zbMATH — the first resource for mathematics

A two-scale model for fluid flow in an unsaturated porous medium with cohesive cracks. (English) Zbl 1154.76053
Summary: We develop a two-scale model for fluid flow in a deforming, unsaturated and progressively fracturing porous medium. At the microscale, the flow in the cohesive crack is modelled using Darcy relation for fluid flow in a porous medium, taking into account changes in the permeability due to the progressive damage evolution inside the cohesive zone. From the micromechanics of the flow in the cavity, identities are derived that couple the local momentum and the mass balances to the governing equations for an unsaturated porous medium, which are assumed to hold on the macroscopic scale. The finite element equations are derived for this two-scale approach and integrated over time. By exploiting the partition-of-unity property of finite element shape functions, the position and direction of fractures are independent of the underlying discretization. The resulting discrete equations are nonlinear due to the cohesive crack model and the nonlinearity of coupling terms. A consistent linearization is given for use within a Newton-Raphson iterative procedure. Finally, examples are given to show the versatility and efficiency of the approach. The calculations indicate that the evolving cohesive cracks can have a significant influence on the fluid flow and vice versa.

76S05 Flows in porous media; filtration; seepage
76M10 Finite element methods applied to problems in fluid mechanics
74R99 Fracture and damage
Full Text: DOI
[1] Abellan MA and Borst R (2006). Wave propagation and localisation in a softening two-phase medium. Comput Methods Appl Mech Eng 195(37–40): 5011–5019 · Zbl 1118.74013 · doi:10.1016/j.cma.2005.05.056
[2] Babuska I and Melenk JM (1997). The partition of unity method. Int J Numer Methods Eng 40: 727–758 · Zbl 0949.65117 · doi:10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N
[3] Belytschko T, Moës N, Usui S and Parimi C (2001). Arbitrary discontinuities in finite elements. Int J Numer Methods Eng 50(4): 993–1013 · Zbl 0981.74062 · doi:10.1002/1097-0207(20010210)50:4<993::AID-NME164>3.0.CO;2-M
[4] Biot MA (1965). Mechanics of incremental deformations. Wiley, Chichester
[5] Black T and Belytschko T (1999). Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45: 601–620 · Zbl 0943.74061 · doi:10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S
[6] Boone T and Ingraffea AR (1990). A numerical procedure for simulation of hydraulically-driven fracture propagation in poroelastic media. Int J Numer Anal Methods Geomech 14(2): 27–47 · doi:10.1002/nag.1610140103
[7] de Borst R, Remmers JJC and Needleman A (2006). Mesh-independent numerical representations of cohesive-zone models. Eng Fracture Mech 173(2): 160–177 · doi:10.1016/j.engfracmech.2005.05.007
[8] de Borst R, Réthoré J and Abellan MA (2006). A numerical approach for arbitrary cracks in a fluid-saturated medium. Arch Appl Mech 75: 595–606 · Zbl 1168.74447 · doi:10.1007/s00419-006-0023-y
[9] Camacho GT and Ortiz M (1996). Computational modeling of impact damage in brittle materials. Int J Solids Struct 33: 1267–1282 · Zbl 0929.74101 · doi:10.1016/0020-7683(95)00255-3
[10] Huyghe HM and Janssen JD (1997). Quadriphasic mechanics of swelling incompressible media. Int J Eng Sci 35: 793–802 · Zbl 0903.73004 · doi:10.1016/S0020-7225(96)00119-X
[11] Jirasek M (1998) Computational modelling of concrete structures, Balkema, Rotterdam, chap Embedded crack models for concrete fracture, pp 291–300
[12] Jouanna P, Abellan MA (1995) Modern issues in non-saturated soils, Springer, Heidelberg, chap Generalized approach to heterogeneous media
[13] Lewis RW and Schrefler BA (1998). The finite element method in the static and dynamic deformation and consolidation of porous media, 2nd edn. Wiley, Chichester · Zbl 0935.74004
[14] van Loon R, Huyghe JM, Wijlaars MW and Baaijens FPT (2003). 3d fe implementation of an incompressible quadriphasic mixture model. Int J Numer Methods Eng 57: 1243–1258 · Zbl 1062.74634 · doi:10.1002/nme.723
[15] Meschke G and Grasberger S (2003). Numerical modeling of coupled hygromechanical degradation of cementitious materials. J Eng Mech 129(4): 383–392 · doi:10.1061/(ASCE)0733-9399(2003)129:4(383)
[16] Moës N, Dolbow J and Belytschko T (1999). A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1): 133–150 · Zbl 0955.74066
[17] Remmers JJC, de Borst R and Needleman A (2003). A cohesive segments method for the simulation of crack growth. Comput Mech 31: 69–77 · Zbl 1038.74679 · doi:10.1007/s00466-002-0394-z
[18] Réthoré J, Gravouil A and Combescure A (2005). A combined space time extended finite element method. Int J Numer Methods Eng 64: 260–284 · Zbl 1138.74403 · doi:10.1002/nme.1368
[19] Réthoré J, Gravouil A and Combescure A (2005). An energy conserving scheme for dynamic crack growth with the extended finite element method. Int J Numer Methods Eng 63: 631–659 · Zbl 1122.74519 · doi:10.1002/nme.1283
[20] Réthoré J, Abellan MA and Borst R (2007). A discrete model for the dynamic propagation of shear bands in a fluid-saturated medium. Int J Numer Anal Methods Geomech 31: 347–370 · Zbl 1112.74055 · doi:10.1002/nag.575
[21] Réthoré J, de Borst R, Abellan MA (2007b) A two-scale approach for fluid flow in fractured porous media. Int J Numer Methods Eng. doi: 10.1002/nme.1962
[22] Schrefler BA, Secchi S and Simoni L (2006). On adaptive refinement techniques in multi-field problems including cohesive fracture. Comput Methods Appl Mech Eng 195(4–6): 444–461 · Zbl 1193.74158 · doi:10.1016/j.cma.2004.10.014
[23] Snijders H, Huyghe JM and Janssen JD (1995). Triphasic finite element model for swelling porous media. Int J Numer Methods Fluids 20: 1039–1046 · Zbl 0850.76337 · doi:10.1002/fld.1650200821
[24] Terzaghi K (1943). Theoretical soil mechanics. Wiley, New York
[25] Wells GN and Sluys LJ (2001). Discontinuous analysis of softening solids under impact loading. Int J Numer Anal Methods Geomech 25: 691–709 · Zbl 1052.74564 · doi:10.1002/nag.148
[26] Wells GN and Sluys LJ (2001). A new method for modeling cohesive cracks using finite elements. Int J Numer Methods Eng 50: 2667–2682 · Zbl 1013.74074 · doi:10.1002/nme.143
[27] Wells GN, de Borst R and Sluys LJ (2002). A consistent geometrically non-linear approach for delamination. Int J Numer Methods Eng 54: 1333–1355 · Zbl 1086.74043 · doi:10.1002/nme.462
[28] Wells GN, Sluys LJ and de Borst R (2002). Simulating the propagation of displacement discontinuities in a regularized strain–softening medium. Int J Numer Methods Eng 53: 1235–1256 · doi:10.1002/nme.375
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.