Technical Papers
Mar 17, 2017

Fracture-Induced Anisotropy of the Stress–Strain Response of Shale at Multiple Scales

Publication: International Journal of Geomechanics
Volume 17, Issue 8

Abstract

This paper investigates deformation and stiffness anisotropy induced by damage propagation in a rock brittle deformation regime. Specifically, a finite-element–based Continuum damage mechanics model is used to capture sample size effects and the influence of intrinsic anisotropy on the stress–strain response of shale. The differential stress-induced damage (DSID) model previously proposed by the authors is calibrated against triaxial compression tests performed on North Dakota Bakken shale samples. Laboratory tests simulated with the FEM reproduce deformation and damage localization phenomena and capture the increase of boundary effects expected in larger samples. Simulations performed for various initial states of damage are used to investigate the role of the dominant fabric anisotropy of the rock: bedding planes in shale are modeled by a smeared damage zone with the DSID model and by a discrete crack plane. The continuum approach successfully captures the development of microcrack propagation and energy dissipation at the early stage of the strain hardening process observed in triaxial compression tests. Additionally, using initial anisotropic damage can effectively account for various types of mechanical anisotropy in shale.

Get full access to this article

View all available purchase options and get full access to this article.

Acknowledgments

This research was performed in the School of Civil and Environmental Engineering at the Georgia Institute of Technology, with the support of ConocoPhillips on a project titled Finite Element Modeling of Hydraulic Fracturing.

