Skip to main content
SpringerLink
Log in

A mesh-independent finite element formulation for modeling crack growth in saturated porous media based on an enriched-FEM technique

  • Original Paper
  • Published:
International Journal of Fracture Aims and scope Submit manuscript

Abstract

In this paper, the crack growth simulation is presented in saturated porous media using the extended finite element method. The mass balance equation of fluid phase and the momentum balance of bulk and fluid phases are employed to obtain the fully coupled set of equations in the framework of \(u{-}p\) formulation. The fluid flow within the fracture is modeled using the Darcy law, in which the fracture permeability is assumed according to the well-known cubic law. The spatial discritization is performed using the extended finite element method, the time domain discritization is performed based on the generalized Newmark scheme, and the non-linear system of equations is solved using the Newton–Raphson iterative procedure. In the context of the X-FEM, the discontinuity in the displacement field is modeled by enhancing the standard piecewise polynomial basis with the Heaviside and crack-tip asymptotic functions, and the discontinuity in the fluid flow normal to the fracture is modeled by enhancing the pressure approximation field with the modified level-set function, which is commonly used for weak discontinuities. Two alternative computational algorithms are employed to compute the interfacial forces due to fluid pressure exerted on the fracture faces based on a ‘partitioned solution algorithm’ and a ‘time-dependent constant pressure algorithm’ that are mostly applicable to impermeable media, and the results are compared with the coupling X-FEM model. Finally, several benchmark problems are solved numerically to illustrate the performance of the X-FEM method for hydraulic fracture propagation in saturated porous media.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  • Akulich AV, Zvyagin AV (2008) Interaction between hydraulic and natural fractures. Fluid Dyn 43:428–435

    Article  Google Scholar 

  • Areias PMA, Belytschko T (2005) Analysis of three-dimensional crack initiation and propagation using the extended finite element method. Int J Numer Methods Eng 63:760–788

    Article  Google Scholar 

  • Barani OR, Khoei AR, Mofid M (2011) Modeling of cohesive crack growth in partially saturated porous media: a study on the permeability of cohesive fracture. Int J Fract 167:15–31

    Article  Google Scholar 

  • Barenblatt GI (1962) The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 7:55–62

    Article  Google Scholar 

  • Bazant ZP, Planas J (1998) Fracture and size effect in concrete and other quasibrittle materials. CRC Press, New York

    Google Scholar 

  • Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45:601–620

    Article  Google Scholar 

  • Belytschko T, Moës N, Usui S, Parimi C (2001) Arbitrary discontinuities in finite elements. Int J Numer Methods Eng 50:993–1013

    Article  Google Scholar 

  • Belytschko T, Chen H, Xu J, Zi G (2003) Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment. Int J Numer Methods Eng 58:1873–1905

    Article  Google Scholar 

  • Belytschko T, Gracie R, Ventura G (2009) A review of extended/generalized finite element methods for material modeling. Model Simul Mater Sci Eng 17:043001

    Article  Google Scholar 

  • Biot MA (1941) General theory of three dimensional consolidation. J Appl Phys 12:155–164

    Article  Google Scholar 

  • Biot MA, Willis PG (1957) The elastic coefficients of the theory consolidation. J Appl Mech 24:594–601

    Google Scholar 

  • Boone TJ, Ingraffea AR (1990) A numerical procedure for simulation of hydraulically driven fracture propagation in poroelastic media. Int J Numer Anal Methods Geomech 14:27–47

    Article  Google Scholar 

  • Bordas S, Legay A (2005) X-FEM mini-course. EPFL, Lausanne

    Google Scholar 

  • Bowen RM (1976) Theory of mixtures in continuum physics. Academic Press, New York

    Google Scholar 

  • Callari C, Armero F, Abati A (2010) Strong discontinuities in partially saturated poroplastic solids. Comp Methods Appl Mech Eng 199:1513–1535

    Article  Google Scholar 

  • Chan AHC (1988) A unified finite element solution to static and dynamic geomechanics problems. Ph.D. dissertation, University of Wales Swansea, Wales, UK

