Crack Growth in Porous Media Using XFEM: Comparison of Modeling Strategies on the Abaqus

Document Type : Research Article

Authors

1 Dept. of Mining, University of Zanjan, Zanjan, Iran

2 Dept. of Materials, University of Zanjan, Zanjan, Iran

10.29252/anm.2020.13220.1426

Abstract

Summary
The recent developments in the Extended Finite Element Method (XFEM) opened new avenues for crack propagation problems. However, its ability to predict the crack path in the micro-scale medium of real porous rock is questionable. In this work, we compare two strategies and introduce one as the best strategy to use the XFEM for such a purpose in Abaqus Software. We demonstrate our claim by comparing numerical results with the analytical solution and experimental test.
 
Introduction
Crack growth has always been one of the major challenges in rock mechanics. Although pores, joints, and fractures are the most critical structures controlling cracks initiation and propagation, their spatial distribution and geometrical effects are not still well-understood. In this study, we aim to numerically model the crack growth in real porous rock.
 
Methodology and Approaches
We use the extended finite element method (XFEM), which has recently attracted more attention due to its ability to estimate the discontinuous deformation field by using special shape functions. Since direct use of the XFEM does not lead to an accurate result, two different strategies are considered for applying the XFEM on a porous model to simulate the crack propagation.
 
Results and Conclusions
Our results showed that applying several different partitions and enrich them individually lead to more logical results than to allocate reduced elastic modulus to porosity. We used this strategy to evaluate the XFEM both analytically and experimentally as a possible numerical solution. Thus, two simple models were constructed, both numerically and experimentally (Granite): i. a sample with one void and one crack, and ii. a sample with two voids and one crack between them. Analytical solutions for the stress intensity factor revealed that the XFEM modeling can compute this parameter with an error of less than 5%. On the other hand, experimental results showed that the XFEM with partitioning strategy can predict the correct crack growth path comparative to the experimental results. Accordingly, digital images of Berea sandstone were used as a real reservoir rock and, then, this method was implemented to simulate multi-crack propagation through the exact medium of rock.

Keywords

Main Subjects


درزه، ترک و تخلخل از جمله پدیده­های متداول در سنگ­ها هستند که به شدت رفتار فیزیکی و مکانیکی محیط­های سنگی را کنترل می­کنند. درک صحیح از رفتـار این محیط­ها به عنوان پیکره حفریات زیرسطحی و شالـوده سازه­های سطحی از اهمیت ویژه­ای در علوم مهندسی برخوردار است. در واقع، تنش برجای سنگ در هنگام آزاد شدن در نوک و گوشـه­های تیز این ناپیوستگی‌ها متمرکز شده و با غلبه بر مقاومت سنگ به صورت ریزترک­هایی بروز پیدا می­کنند. در ادامه، این ریزترک­ها به طور پیوستــه رشد کرده و با پیوستن به یکدیگر صفحه شکستگی را تشکیـل داده که در نهایت موجب شکست و گسیختگی سنگ می­شود. در علم مکانیک سنگ، سازوکار پدیده شکست سنگ با اهداف مختلفی از جمله تونل­سازی، آتشباری، شکست هیدرولیکی، مکانیک زلزله، پایداری شیب و غیره چه به صورت تحلیلی و چه به صورت عددی مطالعه شده است[1-3].