References

Abaqus [Computer software]. Dassault Systèmes, Waltham, MA.
Abu Al-Rub, R. K., and Kim, S. M. (2010). “Computational applications of a coupled plasticity-damage constitutive model for simulating plain concrete fracture.” Eng. Fract. Mech., 77(10), 1577–1603.
Adachi, J., Siebrits, E., Peirce, A., and Desroches, J. (2007). “Computer simulation of hydraulic fractures.” Int. J. Rock Mech. Min. Sci., 44(5), 739–757.
Amendt, D., Busetti, S., and Wenning, Q. (2013). “Mechanical characterization in unconventional reservoirs: A facies-based methodology.” Petrophysics, 54(5), 457–464.
Arson, C. (2009). “Etude théorique et numérique de l’endommagement thermo-hydro-mécanique des milieux poreux non saturés.” Ph.D. thesis, Ecole Nationale des Ponts et Chaussées, Paris.
Arson, C. (2014). “Generalized stress variables in continuum damage mechanics.” Mech. Res. Commun., 60, 81–84.
Arson, C., and Gatmiri, B. (2008). “On damage modelling in unsaturated clay rocks.” Phys. Chem. Earth, 33(S1), S407–S415.
Arson, C., and Pereira, J. M. (2013). “Influence of damage on pore size distribution and permeability of rocks.” Int. J. Numer. Anal. Methods Geomech., 37(8), 810–831.
Arson, C., Xu, H., and Chester, F. (2012). “On the definition of damage in time-dependent healing models for salt rock.” Géotechnique Lett., 2(2), 67–71.
Bahat, D., Rabinovitch, A., and Frid, V. (2005). Tensile fracturing in rocks: Tectonofractographic and electromagnetic radiation method, Springer, Berlin.
Bakhtiary, E., Xu, H., and Arson, C. (2014). “Probabilistic optimization of a continuum mechanics model to predict differential stress-induced damage in claystone.” Int. J. Rock Mech. Min. Sci., 68, 136–149.
Busetti, S., Mish, K., and Reches, Z. (2012). “Damage and plastic deformation of reservoir rocks: Part 1. Damage fracturing.” AAPG Bull., 96(9), 1687–1709.
Busetti, S., Xu, H., and Arson, C. (2014). “Simulation of anisotropic rock damage for geologic fracturing.” Proc., 2014 AGU Fall Meeting, Wiley, Hoboken, NJ.
Butt, S., and Calder, P. (1998). “Experimental procedures to measure volumetric changes and microseismic activity during triaxial compression tests.” Int. J. Rock Mech. Min. Sci., 35(2), 249–254.
Carrier, B., and Granet, S. (2012). “Numerical modeling of hydraulic fracture problem in permeable medium using cohesive zone model.” Eng. Fract. Mech., 79, 312–328.
Chaboche, J. L. (1993). “Development of continuum damage mechanics for elastic solids sustaining anisotropic and unilateral damage.” Int. J. Damage Mech., 2(4), 311–329.
Cicekli, U., Voyiadjis, G. Z., and Abu Al-Rub, R. K. (2007). “A plasticity and anisotropic damage model for plain concrete.” Int. J. Plast., 23(10–11), 1874–1900.
Colovos, J., Brannon, R., and Pinsky, P. (2013). “Reduction of macroscale calibration experiments through constraints on anisotropic elastic stiffnesses.” Proc., 47th U.S. Rock Mechanics/Geomechanics Symp., American Rock Mechanics Association, Alexandria, VA.
Cowin, S. C. (1985). “The relationship between the elasticity tensor and the fabric tensor.” Mech. Mater., 4(2), 137–147.
Crawford, A., and Wylie, D. (1987). “A modified multiple failure state triaxial testing method.” Proc., 28th U.S. Symp. on Rock Mechanics, American Rock Mechanics Association, Alexandria, VA.
Desai, C. S. (2000). Mechanics of materials and interfaces: The disturbed state concept, CRC Press, Boca Raton, FL.
Desai, C. S. (2015). “Constitutive modeling of material and contacts using the disturbed state concept: Part 1. Background and analysis.” Comput. Struct., 146, 214–233.
Desai, C. S., Basaran, C., and Zhang, W. (1997). “Numerical algorithms and mesh dependence in the disturbed state concept.” Int. J. Numer. Methods Eng., 49(16), 3059–3083.
Deudé, V., Dormieux, L., Kondo, D., and Maghous, S. (2002a). “Micromechanical approach to nonlinear poroelasticity: Application to cracked rocks.” J. Eng. Mech., 848–855.
Deudé, V., Dormieux, L., Kondo, D., and Pensée, V. (2002b). “Propriétés élastiques non linéaires d’un milieu mésofissuré.” C.R. Mec., 330, 587–592.
Dormieux, L., Kondo, D., and Ulm, F. (2006). Microporomechanics, John Wiley, Hoboken, NJ.
Dufour, N., Wong, H., Arson, C., Deleruyelle, F., and Pereira, J. M. (2012). “A thermodynamically consistent framework for saturated viscoplastic rock-materials subject to damage.” Mech. Res. Commun., 45, 15–21.
Economides, M., and Valko, P. (1994). “Interpretation and modeling of hydraulic fracturing phenomena with continuum damage mechanics—An application to engineering design.” Computer methods and advances in geomechanics, H. J. Siriwardane and M. M. Zaman, eds., A.A. Balkema, Rotterdam, Netherlands, 1579–1584.
Elleithy, W. M., Al-Gahtani, H. J., and Tanaka, M. (2001). “Iterative coupling of BEM and FEM for the solution of elasto-plastic fracture mechanics problems.” Proceedings of Japan National Symposium on Boundary Element Methods, 18, 43–48.
Faulkner, D. R., Mitchell, T. M., Healy, D., and Heap, M. J. (2006). “Slip on ‘weak’ faults by the rotation of regional stress in the fracture damage zone.” Nature, 444(7121), 922–925.