  • Coussy O (2004) Poromechanics. Wiley, Chichester

    Google Scholar 

  • Daux C, Moës N, Dolbow J, Sukumar N, Belytschko T (2000) Arbitrary branched and intersecting cracks with the extended finite element method. Int J Numer Methods Eng 48:1741–1760

    Article  Google Scholar 

  • De Boer R, Kowalski SJ (1983) A plasticity theory for fluid saturated porous solids. Int J Eng Sci 21:1343–1357

    Article  Google Scholar 

  • De Borst R, Réthoré J, Abellan MA (2006) A numerical approach for arbitrary cracks in a fluid-saturated medium. Arch Appl Mech 75:595–606

    Article  Google Scholar 

  • Derski W (1978) Equations of motion for a fluid-saturated porous solid. Bull Acad Polish Sci Tech 26:11–16

    Google Scholar 

  • Detournay E (2004) Propagation regimes of fluid-driven fractures in impermeable rocks. Int J Geomech 4:35–45

    Article  Google Scholar 

  • Dolbow JE, Moës N, Belytschko T (2001) An extended finite element method for modeling crack growth with frictional contact. Comput Methods Appl Mech Eng 190:6825– 6846

  • Dong CY, de Pater CJ (2001) Numerical implementation of displacement discontinuity method and its application in hydraulic fracturing. Comput Methods Appl Mech Eng 191:745–760

    Article  Google Scholar 

  • Dugdale DS (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8:100–108

    Article  Google Scholar 

  • Emerman SH, Turcotte DL, Spence DA (1986) Transport of magma and hydrothermal solutions by laminar and turbulent fluid fracture. Phys Earth Planet Inter 41:249–259

    Article  Google Scholar 

  • Fries TP, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84:253–304

    Google Scholar 

  • Geertsma J, De Klerk F (1969) A rapid method of predicting width and extent of hydraulically induced fractures. J Pet Technol 21(12):1571–1581

    Article  Google Scholar 

  • Ghaboussi J, Wilson EL (1972) Variational formulation of dynamics of fluid saturated porous elastic solids. J Eng Mech Div 98(EM4):947–963

    Google Scholar 

  • Green AE, Naghdi PM (1969) On basic equations for mixtures. Q J Mech Appl Math 22(4):427–438

    Article  Google Scholar 

  • Jenq Y, Shah SP (1991) Features of mechanics of quasi-brittle crack propagation in concrete. Int J Fract 51:103–120

    Google Scholar 

  • Khoei AR (2014) Extended finite element method, theory and applications. Wiley, London

    Google Scholar 

  • Khoei AR, Haghighat E (2011) Extended finite element modeling of deformable porous media with arbitrary interfaces. Appl Math Model 35:5426–5441

    Article  Google Scholar 

  • Khoei AR, Karimi K (2008) An enriched-FEM model for simulation of localization phenomenon in Cosserat continuum theory. Comput Math Sci 44:733–749

    Article  Google Scholar 

  • Khoei AR, Nikbakht M (2007) An enriched finite element algorithm for numerical computation of contact friction problems. Int J Mech Sci 49:183–199

    Article  Google Scholar 

  • Khoei AR, Azami AR, Haeri SM (2004) Implementation of plasticity based models in dynamic analysis of saturated–unsaturated earth and rockfill dams. Comput Geotech 31:385–410

    Article  Google Scholar 

  • Khoei AR, Anahid M, Shahim K (2008a) An extended arbitrary Lagrangian–Eulerian finite element method for large deformation of solid mechanics. Finite Elem Anal Des 44:401– 416

  • Khoei AR, Azadi H, Moslemi H (2008b) Modeling of crack propagation via an automatic adaptive mesh refinement based on modified superconvergent patch recovery technique. Eng Fract Mech 75:2921–2945

    Article  Google Scholar 

  • Khoei AR, Moslemi H, Ardakany KM, Barani OR, Azadi H (2009) Modeling of cohesive crack growth using an adaptive mesh refinement via the modified-SPR technique. Int J Fract 159:21–41

    Article  Google Scholar 

  • Khoei AR, Barani OR, Mofid M (2011) Modeling of dynamic cohesive fracture propagation in porous saturated media. Int J Numer Anal Methods Geomech 35:1160–1184