[1]           Chang S-H, Lee C-I, Jeon S. Measurement of rock fracture toughness under modes I and II and mixed-mode conditions by using disc-type specimens. Eng Geol 2002;66:79–97. doi:10.1016/S0013-7952(02)00033-9.
[2]           Hoek E, Martin CD. Fracture initiation and propagation in intact rock – A review. J Rock Mech Geotech Eng 2014;6:287–300. doi:10.1016/J.JRMGE.2014.06.001.
[3]           Lisjak A, Kaifosh P, He L, Tatone BSA, Mahabadi OK, Grasselli G. A 2D, fully-coupled, hydro-mechanical, FDEM formulation for modelling fracturing processes in discontinuous, porous rock masses. Comput Geotech 2017;81:1–18. doi:10.1016/J.COMPGEO.2016.07.009.
[4]           Jing L, Hudson JA. Numerical methods in rock mechanics. Int J Rock Mech Min Sci 2002;39:409–27. doi:10.1016/S1365-1609(02)00065-5.
[5]           Cundall PA. A computer model for simulating progressive large-scale movements in blocky rock systems. Proocedings Symp. Int. Soc. Rock Mech. Nancy 2, 1971, p. No. 8. doi:10.1111/j.1469-8137.1986.tb00632.x.
[6]           Lisjak A, Grasselli G. A review of discrete modeling techniques for fracturing processes in discontinuous rock masses. J Rock Mech Geotech Eng 2014;6:301–14. doi:10.1016/J.JRMGE.2013.12.007.
[7]           Bouhala L, Makradi A, Belouettar S. Thermo-anisotropic crack propagation by XFEM. Int J Mech Sci 2015;103:235–46. doi:10.1016/J.IJMECSCI.2015.09.014.
[8]           Sivakumar G, Maji VB. Simulation of crack propagation in rocks by XFEM. Proc. Conf. Recent Adv. Rock Eng. (RARE 2016), Paris, France: Atlantis Press; 2016. doi:10.2991/rare-16.2016.46.
[9]           Yang Y, Ju Y, Sun Y, Zhang D. Numerical study of the stress field during crack growth in porous rocks. Geomech Geophys Geo-Energy Geo-Resources 2015;1:91–101. doi:10.1007/s40948-015-0011-1.
[10]         Abdollahipour A, Marji MF, Bafghi AY, Gholamnejad J. Time-dependent crack propagation in a poroelastic medium using a fully coupled hydromechanical displacement discontinuity method. Int J Fract 2016;199:71–87. doi:10.1007/s10704-016-0095-9.
[11]         Behnia M, Goshtasbi K, Fatehi Marji M, Golshani A. On the crack propagation modeling of hydraulic fracturing by a hybridized displacement discontinuity/boundary collocation method. J Min Environ 2012;2:1–16. doi:10.22044/jme.2012.15.
[12]         Hosseini-Nasab H, Fatehi-Marji M. A semi-infinite higher-order displacement discontinuity method and its application to the quasistatic analysis of radial cracks produced by blasting. J Mech Mater Struct 2007;2:1515–24.
[13]         Haeri H, Khaloo A, Marji MF. A coupled experimental and numerical simulation of rock slope joints behavior. Arab J Geosci 2015;8:7297–308. doi:10.1007/s12517-014-1741-z.
[14]         Rege K, Lemu HG. A review of fatigue crack propagation modelling techniques using FEM and XFEM. IOP Conf Ser Mater Sci Eng 2017;276:012027. doi:10.1088/1757-899X/276/1/012027.
[15]         Giner E, Sukumar N, Denia FD, Fuenmayor FJ. Extended finite element method for fretting fatigue crack propagation. Int J Solids Struct 2008;45:5675–87. doi:10.1016/J.IJSOLSTR.2008.06.009.
[16]         Golewski GL, Golewski P, Sadowski T. Numerical modelling crack propagation under Mode II fracture in plain concretes containing siliceous fly-ash additive using XFEM method. Comput Mater Sci 2012;62:75–8. doi:10.1016/j.commatsci.2012.05.009.
[17]         Dahi-Taleghani A, Olson JE. Numerical Modeling of Multistranded-Hydraulic-Fracture Propagation: Accounting for the Interaction Between Induced and Natural Fractures. SPE J 2011;16:575–81. doi:10.2118/124884-pa.
[18]         Khoei AR, Vahab M, Haghighat E, Moallemi S. A mesh-independent finite element formulation for modeling crack growth in saturated porous media based on an enriched-FEM technique. Int J Fract 2014;188:79–108. doi:10.1007/s10704-014-9948-2.
[19]         Gordeliy E, Peirce A. Enrichment strategies and convergence properties of the XFEM for hydraulic fracture problems. Comput Methods Appl Mech Eng 2015;283:474–502. doi:10.1016/j.cma.2014.09.004.
[20]         Abdollahipour A, Fatehi Marji M, Yarahmadi Bafghi A, Gholamnejad J. Numerical investigation of effect of crack geometrical parameters on hydraulic fracturing process of hydrocarbon reservoirs. J Min Environ 2016;7:205–14. doi:10.22044/jme.2016.532.