Frémond, M., and Nedjar, B. (1996). “Damage, gradient of damage and principle of virtual power.” Int. J. Solids Struct., 33(8), 2294–2306.
Gaede, O., Karrech, A., and Regenauer-Lieb, K. (2013). “Anisotropic damage mechanics as a novel approach to improve pre-and post-failure borehole stability analysis.” Geophys. J. Int., 193, 1095–1109.
Ghabezloo, S., Sulem, J., Guédon, S., Martineau, F., and Saint-Marc, J. (2008). “Poromechanical behaviour of hardened cement paste under isotropic loading.” Cem. Concr. Res., 38(12), 1424–1437.
Goodman, R. E. (1989). Introduction to rock mechanics, 2nd Ed., John Wiley & Sons, New York.
Halm, D., and Dragon, A. (2002). “Modelisation de l’endommagement par mesofissuration du granite.” Rev. Fr. Genie Civi, 17, 21–33.
Hamiel, Y., Liu, Y., Lyakhovsky, V., Ben-Zion, Y., and Lockner, D. (2004a). “A viscoelastic damage model with applications to stable and unstable fracturing.” Geophys. J. Int., 159(3), 1155–1165.
Hamiel, Y., Lyakhovsky, V., and Agnon, A. (2004b). “Coupled evolution of damage and porosity in poroelastic media: Theory and applications to deformation of porous rocks.” Geophys. J. Int., 156(3), 701–713.
Hayakawa, K., and Murakami, S. (1997). “Thermodynamical modeling of elastic-plastic damage and experimental validation of damage potential.” Int. J. Damage Mech., 6(4), 333–363.
Healy, D. (2008). “Damage patterns, stress rotations and pore fluid pressures in strike-slip fault zones.” J. Geophys. Res., 113(B12), 1–16.
Heap, M. J., Faulkner, D. R., Meredith, P. G., and Vinciguerra, S. (2010). “Elastic moduli evolution and accompanying stress changes with increasing crack damage: Implications for stress changes around fault zones and volcanoes during deformation.” Geophys. J. Int., 183(1), 225–236.
Homand-Etienne, F., Hoxha, D., and Shao, J. F. (1998). “A continuum damage constitutive law for brittle rocks.” Comput. Geotech., 22(2), 135–151.
Horii, H., and Nemat-Nasser, S. (1986). “Brittle failure in compression: Splitting, faulting and brittle-ductile transition.” Philos. Trans. R. Soc. London, Ser. A, 319(1549), 337–374.
Jin, W., Xu, H., Arson, C., and Busetti, S. (2016). “Computational model coupling Mode II discrete fracture propagation with continuum damage zone evolution.” Int. J. Numer. Anal. Methods Geomech., 41(2), 223–250.
Kachanov, M. (1992). “Effective elastic properties of cracked solids: Critical review of some basic concepts.” Appl. Mech. Rev., 45(8), 304–335.
Katz, O., and Reches, Z. (2004). “Microfracturing, damage, and failure of brittle granites.” J. Geophys. Res. B: Solid Earth, 109(B1), B01206.
Kondo, D., and Dormieux, L. (2004). “Approche micro-mécanique du couplage permabilité–endommagement.” C.R. Mec., 332(2), 135–140.
Krajcinovic, D. (1996). Damage mechanics, Elsevier, Burlington, MA.
Lee, J., and Fenves, G. (1998). “Plastic-damage model for cyclic loading of concrete structures.” J. Eng. Mech., 892–900.
Lemaître, J., and Desmorat, R. (2005). Engineering damage mechanics: Ductile, creep, fatigue and brittle failure, Springer Verlag, Berlin.
Liu, H. W. (1984). “On the fundamental basis of fracture mechanics.” Eng. Fract. Mech., 17(5), 425–438.
Lu, Y., and Elsworth, D. (2012). “Combined microscopic-macroscopic modeling of rock damage and failure.” Proc., 46th Rock Mechanics Geomechanics Symp., American Rock Mechanics Association, Alexandria, VA. 12–250.
Lubarda, V., and Krajcinovic, D. (1993). “Damage tensors and the crack density distribution.” Int. J. Solids Struct., 30(20), 2859–2877.
Lubliner, J., Oliver, J., Oller, S., and Onate, E. (1989). “A plastic-damage model for concrete.” Int. J. Solids Struct., 25(3), 299–326.
Lyakhovsky, V., Reches, Z., Weinberger, R., and Scott, T. E. (1997). “Non-linear elastic behaviour of damaged rocks.” Geophys. J. Int., 130(1), 157–166.
Lydzba, D., and Shao, J. (2000). “Study of poroelasticity material coefficients as response of microstructure.” Mech. Cohesive Frict. Mater., 5(2), 149–171.
Maleki, K., and Pouya, A. (2010). “Numerical simulation of damage–Permeability relationship in brittle geomaterials.” Comput. Geotech., 37(5), 619–628.
MATLAB [Computer software]. MathWorks, Natick, MA.
Mazars, J. (1986). “A description of micro- and macro scale damage of concrete structures.” Eng. Fract. Mech., 25(5–6), 729–737.
Mazars, J., and Pijaudier-Cabot, G. (1989). “Continuum damage theory–Application to concrete.” J. Eng. Mech., 345–365.
Mohammadi, S. (2007). Extended finite element method for fracture analysis of structures, Blackwell, Oxford, U.K.
Nelson, R. (2001). Geologic analysis of naturally fractured reservoirs, 2nd Ed., Gulf Professional Publishing, Oxford, U.K.
Oda, M. (1984). “Similarity rules of crack geometry in statistically homogeneous rock masses.” Mech. Mater., 3(2), 119–129.
Ortiz, M. (1985). “A constitutive theory for the inelastic behaviour of concrete.” Mech. Mater., 4(1), 67–93.
Pagoulatos, A. (2004). “Evaluation of multistage triaxial testing on Berea sandstone.” M.S. thesis, Univ. of Oklahoma, Oklahoma, Norman, OK.
Paterson, M. (1978). Experimental rock deformation: The brittle field, Springer Verlag, New York.
