraction of the ith phase at Level I and Level II, respectively


Number of cracks per unit volume


Crack radius


Crack width


Crack aspect ratio


Crack density parameter

\(M_{1} ,M_{3}\)

Indentation moduli parallel to the bedding plane and normal to the bedding plane, respectively


Component of stiffness matrix in Voigt’s notation

\(\left\| \cdot \right\|_{\text{F}}\)

Frobenius norm


Bulk modulus


Poisson’s ratio


Shear modulus


Thickness of interfacial transition zone (ITZ)


Coefficient for elastic properties of ITZ


Normalized root mean square deviation

\(\varepsilon , \gamma , \delta^{*}\)

Thomsen’s anisotropy parameters

\(E_{x} , E_{z}\)

Young’s modulus parallel to the bedding plane and normal to the bedding plane, respectively

\(\lambda_{0} , \lambda_{1} , \lambda_{2} ,\lambda^{\text{sol}} ,\lambda_{1}^{\text{sol}} , \ldots ,\lambda_{5}^{\text{sol}}\)

Lagrange multipliers



The authors acknowledge partial funding from the Crisman-Berg Hughes Institute at the Department of Petroleum Engineering of Texas A&M University. Computational resources provided by Texas A&M High Performance Research Computing were utilized to conduct portions of this research. Furthermore, the authors thank the reviewer and the guest editor for providing insightful comments and feedback which improved the quality of the work.