    Article  Google Scholar 

  • Leung KH (1984) Earthquake response of saturated soils and liquefaction. Ph.D. dissertation, University of Wales Swansea, Wales, UK

  • Lewis RW, Rahman NA (1999) Finite element modeling of multiphase immiscible flow in deforming porous media for subsurface systems. Comput Geotech 24:41–63

    Article  Google Scholar 

  • Lewis RW, Schrefler BA (1998) The finite element method in the static and dynamic deformation and consolidation of porous media. Wiley, New York

    Google Scholar 

  • Melenk JM, Babuska I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139:289–314

    Article  Google Scholar 

  • Moës N, Belytschko T (2002) Extended finite element method for cohesive crack growth. Eng Fract Mech 69:813–833

    Article  Google Scholar 

  • Moës N, Dolbow JE, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150

    Article  Google Scholar 

  • Mohammadnejad T, Khoei AR (2013a) Hydro-mechanical modeling of cohesive crack propagation in multiphase porous media using the extended finite element method. Int J Numer Anal Methods Geomech 37:1247–1279

    Article  Google Scholar 

  • Mohammadnejad T, Khoei AR (2013b) An extended finite element method for hydraulic fracture propagation in deformable porous media with the cohesive crack model. Finite Elem Methods Anal Des 73:77–95

    Article  Google Scholar 

  • Morland LW (1972) A simple constitutive theory for fluid saturated porous solids. J Geophys Res 77:890–900

    Article  Google Scholar 

  • Moslemi H, Khoei AR (2009) 3D adaptive finite element modeling of non-planar curved crack growth using the weighted super-convergent patch recovery method. Eng Fract Mech 76:1703–1728

    Article  Google Scholar 

  • Oettl G, Stark RF, Hofstetter G (2004) Numerical simulation of geotechnical problems based on a multi-phase finite element approach. Comput Geotech 31:643–664

    Article  Google Scholar 

  • Preisig M, Prevost JH (2011) Stabilization procedures in coupled poromechanics problem: a critical assessment. Int J Numer Anal Methods Geomech 35:1207–1225

    Article  Google Scholar 

  • Proceedings of 5th ICOLD international benchmark workshop on numerical analysis of dams, Denver, Colorado, 1999

  • Remmers JJC, de Borst R, Needleman A (2008) The simulation of dynamic crack propagation using the cohesive segments method. J Mech Phys Solids 56:70–92

    Article  Google Scholar 

  • Réthoré J, de Borst R, Abellan MA (2007a) A discrete model for the dynamic propagation of shear bands in a fluid-saturated medium. Int J Numer Anal Methods Geomech 31:347– 370

  • Réthoré J, de Borst R, Abellan MA (2007b) A two-scale approach for fluid flow in fractured porous media. Int J Numer Methods Eng 71:780–800

    Article  Google Scholar 

  • Rice JR, Cleary MP (1976) Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents. Rev Geophys Space Phys 14:227–241

    Article  Google Scholar 

  • Schrefler BA, Scotta R (2001) A fully coupled dynamic model for two-phase fluid flow in deformable porous media. Comput Methods Appl Mech Eng 190:3223–3246

    Article  Google Scholar 

  • Schrefler BA, Zhan X (1993) A fully coupled model for waterflow and airflow in deformable porous media. Water Resour Res 29(1):155–167

    Article  Google Scholar 

  • Schrefler BA, Zhan X, Simoni L (1995) A coupled model for water flow, airflow and heat flow in deformable porous media. Int J Heat Fluid Flow 5:531–547

    Article  Google Scholar 

  • Schrefler BA, Secchi S, Simoni L (2006) On adaptive refinement techniques in multi-field problems including cohesive fracture. Comput Methods Appl Mech Eng 195:444–461

    Article  Google Scholar 

  • Secchi S, Simoni L, Schrefler BA (2007) Mesh adaptation and transfer schemes for discrete fracture propagation in porous materials. Int J Numer Anal Methods Geomech 31:331–345

    Article  Google Scholar 

  • Sheng D, Sloan SW, Gens A, Smith DW (2003) Finite element formulation and algorithms for unsaturated soils. Part I: theory. Int J Numer Anal Methods Geomech 27:745–765