Pereira, J. M., and Arson, C. (2013). “Retention and permeability properties of damaged porous rocks.” Comput. Geotech., 48, 272–282.
Ramurthy, M., Barree, R. D., Broacha, E., Longwell, J. D., Kundert, D. P., and Tamayo, C. (2009a). Effects of high process-zone stress in shale stimulation treatment, Rocky Mountain Petroleum Technology, Denver.
Ramurthy, M., Lyons, B., Hendrickson, R. B., Barree, R. D., and Magill, D. P. (2009b). “Effects of high pressure-dependent leakoff and high process-zone stress in coal-stimulation treatments.” SPE J., 24(3), 407–414.
Raveendra, S., and Cruse, T. (2005). “BEM analysis of problems of fracture mechanics.” Industrial applications of boundary element methods, P. K. Banerjee, R. B. Wilson, eds., Elsevier Applied Science, London, 5, 186–204.
Savitski, A., and Detournay, E. (2002). “Propagation of a penny-shaped fluid-driven fracture in an impermeable rock: Asymptotic solutions.” Int. J. Solids Struct., 39(26), 6311–6337.
Shalev, E., and Lyakhovsky, V. (2013). “The processes controlling damage zone propagation induced by wellbore fluid injection.” Geophys. J. Int., 193(1), 209–219.
Shao, J. (1998). “Poroelastic behaviour of brittle rock materials with anisotropic damage.” Mech. Mater., 30(1), 41–53.
Shao, J., and Lydzba, D. (1999). “Un modèle d’endommagement poroélastique pour milieux poreux saturés.” C.R. Acad. Sci., Ser. IIb: Mec., Phys., Chim., Astron., 327(13), 1305–1310.
Shao, J., Zhou, H., and Chau, K. (2005). “Coupling between anisotropic damage and permeability variation in brittle rocks.” Int. J. Numer. Anal. Methods Geomech., 29(12), 1231–1247.
Shen, X. (2012). “Modeling fractures with continuum damage and its numerical application to stimulation estimates.” Proc., 46th Rock Mechanics Geomechanics Symp., American Rock Mechanics Association, Alexandria, VA, 12–196.
Shlyapobersky, J., and Chudnovsky, A. (1994). “Review of recent developments in fracture mechanics with petroleum engineering applications.” Proc., Rock Mechanics in Petroleum Engineering, Society of Petroleum Engineers, Richardson, TX.
Simulia (2013). Abaqus 6.9 documentation, Dassault Systèmes, Waltham, MA.
Smart, K., Ofoegbu, G., Das, K., and Basu, D. (2012). “Geomechanical modeling of hydraulic fracture initiation and propagation in a mechanically stratified geologic system.” Proc., 46th Rock Mechanics Geomechanics Symp., American Rock Mechanics Association, Alexandria, VA, 12–275.
Suzuki, T. (2012). “Understanding of dynamic earthquake slip behavior using damage as a tensor variable: Microcrack distribution, orientation, and mode and secondary faulting.” J. Geophys. Res., 117(B5), 1–20.
Swoboda, G., and Yang, Q. (1999a). “An energy-based damage model of geomaterials. I. Formulation and numerical results.” Int. J. Solids Struct., 36, 1719–1734.
Swoboda, G., and Yang, Q. (1999b). “An energy-based damage model of geomaterials. II. Deduction of damage evolution laws.” Int. J. Solids Struct., 36, 1735–1755.
Swoboda, G., Yang, Q., and Ito, F. (1995). “Damage propagation model and its application to rock engineering problems.” Proc., 8th Int. Congress on Rock Mechanics, International Society of Rock Mechanics, Lisbon, Portugal, 159–163.
Swoboda, G., Yang, Q., and Zhou, W. (1997). “An energy-based damage model and its engineering application.” Proc., 9th Int. Conf. on Computer Methods and Advances in Geomechanics, IACMAG, Newcastle, Australia, 1821–1828.
Tarokh, A., and Fakhimi, A. (2013). “Relationship between grain size and fracture properties of rock.” Proc., 47th US Rock Mechanics/Geomechancis Symp., American Rock Mechanics Association, Alexandria, VA.
Valko, P., and Economides, M. (1993). “A continuum-damage-mechanics model of hydraulic fracturing.” J. Pet. Technol., 45(03), 198–205.
Valko, P., and Economides, M. (1994). “Propagation of hydraulically induced fractures a continuum damage mechanics approach.” Int. J. Rock Mech. Min Sci. Geomech. Abstr., 31(3), 221–229.
Wu, S., and Chudnovsky, A. (1993). “Effect of microcrack array on stress intensity factor of main crack.” Int. J. Fract., 59(1), 41–52.
Xie, N., Zhu, Q. Z., Shao, J. F., and Xu, L. H. (2012). “Micromechanical analysis of damage in saturated quasi brittle materials.” Int. J. Solids Struct., 49(6), 919–928.
Xu, H., and Arson, C. (2014). “Anisotropic damage models for geomaterials: Theoretical and numerical challenges.” Int. J. Comput. Methods, 11(2), 1342007.
Xu, H., and Prévost, J. H. (2016). “Integration of a continuum damage model for shale with the cutting plane algorithm.” Int. J. Anal. Numer. Methods Geomech., 41(4), 471–487.
Yamashita, T. (2000). “Generation of microcracks by dynamic shear rupture and its effects on rupture growth and elastic wave radiation.” Geophys. J. Int., 143(2), 395–406.
Zhou, J., Shao, J., and Xu, W. (2006). “Coupled modeling of damage growth and permeability variation in brittle rocks.” Mech. Res. Commun., 33(4), 450–459.
Zhu, C., and Arson, C. (2014). “A thermo-mechanical damage model for rock stiffness during anisotropic crack opening and closure.” Acta Geotech., 9(5), 874–867.