This study was partially funded by the Crisman-Berg Hughes Institute at the Department of Petroleum Engineering of Texas A&M University (Project no. 2.07.16).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Abedi S, Slim M, Hofmann R, Bryndzia T, Ulm F-J (2016a) Nanochemo-mechanical signature of organic-rich shales: a coupled indentation–EDX analysis. Acta Geotech 11:559–572CrossRefGoogle Scholar
  2. Abedi S, Slim M, Ulm F-J (2016b) Nanomechanics of organic-rich shales: the role of thermal maturity and organic matter content on texture. Acta Geotech 11:775–787CrossRefGoogle Scholar
  3. Ahmadov R, Vanorio T, Mavko G (2009) Confocal laser scanning and atomic-force microscopy in estimation of elastic properties of the organic-rich Bazhenov Formation. Lead Edge 28:18–23CrossRefGoogle Scholar
  4. Aleksandrov K (1961) The elastic properties of rock forming minerals II: layered silicates. Bull Acad Sci USSR Geophys Ser 9:1165–1168Google Scholar
  5. Bažant ZP, Oh BH (1985) Microplane model for progressive fracture of concrete and rock. J Eng Mech 111:559–582CrossRefGoogle Scholar
  6. Bobko CP, Gathier B, Ortega JA, Ulm FJ, Borges L, Abousleiman YN (2011) The nanogranular origin of friction and cohesion in shale—a strength homogenization approach to interpretation of nanoindentation results. Int J Numer Anal Methods Geomech 35:1854–1876CrossRefGoogle Scholar
  7. Bousige C et al (2016) Realistic molecular model of kerogen’s nanostructure. Nat Mater 15:576CrossRefGoogle Scholar
  8. Carol I, Prat PC, Bažant ZP (1992) New explicit microplane model for concrete: theoretical aspects and numerical implementation. Int J Solids Struct 29:1173–1191CrossRefGoogle Scholar
  9. Chen Q, Nezhad MM, Fisher Q, Zhu H (2016) Multi-scale approach for modeling the transversely isotropic elastic properties of shale considering multi-inclusions and interfacial transition zone. Int J Rock Mech Min Sci 84:95–104CrossRefGoogle Scholar
  10. Christensen RM, Lo KH (1979) Solutions for effective shear properties in three phase sphere and cylinder models. J Mech Phys Solids 27:315–330. CrossRefGoogle Scholar
  11. Constantinides G, Ulm FJ, Abousleiman YN (2005) Material invariant poromechanics properties of shales. Taylor & Francis Group, AbingdonGoogle Scholar
  12. Cosenza P, Prêt D, Giraud A, Hedan S (2015) Effect of the local clay distribution on the effective elastic properties of shales. Mech Mater 84:55–74CrossRefGoogle Scholar
  13. Curtis ME, Ambrose RJ, Sondergeld CH (2010) Structural characterization of gas shales on the micro-and nano-scales. In: Canadian unconventional resources and international petroleum conference, Society of Petroleum EngineersGoogle Scholar
  14. Delafargue A, Ulm F-J (2004) Explicit approximations of the indentation modulus of elastically orthotropic solids for conical indenters. Int J Solids Struct 41:7351–7360CrossRefGoogle Scholar
  15. Devroye L (1986) Sample-based non-uniform random variate generation. In: Proceedings of the 18th conference on Winter simulation, ACM, pp 260–265Google Scholar
  16. Dormieux L, Kondo D, Ulm F-J (2006) Microporomechanics. John Wiley & Sons, ChichesterCrossRefGoogle Scholar
  17. Gibbons JD, Chakraborti S (2011) Nonparametric statistical inference. Springer, New YorkCrossRefGoogle Scholar
  18. Giraud A, Huynh QV, Hoxha D, Kondo D (2007a) Application of results on Eshelby tensor to the determination of effective poroelastic properties of anisotropic rocks-like composites. Int J Solids Struct 44:3756–3772CrossRefGoogle Scholar
  19. Giraud A, Huynh QV, Hoxha D, Kondo D (2007b) Effective poroelastic properties of transversely isotropic rock-like composites with arbitrarily oriented ellipsoidal inclusions. Mech Mater 39:1006–1024CrossRefGoogle Scholar
  20. Guilleminot J, Soize C (2013a) On the statistical dependence for the components of random elasticity tensors exhibiting material symmetry properties. J Elast 111:109–130CrossRefGoogle Scholar
  21. Guilleminot J, Soize C (2013b) Stochastic model and generator for random fields with symmetry properties: application to the mesoscopic modeling of elastic random media. Multiscale Model Simul 11:840–870CrossRefGoogle Scholar
  22. Hornby BE, Schwartz LM, Hudson JA (1994) Anisotropic effective-medium modeling of the elastic properties of shales. Geophysics 59:1570–1583CrossRefGoogle Scholar
  23. Jaynes ET (1957) Information theory and statistical mechanics. Phys Rev 106:620CrossRefGoogle Scholar
  24. Jaynes ET (2003) Probability theory: the logic of science. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  25. Jefferson A, Bennett T (2007) Micro-mechanical damage and rough crack closure in cementitious composite materials. Int J Numer Anal Methods Geomech 31:133–146CrossRefGoogle Scholar
  26. Jumarie G (2013) Maximum entropy, information without probability and complex fractals: classical and quantum approach, vol 112. Springer, New YorkGoogle Scholar
  27. Kachanov ML (1982) A microcrack model of rock inelasticity part I: frictional sliding on microcracks. Mech Mater 1:19–27CrossRefGoogle Scholar
  28. Katahara KW (1996) Clay mineral elastic properties. In: SEG technical program expanded abstracts 1996. Society of Exploration Geophysicists, pp 1691–1694Google Scholar
  29. Levin V, Markov M (2005) Elastic properties of inhomogeneous transversely isotropic rocks. Int J Solids Struct 42:393–408CrossRefGoogle Scholar
  30. Mashhadian M, Abedi S, Noshadravan A (2018a) Probabilistic multiscale characterization and modeling of organic-rich shale poroelastic properties. Acta Geotech 13:781–800CrossRefGoogle Scholar
  31. Mashhadian M, Verde A, Sharma P, Abedi S (2018b) Assessing mechanical properties of organic matter in shales: results from coupled nanoindentation/SEM-EDX and micromechanical modeling. J Pet Sci Eng 165:313–324CrossRefGoogle Scholar