    Article  Google Scholar 

  • Shum KM, Hutchinson JW (1990) On toughening by micro-cracks. Mech Math 9:83–91

    Article  Google Scholar 

  • Simoni L, Schrefler BA (1991) A staggered finite element solution for water and gas flow in deforming porous media. Commun Appl Numer Methods 7:213–223

    Article  Google Scholar 

  • Spence DA, Sharp P (1985) Self-similar solutions for elasto-hydrodynamic cavity flow. Proc R Soc Lond A 400:289–313

    Article  Google Scholar 

  • Stelzer R, Hofstetter G (2005) Adaptive finite element analysis of multi-phase problems in geotechnics. Comput Geotech 32:458–481

    Article  Google Scholar 

  • Stolarska M, Chopp DL (2003) Modeling thermal fatigue cracking in integrated circuits by level sets and the extended finite element method. Int J Eng Sci 41(20):2381–2410

    Article  Google Scholar 

  • Sukumar N, Chopp DL, Moës N, Belytschko T (2001) Modeling holes and inclusions by level sets in the extended finite-element method. Comput Methods Appl Mech Eng 190:6183–6200

    Article  Google Scholar 

  • Sun WC, Ostien JT, Salinger AG (2013) A stabilized assumed deformation gradient finite element formulation for strongly coupled poromechanical simulations at finite strain. Int J Numer Anal Methods Geomech 37:2755–2788

    Article  Google Scholar 

  • Terzaghi K (1943) Theoretical soil mechanics. Wiley, New York

    Book  Google Scholar 

  • Ventura G, Budyn E, Belytschko T (2003) Vector level sets for description of propagating cracks in finite elements. Int J Numer Methods Eng 58:1571–1592

    Article  Google Scholar 

  • Witherspoon PA, Wang JSY, Iwai K, Gale JE (1980) Validity of cubic law for fluid flow in a deformable rock fracture. Water Resour Res 16:1016–1024

    Article  Google Scholar 

  • Wu YS, Forsyth PA (2001) On the selection of primary variables in numerical formulation for modeling multiphase flow in porous media. J Contam Hydrol 48:277–304

    Article  Google Scholar 

  • Zhang Z, Ghassemi A (2011) Simulation of hydraulic fracture propagation near a natural fracture using virtual multidimensional internal bonds. Int J Numer Anal Methods Geomech 35:480–495

    Article  Google Scholar 

  • Zienkiewicz OC, Shiomi T (1984) Dynamic behavior of saturated porous media: the generalized Biot formulation and it’s numerical solution. Int J Numer Anal Methods Geomech 8:71–96

    Article  Google Scholar 

  • Zienkiewicz OC, Chan AHC, Pastor M, Paul DK, Shiomi T (1990a) Static and dynamic behavior of geomaterials—a rational approach to quantitative solutions, part I-fully saturated problems. Proc R Soc Lond A429:285–309

    Article  Google Scholar 

  • Zienkiewicz OC, Xie YM, Schrefler BA, Ledesma A, Bicanic N (1990b) Static and dynamic behaviour of soils: a rational approach to quantitative solutions, part 11: semisaturated problems. Proc R Soc Lond A429:310–323

  • Zienkiewicz OC, Chan AHC, Pastor M, Schrefler BA, Shiomi T (1999) Computational geomechanics with special reference to earthquake engineering. Wiley, New York

    Google Scholar 

Download references

Acknowledgments

The authors are grateful for the research support of the Iran National Science Foundation (INSF).

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Center of Excellence in Structures and Earthquake Engineering, Sharif University of Technology, P.O. Box 11365-9313, Tehran, Iran

    A. R. Khoei, M. Vahab, E. Haghighat & S. Moallemi

Authors
  1. A. R. Khoei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Vahab
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. Haghighat
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. S. Moallemi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. R. Khoei.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khoei, A.R., Vahab, M., Haghighat, E. et al. A mesh-independent finite element formulation for modeling crack growth in saturated porous media based on an enriched-FEM technique. Int J Fract 188, 79–108 (2014). https://doi.org/10.1007/s10704-014-9948-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10704-014-9948-2

Keywords

Search

Navigation