[21]         Behnia M, Goshtasbi K, Marji MF, Golshani A. Numerical simulation of crack propagation in layered formations. Arab J Geosci 2014;7:2729–37. doi:10.1007/s12517-013-0885-6.
[22]         Ayatollahi MR, Pavier MJ, Smith DJ. Mode I cracks subjected to large T -stresses 2002:159–74.
[23]         Nasaj Moghaddam H, Keyhani A, Aghayan I. Modelling of Crack Propagation in Layered Structures Using Extended Finite Element Method. Civ Eng J 2016;2:180–8.
[24]         Baydoun M, Fries TP. Crack propagation criteria in three dimensions using the XFEM and an explicit-implicit crack description. Int. J. Fract., vol. 178, Springer Netherlands; 2012, p. 51–70. doi:10.1007/s10704-012-9762-7.
[25]         Karimpouli S, Tahmasebi P. Conditional reconstruction: An alternative strategy in digital rock physics. GEOPHYSICS 2016;81:D465–77. doi:10.1190/geo2015-0260.1.
[26]         Karimpouli S, Tahmasebi P, Saenger EH. Estimating 3D elastic moduli of rock from 2D thin-section images using differential effective medium theory. GEOPHYSICS 2018;83:MR211–9. doi:10.1190/geo2017-0504.1.
[27]         Hillerborg A, Modéer M, Petersson P-E. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res 1976;6:773–81. doi:10.1016/0008-8846(76)90007-7.
[28]         Rezanezhad M, Lajevardi SA, Karimpouli S. Effects of pore-crack relative location on crack propagation in porous media using XFEM method. Theor Appl Fract Mech 2019;103:102241. doi:10.1016/j.tafmec.2019.102241.
[29]         Mohammadi S. Extended Finite Element Method. Oxford, UK: Blackwell Publishing Ltd; 2008. doi:10.1002/9780470697795.
[30]         Giner E, Sukumar N, Tarancón JE, Fuenmayor FJ. An Abaqus implementation of the extended finite element method. Eng Fract Mech 2009;76:347–68. doi:10.1016/j.engfracmech.2008.10.015.
[31]         Rodriguez-Florez N, Carriero A, Shefelbine SJ. The use of XFEM to assess the influence of intra-cortical porosity on crack propagation. Comput Methods Biomech Biomed Engin 2017;20:385–92. doi:10.1080/10255842.2016.1235158.
[32]         Belytschko T, Moes N, Usui S, Parimi C. Arbitrary discontinuities in finite elements. Int J Numer Methods Eng 2001;50:993–1013. doi:10.1002/1097-0207(20010210)50:43.0.CO;2-M.
[33]         Rodriguez-Florez N. Mechanics of cortical bone: exploring the micro- and nano-scale. Imperial College London, 2015.
[34]         Sih GC. Mechanics of fracture, 1. Methods of analysis and solutions of crack problems. Materwiss Werksttech 1973:516. doi:10.1002/mawe.19730040714.
[35]         Arshadnejad S. Analysis of the First Cracks Generating Between Two Holes Under Incremental Static Loading with an Innovation Method by Numerical Modelling. Math Comput Sci 2017;2:120. doi:10.11648/j.mcs.20170206.15.
[36]         Zhang Z. An empirical relation between mode I fracture toughness and the tensile strength of rock. Int J Rock Mech Min Sci 2002;39:401–6. doi:10.1016/S1365-1609(02)00032-1.
[37]         Bazant ZP, Kazemi MT. Size Effect in Fracture of Ceramics and Its Use To Determine Fracture Energy and Effective Process Zone Length. J Am Ceram Soc 1990;73:1841–53. doi:10.1111/j.1151-2916.1990.tb05233.x.
[38]         Bai QS, Tu SH, Zhang C. DEM investigation of the fracture mechanism of rock disc containing hole(s) and its influence on tensile strength. Theor Appl Fract Mech 2016;86:197–216. doi:10.1016/j.tafmec.2016.07.005.
[39]         Andrä H, Combaret N, Dvorkin J, Glatt E, Han J, Kabel M, et al. Digital rock physics benchmarks—Part I: Imaging and segmentation. Comput Geosci 2013;50:25–32. doi:10.1016/j.cageo.2012.09.005.
[40]         Huang J-Q, Huang Q-A, Qin M, Dong W-J, Chen X-W. Experimental study on the dielectrostriction of SiO2 with a micro-fabricated cantilever. 2009 IEEE Sensors, IEEE; 2009, p. 1030–3. doi:10.1109/ICSENS.2009.5398528.
[41]         Karimpouli S, Khoshlesan S, Saenger EH, Koochi HH. Application of alternative digital rock physics methods in a real case study: a challenge between clean and cemented samples. Geophys Prospect 2018;66:767–83. doi:10.1111/1365-2478.12611.