  32. Mavko G, Mukerji T, Dvorkin J (2009) The rock physics handbook: tools for seismic analysis of porous media. Cambridge university press, CambridgeCrossRefGoogle Scholar
  33. McNeil L, Grimsditch M (1993) Elastic moduli of muscovite mica. J Phys Condens Matter 5:1681CrossRefGoogle Scholar
  34. Monfared S, Ulm F-J (2016) A molecular informed poroelastic model for organic-rich, naturally occurring porous geocomposites. J Mech Phys Solids 88:186–203CrossRefGoogle Scholar
  35. Ortega JA, Ulm F-J, Abousleiman Y (2007) The effect of the nanogranular nature of shale on their poroelastic behavior. Acta Geotech 2:155–182CrossRefGoogle Scholar
  36. Ortega JA, Ulm F-J, Abousleiman Y (2009) The nanogranular acoustic signature of shale. Geophysics 74:D65–D84CrossRefGoogle Scholar
  37. Pan Y, Chou T (1976) Point force solution for an infinite transversely isotropic solid ASME. Trans Ser E J Appl Mech 43:608–612CrossRefGoogle Scholar
  38. Pensée V, Kondo D, Dormieux L (2002) Micromechanical analysis of anisotropic damage in brittle materials. J Eng Mech 128:889–897CrossRefGoogle Scholar
  39. Prasad M, Mukerji T, Reinstaedler M, Arnold W (2009) Acoustic signatures, impedance microstructure, textural scales, and anisotropy of kerogen-rich shales. In: SPE annual technical conference and exhibition, Society of Petroleum EngineersGoogle Scholar
  40. Prat PC, Bažant ZP (1997) Tangential stiffness of elastic materials with systems of growing or closing cracks. J Mech Phys Solids 45:611–636CrossRefGoogle Scholar
  41. Qi M, Giraud A, Colliat J-B, Shao J-F (2016a) A numerical damage model for initially anisotropic materials. Int J Solids Struct 100:245–256CrossRefGoogle Scholar
  42. Qi M, Shao J, Giraud A, Zhu Q, Colliat J (2016b) Damage and plastic friction in initially anisotropic quasi brittle materials. Int J Plast 82:260–282CrossRefGoogle Scholar
  43. Robinet JC, Sardini P, Coelho D, Parneix JC, Prêt D, Sammartino S, Boller E, Altmann S (2012) Effects of mineral distribution at mesoscopic scale on solute diffusion in a clay‐rich rock: Example of the Callovo‐Oxfordian mudstone (Bure, France). Water Resour Res 48(5):W05554. CrossRefGoogle Scholar
  44. Sayers C (1994) The elastic anisotrophy of shales. J Geophys Res Solid Earth 99:767–774CrossRefGoogle Scholar
  45. Seo Y-S, Ichikawa Y, Kawamura K (1999) Stress–strain response of rock-forming minerals by molecular dynamics simulation. J Soc Mater Sci Jpn 48:13–20CrossRefGoogle Scholar
  46. Shannon C (2001) A mathematical theory of communication. SIGMOBILE Mob Comput Commun Rev 5(1):3–55CrossRefGoogle Scholar
  47. Sobezyk K, Trȩbicki J (1990) Maximum entropy principle in stochastic dynamics. Probab Eng Mech 5:102–110CrossRefGoogle Scholar
  48. Staber B, Guilleminot J (2015) Approximate solutions of Lagrange multipliers for information-theoretic random field models SIAM/ASA. J Uncertain Quantif 3:599–621CrossRefGoogle Scholar
  49. Thomsen L (1986) Weak elastic anisotropy. Geophysics 51:1954–1966CrossRefGoogle Scholar
  50. Ulm F-J, Delafargue A, Constantinides G (2005) Experimental microporomechanics. In: Dormieux L, Ulm F-J (eds) Applied micromechanics of porous materials. Springer, Vienna, pp 207–288CrossRefGoogle Scholar
  51. Vasin RN, Wenk HR, Kanitpanyacharoen W, Matthies S, Wirth R (2013) Elastic anisotropy modeling of Kimmeridge shale. J Geophys Res Solid Earth 118:3931–3956CrossRefGoogle Scholar
  52. Vaughan MT, Guggenheim S (1986) Elasticity of muscovite and its relationship to crystal structure. J Geophys Res Solid Earth 91:4657–4664CrossRefGoogle Scholar
  53. Vernik L (1993) Microcrack-induced versus intrinsic elastic anisotropy in mature HC-source shales. Geophysics 58:1703–1706CrossRefGoogle Scholar
  54. Vernik L, Liu X (1997) Velocity anisotropy in shales: a petrophysical study. Geophysics 62:521–532CrossRefGoogle Scholar
  55. Walpole L (1984) Fourth-rank tensors of the thirty-two crystal classes: multiplication tables. Proc R Soc Lond A Math Phys Sci 391:149–179CrossRefGoogle Scholar
  56. Wang Z, Wang H, Cates ME (2001) Effective elastic properties of solid clays. Geophysics 66:428–440CrossRefGoogle Scholar
  57. Wang W, Li J, Fan M, Abedi S (2017) Characterization of electrical properties of organic-rich shales at nano/micro scales. Mar Pet Geol 86:563–572CrossRefGoogle Scholar
  58. Xie N, Zhu Q-Z, Xu L, Shao J-F (2011) A micromechanics-based elastoplastic damage model for quasi-brittle rocks. Comput Geotech 38:970–977CrossRefGoogle Scholar
  59. Zaoui A (2002) Continuum micromechanics: survey. J Eng Mech 128:808–816CrossRefGoogle Scholar
  60. Zeszotarski JC, Chromik RR, Vinci RP, Messmer MC, Michels R, Larsen JW (2004) Imaging and mechanical property measurements of kerogen via nanoindentation. Geochim Cosmochim Acta 68:4113–4119CrossRefGoogle Scholar
  61. Zhu Q, Shao J-F (2015) A refined micromechanical damage–friction model with strength prediction for rock-like materials under compression. Int J Solids Struct 60:75–83CrossRefGoogle Scholar
  62. Zhu Q-Z, Kondo D, Shao J (2008a) Micromechanical analysis of coupling between anisotropic damage and friction in quasi brittle materials: role of the homogenization scheme. Int J Solids Struct 45:1385–1405CrossRefGoogle Scholar
  63. Zhu Q, Kondo D, Shao J, Pensee V (2008b) Micromechanical modelling of anisotropic damage in brittle rocks and application. Int J Rock Mech Min Sci 45:467–477CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Zachry Department of Civil EngineeringTexas A&M UniversityCollege StationUSA
  2. 2.Ponce-Fuess Engineering, LLCDallasUSA
  3. 3.Harold Vance Department of Petroleum EngineeringTexas A&M UniversityCollege StationUSA

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