Information & Authors

Information

Published In

Go to International Journal of Geomechanics
International Journal of Geomechanics
Volume 17Issue 8August 2017

History

Received: Sep 3, 2014
Accepted: Dec 7, 2016
Published online: Mar 17, 2017
Published in print: Aug 1, 2017
Discussion open until: Aug 17, 2017

Permissions

Request permissions for this article.

Authors

Affiliations

Ph.D. Student, School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332; Energy Geosciences Division, Earth & Environmental Sciences Area, Lawrence Berkeley National Laboratory, Berkeley, CA 94720. E-mail: [email protected]
Seth Busetti [email protected]
Engineer, Structure & Geomechanics Group, ConocoPhillips, Houston, TX 77079. E-mail: [email protected]
Professor, School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332 (corresponding author). ORCID: https://orcid.org/0000-0002-4477-1072. E-mail: [email protected]

Metrics & Citations

Metrics

Citations

Download citation

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited by

View Options

Get Access

Access content

Please select your options to get access

Log in/Register Log in via your institution (Shibboleth)
ASCE Members: Please log in to see member pricing

Purchase

Save for later Information on ASCE Library Cards
ASCE Library Cards let you download journal articles, proceedings papers, and available book chapters across the entire ASCE Library platform. ASCE Library Cards remain active for 24 months or until all downloads are used. Note: This content will be debited as one download at time of checkout.

Terms of Use: ASCE Library Cards are for individual, personal use only. Reselling, republishing, or forwarding the materials to libraries or reading rooms is prohibited.
ASCE Library Card (5 downloads)
$105.00
Add to cart
ASCE Library Card (20 downloads)
$280.00
Add to cart
Buy Single Article
$35.00
Add to cart

Get Access

Access content

Please select your options to get access

Log in/Register Log in via your institution (Shibboleth)
ASCE Members: Please log in to see member pricing

Purchase

Save for later Information on ASCE Library Cards
ASCE Library Cards let you download journal articles, proceedings papers, and available book chapters across the entire ASCE Library platform. ASCE Library Cards remain active for 24 months or until all downloads are used. Note: This content will be debited as one download at time of checkout.

Terms of Use: ASCE Library Cards are for individual, personal use only. Reselling, republishing, or forwarding the materials to libraries or reading rooms is prohibited.
ASCE Library Card (5 downloads)
$105.00
Add to cart
ASCE Library Card (20 downloads)
$280.00
Add to cart
Buy Single Article
$35.00
Add to cart

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share